https://wiki.swarma.org/api.php?action=feedcontributions&feedformat=atom&user=Lincent集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织 - 用户贡献 [zh-cn]2026-11-02T20:21:14Z用户贡献MediaWiki 1.35.0https://wiki.swarma.org/index.php?title=KS%E6%A3%80%E9%AA%8C&diff=20725KS检验2020-12-31T01:43:19Z<p>Lincent：</p>
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<div>此词条由Jie翻译。由Lincent审校。<br />
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{{Short description|Non-parametric statistical test between two distributions}}<br />
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[[文件:KS_Example.png|缩略图|右|Kolmogorov–Smirnov统计数据的图示。 红线是累积分布函数，蓝线是经验分布函数，黑色箭头是K–S统计量。]]<br />
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In [[statistics]], the '''Kolmogorov–Smirnov test''' ('''K–S test''' or '''KS test''') is a [[nonparametric statistics|nonparametric test]] of the equality of continuous (or discontinuous, see [[#Discrete and mixed null distribution|Section 2.2]]), one-dimensional [[probability distribution]]s that can be used to compare a [[random sample|sample]] with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). It is named after [[Andrey Kolmogorov]] and [[Nikolai Smirnov (mathematician)|Nikolai Smirnov]].<br />
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In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). It is named after Andrey Kolmogorov and Nikolai Smirnov.<br />
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在统计学中，'''<font color="#ff8000"> Kolmogorov–Smirnov检验</font>'''（K-S检验或KS检验）是一种连续（或不连续，请参见第2.2节）的一维概率分布均等性的非参数检验，可用于比较一个样本与一个参考概率分布（单一样本K-S检验），或比较两个样本（两个样本的K-S检验）。它是以安德雷·柯尔莫哥洛夫 Andrey Kolmogorov和尼古莱·斯米尔诺夫 Nikolai Smirnov的名字命名。<br />
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The Kolmogorov–Smirnov statistic quantifies a [[metric (mathematics)|distance]] between the [[empirical distribution function]] of the sample and the [[cumulative distribution function]] of the reference distribution, or between the empirical distribution functions of two samples. The [[null distribution]] of this statistic is calculated under the [[null hypothesis]] that the sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered under the null hypothesis may be continuous (see [[#Kolmogorov distribution|Section 2]]), purely discrete or mixed (see [[#Discrete and mixed null distribution|Section 2.2]]). In the two-sample case (see [[#Two-sample Kolmogorov–Smirnov test|Section 3]]), the distribution considered under the null hypothesis is a continuous distribution but is otherwise unrestricted.<br />
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The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The null distribution of this statistic is calculated under the null hypothesis that the sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered under the null hypothesis may be continuous (see Section 2), purely discrete or mixed (see Section 2.2). In the two-sample case (see Section 3), the distribution considered under the null hypothesis is a continuous distribution but is otherwise unrestricted.<br />
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Kolmogorov-Smirnov统计定量描述了一个样本分布的'''<font color="#ff8000">经验分布函数 Empirical distribution function</font>'''与一个参考分布的'''<font color="#ff8000">累积分布函数 Cumulative distribution function</font>'''之间的距离，或者是两个样本分布的经验分布函数之间的距离。该统计量的'''<font color="#ff8000">零分布 Null distribution</font>'''是基于'''<font color="#ff8000">零假设Null hypothesis</font>'''（或称原始假设）下计算的，可以从参考分布中抽取样本（在单个样本的情况下），或者从相同分布中抽取样本组（在两个样本的情况下）。在单样本情况下，零假设（原假设）考虑的分布可能是连续的（请参阅第2节），纯离散的或混合的（请参阅第2.2节）。然而在考虑两个样本情况下（请参阅第3节），零假设下的分布仅能确定为连续分布，在其他方面并不受限制。<br />
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The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.<br />
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The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.<br />
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K–S双样本检验是比较两个样本分布最有用，也是最通用的非参数方法之一，因为在对比两个样本时，K-S检验对其经验累积分布函数的位置和形状差异具有一定的敏感性。<br />
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The Kolmogorov–Smirnov test can be modified to serve as a [[goodness of fit]] test. In the special case of testing for [[Normal distribution|normality]] of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic (see [[#Test with estimated parameters|Test with estimated parameters]]). Various studies have found that, even in this corrected form, the test is less powerful for testing normality than the [[Shapiro–Wilk test]] or [[Anderson–Darling test]].<br />
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The Kolmogorov–Smirnov test can be modified to serve as a goodness of fit test. In the special case of testing for normality of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic (see Test with estimated parameters). Various studies have found that, even in this corrected form, the test is less powerful for testing normality than the Shapiro–Wilk test or Anderson–Darling test. However, these other tests have their own disadvantages. For instance the Shapiro–Wilk test is known not to work well in samples with many identical values.<br />
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Kolmogorov–Smirnov检验经过修改以后可以作为'''<font color="#ff8000">拟合优度检验 goodness of fit test</font>'''。在测试分布正态性的特殊情况下，将样本先标准化再与标准正态分布进行比较。这相当于将参考分布的均值和方差设置为与样本估计值相等。显然，使用这些值和方差来定义特定参考分布会更改检验统计量的零分布（请参阅使用估算参数进行检验）。各种研究发现，即使采用这种校正形式，该测试也不能像夏皮罗一威尔克 Shapiro-Wilk检验或安德森·达林 Anderson-Darling检验那样有效地检验正态性。当然，这些检验方法也有其自身的缺点。例如，Shapiro–Wilk检验在具有许多相同值的样本中效果并不好。<br />
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== Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计==<br />
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The [[empirical distribution function]] ''F''<sub>''n''</sub> for ''n'' [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) ordered observations ''X<sub>i</sub>'' is defined as<br />
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n个独立且同分布（i.i.d.）的有序观测值Xi的经验分布函数Fn定义为：<br />
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<math>F_n(x)={1 \over n}\sum_{i=1}^n I_{[-\infty,x]}(X_i)</math><br />
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where I_{[-\infty,x]}(X_i) is the indicator function, equal to 1 if X_i \le x and equal to 0 otherwise.<br />
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其中 {\displaystyle I_{[-\infty ,x]}(X_{i})}I_{[-\infty ,x]}(X_{i})是指标函数，如果 {\displaystyle X_{i}\leq x}X_{i}\leq x等于1，否则等于0。<br />
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The Kolmogorov–Smirnov statistic for a given cumulative distribution function F(x) is<br />
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给定累积分布函数F（x）的Kolmogorov–Smirnov统计量为：<br />
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D_n= \sup_x |F_n(x)-F(x)|<br />
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where supx is the supremum of the set of distances. By the Glivenko–Cantelli theorem, if the sample comes from distribution F(x), then Dn converges to 0 almost surely in the limit when n goes to infinity. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see Kolmogorov distribution). Donsker's theorem provides a yet stronger result.<br />
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其中supx是距离集的上限值。根据格利文科·坎泰利 Glivenko-Cantelli定理，如果样本来自分布F（x），则当n变为无穷大时，Dn几乎肯定会收敛于0。科尔莫戈罗夫 Kolmogorov通过有效加入收敛速率来增强此结果（请参阅Kolmogorov分布）。另外Donsker定理提供了更有力的结果。<br />
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In practice, the statistic requires a relatively large number of data points (in comparison to other goodness of fit criteria such as the Anderson–Darling test statistic) to properly reject the null hypothesis.<br />
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在实践中，该统计需要相对大量的数据点（与其他拟合优度标准相比，例如Anderson-Darling检验统计）才能正确地拒绝零假设。<br />
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==Kolmogorov distribution Kolmogorov 分布==<br />
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The Kolmogorov distribution is the distribution of the [[random variable]]<br />
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Kolmogorov分布是随机变量的分布<br />
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<math>K=\sup_{t\in[0,1]}|B(t)|</math><br />
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where B(t) is the Brownian bridge. The cumulative distribution function of K is given by<br />
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其中B（t）是布朗 Brownian桥。K的累积分布函数为<br />
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\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)},<br />
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which can also be expressed by the Jacobi theta function \vartheta_{01}(z=0;\tau=2ix^2/\pi). Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Smirnov. Recurrence relations for the distribution of the test statistic in finite samples are available.<br />
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也可以用雅可比θ Jacobi theta函数{\displaystyle \vartheta _{01}(z=0;\tau =2ix^{2}/\pi )}表示.在零假设下，Andrey Kolmogorov定义并规范了Kolmogorov–Smirnov检验统计量的形式及其渐近分布，Nikolai Smirnov则规范了分布表。这里可以运用有限样本中检验统计量分布的递归关系。<br />
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Under null hypothesis that the sample comes from the hypothesized distribution ''F''(''x''),<br />
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当样本来自假设分布F（x）的零假设下，<br />
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\operatorname{Pr}(K\leq K_\alpha)=1-\alpha.\,<br />
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:<math>\sqrt{n}D_n\xrightarrow{n\to\infty}\sup_t |B(F(t))|</math><br />
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[[convergence of random variables|in distribution]], where ''B''(''t'') is the [[Brownian bridge]].<br />
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在其分布中，B（t）指的是布朗桥。<br />
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If ''F'' is continuous then under the null hypothesis <math>\sqrt{n}D_n</math> converges to the Kolmogorov distribution, which does not depend on ''F''. This result may also be known as the Kolmogorov theorem. The accuracy of this limit as an approximation to the exact cdf of <math>K</math> when <math>n</math> is finite is not very impressive: even when <math>n=1000</math>, the corresponding maximum error is about <math>0.9\%</math>; this error increases to <math>2.6\%</math> when <math>n=100</math> and to a totally unacceptable <math>7\%</math> when <math>n=10</math>. However, a very simple expedient of replacing <math>x</math> by <br />
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如果F是连续的，则在原假设{\displaystyle {\sqrt {n}}D_{n}}下收敛到不依赖于F的Kolmogorov分布。该结果也称为Kolmogorov定理。当n为有限时，此极限的精确度近似为K的确切累积分布函数，效果并不十分令人满意：即使n = 1000，相应的最大误差约为0.9%。在n=100时，此误差增加到2.6%，在n=10时增加到完全不可接受的7%。但是，如果简单地将x替换为<br />
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:<math>x+\frac{1}{6\sqrt{n}}+ \frac{x-1}{4n}</math><br />
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in the argument of the Jacobi theta function reduces these errors to <math>0.003\%</math>, <math>0.027\%</math>, and <math>0.27\%</math> respectively; such accuracy would be usually considered more than adequate for all practical applications.<br />
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在Jacobi theta函数的参数e中，将这些误差分别减小到0.003％，0.027％和0.27％；该精度足以满足现阶段所有实际应用，。<br />
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The ''goodness-of-fit'' test or the Kolmogorov–Smirnov test can be constructed by using the critical values of the Kolmogorov distribution. This test is asymptotically valid when <math>n \to\infty</math>. It rejects the null hypothesis at level <math>\alpha</math> if<br />
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拟合优度检验或Kolmogorov–Smirnov检验可通过使用Kolmogorov分布的临界值来构建。当{\displaystyle n\to \infty }时，该检验是渐近有效的。如果条件为{\displaystyle {\sqrt {n}}D_{n}>K_{\alpha },\,}，它会拒绝{\displaystyle \alpha }等级上的零假设。<br />
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where ''K''<sub>''α''</sub> is found from<br />
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即Kα为：<br />
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The asymptotic [[statistical power|power]] of this test is 1.<br />
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这个检验的渐近幂是1。<br />
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Fast and accurate algorithms to compute the cdf <math>\operatorname{Pr}(D_n \leq x)</math> or its complement for arbitrary <math>n</math> and <math>x</math>, are available from:<br />
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[6] and [7] for continuous null distributions with code in C and Java to be found in [6].<br />
[8] for purely discrete, mixed or continuous null distribution implemented in the KSgeneral package [9] of the R project for statistical computing, which for a given sample also computes the KS test statistic and its p-value. Alternative C++ implementation is available from [8].<br />
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用于计算任意n和x的累积分布函数<math>\operatorname{Pr}(D_n \leq x)</math>或其补数的快速准确的算法可以从以下获取:<br />
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* 统计软件期刊2011年Journal of Statistical Software刊登的Simard R, L'Ecuyer P的文章《计算双向Kolmogorov–Smirnov分布》以及统计与概率通信期刊2017年刊登的Moscovich A, Nadler B 的文章《快速计算泊松过程的边界穿越概率》。关于连续零分布的C和Java代码实现可以在文章《计算双向Kolmogorov–Smirnov分布》中找到。<br />
* 统计软件期刊2019年Journal of Statistical Software刊登的Dimitrova DS, Kaishev VK, Tan S的文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》和Dimitrova, Dimitrina; Kaishev, Vladimir; Tan, Senren.的文章《KSgeneral：计算（离散）连续零分布的K-S检验的P值》。R工程KSgeneral软件包中实现的纯离散，混合或连续零分布，可以进行统计计算，对于给定的样本，它还可以计算KS检验统计量及其p值。或者，可以从文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》中获得替代的C++实现。<br />
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=== Test with estimated parameters 用估计的参数进行测试 ===<br />
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If either the form or the parameters of ''F''(''x'') are determined from the data ''X''<sub>''i''</sub> the critical values determined in this way are invalid. In such cases, [[Monte Carlo method|Monte Carlo]] or other methods may be required, but tables have been prepared for some cases. Details for the required modifications to the test statistic and for the critical values for the [[normal distribution]] and the [[exponential distribution]] have been published, and later publications also include the [[Gumbel distribution]]. The [[Lilliefors test]] represents a special case of this for the normal distribution. The logarithm transformation may help to overcome cases where the Kolmogorov test data does not seem to fit the assumption that it came from the normal distribution.<br />
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如果以数据Xi来确定F（x）的形式或参数，则以这种方式确定的临界值是无效的。在这种情况下，可能需要蒙特卡洛 Monte Carlo或其他方法，不过数据表格已经做了多个情况下的准备。业界目前已经发布了对测试统计量的必要修正细节以及正态分布和指数分布临界值的具体信息，以后的出版物还包括耿贝尔 Gumbel分布。另外莉莉福斯 Lilliefors检验代表正态分布的一种特殊情况。另外为了克服Kolmogorov检验数据可能不符合来自正态分布假设的情况，可以进行对数变换。<br />
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Using estimated parameters, the questions arises which estimation method should be used. Usually this would be the maximum likelihood method, but e.g. for the normal distribution MLE has a large bias error on sigma. Using a moment fit or KS minimization instead has a large impact on the critical values, and also some impact on test power. If we need to decide for Student-T data with df&nbsp;=&nbsp;2 via KS test whether the data could be normal or not, then a ML estimate based on H<sub>0</sub> (data is normal, so using the standard deviation for scale) would give much larger KS distance, than a fit with minimum KS. In this case we should reject H<sub>0</sub>, which is often the case with MLE, because the sample standard deviation might be very large for T-2 data, but with KS minimization we may get still a too low KS to reject&nbsp;H<sub>0</sub>. In the Student-T case, a modified KS test with KS estimate instead of MLE, makes the KS test indeed slightly worse. However, in other cases, such a modified KS test leads to slightly better test power.<br />
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想要使用估计参数值，自然而然会出现应该使用哪种估计方法的问题。通常情况下，采用的是最大似然法，但对于如正态分布，最大似然法在sigma上具有较大的偏差。而使用矩量拟合或KS最小化来替代则对临界值有很大影响，并且对检验功效也有一定影响。如果我们需要通过KS测试来确定df = 2的Student-T数据是否正常，那么基于H0的最大似然率估计（数据是正常的，因此使用标度的标准偏差）会得出更大的KS距离，从而不符合最小KS的拟合。在这种情况下，我们应该拒绝H0，在最大似然法中通常是这样，因为对于T-2数据而言，样本标准偏差可能非常大，但是如果将KS最小化，我们可能会得到太低的KS而无法拒绝H0。在Student-T情况下，用KS估计而不是最大似然法来进行改进的KS检验会使其效果稍差一些。但是在其他情况下，经过改良的KS检测会得到更好的检验功效。<br />
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=== Discrete and mixed null distribution 离散和混合零分布 ===<br />
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Under the assumption that <math>F(x)</math> is non-decreasing and right-continuous, with countable (possibly infinite) number of jumps, the KS test statistic can be expressed as:<br />
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假设F（x）是非递减且右连续的，且具有可数（可能是无限）的跳变次数，则KS检验统计量可表示为：<br />
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:<math>D_n= \sup_x |F_n(x)-F(x)| = \sup_{0 \leq t \leq 1} |F_n(F^{-1}(t)) - F(F^{-1}(t))|. </math><br />
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From the right-continuity of <math>F(x)</math>, it follows that <math>F(F^{-1}(t)) \geq t</math> and <math>F^{-1}(F(x)) \leq x </math> and hence, the distribution of <math>D_{n}</math> depends on the null distribution <math>F(x)</math>, i.e., is no longer distribution-free as in the continuous case. Therefore, a fast and accurate method has been developed to compute the exact and asymptotic distribution of <math>D_{n}</math> when <math>F(x)</math> is purely discrete or mixed , implemented in C++ and in the KSgeneral package of the [[R (programming language)|R language]]. The functions <code>disc_ks_test()</code>, <code>mixed_ks_test()</code> and <code>cont_ks_test()</code> compute also the KS test statistic and p-values for purely discrete, mixed or continuous null distributions and arbitrary sample sizes. The KS test and its p-values for discrete null distributions and small sample sizes are also computed in as part of the dgof package of the R language. Major statistical packages among which [[SAS (software)|SAS]] <code>PROC NPAR1WAY</code> , [[Stata]] <code>ksmirnov</code> implement the KS test under the assumption that <math>F(x)</math> is continuous, which is more conservative if the null distribution is actually not continuous (see [15] [16] [17]).<br />
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从F（x）的右连续性，可以得出<math>F(F^{-1}(t)) \geq t</math>和<math>F^{-1}(F(x)) \leq x </math>，因此Dn的分布取决于零分布F（x），即在连续情况下不再无分布。目前已经开发出一种快速，准确的方法，以C ++和R语言的KSgeneral软件包来实现，当F（x）是纯离散或混合时，可以计算出Dn的精确且渐近分布。函数disc_ks_test（），mixed_ks_test（）和cont_ks_test（）还可以针对纯离散，混合或连续的零分布和任意样本大小，计算出KS检测统计量和p值。另外作为R语言的dgof软件包的一部分，还可以计算出KS检测及其用于离散零分布和小样本量的p值。关于主要统计软件包，其中SAS PROC NPAR1WAY和Stata ksmirnov是假设F（x）是连续的，因此执行KS检验时，如果零分布实际上不是连续的，则该检验更为保守。详情请见：<br />
1. 《关于离散案例中的Kolmogorov统计量的注释Note on the Kolmogorov Statistic in the Discrete Case》<br />
2. 《皮尔逊卡方检验和Kolmogorov拟合优度检验在有效性方面的比较A Comparison of the Pearson Chi-Square and Kolmogorov Goodness-of-Fit Tests with Respect to Validity》<br />
3. 《Kolmogorov–Smirnov的有限概率性质和离散数据的相似统计量Bounded Probability Properties of Kolmogorov–Smirnov and Similar Statistics for Discrete Data》<br />
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== Two-sample Kolmogorov–Smirnov test 双样本Kolmogorov–Smirnov检验 ==<br />
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[[文件:KS2 Example.png|缩略图|右|关于两个样本的Kolmogorov–Smirnov统计量的图解。 红线和蓝线分别对应于经验分布函数，而黑色箭头是两样本的KS统计量。]]<br />
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The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the Kolmogorov–Smirnov statistic is<br />
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Kolmogorov–Smirnov检验也可用于检验两个潜在的一维概率分布是否不同。在这种情况下，Kolmogorov-Smirnov统计量为<br />
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<math>D_{n,m}=\sup_x |F_{1,n}(x)-F_{2,m}(x)|,</math><br />
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where <math>F_{1,n}</math> and <math>F_{2,m}</math> are the [[empirical distribution function]]s of the first and the second sample respectively, and <math>\sup</math> is the [[Infimum and supremum|supremum function]].<br />
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其中F1，n和F2，m分别是第一样本和第二样本的经验分布函数，而sup是上确界函数。<br />
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For large samples, the null hypothesis is rejected at level <math>\alpha</math> if<br />
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对于量大的样本，如果满足以下条件，则原假设在α级被拒绝<br />
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:<math>D_{n,m}>c(\alpha)\sqrt{\frac{n + m}{n\cdot m}}.</math><br />
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Where <math>n</math> and <math>m</math> are the sizes of first and second sample respectively. The value of <math>c({\alpha})</math> is given in the table below for the most common levels of <math>\alpha</math><br />
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其中n和m分别是第一样本和第二样本的大小。下表给出了在α级最常见的α值c（α）：<br />
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| <math>\alpha</math> || 0.20 || 0.15 || 0.10 || 0.05 || 0.025 || 0.01 || 0.005 || 0.001<br />
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| <math>c({\alpha})</math> || 1.073 || 1.138 || 1.224 || 1.358 || 1.48 || 1.628 || 1.731 || 1.949<br />
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|}<br />
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and in general by<br />
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一般而言：<br />
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:<math>c\left(\alpha\right)=\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1}{2}},</math><br />
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so that the condition reads<br />
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以至于条件为：<br />
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:<math>D_{n,m}>\frac{1}{\sqrt{n}}\cdot\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1 + \tfrac{n}{m}}{2}}.</math><br />
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Here, again, the larger the sample sizes, the more sensitive the minimal bound: For a given ratio of sample sizes (e.g. m=n), the minimal bound scales in the size of either of the samples according to its inverse square root. <br />
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同样，样本量越大，最小界限越敏感：对于给定比率的样本大小（例如m = n），最小界限根据其平方根的倒数来缩放两个样本的大小。<br />
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Note that the two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is (e.g. whether it's normal or not normal). Again, tables of critical values have been published. A shortcoming of the Kolmogorov–Smirnov test is that it is not very powerful because it is devised to be sensitive against all possible types of differences between two distribution functions. and showed evidence that the Cucconi test, originally proposed for simultaneously comparing location and scale, is much more powerful than the Kolmogorov–Smirnov test when comparing two distribution functions.<br />
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这里要注意的是两个样本检验出来的数据样本是否来自同一分布。其并未指定该共同分布是什么（例如，它是正常还是不正常）。而且关键值表已经得出。Kolmogorov–Smirnov检验没有那么有效，因为它被设计为对两个分布函数之间所有可能的差异敏感。如刊登在Journal of Nonparametric Statistics2009年刊上Marozzi, Marco (2009)的文章《Some Notes on the Location-Scale Cucconi Test》和刊登在Communications in Statistics – Simulation and Computation2013年刊上同样Marozzi, Marco (2009)的文章《Nonparametric Simultaneous Tests for Location and Scale Testing: a Comparison of Several Methods显示了证据，当比较两个分布函数时，最初建议同时比较位置和比例的Cucconi检验比Kolmogorov-Smirnov检验更有效。<br />
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== Setting confidence limits for the shape of a distribution function 为分布函数的形状设置置信极限 ==<br />
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{{main article|Dvoretzky–Kiefer–Wolfowitz inequality}}<br />
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While the Kolmogorov–Smirnov test is usually used to test whether a given ''F''(''x'') is the underlying probability distribution of ''F''<sub>''n''</sub>(''x''), the procedure may be inverted to give confidence limits on ''F''(''x'') itself. If one chooses a critical value of the test statistic ''D''<sub>''α''</sub> such that P(''D''<sub>''n''</sub>&nbsp;>&nbsp;''D''<sub>''α''</sub>) = ''α'', then a band of width ±''D''<sub>''α''</sub> around ''F''<sub>''n''</sub>(''x'') will entirely contain ''F''(''x'') with probability 1&nbsp;−&nbsp;''α''.<br />
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虽然通常使用Kolmogorov–Smirnov检验法来检验给定的F（x）是否为Fn（x）的潜在概率分布，但可以将过程倒过来给出F（x）本身的置信极限。如果选择检验统计量Dα的临界值，使得P（Dn>Dα）=α，则在Fn（x）周围宽度±Dα内将完全包含概率为1-α的F（x）。<br />
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== The Kolmogorov–Smirnov statistic in more than one dimension 多个维度的Kolmogorov–Smirnov统计 ==<br />
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A distribution-free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997). The test uses a statistic which is built using Rosenblatt's transformation, and an algorithm is developed to compute it in the bivariate case. An approximate test that can be easily computed in any dimension is also presented.<br />
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朱斯特尔 Justel，培尼亚 Peña和扎马 Zamar（1997）提出了无分布的多元Kolmogorov-Smirnov拟合优度检验。该检验使用通过Rosenblatt变换建立的统计量，开发出了一种算法来计算双变量情况。还介绍了可以在任何维度上轻松计算的近似检测法。<br />
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The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to [[multivariate statistics|multivariate data]]. This is not straightforward because the maximum difference between two joint [[cumulative distribution function]]s is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of <math>\Pr(x < X \land y < Y)</math> or <math>\Pr(X < x \land Y > y)</math> or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.<br />
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如果要将类似的检验应用于多元数据，则需要修改Kolmogorov–Smirnov检验统计量。过程略显复杂，因为两个联合累积分布函数之间的最大差异通常与任何互补分布函数中的最大差异都不相同。因此，最大差异将取决于使用<math>\Pr(x < X \land y < Y)</math>或<math>\Pr(X < x \land Y > y)</math>中的哪一个，或者使用其他两种可能分布中的任何一种。当然有可能要求所用的检测结果无关于这样的选择。<br />
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One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In ''d'' dimensions, there are 2<sup>''d''</sup>−1 such orderings. One such variation is due to Peacock(see also Gosset for a 3D version) and another to Fasano and Franceschini (see Lopes et al. for a comparison and computational details). Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.<br />
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在满足以上要求的同时，将Kolmogorov-Smirnov统计量泛化为更高维度的一种方法是，在所有可能的排序中比较两个样本的累积分布函数，并从所得的K-S统计量中取最大。在d维数据中，有2d-1个这样的排序。皮柯克 Peacock得出了一种这样的变化量（有关3D版本，另请参见Gosset），另一种由法萨诺 Fasano和弗朗切斯基尼 Franceschini得出（有关比较和计算细节，请参见Lopes等人）。检测统计量的临界值可以通过仿真获取，但取决于联合分布中的依存关系结构。<br />
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In one dimension, the Kolmogorov–Smirnov statistic is identical to the so-called star discrepancy D, so another native KS extension to higher dimensions would be simply to use D also for higher dimensions. Unfortunately, the star discrepancy is hard to calculate in high dimensions.<br />
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一维的Kolmogorov-Smirnov统计量与所谓的星差D相同，因此，另一个对更高维度的本地KS扩展是将D也用于更高维度。可惜的是，很难从高维度上计算出星差。<br />
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== Implementations 软件实现==<br />
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The Kolmogorov-Smirnov test (one or two sampled test verifies the equality of distributions) is implemented in many software programs:<br />
Kolmogorov-Smirnov检验（一个或两个抽样检验可验证分布是否相等）可以在众多软件程序中实现：<br />
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* [[Mathematica]] has [https://reference.wolfram.com/language/ref/KolmogorovSmirnovTest.html KolmogorovSmirnovTest]<br />
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* [[MATLAB]] has [https://de.mathworks.com/help/stats/kstest.html kstest] in its Statistics Toolbox.<br />
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* The [[R (programming language)|R]] package "KSgeneral"<ref name=KSgeneral/> computes the KS test statistics and its p-values under arbitrary, possibly discrete, mixed or continuous null distribution.<br />
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* [[R (programming language)|R]]'s statistics base-package implements the test as [https://stat.ethz.ch/R-manual/R-patched/library/stats/html/ks.test.html ks.test {stats}] in its "stats" package.<br />
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* [[SAS (software)|SAS]] implements the test in its PROC NPAR1WAY procedure.<br />
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* [[Python (programming language)|Python]] has an implementation of this test provided by [[SciPy]]<ref>{{cite web |url= https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.kstest.html |title=scipy.stats.kstest |work=SciPy SciPy v0.14.0 Reference Guide |publisher=The Scipy community |access-date= 18 June 2019}}</ref> by Statistical functions (scipy.stats)<br />
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* [[SYSTAT (statistics)|SYSTAT]] (SPSS Inc., Chicago, IL)<br />
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* [[Java (programming language)|Java]] has an implementation of this test provided by [[Apache Commons]]<ref>{{cite web |url=https://commons.apache.org/proper/commons-math/javadocs/api-3.5/org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html |title=KolmogorovSmirnovTes|access-date= 18 June 2019}}</ref><br />
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* [[KNIME]] has a node implementing this test based on the above Java implementation<ref>{{cite web |url=https://www.knime.com/whats-new-in-knime-37#new-statistics-nodes |title=New statistics nodes |access-date= 25 June 2020}}</ref><br />
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* [[StatsDirect]] (StatsDirect Ltd, Manchester, UK) implements [https://www.statsdirect.com/help/nonparametric_methods/smirnov.htm all common variants].<br />
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* [[Stata]] (Stata Corporation, College Station, TX) implements the test in ksmirnov (Kolmogorov–Smirnov equality-of-distributions test) command. <ref> {{ cite web |url=https://www.stata.com/manuals15/rksmirnov.pdf|title=ksmirnov — Kolmogorov –Smirnov equality-of-distributions test |access-date= 18 June 2019}} </ref><br />
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* [[PSPP]] implements the test in its [https://www.gnu.org/software/pspp/manual/html_node/KOLMOGOROV_002dSMIRNOV.html KOLMOGOROV-SMIRNOV (or using K-S shortcut] function.<br />
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* [[Microsoft Excel|Excel]] runs the test as KSCRIT and KSPROB<br />
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* 科学计算软件Wolfram Mathematica内含Kolmogorov Smirnov Test。<br />
* MATLAB在其统计工具箱中含有K-S test。<br />
* R语言包“KSgeneral”可以在任意，可能离散，混合或连续的零分布下计算KS检验统计信息及其p值。<br />
* R语言的统计基本程序包在其“stats”程序包中可以运行检验，命令为ks.test {stats}。<br />
* 统计软件SAS在其PROC NPAR1WAY程序中可以实现检验。<br />
* Python通过SciPy中的统计功能（scipy.stats）可以实现检验。<br />
* SYSTAT（SPSS Inc.，伊利诺伊州芝加哥）<br />
* 基于Java语言开发的Apache Commons可以实现检验。<br />
* KNIME是一个数据分析平台，基于上述Java语言，可以通过一个节点Node来实现检验。<br />
* StatsDirect（StatsDirect Ltd，英国曼彻斯特）包含所有常见的变体。<br />
* Stata（德克萨斯州大学城Stata公司）在ksmirnov（Kolmogorov–Smirnov分配均等测试）中执行检验命令。<br />
* PSPP通过其Kolmogorov–Smirnov（或使用K-S快捷功能）实施检验。<br />
* Excel以KSCRIT和KSPROB的形式运行检验。<br />
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== See also 其他参考资料 ==<br />
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*[[Lepage test]]<br />
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*[[Cucconi test]]<br />
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*[[Kuiper's test]]<br />
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*[[Shapiro–Wilk test]]<br />
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*[[Anderson–Darling test]]<br />
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*[[Cramér–von Mises criterion|Cramér–von Mises test]]<br />
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* 勒帕热 Lepage检验<br />
* Cucconi检验<br />
* 柯伊伯 Kuiper检验<br />
* 夏皮罗-威尔克 Shapiro–Wilk检验<br />
* 安德森-达林 Anderson–Darling检验<br />
* 克莱姆·冯·米塞斯 Cramér–von Mises检验<br />
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== References 参考文献 ==<br />
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{{Reflist|30em}}<br />
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== Further reading 其他资料==<br />
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* Daniel, Wayne W. (1990). "Kolmogorov–Smirnov one-sample test". Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 319–330. ISBN 978-0-534-91976-4.<br />
* Eadie, W.T.; D. Drijard; F.E. James; M. Roos; B. Sadoulet (1971). Statistical Methods in Experimental Physics. Amsterdam: North-Holland. pp. 269–271. ISBN 978-0-444-10117-4.<br />
* Stuart, Alan; Ord, Keith; Arnold, Steven [F.] (1999). Classical Inference and the Linear Model. Kendall's Advanced Theory of Statistics. 2A (Sixth ed.). London: Arnold. pp. 25.37–25.43. ISBN 978-0-340-66230-4. MR 1687411.<br />
* Corder, G. W.; Foreman, D. I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley. ISBN 978-1118840313.<br />
* Stephens, M. A. (1979). "Test of fit for the logistic distribution based on the empirical distribution function". Biometrika. 66 (3): 591–595. doi:10.1093/biomet/66.3.591.<br />
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* Daniel, Wayne W. (1990). " Kolmogorov–Smirnov单样本检验. Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 319–330. ISBN 978-0-534-91976-4.<br />
* Eadie, W.T.; D. Drijard; F.E. James; M. Roos; B. Sadoulet (1971). 实验物理中的统计方法. Amsterdam: North-Holland. pp. 269–271. ISBN 978-0-444-10117-4.<br />
* Stuart, Alan; Ord, Keith; Arnold, Steven [F.] (1999). 经典推论和线性模型。肯德尔Kendall高级统计理论. 2A (Sixth ed.). London: Arnold. pp. 25.37–25.43. ISBN 978-0-340-66230-4. MR 1687411.<br />
* Corder, G. W.; Foreman, D. I. (2014). 非参数统计：分步法. Wiley. ISBN 978-1118840313.<br />
* Stephens, M. A. (1979). "基于经验分布函数的逻辑分布拟合检验". Biometrika. 66 (3): 591–595. doi:10.1093/biomet/66.3.591.<br />
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== External links 相关链接 ==<br />
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* "Kolmogorov–Smirnov test", Encyclopedia of Mathematics, EMS Press, 2001 [1994]<br />
* Short introduction<br />
* KS test explanation<br />
* JavaScript implementation of one- and two-sided tests<br />
* Online calculator with the KS test<br />
* Open-source C++ code to compute the Kolmogorov distribution and perform the KS test<br />
* Paper on Evaluating Kolmogorov’s Distribution; contains C implementation. This is the method used in Matlab.<br />
* Paper on Computing the Two-Sided Kolmogorov–Smirnov Distribution; computing the cdf of the KS statistic in C or Java.<br />
* Paper powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions; Jeff Alstott, Ed Bullmore, Dietmar Plenz. Among others, it also performs the Kolmogorov–Smirnov test. Source code and installers of powerlaw package are available at PyPi<br />
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* “Kolmogorov–Smirnov测验”，数学百科全书，EMS出版社，2001[1994]<br />
* KS简短介绍<br />
* KS检验说明<br />
* 单边和双边检验的JavaScript实现<br />
* KS检验在线计算器<br />
* 开源C++代码可计算Kolmogorov分布并执行KS检验<br />
* 关于评估Kolmogorov分布的论文，包含C实现。这是Matlab中常使用的方法。<br />
* 计算双向Kolmogorov-Smirnov分布的论文；用C或Java计算KS统计信息的累积分布函数。<br />
* Paper powerlaw：用于分析重尾分布的Python包； Jeff Alstott，Ed Bullmore和Dietmar Plenz。除此之外，它还执行Kolmogorov-Smirnov检验。PyPi提供了powerlaw软件包的源代码和安装程序。</div>Lincenthttps://wiki.swarma.org/index.php?title=KS%E6%A3%80%E9%AA%8C&diff=20724KS检验2020-12-31T01:38:37Z<p>Lincent：</p>
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<div>此词条由Jie翻译。由Lincent审校。<br />
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{{Short description|Non-parametric statistical test between two distributions}}<br />
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[[文件:KS_Example.png|缩略图|右|Kolmogorov–Smirnov统计数据的图示。 红线是累积分布函数，蓝线是经验分布函数，黑色箭头是K–S统计量。]]<br />
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In [[statistics]], the '''Kolmogorov–Smirnov test''' ('''K–S test''' or '''KS test''') is a [[nonparametric statistics|nonparametric test]] of the equality of continuous (or discontinuous, see [[#Discrete and mixed null distribution|Section 2.2]]), one-dimensional [[probability distribution]]s that can be used to compare a [[random sample|sample]] with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). It is named after [[Andrey Kolmogorov]] and [[Nikolai Smirnov (mathematician)|Nikolai Smirnov]].<br />
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In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). It is named after Andrey Kolmogorov and Nikolai Smirnov.<br />
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在统计学中，'''<font color="#ff8000"> Kolmogorov–Smirnov检验</font>'''（K-S检验或KS检验）是一种连续（或不连续，请参见第2.2节）的一维概率分布均等性的非参数检验，可用于比较一个样本与一个参考概率分布（单一样本K-S检验），或比较两个样本（两个样本的K-S检验）。它是以安德雷·柯尔莫哥洛夫 Andrey Kolmogorov和尼古莱·斯米尔诺夫 Nikolai Smirnov的名字命名。<br />
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The Kolmogorov–Smirnov statistic quantifies a [[metric (mathematics)|distance]] between the [[empirical distribution function]] of the sample and the [[cumulative distribution function]] of the reference distribution, or between the empirical distribution functions of two samples. The [[null distribution]] of this statistic is calculated under the [[null hypothesis]] that the sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered under the null hypothesis may be continuous (see [[#Kolmogorov distribution|Section 2]]), purely discrete or mixed (see [[#Discrete and mixed null distribution|Section 2.2]]). In the two-sample case (see [[#Two-sample Kolmogorov–Smirnov test|Section 3]]), the distribution considered under the null hypothesis is a continuous distribution but is otherwise unrestricted.<br />
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The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The null distribution of this statistic is calculated under the null hypothesis that the sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered under the null hypothesis may be continuous (see Section 2), purely discrete or mixed (see Section 2.2). In the two-sample case (see Section 3), the distribution considered under the null hypothesis is a continuous distribution but is otherwise unrestricted.<br />
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Kolmogorov-Smirnov统计定量描述了一个样本分布的'''<font color="#ff8000">经验分布函数 Empirical distribution function</font>'''与一个参考分布的'''<font color="#ff8000">累积分布函数 Cumulative distribution function</font>'''之间的距离，或者是两个样本分布的经验分布函数之间的距离。该统计量的'''<font color="#ff8000">零分布 Null distribution</font>'''是基于'''<font color="#ff8000">零假设Null hypothesis</font>'''（或称原始假设）下计算的，可以从参考分布中抽取样本（在单个样本的情况下），或者从相同分布中抽取样本组（在两个样本的情况下）。在单样本情况下，零假设（原假设）考虑的分布可能是连续的（请参阅第2节），纯离散的或混合的（请参阅第2.2节）。然而在考虑两个样本情况下（请参阅第3节），零假设下的分布仅能确定为连续分布，在其他方面并不受限制。<br />
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The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.<br />
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The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.<br />
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K–S双样本检验是比较两个样本分布最有用，也是最通用的非参数方法之一，因为在对比两个样本时，K-S检验对其经验累积分布函数的位置和形状差异具有一定的敏感性。<br />
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The Kolmogorov–Smirnov test can be modified to serve as a [[goodness of fit]] test. In the special case of testing for [[Normal distribution|normality]] of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic (see [[#Test with estimated parameters|Test with estimated parameters]]). Various studies have found that, even in this corrected form, the test is less powerful for testing normality than the [[Shapiro–Wilk test]] or [[Anderson–Darling test]].<br />
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The Kolmogorov–Smirnov test can be modified to serve as a goodness of fit test. In the special case of testing for normality of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic (see Test with estimated parameters). Various studies have found that, even in this corrected form, the test is less powerful for testing normality than the Shapiro–Wilk test or Anderson–Darling test. However, these other tests have their own disadvantages. For instance the Shapiro–Wilk test is known not to work well in samples with many identical values.<br />
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Kolmogorov–Smirnov检验经过修改以后可以作为'''<font color="#ff8000">拟合优度检验 goodness of fit test</font>'''。在测试分布正态性的特殊情况下，将样本先标准化再与标准正态分布进行比较。这相当于将参考分布的均值和方差设置为与样本估计值相等。显然，使用这些值和方差来定义特定参考分布会更改检验统计量的零分布（请参阅使用估算参数进行检验）。各种研究发现，即使采用这种校正形式，该测试也不能像夏皮罗一威尔克 Shapiro-Wilk检验或安德森·达林 Anderson-Darling检验那样有效地检验正态性。当然，这些检验方法也有其自身的缺点。例如，Shapiro–Wilk检验在具有许多相同值的样本中效果并不好。<br />
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== Kolmogorov–Smirnov statistic Kolmogorov-Smirnov统计==<br />
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The [[empirical distribution function]] ''F''<sub>''n''</sub> for ''n'' [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) ordered observations ''X<sub>i</sub>'' is defined as<br />
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n个独立且同分布（i.i.d.）的有序观测值Xi的经验分布函数Fn定义为：<br />
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<math>F_n(x)={1 \over n}\sum_{i=1}^n I_{[-\infty,x]}(X_i)</math><br />
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where I_{[-\infty,x]}(X_i) is the indicator function, equal to 1 if X_i \le x and equal to 0 otherwise.<br />
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其中 {\displaystyle I_{[-\infty ,x]}(X_{i})}I_{[-\infty ,x]}(X_{i})是指标函数，如果 {\displaystyle X_{i}\leq x}X_{i}\leq x等于1，否则等于0。<br />
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The Kolmogorov–Smirnov statistic for a given cumulative distribution function F(x) is<br />
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给定累积分布函数F（x）的Kolmogorov–Smirnov统计量为：<br />
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D_n= \sup_x |F_n(x)-F(x)|<br />
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where supx is the supremum of the set of distances. By the Glivenko–Cantelli theorem, if the sample comes from distribution F(x), then Dn converges to 0 almost surely in the limit when n goes to infinity. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see Kolmogorov distribution). Donsker's theorem provides a yet stronger result.<br />
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其中supx是距离集的上限值。根据格利文科·坎泰利 Glivenko-Cantelli定理，如果样本来自分布F（x），则当n变为无穷大时，Dn几乎肯定会收敛于0。科尔莫戈罗夫 Kolmogorov通过有效加入收敛速率来增强此结果（请参阅Kolmogorov分布）。另外Donsker定理提供了更有力的结果。<br />
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In practice, the statistic requires a relatively large number of data points (in comparison to other goodness of fit criteria such as the Anderson–Darling test statistic) to properly reject the null hypothesis.<br />
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在实践中，该统计需要相对大量的数据点（与其他拟合优度标准相比，例如Anderson-Darling检验统计）才能正确地拒绝零假设。<br />
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==Kolmogorov distribution Kolmogorov 分布==<br />
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The Kolmogorov distribution is the distribution of the [[random variable]]<br />
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Kolmogorov分布是随机变量的分布<br />
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<math>K=\sup_{t\in[0,1]}|B(t)|</math><br />
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where B(t) is the Brownian bridge. The cumulative distribution function of K is given by<br />
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其中B（t）是布朗 Brownian桥。K的累积分布函数为<br />
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\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)},<br />
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which can also be expressed by the Jacobi theta function \vartheta_{01}(z=0;\tau=2ix^2/\pi). Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Smirnov. Recurrence relations for the distribution of the test statistic in finite samples are available.<br />
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也可以用雅可比θ Jacobi theta函数{\displaystyle \vartheta _{01}(z=0;\tau =2ix^{2}/\pi )}表示.在零假设下，Andrey Kolmogorov定义并规范了Kolmogorov–Smirnov检验统计量的形式及其渐近分布，Nikolai Smirnov则规范了分布表。这里可以运用有限样本中检验统计量分布的递归关系。<br />
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Under null hypothesis that the sample comes from the hypothesized distribution ''F''(''x''),<br />
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当样本来自假设分布F（x）的零假设下，<br />
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\operatorname{Pr}(K\leq K_\alpha)=1-\alpha.\,<br />
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:<math>\sqrt{n}D_n\xrightarrow{n\to\infty}\sup_t |B(F(t))|</math><br />
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[[convergence of random variables|in distribution]], where ''B''(''t'') is the [[Brownian bridge]].<br />
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在其分布中，B（t）指的是布朗桥。<br />
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If ''F'' is continuous then under the null hypothesis <math>\sqrt{n}D_n</math> converges to the Kolmogorov distribution, which does not depend on ''F''. This result may also be known as the Kolmogorov theorem. The accuracy of this limit as an approximation to the exact cdf of <math>K</math> when <math>n</math> is finite is not very impressive: even when <math>n=1000</math>, the corresponding maximum error is about <math>0.9\%</math>; this error increases to <math>2.6\%</math> when <math>n=100</math> and to a totally unacceptable <math>7\%</math> when <math>n=10</math>. However, a very simple expedient of replacing <math>x</math> by <br />
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如果F是连续的，则在原假设{\displaystyle {\sqrt {n}}D_{n}}下收敛到不依赖于F的Kolmogorov分布。该结果也称为Kolmogorov定理。当n为有限时，此极限的精确度近似为K的确切累积分布函数，效果并不十分令人满意：即使n = 1000，相应的最大误差约为0.9%。在n=100时，此误差增加到2.6%，在n=10时增加到完全不可接受的7%。但是，如果简单地将x替换为<br />
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:<math>x+\frac{1}{6\sqrt{n}}+ \frac{x-1}{4n}</math><br />
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in the argument of the Jacobi theta function reduces these errors to <math>0.003\%</math>, <math>0.027\%</math>, and <math>0.27\%</math> respectively; such accuracy would be usually considered more than adequate for all practical applications.<br />
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在Jacobi theta函数的参数e中，将这些误差分别减小到0.003％，0.027％和0.27％；该精度足以满足现阶段所有实际应用，。<br />
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The ''goodness-of-fit'' test or the Kolmogorov–Smirnov test can be constructed by using the critical values of the Kolmogorov distribution. This test is asymptotically valid when <math>n \to\infty</math>. It rejects the null hypothesis at level <math>\alpha</math> if<br />
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拟合优度检验或Kolmogorov–Smirnov检验可通过使用Kolmogorov分布的临界值来构建。当{\displaystyle n\to \infty }时，该检验是渐近有效的。如果条件为{\displaystyle {\sqrt {n}}D_{n}>K_{\alpha },\,}，它会拒绝{\displaystyle \alpha }等级上的零假设。<br />
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where ''K''<sub>''α''</sub> is found from<br />
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即Kα为：<br />
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The asymptotic [[statistical power|power]] of this test is 1.<br />
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这个检验的渐近幂是1。<br />
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Fast and accurate algorithms to compute the cdf <math>\operatorname{Pr}(D_n \leq x)</math> or its complement for arbitrary <math>n</math> and <math>x</math>, are available from:<br />
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[6] and [7] for continuous null distributions with code in C and Java to be found in [6].<br />
[8] for purely discrete, mixed or continuous null distribution implemented in the KSgeneral package [9] of the R project for statistical computing, which for a given sample also computes the KS test statistic and its p-value. Alternative C++ implementation is available from [8].<br />
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用于计算任意n和x的累积分布函数<math>\operatorname{Pr}(D_n \leq x)</math>或其补数的快速准确的算法可以从以下获取:<br />
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* 统计软件期刊2011年Journal of Statistical Software刊登的Simard R, L'Ecuyer P的文章《计算双向Kolmogorov–Smirnov分布》以及统计与概率通信期刊2017年刊登的Moscovich A, Nadler B 的文章《快速计算泊松过程的边界穿越概率》。关于连续零分布的C和Java代码实现可以在文章《计算双向Kolmogorov–Smirnov分布》中找到。<br />
* 统计软件期刊2019年Journal of Statistical Software刊登的Dimitrova DS, Kaishev VK, Tan S的文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》和Dimitrova, Dimitrina; Kaishev, Vladimir; Tan, Senren.的文章《KSgeneral：计算（离散）连续零分布的K-S检验的P值》。R工程KSgeneral软件包中实现的纯离散，混合或连续零分布，可以进行统计计算，对于给定的样本，它还可以计算KS检验统计量及其p值。或者，可以从文章《当潜在累积分布函数是完全离散，混合或连续时，计算Kolmogorov–Smirnov分布》中获得替代的C++实现。<br />
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=== Test with estimated parameters 用估计的参数进行测试 ===<br />
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If either the form or the parameters of ''F''(''x'') are determined from the data ''X''<sub>''i''</sub> the critical values determined in this way are invalid. In such cases, [[Monte Carlo method|Monte Carlo]] or other methods may be required, but tables have been prepared for some cases. Details for the required modifications to the test statistic and for the critical values for the [[normal distribution]] and the [[exponential distribution]] have been published, and later publications also include the [[Gumbel distribution]]. The [[Lilliefors test]] represents a special case of this for the normal distribution. The logarithm transformation may help to overcome cases where the Kolmogorov test data does not seem to fit the assumption that it came from the normal distribution.<br />
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如果以数据Xi来确定F（x）的形式或参数，则以这种方式确定的临界值是无效的。在这种情况下，可能需要蒙特卡洛 Monte Carlo或其他方法，不过数据表格已经做了多个情况下的准备。业界目前已经发布了对测试统计量的必要修正细节以及正态分布和指数分布临界值的具体信息，以后的出版物还包括耿贝尔 Gumbel分布。另外莉莉福斯 Lilliefors检验代表正态分布的一种特殊情况。另外为了克服Kolmogorov检验数据可能不符合来自正态分布假设的情况，可以进行对数变换。<br />
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Using estimated parameters, the questions arises which estimation method should be used. Usually this would be the maximum likelihood method, but e.g. for the normal distribution MLE has a large bias error on sigma. Using a moment fit or KS minimization instead has a large impact on the critical values, and also some impact on test power. If we need to decide for Student-T data with df&nbsp;=&nbsp;2 via KS test whether the data could be normal or not, then a ML estimate based on H<sub>0</sub> (data is normal, so using the standard deviation for scale) would give much larger KS distance, than a fit with minimum KS. In this case we should reject H<sub>0</sub>, which is often the case with MLE, because the sample standard deviation might be very large for T-2 data, but with KS minimization we may get still a too low KS to reject&nbsp;H<sub>0</sub>. In the Student-T case, a modified KS test with KS estimate instead of MLE, makes the KS test indeed slightly worse. However, in other cases, such a modified KS test leads to slightly better test power.<br />
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想要使用估计参数值，自然而然会出现应该使用哪种估计方法的问题。通常情况下，采用的是最大似然法，但对于如正态分布，最大似然法在sigma上具有较大的偏差。而使用矩量拟合或KS最小化来替代则对临界值有很大影响，并且对检验功效也有一定影响。如果我们需要通过KS测试来确定df = 2的Student-T数据是否正常，那么基于H0的最大似然率估计（数据是正常的，因此使用标度的标准偏差）会得出更大的KS距离，从而不符合最小KS的拟合。在这种情况下，我们应该拒绝H0，在最大似然法中通常是这样，因为对于T-2数据而言，样本标准偏差可能非常大，但是如果将KS最小化，我们可能会得到太低的KS而无法拒绝H0。在Student-T情况下，用KS估计而不是最大似然法来进行改进的KS检验会使其效果稍差一些。但是在其他情况下，经过改良的KS检测会得到更好的检验功效。<br />
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=== Discrete and mixed null distribution 离散和混合零分布 ===<br />
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Under the assumption that <math>F(x)</math> is non-decreasing and right-continuous, with countable (possibly infinite) number of jumps, the KS test statistic can be expressed as:<br />
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假设F（x）是非递减且右连续的，且具有可数（可能是无限）的跳变次数，则KS检验统计量可表示为：<br />
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:<math>D_n= \sup_x |F_n(x)-F(x)| = \sup_{0 \leq t \leq 1} |F_n(F^{-1}(t)) - F(F^{-1}(t))|. </math><br />
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From the right-continuity of <math>F(x)</math>, it follows that <math>F(F^{-1}(t)) \geq t</math> and <math>F^{-1}(F(x)) \leq x </math> and hence, the distribution of <math>D_{n}</math> depends on the null distribution <math>F(x)</math>, i.e., is no longer distribution-free as in the continuous case. Therefore, a fast and accurate method has been developed to compute the exact and asymptotic distribution of <math>D_{n}</math> when <math>F(x)</math> is purely discrete or mixed , implemented in C++ and in the KSgeneral package of the [[R (programming language)|R language]]. The functions <code>disc_ks_test()</code>, <code>mixed_ks_test()</code> and <code>cont_ks_test()</code> compute also the KS test statistic and p-values for purely discrete, mixed or continuous null distributions and arbitrary sample sizes. The KS test and its p-values for discrete null distributions and small sample sizes are also computed in as part of the dgof package of the R language. Major statistical packages among which [[SAS (software)|SAS]] <code>PROC NPAR1WAY</code> , [[Stata]] <code>ksmirnov</code> implement the KS test under the assumption that <math>F(x)</math> is continuous, which is more conservative if the null distribution is actually not continuous (see [15] [16] [17]).<br />
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从F（x）的右连续性，可以得出<math>F(F^{-1}(t)) \geq t</math>和<math>F^{-1}(F(x)) \leq x </math>，因此Dn的分布取决于零分布F（x），即在连续情况下不再无分布。目前已经开发出一种快速，准确的方法，以C ++和R语言的KSgeneral软件包来实现，当F（x）是纯离散或混合时，可以计算出Dn的精确且渐近分布。函数disc_ks_test（），mixed_ks_test（）和cont_ks_test（）还可以针对纯离散，混合或连续的零分布和任意样本大小，计算出KS检测统计量和p值。另外作为R语言的dgof软件包的一部分，还可以计算出KS检测及其用于离散零分布和小样本量的p值。关于主要统计软件包，其中SAS PROC NPAR1WAY和Stata ksmirnov是假设F（x）是连续的，因此执行KS检验时，如果零分布实际上不是连续的，则该检验更为保守。详情请见：<br />
1. 《关于离散案例中的Kolmogorov统计量的注释Note on the Kolmogorov Statistic in the Discrete Case》<br />
2. 《皮尔逊卡方检验和Kolmogorov拟合优度检验在有效性方面的比较A Comparison of the Pearson Chi-Square and Kolmogorov Goodness-of-Fit Tests with Respect to Validity》<br />
3. 《Kolmogorov–Smirnov的有限概率性质和离散数据的相似统计量Bounded Probability Properties of Kolmogorov–Smirnov and Similar Statistics for Discrete Data》<br />
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== Two-sample Kolmogorov–Smirnov test 双样本Kolmogorov–Smirnov检验 ==<br />
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[[文件:KS2 Example.png|缩略图|右|关于两个样本的Kolmogorov–Smirnov统计量的图解。 红线和蓝线分别对应于经验分布函数，而黑色箭头是两样本的KS统计量。]]<br />
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The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the Kolmogorov–Smirnov statistic is<br />
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Kolmogorov–Smirnov检验也可用于检验两个潜在的一维概率分布是否不同。在这种情况下，Kolmogorov-Smirnov统计量为<br />
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<math>D_{n,m}=\sup_x |F_{1,n}(x)-F_{2,m}(x)|,</math><br />
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where <math>F_{1,n}</math> and <math>F_{2,m}</math> are the [[empirical distribution function]]s of the first and the second sample respectively, and <math>\sup</math> is the [[Infimum and supremum|supremum function]].<br />
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其中F1，n和F2，m分别是第一样本和第二样本的经验分布函数，而sup是上确界函数。<br />
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For large samples, the null hypothesis is rejected at level <math>\alpha</math> if<br />
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对于量大的样本，如果满足以下条件，则原假设在α级被拒绝<br />
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:<math>D_{n,m}>c(\alpha)\sqrt{\frac{n + m}{n\cdot m}}.</math><br />
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Where <math>n</math> and <math>m</math> are the sizes of first and second sample respectively. The value of <math>c({\alpha})</math> is given in the table below for the most common levels of <math>\alpha</math><br />
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其中n和m分别是第一样本和第二样本的大小。下表给出了在α级最常见的α值c（α）：<br />
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| <math>\alpha</math> || 0.20 || 0.15 || 0.10 || 0.05 || 0.025 || 0.01 || 0.005 || 0.001<br />
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| <math>c({\alpha})</math> || 1.073 || 1.138 || 1.224 || 1.358 || 1.48 || 1.628 || 1.731 || 1.949<br />
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|}<br />
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and in general by<br />
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一般而言：<br />
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:<math>c\left(\alpha\right)=\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1}{2}},</math><br />
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so that the condition reads<br />
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以至于条件为：<br />
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:<math>D_{n,m}>\frac{1}{\sqrt{n}}\cdot\sqrt{-\ln\left(\tfrac{\alpha}{2}\right)\cdot \tfrac{1 + \tfrac{n}{m}}{2}}.</math><br />
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Here, again, the larger the sample sizes, the more sensitive the minimal bound: For a given ratio of sample sizes (e.g. m=n), the minimal bound scales in the size of either of the samples according to its inverse square root. <br />
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同样，样本量越大，最小界限越敏感：对于给定比率的样本大小（例如m = n），最小界限根据其平方根的倒数来缩放两个样本的大小。<br />
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Note that the two-sample test checks whether the two data samples come from the same distribution. This does not specify what that common distribution is (e.g. whether it's normal or not normal). Again, tables of critical values have been published. A shortcoming of the Kolmogorov–Smirnov test is that it is not very powerful because it is devised to be sensitive against all possible types of differences between two distribution functions. and showed evidence that the Cucconi test, originally proposed for simultaneously comparing location and scale, is much more powerful than the Kolmogorov–Smirnov test when comparing two distribution functions.<br />
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这里要注意的是两个样本检验出来的数据样本是否来自同一分布。其并未指定该共同分布是什么（例如，它是正常还是不正常）。而且关键值表已经得出。Kolmogorov–Smirnov检验没有那么有效，因为它被设计为对两个分布函数之间所有可能的差异敏感。如刊登在Journal of Nonparametric Statistics2009年刊上Marozzi, Marco (2009)的文章《Some Notes on the Location-Scale Cucconi Test》和刊登在Communications in Statistics – Simulation and Computation2013年刊上同样Marozzi, Marco (2009)的文章《Nonparametric Simultaneous Tests for Location and Scale Testing: a Comparison of Several Methods显示了证据，当比较两个分布函数时，最初建议同时比较位置和比例的Cucconi检验比Kolmogorov-Smirnov检验更有效。<br />
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== Setting confidence limits for the shape of a distribution function 为分布函数的形状设置置信极限 ==<br />
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{{main article|Dvoretzky–Kiefer–Wolfowitz inequality}}<br />
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While the Kolmogorov–Smirnov test is usually used to test whether a given ''F''(''x'') is the underlying probability distribution of ''F''<sub>''n''</sub>(''x''), the procedure may be inverted to give confidence limits on ''F''(''x'') itself. If one chooses a critical value of the test statistic ''D''<sub>''α''</sub> such that P(''D''<sub>''n''</sub>&nbsp;>&nbsp;''D''<sub>''α''</sub>) = ''α'', then a band of width ±''D''<sub>''α''</sub> around ''F''<sub>''n''</sub>(''x'') will entirely contain ''F''(''x'') with probability 1&nbsp;−&nbsp;''α''.<br />
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虽然通常使用Kolmogorov–Smirnov检验法来检验给定的F（x）是否为Fn（x）的潜在概率分布，但可以将过程倒过来给出F（x）本身的置信极限。如果选择检验统计量Dα的临界值，使得P（Dn>Dα）=α，则在Fn（x）周围宽度±Dα内将完全包含概率为1-α的F（x）。<br />
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== The Kolmogorov–Smirnov statistic in more than one dimension 多个维度的Kolmogorov–Smirnov统计 ==<br />
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A distribution-free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997). The test uses a statistic which is built using Rosenblatt's transformation, and an algorithm is developed to compute it in the bivariate case. An approximate test that can be easily computed in any dimension is also presented.<br />
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Justel，Peña和Zamar（1997）提出了无分布的多元Kolmogorov-Smirnov拟合优度检验。该检验使用通过Rosenblatt变换建立的统计量，开发出了一种算法来计算双变量情况。还介绍了可以在任何维度上轻松计算的近似检测法。<br />
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The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to [[multivariate statistics|multivariate data]]. This is not straightforward because the maximum difference between two joint [[cumulative distribution function]]s is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of <math>\Pr(x < X \land y < Y)</math> or <math>\Pr(X < x \land Y > y)</math> or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.<br />
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如果要将类似的检验应用于多元数据，则需要修改Kolmogorov–Smirnov检验统计量。过程略显复杂，因为两个联合累积分布函数之间的最大差异通常与任何互补分布函数中的最大差异都不相同。因此，最大差异将取决于使用<math>\Pr(x < X \land y < Y)</math>或<math>\Pr(X < x \land Y > y)</math>中的哪一个，或者使用其他两种可能分布中的任何一种。当然有可能要求所用的检测结果无关于这样的选择。<br />
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One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In ''d'' dimensions, there are 2<sup>''d''</sup>−1 such orderings. One such variation is due to Peacock(see also Gosset for a 3D version) and another to Fasano and Franceschini (see Lopes et al. for a comparison and computational details). Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.<br />
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在满足以上要求的同时，将Kolmogorov-Smirnov统计量泛化为更高维度的一种方法是，在所有可能的排序中比较两个样本的累积分布函数，并从所得的K-S统计量中取最大。在d维数据中，有2d-1个这样的排序。Peacock得出了一种这样的变化量（有关3D版本，另请参见Gosset），另一种由Fasano和Franceschini得出（有关比较和计算细节，请参见Lopes等人）。检测统计量的临界值可以通过仿真获取，但取决于联合分布中的依存关系结构。<br />
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In one dimension, the Kolmogorov–Smirnov statistic is identical to the so-called star discrepancy D, so another native KS extension to higher dimensions would be simply to use D also for higher dimensions. Unfortunately, the star discrepancy is hard to calculate in high dimensions.<br />
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一维的Kolmogorov-Smirnov统计量与所谓的星差D相同，因此，另一个对更高维度的本地KS扩展是将D也用于更高维度。可惜的是，很难从高维度上计算出星差。<br />
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== Implementations 软件实现==<br />
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The Kolmogorov-Smirnov test (one or two sampled test verifies the equality of distributions) is implemented in many software programs:<br />
Kolmogorov-Smirnov检验（一个或两个抽样检验可验证分布是否相等）可以在众多软件程序中实现：<br />
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* [[Mathematica]] has [https://reference.wolfram.com/language/ref/KolmogorovSmirnovTest.html KolmogorovSmirnovTest]<br />
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* [[MATLAB]] has [https://de.mathworks.com/help/stats/kstest.html kstest] in its Statistics Toolbox.<br />
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* The [[R (programming language)|R]] package "KSgeneral"<ref name=KSgeneral/> computes the KS test statistics and its p-values under arbitrary, possibly discrete, mixed or continuous null distribution.<br />
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* [[R (programming language)|R]]'s statistics base-package implements the test as [https://stat.ethz.ch/R-manual/R-patched/library/stats/html/ks.test.html ks.test {stats}] in its "stats" package.<br />
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* [[SAS (software)|SAS]] implements the test in its PROC NPAR1WAY procedure.<br />
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* [[Python (programming language)|Python]] has an implementation of this test provided by [[SciPy]]<ref>{{cite web |url= https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.kstest.html |title=scipy.stats.kstest |work=SciPy SciPy v0.14.0 Reference Guide |publisher=The Scipy community |access-date= 18 June 2019}}</ref> by Statistical functions (scipy.stats)<br />
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* [[SYSTAT (statistics)|SYSTAT]] (SPSS Inc., Chicago, IL)<br />
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* [[Java (programming language)|Java]] has an implementation of this test provided by [[Apache Commons]]<ref>{{cite web |url=https://commons.apache.org/proper/commons-math/javadocs/api-3.5/org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html |title=KolmogorovSmirnovTes|access-date= 18 June 2019}}</ref><br />
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* [[KNIME]] has a node implementing this test based on the above Java implementation<ref>{{cite web |url=https://www.knime.com/whats-new-in-knime-37#new-statistics-nodes |title=New statistics nodes |access-date= 25 June 2020}}</ref><br />
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* [[StatsDirect]] (StatsDirect Ltd, Manchester, UK) implements [https://www.statsdirect.com/help/nonparametric_methods/smirnov.htm all common variants].<br />
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* [[Stata]] (Stata Corporation, College Station, TX) implements the test in ksmirnov (Kolmogorov–Smirnov equality-of-distributions test) command. <ref> {{ cite web |url=https://www.stata.com/manuals15/rksmirnov.pdf|title=ksmirnov — Kolmogorov –Smirnov equality-of-distributions test |access-date= 18 June 2019}} </ref><br />
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* [[PSPP]] implements the test in its [https://www.gnu.org/software/pspp/manual/html_node/KOLMOGOROV_002dSMIRNOV.html KOLMOGOROV-SMIRNOV (or using K-S shortcut] function.<br />
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* [[Microsoft Excel|Excel]] runs the test as KSCRIT and KSPROB<br />
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* 科学计算软件Wolfram Mathematica内含Kolmogorov Smirnov Test。<br />
* MATLAB在其统计工具箱中含有K-S test。<br />
* R语言包“KSgeneral”可以在任意，可能离散，混合或连续的零分布下计算KS检验统计信息及其p值。<br />
* R语言的统计基本程序包在其“stats”程序包中可以运行检验，命令为ks.test {stats}。<br />
* 统计软件SAS在其PROC NPAR1WAY程序中可以实现检验。<br />
* Python通过SciPy中的统计功能（scipy.stats）可以实现检验。<br />
* SYSTAT（SPSS Inc.，伊利诺伊州芝加哥）<br />
* 基于Java语言开发的Apache Commons可以实现检验。<br />
* KNIME是一个数据分析平台，基于上述Java语言，可以通过一个节点Node来实现检验。<br />
* StatsDirect（StatsDirect Ltd，英国曼彻斯特）包含所有常见的变体。<br />
* Stata（德克萨斯州大学城Stata公司）在ksmirnov（Kolmogorov–Smirnov分配均等测试）中执行检验命令。<br />
* PSPP通过其Kolmogorov–Smirnov（或使用K-S快捷功能）实施检验。<br />
* Excel以KSCRIT和KSPROB的形式运行检验。<br />
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== See also 其他参考资料 ==<br />
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*[[Lepage test]]<br />
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*[[Cucconi test]]<br />
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*[[Kuiper's test]]<br />
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*[[Shapiro–Wilk test]]<br />
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*[[Anderson–Darling test]]<br />
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*[[Cramér–von Mises criterion|Cramér–von Mises test]]<br />
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* 勒帕热 Lepage检验<br />
* Cucconi检验<br />
* 柯伊伯 Kuiper检验<br />
* 夏皮罗-威尔克 Shapiro–Wilk检验<br />
* 安德森-达林 Anderson–Darling检验<br />
* 克莱姆·冯·米塞斯 Cramér–von Mises检验<br />
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== References 参考文献 ==<br />
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{{Reflist|30em}}<br />
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== Further reading 其他资料==<br />
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* Daniel, Wayne W. (1990). "Kolmogorov–Smirnov one-sample test". Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 319–330. ISBN 978-0-534-91976-4.<br />
* Eadie, W.T.; D. Drijard; F.E. James; M. Roos; B. Sadoulet (1971). Statistical Methods in Experimental Physics. Amsterdam: North-Holland. pp. 269–271. ISBN 978-0-444-10117-4.<br />
* Stuart, Alan; Ord, Keith; Arnold, Steven [F.] (1999). Classical Inference and the Linear Model. Kendall's Advanced Theory of Statistics. 2A (Sixth ed.). London: Arnold. pp. 25.37–25.43. ISBN 978-0-340-66230-4. MR 1687411.<br />
* Corder, G. W.; Foreman, D. I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley. ISBN 978-1118840313.<br />
* Stephens, M. A. (1979). "Test of fit for the logistic distribution based on the empirical distribution function". Biometrika. 66 (3): 591–595. doi:10.1093/biomet/66.3.591.<br />
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* Daniel, Wayne W. (1990). " Kolmogorov–Smirnov单样本检验. Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 319–330. ISBN 978-0-534-91976-4.<br />
* Eadie, W.T.; D. Drijard; F.E. James; M. Roos; B. Sadoulet (1971). 实验物理中的统计方法. Amsterdam: North-Holland. pp. 269–271. ISBN 978-0-444-10117-4.<br />
* Stuart, Alan; Ord, Keith; Arnold, Steven [F.] (1999). 经典推论和线性模型。肯德尔Kendall高级统计理论. 2A (Sixth ed.). London: Arnold. pp. 25.37–25.43. ISBN 978-0-340-66230-4. MR 1687411.<br />
* Corder, G. W.; Foreman, D. I. (2014). 非参数统计：分步法. Wiley. ISBN 978-1118840313.<br />
* Stephens, M. A. (1979). "基于经验分布函数的逻辑分布拟合检验". Biometrika. 66 (3): 591–595. doi:10.1093/biomet/66.3.591.<br />
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== External links 相关链接 ==<br />
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* "Kolmogorov–Smirnov test", Encyclopedia of Mathematics, EMS Press, 2001 [1994]<br />
* Short introduction<br />
* KS test explanation<br />
* JavaScript implementation of one- and two-sided tests<br />
* Online calculator with the KS test<br />
* Open-source C++ code to compute the Kolmogorov distribution and perform the KS test<br />
* Paper on Evaluating Kolmogorov’s Distribution; contains C implementation. This is the method used in Matlab.<br />
* Paper on Computing the Two-Sided Kolmogorov–Smirnov Distribution; computing the cdf of the KS statistic in C or Java.<br />
* Paper powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions; Jeff Alstott, Ed Bullmore, Dietmar Plenz. Among others, it also performs the Kolmogorov–Smirnov test. Source code and installers of powerlaw package are available at PyPi<br />
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* “Kolmogorov–Smirnov测验”，数学百科全书，EMS出版社，2001[1994]<br />
* KS简短介绍<br />
* KS检验说明<br />
* 单边和双边检验的JavaScript实现<br />
* KS检验在线计算器<br />
* 开源C++代码可计算Kolmogorov分布并执行KS检验<br />
* 关于评估Kolmogorov分布的论文，包含C实现。这是Matlab中常使用的方法。<br />
* 计算双向Kolmogorov-Smirnov分布的论文；用C或Java计算KS统计信息的累积分布函数。<br />
* Paper powerlaw：用于分析重尾分布的Python包； Jeff Alstott，Ed Bullmore和Dietmar Plenz。除此之外，它还执行Kolmogorov-Smirnov检验。PyPi提供了powerlaw软件包的源代码和安装程序。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E4%B8%89%E4%BD%93%E9%97%AE%E9%A2%98&diff=20399三体问题2020-12-22T10:39:55Z<p>Lincent：</p>
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<div>已由Lincent审校。<br />
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In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.<br />
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在物理学和经典力学领域中，三体问题是根据牛顿运动定律和牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）求出它们随后的运动的问题。三体是N体问题中的一个特例。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。<br />
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Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.<br />
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历史上看，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以是经典力学或量子力学中模拟三个粒子运动的任何问题。<br />
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==数学描述==<br />
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三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：<br />
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<math>\begin{align}<br />
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.<br />
\end{align}</math><br />
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其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程构成的方程组。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置<math>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：<br />
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:<math><br />
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},<br />
</math><br />
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其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：<br />
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:<math><br />
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.<br />
</math><br />
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这种情况下，<math>\mathcal{H}</math>仅仅是系统的总能量，重力加上动能。<br />
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==受限制的三题问题==<br />
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In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.<br />
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在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的天体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。<br />
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The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<br />
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受限制的三体问题比完全的三体问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳的系统，这也是在三体问题的历史发展中有重要地位的一个典型。<br />
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Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by<br />
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在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(x_2, y_2)</math>分别为小行星的坐标。简单起见，选择的单位应该要确保两大质量天体的距离和重力常数都等于1。则小行星的运动可以用公式描述为：<br />
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:<math><br />
\begin{align}<br />
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\<br />
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},<br />
\end{align}<br />
</math><br />
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其中<math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>，在这种形式下，运动方程通过坐标具有明确的时间依赖性<math>x_i(t), y_i(t)</math>。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。<br />
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==求解==<br />
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There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]<br />
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由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。<br />
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On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)<br />
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另一方面，1912年芬兰数学家 卡尔·弗里蒂奥夫·桑德曼 Karl Fritiof Sundman 证明了三体问题存在一个 {{math|''t''<sup>1/3</sup>}}幂次方的级数解。除了对应于角动量为零的初始条件外，这个级数对所有实数t都收敛。<br />
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An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).<br />
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证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。<br />
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Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:<br />
<br />
无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以对相应的解避免碰撞。因此，**Sundman**的求解方法包括以下步骤:<br />
<br />
<br />
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.<br />
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.<br />
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).<br />
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by<br />
<br />
<br />
1. 使用适当的变量变化来继续分析二元碰撞之外的解，这个过程称为正则化。<br />
<br />
2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实数奇点。<br />
<br />
3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过对微分方程使用柯西存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，一个条带区域（取决于L的值）中不存在复奇点。<br />
<br />
4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：<br />
<br />
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math><br />
<br />
This finishes the proof of Sundman's theorem.<br />
上述即为完整的Sundman定律的证明。<br />
<br />
Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]<br />
<br />
但不幸运的是，对应的级数收敛得非常慢。也就是说，为了获得一定精度的值需要很多级数项，这样的解法并没有什么实际用途。的确，在1930年，大卫·贝洛里奇 David Beloriszky计算出，如果将Sundman级数用于天文观测，则计算将至少涉及10<sup>{{val|8000000}}</sup>项。<br />
<br />
==特殊的求解方法==<br />
<br />
In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.<br />
<br />
1767年， 莱昂哈德·欧拉 Leonhard Euler提出了三个周期解系列，其中三个质量在每个瞬间共线。<br />
<br />
1772年，拉格朗日 Lagrange找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与欧拉的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>和L<sub>5</sub>，并且叫做拉格朗日点，其中L<sub>3</sub>, L<sub>4</sub>是拉格朗日的对称解的实例。<br />
<br />
在1892年至1899年的工作中，亨利·波因加 Henri Poincaré建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。<br />
<br />
1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。布鲁 Burrau在1913年进一步研究了这个问题。1967年，维克多·塞贝赫利 Victor Szebehely和 C、 弗雷德里克·彼得斯 C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。<br />
<br />
20世纪70年代，米歇尔·赫农 Michel Hénon和 罗杰A.布鲁克 Roger A. Broucke各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: 布鲁克-赫农-哈德吉德梅特里奥<br />
Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在布鲁克的一些解中，两个物体遵循同样的路径。<br />
<br />
<br />
1993年，[[圣塔菲研究所]]的物理学家克里斯摩尔 Cris Moore提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家'''阿兰·契纳 Alain Chenciner'''和'''理查德·蒙哥马利 Richard Montgomery'''证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。<br />
<br />
In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2013年，贝尔格莱德物理研究所的物理学家 米洛万·乌瓦科夫 Milovan uvakov 和 维利科·德米特拉·伊诺维 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。<br />
<br />
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2015年，物理学家 安娜·胡多马尔 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。<br />
<br />
In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]<br />
<br />
2017年，研究人员 李晓明 Xiaoming Li 和 廖世俊 Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。<br />
<br />
In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".<br />
<br />
<br />
2018年，李晓明 Li 和 廖世俊 Liao 提出了234个不等质量“自由落体”三体问题的解。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。<br />
<br />
===数值方法===<br />
<br />
Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]<br />
<br />
尽管高精度需要大量的CPU时间，但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年，布林 Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。<br />
<br />
==历史==<br />
<br />
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.<br />
<br />
传统意义上的三个物体的引力问题可以追溯到1687年，当时 艾萨克·牛顿 Isaac Newton 发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也迈出了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。<br />
<br />
The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.<br />
<br />
亚美利哥·韦斯普奇 Amerigo Vespucci和随后的 伽利略·伽利雷 Galileo Galilei提出了三体问题; 1499年，韦斯普奇 Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是在海上确定经度，1720年代该方法变得非常技术实用。事实上确定经度的问题被 约翰·哈里森 John Harrison发明的航海经线仪所解决。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。<br />
<br />
<br />
Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<br />
<br />
建立了长期竞争关系的 让·勒朗·达朗贝尔 Jean le Rond d'Alembert 和 亚历克西斯·克莱奥特 Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始被普遍使用，与他们的研究有关。让·勒朗·达朗贝尔 Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。<br />
<br />
<br />
==其他涉及三体的问题==<br />
<br />
The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.<br />
三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。<br />
<br />
A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]<br />
<br />
<br />
氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用。就像引力三体问题一样，氦原子的三体问题没有精确解。<br />
<br />
In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]<br />
<br />
<br />
然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。<br />
<br />
The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]<br />
<br />
引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。<br />
<br />
==N体问题==<br />
<br />
The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.<br />
<br />
三体问题是N体问题的一个特例，它描述了n个物体在其中一种物理力（如重力）下如何运动。这些问题具有收敛幂级数形式的全局解析解，比如，Karl F.Sundman证明n=3的情况，qaudong Wang证明n>3的情况。然而，Sundman级数和Wang级数收敛速度太慢，无法用于实际目的；因此，目前有必要通过数值分析以数值积分的形式来近似解，或者在某些情况下，采用经典三角级数近似。原子系统，例如原子、离子和分子，可以用量子N体问题来处理。在经典物理系统中，N体问题通常是指一个星系或一个星系团；行星系统，如恒星、行星及其卫星，也可以被视为N体系统。一些应用可以方便地用扰动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。<br />
<br />
==三体小说==<br />
<br />
The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole<br />
<br />
三体问题被中国作家刘慈欣用于科幻小说三部曲中的情节设计，也被用于作为第一卷和整个三部曲的书名。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E4%B8%89%E4%BD%93%E9%97%AE%E9%A2%98&diff=20398三体问题2020-12-22T10:38:23Z<p>Lincent：</p>
<hr />
<div>已由Lincent审校。<br />
<br />
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.<br />
<br />
在物理学和经典力学领域中，三体问题是根据牛顿运动定律和牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）求出它们随后的运动的问题。三体是N体问题中的一个特例。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。<br />
<br />
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.<br />
<br />
历史上看，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以是经典力学或量子力学中模拟三个粒子运动的任何问题。<br />
<br />
==数学描述==<br />
<br />
三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：<br />
<br />
<math>\begin{align}<br />
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.<br />
\end{align}</math><br />
<br />
其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程构成的方程组。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置<math>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：<br />
<br />
:<math><br />
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},<br />
</math><br />
<br />
其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：<br />
<br />
:<math><br />
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.<br />
</math><br />
<br />
这种情况下，<math>\mathcal{H}</math>仅仅是系统的总能量，重力加上动能。<br />
<br />
==受限制的三题问题==<br />
<br />
In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.<br />
<br />
在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的天体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。<br />
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The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<br />
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受限制的三体问题比完全的三体问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳的系统，这也是在三体问题的历史发展中有重要地位的一个典型。<br />
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Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by<br />
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在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(x_2, y_2)</math>分别为小行星的坐标。简单起见，选择的单位应该要确保两大质量天体的距离和重力常数都等于1。则小行星的运动可以用公式描述为：<br />
<br />
:<math><br />
\begin{align}<br />
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\<br />
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},<br />
\end{align}<br />
</math><br />
<br />
其中<math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>，在这种形式下，运动方程通过坐标具有明确的时间依赖性<math>x_i(t), y_i(t)</math>。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。<br />
<br />
==求解==<br />
<br />
There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]<br />
<br />
由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。<br />
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On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)<br />
<br />
另一方面，1912年芬兰数学家 卡尔·弗里蒂奥夫·桑德曼 Karl Fritiof Sundman 证明了三体问题存在一个 {{math|''t''<sup>1/3</sup>}}幂次方的级数解。除了对应于角动量为零的初始条件外，这个级数对所有实数t都收敛。<br />
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An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).<br />
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证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。<br />
<br />
Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:<br />
<br />
无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以对相应的解避免碰撞。因此，**Sundman**的求解方法包括以下步骤:<br />
<br />
<br />
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.<br />
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.<br />
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).<br />
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by<br />
<br />
<br />
1. 使用适当的变量变化来继续分析二元碰撞之外的解，这个过程称为正则化。<br />
<br />
2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实数奇点。<br />
<br />
3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过对微分方程使用柯西存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，一个条带区域（取决于L的值）中不存在复奇点。<br />
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4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：<br />
<br />
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math><br />
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This finishes the proof of Sundman's theorem.<br />
上述即为完整的Sundman定律的证明。<br />
<br />
Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]<br />
<br />
但不幸运的是，对应的级数收敛得非常慢。也就是说，为了获得一定精度的值需要很多级数项，这样的解法并没有什么实际用途。的确，在1930年，大卫·贝洛里奇 David Beloriszky计算出，如果将Sundman级数用于天文观测，则计算将至少涉及10<sup>{{val|8000000}}</sup>项。<br />
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==特殊的求解方法==<br />
<br />
In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.<br />
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1767年， 莱昂哈德·欧拉 Leonhard Euler提出了三个周期解系列，其中三个质量在每个瞬间共线。<br />
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1772年，拉格朗日 Lagrange找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与欧拉的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>和L<sub>5</sub>，并且叫做拉格朗日点，其中L<sub>3</sub>, L<sub>4</sub>是拉格朗日的对称解的实例。<br />
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在1892年至1899年的工作中，亨利·波因加 Henri Poincaré建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。<br />
<br />
1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。布鲁 Burrau在1913年进一步研究了这个问题。1967年，维克多·塞贝赫利 Victor Szebehely和 C、 弗雷德里克·彼得斯 C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。<br />
<br />
20世纪70年代，米歇尔·赫农 Michel Hénon和 罗杰A.布鲁克 Roger A. Broucke各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: 布鲁克-赫农-哈德吉德梅特里奥<br />
Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在布鲁克的一些解中，两个物体遵循同样的路径。<br />
<br />
<br />
1993年，[[圣塔菲研究所]]的物理学家克里斯摩尔 Cris Moore提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家'''阿兰·契纳 Alain Chenciner'''和'''理查德·蒙哥马利 Richard Montgomery'''证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。<br />
<br />
In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2013年，贝尔格莱德物理研究所的物理学家 米洛万·乌瓦科夫 Milovan uvakov 和 维利科·德米特拉·伊诺维 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。<br />
<br />
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2015年，物理学家 安娜·胡多马尔 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。<br />
<br />
In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]<br />
<br />
2017年，研究人员 李晓明 Xiaoming Li 和 廖世俊 Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。<br />
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In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".<br />
<br />
<br />
2018年，李晓明 Li 和 廖世俊 Liao 提出了234个不等质量“自由落体”三体问题的解。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。<br />
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===数值方法===<br />
<br />
Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]<br />
<br />
尽管高精度需要大量的CPU时间，但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年，布林 Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。<br />
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==历史==<br />
<br />
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.<br />
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传统意义上的三个物体的引力问题可以追溯到1687年，当时 艾萨克·牛顿 Isaac Newton 发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也迈出了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。<br />
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The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.<br />
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亚美利哥·韦斯普奇 Amerigo Vespucci和随后的 伽利略·伽利雷 Galileo Galilei提出了三体问题; 1499年，韦斯普奇 Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是在海上确定经度，1720年代该方法变得非常技术实用。事实上确定经度的问题被 约翰·哈里森 John Harrison发明的航海经线仪所解决。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。<br />
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<br />
Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<br />
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建立了长期竞争关系的 让·勒朗·达朗贝尔 Jean le Rond d'Alembert 和 亚历克西斯·克莱奥特 Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始被普遍使用，与他们的研究有关。让·勒朗·达朗贝尔 Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。<br />
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<br />
==其他涉及三体的问题==<br />
<br />
The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.<br />
三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。<br />
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A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]<br />
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<br />
氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用。就像引力三体问题一样，氦原子的三体问题没有精确解。<br />
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In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]<br />
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<br />
然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。<br />
<br />
The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]<br />
<br />
引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。<br />
<br />
==n体问题==<br />
<br />
The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.<br />
<br />
三体问题是N体问题的一个特例，它描述了n个物体在其中一种物理力（如重力）下如何运动。这些问题具有收敛幂级数形式的全局解析解，比如，Karl F.Sundman证明n=3的情况，qaudong Wang证明n>3的情况。然而，Sundman级数和Wang级数收敛速度太慢，无法用于实际目的；因此，目前有必要通过数值分析以数值积分的形式来近似解，或者在某些情况下，采用经典三角级数近似。原子系统，例如原子、离子和分子，可以用量子N体问题来处理。在经典物理系统中，N体问题通常是指一个星系或一个星系团；行星系统，如恒星、行星及其卫星，也可以被视为N体系统。一些应用可以方便地用扰动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。<br />
<br />
==三体小说==<br />
<br />
The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole<br />
<br />
三体问题被中国作家刘慈欣用于科幻小说三部曲中的情节设计，也被用于作为第一卷和整个三部曲的书名。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E4%B8%89%E4%BD%93%E9%97%AE%E9%A2%98&diff=20397三体问题2020-12-22T10:19:54Z<p>Lincent：</p>
<hr />
<div>已由Lincent审校。<br />
<br />
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.<br />
<br />
在物理学和经典力学领域中，三体问题是根据牛顿运动定律和牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）求出它们随后的运动的问题。三体是N体问题中的一个特例。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。<br />
<br />
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.<br />
<br />
历史上看，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以是经典力学或量子力学中模拟三个粒子运动的任何问题。<br />
<br />
==数学描述==<br />
<br />
三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：<br />
<br />
<math>\begin{align}<br />
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.<br />
\end{align}</math><br />
<br />
其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程构成的方程组。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置<math>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：<br />
<br />
:<math><br />
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},<br />
</math><br />
<br />
其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：<br />
<br />
:<math><br />
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.<br />
</math><br />
<br />
这种情况下，<math>\mathcal{H}</math>仅仅是系统的总能量，重力加上动能。<br />
<br />
==受限制的三题问题==<br />
<br />
In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.<br />
<br />
在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的天体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。<br />
<br />
The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<br />
<br />
受限制的三体问题比完全的三体问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳的系统，这也是在三体问题的历史发展中有重要地位的一个典型。<br />
<br />
Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by<br />
<br />
在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(x_2, y_2)</math>分别为小行星的坐标。简单起见，选择的单位应该要确保两大质量天体的距离和重力常数都等于1。则小行星的运动可以用公式描述为：<br />
<br />
:<math><br />
\begin{align}<br />
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\<br />
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},<br />
\end{align}<br />
</math><br />
<br />
其中<math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>，在这种形式下，运动方程通过坐标具有明确的时间依赖性<math>x_i(t), y_i(t)</math>。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。<br />
<br />
==求解==<br />
<br />
There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]<br />
<br />
由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。<br />
<br />
<br />
On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)<br />
<br />
另一方面，1912年芬兰数学家**Karl Fritiof Sundman**证明了三体问题存在一个 {{math|''t''<sup>1/3</sup>}}幂次方的级数解。除了对应于角动量为零的初始条件外，这个级数对所有实数t都收敛。<br />
<br />
An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).<br />
<br />
证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。<br />
<br />
Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:<br />
<br />
无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以对相应的解避免碰撞。因此，**Sundman**的求解方法包括以下步骤:<br />
<br />
<br />
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.<br />
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.<br />
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).<br />
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by<br />
<br />
<br />
1. 使用适当的变量变化来继续分析二元碰撞之外的解，这个过程称为正则化。<br />
<br />
2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实数奇点。<br />
<br />
3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过对微分方程使用柯西存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，一个条带区域（取决于L的值）中不存在复奇点。<br />
<br />
4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：<br />
<br />
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math><br />
<br />
This finishes the proof of Sundman's theorem.<br />
上述即为完整的Sundman定律的证明。<br />
<br />
Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]<br />
<br />
但不幸运的是，对应的级数收敛得非常慢。也就是说，为了获得一定精度的值需要很多级数项，这样的解法并没有什么实际用途。的确，在1930年，David Beloriszky计算出，如果将Sundman级数用于天文观测，则计算将至少涉及10<sup>{{val|8000000}}</sup>项。<br />
<br />
==特殊的求解方法==<br />
<br />
In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.<br />
<br />
1767年，**Leonhard Euler**提出了三个周期解系列，其中三个质量在每个瞬间共线。<br />
<br />
1772年，拉格朗日（Lagrange）找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与Euler的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>和L<sub>5</sub>，并且叫做拉格朗日点，其中L<sub>3</sub>, L<sub>4</sub>是拉格朗日的对称解的实例。<br />
<br />
在1892年至1899年的工作中，**Henri Poincaré**建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。<br />
<br />
1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。Burrau在1913年进一步研究了这个问题。1967年，Victor Szebehely和C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。<br />
<br />
20世纪70年代，**Michel Hénon**和 '''Roger A. Broucke'''各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在、Broucke的一些解中，两个物体遵循同样的路径。<br />
<br />
<br />
1993年，[[圣塔菲研究所]]的物理学家'''Cris Moore'''提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家'''Alain Chenciner'''和'''Richard Montgomery'''证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔（Cris Moore）在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。<br />
<br />
In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2013年，贝尔格莱德物理研究所的物理学家Milovan uvakov 和 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。<br />
<br />
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2015年，物理学家 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。<br />
<br />
In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]<br />
<br />
2017年，研究人员Xiaoming Li 和Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。<br />
<br />
In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".<br />
<br />
<br />
2018年，Li和Liao提出了234个不等质量“自由落体”三体问题的解。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。<br />
<br />
===数值方法===<br />
<br />
Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]<br />
<br />
尽管高精度需要大量的CPU时间，但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年，Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。<br />
<br />
==历史==<br />
<br />
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.<br />
<br />
传统意义上的三个物体的引力问题可以追溯到1687年，当时牛顿发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也迈出了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。<br />
<br />
The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.<br />
<br />
Amerigo Vespucci和随后的Galileo Galilei提出了三体问题; 1499年，Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是在海上确定经度，1720年代该方法变得非常技术实用。事实上确定经度的问题被John Harrison发明的航海经线仪所解决。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。<br />
<br />
<br />
Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<br />
<br />
建立了长期竞争关系的Jean le Rond d'Alembert和Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始被普遍使用，与他们的研究有关。Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。<br />
<br />
<br />
==其他涉及三体的问题==<br />
<br />
The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.<br />
三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。<br />
<br />
A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]<br />
<br />
<br />
氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用。就像引力三体问题一样，氦原子的三体问题没有精确解。<br />
<br />
In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]<br />
<br />
<br />
然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。<br />
<br />
The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]<br />
<br />
引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。<br />
<br />
==n体问题==<br />
<br />
The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.<br />
<br />
三体问题是N体问题的一个特例，它描述了n个物体在其中一种物理力（如重力）下如何运动。这些问题具有收敛幂级数形式的全局解析解，比如，Karl F.Sundman证明n=3的情况，qaudong Wang证明n>3的情况。然而，Sundman级数和Wang级数收敛速度太慢，无法用于实际目的；因此，目前有必要通过数值分析以数值积分的形式来近似解，或者在某些情况下，采用经典三角级数近似。原子系统，例如原子、离子和分子，可以用量子N体问题来处理。在经典物理系统中，N体问题通常是指一个星系或一个星系团；行星系统，如恒星、行星及其卫星，也可以被视为N体系统。一些应用可以方便地用扰动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。<br />
<br />
==三体小说==<br />
<br />
The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole<br />
<br />
三体问题被中国作家刘慈欣用于科幻小说三部曲中的情节设计，也被用于作为第一卷和整个三部曲的书名。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E4%B8%89%E4%BD%93%E9%97%AE%E9%A2%98&diff=20365三体问题2020-12-21T07:38:57Z<p>Lincent：</p>
<hr />
<div>。lincent审校。<br />
<br />
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.<br />
<br />
在物理学和经典力学领域中，三体问题是根据牛顿运动定律和牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）求出它们随后的运动的问题。三体是n体问题中的一个特例。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。<br />
<br />
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.<br />
<br />
历史上看，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以是经典力学或量子力学中模拟三个粒子运动的任何问题。<br />
<br />
==数学描述==<br />
<br />
三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：<br />
<br />
<math>\begin{align}<br />
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.<br />
\end{align}</math><br />
<br />
其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程构成的方程组。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置<math>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：<br />
<br />
:<math><br />
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},<br />
</math><br />
<br />
其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：<br />
<br />
:<math><br />
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.<br />
</math><br />
<br />
这种情况下，<math>\mathcal{H}</math>仅仅是系统的总能量，重力加上动能。<br />
<br />
==受限制的三题问题==<br />
<br />
In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.<br />
<br />
在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的天体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。<br />
<br />
The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<br />
<br />
受限制的三体问题比完全的三体问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳的系统，这也是在三体问题的历史发展中有重要地位的一个典型。<br />
<br />
Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by<br />
<br />
在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(x_2, y_2)</math>分别为小行星的坐标。简单起见，选择的单位应该要确保两大质量天体的距离和重力常数都等于1。则小行星的运动可以用公式描述为：<br />
<br />
:<math><br />
\begin{align}<br />
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\<br />
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},<br />
\end{align}<br />
</math><br />
<br />
其中<math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>，在这种形式下，运动方程通过坐标具有明确的时间依赖性<math>x_i(t), y_i(t)</math>。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。<br />
<br />
==求解==<br />
<br />
There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]<br />
<br />
由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。<br />
<br />
On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)<br />
<br />
另一方面，1912年芬兰数学家**Karl Fritiof Sundman**证明了三体问题存在一个 {{math|''t''<sup>1/3</sup>}}幂次方的级数解。除了对应于角动量为零的初始条件外，这个级数对所有实数t都收敛。<br />
<br />
An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).<br />
<br />
证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。<br />
<br />
Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:<br />
<br />
无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以对相应的解避免碰撞。因此，**Sundman**的求解方法包括以下步骤:<br />
<br />
<br />
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.<br />
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.<br />
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).<br />
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by<br />
<br />
<br />
1. 使用适当的变量变化来继续分析二元碰撞之外的解，这个过程称为正则化。<br />
<br />
2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实数奇点。<br />
<br />
3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过对微分方程使用柯西存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，一个条带区域（取决于L的值）中不存在复奇点。<br />
<br />
4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：<br />
<br />
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math><br />
<br />
This finishes the proof of Sundman's theorem.<br />
上述即为完整的Sundman定律的证明。<br />
<br />
Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]<br />
<br />
但不幸运的是，对应的级数收敛得非常慢。也就是说，为了获得一定精度的值需要很多级数项，这样的解法并没有什么实际用途。的确，在1930年，David Beloriszky计算出，如果将Sundman级数用于天文观测，则计算将至少涉及10<sup>{{val|8000000}}</sup>项。<br />
<br />
==特殊的求解方法==<br />
<br />
In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.<br />
<br />
1767年，**Leonhard Euler**提出了三个周期解系列，其中三个质量在每个瞬间共线。<br />
<br />
1772年，拉格朗日（Lagrange）找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与Euler的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>和L<sub>5</sub>，并且叫做拉格朗日点，其中L<sub>3</sub>, L<sub>4</sub>是拉格朗日的对称解的实例。<br />
<br />
在1892年至1899年的工作中，**Henri Poincaré**建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。<br />
<br />
1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。Burrau在1913年进一步研究了这个问题。1967年，Victor Szebehely和C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。<br />
<br />
20世纪70年代，**Michel Hénon**和 '''Roger A. Broucke'''各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在、Broucke的一些解中，两个物体遵循同样的路径。<br />
<br />
<br />
1993年，[[圣塔菲研究所]]的物理学家'''Cris Moore'''提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家'''Alain Chenciner'''和'''Richard Montgomery'''证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔（Cris Moore）在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。<br />
<br />
In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2013年，贝尔格莱德物理研究所的物理学家Milovan uvakov 和 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。<br />
<br />
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2015年，物理学家 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。<br />
<br />
In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]<br />
<br />
2017年，研究人员Xiaoming Li 和Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。<br />
<br />
In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".<br />
<br />
<br />
2018年，Li和Liao提出了234个不等质量“自由落体”三体问题的解。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。<br />
<br />
===数值方法===<br />
<br />
Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]<br />
<br />
尽管高精度需要大量的CPU时间，但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年，Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。<br />
<br />
==历史==<br />
<br />
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.<br />
<br />
传统意义上的三个物体的引力问题可以追溯到1687年，当时牛顿发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也迈出了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。<br />
<br />
The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.<br />
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Amerigo Vespucci和随后的Galileo Galilei提出了三体问题; 1499年，Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是在海上确定经度，1720年代该方法变得非常技术实用。事实上确定经度的问题被John Harrison发明的航海经线仪所解决。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。<br />
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Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<br />
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建立了长期竞争关系的Jean le Rond d'Alembert和Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始被普遍使用，与他们的研究有关。Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。<br />
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<br />
==其他涉及三体的问题==<br />
<br />
The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.<br />
三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。<br />
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A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]<br />
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<br />
氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用，这种相互作用称为经典力学中的三体问题。就像重力三体问题一样，氦原子的三体问题没有精确解。<br />
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In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]<br />
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<br />
然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。<br />
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The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]<br />
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引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。<br />
<br />
==n体问题==<br />
<br />
The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.<br />
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三体问题是n体问题的一个特例，它描述了n个物体在其中一种物理力（如重力）下如何运动。这些问题具有收敛幂级数形式的全局解析解，比如，Karl F.Sundman证明n=3的情况，qaudong Wang证明n>3的情况。然而，Sundman级数和Wang级数收敛速度太慢，无法用于实际目的；因此，目前有必要通过数值分析以数值积分的形式来近似解，或者在某些情况下，采用经典三角级数近似。原子系统，例如原子、离子和分子，可以用量子n体问题来处理。在经典物理系统中，n体问题通常是指一个星系或一个星系团；行星系统，如恒星、行星及其卫星，也可以被视为n体系统。一些应用可以方便地用扰动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。<br />
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==三体小说==<br />
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The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole<br />
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三体问题是中国作家刘慈欣的科幻三部曲中有所提及的模拟游戏，也被用于第一卷和整个三部曲的书名。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E4%B8%89%E4%BD%93%E9%97%AE%E9%A2%98&diff=20364三体问题2020-12-21T07:33:02Z<p>Lincent：</p>
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<div><br />
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.<br />
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在物理学和经典力学领域中，三体问题是根据牛顿运动定律和牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）求出它们随后的运动的问题。三体是n体问题中的一个特例。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。<br />
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Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.<br />
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历史上看，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以是经典力学或量子力学中模拟三个粒子运动的任何问题。<br />
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==数学描述==<br />
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三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：<br />
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<math>\begin{align}<br />
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.<br />
\end{align}</math><br />
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其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程构成的方程组。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置<math>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：<br />
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:<math><br />
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},<br />
</math><br />
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其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：<br />
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:<math><br />
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.<br />
</math><br />
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这种情况下，<math>\mathcal{H}</math>仅仅是系统的总能量，重力加上动能。<br />
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==受限制的三题问题==<br />
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In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.<br />
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在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的天体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。<br />
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The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<br />
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受限制的三体问题比完全的三体问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳的系统，这也是在三体问题的历史发展中有重要地位的一个典型。<br />
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Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by<br />
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在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(x_2, y_2)</math>分别为小行星的坐标。简单起见，选择的单位应该要确保两大质量天体的距离和重力常数都等于1。则小行星的运动可以用公式描述为：<br />
<br />
:<math><br />
\begin{align}<br />
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\<br />
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},<br />
\end{align}<br />
</math><br />
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其中<math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>，在这种形式下，运动方程通过坐标具有明确的时间依赖性<math>x_i(t), y_i(t)</math>。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。<br />
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==求解==<br />
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There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]<br />
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由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。<br />
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On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)<br />
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另一方面，1912年芬兰数学家**Karl Fritiof Sundman**证明了三体问题存在一个 {{math|''t''<sup>1/3</sup>}}幂次方的级数解。除了对应于角动量为零的初始条件外，这个级数对所有实数t都收敛。<br />
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An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).<br />
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证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。<br />
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Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:<br />
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无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以对相应的解避免碰撞。因此，**Sundman**的求解方法包括以下步骤:<br />
<br />
<br />
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.<br />
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.<br />
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).<br />
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by<br />
<br />
<br />
1. 使用适当的变量变化来继续分析二元碰撞之外的解，这个过程称为正则化。<br />
<br />
2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实数奇点。<br />
<br />
3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过对微分方程使用柯西存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，一个条带区域（取决于L的值）中不存在复奇点。<br />
<br />
4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：<br />
<br />
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math><br />
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This finishes the proof of Sundman's theorem.<br />
上述即为完整的Sundman定律的证明。<br />
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Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]<br />
<br />
但不幸运的是，对应的级数收敛得非常慢。也就是说，为了获得一定精度的值需要很多级数项，这样的解法并没有什么实际用途。的确，在1930年，David Beloriszky计算出，如果将Sundman级数用于天文观测，则计算将至少涉及10<sup>{{val|8000000}}</sup>项。<br />
<br />
==特殊的求解方法==<br />
<br />
In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.<br />
<br />
1767年，**Leonhard Euler**提出了三个周期解系列，其中三个质量在每个瞬间共线。<br />
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1772年，拉格朗日（Lagrange）找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与Euler的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>和L<sub>5</sub>，并且叫做拉格朗日点，其中L<sub>3</sub>, L<sub>4</sub>是拉格朗日的对称解的实例。<br />
<br />
在1892年至1899年的工作中，**Henri Poincaré**建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。<br />
<br />
1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。Burrau在1913年进一步研究了这个问题。1967年，Victor Szebehely和C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。<br />
<br />
20世纪70年代，**Michel Hénon**和 '''Roger A. Broucke'''各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在、Broucke的一些解中，两个物体遵循同样的路径。<br />
<br />
<br />
1993年，[[圣塔菲研究所]]的物理学家'''Cris Moore'''提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家'''Alain Chenciner'''和'''Richard Montgomery'''证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔（Cris Moore）在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。<br />
<br />
In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2013年，贝尔格莱德物理研究所的物理学家Milovan uvakov 和 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。<br />
<br />
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
<br />
2015年，物理学家 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。<br />
<br />
In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]<br />
<br />
2017年，研究人员Xiaoming Li 和Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。<br />
<br />
In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".<br />
<br />
<br />
2018年，Li和Liao提出了234个不等质量“自由落体”三体问题的解。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。<br />
<br />
===数值方法===<br />
<br />
Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]<br />
<br />
尽管高精度需要大量的CPU时间，但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年，Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。<br />
<br />
==历史==<br />
<br />
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.<br />
<br />
传统意义上的三个物体的引力问题可以追溯到1687年，当时牛顿发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也迈出了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。<br />
<br />
The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.<br />
<br />
Amerigo Vespucci和随后的Galileo Galilei提出了三体问题; 1499年，Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是在海上确定经度，1720年代该方法变得非常技术实用。事实上确定经度的问题被John Harrison发明的航海经线仪所解决。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。<br />
<br />
<br />
Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<br />
<br />
建立了长期竞争关系的Jean le Rond d'Alembert和Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始被普遍使用，与他们的研究有关。Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。<br />
<br />
<br />
==其他涉及三体的问题==<br />
<br />
The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.<br />
三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。<br />
<br />
A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]<br />
<br />
<br />
氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用，这种相互作用称为经典力学中的三体问题。就像重力三体问题一样，氦原子的三体问题没有精确解。<br />
<br />
In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]<br />
<br />
<br />
然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。<br />
<br />
The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]<br />
<br />
引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。<br />
<br />
==n体问题==<br />
<br />
The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.<br />
<br />
三体问题是n体问题的一个特例，它描述了n个物体在其中一种物理力（如重力）下如何运动。这些问题具有收敛幂级数形式的全局解析解，比如，Karl F.Sundman证明n=3的情况，qaudong Wang证明n>3的情况。然而，Sundman级数和Wang级数收敛速度太慢，无法用于实际目的；因此，目前有必要通过数值分析以数值积分的形式来近似解，或者在某些情况下，采用经典三角级数近似。原子系统，例如原子、离子和分子，可以用量子n体问题来处理。在经典物理系统中，n体问题通常是指一个星系或一个星系团；行星系统，如恒星、行星及其卫星，也可以被视为n体系统。一些应用可以方便地用扰动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。<br />
<br />
==三体小说==<br />
<br />
The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole<br />
<br />
三体问题是中国作家刘慈欣的科幻三部曲中有所提及的模拟游戏，也被用于第一卷和整个三部曲的书名。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E4%B8%89%E4%BD%93%E9%97%AE%E9%A2%98&diff=20363三体问题2020-12-21T07:32:30Z<p>Lincent：</p>
<hr />
<div><br />
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.<br />
<br />
在物理学和经典力学领域中，三体问题是根据牛顿运动定律和牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）求出它们随后的运动的问题。三体是n体问题中的一个特例。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。<br />
<br />
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.<br />
<br />
历史上看，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以是经典力学或量子力学中模拟三个粒子运动的任何问题。<br />
<br />
==数学描述==<br />
<br />
三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：<br />
<br />
<math>\begin{align}<br />
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\<br />
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.<br />
\end{align}</math><br />
<br />
其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程构成的方程组。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置<math>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：<br />
<br />
:<math><br />
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},<br />
</math><br />
<br />
其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：<br />
<br />
:<math><br />
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.<br />
</math><br />
<br />
这种情况下，<math>\mathcal{H}</math>仅仅是系统的总能量，重力加上动能。<br />
<br />
==受限制的三题问题==<br />
<br />
In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.<br />
<br />
在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的天体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。<br />
<br />
The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<br />
<br />
受限制的三体问题比完全的三体问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳的系统，这也是在三体问题的历史发展中有重要地位的一个典型。<br />
<br />
Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by<br />
<br />
在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(x_2, y_2)</math>分别为小行星的坐标。简单起见，选择的单位应该要确保两大质量天体的距离和重力常数都等于1。则小行星的运动可以用公式描述为：<br />
<br />
:<math><br />
\begin{align}<br />
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\<br />
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},<br />
\end{align}<br />
</math><br />
<br />
其中<math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>，在这种形式下，运动方程通过坐标具有明确的时间依赖性<math>x_i(t), y_i(t)</math>。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。<br />
<br />
==求解==<br />
<br />
There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]<br />
<br />
由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。<br />
<br />
On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)<br />
<br />
另一方面，1912年芬兰数学家**Karl Fritiof Sundman**证明了三体问题存在一个 {{math|''t''<sup>1/3</sup>}}幂次方的级数解。除了对应于角动量为零的初始条件外，这个级数对所有实数t都收敛。<br />
<br />
An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).<br />
<br />
证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。<br />
<br />
Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:<br />
<br />
无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以对相应的解避免碰撞。因此，**Sundman**的求解方法包括以下步骤:<br />
<br />
<br />
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.<br />
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.<br />
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).<br />
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by<br />
<br />
<br />
1. 使用适当的变量变化来继续分析二元碰撞之外的解，这个过程称为正则化。<br />
<br />
2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实数奇点。<br />
<br />
3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过对微分方程使用柯西存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，一个条带区域（取决于L的值）中不存在复奇点。<br />
<br />
4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：<br />
<br />
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math><br />
<br />
This finishes the proof of Sundman's theorem.<br />
上述即为完整的Sundman定律的证明。<br />
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Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]<br />
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但不幸运的是，对应的级数收敛得非常慢。也就是说，为了获得一定精度的值需要很多级数项，这样的解法并没有什么实际用途。的确，在1930年，David Beloriszky计算出，如果将Sundman级数用于天文观测，则计算将至少涉及10<sup>{{val|8000000}}</sup>项。<br />
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==特殊的求解方法==<br />
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In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.<br />
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1767年，**Leonhard Euler**提出了三个周期解系列，其中三个质量在每个瞬间共线。<br />
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1772年，拉格朗日（Lagrange）找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与Euler的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>和L<sub>5</sub>，并且叫做拉格朗日点，其中L<sub>3</sub>, L<sub>4</sub>是拉格朗日的对称解的实例。<br />
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在1892年至1899年的工作中，**Henri Poincaré**建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。<br />
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1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。Burrau在1913年进一步研究了这个问题。1967年，Victor Szebehely和C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。<br />
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20世纪70年代，**Michel Hénon**和 '''Roger A. Broucke'''各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在、Broucke的一些解中，两个物体遵循同样的路径。<br />
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1993年，[[圣塔菲研究所]]的物理学家'''Cris Moore'''提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家'''Alain Chenciner'''和'''Richard Montgomery'''证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔（Cris Moore）在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。<br />
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In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
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2013年，贝尔格莱德物理研究所的物理学家Milovan uvakov 和 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。<br />
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In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<br />
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2015年，物理学家 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。<br />
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In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]<br />
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2017年，研究人员Xiaoming Li 和Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。<br />
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In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".<br />
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2018年，Li和Liao提出了234个不等质量“自由落体”三体问题的解。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。<br />
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===数值方法===<br />
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Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]<br />
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尽管高精度需要大量的CPU时间，但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年，Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。<br />
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==历史==<br />
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The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.<br />
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传统意义上的三个物体的引力问题可以追溯到1687年，当时牛顿发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也迈出了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。<br />
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The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.<br />
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Amerigo Vespucci和随后的Galileo Galilei提出了三体问题; 1499年，Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是在海上确定经度，1720年代该方法变得非常技术实用。事实上确定经度的问题被John Harrison发明的航海经线仪所解决。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。<br />
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Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<br />
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建立了长期竞争关系的Jean le Rond d'Alembert和Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始被普遍使用，与他们的研究有关。Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。<br />
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==其他涉及三体的问题==<br />
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The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.<br />
三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。<br />
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A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]<br />
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氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用，这种相互作用称为经典力学中的三体问题。就像重力三体问题一样，氦原子的三体问题没有精确解。<br />
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In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]<br />
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然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。<br />
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The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]<br />
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引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。<br />
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==n体问题==<br />
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The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.<br />
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三体问题是n体问题的一个特例，它描述了n个物体在其中一种物理力（如重力）下如何运动。这些问题具有收敛幂级数形式的全局解析解，比如，Karl F.Sundman证明n=3的情况，qaudong Wang证明n>3的情况。然而，Sundman级数和Wang级数收敛速度太慢，无法用于实际目的；因此，目前有必要通过数值分析以数值积分的形式来近似解，或者在某些情况下，采用经典三角级数近似。原子系统，例如原子、离子和分子，可以用量子n体问题来处理。在经典物理系统中，n体问题通常是指一个星系或一个星系团；行星系统，如恒星、行星及其卫星，也可以被视为n体系统。一些应用可以方便地用扰动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。<br />
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==三体小说==<br />
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The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole<br />
三体问题是中国作家刘慈欣的科幻三部曲中有所提及的模拟游戏，也被用于第一卷和整个三部曲的书名。</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=20099蚁群优化算法2020-12-15T03:34:38Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临从两条不同的路径去寻找并抵达食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种解决计算问题的<font color="#ff8000">概率</font>化方法，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚁群代表了受真实蚂蚁行为启发的多主体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信方法常被作为一种典型范式<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造”蚂蚁“（例如:仿真模拟的主体）通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的仿真迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫——蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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在<font color="#ff8000">群智能</font>方法中,这个算法是蚁群算法家族的一员。它构成了某些<font color="#ff8000">元启发式</font>的优化。第一个蚁群算法最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样化以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法执行基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素的路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对后面的蚂蚁来说过度有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中非常关键<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁沿着同一条路径。蚁群算法的思想是通过“仿真蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是集中的而可以在所有微小的物体中找到，所以需要引入一些新的概念。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和算力是集中的。这些集中的单元功能不断地提高，并可以与人类大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似于大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物在遵循同样的基本规则时，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了<font color="#32CD32">许多可以应用到环境对象的网络中的特性</font>。蚂蚁群体具有很高的适应环境变化的能力，并且有很强的能力处理某项个体无法完成的任务。这种灵活性对于不断发展的移动网络也是非常有用的,从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
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2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
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该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
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==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
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In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
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在蚁群算法中，人工蚂蚁是一种简单的<font color="#32CD32">计算代理</font>，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
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'''procedure''' ACO_MetaHeuristic '''is'''<br />
'''while''' not_termination '''do'''<br />
generateSolutions()<br />
daemonActions()<br />
pheromoneUpdate()<br />
'''repeat'''<br />
end procedure<br />
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===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
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Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
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每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式集合<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#32cd32">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font><br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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一般来说，蚂蚁k从状态x移动到状态y的概率<br />
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<math><br />
p_{xy}^k =\frac{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }</math><br />
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where<br />
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where<br />
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其中<br />
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<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
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<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
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<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 转移的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#32cd32"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font><br />
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===Pheromone update===<br />
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信息素更新<br />
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Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
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Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
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当所有的蚂蚁都完成了求解过程，路径通常都被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
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<math><br />
\tau_{xy} \leftarrow (1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
</math><br />
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<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
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式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
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<math><br />
\Delta \tau^{k}_{xy} =\begin{cases}<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
0 & \mbox{otherwise}<br />
\end{cases}<br />
</math><br />
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<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
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where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
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其 Lk是蚂蚁k移动的代价(通常是长度)，Q是一个常数。<br />
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==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
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Here are some of the most popular variations of ACO algorithms.<br />
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下面是一些常见的 ACO 算法变体。<br />
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===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
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The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
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蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
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===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
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在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32cd32">开发</font>'''（即偏向选择具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁将信息素沉积在该路径上(即使这条路径没有被重访问过)。<br />
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===Max-Min Ant System (MMAS)===<br />
最大-最小蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
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该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素。<br />
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===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
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COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#32cd32">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
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它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#32cd32">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
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For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
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==Applications==<br />
应用<br />
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[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
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Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
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蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
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It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题。旅行商问题的目标是找到最短的往返路线连接一系列城市。通常的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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:1.It must visit each city exactly once;<br />
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每个城市必须只到达一次;<br />
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:2.A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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:3.The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上留下的信息素轨迹越强烈，这条边被选择的概率就越大;<br />
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:4.Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
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:5.After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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===Scheduling problem===<br />
规划问题<br />
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*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
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*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
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*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
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*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 <ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
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*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
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*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
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*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
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*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
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*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer,, pp.111-138, 2008.</ref><br />
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===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
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*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
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*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
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*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
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*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
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*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
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*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
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*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
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*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
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===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
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*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
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*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
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*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
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===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权的l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解。<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#32cd32">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>10×10的回送和卸载振动器<ref name=slyusarant1/><br />
<br />
<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
在图像处理中，蚁群算法可以用于进行边缘检测与与边缘连接。<br />
<br />
* '''Edge detection:'''<br />
边缘检测<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math>上随机放置蚂蚁<math>k</math >，其中<math>k=(m_1*m_2)^tfrac{1}</math> 。信息素矩阵<math>tau_{(i，j)}</math>由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
其中 <math>I</math> 是尺寸为 <math>M_1*M_2</math>的图像<br /><br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, 是一个归一化因子<br />
<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
<br />
<math>f(\cdot)</math>可以用以下函数计算: <br />
<br /><br />
<math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><br />
<math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)}</math><br /><br />
<math>f(x) =\begin{cases}<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
0, & \text{else}<br />
\end{cases}</math><br /><br />
<math>f(x) =\begin{cases}\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
0, & \text{else}\end{cases}</math><br />
<br />
The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br /><br />
上面每个函数中的参数<math>\lambda</math>用来调整函数各自的形状。<br />
<br />
Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br/>Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br />
<br />
步骤2构建过程:<br />
蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程<math>P_{x,y}</math>给出蚂蚁移动的概率<br/><br />
<br />
步骤3与步骤5更新过程<br/>信息素矩阵更新两次。在步骤3中，蚂蚁(由<math>\tau_{(x,y)}</math>给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。<br />
<br/><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br /><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, 其中 <math>\psi</math> 是信息素衰减系数 <math>0< \tau <1</math><br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
<br />
边缘连接：<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref><br />
ACO算法在边缘连接算法中也是证明有效的。<br />
<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
| last1 = Warner | first1 = Lars<br />
| last2 = Vogel | first2 = Ute<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
| date = 2008<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
| publisher = Shaker Verlag<br />
| location = Aachen, Germany<br />
| isbn = 978-3-8322-7313-2<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
| access-date = 2018-10-09<br />
}}<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
<br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
<br />
<br />
==Definition difficulty==<br />
定义困难<br />
[[File:Aco shortpath.svg|thumb]]<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence". <br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref>要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref>蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> 在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。各种算法(无论是优化还是非优化)寻找生物系统中的自组织促进了“群体智能”的概念。<br />
<br />
<br />
<br />
==Stigmergy algorithms==<br />
Stigmergy算法 <br />
<br />
There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> <br />
<br />
在实践中，有大量的算法声称是“蚁群”，但并不总是拥有典型蚁群优化的一般框架。<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> <br />
<br />
In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
<font color="#32CD32">在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。</font><ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
<font color="#32CD32">实际上，通过蚂蚁之间通过环境交换信息的行为(被称为“stigmergy”原理)足以使一种算法属于蚁群算法的一类。这一原则促使一些作者创造“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。</font><ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
==Related methods==<br />
<br />
<br />
*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护不是一个而是一组解。寻找更好的解的过程模仿了进化的过程，解被组合或变异以改变解空间，较差的解被丢弃。 <br />
<br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
<br />
<br />
分布估计算法（EDA）是一种进化算法，它用模型引导的算子代替传统的复制算子。这类模型通过机器学习技术从群体中学习，并以概率图形模型表示，从中可以通过抽样<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref>或引导交叉<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref>生成新的解。<br />
<br />
<br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
塔布搜索（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，塔布搜索则产生许多变异解，并转移到适应度最低的解。为了防止循环并鼓励在解空间中进行更大的移动，塔布列表保留了部分或完整的解。禁止移动到包含塔布列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
人工免疫系统（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
粒子群优化，一种群智能方法。<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
智能水滴（IWD），一种基于自然水滴在河流中流动的群优化算法<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
随机扩散搜索（SDS），一种基于主体的概率全局搜索和优化技术，最适合于目标函数可分解为多个独立的部分函数的问题<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中<br />
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==References==<br />
参考<br />
{{Reflist|30em}}<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=20098蚁群优化算法2020-12-15T03:31:26Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临从两条不同的路径去寻找并抵达食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种解决计算问题的<font color="#ff8000">概率</font>化方法，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚁群代表了受真实蚂蚁行为启发的多主体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信方法常被作为一种典型范式<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造”蚂蚁“（例如:仿真模拟的主体）通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的仿真迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫——蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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在<font color="#ff8000">群智能</font>方法中,这个算法是蚁群算法家族的一员。它构成了某些<font color="#ff8000">元启发式</font>的优化。第一个蚁群算法最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样化以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法执行基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素的路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对后面的蚂蚁来说过度有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中非常关键<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁沿着同一条路径。蚁群算法的思想是通过“仿真蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是集中的而可以在所有微小的物体中找到，所以需要引入一些新的概念。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和算力是集中的。这些集中的单元功能不断地提高，并可以与人类大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似于大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物在遵循同样的基本规则时，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了<font color="#32CD32">许多可以应用到环境对象的网络中的特性</font>。蚂蚁群体具有很高的适应环境变化的能力，并且有很强的能力处理某项个体无法完成的任务。这种灵活性对于不断发展的移动网络也是非常有用的,从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
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2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
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该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
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<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
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在蚁群算法中，人工蚂蚁是一种简单的<font color="#32CD32">计算代理</font>，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
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'''procedure''' ACO_MetaHeuristic '''is'''<br />
'''while''' not_termination '''do'''<br />
generateSolutions()<br />
daemonActions()<br />
pheromoneUpdate()<br />
'''repeat'''<br />
end procedure<br />
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<br />
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<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
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每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式集合<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#32cd32">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font><br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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一般来说，蚂蚁k从状态x移动到状态y的概率<br />
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<br />
<math><br />
p_{xy}^k =\frac{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
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其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
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<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 转移的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#32cd32"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font><br />
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===Pheromone update===<br />
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信息素更新<br />
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Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
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当所有的蚂蚁都完成了求解过程，路径通常都被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
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<br />
<math><br />
\tau_{xy} \leftarrow (1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
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式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
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<br />
<math><br />
\Delta \tau^{k}_{xy} =\begin{cases}<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
0 & \mbox{otherwise}<br />
\end{cases}<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，Q是一个常数。<br />
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==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
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Here are some of the most popular variations of ACO algorithms.<br />
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下面是一些常见的 ACO 算法变体。<br />
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===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
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The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
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蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
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===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
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在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32cd32">开发</font>'''（即偏向选择具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁将信息素沉积在该路径上(即使这条路径没有被重访问过)。<br />
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===Max-Min Ant System (MMAS)===<br />
最大-最小蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
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该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素。<br />
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===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
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COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#32cd32">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
<br />
它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#32cd32">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
<br />
对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
<br />
<br />
==Applications==<br />
应用<br />
<br />
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
<br />
Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
<br />
Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
<br />
蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
<br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
<br />
第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题。旅行商问题的目标是找到最短的往返路线连接一系列城市。通常的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
<br />
<br />
:1.It must visit each city exactly once;<br />
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每个城市必须只到达一次;<br />
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:2.A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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<br />
:3.The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上留下的信息素轨迹越强烈，这条边被选择的概率就越大;<br />
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:4.Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
<br />
:5.After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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<br />
[[File:Aco TSP.svg|thumb|600px|center]]<br />
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<br />
===Scheduling problem===<br />
规划问题<br />
<br />
*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
<br />
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
<br />
*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 <ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
<br />
*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
<br />
*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
<br />
*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
<br />
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
<br />
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer,, pp.111-138, 2008.</ref><br />
<br />
<br />
===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
<br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
<br />
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
<br />
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
<br />
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
<br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
<br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
<br />
<br />
===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
<br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
<br />
<br />
<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权的l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解。<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#32cd32">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>10×10的回送和卸载振动器<ref name=slyusarant1/><br />
<br />
<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
在图像处理中，蚁群算法可以用于进行边缘检测与与边缘连接。<br />
<br />
* '''Edge detection:'''<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math>上随机放置蚂蚁<math>k</math >，其中<math>k=(m_1*m_2)^tfrac{1}</math> 。信息素矩阵<math>tau_{(i，j)}</math>由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
其中 <math>I</math> 是尺寸为 <math>M_1*M_2</math>的图像<br /><br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, 是一个归一化因子<br />
<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
<br />
<math>f(\cdot)</math>可以用以下函数计算: <br />
<br /><br />
<math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><br />
<math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)}</math><br /><br />
<math>f(x) =\begin{cases}<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
0, & \text{else}<br />
\end{cases}</math><br /><br />
<math>f(x) =\begin{cases}\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
0, & \text{else}\end{cases}</math><br />
<br />
The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br /><br />
上面每个函数中的参数<math>\lambda</math>用来调整函数各自的形状。<br />
<br />
Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br/>Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
步骤2构建过程: <br/> <br />
蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程<math>P_{x,y}</math>给出蚂蚁移动的概率<br/><br />
<br />
步骤3与步骤5更新过程<br/>信息素矩阵更新两次。在步骤3中，蚂蚁(由<math>\tau_{(x,y)}</math>给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。<br />
<br/><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br /><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, 其中 <math>\psi</math> 是信息素衰减系数 <math>0< \tau <1</math><br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
<br />
边缘连接：<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref><br />
ACO算法在边缘连接算法中也是证明有效的。<br />
<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
| last1 = Warner | first1 = Lars<br />
| last2 = Vogel | first2 = Ute<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
| date = 2008<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
| publisher = Shaker Verlag<br />
| location = Aachen, Germany<br />
| isbn = 978-3-8322-7313-2<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
| access-date = 2018-10-09<br />
}}<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
<br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
<br />
<br />
==Definition difficulty==<br />
定义困难<br />
[[File:Aco shortpath.svg|thumb]]<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence". <br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref>要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref>蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> 在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。各种算法(无论是优化还是非优化)寻找生物系统中的自组织促进了“群体智能”的概念。<br />
<br />
<br />
<br />
==Stigmergy algorithms==<br />
Stigmergy算法 <br />
<br />
There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> <br />
<br />
在实践中，有大量的算法声称是“蚁群”，但并不总是拥有典型蚁群优化的一般框架。<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> <br />
<br />
In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
<font color="#32CD32">在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。</font><ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
<font color="#32CD32">实际上，通过蚂蚁之间通过环境交换信息的行为(被称为“stigmergy”原理)足以使一种算法属于蚁群算法的一类。这一原则促使一些作者创造“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。</font><ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
==Related methods==<br />
<br />
<br />
*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护不是一个而是一组解。寻找更好的解的过程模仿了进化的过程，解被组合或变异以改变解空间，较差的解被丢弃。 <br />
<br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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分布估计算法（EDA）是一种进化算法，它用模型引导的算子代替传统的复制算子。这类模型通过机器学习技术从群体中学习，并以概率图形模型表示，从中可以通过抽样<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref>或引导交叉<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref>生成新的解。<br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
塔布搜索（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，塔布搜索则产生许多变异解，并转移到适应度最低的解。为了防止循环并鼓励在解空间中进行更大的移动，塔布列表保留了部分或完整的解。禁止移动到包含塔布列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
人工免疫系统（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
粒子群优化，一种群智能方法。<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
智能水滴（IWD），一种基于自然水滴在河流中流动的群优化算法<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
随机扩散搜索（SDS），一种基于主体的概率全局搜索和优化技术，最适合于目标函数可分解为多个独立的部分函数的问题<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中<br />
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==References==<br />
参考<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=20097蚁群优化算法2020-12-15T03:25:55Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临从两条不同的路径去寻找并抵达食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种解决计算问题的<font color="#ff8000">概率</font>化方法，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚁群代表了受真实蚂蚁行为启发的多主体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信方法常被作为一种典型范式<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造”蚂蚁“（例如:仿真模拟的主体）通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的仿真迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫——蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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在<font color="#ff8000">群智能</font>方法中,这个算法是蚁群算法家族的一员。它构成了某些<font color="#ff8000">元启发式</font>的优化。第一个蚁群算法最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样化以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法执行基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素的路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对后面的蚂蚁来说过度有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中非常关键<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁沿着同一条路径。蚁群算法的思想是通过“仿真蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是集中的而可以在所有微小的物体中找到，所以需要引入一些新的概念。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和算力是集中的。这些集中的单元功能不断地提高，并可以与人类大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似于大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物在遵循同样的基本规则时，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了<font color="#32CD32">许多可以应用到环境对象的网络中的特性</font>。蚂蚁群体具有很高的适应环境变化的能力，并且有很强的能力处理某项个体无法完成的任务。这种灵活性对于不断发展的移动网络也是非常有用的,从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
<br />
2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
<br />
该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
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<br />
<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
在蚁群算法中，人工蚂蚁是一种简单的<font color="#32CD32">计算代理</font>，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
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<br />
'''procedure''' ACO_MetaHeuristic '''is'''<br />
'''while''' not_termination '''do'''<br />
generateSolutions()<br />
daemonActions()<br />
pheromoneUpdate()<br />
'''repeat'''<br />
end procedure<br />
<br />
<br />
<br />
<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式集合<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#32cd32">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font><br />
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<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
p_{xy}^k =\frac{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 转移的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#32cd32"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font><br />
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<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常都被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
\tau_{xy} \leftarrow (1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
\Delta \tau^{k}_{xy} =\begin{cases}<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
0 & \mbox{otherwise}<br />
\end{cases}<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，Q是一个常数。<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
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===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
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===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
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在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32cd32">开发</font>'''（即偏向选择具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁将信息素沉积在该路径上(即使这条路径没有被重访问过)。<br />
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<br />
===Max-Min Ant System (MMAS)===<br />
最大-最小蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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<br />
<br />
<br />
===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素。<br />
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<br />
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<br />
===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
<br />
COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#32cd32">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
<br />
<br />
===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
<br />
它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#32cd32">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
<br />
<br />
<br />
<br />
==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
<br />
对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
<br />
<br />
==Applications==<br />
应用<br />
<br />
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
<br />
Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
<br />
Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
<br />
蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
<br />
It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
<br />
它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
<br />
[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
<br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
<br />
第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题。旅行商问题的目标是找到最短的往返路线连接一系列城市。通常的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
<br />
<br />
:1.It must visit each city exactly once;<br />
<br />
每个城市必须只到达一次;<br />
<br />
:2.A distant city has less chance of being chosen (the visibility);<br />
<br />
较远的城市被选中的机会较少(可见度) ;<br />
<br />
<br />
:3.The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
<br />
在两个城市之间的边上留下的信息素轨迹越强烈，这条边被选择的概率就越大;<br />
<br />
:4.Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
<br />
旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
<br />
:5.After each iteration, trails of pheromones evaporate.<br />
<br />
每次迭代后，信息素踪迹会蒸发。<br />
<br />
<br />
[[File:Aco TSP.svg|thumb|600px|center]]<br />
<br />
<br />
<br />
===Scheduling problem===<br />
规划问题<br />
<br />
*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
<br />
*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
<br />
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
<br />
*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 <ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
<br />
*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
<br />
*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
<br />
*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
<br />
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
<br />
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer,, pp.111-138, 2008.</ref><br />
<br />
<br />
===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
<br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
<br />
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
<br />
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
<br />
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
<br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
<br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
<br />
<br />
===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
<br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
<br />
<br />
<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权的l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解。<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#32cd32">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>10×10的回送和卸载振动器<ref name=slyusarant1/><br />
<br />
<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
在图像处理中，蚁群算法可以用于进行边缘检测与与边缘连接。<br />
<br />
* '''Edge detection:'''<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math>上随机放置蚂蚁<math>k</math >，其中<math>k=(m_1*m_2)^tfrac{1}</math> 。信息素矩阵<math>tau_{(i，j)}</math>由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
其中 <math>I</math> 是尺寸为 <math>M_1*M_2</math>的图像<br /><br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, 是一个归一化因子<br />
<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
<br />
<math>f(\cdot)</math>可以用以下函数计算: <br />
<br /><br />
<math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><br />
<math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)}</math><br /><br />
<math>f(x) =\begin{cases}<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
0, & \text{else}<br />
\end{cases}</math><br /><br />
<math>f(x) =\begin{cases}\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
0, & \text{else}\end{cases}</math><br />
<br />
The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br /><br />
上面每个函数中的参数<math>\lambda</math>用来调整函数各自的形状。<br />
<br />
Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
步骤2构建过程: <br/> <br />
蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程<math>P_{x,y}</math>给出蚂蚁移动的概率<br /><br />
步骤3与步骤5更新过程<br />信息素矩阵更新两次。在步骤3中，蚂蚁(由<math>\tau_{(x,y)}</math>给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。<br />
<br /><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br /><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, 其中 <math>\psi</math> 是信息素衰减系数 <math>0< \tau <1</math><br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
<br />
边缘连接：<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref><br />
ACO算法在边缘连接算法中也是证明有效的。<br />
<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
| last1 = Warner | first1 = Lars<br />
| last2 = Vogel | first2 = Ute<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
| date = 2008<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
| publisher = Shaker Verlag<br />
| location = Aachen, Germany<br />
| isbn = 978-3-8322-7313-2<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
| access-date = 2018-10-09<br />
}}<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
<br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
<br />
<br />
==Definition difficulty==<br />
定义困难<br />
[[File:Aco shortpath.svg|thumb]]<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence". <br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref>要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref>蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> 在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。各种算法(无论是优化还是非优化)寻找生物系统中的自组织促进了“群体智能”的概念。<br />
<br />
<br />
<br />
==Stigmergy algorithms==<br />
Stigmergy算法 <br />
<br />
There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> <br />
<br />
在实践中，有大量的算法声称是“蚁群”，但并不总是拥有典型蚁群优化的一般框架。<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> <br />
<br />
In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
<font color="#32CD32">在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。</font><ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
<font color="#32CD32">实际上，通过蚂蚁之间通过环境交换信息的行为(被称为“stigmergy”原理)足以使一种算法属于蚁群算法的一类。这一原则促使一些作者创造“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。</font><ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
<br />
==Related methods==<br />
<br />
<br />
*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护不是一个而是一组解。寻找更好的解的过程模仿了进化的过程，解被组合或变异以改变解空间，较差的解被丢弃。 <br />
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* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled or generated from guided-crossover.<br />
分布估计算法（EDA）是一种进化算法，它用模型引导的算子代替传统的复制算子。这类模型通过机器学习技术从群体中学习，并以概率图形模型表示，从中可以通过抽样或引导交叉生成新的解。<br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
塔布搜索（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，塔布搜索则产生许多变异解，并转移到适应度最低的解。为了防止循环并鼓励在解空间中进行更大的移动，塔布列表保留了部分或完整的解。禁止移动到包含塔布列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
人工免疫系统（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
粒子群优化，一种群智能方法。<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
智能水滴（IWD），一种基于自然水滴在河流中流动的群优化算法<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
随机扩散搜索（SDS），一种基于主体的概率全局搜索和优化技术，最适合于目标函数可分解为多个独立的部分函数的问题<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中<br />
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==References==<br />
参考<br />
{{Reflist|30em}}<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=20096蚁群优化算法2020-12-15T02:03:24Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临从两条不同的路径去寻找并抵达食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种解决计算问题的<font color="#ff8000">概率</font>化方法，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚁群代表了受真实蚂蚁行为启发的多主体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信方法常被作为一种典型范式<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造”蚂蚁“（例如:仿真模拟的主体）通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的仿真迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫——蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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在<font color="#ff8000">群智能</font>方法中,这个算法是蚁群算法家族的一员。它构成了某些<font color="#ff8000">元启发式</font>的优化。第一个蚁群算法最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样化以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法执行基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素的路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对后面的蚂蚁来说过度有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中非常关键<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁沿着同一条路径。蚁群算法的思想是通过“仿真蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是集中的而可以在所有微小的物体中找到，所以需要引入一些新的概念。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和算力是集中的。这些集中的单元功能不断地提高，并可以与人类大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似于大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物在遵循同样的基本规则时，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了<font color="#32CD32">许多可以应用到环境对象的网络中的特性</font>。蚂蚁群体具有很高的适应环境变化的能力，并且有很强的能力处理某项个体无法完成的任务。这种灵活性对于不断发展的移动网络也是非常有用的,从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
<br />
<br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
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2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
<br />
该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
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<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
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在蚁群算法中，人工蚂蚁是一种简单的<font color="#32CD32">计算代理</font>，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
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'''procedure''' ACO_MetaHeuristic '''is'''<br />
'''while''' not_termination '''do'''<br />
generateSolutions()<br />
daemonActions()<br />
pheromoneUpdate()<br />
'''repeat'''<br />
end procedure<br />
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<br />
<br />
<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
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每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式集合<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#32cd32">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font><br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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一般来说，蚂蚁k从状态x移动到状态y的概率<br />
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<br />
<br />
<math><br />
p_{xy}^k =\frac{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
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<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 转移的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#32cd32"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font><br />
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===Pheromone update===<br />
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信息素更新<br />
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Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
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当所有的蚂蚁都完成了求解过程，路径通常都被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
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<br />
<br />
<math><br />
\tau_{xy} \leftarrow (1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
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<br />
<math><br />
\Delta \tau^{k}_{xy} =\begin{cases}<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
0 & \mbox{otherwise}<br />
\end{cases}<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，Q是一个常数。<br />
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==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
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Here are some of the most popular variations of ACO algorithms.<br />
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下面是一些常见的 ACO 算法变体。<br />
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===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
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The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
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蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
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===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
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在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32cd32">开发</font>'''（即偏向选择具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁将信息素沉积在该路径上(即使这条路径没有被重访问过)。<br />
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===Max-Min Ant System (MMAS)===<br />
最大-最小蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素。<br />
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===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
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COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#32cd32">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
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它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#32cd32">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
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For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
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==Applications==<br />
应用<br />
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[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
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Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
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蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
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It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题。旅行商问题的目标是找到最短的往返路线连接一系列城市。通常的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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:1.It must visit each city exactly once;<br />
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每个城市必须只到达一次;<br />
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:2.A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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:3.The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上留下的信息素轨迹越强烈，这条边被选择的概率就越大;<br />
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:4.Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
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:5.After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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===Scheduling problem===<br />
规划问题<br />
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*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
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*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
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*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
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*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 <ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
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*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
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*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
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*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
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*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
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*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer,, pp.111-138, 2008.</ref><br />
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===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
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*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
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*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
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*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
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*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
<br />
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
<br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
<br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
<br />
<br />
===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
<br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
<br />
<br />
<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权的l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
<br />
采用蚁群算法合成回路振子10 × 10<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#32cd32">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
* '''Edge detection:'''<br />
<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor 是一个归一化因子<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
<br />
<math>f(\cdot)</math>可以用以下函数计算: <br />
<br /><br />
<math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><br />
<math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)}</math><br /><br />
<math>f(x) =\begin{cases}<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
0, & \text{else}<br />
\end{cases}</math><br /><br />
<math>f(x) =\begin{cases}\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
0, & \text{else}\end{cases}</math><br />
<br />
The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
上面每个函数中的参数<math>lambda</math>用来调整函数各自的形状。<br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程<math>P_{x,y}</math>给出蚂蚁移动的概率<br />步骤3与步骤5更新过程<br />信息素矩阵更新两次。在步骤3中，蚂蚁(由<math>\tau_{(x,y)}</math>给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。<br />
<br /><math><br />
\tau_{new} \leftarrow<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
这里是信息素衰减系数。<br />
<br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
| last1 = Warner | first1 = Lars<br />
| last2 = Vogel | first2 = Ute<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
| date = 2008<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
| publisher = Shaker Verlag<br />
| location = Aachen, Germany<br />
| isbn = 978-3-8322-7313-2<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
| access-date = 2018-10-09<br />
}}<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
<br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
<br />
<br />
==Definition difficulty==<br />
定义困难<br />
<br />
<br />
[[File:Aco shortpath.svg|thumb]]<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
==Stigmergy algorithms==<br />
Stigmergy算法 <br />
<br />
There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
<br />
==Related methods==<br />
<br />
<br />
*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中<br />
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==References==<br />
参考<br />
{{Reflist|30em}}<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E6%9D%A1%E4%BB%B6%E7%86%B5&diff=19626条件熵2020-12-03T11:10:01Z<p>Lincent：</p>
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<div>此词条由Jie翻译。由Lincent审校。<br />
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{{Short description|Measure of relative information in probability theory}}<br />
{{Information theory}}<br />
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[[文件:Entropy-mutual-information-relative-entropy-relation-diagram.svg|缩略图|右|该图表示在变量X、Y相关联的各种信息量之间，进行加减关系的维恩图。两个圆所包含的区域是联合熵H(X,Y)。左侧的圆圈（红色和紫色）是单个熵H（X），红色是条件熵H（X ǀ Y）。右侧的圆圈（蓝色和紫色）为H（Y），蓝色为H（Y ǀ X）。中间紫色的是相互信息i（X; Y）。]]<br />
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In [[information theory]], the '''conditional entropy''' quantifies the amount of information needed to describe the outcome of a [[random variable]] <math>Y</math> given that the value of another random variable <math>X</math> is known. Here, information is measured in [[Shannon (unit)|shannon]]s, [[Nat (unit)|nat]]s, or [[Hartley (unit)|hartley]]s. The ''entropy of <math>Y</math> conditioned on <math>X</math>'' is written as H（X ǀ Y）.<br />
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在'''<font color="#ff8000"> 信息论Information theory</font>'''中，假设随机变量<math>X</math>的值已知，那么'''<font color="#ff8000"> 条件熵Conditional entropy</font>'''则用于去定量描述随机变量<math>Y</math>表示的信息量。此时，信息以'''<font color="#ff8000"> 香农Shannon </font>'''，'''<font color="#ff8000"> 奈特nat</font>'''或'''<font color="#ff8000"> 哈特莱hartley</font>'''来衡量。已知<math>X</math>的条件下<math>Y</math>的熵记为<math>H（X ǀ Y）</math>。<br />
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== Definition 定义 ==<br />
The conditional entropy of <math>Y</math> given <math>X</math> is defined as<br />
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在给定<math>X</math>的情况下，<math>Y</math>的条件熵定义为：<br />
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{{Equation box 1<br />
|indent =<br />
|title=<br />
|equation = {{NumBlk||<math>\Eta(Y|X)\ = -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)}</math>|{{EquationRef|Eq.1}}}}<br />
|cellpadding= 6<br />
|border<br />
|border colour = #0073CF<br />
|background colour=#F5FFFA}}<br />
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where <math>\mathcal X</math> and <math>\mathcal Y</math> denote the [[Support (mathematics)|support sets]] of <math>X</math> and <math>Y</math>.<br />
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其中<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的<font color="#32cd32">支撑集</font>。<br />
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''Note:'' It is conventioned that the expressions <math>0 \log 0</math> and <math>0 \log c/0</math> for fixed <math>c > 0</math> should be treated as being equal to zero. This is because <math>\lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0</math> and <math>\lim_{\theta\to0^+} \theta\, \log \theta = 0</math><ref>{{Cite web|url=http://www.inference.org.uk/mackay/itprnn/book.html|title=David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book|website=www.inference.org.uk|access-date=2019-10-25}}</ref> <!-- because p(x,y) could still equal 0 even if p(x) != 0 and p(y) != 0. What about p(x,y)=p(x)=0? --><br />
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注意：约定<math>c > 0</math>始终成立时，表达式<math>0 \log 0</math>和<math>0 \log c/0</math>视为等于零。这是因为<math>\lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0</math>，而且<math>\lim_{\theta\to0^+} \theta\, \log \theta = 0</math>><ref>{{Cite web|url=http://www.inference.org.uk/mackay/itprnn/book.html|title=David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book|website=www.inference.org.uk|access-date=2019-10-25}}</ref> <!-- because p(x,y) could still equal 0 even if p(x) != 0 and p(y) != 0. What about p(x,y)=p(x)=0? --><br />
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Intuitive explanation of the definition : According to the definition, <math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math> where <math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) .</math> <math>\displaystyle f</math> associates to <math>\displaystyle ( x,y)</math> the information content of <math>\displaystyle ( Y=y)</math> given <math>\displaystyle (X=x)</math>, which is the amount of information needed to describe the event <math>\displaystyle (Y=y)</math> given <math>(X=x)</math>. According to the law of large numbers, <math>\displaystyle H(Y|X)</math> is the arithmetic mean of a large number of independent realizations of <math>\displaystyle f(X,Y)</math>.<br />
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对该定义的直观解释是：根据定义<math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math>，其中<math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) </math>. <math>\displaystyle f</math>将给定<math>\displaystyle (X=x)</math>时的<math>\displaystyle ( Y=y)</math>的信息内容与<math>\displaystyle ( x,y)</math>相关联，这是描述在给定<math>(X=x)</math>条件下的事件<math>\displaystyle (Y=y)</math>所需的信息量。根据大数定律，<math>H（Y ǀ X）</math>是大量<math>\displaystyle f(X,Y)</math>独立实验结果的算术平均值。<br />
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== Motivation 动机 ==<br />
Let <math>H（Y ǀ X = x）</math> be the [[Shannon Entropy|entropy]] of the discrete random variable <math>Y</math> conditioned on the discrete random variable <math>X</math> taking a certain value <math>x</math>. Denote the support sets of <math>X</math> and <math>Y</math> by <math>\mathcal X</math> and <math>\mathcal Y</math>. Let <math>Y</math> have [[probability mass function]] <math>p_Y{(y)}</math>. The unconditional entropy of <math>Y</math> is calculated as <math>H（Y）:=E[I（Y）</math>, i.e.<br />
<br />
设<math>H（Y ǀ X = x）</math>为离散随机变量<math>Y</math>在离散随机变量<math>X</math>取定值<math>x</math>时的熵。用<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的支撑集。令<math>Y</math>的概率密度函数为<math>p_Y{(y)}</math>。<math>Y</math>的无条件熵计算为<math>H（Y）:=E[I（Y）</math>。<br />
<br />
<br />
:<math>H(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)} <br />
= -\sum_{y\in\mathcal Y} {p_Y(y) \log_2{p_Y(y)}},</math><br />
<br />
<br />
where <math>\operatorname{I}(y_i)</math> is the [[information content]] of the [[Outcome (probability)|outcome]] of <math>Y</math> taking the value <math>y_i</math>. The entropy of <math>Y</math> conditioned on <math>X</math> taking the value <math>x</math> is defined analogously by [[conditional expectation]]: <br />
<br />
当<math>Y</math>取值为<math>y_i</math>时，<math>\operatorname{I}(y_i)</math>是其结果<math>Y</math>的信息内容。类似地，当<math>X</math>值为<math>x</math>时以<math>X</math>为条件的<math>Y</math>的熵，也可以通过条件期望来定义：<br />
<br />
<br />
:<math>H(Y|X=x)<br />
= -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}.</math><br />
<br />
<br />
Note that<math> H（Y ǀ X）</math> is the result of averaging <math>H（Y ǀ X = x）</math> over all possible values <math>x</math> that <math>X</math> may take. Also, if the above sum is taken over a sample <math>y_1, \dots, y_n</math>, the expected value <math>E_X[ H(y_1, \dots, y_n \mid X = x)]</math> is known in some domains as '''equivocation'''.<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref><br />
<br />
注意，<math> H（Y ǀ X）</math>是在<math>X</math>可能取的所有可能值<math>x</math>时对<math>H（Y ǀ X = x）</math>求平均值的结果。同样，如果上述和取自样本<math>y_1, \dots, y_n</math>上，则期望值<math>E_X[ H(y_1, \dots, y_n \mid X = x)]</math><font color="#32cd32"> 在某些领域中认为是模糊值</font>。<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref><br />
<br />
<br />
<br />
Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>H(Y|X=x)</math> for each possible value of <math>x</math>, using <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}<br />
<br />
给定具有像<math>\mathcal X</math>的离散随机变量<math>X</math>和具有像<math>\mathcal Y</math>的离散随机变量<math>Y</math>，将给定<math>X</math>的<math>Y</math>的条件熵定义为以<math>p(x)</math>作为权重，对<math>x</math>的每个可能取值得到的<math>H(Y|X=x)</math>的加权和。其表达式如下：<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}<br />
<br />
<br />
:<math><br />
\begin{align}<br />
H(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,H(Y|X=x)\\<br />
& =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, p(y|x)\\<br />
& =-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\<br />
& =-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log\,p(y|x)\\<br />
& =-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)}. \\<br />
& = \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}. \\<br />
\end{align}<br />
</math><br />
<br />
<!-- This paragraph is incorrect; the last line is not the KL divergence between any two distributions, since p(x) is [in general] not a valid distribution over the domains of X and Y. The last formula above is the [[Kullback-Leibler divergence]], also known as relative entropy. Relative entropy is always positive, and vanishes if and only if <math>p(x,y) = p(x)</math>. This is when knowing <math>x</math> tells us everything about <math>y</math>. ADDED: Could this comment be out of date since the KL divergence is not mentioned above? November 2014 --><br />
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<br />
<br />
== Properties 属性 ==<br />
=== Conditional entropy equals zero 条件熵等于零 ===<br />
<math>H(Y|X)=0</math> if and only if the value of <math>Y</math> is completely determined by the value of <math>X</math>.<br />
<br />
当且仅当<math>Y</math>的值完全由<math>X</math>的值确定时，<math>H(Y|X)=0</math>。<br />
<br />
<br />
<br />
=== Conditional entropy of independent random variables 独立随机变量的条件熵 ===<br />
Conversely, <math>H(Y|X) = H(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].<br />
<br />
相反，当且仅当<math>Y</math>和<math>X</math>是互相独立的随机变量时，则<math>H(Y|X) =H(Y)</math>。<br />
<br />
<br />
<br />
=== Chain rule 链式法则 ===<br />
Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>H(X,Y)</math>, that is, we need <math>H(X,Y)</math> bits of information on average to describe its exact state. Now if we first learn the value of <math>X</math>, we have gained <math>H(X)</math> bits of information. Once <math>X</math> is known, we only need <math>H(X,Y)-H(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>H(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:<br />
<br />
假设由两个随机变量<math>X</math>和<math>Y</math>确定的组合系统具有联合熵<math>H(X,Y)</math>，也就是说，我们通常需要<math>H(X,Y)</math>位信息来描述其确切状态。现在，如果我们首先尝试获得<math>X</math>的值，我们将知晓<math>H(X)</math>位信息。一旦<math>X</math>的值确定，我们就可以通过<math>H(X,Y)</math>-<math>H(X)</math>位来描述整个系统的状态。这个数量恰好是<math>H(Y|X)</math>，它给出了条件熵的链式法则：<br />
<br />
<br />
:<math>H(Y|X)\, = \, H(X,Y)- H(X).</math><ref name=cover1991 />{{rp|17}}<br />
<br />
<br />
The chain rule follows from the above definition of conditional entropy:<br />
<br />
链式法则遵循以上条件熵的定义：<br />
<br />
<br />
:<math>\begin{align} <br />
H(Y|X) &= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \left(\frac{p(x)}{p(x,y)} \right) \\[4pt]<br />
&= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)(\log (p(x))-\log (p(x,y))) \\[4pt]<br />
&= -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log (p(x,y)) + \sum_{x\in\mathcal X, y\in\mathcal Y}{p(x,y)\log(p(x))} \\[4pt]<br />
& = H(X,Y) + \sum_{x \in \mathcal X} p(x)\log (p(x) ) \\[4pt]<br />
& = H(X,Y) - H(X). <br />
\end{align}</math><br />
<br />
<br />
In general, a chain rule for multiple random variables holds:<br />
<br />
通常情况下，多个随机变量的链式法则表示为：<br />
<br />
<br />
:<math> H(X_1,X_2,\ldots,X_n) =<br />
\sum_{i=1}^n H(X_i | X_1, \ldots, X_{i-1}) </math><ref name=cover1991 />{{rp|22}}<br />
<br />
<br />
It has a similar form to [[Chain rule (probability)|chain rule]] in probability theory, except that addition instead of multiplication is used.<br />
<br />
除了使用加法而不是乘法之外，它具有与概率论中的链式法则类似的形式。<br />
<br />
<br />
<br />
=== Bayes' rule 贝叶斯法则 ===<br />
[[Bayes' rule]] for conditional entropy states<br />
<br />
条件熵状态的贝叶斯法则<br />
<br />
<br />
:<math>H(Y|X) \,=\, H(X|Y) - H(X) + H(Y).</math><br />
<br />
<br />
''Proof.'' <math>H(Y|X) = H(X,Y) - H(X)</math> and <math>H(X|Y) = H(Y,X) - H(Y)</math>. Symmetry entails <math>H(X,Y) = H(Y,X)</math>. Subtracting the two equations implies Bayes' rule.<br />
<br />
证明，<math>H(Y|X) = H(X,Y) - H(X)</math> 和 <math>H(X|Y) = H(Y,X) - H(Y)</math>。对称性要求<math>H(X,Y) = H(Y,X)</math>。将两个方程式相减就意味着贝叶斯定律。<br />
<br />
<br />
<br />
If <math>Y</math> is [[Conditional independence|conditionally independent]] of <math>Z</math> given <math>X</math> we have:<br />
<br />
如果给定<math>X</math>，<math>Y</math>有条件地独立于<math>Z</math>，则我们有：<br />
<br />
<br />
:<math>H(Y|X,Z) \,=\, H(Y|X).</math><br />
<br />
<br />
<br />
=== Other properties 其他性质 ===<br />
For any <math>X</math> and <math>Y</math>:<br />
<br />
对于任何<math>X</math>和<math>Y</math>：<br />
<br />
<br />
:<math display="block">\begin{align}<br />
H(Y|X) &\le H(Y) \, \\<br />
H(X,Y) &= H(X|Y) + H(Y|X) + \operatorname{I}(X;Y),\qquad \\<br />
H(X,Y) &= H(X) + H(Y) - \operatorname{I}(X;Y),\, \\<br />
\operatorname{I}(X;Y) &\le H(X),\,<br />
\end{align}</math><br />
<br />
where <math>\operatorname{I}(X;Y)</math> is the [[mutual information]] between <math>X</math> and <math>Y</math>.<br />
<br />
其中<math>\operatorname{I}(X;Y)</math>是<math>X</math>和<math>Y</math>之间的<font color="#ff8000"> 互信息</font>。<br />
<br />
<br />
<br />
For independent <math>X</math> and <math>Y</math>:<br />
<br />
对于独立的X和Y：<br />
<br />
<br />
:<math>H(Y|X) = H(Y) </math> and <math>H(X|Y) = H(X) \, </math><br />
<br />
<br />
Although the specific-conditional entropy <math>H(X|Y=y)</math> can be either less or greater than <math>H(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>H(X|Y)</math> can never exceed <math>H(X)</math>.<br />
<br />
对于给定随机变量<math>Y</math>的值<math>y</math>，尽管特定条件熵<math>H(X|Y=y)</math>可以小于或大于<math>H(X)</math>，但<math>H(X|Y)</math>永远不会超过<math>H(X)</math>。<br />
<br />
<br />
<br />
== Conditional differential entropy 条件微分熵 ==<br />
=== Definition 定义 ===<br />
The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called ''conditional differential (or continuous) entropy''. Let <math>X</math> and <math>Y</math> be a continuous random variables with a [[joint probability density function]] <math>f(x,y)</math>. The differential conditional entropy <math>h(X|Y)</math> is defined as<ref name=cover1991 />{{rp|249}}<br />
<br />
上面的定义是针对离散随机变量的。离散条件熵的连续形式称为'''<font color="#ff8000"> 条件微分（或连续）熵Conditional differential (or continuous) entropy </font>'''。 令<math>X</math>和<math>Y</math>为具有联合概率密度函数<math>f(x,y)</math>的连续随机变量。则微分条件熵<math>h(X|Y)</math>定义为：<ref name=cover1991 />{{rp|249}}<br />
<br />
{{Equation box 1<br />
|indent =<br />
|title=<br />
|equation = {{NumBlk||<math>h(X|Y) = -\int_{\mathcal X, \mathcal Y} f(x,y)\log f(x|y)\,dx dy</math>|{{EquationRef|Eq.2}}}}<br />
|cellpadding= 6<br />
|border<br />
|border colour = #0073CF<br />
|background colour=#F5FFFA}}<br />
<br />
<br />
<br />
=== Properties 属性 ===<br />
In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative.<br />
<br />
与离散随机变量的条件熵相比，条件微分熵可能为负。<br />
<br />
<br />
<br />
As in the discrete case there is a chain rule for differential entropy:<br />
<br />
与离散情况一样，微分熵也有链式法则：<br />
<br />
<br />
:<math>h(Y|X)\,=\,h(X,Y)-h(X)</math><ref name=cover1991 />{{rp|253}}<br />
<br />
<br />
Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.<br />
<br />
但是请注意，如果所涉及的微分熵不存在或无限，则此法则可能不成立。<br />
<br />
<br />
<br />
Joint differential entropy is also used in the definition of the [[mutual information]] between continuous random variables:<br />
<br />
联合微分熵也用于定义连续随机变量之间的互信息：<br />
<br />
<br />
:<math>\operatorname{I}(X,Y)=h(X)-h(X|Y)=h(Y)-h(Y|X)</math><br />
<br />
<br />
<math>h(X|Y) \le h(X)</math> with equality if and only if <math>X</math> and <math>Y</math> are independent.<ref name=cover1991 />{{rp|253}}<br />
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当且仅当X和Y是独立的，<math>h(X|Y) \le h(X)</math>等号成立。<br />
<br />
<br />
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=== Relation to estimator error 与估计量误差的关系 ===<br />
The conditional differential entropy yields a lower bound on the expected squared error of an [[estimator]]. For any random variable <math>X</math>, observation <math>Y</math> and estimator <math>\widehat{X}</math> the following holds:<ref name=cover1991 />{{rp|255}}<br />
<br />
条件微分熵在估计量的期望平方误差上有一个下限。对于任何随机变量<math>X</math>，观察值<math>Y</math>和估计量<math>\widehat{X}</math>，以下条件成立：<br />
<br />
<br />
:<math display="block">\mathbb{E}\left[\bigl(X - \widehat{X}{(Y)}\bigr)^2\right] <br />
\ge \frac{1}{2\pi e}e^{2h(X|Y)}</math><br />
<br />
<br />
This is related to the [[uncertainty principle]] from [[quantum mechanics]].<br />
<br />
这与来自量子力学的不确定性原理有关。<br />
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<br />
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== Generalization to quantum theory 量子理论泛化 ==<br />
In [[quantum information theory]], the conditional entropy is generalized to the [[conditional quantum entropy]]. The latter can take negative values, unlike its classical counterpart.<br />
<br />
在量子信息论中，条件熵被广义化为条件量子熵。后者可以采用负值，这与经典方法不同。<br />
<br />
<br />
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== See also 其他参考资料 ==<br />
* [[Entropy (information theory)]]<br />
* [[Mutual information]]<br />
* [[Conditional quantum entropy]]<br />
* [[Variation of information]]<br />
* [[Entropy power inequality]]<br />
* [[Likelihood function]]<br />
<br />
* '''<font color="#ff8000"> 熵（信息论）Entropy (information theory)</font>'''<br />
* '''<font color="#ff8000"> 互信息Mutual information</font>'''<br />
* '''<font color="#ff8000"> 条件量子熵Conditional quantum entropy</font>'''<br />
* '''<font color="#ff8000"> 信息差异Variation of information</font>'''<br />
* '''<font color="#ff8000"> 熵幂不等式Entropy power inequality</font>'''<br />
* '''<font color="#ff8000"> 似然函数Likelihood function</font>'''<br />
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== References 参考文献 ==<br />
{{Reflist}}<br />
<br />
[[Category:Entropy and information]]<br />
[[Category:Information theory]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=19571蚁群优化算法2020-12-01T11:54:15Z<p>Lincent：</p>
<hr />
<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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{{multiple issues|<br />
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{{multiple issues|<br />
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{多个问题 |<br />
<br />
{{Original research|date=August 2018}}<br />
<br />
{{More footnotes|date=August 2018}}<br />
<br />
}}<br />
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}}<br />
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}}<br />
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<br />
<br />
[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
<br />
Ant behavior was the inspiration for the metaheuristic optimization technique<br />
<br />
蚂蚁行为是元启发式优化技术的灵感来源<br />
<br />
[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
<br />
When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
<br />
当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
<br />
In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
<br />
在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于<font color="#ff8000"> 概率</font>解决计算问题的技术，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚂蚁群代表了受真实蚂蚁行为启发的多主体方法。<br />
<br />
The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
<br />
The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
<br />
基于信息素的蚂蚁通信常被作为典型范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造「蚂蚁」(例如:模拟仿真代替)通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的模拟迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫————蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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这个算法是蚁群算法家族的一员，在<font color="#ff8000">群体智能</font>方法中，它构成了一些<font color="#ff8000">元启发式</font>的优化。最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，第一个蚁群算法的目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法实现了基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这样一条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，从而对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对于后面的蚂蚁来说就会非常有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中具有重要意义<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是中心化的而可以在所有微小的物体中找到，所以需要一些新的概念来描述。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和计算力量是集中的。这些集中的单元功能不断地提高，并可以与人类的大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比较的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物体如果都遵循同样的基本规则，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了许多可以应用到环境对象的网络中的特性。蚂蚁群体具有很高的适应环境变化的能力，并且在处理某项个体无法完成的任务时具有很大的能力。这种灵活性对于不断发展的移动网络也是非常有用的。从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
<br />
<br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
<br />
2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
<br />
该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
<br />
<br />
<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
<br />
'''<font color="#32CD32"><br />
<br />
'''procedure''' ACO_MetaHeuristic '''is'''<br />
<br />
procedure ACO_MetaHeuristic is<br />
<br />
过程 ACO_ 元启发式是<br />
<br />
'''while''' not_termination '''do'''<br />
<br />
while not_termination do<br />
<br />
而不是终止合同<br />
<br />
generateSolutions()<br />
<br />
generateSolutions()<br />
<br />
概要解答()<br />
<br />
daemonActions()<br />
<br />
daemonActions()<br />
<br />
daemonActions ()<br />
<br />
pheromoneUpdate()<br />
<br />
pheromoneUpdate()<br />
<br />
信息素更新()<br />
<br />
'''repeat'''<br />
<br />
repeat<br />
<br />
重复<br />
<br />
end procedure<br />
<br />
end procedure<br />
<br />
结束过程<br />
<br />
</font>'''<br />
<br />
<br />
<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#32cd32">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font><br />
<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
p_{xy}^k =<br />
<br />
p_{xy}^k =<br />
<br />
1 = 0 = 0 = 0<br />
<br />
\frac<br />
<br />
\frac<br />
<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })}<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })}<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#32cd32"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font><br />
<br />
<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常会被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy }<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
3. d<br />
<br />
\begin{cases}<br />
<br />
\begin{cases}<br />
<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
<br />
0 & \mbox{otherwise}<br />
<br />
0 & \mbox{otherwise}<br />
<br />
<br />
\end{cases}<br />
<br />
\end{cases}<br />
<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。<br />
<br />
<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
<br />
<br />
<br />
===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
<br />
<br />
<br />
<br />
===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
<br />
在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32cd32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁一起将信息素释放在该路径上(即使这条路径没有被重访问过)。<br />
<br />
<br />
<br />
===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
<br />
<br />
<br />
===Max-Min Ant System (MMAS)===<br />
最大-最小 蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素<br />
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===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
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COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#32cd32">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
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它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#32cd32">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
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==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
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[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
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Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
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蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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==Applications==<br />
应用<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现 <br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题，其中的目标是找到最短的往返路线连接一系列城市。一般的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
它还被用于产生[[旅行商问题]]的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的[[模拟退火]]和[[遗传算法]]方法的优势；蚁群算法可以连续运行，实时适应变化。这对[[网络路由]]和城市交通系统很有意义 <br />
It must visit each city exactly once;<br />
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它必须每个城市只到达一次;<br />
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A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上留下的信息素踪迹越强烈，这条边被选择的概率就越大;<br />
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# It must visit each city exactly once;<br />
它必须访问每座城市一次<br />
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
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# A distant city has less chance of being chosen (the visibility);<br />
远的城市只有很小的机会被选中（可视性）<br />
After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
在两个城市之间的边上，信息素的踪迹越密集，被选中的可能性就越大 <br />
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
蚂蚁完成了它的旅程，如果旅程很短的话，它会在经过的所有边缘上沉积更多的信息素 <br />
center<br />
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中心<br />
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# After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素轨迹蒸发<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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===Scheduling problem===<br />
规划难题<br />
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*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
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*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
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*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
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*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
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*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
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*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
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*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
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*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
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*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
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===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
<br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
<br />
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
<br />
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
<br />
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
<br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
<br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
<br />
<br />
===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
<br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
<br />
<br />
<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
<br />
采用蚁群算法合成回路振子10 × 10<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
<br />
[文件: ANT antenna 2. jpg | thumb | Unloopback 振子10 × 10，由 ACO 算法回环和未回环振子10 × 10合成<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#32cd32">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
* '''Edge detection:'''<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math > <br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，是一个归一化因子<br />
<br />
<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
< math > begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math > <br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) = <br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
结束{ cases } </math > < br/> < math > f (x) = <br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]<br />
<br />
\begin{cases}<br />
<br />
\tau_{new} \leftarrow<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
(1-psi) tau _ { old } + psi tau _ {0}<br />
<br />
0, & \text{else}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
这里是信息素衰减系数。<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
\begin{cases}<br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
0, & \text{else}<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
thumb<br />
<br />
拇指<br />
<br />
\tau_{new} \leftarrow<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br />
<br />
''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].<br />
<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
<br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
<br />
<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
<br />
thumb<br />
<br />
拇指<br />
<br />
| last1 = Warner | first1 = Lars<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
| last2 = Vogel | first2 = Ute<br />
<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
<br />
| date = 2008<br />
<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
<br />
| publisher = Shaker Verlag<br />
<br />
| location = Aachen, Germany<br />
<br />
| isbn = 978-3-8322-7313-2<br />
<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
<br />
| access-date = 2018-10-09<br />
<br />
}}<br />
<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
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* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
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* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
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The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
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发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
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* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
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==Definition difficulty==<br />
定义困难那<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
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According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of [[social insects]] remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "[[swarm intelligence]]",<ref name="Waldner 2008 214"/> which is a very general framework in which ant colony algorithms fit.<br />
根据一些作者的观点，ACO算法与其他相关算法（如<font color="#ff8000">分布估计算法</font>或<font color="#ff8000">粒子群优化算法</font>）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。社会性昆虫的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 <br />
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==Stigmergy algorithms==<br />
Stigmergy算法 <br />
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There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
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==Related methods==<br />
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*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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*Gravitational search algorithm (GSA), a [[swarm intelligence]] method<br />
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*Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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蚁群算法年表。<br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中 <br />
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==References==<br />
参考<br />
{{Reflist|30em}}<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=19570蚁群优化算法2020-12-01T11:48:35Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于<font color="#ff8000"> 概率</font>解决计算问题的技术，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚂蚁群代表了受真实蚂蚁行为启发的多主体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信常被作为典型范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造「蚂蚁」(例如:模拟仿真代替)通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的模拟迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫————蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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这个算法是蚁群算法家族的一员，在<font color="#ff8000">群体智能</font>方法中，它构成了一些<font color="#ff8000">元启发式</font>的优化。最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，第一个蚁群算法的目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法实现了基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这样一条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，从而对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对于后面的蚂蚁来说就会非常有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中具有重要意义<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是中心化的而可以在所有微小的物体中找到，所以需要一些新的概念来描述。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和计算力量是集中的。这些集中的单元功能不断地提高，并可以与人类的大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比较的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物体如果都遵循同样的基本规则，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了许多可以应用到环境对象的网络中的特性。蚂蚁群体具有很高的适应环境变化的能力，并且在处理某项个体无法完成的任务时具有很大的能力。这种灵活性对于不断发展的移动网络也是非常有用的。从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
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2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
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该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
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==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
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In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
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在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
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'''<font color="#32CD32"><br />
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'''procedure''' ACO_MetaHeuristic '''is'''<br />
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procedure ACO_MetaHeuristic is<br />
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过程 ACO_ 元启发式是<br />
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'''while''' not_termination '''do'''<br />
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while not_termination do<br />
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而不是终止合同<br />
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generateSolutions()<br />
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generateSolutions()<br />
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概要解答()<br />
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daemonActions()<br />
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daemonActions()<br />
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daemonActions ()<br />
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pheromoneUpdate()<br />
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pheromoneUpdate()<br />
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信息素更新()<br />
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'''repeat'''<br />
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repeat<br />
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重复<br />
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end procedure<br />
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end procedure<br />
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结束过程<br />
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</font>'''<br />
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===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#ff8000">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font><br />
<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
p_{xy}^k =<br />
<br />
p_{xy}^k =<br />
<br />
1 = 0 = 0 = 0<br />
<br />
\frac<br />
<br />
\frac<br />
<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })}<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })}<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#ff8000"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font><br />
<br />
<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常会被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy }<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
3. d<br />
<br />
\begin{cases}<br />
<br />
\begin{cases}<br />
<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
<br />
0 & \mbox{otherwise}<br />
<br />
0 & \mbox{otherwise}<br />
<br />
<br />
\end{cases}<br />
<br />
\end{cases}<br />
<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。<br />
<br />
<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
<br />
<br />
<br />
===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
<br />
<br />
<br />
<br />
===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
<br />
在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁一起将信息素释放在该路径上(即使这条路径没有被重访问过)。<br />
<br />
<br />
<br />
===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
<br />
<br />
<br />
===Max-Min Ant System (MMAS)===<br />
最大-最小 蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素<br />
<br />
<br />
===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
<br />
COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#ff8000">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
<br />
<br />
===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
<br />
它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#ff8000">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
<br />
<br />
===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
<br />
对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
<br />
<br />
<br />
==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
<br />
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
<br />
<br />
<br />
Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
<br />
蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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==Applications==<br />
应用<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现 <br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题，其中的目标是找到最短的往返路线连接一系列城市。一般的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
它还被用于产生[[旅行商问题]]的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的[[模拟退火]]和[[遗传算法]]方法的优势；蚁群算法可以连续运行，实时适应变化。这对[[网络路由]]和城市交通系统很有意义 <br />
It must visit each city exactly once;<br />
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它必须每个城市只到达一次;<br />
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A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上留下的信息素踪迹越强烈，这条边被选择的概率就越大;<br />
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# It must visit each city exactly once;<br />
它必须访问每座城市一次<br />
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
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# A distant city has less chance of being chosen (the visibility);<br />
远的城市只有很小的机会被选中（可视性）<br />
After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
在两个城市之间的边上，信息素的踪迹越密集，被选中的可能性就越大 <br />
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
蚂蚁完成了它的旅程，如果旅程很短的话，它会在经过的所有边缘上沉积更多的信息素 <br />
center<br />
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中心<br />
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# After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素轨迹蒸发<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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===Scheduling problem===<br />
规划难题<br />
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*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
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*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
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*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
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*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
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*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
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*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
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*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
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*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
<br />
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
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===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
<br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
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*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
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*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
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*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
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*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
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*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
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*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
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===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
<br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
<br />
<br />
<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
<br />
采用蚁群算法合成回路振子10 × 10<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
<br />
[文件: ANT antenna 2. jpg | thumb | Unloopback 振子10 × 10，由 ACO 算法回环和未回环振子10 × 10合成<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#ff8000">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
* '''Edge detection:'''<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math > <br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，是一个归一化因子<br />
<br />
<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
< math > begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math > <br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) = <br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
结束{ cases } </math > < br/> < math > f (x) = <br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]<br />
<br />
\begin{cases}<br />
<br />
\tau_{new} \leftarrow<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
(1-psi) tau _ { old } + psi tau _ {0}<br />
<br />
0, & \text{else}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
这里是信息素衰减系数。<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
\begin{cases}<br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
0, & \text{else}<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
thumb<br />
<br />
拇指<br />
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\tau_{new} \leftarrow<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br />
<br />
''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].<br />
<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
<br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
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<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
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<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
<br />
thumb<br />
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拇指<br />
<br />
| last1 = Warner | first1 = Lars<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
| last2 = Vogel | first2 = Ute<br />
<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
<br />
| date = 2008<br />
<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
<br />
| publisher = Shaker Verlag<br />
<br />
| location = Aachen, Germany<br />
<br />
| isbn = 978-3-8322-7313-2<br />
<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
<br />
| access-date = 2018-10-09<br />
<br />
}}<br />
<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
<br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
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{{image frame|content=<br />
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{图片框架 | 内容 = <br />
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==Definition difficulty==<br />
定义困难那<br />
<timeline><br />
<br />
< 时间表 > <br />
<br />
[[File:Aco shortpath.svg|thumb]]<br />
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ImageSize = width:210 height:300<br />
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210 height: 300<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
<br />
According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of [[social insects]] remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "[[swarm intelligence]]",<ref name="Waldner 2008 214"/> which is a very general framework in which ant colony algorithms fit.<br />
根据一些作者的观点，ACO算法与其他相关算法（如<font color="#ff8000">分布估计算法</font>或<font color="#ff8000">粒子群优化算法</font>）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。社会性昆虫的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 <br />
PlotArea = width:170 height:280 left:40 bottom:10<br />
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==Stigmergy algorithms==<br />
Stigmergy算法 <br />
DateFormat = yyyy<br />
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There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
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==Related methods==<br />
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*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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*Gravitational search algorithm (GSA), a [[swarm intelligence]] method<br />
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*Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
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蚁群算法年表。<br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中 <br />
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==References==<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=19564蚁群优化算法2020-12-01T02:56:09Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些使用较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于<font color="#ff8000"> 概率</font>解决计算问题的技术，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚂蚁代表了受真蚂蚁行为启发的多智能体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信被用来作为非常典型的范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业用途。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造「蚂蚁」(例如:模拟仿真代替)通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生<font color="#ff8000">信息素</font>来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的模拟迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫————蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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这个算法是蚁群算法家族的一员，在<font color="#ff8000">群体智能</font>方法中，它构成了一些<font color="#ff8000">元启发式</font>的优化。最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，第一个蚁群算法的目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法实现了基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这样一条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，从而对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对于后面的蚂蚁来说就会非常有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中具有重要意义<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是中心化的而可以在所有微小的物体中找到，所以需要一些新的概念来描述。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和计算力量是集中的。这些集中的单元功能不断地提高，并可以与人类的大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境网络，以及基于<font color="#ff8000">纳米技术</font>的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比较的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物体如果都遵循同样的基本规则，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了许多可以应用到环境对象的网络中的特性。蚂蚁群体具有很高的适应环境变化的能力，并且在处理某项个体无法完成的任务时具有很大的能力。这种灵活性对于不断发展的移动网络也是非常有用的。从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
<br />
bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
<br />
bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
<br />
蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
<br />
<br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
<br />
2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
<br />
该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
<br />
<br />
<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
<br />
'''<font color="#32CD32"><br />
<br />
'''procedure''' ACO_MetaHeuristic '''is'''<br />
<br />
procedure ACO_MetaHeuristic is<br />
<br />
过程 ACO_ 元启发式是<br />
<br />
'''while''' not_termination '''do'''<br />
<br />
while not_termination do<br />
<br />
而不是终止合同<br />
<br />
generateSolutions()<br />
<br />
generateSolutions()<br />
<br />
概要解答()<br />
<br />
daemonActions()<br />
<br />
daemonActions()<br />
<br />
daemonActions ()<br />
<br />
pheromoneUpdate()<br />
<br />
pheromoneUpdate()<br />
<br />
信息素更新()<br />
<br />
'''repeat'''<br />
<br />
repeat<br />
<br />
重复<br />
<br />
end procedure<br />
<br />
end procedure<br />
<br />
结束过程<br />
<br />
</font>'''<br />
<br />
<br />
<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。<br />
<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
p_{xy}^k =<br />
<br />
p_{xy}^k =<br />
<br />
1 = 0 = 0 = 0<br />
<br />
\frac<br />
<br />
\frac<br />
<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })}<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })}<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力<br />
<br />
<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常会被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy }<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
3. d<br />
<br />
\begin{cases}<br />
<br />
\begin{cases}<br />
<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
<br />
0 & \mbox{otherwise}<br />
<br />
0 & \mbox{otherwise}<br />
<br />
<br />
\end{cases}<br />
<br />
\end{cases}<br />
<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。<br />
<br />
<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
<br />
<br />
<br />
===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
<br />
<br />
<br />
<br />
===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
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在蚁群系统算法中，原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
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In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁一起将信息素释放在该路径上(即使这条路径没有被重访问过)。<br />
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===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
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This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
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该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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===Max-Min Ant System (MMAS)===<br />
最大-最小 蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
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All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素<br />
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===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
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The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
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COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用正交化设计法，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
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It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
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它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，并将其中最好的几个子域提升到下一个等级。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
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==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
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[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
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Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
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蚁群算法已经被应用于许多组合优化问题，从<font color="#ff8000">二次分配</font>到<font color="#ff8000">蛋白质折叠</font>或<font color="#ff8000">路径选择</font>问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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==Applications==<br />
应用<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生<font color="#ff8000">旅行商问题</font>的近似最优解。当图形可能发生动态变化时，它们比<font color="#ff8000">模拟退火算法</font>和<font color="#ff8000">遗传算法</font>具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现 <br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题，其中的目标是找到最短的往返路线连接一系列城市。一般的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
它还被用于产生[[旅行商问题]]的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的[[模拟退火]]和[[遗传算法]]方法的优势；蚁群算法可以连续运行，实时适应变化。这对[[网络路由]]和城市交通系统很有意义 <br />
It must visit each city exactly once;<br />
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它必须每个城市只到达一次;<br />
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A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上布置的信息素踪迹越强烈，那条边被选择的概率就越大;<br />
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# It must visit each city exactly once;<br />
它必须访问每座城市一次<br />
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
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# A distant city has less chance of being chosen (the visibility);<br />
远的城市只有很小的机会被选中（可视性）<br />
After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
在两个城市之间的边上，信息素的踪迹越密集，被选中的可能性就越大 <br />
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
蚂蚁完成了它的旅程，如果旅程很短的话，它会在经过的所有边缘上沉积更多的信息素 <br />
center<br />
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中心<br />
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# After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素轨迹蒸发<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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===Scheduling problem===<br />
规划难题<br />
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*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
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*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
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*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
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*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
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*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
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*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
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*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
<br />
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
<br />
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
<br />
<br />
===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
<br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
<br />
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
<br />
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
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*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
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*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
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*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
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*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
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===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
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*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
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*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
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*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
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<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
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*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
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*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
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*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
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*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
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<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
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采用蚁群算法合成回路振子10 × 10<br />
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* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
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[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
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[文件: ANT antenna 2. jpg | thumb | Unloopback 振子10 × 10，由 ACO 算法回环和未回环振子10 × 10合成<br />
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===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
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这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
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[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
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[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
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The following are the steps involved in edge detection using ACO:<br />
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以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线<font color="#ff8000">RFID</font>标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
* '''Edge detection:'''<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math > <br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，是一个归一化因子<br />
<br />
<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
< math > begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math > <br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) = <br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
结束{ cases } </math > < br/> < math > f (x) = <br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]<br />
<br />
\begin{cases}<br />
<br />
\tau_{new} \leftarrow<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
(1-psi) tau _ { old } + psi tau _ {0}<br />
<br />
0, & \text{else}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
这里是信息素衰减系数。<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
\begin{cases}<br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
0, & \text{else}<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
thumb<br />
<br />
拇指<br />
<br />
\tau_{new} \leftarrow<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br />
<br />
''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].<br />
<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
<br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
<br />
<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
<br />
thumb<br />
<br />
拇指<br />
<br />
| last1 = Warner | first1 = Lars<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
| last2 = Vogel | first2 = Ute<br />
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| title = Optimization of energy supply networks using ant colony optimization<br />
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| date = 2008<br />
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| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
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| publisher = Shaker Verlag<br />
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| location = Aachen, Germany<br />
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| isbn = 978-3-8322-7313-2<br />
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| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
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| access-date = 2018-10-09<br />
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}}<br />
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</ref><br />
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* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
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* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
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* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
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* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
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* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
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==Definition difficulty==<br />
定义困难那<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
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According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of [[social insects]] remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "[[swarm intelligence]]",<ref name="Waldner 2008 214"/> which is a very general framework in which ant colony algorithms fit.<br />
根据一些作者的观点，ACO算法与其他相关算法（如<font color="#ff8000">分布估计算法</font>或<font color="#ff8000">粒子群优化算法</font>）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。社会性昆虫的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 <br />
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==Stigmergy algorithms==<br />
Stigmergy算法 <br />
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There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
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==Related methods==<br />
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*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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*Gravitational search algorithm (GSA), a [[swarm intelligence]] method<br />
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*Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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from:1989 till:1989 shift:($dx,$dy) text:studies of collective behavior<br />
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from:1996 till:1996 shift:($dx,$dy) text:ant colony system (ACS)<br />
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Chronology of ant colony optimization algorithms.<br />
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蚁群算法年表。<br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中 <br />
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==References==<br />
参考<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=19563蚁群优化算法2020-12-01T02:13:50Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些使用较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于概率解决计算问题的技术，它可以简化为通过图来寻找最优路径。人工蚂蚁代表了受真蚂蚁行为启发的多智能体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信被用来作为非常典型的范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种图，例如车辆路径和互联网路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业用途。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造「蚂蚁」(例如:模拟仿真代替)通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生信息素来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的模拟迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫————蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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这个算法是蚁群算法家族的一员，在群体智能方法中，它构成了一些元启发式的优化。最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，第一个蚁群算法的目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法实现了基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这样一条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，从而对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对于后面的蚂蚁来说就会非常有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中具有重要意义<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是中心化的而可以在所有微小的物体中找到，所以需要一些新的概念来描述。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和计算力量是集中的。这些集中的单元功能不断地提高，并可以与人类的大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境的网络，以及基于纳米技术的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比较的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物体如果都遵循同样的基本规则，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> 他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了许多可以应用到环境对象的网络中的特性。蚂蚁群体具有很高的适应环境变化的能力，并且在处理某项个体无法完成的任务时具有很大的能力。这种灵活性对于不断发展的移动网络也是非常有用的。从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学<ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref>或物理(RFID标签<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref>、光<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref>、声<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
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2007年，Garnier，Simon 等人在一篇IEEE论文<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> 中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref>，该方法基于精确快速的视觉定位。<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
<br />
该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
<br />
<br />
<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
<br />
'''<font color="#32CD32"><br />
<br />
'''procedure''' ACO_MetaHeuristic '''is'''<br />
<br />
procedure ACO_MetaHeuristic is<br />
<br />
过程 ACO_ 元启发式是<br />
<br />
'''while''' not_termination '''do'''<br />
<br />
while not_termination do<br />
<br />
而不是终止合同<br />
<br />
generateSolutions()<br />
<br />
generateSolutions()<br />
<br />
概要解答()<br />
<br />
daemonActions()<br />
<br />
daemonActions()<br />
<br />
daemonActions ()<br />
<br />
pheromoneUpdate()<br />
<br />
pheromoneUpdate()<br />
<br />
信息素更新()<br />
<br />
'''repeat'''<br />
<br />
repeat<br />
<br />
重复<br />
<br />
end procedure<br />
<br />
end procedure<br />
<br />
结束过程<br />
<br />
</font>'''<br />
<br />
<br />
<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。<br />
<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
p_{xy}^k =<br />
<br />
p_{xy}^k =<br />
<br />
1 = 0 = 0 = 0<br />
<br />
\frac<br />
<br />
\frac<br />
<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })}<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })}<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力<br />
<br />
<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常会被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy }<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，<math>\tau_{xy}</math>是状态转换<math>xy</math>的信息素沉积量，<math>\rho</math>是信息素蒸发系数，<math>\Delta \tau^{k}_{xy}</math>是第<math>k</math>只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
3. d<br />
<br />
\begin{cases}<br />
<br />
\begin{cases}<br />
<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
<br />
0 & \mbox{otherwise}<br />
<br />
0 & \mbox{otherwise}<br />
<br />
<br />
\end{cases}<br />
<br />
\end{cases}<br />
<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。<br />
<br />
<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
<br />
<br />
<br />
===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<ref name="Ant system" /><br />
<br />
<br />
<br />
<br />
===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
<br />
在蚁群系统算法中，原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁一起将信息素释放在该路径上(即使这条路径没有被重访问过)。<br />
<br />
<br />
<br />
===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ<sub>max</sub>,τ<sub>min</sub>]。所有边都被初始化为τ<sub>max</sub>，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ<sub>max</sub>。<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
<br />
<br />
<br />
===Max-Min Ant System (MMAS)===<br />
最大-最小 蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素<br />
<br />
<br />
===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
<br />
COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用正交化设计法，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
<br />
<br />
===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
<br />
它是蚂蚁系统的一种递归形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，并将其中最好的几个子域提升到下一个等级。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
<br />
<br />
===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
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For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 在2004年，Zlochin 和他的同事<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref>们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
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==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
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[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
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Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
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蚁群算法已经被应用于许多组合优化问题，从二次分配到蛋白质折叠或路径选择问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
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==Applications==<br />
应用<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
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它也被用来产生旅行商问题的近似最优解。当图形可能发生动态变化时，它们比模拟退火算法和遗传算法具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现 <br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref>，它的目标是解决旅行商问题，其中的目标是找到最短的往返路线连接一系列城市。一般的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
它还被用于产生[[旅行商问题]]的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的[[模拟退火]]和[[遗传算法]]方法的优势；蚁群算法可以连续运行，实时适应变化。这对[[网络路由]]和城市交通系统很有意义 <br />
It must visit each city exactly once;<br />
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它必须每个城市只到达一次;<br />
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A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
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The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上布置的信息素踪迹越强烈，那条边被选择的概率就越大;<br />
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# It must visit each city exactly once;<br />
它必须访问每座城市一次<br />
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
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# A distant city has less chance of being chosen (the visibility);<br />
远的城市只有很小的机会被选中（可视性）<br />
After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素踪迹会蒸发。<br />
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# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
在两个城市之间的边上，信息素的踪迹越密集，被选中的可能性就越大 <br />
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
蚂蚁完成了它的旅程，如果旅程很短的话，它会在经过的所有边缘上沉积更多的信息素 <br />
center<br />
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中心<br />
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# After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素轨迹蒸发<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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===Scheduling problem===<br />
规划难题<br />
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*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
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*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
工序车间调度问题<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
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*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
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*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
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*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
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*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
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*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
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*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
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*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
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*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
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===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
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*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
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*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
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*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
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*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
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*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
<br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
<br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
<br />
<br />
===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
<br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
<br />
<br />
<br />
===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
<br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
<br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
<br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权l-基数树问题 <ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
<br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
<br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
<br />
采用蚁群算法合成回路振子10 × 10<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
<br />
[文件: ANT antenna 2. jpg | thumb | Unloopback 振子10 × 10，由 ACO 算法回环和未回环振子10 × 10合成<br />
<br />
<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线RFID标签, <ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref>loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
<br />
<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
* '''Edge detection:'''<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math > <br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，是一个归一化因子<br />
<br />
<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
< math > begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math > <br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) = <br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
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结束{ cases } </math > < br/> < math > f (x) = <br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]<br />
<br />
\begin{cases}<br />
<br />
\tau_{new} \leftarrow<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
(1-psi) tau _ { old } + psi tau _ {0}<br />
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0, & \text{else}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
这里是信息素衰减系数。<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
\begin{cases}<br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
<br />
0, & \text{else}<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
thumb<br />
<br />
拇指<br />
<br />
\tau_{new} \leftarrow<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br />
<br />
''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].<br />
<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
<br />
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
<br />
<br />
<br />
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
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<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
<br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <ref name="warner-and-vogel-2008"><br />
{{cite conference<br />
<br />
thumb<br />
<br />
拇指<br />
<br />
| last1 = Warner | first1 = Lars<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
| last2 = Vogel | first2 = Ute<br />
<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
<br />
| date = 2008<br />
<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
<br />
| publisher = Shaker Verlag<br />
<br />
| location = Aachen, Germany<br />
<br />
| isbn = 978-3-8322-7313-2<br />
<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
<br />
| access-date = 2018-10-09<br />
<br />
}}<br />
<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
<br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
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* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
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The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
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发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
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* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
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==Definition difficulty==<br />
定义困难那<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> <br />
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According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of [[social insects]] remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "[[swarm intelligence]]",<ref name="Waldner 2008 214"/> which is a very general framework in which ant colony algorithms fit.<br />
根据一些作者的观点，ACO算法与其他相关算法（如分布估计算法或粒子群优化算法）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。[[群居昆虫]的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 <br />
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==Stigmergy algorithms==<br />
Stigmergy算法 <br />
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There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
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==Related methods==<br />
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*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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*Gravitational search algorithm (GSA), a [[swarm intelligence]] method<br />
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*Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
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蚁群算法年表。<br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中 <br />
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==References==<br />
参考<br />
{{Reflist|30em}}<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=19513蚁群优化算法2020-11-30T10:15:27Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些使用较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于概率解决计算问题的技术，它可以简化为通过图来寻找最优路径。人工蚂蚁代表了受真蚂蚁行为启发的多智能体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信被用来作为非常典型的范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种图，例如车辆路径和互联网路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业用途。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造「蚂蚁」(例如:模拟仿真代替)通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生信息素来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的模拟迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫————蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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这个算法是蚁群算法家族的一员，在群体智能方法中，它构成了一些元启发式的优化。最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，第一个蚁群算法的目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法实现了基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这样一条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，从而对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对于后面的蚂蚁来说就会非常有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中具有重要意义<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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<br />
===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
<br />
因为“智能”不再是中心化的而可以在所有微小的物体中找到，所以需要一些新的概念来描述。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和计算力量是集中的。这些集中的单元功能不断地提高，并可以与人类的大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境的网络，以及基于纳米技术的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比较的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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<br />
Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
<br />
Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
<br />
大自然中存在很多例子，说明了微小的生物体如果都遵循同样的基本规则，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了许多可以应用到环境对象的网络中的特性。蚂蚁群体具有很高的适应环境变化的能力，并且在处理某项个体无法完成的任务时具有很大的能力。这种灵活性对于不断发展的移动网络也是非常有用的。从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
<br />
基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
<br />
bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
<br />
bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
<br />
蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学或物理(RFID标签、光、声)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
<br />
<br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
<br />
2007年，Garnier，Simon 等人在一篇IEEE论文中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ，该方法基于精确快速的视觉定位。<br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
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The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
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该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
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<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
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'''<font color="#32CD32"><br />
<br />
'''procedure''' ACO_MetaHeuristic '''is'''<br />
<br />
procedure ACO_MetaHeuristic is<br />
<br />
过程 ACO_ 元启发式是<br />
<br />
'''while''' not_termination '''do'''<br />
<br />
while not_termination do<br />
<br />
而不是终止合同<br />
<br />
generateSolutions()<br />
<br />
generateSolutions()<br />
<br />
概要解答()<br />
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daemonActions()<br />
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daemonActions()<br />
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daemonActions ()<br />
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pheromoneUpdate()<br />
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pheromoneUpdate()<br />
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信息素更新()<br />
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'''repeat'''<br />
<br />
repeat<br />
<br />
重复<br />
<br />
end procedure<br />
<br />
end procedure<br />
<br />
结束过程<br />
<br />
</font>'''<br />
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<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。<br />
<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
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一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
p_{xy}^k =<br />
<br />
p_{xy}^k =<br />
<br />
1 = 0 = 0 = 0<br />
<br />
\frac<br />
<br />
\frac<br />
<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })}<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })}<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
τxy是从状态x到y的信息素积累量，0≤α是控制τxy影响的参数，ηxy是状态转换xy的期望值（一种先验信息，通常为1/dxy，其中d是距离），β≥1是控制ηxy影响的参数。τxz和ηxz代表了其他可能的状态转移的轨迹浓度等级和吸引力<br />
<br />
<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常会被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy }<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，τxy是状态转换xy的信息素沉积量，ρ是信息素蒸发系数，Δτkxy是第k只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
3. d<br />
<br />
\begin{cases}<br />
<br />
\begin{cases}<br />
<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
<br />
0 & \mbox{otherwise}<br />
<br />
0 & \mbox{otherwise}<br />
<br />
<br />
\end{cases}<br />
<br />
\end{cases}<br />
<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。<br />
<br />
<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
<br />
<br />
<br />
===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<br />
<br />
<br />
<br />
===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
<br />
在蚁群系统算法中，原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<br />
<br />
<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁一起将信息素释放在该路径上(即使这条路径没有被重访问过)。<br />
<br />
<br />
<br />
===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τmax,τmin]。所有边都被初始化为τmax，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τmax 。<br />
<br />
<br />
<br />
===Max-Min Ant System (MMAS)===<br />
最大-最小 蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素<br />
<br />
<br />
===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
<br />
COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用正交化设计法，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<br />
<br />
<br />
<br />
===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
<br />
它是蚂蚁系统的一种递归形式，将整个搜索域划分为若干子域，并在这些子域上求解。对所有子域的结果进行比较，并将其中最好的几个子域提升到下一个等级。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<br />
<br />
<br />
<br />
===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
<br />
对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。在2004年，Zlochin 和他的同事们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
<br />
<br />
<br />
==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
<br />
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
<br />
<br />
<br />
Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
<br />
蚁群算法已经被应用于许多组合优化问题，从二次分配到蛋白质折叠或路径选择问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
<br />
==Applications==<br />
应用<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
<br />
它也被用来产生旅行商问题的近似最优解。当图形可能发生动态变化时，它们比模拟退火算法和遗传算法具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
<br />
[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现 <br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
<br />
第一个蚁群算法被称为蚂蚁系统，它的目标是解决旅行商问题，其中的目标是找到最短的往返路线连接一系列城市。一般的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
<br />
It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
它还被用于产生[[旅行商问题]]的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的[[模拟退火]]和[[遗传算法]]方法的优势；蚁群算法可以连续运行，实时适应变化。这对[[网络路由]]和城市交通系统很有意义 <br />
It must visit each city exactly once;<br />
<br />
它必须每个城市只到达一次;<br />
<br />
<br />
<br />
A distant city has less chance of being chosen (the visibility);<br />
<br />
较远的城市被选中的机会较少(可见度) ;<br />
<br />
The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
<br />
The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
<br />
在两个城市之间的边上布置的信息素踪迹越强烈，那条边被选择的概率就越大;<br />
<br />
# It must visit each city exactly once;<br />
它必须访问每座城市一次<br />
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
<br />
旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
<br />
# A distant city has less chance of being chosen (the visibility);<br />
远的城市只有很小的机会被选中（可视性）<br />
After each iteration, trails of pheromones evaporate.<br />
<br />
每次迭代后，信息素踪迹会蒸发。<br />
<br />
# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
在两个城市之间的边上，信息素的踪迹越密集，被选中的可能性就越大 <br />
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
蚂蚁完成了它的旅程，如果旅程很短的话，它会在经过的所有边缘上沉积更多的信息素 <br />
center<br />
<br />
中心<br />
<br />
# After each iteration, trails of pheromones evaporate.<br />
<br />
每次迭代后，信息素轨迹蒸发<br />
<br />
[[File:Aco TSP.svg|thumb|600px|center]]<br />
<br />
<br />
<br />
===Scheduling problem===<br />
规划难题<br />
*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <br />
*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.<br />
工序车间调度问题<br />
</ref><br />
<br />
*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<br />
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <br />
*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 <br />
*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <br />
*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<br />
*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<br />
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <br />
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <br />
<br />
<br />
===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <br />
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<br />
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <br />
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
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===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <br />
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===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权l-基数树问题 <br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<br />
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<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
<br />
采用蚁群算法合成回路振子10 × 10<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
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[文件: ANT antenna 2. jpg | thumb | Unloopback 振子10 × 10，由 ACO 算法回环和未回环振子10 × 10合成<br />
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<br />
<br />
===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
<br />
这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
<br />
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
<br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
<br />
The following are the steps involved in edge detection using ACO:<br />
<br />
以下是使用 ACO 进行边缘检测的步骤:<br />
<br />
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
为了优化天线的形状，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线RFID标签 <br />
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<br />
Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
<br />
===Image processing===<br />
<br />
The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
<br />
* '''Edge detection:'''<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
像素(i，j)的局部统计数据。<br />
<br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
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<br />
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
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< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math > <br />
<br />
The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
<br />
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
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[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，是一个归一化因子<br />
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<br />
<br />
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
<br />
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
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< math > begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\<br />
<br />
the local statistics at the pixel position (i,j).<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
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<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
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左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math > <br />
<br />
<br />
<br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
<br />
<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) = <br />
<br />
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}<br />
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<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
<br />
0, & \text{else}<br />
<br />
0，& text { else }<br />
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& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
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结束{ cases } </math > < br/> < math > f (x) = <br />
<br />
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
<br />
\begin{cases}<br />
<br />
开始{ cases }<br />
<br />
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
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<br />
0, & \text{else}<br />
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0，& text { else }<br />
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<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
<br />
\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
<br />
上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]<br />
<br />
\begin{cases}<br />
<br />
\tau_{new} \leftarrow<br />
<br />
<br />
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
(1-psi) tau _ { old } + psi tau _ {0}<br />
<br />
0, & \text{else}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
这里是信息素衰减系数。<br />
<br />
\end{cases}</math><br /><math>f(x) =<br />
<br />
\begin{cases}<br />
<br />
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
<br />
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
<br />
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
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0, & \text{else}<br />
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Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />
<br />
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
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\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
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thumb<br />
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\tau_{new} \leftarrow<br />
<br />
(1-\psi)\tau_{old} + \psi \tau_{0}<br />
<br />
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
<br />
<br />
<br />
''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].<br />
<br />
<br />
<br />
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
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[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
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* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
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<br />
<br />
=== Other applications ===<br />
<br />
其他应用<br />
<br />
* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <br />
{{cite conference<br />
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thumb<br />
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| last1 = Warner | first1 = Lars<br />
<br />
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
<br />
采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
| last2 = Vogel | first2 = Ute<br />
<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
<br />
| date = 2008<br />
<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
<br />
| publisher = Shaker Verlag<br />
<br />
| location = Aachen, Germany<br />
<br />
| isbn = 978-3-8322-7313-2<br />
<br />
| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
<br />
| access-date = 2018-10-09<br />
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}}<br />
<br />
</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<br />
<br />
<br />
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==Definition difficulty==<br />
定义困难那<br />
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[[File:Aco shortpath.svg|thumb]]<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of [[social insects]] remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "[[swarm intelligence]]",<ref name="Waldner 2008 214"/> which is a very general framework in which ant colony algorithms fit.<br />
根据一些作者的观点，ACO算法与其他相关算法（如分布估计算法或粒子群优化算法）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。[[群居昆虫]的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 <br />
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==Stigmergy algorithms==<br />
Stigmergy算法 <br />
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There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
Period = from:1985 till:2005<br />
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==Related methods==<br />
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*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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*Gravitational search algorithm (GSA), a [[swarm intelligence]] method<br />
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*Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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Chronology of ant colony optimization algorithms.<br />
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蚁群算法年表。<br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中 <br />
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==References==<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Lincenthttps://wiki.swarma.org/index.php?title=%E8%9A%81%E7%BE%A4%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95&diff=19512蚁群优化算法2020-11-30T10:13:43Z<p>Lincent：</p>
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<div>此词条暂由Henry翻译。 由Lincent审校。<br />
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[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]<br />
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Ant behavior was the inspiration for the metaheuristic optimization technique<br />
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蚂蚁行为是元启发式优化技术的灵感来源<br />
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[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>]]<br />
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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<br />
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当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些使用较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 225}}</ref>。<br />
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In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] ('''ACO''') is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. '''Artificial Ants''' stand for [[multi-agent]] methods inspired by the behavior of real ants. <br />
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In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. <br />
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在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于概率解决计算问题的技术，它可以简化为通过图来寻找最优路径。人工蚂蚁代表了受真蚂蚁行为启发的多智能体方法。<br />
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The pheromone-based communication of biological [[ant]]s is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref> Combinations of Artificial Ants and [[local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[vehicle routing problem|vehicle routing]] and internet [[routing]]. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as [[AntOptima]].<br />
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The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima.<br />
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基于信息素的蚂蚁通信被用来作为非常典型的范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick |title = Artificial Ants |publisher = Wiley-ISTE |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种图，例如车辆路径和互联网路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业用途。<br />
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As an example, Ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref> is a class of [[optimization (computer science)|optimization]] [[algorithm]]s modeled on the actions of an [[ant colony]]. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[parameter space]] representing all possible solutions. Real ants lay down [[pheromone]]s directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref> One variation on this approach is [[bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[honey bee]], another social insect.<br />
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As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.<br />
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以蚁群算法<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |authorlink = M. Dorigo & L. M. Gambardella |title = Learning Approach to the Traveling Salesman Problem |publisher = IEEE Transactions on Evolutionary Computation, 1 (1) |page=214 |year = 1997}}</ref>为例，它是一种模拟蚁群行为的优化算法。人造「蚂蚁」(例如:模拟仿真代替)通过在代表所有可能解的参数空间移动来求取最优解。真实蚂蚁们在探索环境时，会产生信息素来指引彼此寻找资源。模拟的“蚂蚁”会类似地记录它们的位置和求解结果的好坏，以便在随后的模拟迭代计算中有更多的蚂蚁找到更好的解。<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>蚁群算法的一个变种是蜜蜂算法，它更类似于另一种社会性昆虫————蜜蜂的觅食模式。<br />
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This algorithm is a member of the '''ant colony algorithms''' family, in [[swarm intelligence]] methods, and it constitutes some [[metaheuristic]] optimizations. Initially proposed by [[Marco Dorigo]] in 1992 in his PhD thesis,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of [[ants]] seeking a path between their [[ant colony|colony]] and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[estimation of distribution algorithm]]s.<br />
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This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with estimation of distribution algorithms.<br />
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这个算法是蚁群算法家族的一员，在群体智能方法中，它构成了一些元启发式的优化。最初由 Marco Dorigo 在1992年的博士论文中提出<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref>，第一个蚁群算法的目的是基于蚂蚁在它们的聚居地和食物之间寻找一条路径的行为，在一张图中寻找最佳路径。借鉴了蚂蚁行为的各个方面，最初的蚁群算法已经变得多样以解决更多类型的数值问题，并且也出现了一些问题。从更高的角度来看，蚁群算法实现了基于模型的搜索<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> ，并与分布估计算法具有一定的相似性。<br />
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==Overview==<br />
概览<br />
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In the natural world, ants of some species (initially) wander [[random]]ly, and upon finding food return to their colony while laying down [[pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref><br />
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In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).<br />
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在自然世界中，某些种类的蚂蚁(最初)随机徘徊，一旦发现食物便回到群落，同时留下信息素的轨迹。如果其他蚂蚁找到了这样一条路径，它们可能不会继续随机行走，而是沿着信息素轨迹前进。如果它们最终找到了食物，就会返回并加强这一信息素路径(参见蚂蚁通讯)<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>。<br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref><br />
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Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<br />
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然而，随着时间的推移，信息素的轨迹开始蒸发，从而对蚂蚁的吸引力减小。蚂蚁沿着路径往返需要的时间越长，信息素蒸发所需的时间就越长。相比之下，蚂蚁更频繁地穿越短路径，因此短路径上的信息素密度比长路径上更高。在算法中，信息素蒸发还具有避免收敛到局部最优解的优点。如果根本没有蒸发，那么第一批蚂蚁选择的路径对于后面的蚂蚁来说就会非常有吸引力。在这种情况下，算法对解空间的探索范围将受到限制。信息素蒸发对实际蚂蚁系统的影响尚不清楚，但在人工系统中具有重要意义<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>。<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and [[positive feedback]] eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.<br />
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总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。<br />
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===Ambient networks of intelligent objects===<br />
智能物体的周围网络<br />
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref><br />
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New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<br />
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因为“智能”不再是中心化的而可以在所有微小的物体中找到，所以需要一些新的概念来描述。以人为中心的概念被认为引导了 IT 系统的产生，其中数据处理、控制单元和计算力量是集中的。这些集中的单元功能不断地提高，并可以与人类的大脑相比较。大脑模型已经成为计算机的终极愿景。由智能物体构成的环境的网络，以及基于纳米技术的新一代信息系统，迟早会深刻地改变这种概念。可以与昆虫相比较的小型设备本身并不具备高智能。事实上，他们的智能程度相当有限。例如，不可能把一个能解决任何数学问题的高性能计算器集成到可以植入人体的生物芯片中，或者集成到一个用于跟踪商品的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题下，这种类型的智能可能优于类似大脑的集中系统的推理<ref name="Waldner 2008 214">{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}</ref>。<br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}</ref><br />
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Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of collective intelligence on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior. They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<br />
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大自然中存在很多例子，说明了微小的生物体如果都遵循同样的基本规则，是如何创造出一种宏观层面的集体智慧的。社会性昆虫的群落完美地说明了这个与人类社会有很大不同的模型。该模型基于具有简单和不可预测行为的独立单元之间的合作。他们穿过周围地区执行某些任务，但只掌握了非常有限的信息。例如，一群蚂蚁代表了许多可以应用到环境对象的网络中的特性。蚂蚁群体具有很高的适应环境变化的能力，并且在处理某项个体无法完成的任务时具有很大的能力。这种灵活性对于不断发展的移动网络也是非常有用的。从一台计算机到一个数字化对象的信息包和蚂蚁的行为一样。它们穿过网络并从一个节点传到下一个节点，目的是尽快到达终点。<br />
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===Artificial pheromone system===<br />
人工信息素系统<br />
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as<br />
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基于信息素的通信是自然界中普遍存在的、最有效的通信方式之一。信息素被社会性的昆虫所利用，如<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.<br />
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蜜蜂、蚂蚁和白蚁; 既用于主体之间的通信也用于主体与群体之间的通信。由于其可行性，人工信息素已被应用于多机器人和群机器人系统中。基于信息素的通信是通过化学或物理(RFID标签、光、声)方式实现的。然而，这些方式并不能复制信息素在自然中呈现的所有方面。<br />
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Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, ''COSΦ'',<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref> for a swarm robotic system is based on precise and fast visual localization.<ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref> <br />
<br />
Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization. <br />
<br />
2007年，Garnier，Simon 等人在一篇IEEE论文中提出了使用投射光作为研究基于信息素的微型自主机器人通信的实验装置。另一项针对群机器人系统的研究提出了一种新颖的信息素通信方法 cosφ，该方法基于精确快速的视觉定位。<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}<br />
<br />
The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.<br />
<br />
该系统可以模拟几乎无限数量的不同信息素，并在机器人移动的水平 LCD 屏幕上以灰度图像的形式提供它们相互作用的结果。为了演示信息素通信方法，自主微型机器人Colias被作为群体机器人平台。<br />
<br />
<br />
<br />
==Algorithm and formulae==<br />
算法和公式<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.<br />
<br />
在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定优化问题寻找最优解。为了应用蚁群算法，需要将优化问题转化为在加权图上寻找最短路径的问题。在每次迭代的第一步，每个蚂蚁随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较不同蚂蚁发现的路径。最后一步是更新每个边上的信息素水平。<br />
<br />
'''<font color="#32CD32"><br />
<br />
'''procedure''' ACO_MetaHeuristic '''is'''<br />
<br />
procedure ACO_MetaHeuristic is<br />
<br />
过程 ACO_ 元启发式是<br />
<br />
'''while''' not_termination '''do'''<br />
<br />
while not_termination do<br />
<br />
而不是终止合同<br />
<br />
generateSolutions()<br />
<br />
generateSolutions()<br />
<br />
概要解答()<br />
<br />
daemonActions()<br />
<br />
daemonActions()<br />
<br />
daemonActions ()<br />
<br />
pheromoneUpdate()<br />
<br />
pheromoneUpdate()<br />
<br />
信息素更新()<br />
<br />
'''repeat'''<br />
<br />
repeat<br />
<br />
重复<br />
<br />
end procedure<br />
<br />
end procedure<br />
<br />
结束过程<br />
<br />
</font>'''<br />
<br />
<br />
<br />
===Edge selection===<br />
边的选择<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The ''trail level'' represents a posteriori indication of the desirability of that move.<br />
<br />
Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.<br />
<br />
每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。<br />
<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability<br />
<br />
一般来说，蚂蚁k从状态x移动到状态y的概率<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
p_{xy}^k =<br />
<br />
p_{xy}^k =<br />
<br />
1 = 0 = 0 = 0<br />
<br />
\frac<br />
<br />
\frac<br />
<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }<br />
<br />
{(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })}<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }<br />
<br />
{ sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })}<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where<br />
<br />
where<br />
<br />
其中<br />
<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
<math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (a priori knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.<br />
<br />
τxy是从状态x到y的信息素积累量，0≤α是控制τxy影响的参数，ηxy是状态转换xy的期望值（一种先验信息，通常为1/dxy，其中d是距离），β≥1是控制ηxy影响的参数。τxz和ηxz代表了其他可能的状态转移的轨迹浓度等级和吸引力<br />
<br />
<br />
<br />
===Pheromone update===<br />
<br />
信息素更新<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is<br />
<br />
当所有的蚂蚁都完成了求解过程，路径通常会被更新，通过分别增加或减少路径浓度水平对应的“好”或者“坏”的移动。全局信息素更新规则的一个例子是<br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
\tau_{xy} \leftarrow<br />
<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}<br />
<br />
(1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy }<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by<br />
<br />
where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the pheromone evaporation coefficient and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by<br />
<br />
式中，τxy是状态转换xy的信息素沉积量，ρ是信息素蒸发系数，Δτkxy是第k只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出： <br />
<br />
<br />
<br />
<math><br />
<br />
<math><br />
<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
\Delta \tau^{k}_{xy} =<br />
<br />
3. d<br />
<br />
\begin{cases}<br />
<br />
\begin{cases}<br />
<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\<br />
<br />
<br />
0 & \mbox{otherwise}<br />
<br />
0 & \mbox{otherwise}<br />
<br />
<br />
\end{cases}<br />
<br />
\end{cases}<br />
<br />
<br />
</math><br />
<br />
</math><br />
<br />
<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.<br />
<br />
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。<br />
<br />
<br />
<br />
==Common extensions==<br />
常见延展<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
Here are some of the most popular variations of ACO algorithms.<br />
<br />
下面是一些常见的 ACO 算法变体。<br />
<br />
<br />
<br />
===Ant System (AS)===<br />
蚂蚁系统<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" /><br />
<br />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo. <br />
<br />
蚂蚁系统是第一种蚁群算法。此算法与前述算法相对应。它是由 Dorigo 开发的。<br />
<br />
<br />
<br />
===Ant Colony System (ACS)===<br />
蚁群系统<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref> <br />
<br />
<br />
In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<br />
<br />
在蚁群系统算法中，原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<br />
<br />
<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
这种算法中，在每次迭代之后，全局最优解的蚂蚁与其他所有蚂蚁一起将信息素释放在该路径上(即使这条路径没有被重访问过)。<br />
<br />
<br />
<br />
===Elitist Ant System===<br />
精英蚂蚁系统<br />
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.<br />
<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<br />
<br />
该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τmax,τmin]。所有边都被初始化为τmax，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τmax 。<br />
<br />
<br />
<br />
===Max-Min Ant System (MMAS)===<br />
最大-最小 蚂蚁系统<br />
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref><br />
<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
所有解都是根据长度排序的。只有固定个数的最佳蚂蚁可以在迭代中更新他们的轨迹。信息素沉积量对每个解进行加权，使得路径较短的解比路径较长的解沉积更多的信息素<br />
<br />
<br />
===Rank-based Ant System (ASrank)===<br />
基于排序的蚂蚁系统<br />
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.<br />
<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<br />
<br />
COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用正交化设计法，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<br />
<br />
<br />
<br />
===Continuous Orthogonal Ant Colony (COAC)===<br />
连续正交蚁群<br />
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref><br />
<br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains. The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<br />
<br />
它是蚂蚁系统的一种递归形式，将整个搜索域划分为若干子域，并在这些子域上求解。对所有子域的结果进行比较，并将其中最好的几个子域提升到下一个等级。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<br />
<br />
<br />
<br />
===Recursive Ant Colony Optimization===<br />
递归蚁群优化 <br />
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref><br />
<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".<br />
<br />
对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。在2004年，Zlochin 和他的同事们展示了COAC类型的算法在交叉熵和分布估计算法中可以同化为随机梯度下降方法。他们提出这些元启发式算法作为一个“基于研究的模型”。<br />
<br />
<br />
<br />
==Convergence==<br />
收敛<br />
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".<br />
<br />
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]<br />
<br />
<br />
<br />
Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.<br />
<br />
蚁群算法已经被应用于许多组合优化问题，从二次分配到蛋白质折叠或路径选择问题，以及许多派生方法已经被应用于实变量的动态问题、随机问题、多目标和并行方法。<br />
<br />
==Applications==<br />
应用<br />
It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.<br />
<br />
它也被用来产生旅行商问题的近似最优解。当图形可能发生动态变化时，它们比模拟退火算法和遗传算法具有优势; 蚁群算法可以连续运行并实时适应变化。这是网络路由和城市交通系统中很感兴趣的内容。<br />
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[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]<br />
<br />
Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.<br />
蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现 <br />
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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第一个蚁群算法被称为蚂蚁系统，它的目标是解决旅行商问题，其中的目标是找到最短的往返路线连接一系列城市。一般的算法相对简单，可以基于一组蚂蚁，每个蚂蚁沿着城市进行一次可能的往返旅行。在每个阶段，蚂蚁都会根据一些规则从一个城市迁移到另一个城市:<br />
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.<br />
它还被用于产生[[旅行商问题]]的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的[[模拟退火]]和[[遗传算法]]方法的优势；蚁群算法可以连续运行，实时适应变化。这对[[网络路由]]和城市交通系统很有意义 <br />
It must visit each city exactly once;<br />
<br />
它必须每个城市只到达一次;<br />
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A distant city has less chance of being chosen (the visibility);<br />
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较远的城市被选中的机会较少(可见度) ;<br />
<br />
The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant system: optimization by a colony of cooperating agents]'', IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.</ref> and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:<br />
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The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
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在两个城市之间的边上布置的信息素踪迹越强烈，那条边被选择的概率就越大;<br />
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# It must visit each city exactly once;<br />
它必须访问每座城市一次<br />
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
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旅程结束后，如果旅程很短，蚂蚁会在所经过的所有边上释放更多的信息素;<br />
<br />
# A distant city has less chance of being chosen (the visibility);<br />
远的城市只有很小的机会被选中（可视性）<br />
After each iteration, trails of pheromones evaporate.<br />
<br />
每次迭代后，信息素踪迹会蒸发。<br />
<br />
# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;<br />
在两个城市之间的边上，信息素的踪迹越密集，被选中的可能性就越大 <br />
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;<br />
蚂蚁完成了它的旅程，如果旅程很短的话，它会在经过的所有边缘上沉积更多的信息素 <br />
center<br />
<br />
中心<br />
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# After each iteration, trails of pheromones evaporate.<br />
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每次迭代后，信息素轨迹蒸发<br />
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[[File:Aco TSP.svg|thumb|600px|center]]<br />
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<br />
===Scheduling problem===<br />
规划难题<br />
*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref><br />
顺序排序问题 <br />
*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.<br />
工序车间调度问题<br />
</ref><br />
<br />
*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref><br />
开放车间调度问题<br />
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref><br />
排列流车间问题 <br />
*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref><br />
单机总延迟问题 <br />
*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref><br />
单机总加权延迟调度问题 <br />
*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref><br />
资源受限的项目调度问题<br />
*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref><br />
组车间调度问题<br />
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref><br />
具有序列依赖设置时间的单机总延迟问题 <br />
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref><br />
具有序列依赖设置/转换时间的多级流车间调度问题 <br />
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<br />
===Vehicle routing problem===<br />
车辆路线问题<br />
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref><br />
容量受限车辆路径问题 <br />
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref><br />
多基地车辆路径问题<br />
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref><br />
周期车辆路径问题 <br />
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref><br />
分批配送车辆路径问题<br />
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref><br />
随机车辆路径问题 <br />
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref><br />
带取送的车辆路径问题 <br />
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref><br />
带时间窗的车辆路径问题 <br />
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref><br />
带时间窗的时间相关的车辆路径问题<br />
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)<br />
带时间窗和多个服务工人的车辆路径问题 <br />
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===Assignment problem===<br />
分配问题<br />
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref><br />
二次分配问题<br />
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref><br />
广义分配问题<br />
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref><br />
频率分配问题 <br />
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref><br />
冗余分配问题 <br />
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===Set problem===<br />
集合问题<br />
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref><br />
集合覆盖问题 <br />
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref><br />
划分问题 <br />
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref><br />
权约束的图树划分问题 <br />
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref><br />
弧加权l-基数树问题 <br />
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref><br />
多重背包问题 <br />
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref><br />
最大独立集问题<br />
<br />
<br />
===Device sizing problem in nanoelectronics physical design===<br />
纳米电子学物理设计中的器件尺寸问题 <br />
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref><br />
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解 <br />
Loopback vibrators 10×10, synthesized by means of ACO algorithm<br />
<br />
采用蚁群算法合成回路振子10 × 10<br />
<br />
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref><br />
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率 <br />
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10<br />
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[文件: ANT antenna 2. jpg | thumb | Unloopback 振子10 × 10，由 ACO 算法回环和未回环振子10 × 10合成<br />
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===Antennas optimization and synthesis===<br />
天线优化与综合 <br />
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
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这里的图是二维图像,并且蚂蚁从一个像素的沉积信息素遍历。蚂蚁从一个像素移动到另一个是由图像灰度值的局部变化引导的。这样的移动导致最高密度的信息素沉积在边缘上。<br />
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[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1>Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]<br />
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[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]<br />
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The following are the steps involved in edge detection using ACO:<br />
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以下是使用 ACO 进行边缘检测的步骤:<br />
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To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/><br />
为了优化天线的形状，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线RFID标签 <br />
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Step1: Initialization:<br />Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
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Step1: 初始化: 在图像<math>I_{M_1 M_2}</math> 上随机放置蚂蚁 < math > k </math >，其中 < math > k = (m _ 1 * m _ 2) ^ tfrac {1} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要困难是确定启发式矩阵。<br />
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===Image processing===<br />
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The ACO algorithm is used in image processing for image edge detection and edge linking.<ref>S. Meshoul and M Batouche, "[https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref><br />
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There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
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确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据局部统计数据计算出来的:<br />
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* '''Edge detection:'''<br />
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the local statistics at the pixel position (i,j).<br />
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像素(i，j)的局部统计数据。<br />
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The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.<br />
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<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
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< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math > <br />
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The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|title = Edge detection using ant colony search algorithm and multiscale contrast enhancement|journal = IEEE International Conference on Systems, Man and Cybernetics, 2009. SMC 2009|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina|year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref><br />
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Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
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大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里<br />
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''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.<br />
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<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
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[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，是一个归一化因子<br />
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There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:<br />
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<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
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< math > begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\<br />
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the local statistics at the pixel position (i,j).<br />
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& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
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& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
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& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
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& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
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<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math><br />
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& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
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左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math > <br />
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Where <math>I</math> is the image of size <math>M_1*M_2</math><br /><br />
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<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
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< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) = <br />
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<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor<br />
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\begin{cases}<br />
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开始{ cases }<br />
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\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
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Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}<br />
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<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\<br />
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0, & \text{else}<br />
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0，& text { else }<br />
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& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\<br />
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\end{cases}</math><br /><math>f(x) =<br />
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结束{ cases } </math > < br/> < math > f (x) = <br />
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& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\<br />
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\begin{cases}<br />
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开始{ cases }<br />
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& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math><br />
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
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0, & \text{else}<br />
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0，& text { else }<br />
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<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} </math><br /><math>f(x) =<br />
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\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
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上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]<br />
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\begin{cases}<br />
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\tau_{new} \leftarrow<br />
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\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\<br />
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(1-\psi)\tau_{old} + \psi \tau_{0}<br />
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(1-psi) tau _ { old } + psi tau _ {0}<br />
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0, & \text{else}<br />
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</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
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这里是信息素衰减系数。<br />
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\end{cases}</math><br /><math>f(x) =<br />
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\begin{cases}<br />
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Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.<br />
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步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。<br />
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\<br />
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0, & \text{else}<br />
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Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />
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使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。<br />
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\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math><br />
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\tau_{new} \leftarrow<br />
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(1-\psi)\tau_{old} + \psi \tau_{0}<br />
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</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math><br />
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''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].<br />
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Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref><br />
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[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]<br />
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* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also been proven effective in edge linking algorithms too.<br />
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=== Other applications ===<br />
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其他应用<br />
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* [[Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref><br />
破产预测<br />
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref><br />
分类<br />
* Connection-oriented [[network routing]]<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref><br />
面向连接的网络路由<br />
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf AntNet: a mobile agents approach to adaptive routing]," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf Two ant colony algorithms for best-effort routing in datagram networks]," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref><br />
无连接网络路由 <br />
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf Editorial Survey: Swarm Intelligence for Data Mining]," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf An ant colony algorithm for classification rule discovery]," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref><br />
数据挖掘<br />
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref><br />
项目调度中的折现现金流 <br />
* [[distributed computing|Distributed]] [[information retrieval]]<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref><br />
分布式计算 <br />
* Energy and electricity network design<ref name="warner-and-vogel-2008"><br />
能源与电力网络设计 <br />
{{cite conference<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as probabilistic multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence", In practice, the use of an exchange of information between ants via the environment (a principle called "stigmergy") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<br />
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采用蚁群优化算法，图中两点 a 和 b 之间的最短路径是由多条路径组合建立的。要准确定义算法是不是蚁群算法并不容易，因为其定义可能因作者和用途而有所不同。广义地说，蚁群算法被认为是一种填充的元启发式算法，每个解由一个在搜索空间中移动的蚂蚁表示。蚂蚁标记最优解，并考虑到以前的标记来优化搜索。它们可以被看作是概率化多智能体算法，使用概率分布进行每次迭代之间的转换。在用于解决组合问题的蚁群算法版本中，使用了一种解的迭代构造方法。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布的算法或粒子群优化算法)的是蚁群算法的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，最终可能会找到最好的解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最优解中最强的部分建立起来。然而，在实变量中没有“相邻”这样的结构存在，所以这一定义在实变量问题的情况下可能是有问题的。群居昆虫的集体行为仍然是研究人员的灵感来源。在生物系统中寻找自我组织的各种算法(无论是优化还是非优化)促进了“群体智能”的概念，实际上，利用蚂蚁之间通过环境交换信息的行为(一个被称为“暂时能力”的原则)被认为足以使算法属于蚁群算法的一类。这一原则促使一些作者创造了“价值”这个词来组织基于寻找食物，幼虫分类，分工和合作运输的方法和行为。<br />
<br />
| last2 = Vogel | first2 = Ute<br />
<br />
| title = Optimization of energy supply networks using ant colony optimization<br />
<br />
| date = 2008<br />
<br />
| conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection<br />
<br />
| publisher = Shaker Verlag<br />
<br />
| location = Aachen, Germany<br />
<br />
| isbn = 978-3-8322-7313-2<br />
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| url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf<br />
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| access-date = 2018-10-09<br />
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}}<br />
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</ref><br />
<br />
* Grid workflow scheduling problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref><br />
网格工作流调度问题 <br />
* Inhibitory peptide design for [[protein protein interaction]]s<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref><br />
蛋白质-蛋白质相互作用的抑制肽设计<br />
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method]", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref><br />
智能测试系统 <br />
* Power [[electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design]", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref><br />
电源电子电路设计 <br />
* [[Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach] ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. Of ACM SAC 2013|year=2013|pages=1320–1327 |doi= 10.1145/2480362.2480611|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref><br />
蛋白质折叠<br />
The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.<br />
<br />
发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。<br />
<br />
* System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp. 401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref><br />
系统辨识<br />
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==Definition difficulty==<br />
定义困难那<br />
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.}}</ref> It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as [[people|populated]] [[metaheuristics]] with each solution represented by an ant moving in the search space.<ref>Fred W. Glover,Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&pg=PA276&lpg=PA276&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&source=bl&ots=4kyU_bZLpg&sig=zSrzu89MRED00H8QWjixBMkw11k&hl=fr&sa=X&ved=0ahUKEwjm4_2ysurTAhUHZ1AKHabGAZIQ6AEIODAE#v=onepage&q=aco%20algorithms%20with%20guaranteed%20convergence%20to%20the%20optimal%20solution%20metaheuristics&f=false], Springer (2003)</ref> Ants mark the best solutions and take account of previous markings to optimize their search. They can be seen as [[probabilistic]] [[multi-agent]] algorithms using a [[probability distribution]] to make the transition between each [[iteration]].<ref>http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr , ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists. The collective behaviour of [[social insects]] remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "[[swarm intelligence]]",<ref name="Waldner 2008 214"/> which is a very general framework in which ant colony algorithms fit.<br />
根据一些作者的观点，ACO算法与其他相关算法（如分布估计算法或粒子群优化算法）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。[[群居昆虫]的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 <br />
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==Stigmergy algorithms==<br />
Stigmergy算法 <br />
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There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.<ref>Santpal Singh Dhillon , ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&pg=PA33&dq=Stigmergy+algorithms&hl=fr&sa=X&ved=0ahUKEwjwjfaAtOrTAhWnLsAKHVPkCjYQ6AEIKTAB#v=onepage&q=Stigmergy%20algorithms&f=false], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[stigmergy]]") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. {{ISBN|978-3-540-34689-0}}</ref><br />
在实践中，蚂蚁之间通过环境交换信息（称为“[[stigmergy]]”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 <br />
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==Related methods==<br />
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*[[Genetic algorithm]]s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.<br />
基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。 <br />
* An [[estimation of distribution algorithm]] (EDA) is an [[evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|journal=Parallel Problem Solving from Nature, PPSN XI|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref><br />
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*[[Simulated annealing]] (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.<br />
模拟退火（SA）是一种相关的全局优化技术，它通过生成当前解的相邻解来遍历搜索空间。更好的的相邻解总是被接受的。根据质量和温度参数的差异概率地接受较差的相邻解。随着算法的进行，温度参数会被修改，以改变搜索的性质。<br />
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* Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.<br />
反应式搜索优化是将机器学习与优化相结合，通过添加一个内部反馈回路，根据问题、实例和当前解周围的局部情况，对算法的自由参数进行自调整。 <br />
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*[[Tabu search]] (TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space.<br />
[[塔布研究]]（TS）类似于模拟退火，两者都通过测试单个解的突变来遍历解空间。模拟退火只产生一个变异解，禁忌搜索则产生许多变异解，并移到适应度最低的解。为了防止循环并鼓励在解决方案空间中进行更大的移动，禁忌列表保留了部分或完整解决方案。禁止移动到包含禁忌列表元素的解，该列表会随着解遍历解空间而更新<br />
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*[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.<br />
[[人工免疫系统]]（AIS）算法是以脊椎动物免疫系统为模型的。<br />
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*[[Particle swarm optimization]] (PSO), a [[swarm intelligence]] method<br />
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*[[Intelligent Water Drops|Intelligent water drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers<br />
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*Gravitational search algorithm (GSA), a [[swarm intelligence]] method<br />
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*Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.<br />
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* [[Stochastic diffusion search]] (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions<br />
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==History==<br />
历史<br />
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The inventors are [[Frans Moyson]] and [[Bernard Manderick]]. Pioneers of the field include [[Marco Dorigo]], [[Luca Maria Gambardella]].<ref>{{cite journal|last = Manderick, Moyson |first = Bernard, Frans |authorlink = Manderick, Bernard, and Moyson, Frans |title = The collective behavior of ants: An example of self-organization in massive parallelism. |publisher = Proceedings of the AAAI Spring Symposium on Parallel Models of Intelligence |place = Stanford |year = 1988}}</ref><br />
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from:1989 till:1989 shift:($dx,$dy) text:studies of collective behavior<br />
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Chronology of ant colony optimization algorithms.<br />
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蚁群算法年表。<br />
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Chronology of ant colony optimization algorithms.<br />
蚁群优化算法年表 <br />
* 1959, [[Pierre-Paul Grassé]] invented the theory of [[stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref><br />
1959年，[[Pierre Paul grasse]发明了[[stigmergy]]理论来解释[[白蚁]筑巢的行为 <br />
* 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref><br />
1983年，Deneubourg和他的同事研究了[[蚂蚁]]的[[集体行为]] <br />
* 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref><br />
1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章 <br />
* 1989, the work of Goss, Aron, Deneubourg and Pasteels on the '''collective behavior of Argentine ants''', which will give the idea of ant colony optimization algorithms;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref><br />
1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路 <br />
* 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,''An Ant Foraging Model Implemented on the Time Warp Operating System'', Proceedings of the SCS Multiconference on Distributed Simulation, 1989</ref><br />
1989年，Ebling 和他的同事们实施了一种食物行为模式 <br />
* 1991, M. Dorigo proposed the '''ant system''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni<ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" /><br />
1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统” <br />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to [[telecommunications]] networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref><br />
1994年，英国电信公司的Appleby和Steward发布了第一个应用于[[电信]]网络的应用程序 <br />
* 1995, Gambardella and Dorigo proposed '''ant-q''', <ref> L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995 </ref> the preliminary version of ant colony system as first estension of ant system; <ref name="Ant system" />.<br />
1995年，Gambardella和Dorigo提议“蚂蚁-q” <br />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref> L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20-22, pp. 622-627, 1996; </ref> <br />
1996年，Gambardella和Dorigo提议“蚁群系统”<br />
* 1996, publication of the article on ant system;<ref name="Ant system" /><br />
1996年，发表关于蚂蚁系统的文章 <br />
* 1996, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" /><br />
1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”<br />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" /><br />
1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统 <br />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref><br />
1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到[[电信]]网络 <br />
* 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref><br />
1998年，Dorigo召开了第一次专门讨论ACO算法的会议<br />
* 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref><br />
1998年，Stützle提出最初的“并行实现” <br />
*1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows, <ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><br />
1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统<br />
* 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants<ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref><br />
1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社<br />
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref><br />
2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊 <br />
* 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and the [[constraint satisfaction|satisfaction of constraints]];<br />
2000年，首次应用于[[调度算法|调度]]，调度序列和[[约束满足|满足约束 <br />
* 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref><br />
2000年，Gutjahr为蚁群算法提供了[[序列极限|收敛性]]的第一个证据 <br />
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);<br />
2001年，公司首次使用COA算法<br />
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 Bi-Criterion Optimization with Multi Colony Ant Algorithms]'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref><br />
2001年，Iredi和他的同事发表了第一个“多目标”算法<br />
* 2002, first applications in the design of schedule, Bayesian networks;<br />
2002年，贝叶斯网络在进度计划设计中的首次应用<br />
* 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf An ant colony optimization approach to the probabilistic traveling salesman problem]'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref><br />
2002年，Bianchi和她的同事提出了[[随机]]问题的第一个算法；<br />
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref><br />
2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书 <br />
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy method]] and [[algorithms to estimate distribution]]<ref name="Zlochin model-based search"/><br />
2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、[[交叉熵方法]]和[[估计分布的算法]] <br />
* 2005, first applications to [[protein folding]] problems.<br />
2005年，首次应用于[[蛋白质折叠]]问题 <br />
* 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]].<ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref><br />
2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与[[传输控制协议]]进行了比较<br />
* 2016, first application to peptide sequence design.<ref name=":0" /><br />
2016年，首次应用于肽序列设计。 <br />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref><br />
2017年，多准则决策方法PROMETHEE成功集成到ACO算法中 <br />
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==References==<br />
参考<br />
{{Reflist|30em}}<br />
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==Publications (selected)==<br />
出版物<br />
Category:Articles which contain graphical timelines<br />
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类别: 包含图形时间表的文章<br />
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* [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.<br />
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Category:Nature-inspired metaheuristics<br />
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类别: 自然启发的启发式元推理<br />
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* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.<br />
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Category:Optimization algorithms and methods<br />
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类别: 优化算法和方法<br />
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<small>This page was moved from [[wikipedia:en:Ant colony optimization algorithms]]. Its edit history can be viewed at [[蚁群优化算法/edithistory]]</small></noinclude><br />
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