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第12行： − 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>。+ 第20行：
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第26行： − 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。+ 第34行：
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第42行： − 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。+ 第54行：
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第62行： − 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>。+ 第70行：
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第76行： + − 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>。+ 第89行：
第90行： − 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>+ 第129行：
第130行： − 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。+ 第149行：
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第190行： − 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+ 第220行：
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第251行： − + − 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. − 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)。 第263行：
第262行： − 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.+ + + + − 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> 第271行：
第272行： − + − 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.+ + + + − 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 第283行：
第286行： − 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.+ + + + − 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> 第293行：
第298行： − 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.+ − 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>+ 第303行：
第308行： − 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".+ + + + − 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”。 第312行：
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第338行： − ==Applications==+ − 应用+ 第331行：
第346行： − + − 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 + − + + − 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. − 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 − 它必须每个城市只到达一次;+ 第347行：
第361行： − 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: − 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;+ 第367行：
第380行： − 规划难题+ 第382行：
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第471行： − 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>+ 第479行：
第492行： − − [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 第486行：
第497行： − 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. − − 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。 第494行：
第502行： − The following are the steps involved in edge detection using ACO: − − 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> 第503行：
第508行： − 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. − − 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。 第516行：
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无编辑摘要
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.
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.
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>。
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.
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.
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。
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。
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]].
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]].
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.
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.
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。
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.
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.
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。
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。
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.
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.
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。
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).
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).
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>。
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>。
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.
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.
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>。
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.
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.
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。
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。
===Ambient networks of intelligent objects===
===Ambient networks of intelligent objects===
智能物体的周围网络
智能物体的周围网络
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>
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>
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.
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.
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>。
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.
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.
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>
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.
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.
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。
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.
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.
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>
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>
<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.
<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.
<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>
<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>
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
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
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
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.
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.
其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。
其 Lk是蚂蚁k移动的代价(通常是长度)，Q是一个常数。
==Common extensions==
==Common extensions==
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.
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.
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>
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>
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.
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.
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.
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)。
===Max-Min Ant System (MMAS)===
===Max-Min Ant System (MMAS)===
最大-最小 蚂蚁系统
最大-最小蚂蚁系统
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>
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>
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.
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>
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.
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.
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.
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。
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>
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>
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.
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>
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>
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>
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.
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>
收敛
收敛
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]]".
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]]".
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".
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”。
==Applications==
应用
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]
[[Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]
[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]
[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]
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.
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.
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.
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。
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。
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.
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.
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.
[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]
[[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]
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.
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:
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:
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:
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:
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:
:1.It must visit each city exactly once;
:1.It must visit each city exactly once;
每个城市必须只到达一次;
:2.A distant city has less chance of being chosen (the visibility);
:2.A distant city has less chance of being chosen (the visibility);
较远的城市被选中的机会较少(可见度) ;
较远的城市被选中的机会较少(可见度) ;
: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;
: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;
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;
:4.Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
:4.Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
===Scheduling problem===
===Scheduling problem===
规划问题
*[[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>
*[[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>
*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>
*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>
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>
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>
*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>
*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>
*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>
*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>
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>
*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>
*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>
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10
===Antennas optimization and synthesis===
===Antennas optimization and synthesis===
天线优化与综合
天线优化与综合
[[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>]]
[[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>]]
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]
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/>
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/>
===Image processing===
===Image processing===
* '''Edge detection:'''
* '''Edge detection:'''
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.
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。
The following are the steps involved in edge detection using ACO:
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>
the local statistics at the pixel position (i,j).
the local statistics at the pixel position (i,j).
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.
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.
''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.
''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.
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.
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。
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics: