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第1行：
第1行： − + 第29行：
第29行： − 当一群蚂蚁面临选择通过两条不同的路径获取食物时，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，那些使用较短的路线到达食物更快，因此往返于蚁丘和食物之间的频率更高。+ 第37行：
第37行： − + 第43行：
第43行： − 基于信息素的生物蚂蚁通信往往是主要的范例使用。人工蚁群算法和局部搜索算法的结合已经成为许多优化任务的一种选择方法，这些优化任务涉及到一些图形，例如车辆路径和互联网路由。这一领域蓬勃发展的活动导致了专门讨论人工蚂蚁的会议，以及诸如 AntOptima 等专业公司的大量商业应用。+ 第51行：
第51行： − 以蚁群算法为例，它是一种提出了一类以蚁群行为为模型的优化算法。人造「蚂蚁」(例如:模拟代理)通过移动的参数空间代表所有可能的解决方案定位最优解。真正的蚂蚁在探索自己的环境时，会产生信息素，互相指引对方寻找资源。模拟的“蚂蚁”类似地记录它们的位置和解决方案的质量，这样在随后的模拟迭代中，更多的蚂蚁找到更好的解决方案。这种方法的一个变种是蜜蜂算法，它更类似于另一种社会昆虫---- 蜜蜂的觅食模式。+ 第59行：
第59行： − 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优化。最初由 Marco Dorigo 在1992年的博士论文中提出，第一个算法的目的是在图中寻找最佳路径，基于蚂蚁的行为寻找一条在它们的群体和食物来源之间的路径。最初的想法已经多样化，以解决更广泛的数值问题，结果，出现了几个问题，借鉴了蚂蚁行为的各个方面。从更广泛的角度来看，蚁群算法实现了基于模型的搜索，并与分布估计算法具有一定的相似性。+ 第71行：
第71行： − 在自然世界中，一些物种的蚂蚁(最初)随机徘徊，一发现食物便回到它们的群落，同时产生信息素的踪迹。如果其他蚂蚁找到了这样一条路径，它们很可能不会继续随机行走，而是沿着这条路径前进，如果它们最终找到了食物，就会返回并加固这条路径(参见蚂蚁通讯)。+ 第79行：
第79行： − 然而，随着时间的推移，信息素的踪迹开始蒸发，从而减少了它的吸引力。蚂蚁沿着路径往返需要的时间越长，信息素蒸发的时间就越长。相比之下，短路径更频繁地穿越，因此短路径上的信息素密度比长路径上的高。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，对解空间的探索将受到限制。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。+ 第87行：
第87行： − + 第97行：
第97行： − 我们需要新的概念，因为“智能”不再是集中的，而是可以在所有微小的物体中找到。以人为中心的概念已经被认为导致了 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。这些集中的单位不断地提高他们的表现，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，以及基于纳米技术的新一代信息系统，迟早会深刻地改变这一概念。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，他们的智力可以归类为相当有限。例如，不可能将一个有能力解决任何数学问题的高性能计算器集成到植入人体的生物芯片中，或者集成到一个用于跟踪商业文章的智能标签中。然而，一旦这些物体互相连接起来，它们就拥有了一种可以与一群蚂蚁或蜜蜂相提并论的智慧。在某些问题的情况下，这种类型的智能可能优于类似大脑的集中式系统的推理。+ + + + + 第105行：
第109行： − 大自然提供了几个例子，说明如果微小的生物体都遵循同样的基本规律，它们是如何在宏观层面上创造出一种集体智慧的。社会性昆虫的群落完美地说明了这个模型，它与人类社会有很大的不同。该模型基于独立单元之间的合作，具有简单和不可预测的行为。他们穿过周围地区执行某些任务，但只掌握非常有限的信息。例如，一群蚂蚁代表了许多特性，这些特性也可以应用到环境对象的网络中。蚂蚁群体具有很高的适应环境变化的能力，并且在处理个体无法完成某项任务的情况时具有巨大的力量。这种灵活性对于不断发展的移动对象网络也是非常有用的。从计算机到数字物体的信息包和蚂蚁的行为一样。它们穿过网络，从一个节点传到下一个节点，目的是尽快到达最终目的地。+ 第115行：
第119行： − 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。信息素用于社会昆虫，如+ 第121行：
第125行： − + 第129行：
第133行： − 2007年，由 Garnier，Simon 等人在 IEEE 的一篇论文中提出了使用投射光作为研究信息素与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。+ 第135行：
第139行： − + 第145行：
第149行： − 在蚁群算法中，人工蚂蚁是一种简单的计算代理，可以为给定的最佳化问题寻找好的解决方案。为了应用蚁群算法，需要将最佳化问题问题转化为在加权图上寻找最短路径的问题。在每个迭代的第一步，每个随机构造一个解，即图中的边应遵循的顺序。在第二步中，比较了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153行：
第157行： − + 第190行：
第194行： + + − 边际选择+ − 每个蚂蚁都需要构造一个解决方案来遍历图。为了在巡游中选择下一条边，蚂蚁将考虑从其当前位置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x移动到状态y，对应于一个更完整的中间解。因此，每个蚂蚁k在每次迭代中计算一组可行展开式的Ak（x），并以概率移动到其中一个。对于蚂蚁k，从状态x移动到状态y的概率pkxy取决于两个值的组合，即移动的吸引力ηxy，这是由某种启发式计算得出的，表示该移动的先验期望值和该移动的跟踪水平τxy，这表明它在过去进行该特定移动的熟练程度。审判级别代表了这一行动的可取性的后验指标。+ 第213行：
第219行： − − 《数学》 第226行：
第230行： − 压裂 第243行：
第246行： − − 数学 第252行：
第253行： − 在哪里+ − 第260行：
第260行： − τxy是从状态x到y的信息素沉积量，0≤α是控制τxy影响的参数，ηxy是状态转换xy的期望值（先验知识，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跃迁的轨迹能级和吸引力+ 第272行：
第272行： − 当所有的蚂蚁都完成了他们的解决方案，路径通常更新，增加或减少路径的水平对应的移动，分别是“好”或“坏”的解决方案的一部分。全局信息素更新规则的一个例子是+ 第280行：
第280行： − 《数学》 第286行：
第285行： − 左眼 第297行：
第295行： − − 数学 第306行：
第302行： − 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kth ant沉积的信息素量，通常针对TSP问题（移动量对应于图形的弧）给出： + 第314行：
第310行： − 《数学》 第326行：
第321行： − 开始{ cases } 第332行：
第326行： − Q/L _ k & mbox { if ant } k mbox { use curve } xy mbox { in its tour } 第338行：
第331行： − 0 & mbox { otherwise } 第344行：
第336行： − 结束{ cases } − − 数学 第358行：
第347行： − + 第368行：
第357行： − 下面是一些最流行的 ACO 算法变体。+ 第378行：
第367行： − 蚁群算法是第一种蚁群算法。此算法与上述算法相对应。它是由 Dorigo 开发的。+ 第385行：
第374行： + + + + + + − 在这种算法中，全局最优解在每次迭代之后(即使这条路径没有被重新访问)与其他所有蚂蚁一起将信息素存放在它的路径上。+ 第398行：
第393行： − 该算法控制每条路线上信息素的最大和最小数量。只有全局最优旅行或迭代最优旅行才允许在其轨迹中添加信息素。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，以迫使更高的解决方案探索。当接近停滞状态时，轨迹重新初始化为 τmax 。+ 第408行：
第403行： − 所有的解决方案都是根据长度排序的。在这个迭代中，只有固定数量的最佳蚂蚁可以更新他们的试验。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417行：
第412行： − COAC 的信息素存储机制是使蚂蚁能够协同有效地寻找解决方案。利用正交设计方法，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。+ 第427行：
第422行： − 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目标。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级别。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。+ 第437行：
第432行： − + 第453行：
第448行： − 从二次赋值到蛋白质折叠或路径选择问题，蚁群算法已经被应用于许多组合优化问题，许多派生方法已经被应用于实变量、随机问题、多目标和并行实现的动态问题。+ 第459行：
第454行： − 它也被用来产生接近最佳的解决方案的旅行推销员问题。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。这是在网络路由和城市交通系统的兴趣。+ 第467行：
第462行： − 第一个蚁群算法被称为蚂蚁系统，它的目标是解决旅行推销员问题，其中的目标是找到最短的往返连接一系列城市。一般算法相对简单，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473行：
第468行： − 它必须每个城市只参观一次;+ 第479行：
第474行： − 一个遥远的城市被选中的机会较少(能见度) ;+ 第485行：
第480行： − 在两个城市之间的边缘上布置的信息素踪迹越强烈，那条边缘被选择的概率就越大;+ 第491行：
第486行： − 旅程结束后，如果旅程很短，蚂蚁会在所经过的边缘释放更多的信息素;+ − 远的城市有更小的机会被选中（可视性）+ − 每次迭代后，信息素的踪迹蒸发。+ − 在两个城市之间的边缘上，信息素的踪迹越密集，被选中的可能性就越大 + 第520行：
第515行： − 作业车间调度问题+ 第526行：
第521行： − 置换流水车间问题 + − 单机总拖期问题 + − 单机总加权拖期问题 + − 成组车间调度问题+ − 具有顺序相关设置时间的单机总拖期问题 + − 具有序列相关设置/转换时间的多级flowshop调度问题 + 第554行：
第549行： − 带取车和送货的车辆路径问题 + − 具有时间窗的时变车辆路径问题+ − 具有时间窗和多个服务工人的车辆路径问题 + − 分配难题+ − 广义指派问题+ 第576行：
第571行： − 设置问题+ − 设置覆盖问题 + − 分区问题 + − 权约束图树划分问题 + 第592行：
第587行： − 米电子学物理设计中的器件尺寸问题 + 第600行：
第595行： − 基于蚁群优化（ACO）的可逆电路综合可以显著提高效率 + + + + + + + + 第611行：
第613行： − 这里的图形是二维图像和蚂蚁遍历从一个像素沉积信息素。蚂蚁从一个像素移动到另一个像素是由图像灰度值的局部变化引导的。这一运动导致最高密度的信息素沉积在边缘。+ 第627行：
第629行： − + 第635行：
第637行： − 确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据当地统计数据计算出来的:+ 第641行：
第643行： − 像素位置(i，j)的局部统计。+ 第657行：
第659行： − 大小如下的图像在哪里+ 第663行：
第665行： − + 第736行：
第738行： + + 第745行：
第749行： − + 第751行：
第755行： − 左塔罗 第763行：
第766行： − 这里是信息素衰变系数+ 第769行：
第772行： − + − + 第818行：
第821行： − 面向联系的网络路由+ 第824行：
第827行： − 项目进度计划中的折现现金流 + 第839行：
第842行： − 采用蚁群优化算法，由多条路径组合构造图中两点 a 和 b 之间的最短路径。要准确定义什么算法是蚁群算法，什么算法不是蚁群算法并不容易，因为蚁群算法的定义可能因作者和用途的不同而有所不同。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。蚂蚁标记最好的解决方案，并考虑到以前的标记，以优化他们的搜索。它们可以被看作是概率多智能体算法，使用概率分布进行每次迭代之间的转换。在他们解决组合问题的版本中，他们使用了迭代构造的解决方案。根据一些作者的观点，蚁群算法区别于其他相关算法(比如估计分布或粒子群优化的算法)的正是蚁群算法的建设性方面。在组合问题中，最终可能找到最好的解决方案，即使没有蚂蚁被证明是有效的。因此，在旅行推销员问题的例子中，蚂蚁实际上并不需要走最短的路线: 最短的路线可以从最好的解决方案中最强的部分建立起来。然而，这一定义在实变量问题的情况下可能是有问题的，在实变量中没有“邻居”的结构存在。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算法种类繁多(无论是优化还是非优化)已经导致了“群体智能”的概念，在实践中，蚂蚁之间通过环境交换信息的使用(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。这个原则促使一些作者创造了“价值”这个词来组织方法和行为，基于寻找食物，分类幼虫，分工和合作运输。+ 第878行：
第881行： − 系统标识+ 第938行：
第941行： − 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。优越的邻居总是被接受的。根据质量和温度参数的差异概率地接受次邻居。随着算法的进展，温度参数会被修改，以改变搜索的性质 + 第944行：
第947行： − 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、实例和当前解决方案周围的局部情况，对算法的自由参数进行自调整。 + 第950行：
第953行： − +
无编辑摘要
此词条暂由Henry翻译。
此词条暂由Henry翻译。 由Lincent审校。
{{multiple issues|
{{multiple issues|
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>。
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)是一种用于解决计算问题的概率技术，它可以简化为通过图表来寻找好的路径。人工蚂蚁是受真蚂蚁行为启发的多智能体方法。
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。
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。
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。
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.
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。
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>。
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，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>。
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，其他蚂蚁更有可能遵循这条路径，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“模拟蚂蚁”在代表要解决问题的图上走来模拟这种行为。
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。
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>。
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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.
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。
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
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
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
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.
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.
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.
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.
蜜蜂、蚂蚁和白蚁; 都用于内部代理和代理群通信。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，这些实现并不能复制信息素的所有方面。
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。
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.
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.
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。
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}}
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}}
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.
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.
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可调动为群体机器人平台。
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。
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。
'''<font color="#32CD32">
'''procedure''' ACO_MetaHeuristic '''is'''
'''procedure''' ACO_MetaHeuristic '''is'''
procedure ACO_MetaHeuristic is
procedure ACO_MetaHeuristic is
过程 ACO _ 元启发式是
过程 ACO_ 元启发式是
'''while''' not_termination '''do'''
'''while''' not_termination '''do'''
结束过程
结束过程
</font>'''
===Edge selection===
===Edge selection===
边的选择
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.
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，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。
<math>
<math>
p_{xy}^k =
p_{xy}^k =
\frac
\frac
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }
</math>
</math>
where
where
其中
<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.
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
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
<math>
<math>
\tau_{xy} \leftarrow
\tau_{xy} \leftarrow
\tau_{xy} \leftarrow
\tau_{xy} \leftarrow
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}
(1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}
</math>
</math>
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
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
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：
<math>
<math>
\Delta \tau^{k}_{xy} =
\Delta \tau^{k}_{xy} =
\begin{cases}
\begin{cases}
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\
0 & \mbox{otherwise}
0 & \mbox{otherwise}
0 & \mbox{otherwise}
0 & \mbox{otherwise}
\end{cases}
\end{cases}
\end{cases}
\end{cases}
</math>
</math>
</math>
</math>
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旅行的费用(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。
Here are some of the most popular variations of ACO algorithms.
Here are some of the most popular variations of ACO algorithms.
下面是一些常见的 ACO 算法变体。
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.
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。
蚁群系统
蚁群系统
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>
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>
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，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。
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)。
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.
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[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 。
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.
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。
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.
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。
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".
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，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。通过对连续蚁群算法各参数(边缘选择策略、距离测度和信息素蒸发率)的性能分析，表明蚁群算法的性能和收敛速度对参数的选择，特别是信息素蒸发率的选择非常敏感。在2004年，Zlochin 和他的同事们展示了COAC类型的算法可以同化随机梯度下降的方法，比如交叉熵和分布估计算法。他们提出这些元启发式作为一个“基于研究的模型”。
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”。
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。
==Applications==
==Applications==
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.
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。
[[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]]
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，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:
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 must visit each city exactly once;
It must visit each city exactly once;
它必须每个城市只到达一次;
A distant city has less chance of being chosen (the visibility);
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) ;
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<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 more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;
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布置的信息素踪迹越强烈，那条边被选择的概率就越大;
# It must visit each city exactly once;
# It must visit each city exactly once;
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
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;
# A distant city has less chance of being chosen (the visibility);
# A distant city has less chance of being chosen (the visibility);
远的城市只有很小的机会被选中（可视性）
After each iteration, trails of pheromones evaporate.
After each iteration, trails of pheromones evaporate.
每次迭代后，信息素踪迹会蒸发。
# The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;
# 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，信息素的踪迹越密集，被选中的可能性就越大
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
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
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
顺序排序问题
顺序排序问题
*[[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.
*[[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.
工序车间调度问题
</ref>
</ref>
开放车间调度问题
开放车间调度问题
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref>
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref>
排列流车间问题
*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
*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>
单机总加权延迟调度问题
*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>
*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>
资源受限的项目调度问题
资源受限的项目调度问题
*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>
*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>
组车间调度问题
*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>
*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>
具有序列依赖设置时间的单机总延迟问题
*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>
*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>
具有序列依赖设置/转换时间的多级流车间调度问题
随机车辆路径问题
随机车辆路径问题
*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>
*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>
带取送的车辆路径问题
*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>
*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>
带时间窗的车辆路径问题
带时间窗的车辆路径问题
*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>
*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>
带时间窗的时间相关的车辆路径问题
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)
带时间窗和多个服务工人的车辆路径问题
===Assignment problem===
===Assignment problem===
分配问题
*[[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>
*[[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>
二次分配问题
二次分配问题
*[[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>
*[[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>
广义分配问题
*[[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>
*[[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>
频率分配问题
频率分配问题
===Set problem===
===Set problem===
集合问题
*[[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>
*[[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>
集合覆盖问题
*[[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>
*[[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>
划分问题
*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>
*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>
权约束的图树划分问题
*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l-基数树问题
弧加权l-基数树问题
===Device sizing problem in nanoelectronics physical design===
===Device sizing problem in nanoelectronics physical design===
纳米电子学物理设计中的器件尺寸问题
* Ant colony optimization (ACO) based optimization of 45 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>
* Ant colony optimization (ACO) based optimization of 45 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>
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
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
* 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>
* 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>
基于蚁群优化（ACO）的可逆电路合成可以显著提高效率
[[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
[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
[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
#### ###########################################################################11 28， 21：16
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.
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。
[[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>]]
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: 随机地点 < math > k </math > 图像上的蚂蚁 < math > i _ { m _ 1 m _ 2} </math > where < 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挑战是确定启发式矩阵。
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。
===Image processing===
===Image processing===
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:
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:
* '''Edge detection:'''
* '''Edge detection:'''
the local statistics at the pixel position (i,j).
the local statistics at the pixel position (i,j).
像素(i，j)的局部统计数据。
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.
Where <math>I</math> is the image of size <math>M_1*M_2</math><br />
Where <math>I</math> is the image of size <math>M_1*M_2</math><br />
大小<math>M_1*M_2</math>的图像<math>I</math> 在哪里
''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.
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor
<math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor
[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，这是一个归一化因子
[ 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
##### ################################################################################################################### 11 29
0, & \text{else}
0, & \text{else}
\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>
\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>
上面每个函数中的参数 < math > lambda </math > 调整函数各自的形状。< br/> 步骤2构建过程: < br/> 蚂蚁的移动是基于4个连接的像素或8个连接的像素。根据概率方程 < math > p _ { x，y } </math > < br/> 步骤3和步骤5更新过程给出蚂蚁移动的概率: < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁的踪迹(由 < math > tau _ {(x，y)} </math > 给出)被更新，在步骤5中，蚂蚁的踪迹蒸发率被更新，这个更新是由下面的方程给出的。[数学]
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]
\begin{cases}
\begin{cases}
\tau_{new} \leftarrow
\tau_{new} \leftarrow
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>
这里是信息素衰减系数。
\end{cases}</math><br /><math>f(x) =
\end{cases}</math><br /><math>f(x) =
\begin{cases}
\begin{cases}
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.
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.
步骤7决策过程: 一旦 k 蚂蚁在 n 次迭代中移动了一个固定的距离 l，判断它是否是一个边缘是基于信息素矩阵 τ 上的阈值 t。下面例子的阈值是根据 Otsu 的方法计算的。
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。
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\
分类
分类
* 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>
* 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>
面向连接的网络路由
* 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>
* 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>
无连接网络路由
无连接网络路由
数据挖掘
数据挖掘
* 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>
* 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>
项目调度中的折现现金流
* [[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>
* [[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>
分布式计算
分布式计算
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.
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.
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。
| last2 = Vogel | first2 = Ute
| last2 = Vogel | first2 = Ute
* 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>
* 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>
系统辨识
*[[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.
*[[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.
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。
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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.
* 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.
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。
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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.
*[[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.
[[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，保留了部分或完整解决方案的禁忌列表。禁止移动到包含禁忌列表元素的解决方案，该列表会随着解决方案遍历解决方案空间而更新
[[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
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