蚁群优化算法

此词条暂由Henry翻译。 由Lincent审校。

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Ant behavior was the inspiration for the metaheuristic optimization technique

Ant behavior was the inspiration for the metaheuristic optimization technique

蚂蚁行为是元启发式优化技术的灵感来源

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。

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， 蚁群优化算法 ant colony optimization algorithm(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 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。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。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，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 ，并与分布估计算法具有一定的相似性。

Overview

概览

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)。

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。

The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.

The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.

总的结果是，当一只蚂蚁找到一条从蚁群到食物来源的好的(即短的)路径时，其他蚂蚁更有可能沿着这条路径，正反馈最终导致许多蚂蚁走同一条路径。蚁群算法的思想是通过“模拟的蚂蚁”在代表待解决问题的图上移动来模仿蚂蚁的行为。

Ambient networks of intelligent objects

智能物体的周围网络 New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.

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。

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。

Artificial pheromone system

人工信息素系统 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 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; 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.

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被作为群体机器人平台。

Algorithm and formulae

算法和公式 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。

procedure ACO_MetaHeuristic is

procedure ACO_MetaHeuristic is

过程 ACO_ 元启发式是

    while not_termination do

    while not_termination do

而不是终止合同

        generateSolutions()

        generateSolutions()

概要解答()

        daemonActions()

        daemonActions()

daemonActions ()

        pheromoneUpdate()

        pheromoneUpdate()

信息素更新()

    repeat

    repeat

重复

end procedure

end procedure

结束过程

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]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math], corresponding to a more complete intermediate solution. Thus, each ant [math]\displaystyle{ k }[/math] computes a set [math]\displaystyle{ 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]\displaystyle{ k }[/math], the probability [math]\displaystyle{ p_{xy}^k }[/math] of moving from state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math] depends on the combination of two values, the attractiveness [math]\displaystyle{ \eta_{xy} }[/math] of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level [math]\displaystyle{ \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]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math], corresponding to a more complete intermediate solution. Thus, each ant [math]\displaystyle{ k }[/math] computes a set [math]\displaystyle{ 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]\displaystyle{ k }[/math], the probability [math]\displaystyle{ p_{xy}^k }[/math] of moving from state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math] depends on the combination of two values, the attractiveness [math]\displaystyle{ \eta_{xy} }[/math] of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level [math]\displaystyle{ \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]\displaystyle{ x }[/math]移动到状态[math]\displaystyle{ y }[/math]，状态[math]\displaystyle{ y }[/math]对应一个更完整的中间解。因此，每个蚂蚁[math]\displaystyle{ k }[/math] 在每次迭代中计算一组可行展开式[math]\displaystyle{ A_k(x) }[/math]，并以一定概率移动到其中一个。对于蚂蚁[math]\displaystyle{ k }[/math]，从状态[math]\displaystyle{ x }[/math]移动到状态[math]\displaystyle{ y }[/math]的概率[math]\displaystyle{ p_{xy}^k }[/math]取决于两个值的组合，即移动的吸引力[math]\displaystyle{ \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]\displaystyle{ \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。

In general, the [math]\displaystyle{ k }[/math]th ant moves from state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math] with probability

In general, the [math]\displaystyle{ k }[/math]th ant moves from state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math] with probability

一般来说，蚂蚁k从状态x移动到状态y的概率

[math]\displaystyle{ \lt math\gt p_{xy}^k = p_{xy}^k = 1 = 0 = 0 = 0 \frac \frac { (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) } { (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) } {(tau _ { xy } ^ { alpha })(eta _ { xy } ^ { beta })} { \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) } { \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) } { sum _ { z in mathrm { allowed } _ x }(tau _ { xz } ^ { alpha })(eta _ { xz } ^ { beta })} }[/math]

</math>

where

where

其中

[math]\displaystyle{ \tau_{xy} }[/math] is the amount of pheromone deposited for transition from state [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math], 0 ≤ [math]\displaystyle{ \alpha }[/math] is a parameter to control the influence of [math]\displaystyle{ \tau_{xy} }[/math], [math]\displaystyle{ \eta_{xy} }[/math] is the desirability of state transition [math]\displaystyle{ xy }[/math] (a priori knowledge, typically [math]\displaystyle{ 1/d_{xy} }[/math], where [math]\displaystyle{ d }[/math] is the distance) and [math]\displaystyle{ \beta }[/math] ≥ 1 is a parameter to control the influence of [math]\displaystyle{ \eta_{xy} }[/math]. [math]\displaystyle{ \tau_{xz} }[/math] and [math]\displaystyle{ \eta_{xz} }[/math] represent the trail level and attractiveness for the other possible state transitions.

[math]\displaystyle{ \tau_{xy} }[/math] is the amount of pheromone deposited for transition from state [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math], 0 ≤ [math]\displaystyle{ \alpha }[/math] is a parameter to control the influence of [math]\displaystyle{ \tau_{xy} }[/math], [math]\displaystyle{ \eta_{xy} }[/math] is the desirability of state transition [math]\displaystyle{ xy }[/math] (a priori knowledge, typically [math]\displaystyle{ 1/d_{xy} }[/math], where [math]\displaystyle{ d }[/math] is the distance) and [math]\displaystyle{ \beta }[/math] ≥ 1 is a parameter to control the influence of [math]\displaystyle{ \eta_{xy} }[/math]. [math]\displaystyle{ \tau_{xz} }[/math] and [math]\displaystyle{ \eta_{xz} }[/math] represent the trail level and attractiveness for the other possible state transitions.

[math]\displaystyle{ \tau_{xy} }[/math] 是从状态[math]\displaystyle{ x }[/math] 到[math]\displaystyle{ y }[/math] 的信息素积累量，0 ≤ [math]\displaystyle{ \alpha }[/math]是控制[math]\displaystyle{ \tau_{xy} }[/math] 影响的参数，[math]\displaystyle{ \eta_{xy} }[/math] 是状态转换[math]\displaystyle{ xy }[/math] 的期望值（一种先验信息，通常为[math]\displaystyle{ 1/d_{xy} }[/math]，其中d是距离），[math]\displaystyle{ \beta }[/math]≥1是控制[math]\displaystyle{ \eta_{xy} }[/math]影响的参数。[math]\displaystyle{ \tau_{xz} }[/math]和[math]\displaystyle{ \eta_{xz} }[/math]代表了其他可能的状态转移的轨迹浓度等级和吸引力

Pheromone update

信息素更新

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]\displaystyle{ \lt math\gt \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} (1-rho) tau _ { xy } + sum _ { k } Delta tau ^ { k } _ { xy } }[/math]

</math>

where [math]\displaystyle{ \tau_{xy} }[/math] is the amount of pheromone deposited for a state transition [math]\displaystyle{ xy }[/math], [math]\displaystyle{ \rho }[/math] is the pheromone evaporation coefficient and [math]\displaystyle{ \Delta \tau^{k}_{xy} }[/math] is the amount of pheromone deposited by [math]\displaystyle{ k }[/math]th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by

where [math]\displaystyle{ \tau_{xy} }[/math] is the amount of pheromone deposited for a state transition [math]\displaystyle{ xy }[/math], [math]\displaystyle{ \rho }[/math] is the pheromone evaporation coefficient and [math]\displaystyle{ \Delta \tau^{k}_{xy} }[/math] is the amount of pheromone deposited by [math]\displaystyle{ k }[/math]th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by

式中，[math]\displaystyle{ \tau_{xy} }[/math]是状态转换[math]\displaystyle{ xy }[/math]的信息素沉积量，[math]\displaystyle{ \rho }[/math]是信息素蒸发系数，[math]\displaystyle{ \Delta \tau^{k}_{xy} }[/math]是第[math]\displaystyle{ k }[/math]只蚂蚁沉积的信息素量，通常针对TSP问题（移动对应于图的弧）给出：

[math]\displaystyle{ \lt math\gt \Delta \tau^{k}_{xy} = \Delta \tau^{k}_{xy} = 3. d \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} \\ 0 & \mbox{otherwise} 0 & \mbox{otherwise} \end{cases} \end{cases} }[/math]

</math>

where [math]\displaystyle{ L_k }[/math] is the cost of the [math]\displaystyle{ k }[/math]th ant's tour (typically length) and [math]\displaystyle{ Q }[/math] is a constant.

where [math]\displaystyle{ L_k }[/math] is the cost of the [math]\displaystyle{ k }[/math]th ant's tour (typically length) and [math]\displaystyle{ Q }[/math] is a constant.

其 Lk是蚂蚁k移动的代价(通常是长度)，q是一个常数。

Common extensions

常见延展 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。

Ant System (AS)

蚂蚁系统 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。

Ant Colony System (ACS)

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.

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）边的选择偏向于开发。（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.

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)。

Elitist Ant System

精英蚂蚁系统 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.

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_max,τ_min]. All edges are initialized to τ_max to force a higher exploration of solutions. The trails are reinitialized to τ_max when nearing stagnation.

该算法控制每条路线上信息素的最大和最小数量。只有全局最优或迭代最优才允许在其路线中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τ_max,τ_min]。所有边都被初始化为τ_max，以进行最多的求解。当算法接近停滞状态时，路线被重新初始化为τ_max。

Max-Min Ant System (MMAS)

最大-最小 蚂蚁系统 This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ_max,τ_min]. All edges are initialized to τ_max to force a higher exploration of solutions. The trails are reinitialized to τ_max when nearing stagnation.

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

Rank-based Ant System (ASrank)

基于排序的蚂蚁系统 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.

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。

Continuous Orthogonal Ant Colony (COAC)

连续正交蚁群 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.

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。

Recursive Ant Colony Optimization

递归蚁群优化 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.

For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".

对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，然后是基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。对连续蚁群算法相关各参数(边的选择策略、距离测度方法和信息素蒸发率)的性能分析表明，蚁群算法的性能和收敛速度对参数值，特别是信息素蒸发率的参数选择非常敏感。 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”。

Convergence

收敛 For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most 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".

[背包问题: 相对于数量更多但营养更少的糖，蚂蚁更喜欢小滴的蜂蜜]

Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or 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

应用 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。

Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar

Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations. 蚁群优化算法已被应用于许多组合优化问题，从二次分配到蛋白质折叠或路径车辆等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和并行实现 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. 它还被用于产生旅行商问题的近似最优解。在图形动态变化的情况下，蚁群算法具有类似问题的模拟退火和遗传算法方法的优势；蚁群算法可以连续运行，实时适应变化。这对网络路由和城市交通系统很有意义

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);

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 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;

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;

1. It must visit each city exactly once;

它必须访问每座城市一次

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;

1. 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.

每次迭代后，信息素踪迹会蒸发。

1. 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

1. 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 center

中心

1. After each iteration, trails of pheromones evaporate.

每次迭代后，信息素轨迹蒸发

Scheduling problem

规划难题

• Sequential Ordering Problem (SOP)

顺序排序问题

• Job-shop scheduling problem (JSP)

工序车间调度问题

• Open-shop scheduling problem (OSP)

开放车间调度问题

• Permutation flow shop problem (PFSP)

排列流车间问题

• Single machine total tardiness problem (SMTTP)

单机总延迟问题 ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref>

• Single machine total weighted tardiness problem (SMTWTP)

单机总加权延迟调度问题

• Resource-constrained project scheduling problem (RCPSP)

资源受限的项目调度问题

• Group-shop scheduling problem (GSP)

组车间调度问题

• Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)

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

具有序列依赖设置/转换时间的多级流车间调度问题

Vehicle routing problem

车辆路线问题

• Capacitated vehicle routing problem (CVRP)

容量受限车辆路径问题

• Multi-depot vehicle routing problem (MDVRP)

多基地车辆路径问题

• Period vehicle routing problem (PVRP)

周期车辆路径问题

• Split delivery vehicle routing problem (SDVRP)

分批配送车辆路径问题

• Stochastic vehicle routing problem (SVRP)

随机车辆路径问题

• Vehicle routing problem with pick-up and delivery (VRPPD)

带取送的车辆路径问题

• Vehicle routing problem with time windows (VRPTW)

带时间窗的车辆路径问题

• Time dependent vehicle routing problem with time windows (TDVRPTW)

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)

带时间窗和多个服务工人的车辆路径问题

Assignment problem

分配问题

• Quadratic assignment problem (QAP)

二次分配问题

• Generalized assignment problem (GAP)

广义分配问题

• Frequency assignment problem (FAP)

频率分配问题

• Redundancy allocation problem (RAP)

冗余分配问题

Set problem

集合问题

• Set cover problem (SCP)

集合覆盖问题

• Partition problem (SPP)

划分问题

• Weight constrained graph tree partition problem (WCGTPP)

权约束的图树划分问题

• Arc-weighted l-cardinality tree problem (AWlCTP)

弧加权l-基数树问题

• Multiple knapsack problem (MKP)

多重背包问题

• Maximum independent set problem (MIS)

最大独立集问题

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.

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 Loopback vibrators 10×10, synthesized by means of ACO algorithm

采用蚁群算法合成回路振子10 × 10

• Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.

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

[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

Antennas optimization and synthesis

天线优化与综合 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。

Loopback vibrators 10×10, synthesized by means of ACO algorithm

Unloopback vibrators 10×10, synthesized by means of ACO algorithm

The following are the steps involved in edge detection using ACO:

以下是使用 ACO 进行边缘检测的步骤:

To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO), loopback and unloopback vibrators 10×10

为了优化天线的结构，可以使用蚁群算法。例如，可以考虑基于蚁群算法（ACO）的天线RFID标签, loopback and unloopback vibrators 10×10

Step1: Initialization:
Randomly place [math]\displaystyle{ K }[/math] ants on the image [math]\displaystyle{ I_{M_1 M_2} }[/math] where [math]\displaystyle{ K= (M_1*M_2)^\tfrac{1}{2} }[/math] . Pheromone matrix [math]\displaystyle{ \tau_{(i,j)} }[/math] are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.

Step1: 初始化: 在图像[math]\displaystyle{ 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

The ACO algorithm is used in image processing for image edge detection and edge linking.

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:

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.

[math]\displaystyle{ \eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)} }[/math]

< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math >

The following are the steps involved in edge detection using ACO:

Where [math]\displaystyle{ I }[/math] is the image of size [math]\displaystyle{ M_1*M_2 }[/math]

大小[math]\displaystyle{ M_1*M_2 }[/math]的图像[math]\displaystyle{ I }[/math] 在哪里

Step1: Initialization:
Randomly place [math]\displaystyle{ K }[/math] ants on the image [math]\displaystyle{ I_{M_1 M_2} }[/math] where [math]\displaystyle{ K= (M_1*M_2)^\tfrac{1}{2} }[/math] . Pheromone matrix [math]\displaystyle{ \tau_{(i,j)} }[/math] are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.

[math]\displaystyle{ 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 > ，是一个归一化因子

There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:

[math]\displaystyle{ \begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\ \lt math \gt begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\ the local statistics at the pixel position (i,j). & +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\ & +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\ & +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\ & +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\ \lt math\gt \eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)} }[/math]

& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math>

左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math >

Where [math]\displaystyle{ I }[/math] is the image of size [math]\displaystyle{ M_1*M_2 }[/math]

[math]\displaystyle{ f(\cdot) }[/math] can be calculated using the following functions:
[math]\displaystyle{ f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} }[/math]
[math]\displaystyle{ f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} }[/math]
[math]\displaystyle{ f(x) = \lt math \gt f (cdot) }[/math] 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) =

[math]\displaystyle{ Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j}) }[/math], which is a normalization factor

\begin{cases}

开始{ cases }

\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]\displaystyle{ \begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\ 0, & \text{else} 0，& text { else } & +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\ \end{cases} }[/math]
[math]\displaystyle{ f(x) = 结束{ cases } }[/math] < br/> < math > f (x) =

& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\

\begin{cases}

开始{ cases }

& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math>

\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)} \\

0, & \text{else}

0，& text { else }

[math]\displaystyle{ f(\cdot) }[/math] can be calculated using the following functions:
[math]\displaystyle{ f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} }[/math]
[math]\displaystyle{ f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} }[/math]
[math]\displaystyle{ f(x) = \end{cases} }[/math]
The parameter [math]\displaystyle{ \lambda }[/math] in each of above functions adjusts the functions’ respective shapes.
Step 2 Construction process:
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]\displaystyle{ P_{x,y} }[/math]
Step 3 and Step 5 Update process:
The pheromone matrix is updated twice. in step 3 the trail of the ant (given by [math]\displaystyle{ \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.
[math]\displaystyle{ 上面每个函数中的参数 \lt math \gt lambda }[/math] 用来调整函数各自的形状。< br/> 步骤2构建过程:
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}

\tau_{new} \leftarrow

\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\

(1-\psi)\tau_{old} + \psi \tau_{0}

(1-psi) tau _ { old } + psi tau _ {0}

0, & \text{else}

</math>, where [math]\displaystyle{ \psi }[/math] is the pheromone decay coefficient [math]\displaystyle{ 0\lt \tau \lt 1 }[/math]

这里是信息素衰减系数。

\end{cases}</math>
[math]\displaystyle{ f(x) = \begin{cases} Step 7 Decision Process:\lt br /\gt 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 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)} \\ 0, & \text{else} Image Edge detected using ACO:\lt br /\gt The images below are generated using different functions given by the equation (1) to (4). 使用 ACO 检测图像边缘: \lt br/\gt 下面的图像是使用方程(1)至(4)给出的不同函数生成的。 \end{cases} }[/math]
The parameter [math]\displaystyle{ \lambda }[/math] in each of above functions adjusts the functions’ respective shapes.
Step 2 Construction process:
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]\displaystyle{ P_{x,y} }[/math]
Step 3 and Step 5 Update process:
The pheromone matrix is updated twice. in step 3 the trail of the ant (given by [math]\displaystyle{ \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.
[math]\displaystyle{ thumb 拇指 \tau_{new} \leftarrow (1-\psi)\tau_{old} + \psi \tau_{0} }[/math], where [math]\displaystyle{ \psi }[/math] is the pheromone decay coefficient [math]\displaystyle{ 0\lt \tau \lt 1 }[/math]

Step 7 Decision Process:
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.

Image Edge detected using ACO:
The images below are generated using different functions given by the equation (1) to (4).

• Edge linking: ACO has also been proven effective in edge linking algorithms too.

Other applications

其他应用

• Bankruptcy prediction

破产预测

• Classification

分类

• Connection-oriented network routing

面向连接的网络路由

• Connectionless network routing

无连接网络路由

• Data mining

数据挖掘

• Discounted cash flows in project scheduling

项目调度中的折现现金流

• Distributed information retrieval

分布式计算

• Energy and electricity network design

• Grid workflow scheduling problem

网格工作流调度问题

• Inhibitory peptide design for protein protein interactions

蛋白质-蛋白质相互作用的抑制肽设计

• Intelligent testing system

智能测试系统

• Power electronic circuit design

电源电子电路设计

• Protein folding

蛋白质折叠

The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.

发明者是 Frans Moyson 和 Bernard Manderick。这一领域的先驱包括 Marco Dorigo, Luca Maria Gambardell。

• System identification

系统辨识

{{image frame|content=

{图片框架 | 内容 =

Definition difficulty

定义困难那 <timeline>

< 时间表 >

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210 height: 300

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", which is a very general framework in which ant colony algorithms fit. 根据一些作者的观点，ACO算法与其他相关算法（如分布估计算法或粒子群优化算法）的区别正是它们的建设性方面。在组合问题中，即使没有蚂蚁被证明是有效的，也有可能最终找到最佳解。因此，在旅行商问题的例子中，蚂蚁实际上并不需要走最短的路线：最短的路线可以从最佳解的最强部分建立起来。然而，在实际变量中存在问题的情况下，这个定义可能会有问题，因为不存在“邻居”的结构。[[群居昆虫]的集体行为仍然是研究人员灵感的源泉。生物系统中寻求自组织的各种各样的算法（无论是优化还是非优化）导致了“群体智能”的概念 PlotArea = width:170 height:280 left:40 bottom:10

170 height: 280 left: 40 bottom: 10

Stigmergy algorithms

Stigmergy算法 DateFormat = yyyy

日期/格式 = yyyy

There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies. 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. 在实践中，蚂蚁之间通过环境交换信息（称为“stigmergy”的原理）被认为足以使算法属于蚁群算法。基于这种“合作性”的分类原则，作者提出了一些“合作式”的分类和运输方式。 Period = from:1985 till:2005

时间 = 从1985年到2005年

TimeAxis = orientation:vertical

时间轴 = 方向: 垂直

Related methods

ScaleMajor = unit:year increment:5 start:1985

单元: year increment: 5 start: 1985

• Genetic algorithms (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.

基因算法（GA）维护一个解决方案池，而不仅仅是一个。寻找更好的解决方案的过程模仿了进化的过程，解决方案被组合或变异以改变解决方案池，劣质的解决方案被丢弃。

• An estimation of distribution algorithm (EDA) is an evolutionary algorithm that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled or generated from guided-crossover.

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颜色 =

• 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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Id: fond value: white # rgb (0.95,0.95,0.98)

• 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。

  id:marque   value:rgb(1,0,0)

Id: marque value: rgb (1,0,0)

• 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，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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Id: marque _ fond value: rgb (1,0.9,0.9)

• Artificial immune system (AIS) algorithms are modeled on vertebrate immune systems.

人工免疫系统（AIS）算法是以脊椎动物免疫系统为模型的。 BackgroundColors = canvas:fond

背景颜色 = 画布: 喜欢

• Particle swarm optimization (PSO), a swarm intelligence method

• Intelligent water drops (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers

Define $dx = 7 # décalage du texte à droite de la barre

Define $dx = 7 # décalage du texte à droite de la barre

• Gravitational search algorithm (GSA), a swarm intelligence method

Define $dy = -3 # décalage vertical

Define $dy = -3 # décalage vertical

• Ant colony clustering method (ACCM), a method that make use of clustering approach, extending the ACO.

Define $dy2 = 6 # décalage vertical pour double texte

Define $dy2 = 6 # décalage vertical pour double texte

• Stochastic diffusion search (SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions

PlotData=

绘图数据 =

History

历史

 bar:Leaders color:marque_fond width:5 mark:(line,marque) align:left fontsize:S

条: 领袖颜色: marque _ fond 宽度: 5马克: (线，马克)对齐: 左方块大小: s

The inventors are Frans Moyson and Bernard Manderick. Pioneers of the field include Marco Dorigo, Luca Maria Gambardella.

 from:1989  till:1989 shift:($dx,$dy)    text:studies of collective behavior

从: 1989年到: 1989年转移: ($dx，$dy)文本: 集体行为的研究

}}

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Chronology of ant colony optimization algorithms.

蚁群算法年表。

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Define $dx = 7 # décalage du texte à droite de la barre

Define $dy = -3 # décalage vertical

Define $dy2 = 6 # décalage vertical pour double texte

PlotData=

 bar:Leaders color:marque_fond width:5 mark:(line,marque) align:left fontsize:S

 from:1989  till:1989 shift:($dx,$dy)    text:studies of collective behavior

 from:1991  till:1992 shift:($dx,$dy)    text:ant system (AS)

 from:1995  till:1995 shift:($dx,$dy)    text:continuous problem (CACO)

 from:1996  till:1996 shift:($dx,$dy)    text:ant colony system (ACS)

 from:1996  till:1996 shift:($dx,$dy2)   text:max-min ant system (MMAS)

 from:2000  till:2000 shift:($dx,$dy)   text:proof to convergence (GBAS)

 from:2001  till:2001 shift:($dx,$dy)   text:multi-objective algorithm

</timeline>|caption=Chronology of COA algorithms

}}

Chronology of ant colony optimization algorithms. 蚁群优化算法年表

• 1959, Pierre-Paul Grassé invented the theory of stigmergy to explain the behavior of nest building in termites;

1959年，[[Pierre Paul grasse]发明了stigmergy理论来解释[[白蚁]筑巢的行为

• 1983, Deneubourg and his colleagues studied the collective behavior of ants;

1983年，Deneubourg和他的同事研究了蚂蚁的集体行为

• 1988, and Moyson Manderick have an article on self-organization among ants;

1988年，Moyson Manderick发表了一篇关于蚂蚁“自组织”的文章

• 1989, the work of Goss, Aron, Deneubourg and Pasteels on the collective behavior of Argentine ants, which will give the idea of ant colony optimization algorithms;

1989年，Goss，Aron，Deneubourg和pastels关于“阿根廷蚂蚁的集体行为”的研究，这将为蚁群优化算法提供思路

• 1989, implementation of a model of behavior for food by Ebling and his colleagues;

1989年，Ebling 和他的同事们实施了一种食物行为模式

• 1991, M. Dorigo proposed the ant system in his doctoral thesis (which was published in 1992). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni was published five years later;

1991年，M.Dorigo在他的博士论文（发表于1992年）中提出了“蚂蚁系统”

• 1994, Appleby and Steward of British Telecommunications Plc published the first application to telecommunications networks

1994年，英国电信公司的Appleby和Steward发布了第一个应用于电信网络的应用程序

• 1995, Gambardella and Dorigo proposed ant-q, the preliminary version of ant colony system as first estension of ant system; .

1995年，Gambardella和Dorigo提议“蚂蚁-q”

• 1996, Gambardella and Dorigo proposed ant colony system

1996年，Gambardella和Dorigo提议“蚁群系统”

• 1996, publication of the article on ant system;

1996年，发表关于蚂蚁系统的文章

• 1996, Hoos and Stützle invent the max-min ant system;

1996年，Hoos和Stützle发明了“最大最小蚂蚁系统”

• 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;

1997年，Dorigo和Gambardella提出了结合局部搜索的蚁群系统

• 1997, Schoonderwoerd and his colleagues published an improved application to telecommunication networks;

1997年，Schoonderwoerd和他的同事发表了一个改进的应用程序到电信网络

• 1998, Dorigo launches first conference dedicated to the ACO algorithms;

1998年，Dorigo召开了第一次专门讨论ACO算法的会议

• 1998, Stützle proposes initial parallel implementations;

1998年，Stützle提出最初的“并行实现”

• 1999, Gambardella, Taillard and Agazzi proposed macs-vrptw, first multi ant colony system applied to vehicle routing problems with time windows,

1999年，Gambardella、Taillard和Agazzi提出了macs-vrptw，这是第一个应用于带时间窗的车辆路径问题的多蚁群系统

• 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants

1999年，博纳博、多里戈和塞拉兹出版了一本主要讨论人工蚂蚁的书。博纳博，M.Dorigo et G.Theraulaz，“群体智能”，牛津大学出版社

• 2000, special issue of the Future Generation Computer Systems journal on ant algorithms

2000年，《未来一代计算机系统杂志》关于蚂蚁算法的特刊

• 2000, first applications to the scheduling, scheduling sequence and the satisfaction of constraints;

2000年，首次应用于调度，调度序列和[[约束满足|满足约束

• 2000, Gutjahr provides the first evidence of convergence for an algorithm of ant colonies

2000年，Gutjahr为蚁群算法提供了收敛性的第一个证据

• 2001, the first use of COA algorithms by companies (Eurobios and AntOptima);

2001年，公司首次使用COA算法

• 2001, Iredi and his colleagues published the first multi-objective algorithm

2001年，Iredi和他的同事发表了第一个“多目标”算法

• 2002, first applications in the design of schedule, Bayesian networks;

2002年，贝叶斯网络在进度计划设计中的首次应用

• 2002, Bianchi and her colleagues suggested the first algorithm for stochastic problem;

2002年，Bianchi和她的同事提出了随机问题的第一个算法；

• 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press

2004年，Dorigo和Stützle与麻省理工学院出版社出版了《蚁群优化》一书

• 2004, Zlochin and Dorigo show that some algorithms are equivalent to the stochastic gradient descent, the cross-entropy method and algorithms to estimate distribution

2004年，Zlochin和Dorigo证明了一些算法等价于[[随机梯度下降]、交叉熵方法和估计分布的算法

• 2005, first applications to protein folding problems.

2005年，首次应用于蛋白质折叠问题

• 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the Transmission Control Protocol.

2012年，Prabhakar和他的同事们发表了一项研究，内容涉及单个蚂蚁在没有信息素的情况下进行串联通信，这反映了计算机网络组织的原理。将通信模型与传输控制协议进行了比较

• 2016, first application to peptide sequence design.

2016年，首次应用于肽序列设计。

• 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm (HUMANT algorithm).

2017年，多准则决策方法PROMETHEE成功集成到ACO算法中

References

参考

Publications (selected)

出版物 Category:Articles which contain graphical timelines

类别: 包含图形时间表的文章

• M. Dorigo, 1992. Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italy.

Category:Nature-inspired metaheuristics

类别: 自然启发的启发式元推理

• M. Dorigo, V. Maniezzo & A. Colorni, 1996. "Ant System: Optimization by a Colony of Cooperating Agents", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.

Category:Optimization algorithms and methods

类别: 优化算法和方法

This page was moved from wikipedia:en:Ant colony optimization algorithms. Its edit history can be viewed at 蚁群优化算法/edithistory