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When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.

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>。

当一群蚂蚁面临着可以在两条不同的路径中选择获取食物，而其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些通过较短路线的蚂蚁到达食物更快，因此往返于蚁丘和食物之间的频率更高<ref>{{cite book |last = Waldner  |first = Jean-Baptiste  |authorlink = Jean-Baptiste Waldner  |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] [[John Wiley & Sons]] |place = London |year = 2008 |isbn = 978-1-84704-002-2  | page = 225}}</ref>。

In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants.  

In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants.  

在计算机科学和运筹学中，<font color="#ff8000"> 蚁群优化算法 ant colony optimization algorithm</font>(ACO)是一种基于<font color="#ff8000"> 概率</font>解决计算问题的技术，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚂蚁代表了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<font color="#ff8000"> 概率</font>解决计算问题的技术，它可以简化为通过<font color="#ff8000">图</font>来寻找最优路径。人工蚂蚁群代表了受真实蚂蚁行为启发的多主体方法。

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<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业用途。

基于信息素的蚂蚁通信常被作为典型范例<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick  |title = Artificial Ants |publisher = Wiley-ISTE  |year = 2010 |isbn = 978-1-84821-194-0}}</ref>。人工蚁群和局部搜索算法的组合已经是许多优化任务的一种求解方法，这些优化任务往往涉及某种<font color="#ff8000">图</font>，例如<font color="#ff8000">车辆路径</font>和<font color="#ff8000">互联网</font>路由的问题。这一领域的蓬勃发展催生了专门讨论人工蚂蚁的学术会议，以及诸如 AntOptima 等专业公司的大量商业应用。

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。

每个蚂蚁都需要构造一个解来遍历图。为了在遍历过程中选择下一条边，蚂蚁将考虑从其当前位置可以获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态<math>x</math>移动到状态<math>y</math>，状态<math>y</math>对应一个更完整的中间解。因此，每个蚂蚁<math>k</math> 在每次迭代中计算一组可行展开式<math>A_k(x)</math>，并以一定概率移动到其中一个。对于蚂蚁<math>k</math>，从状态<math>x</math>移动到状态<math>y</math>的概率<math>p_{xy}^k</math>取决于两个值的组合，<font color="#ff8000">即移动的吸引力<math>\eta_{xy}</math>，这是由某种启发式计算得出的，表示移动的先验期望值和本次移动的信息素踪迹浓度等级<math>\tau_{xy}</math>，这表明它在过去进行该特定移动的熟练程度。踪迹浓度等级代表了本次移动期望的后验指标。</font>

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

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

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

<math>\tau_{xy}</math> 是从状态<math>x</math> 到<math>y</math> 的信息素积累量，0 ≤ <math>\alpha</math>是控制<math>\tau_{xy}</math> 影响的参数，<math>\eta_{xy}</math> 是状态转换<math>xy</math> 的期望值（一种先验信息，通常为<math>1/d_{xy}</math>，其中d是距离），<math>\beta</math>≥1是控制<math>\eta_{xy}</math>影响的参数。<font color="#ff8000"> <math>\tau_{xz}</math>和<math>\eta_{xz}</math>代表了其他可能的状态转移的轨迹浓度等级和吸引力</font>

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'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref>  

在蚁群系统算法中，对原来的蚁群算法在三个方面进行了改进：（一）边的选择偏向于'''<font color="#32CD32">开发</font>'''。（即偏向具有大量信息素的最短边的概率）；（二）在求解过程中，蚂蚁通过应用局部信息素更新规则来改变所选边的信息素水平；（iii）在每次迭代结束时，只允许最优的蚂蚁通过改进的全局信息素更新规则来更新路径。<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem]'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref>  

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。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref>

COAC的信息素沉积机制使蚂蚁能够协同有效地寻找解决方案。利用<font color="#ff8000">正交化设计法</font>，可行域中的蚂蚁可以快速有效地搜索所选区域，提高了全局搜索能力和准确性。正交设计法和自适应半径调整法也可以推广到其他优化算法中，在解决实际问题时具有更广泛的优势。<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref>

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<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，并将其中最好的几个子域提升到下一个等级。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref>

它是蚂蚁系统的一种<font color="#ff8000">递归</font>形式，将整个搜索域划分为若干子域，并在这些子域上求解。<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref>对所有子域的结果进行比较，<font color="#ff8000">并将其中最好的几个子域提升到下一个等级</font>。与选定结果相对应的子区域被进一步细分，并重复这个过程，直到获得所需精度的输出。该方法在不适定地球物理反演问题中得到了验证，效果良好。<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref>

  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;

在两个城市之间的边上留下的信息素踪迹越强烈，这条边被选择的概率就越大;

# It must visit each city exactly once;

# It must visit each city exactly once;