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Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, the attractiveness <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.
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.
每只蚂蚁都需要构造一个解来移动图形。为了在它的旅行中选择下一条边，蚂蚁会考虑每条边从它当前位置可以获得的长度，以及相应的信息素水平。在算法的每个步骤中，每个蚂蚁从一个状态 x移动到状态y，对应于一个更完整的中间解。因此，每个蚂蚁在每次迭代中计算一组可行展开式到它的当前状态，并移动到其中一个概率上。对于蚂蚁k来说，从状态x到数学状态y的概率取决于两个值的组合---- 吸引力，数状态到数学状态的概率。试验水平代表了一个后验指示的愿望，这一举动。
每只蚂蚁都需要构造一个解来移动图形。为了在它的旅行中选择下一条边，蚂蚁会考虑每条边从它当前位置可以获得的长度，以及相应的信息素水平。在算法的每个步骤中，每个蚂蚁从一个状态 x移动到状态y，对应于一个更完整的中间解。因此，每个蚂蚁在每次迭代中计算一组可行展开式到它的当前状态，并移动到其中一个概率上。对于蚂蚁k来说，从状态x到数学状态y的概率取决于两个值的组合---- 吸引力。该启发式表示该移动的先验可取性和移动，表明它在过去是如何熟练地作出这一特定的移动。审判级别代表了这一行动的可取性的后验指标。
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability
In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability
一般来说，蚂蚁k从状态x移动到状态y的概率
===Pheromone update===
===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
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
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
其中，tau { xy } </math > 是一个状态转换的信息素沉积量 < math > xy </math > ，< math > rho </math > 是信息素蒸发系数，< math > Delta tau ^ { xy } </math > 是数学 < k </math > 蚂蚁沉积的信息素沉积量，通常用于 TSP 问题(与图中弧线相对应的移动)
其中，tau { xy }是一个状态转换的信息素沉积量 < math > xy </math > ，< math > rho </math > 是信息素蒸发系数，< math > Delta tau ^ { xy } </math > 是 < k </math > 蚂蚁沉积的信息素沉积量，通常用于 TSP 问题(与图中弧线相对应的移动)
==Common extensions==
==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.
===Ant System (AS)===
===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.<ref name="Ant system" />
The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" />
===Ant Colony System (ACS)===
===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.<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>
===Elitist Ant System===
===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.
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.
===Max-Min Ant System (MMAS)===
===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<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref>
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions. The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref>
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
===Rank-based Ant System (ASrank)===
===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.
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.
===Continuous Orthogonal Ant Colony (COAC)===
===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.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref>
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy. The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref>
===Recursive Ant Colony Optimization===
===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.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref>
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics , vol. 11, no. 3, pp.325-339</ref>