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蚁群优化算法 (查看源代码)

2020年11月3日 (二) 09:55的版本

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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.
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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。
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每个蚂蚁都需要构造一个解决方案来遍历图。为了在巡游中选择下一条边，蚂蚁将考虑从其当前位置获得的每条边的长度，以及相应的信息素水平。在算法的每一步，每一个蚂蚁都从状态x移动到状态y，对应于一个更完整的中间解。因此，每个蚂蚁k在每次迭代中计算一组可行展开式的Ak（x），并以概率移动到其中一个。对于蚂蚁k，从状态x移动到状态y的概率pkxy取决于两个值的组合，即移动的吸引力ηxy，这是由某种启发式计算得出的，表示该移动的先验期望值和该移动的跟踪水平τxy，这表明它在过去进行该特定移动的熟练程度。审判级别代表了这一行动的可取性的后验指标。
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<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.
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是从状态 < math > x </math > 到 < math > y </math > 的信息素沉积量，0≤ < math > alpha </math > 是一个控制 < math > tau { xy > </math > 影响的参数,状态转换的可取性是指状态转换对数学的可取性(先验知识，通常是数学知识，其中数学知识是距离)和数学知识是控制数学知识影响的参数。< math > tau _ { xz } </math > 和 < math > eta _ { xz } </math > 表示其他可能的状态转换的轨迹级别和吸引力。
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τxy是从状态x到y的信息素沉积量，0≤α是控制τxy影响的参数，ηxy是状态转换xy的期望值（先验知识，通常为1/dxy，其中d是距离），β≥1是控制ηxy影响的参数。τxz和ηxz代表了其他可能的状态跃迁的轨迹能级和吸引力
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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
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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)
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式中，τxy是状态转换xy的信息素沉积量，ρ是信息素蒸发系数，Δτkxy是kth ant沉积的信息素量，通常针对TSP问题（移动量对应于图形的弧）给出：
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where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.
 
where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.
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其中 l _ k </math > 是蚂蚁旅行的费用(通常是长度) ，q </math > 是一个常数。
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其 Lk是蚂蚁k旅行的费用(通常是长度) ，q 是一个常数。
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This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions.  The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.
 
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions.  The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.
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该算法控制每条路线上信息素的最大和最小数量。只有全局最优旅行或迭代最优旅行才允许在其轨迹中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[ τ < sub > max </sub > ，τ < sub > min </sub > ]。所有边都被初始化为 τ < sub > > max </sub > ，以迫使更高的解决方案探索。当接近停滞状态时，轨迹重新初始化为 τ < sub > max </sub > 。
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该算法控制每条路线上信息素的最大和最小数量。只有全局最优旅行或迭代最优旅行才允许在其轨迹中添加信息素。为了避免搜索算法的停滞不前，每条路径上可能的信息素数量范围被限制在一个区间[τmax,τmin].。所有边都被初始化为τmax，以迫使更高的解决方案探索。当接近停滞状态时，轨迹重新初始化为 τmax 。
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For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".
 
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate. In 2004, Zlochin and his colleagues showed that COAC-type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristics as a "research-based model".
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。通过对连续蚁群算法各参数(边缘选择策略、距离测度和信息素蒸发率)的性能分析，表明蚁群算法的性能和收敛速度对参数的选择，特别是信息素蒸发率的选择非常敏感。在2004年，Zlochin 和他的同事们展示了 coac 类型的算法可以同化随机梯度下降的方法，比如交叉熵和分布估计算法。他们提出这些元启发式作为一个“基于研究的模型”。
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对于某些版本的算法，可以证明它是收敛的(也就是说，它能在有限时间内找到全局最优解)。蚁群算法的收敛性在2000年首次得到证实，基于图的蚂蚁系统算法，以及后来的 ACS 和 MMAS 算法。和大多数启发式算法一样，很难估计理论上的收敛速度。通过对连续蚁群算法各参数(边缘选择策略、距离测度和信息素蒸发率)的性能分析，表明蚁群算法的性能和收敛速度对参数的选择，特别是信息素蒸发率的选择非常敏感。在2004年，Zlochin 和他的同事们展示了COAC类型的算法可以同化随机梯度下降的方法，比如交叉熵和分布估计算法。他们提出这些元启发式作为一个“基于研究的模型”。
          
==Convergence==
 
==Convergence==
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收敛
 
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".
 
For some versions of the algorithm, it is possible to prove that it is convergent (i.e., it is able to find the global optimum in finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> showed that COAC-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".
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==Applications==
 
==Applications==
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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.
 
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.
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Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.
 
Ant colony optimization algorithms have been applied to many [[combinatorial optimization]] problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and [[parallel computing|parallel]] implementations.
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蚁群优化算法已被应用于许多[[组合优化]]问题，从二次分配到[[蛋白质]]折叠或[[车辆路径问题|路径车辆]]等，许多衍生的方法已适用于实际变量中的动态问题，随机问题，多目标和[[并行计算|并行]]实现
 
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:
 
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities. The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:
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It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.
 
It has also been used to produce near-optimal solutions to the [[travelling salesman problem]]. They have an advantage over [[simulated annealing]] and [[genetic algorithm]] approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in [[network routing]] and urban transportation systems.
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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;
 
  It must visit each city exactly once;
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# It must visit each city exactly once;
 
# It must visit each city exactly once;
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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;
 
  Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
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# A distant city has less chance of being chosen (the visibility);
 
# A distant city has less chance of being chosen (the visibility);
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远的城市有更小的机会被选中（可视性）
 
  After each iteration, trails of pheromones evaporate.
 
  After each iteration, trails of pheromones evaporate.
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# 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;
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在两个城市之间的边缘上，信息素的踪迹越密集，被选中的可能性就越大
 
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
 
# Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
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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
 
center
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# After each iteration, trails of pheromones evaporate.
 
# After each iteration, trails of pheromones evaporate.
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每次迭代后，信息素轨迹蒸发
    
[[File:Aco TSP.svg|thumb|600px|center]]
 
[[File:Aco TSP.svg|thumb|600px|center]]
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===Scheduling problem===
 
===Scheduling problem===
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规划难题
 
*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref>
 
*[[Sequential Ordering Problem]] (SOP) <ref> L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref>
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顺序排序问题
 
*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.
 
*[[Job-shop scheduling]] problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.
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作业车间调度问题
 
</ref>
 
</ref>
    
*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref>
 
*[[Open-shop scheduling]] problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref>
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开放车间调度问题
 
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref>
 
*Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref>
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置换流水车间问题
 
*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref>
 
*Single machine total tardiness problem (SMTTP)<ref>A. Bauer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref>
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单机总拖期问题
 
*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref>
 
*Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of [[Lecture Notes in Computer Science]], pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf An ant algorithm with a new pheromone evaluation rule for total tardiness problems]," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref>
−
 
+
单机总加权拖期问题
 
*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref>
 
*Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref>
−
 
+
资源受限的项目调度问题
 
*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref>
 
*Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref>
−
 
+
成组车间调度问题
 
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref>
 
*Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times]," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref>
−
 
+
具有顺序相关设置时间的单机总拖期问题
 
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref>
 
*Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, {{ISBN|978-3-540-72959-4}}, pp.111-138, 2008.</ref>
−
 
+
具有序列相关设置/转换时间的多级flowshop调度问题
       
===Vehicle routing problem===
 
===Vehicle routing problem===
−
 
+
车辆路线问题
 
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref>
 
*Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele}}</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref>
−
 
+
容量受限车辆路径问题
 
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref>
 
*Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref>
−
 
+
多基地车辆路径问题
 
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref>
 
*Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|author2-link= M. Grazia Speranza}}</ref>
−
 
+
周期车辆路径问题
 
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref>
 
*Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref>
−
 
+
分批配送车辆路径问题
 
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref>
 
*Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref>
−
 
+
随机车辆路径问题
 
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref>
 
*Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf Solving the pickup and delivery problem with time windows using reactive tabu search]," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows]," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref>
−
 
+
带取车和送货的车辆路径问题
 
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref>
 
*Vehicle routing problem with time windows (VRPTW)<ref> L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref>
−
 
+
带时间窗的车辆路径问题
 
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref>
 
*Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf Time Dependent Vehicle Routing Problem with a Multi Ant Colony System]", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref>
−
 
+
具有时间窗的时变车辆路径问题
 
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)
 
*Vehicle routing problem with time windows and multiple service workers (VRPTWMS)
−
 
+
具有时间窗和多个服务工人的车辆路径问题
       
===Assignment problem===
 
===Assignment problem===
−
 
+
分配难题
 
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref>
 
*[[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref>
−
 
+
二次分配问题
 
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref>
 
*[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf Adaptive search heuristics for the generalized assignment problem]," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref>
−
 
+
广义指派问题
 
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref>
 
*[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, [[S. P. M. van Hoesel]], A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref>
−
 
+
频率分配问题
 
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref>
 
*[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref>
−
 
+
冗余分配问题
       
===Set problem===
 
===Set problem===
−
 
+
设置问题
 
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2,  pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref>
 
*[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf A new version of ant system for subset problems]," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2,  pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref>
−
 
+
设置覆盖问题
 
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref>
 
*[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref>
−
 
+
分区问题
 
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref>
 
*Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref>
−
 
+
权约束图树划分问题
 
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref>
 
*Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref>
−
 
+
弧加权l-基数树问题
 
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref>
 
*Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref>
−
 
+
多重背包问题
 
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref>
 
*Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 An ant system for the maximum independent set problem]," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref>
−
 
+
最大独立集问题
       
===Device sizing problem in nanoelectronics physical design===
 
===Device sizing problem in nanoelectronics physical design===
−
 
+
米电子学物理设计中的器件尺寸问题
 
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref>
 
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref>
−
 
+
基于蚁群优化（ACO）的45 nm CMOS感放电路优化可以在非常短的时间内收敛到最优解
 
Loopback vibrators 10×10, synthesized by means of ACO algorithm
 
Loopback vibrators 10×10, synthesized by means of ACO algorithm
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* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref>
 
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref>
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基于蚁群优化（ACO）的可逆电路综合可以显著提高效率
 
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10
 
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm loopback and unloopback vibrators 10×10
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===Antennas optimization and synthesis===
 
===Antennas optimization and synthesis===
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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.
 
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.
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To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/>
 
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/>
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为了优化天线的形状，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
     
Henry
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