求解代价矩阵不精确的旅行商问题的改进粒子群算法

Indadul Khan, Sova Pal, M. Maiti
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引用次数: 9

摘要

本文提出了一种改进的粒子群优化算法(PSO),利用交换顺序、交换操作和不同的速度更新规则来解决成本矩阵清晰或模糊的旅行商问题。在提出的方法中,TSP的潜在解决方案由一个节点的顺序表示,其中销售人员访问所有节点,并命名为路径。交换任意路径上任意两个节点的位置称为交换操作(SO)。按特定顺序应用在路径上的一组SOs被定义为交换序列(SS)。利用这种可能解的表示结构,利用SO和SS对PSO的不同摄动规则进行修正,以找出清晰和模糊环境中tsp的解。在搜索过程中,为了改进任意静态路径,采用了强摄动技术(K-Opt)。提出了五种不同的规则来确定算法中解的更新速度。遵循轮盘赌轮盘选择过程,每次从规则集中随机选择一条规则,用于更新算法中的一个解。在算法的最后再次使用K-opt操作,以可能改进到目前为止找到的最佳路径。该算法能够求解任何具有清晰或模糊代价矩阵的TSP问题。利用TSPLIB中不同规模的基准测试问题来测试算法的性能。实验结果表明,对于具有相当大的脆代价矩阵的tsp,该算法的成功率接近100%。由于文献中没有关于带有模糊代价矩阵的TSP的测试问题,因此本文将TSPLIB的一些较为清晰的测试问题随机生成一些此类问题,并用于测试算法解决此类问题的能力。数值实验表明,该算法在求解对称型和非对称型TSP问题的精度和运行时间上都优于已有的TSP算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A modified particle swarm optimization algorithm for solving traveling salesman problem with imprecise cost matrix
Here, using swap sequence, swap operation and different velocity update rules a modified form of Particle Swarm Optimization(PSO) technique is outlined to solve Traveling Salesman Problem (TSP) having crisp or fuzzy cost matrix. In the propose approach, a potential solution of a TSP is represented by an order of nodes in which a salesman visits all the nodes and named as path. Interchange of the positions of any two nodes of any path is named as swap operation (SO). A set of SOs applied on a path in a particular order is defined as swap sequence(SS). Using this structure of representation of a probable solution, different perturbed rules of PSO are modified using SO and SS to find out the solutions of TSPs in crisp and fuzzy environments. During search process, to improve any static path a strong perturbation technique (K-Opt) is applied. Five different rules are proposed to determine update velocities of a solution in the algorithm. Following roulette wheel selection process, one rule from the rule set is randomly selected each time and is used for updating a solution in the algorithm. K-opt operation is again used at the end of the algorithm for the possible improvement of the best path found so far. The algorithm is capable of solving any TSP with crisp or fuzzy cost matrix. Different sizes benchmark test problems available in TSPLIB are used to test the capability of the algorithm. It is found that success rate of the algorithm is nearly 100% for TSPs with considerably large sizes crisp cost matrix. Due to the absence of test problems on TSP with fuzzy cost matrix in the literature, here, few such problems are randomly generated from some crisp test problems of TSPLIB and are applied to test the capability of the algorithm for solving such problems. It is found form numerical experiments that, efficiency of the said algorithm for solving such problems (Symmetric TSP and Asymmetric TSP) in respect of accuracy and run time is better than some other well established algorithms for TSPs.
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