A modified particle swarm optimization algorithm for solving traveling salesman problem with imprecise cost matrix

Abstract

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.