Particle Swarm Optimization Algorithm for the Traveling Salesman Problem

Particle swarm optimization, PSO, is an evolutionary computation technique inspired in the behavior of bird flocks. PSO algorithms were first introduced by Kennedy & Eberhart (1995) for optimizing continuous nonlinear functions. The fundamentals of this metaheuristic approach rely on researches where the movements of social creatures were simulated by computers (Reeves, 1983; Reynolds, 1987; Heppner & Grenander, 1990). The research in PSO algorithms has significantly grown in the last few years and a number of successful applications concerning single and multi-objective optimization have been presented (Kennedy& Eberhart, 2001; Coello et al., 2004). This popularity is partially due to the fact that in the canonical PSO algorithm only a small number of parameters have to be tuned and also due to the easiness of implementation of the algorithms based on this technique. Motivated by the success of PSO algorithms with continuous problems, researchers that deal with discrete optimization problems have investigated ways to adapt the original proposal to the discrete case. In many of those researches, the new approaches are illustrated with the Traveling Salesman Problem, TSP, once it has been an important test ground for most algorithmic ideas. Given a graph G = (N,E), where N = {1,...,n} and E = {1,...,m}, and costs, cij, associated with each edge linking vertices i and j, the TSP consists in finding the minimal total length Hamiltonian cycle of G. The length is calculated by the summation of the costs of the edges in the considered cycle. If for all pairs of nodes {i,j}, the costs cij and cji are equal then the problem is said to be symmetric, otherwise it is said to be asymmetric. The main importance of TSP regarding applicability is due to its variations, nevertheless some applications of the basic problem in real world problems are reported for different areas such as VLSI chip fabrication, X-ray crystallography, genome map and broadcast schedule, among others. Although, a great research effort has been done to accomplish the task of adapting PSO to discrete problems, many approaches still obtain results very far from the best results known for the TSP. Some of those works are summarized in section 2. An effective PSO approach for the TSP is presented by Goldbarg et al. (2006a), where distinct types of velocity operators are considered, each of them concerning one movement the particles are allowed to do. This proposal is presented and extended in this chapter, where search strategies for Combinatorial Optimization problems are associated with the velocity operators. Rather than a metaheuristic technique, the PSO approach in this context O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m