A new metric function and harmonic crossover for symmetric and asymmetric traveling salesman problems

For the successful application of a genetic algorithm (GA) to the traveling salesman problem (TSP), a suitable distance between two Hamiltonian circuits on a complete graph is useful to estimate the problem landscape. This paper presents a new distance between two Hamiltonian circuits, or phenotypes. The phenotypic distance is defined by the least Hamming distance between isomorphic genotypes. Therefore, it is convenient to analyze and control the behavior of genotypes in the search space. In this paper, a new crossover technique based on the phenotypic distance is also proposed. The crossover technique works together the conventional crossovers arranged for the TSP such as partially mapped (PMX), order (OX) and cycle (CX) crossovers. Because a new child is sure to be located between two parents in the problem space, the local search performance of the conventional crossovers is enhanced with the proposed crossover technique.