Genetic Algorithm with Adapted Crossover Operators for Multiple Traveling Salesmen Problem with Visiting Constraints

Multiple traveling salesmen problem with visiting constraints (VCMTSP) is a general version of the classical multiple traveling salesmen problem (MTSP), where each city can be only accessed by a number of salesmen. To cope with this new problem, we adapt the genetic algorithm (GA) for MTSP by using a dual-chromosome representation scheme with one chromosome denoting the visiting sequence of cities and the other representing the assignment of cities to salesmen. To further promote the effectiveness of GA in solving VCMTSP, we modify three popular crossover operators, namely the cycle crossover (CX), the order crossover (OX), and the partially mapped crossover (PMX). Similar to the execution for traditional TSP, the three crossover operators are all executed on the city sequence chromosome, while the adaption of them lies in the modification of the salesman assignment in the second chromosome. To this end, a correction mechanism according to the accessibility matrix is conducted to make the generated solutions after crossover feasible. Extensive experiments conducted on totally 16 VCMTSP instances generated from the benchmark TSPLIB set demonstrate that the adapted GA could effectively cope with VCMTSP, and the GA with the modified PMX achieves the best overall performance.