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The multiple Traveling Salesmen Problem (mTSP) is a general extension of thefamous NP-hard Traveling Salesmen Problem (TSP), that there are m (m > 1)salesmen to visit the cities. In this paper, we address the mTSP with both theminsum objective and minmax objective, which aims at minimizing the totallength of the $m$ tours and the length of the longest tour among all the mtours, respectively. We propose an iterated two-stage heuristic algorithmcalled ITSHA for the mTSP. Each iteration of ITSHA consists of aninitialization stage and an improvement stage. The initialization stage aims togenerate high-quality and diverse initial solutions. The improvement stagemainly applies the variable neighborhood search (VNS) approach based on ourproposed effective local search neighborhoods to optimize the initial solution.Moreover, some local optima escaping approaches are employed to enhance thesearch ability of the algorithm. Extensive experimental results on a wide rangeof public benchmark instances show that ITSHA significantly outperforms thestate-of-the-art heuristic algorithms in solving the mTSP on both theobjectives.

An Effective Iterated Two-stage Heuristic Algorithm for the Multiple Traveling Salesmen Problem