Ant Colony Optimization for Multiple Travelling Salesmen Problem with Pivot Cities

Multiple travelling salesmen problem with pivot cities (PCMTSP) is a variant of the multiple travelling salesmen problem (MTSP). The same with MTSP, in PCMTSP, multiple salesmen visit all cities together. However, different from MTSP, in PCMTSP, some cities can be visited multiple times by multiple salesmen. Such cities are called pivot cities. The objective of this new problem is to minimize both the total travelling cost of all salesmen and the path difference among salesmen on the condition that only the pivot cities are visited by multiple travelers, while the other cities are only visited once by only one salesman. To tackle this new problem, this paper adapts ant colony optimization (ACO) by maintaining ant groups to find the shortest paths of all salesmen. In particular, each ant group contains the same number of ants with that of the salesmen, and each ant is employed to build the path of one salesman. To maintain a good balance among the paths of all salesmen, four ant selection strategies are further devised to build the paths of all salesmen city by city, namely In-turn Selection (IS), Shortest City Distance Biased Selection (SCDBS), Shortest Path Biased Selection (SPBS) and City First Path Second Based Selection (CFPSBS). To further promote the solution accuracy, the 2-opt operator is employed to locally optimize the paths of all salesmen. At last, experiments are conducted on four TSPLIB benchmark instances with different numbers of travelling salesmen. Experimental results have demonstrated the effectiveness of the four proposed selection schemes. Particularly, among the devised four ant selection strategies, SPBS assists ACO to attain the best optimization performance in coping with PCMTSP.