Modeling and optimization of multiple traveling salesmen problems: An evolution strategy approach

Abstract

The multiple traveling salesmen problems (mTSP) are variants of the well-known traveling salesmen problems, in which n cities are to be assigned to m salespeople. In this paper, we propose an evolution strategy (ES) approach for solving the mTSP with minsum and minmax objectives. The ES employs a self-adaptive Ruin and Recreate (RR) heuristic to generate an offspring population. In the RR heuristic, some solution components are removed from a solution and the removed components are reinserted into the partial solution to obtain a new complete solution again. We employ the Multiple Insertion Heuristic (MIH) when carrying out insertions with the speed-up method based on the nearest neighbor approach in the recreate procedure. Through the ES, the ruin size parameter of the RR heuristic and the selection probabilities of applying a random search algorithm consisting of four different moves are self-adapted. 3Opt local search is also embedded in the ES to further enhance the solution quality. A Boolean flag is developed whether or not to apply 3Opt local search, which is computationally expensive, on non-improving tours. Moreover, new constructive heuristics are presented for both minsum and minmaxmTSP. Computational experiments show that the proposed ES algorithm is very competitive or superior to the best performing algorithms from the literature for both objectives. Ultimately, 21 new best-solutions are presented for the mTSP with minsum and minmax objectives for the first time in the literature. In this paper, the multi-depot mTSP with non-predetermined depots (M-mTSP) is also studied as an extension of the mTSP. Two new MILP models are proposed for both minsum and minmaxM-mTSP as well as lower bounds, and optimal results are obtained for small instances. The computational results show that the ES algorithm can find the optimal solution for all the small-sized instances. Furthermore, we derive new large-sized M-mTSP benchmarks from the well-known TSPLIB, which have nodes ranging from 52 and 442. We report the results for both minsum and minmax objectives for these instances as new M-mTSP benchmarks, considering four different m settings.