An Algorithm for the Euclidean Bounded Multiple Traveling Salesman Problem

Abstract

In the Bounded Multiple Traveling Salesman Problem (BMTSP), a tour for each salesman, that starts and ends at the depot and that respects the bounds on the number of cities that a feasible salesman tour should satisfy, is to be constructed. The objective is to minimize the total length of all tours. Already Euclidean traveling salesman problem is NP-hard. We propose a 3-Phase heuristic algorithm for the Euclidean BMTSP. We tested the algorithm for the 22 benchmark instances and 168 new problem instances that we created. We report 19 best known solutions for the 22 benchmark instances including the 12 largest ones. For the newly created instances, we compared the performance of our algorithm with that of an ILP-solver CPLEX, which was able to construct a feasible solution for 71% of the instances within the time limit of two hours imposed by us. For about 10% of the smallest new instances, CPLEX delivered slightly better solutions, where our algorithm took less than 180 seconds for the largest of these instances. For the remaining 61% of the instances solved by CPLEX, the solutions by our heuristic were, on average, about 21.5% better than those obtained by CPLEX.