Bi-Objective Colored Traveling Salesman Problems

As a generalization of the well-known multiple traveling salesman problem, a colored traveling salesman problem (CTSP) utilizes colors to describe the accessibility of individual cities to salesmen. To expand its application scope, this work presents a bi-objective CTSP (BCTSP) over hypergraphs, taking into account the balance of workload among salesmen. To solve it, a bi-objective variable neighborhood search (BVNS) is proposed as a solution framework. In BVNS, we exploit a two-stage initialization and a probability-based insertion to produce feasible solutions and generate their neighborhood. Next, population-based multi-insertion, color-preserving exchange and 2-opt constitute a powerful local search procedure, where color-preserving exchange improves the solutions by modifying the route intersections among distinct salesmen. Besides, Delaunay triangulation is utilized to prepare candidates for multi-insertion, thereby increasing the possibility to find an optimal route by shortening links among vertices. Extensive experiments are conducted on 20 cases. To make a comprehensive comparison, four existing methods, i.e., two genetic algorithms and two variable neighborhood search methods, are adapted by exploiting an elitist non-dominated sorting for solving BCTSP instances. The experimental results show that BVNS is superior to its peers in achieving Pareto optima in terms of three popular performance metrics, i.e., hypervolume, inverted generational distance, and C-metric. In addition, the study of four BVNS variants reveals that probability-based insertion, population-based multi-insertion, and color-preserving exchange play significant roles in BVNS's high performance.