An exact image-constraint method for bi-objective combinatorial optimization problems: Application to the Traveling Salesman Problem with Profits

This paper describes an exact ϵϵ-constraint method for bi-objective combinatorial optimization problems with integer objective values. This method tackles multi-objective optimization problems by solving a series of single objective subproblems, where all but one objectives are transformed into constraints. We show in this paper that the Pareto front of bi-objective problems can be efficiently generated with the ϵϵ-constraint method. Furthermore, we describe heuristics based on information gathered from previous subproblems that significantly speed up the method. This approach is used to find the exact Pareto front of the Traveling Salesman Problem with Profits, a variant of the Traveling Salesman Problem in which a profit or prize value is associated with each vertex. The goal here is to visit a subset of vertices while addressing two conflicting objectives: maximize the collected prize and minimize the travel costs. We report the first exact results for this problem on instances derived from classical Vehicle Routing and Traveling Salesman Problem instances with up to 150 vertices. Results on approximations of the Pareto front obtained from a variant of our exact algorithm are also reported.