A branch‐and‐cut algorithm for the undirected prize collecting traveling salesman problem

Abstract

Given an undirected graph with edge costs and vertex prizes, the aim of the Prize Collecting Traveling Salesman Problem (PCTSP) is to find a simple cycle minimizing the total edge cost while collecting at least a minimum amount of prizes. In this article, we present a branchand-cut algorithm to solve the PCTSP. We have adapted and implemented some classical polyhedral results for the PCTSP and derived inequalities from cuts designed for the Orienteering Problem. Computational results on instances with more than 500 vertices are reported. The Prize Collecting Traveling Salesman Problem (PCTSP) was introduced as a model for scheduling the daily operations of a steel rolling mill [7]. Given an undirected graph with edge costs and vertex prizes, the aim of the PCTSP is to find a simple cycle minimizing the total edge cost while collecting at least a minimum total prize. Also known as the Quota TSP [5], the PCTSP has been classified as a Traveling Salesman Problem with Profits [14], along with the the Profitable Tour Problem (PTP) and the Orienteering Problem (OP). The PTP, introduced in [13], is aimed at maximizing the difference between the collected prizes and the travel costs; it is also known as the Simple Cycle Problem [16]. In the OP, one must find a tour that maximizes the total collected prize while maintaining the costs under a fixed value. It has been introduced in a study on orienteering competitions To our knowledge, the only exact approach for the PCTSP is a branch-and-bound algorithm given by Fischetti and Toth [17]. However, there are a number of publications on exact methods to solve the OP [14]. In particular, the two branch-and-cut algorithms devised for the symmetric OP in [15,19] are currently the most effective procedures. Our goal is thus (1) to solve the PCTSP with an exact branch-andprice algorithm by exploiting valid inequalities derived from classical studies on the knapsack and Traveling Salesman (TS) polytopes and (2) to analyze, through extensive computational results, how these classical inequalities behave in practice. Although our algorithm does not bring any new polyhedral results, the implementation of these inequalities in the context of the PCTSP has allowed us to get some interesting insights about their respective strengths and weaknesses. We believe that these results will be helpful in the development of future exact algorithms for the PCTSP.The article is organized as follows. In Section 1, the mathematical model is presented, along with some notation. Section 2 describes the valid inequalities, while separation procedures are discussed in Section 3. The overall branchand-cut algorithm is described in Section 4. Finally, Section 5 reports computational results with different application orders for the separation procedures, using instances generated from the Traveling Salesman Library TSPLIB [30]. Note that throughout this text, we assume that the reader is familiar with the branch-and-cut methodology, as described in [22]. MATHEMATICAL MODELLet G = (V , E) be a complete undir...

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

“…We transformed VRP and TSP instances of the TSPLIB [25] into TSPP instances using the rules provided in [5] and [12]. We considered instances for which the node coordinates were available.…”

“…This appendix summarizes the valid inequalities used by the branch-and-cut algorithm presented in [5] for the model defined by Eqs. (14) to (20).…”

“…Compute a feasible solutionx s with the heuristic algorithm used in [5] to find an initial feasible solution. 2.…”

“…On the other hand, the point (c N ; p I ) is associated with the PCTSP with p ¼ pðV Þ, which is a classical TSP. The PCTSPs are efficiently solved through the branchand-cut procedure described in [5]. The latter has been modified to generate a solution with the greatest collected prize among the optimal solutions of P i ð j Þ, as explained under Handling dominated points in Section 1.…”

Heuristics for Solving Problem of Evacuating Non-Ambulatory People in a Short-Notice Disaster

“…This method has been very successful in finding optimal solutions of large instances of the closely related Symmetric TSP (STSP), as well as Prize Collected TSP (PCTSP) (Bérubé, Gendreau, & Potvin, 2009). …”