A Branch & Cut algorithm for the prize-collecting capacitated location routing problem

The Capacitated location routing problem (CLRP) is the combination of two well-studied problems in Operations Research: the capacitated facility location problem (CFLP) and the multiple depots vehicle routing problem (MDVRP). Given a set of available locations and a fleet of vehicles, the aim is to determine a set of depots to open and routes of the vehicles to satisfy the customers demands. The objective of the CLRP is to minimize the total cost, that is the cost of the opened depots, the fixed cost of the vehicles and the cost of the routes while satisfying vehicle and depot capacity constraints. In this work the prize-collecting capacitated location routing problem (PC-CLRP), a new variant of the CLRP is presented. In this case it is possible to leave some customers unvisited and if a customer is visited it gives a gain. The objective is to maximize the overall benefit. A two-index formulation for the PC-CLRP and a Branch & Cut algorithm based on this model are proposed. Valid inequalities for the CLRP are adapted for the PC-CLRP. Also new valid inequalities and optimality cuts are proposed together with their corresponding separation algorithms. A hierarchical branching strategy is also included. The initial solution was provided by an Ant Colony Algorithm. The algorithm is tested on a set of instances specially designed for the new problem. Computational results showed very promising. We also compare the performance of the algorithm with recent work for CLRP on instances from the literature.