The Prize-Collecting Vehicle Routing Problem with Single and Multiple Depots and Non-Linear Cost

In this paper, we propose a new routing problem to model a highly relevant planning task in small package shipping. We consider the Prize-Collecting Vehicle Routing Problem with Non-Linear cost (PCVRPNL) in its single and multi-depot version, which integrates the option of outsourcing customers to subcontractors instead of serving them with the private fleet. Thereby, a lower bound on the total customer demand to be served by the private fleet guarantees a high utilization of the fleet capacity. To represent the practical situation, where a discount is given by a subcontractor if larger amounts of packages are outsourced, subcontracting costs follow a non-linear function. The considered problem is NP-hard and we propose an Adaptive Variable Neighborhood Search algorithm to solve instances of realistic size. We propose new benchmark sets for the single and the multi-depot problem, which are adapted from test instances of the capacitated VRP and the closely related Multi-Depot VRP with Private fleet and Common carrier (MDVRPPC). In numerical studies, we investigate the performance of our algorithm on the newly generated test instances and on standard benchmark problems of related problems. Moreover, we study the effect of different cost functions and different values of the minimal demand to be served by the private fleet on the routing solutions obtained.