A matheuristic for the vehicle routing problem with drones and its variants

In this work, we are interested in studying the Vehicle Routing Problem with Drones (VRPD). Given a fleet of trucks, where each truck carries a given number of drones, the objective consists in designing feasible routes and drone operations such that all customers are served and minimal makespan is achieved. We formulate the VRPD as a Mixed Integer Linear Program (MILP), which can be solved by any standard MILP solver. Moreover, with the aim of improving the performance of solvers, we introduce several sets of valid inequalities (VIEQ). Due to limited performance of the solvers in addressing large instances, we propose a matheuristic approach that effectively exploits the problem structure of the VRPD. Integral to this approach, we propose the Drone Assignment and Scheduling Problem (DASP) that, given an existing routing of trucks, looks for an optimal assignment and schedule of drones such that the makespan is minimized. In this context, we propose two MILP formulations for the DASP. In order to evaluate the performance of a state-of-the-art solver in tackling the MILP formulation of the VRPD, the benefit of the proposed VIEQs, and the performance of the matheuristic, we carried out extensive computational experiments. According to the numerical results, the use of drones can significantly reduce the makespan and the proposed VIEQ as well as the matheuristic approach have a significant contribution in solving the VRPD effectively.