Exact Solution of the Vehicle Routing Problem with Drones

The vehicle routing problem with drones (VRP-D) that we consider is an extension of the capacitated vehicle routing problem in which the fleet consists of trucks equipped with one drone each. A truck and its drone can either move together or separately. A truck can release its drone at the depot or at a customer location and must pick it up later at another customer or the depot location. In this way, both trucks and drones deliver goods to customers working together as synchronized working units. A feasible route has to satisfy the capacity constraints of both the truck and the drone. A feasible solution to the VRP-D is a set of feasible routes such that each customer is served exactly once by either a truck or a drone. We investigate two standard objectives considered in the literature, that is, the minimization of the total routing cost and the sum of the routes’ durations. To solve the VRP-D exactly, we develop a branch-price-and-cut (BPC) algorithm. In particular, we present a new forward and implicit bidirectional labeling algorithm defined over an artificial network to solve the column-generation subproblems. The new bidirectional labeling algorithm substantially accelerates the solution process compared with its monodirectional counterpart. The time needed to solve the pricing problems is reduced by 55% on average when minimizing routing costs and by 30% when minimizing the sum of the routes’ durations. In further computational experiments, we analyze algorithmic components of the BPC algorithm, compare the cost and duration objectives, and highlight the impact of the drones’ speed on the structure of VRP-D solutions. For the routing-cost minimization objective, our BPC algorithm is able to solve several VRP-D instances with 50 vertices to proven optimality within one hour of computation time. The same instances with duration minimization are more difficult, and the BPC algorithm provides only heuristic solutions with an average gap not exceeding 3%. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Project 418727865, Grant IR 122/10-1]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0544 .