Exact Solution of the Vehicle Routing Problem With Drones

The vehicle routing problem with drones (VRP-D) 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. To operate alone, a truck can release its drone at the depot or at a customer location and likewise pick it up at a later location visited by the same truck. 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. The VRP-D consists of finding a minimum-cost set of feasible routes such that each customer is served exactly once by either a truck or a drone. We develop a branch-price-and-cut (BPC) algorithm to solve the VRP-D exactly for both standard objectives considered in the literature, i.e., the minimization of the total routing cost and the sum of the routes' durations. To solve the column-generation subproblems, we present a new forward and implicit bidirectional labeling algorithm defined over an artificial network. The new bidirectional labeling algorithm substantially accelerates the solution process compared its monodirectional counterpart. In several 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. The final version of the BPC algorithm is able to solve VRP-D instances with up to 50 vertices to proven optimality within one hour of computation time.