A branch-and-price-and-cut algorithm for the vehicle routing problem with load-dependent drones

In this paper, we consider the vehicle routing problem with load-dependent drones (VRPLD), in which the energy consumption of drones is load-dependent and represented by a nonlinear function. To strengthen the collaboration between trucks and drones, a kind of facility called the docking hub is introduced to extend the service coverage of drones. When a truck visits the hub, a part number of parcels are transferred to the drones departing from the hub to serve the designated customers. We propose a mixed-integer model for the problem, which is nonlinear due to the load-dependent energy consumption. To solve the model, we develop a branch-and-price-and-cut algorithm based on the Danzig–Wolfe decomposition framework, and propose a series of acceleration strategies, including two valid inequalities, to expedite the convergence of the exact algorithm. Computational results on a set of randomly generated instances reflect that the proposed algorithm outperforms Gurobi in terms of both efficiency and effectiveness. Compared with VRPLD, the vehicle routing problem with drones (VRPD) which ignores the load-dependent constraints underestimates the total travel cost by 6.83%. Another drawback of VRPD is that some results may become infeasible when considering the load-dependent energy consumption. The results under VRPLD further reveal that a more accurate description of the energy consumption makes the drones rely more on services from auxiliary facilities. We also conduct sensitivity analysis to draw some managerial insights that setting the hub at a reasonable location can significantly reduce the delivery cost and improve truck and drone cooperation efficiency.