A branch-and-price approach for the parallel drone scheduling vehicle routing problem

This paper considers a delivery system in which a fleet of trucks and unmanned aerial vehicles (or drones) operate independently to serve a set of customers in the minimum possible time. Three mixed integer linear programming (MILP) formulations are proposed for this problem, one of which is an arc-based formulation and the other two are set covering based formulations based on column generation. Owing to the strength of set covering based formulations in producing good linear relaxations, a branch-and-price (BAP) algorithm based on the second set covering based formulation is designed. To solve large-scale problems, a heuristic version of the BAP algorithm is proposed as well. Results from extensive numerical testing indicate that the performances of both the exact and heuristic versions of the BAP algorithm are comparable to the arc-based formulation in solving small-scale instances. Moreover, the heuristic version of the BAP algorithm significantly outperforms the arc-based formulation both in terms of solution quality and runtime for larger problems. Finally, the impact of a UAV power consumption model, according to which, power consumption depends on both speed and payload, is analyzed in the context of this problem. Specifically, a detailed study is performed to show the benefits of optimizing UAV speeds in reducing delivery makespan when UAV endurance is estimated using this power consumption model.