Exact solution approaches for the minimum total cost traveling salesman problem with multiple drones

Deployment of drones in delivery operations has been attracting growing interest from the commercial sector due to its prospective advantages for a range of distribution systems. Motivated by the widespread adoption of drones in last-mile delivery, we introduce the minimum cost traveling salesman problem with multiple drones, where a truck and multiple drones work in synchronization to deliver parcels to customers. In this problem, we aim to find an optimal delivery plan for the truck and drones operating in tandem with the objective of minimizing the total operational cost including the vehicles' operating and waiting costs. Unlike most studies in the literature where the objective is to minimize completion time, which means one needs to know only the arrival time of the latest arriving vehicle (truck or drone) at each synchronization point, we need to keep track of all the individual waiting times of the truck and the drones to properly account for waiting costs, which makes it more challenging to handle the synchronization. We provide a flow based and two cut based mixed integer linear programming formulations strengthened with valid inequalities. For non-compact models, we devise a variety of branch-and-cut schemes to solve our problem to optimality. To compare our formulations/algorithms and to demonstrate their competitiveness, we conduct computational experiments on a range of instances. The results indicate the superiority of utilizing branch-and-cut methodology over a flow based formulation. We also use our model to conduct sensitivity analyses with several problem parameters and to explore the benefits of launch and retrieval at the same node, the tradeoff between the number of drones and the operational cost, and the special case with a minimize completion objective with one drone. We also document very low waiting times for drones in optimal solutions and show solutions from minimizing cost have much lower cost than those from minimizing makespan.