Optimization of last-mile deliveries with synchronous truck and drones

Abstract

Deployment of drones in delivery operations has been attracting a growing interest from commercial sector due to its prospective advantages for the 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. We provide flow based and cut based mixed integer linear programming formulations along with valid inequalities. Since the connectivity constraints in the cut based formulation and the proposed valid inequalities are exponential in the size of the problem, we devise different branch-and-cut schemes to solve our problem. We also provide an alternative solution methodology using another cut based formulation with undirected route variables. To compare our formulations/algorithms and to demonstrate their competitiveness, we conduct computational experiments on a set of instances. The results indicate the superiority of utilizing branch-and-cut methodology over flow based formulation and good computational performance of the proposed algorithms in comparison to existing exact solution approaches in the literature. We also conduct sensitivity analyses on problem parameters and discuss their effects on the optimal solutions.