The Multi-visit Traveling Salesman Problem with Multi-Drones

The use of drones for parcel delivery has recently attracted wide attention due to its potential in improving efficiency of the last-mile delivery. Though attempts have been made on combined truck-drone delivery to deploy multiple drones that can deliver multiple packages per trip, many placed extra assumptions to simplify the problem. This paper investigates the multi-visit traveling salesman problem with multi-drones (MTSP-MD), whose objective is to minimize the time (makespan) required by the truck and the drones to serve all customers together. The energy consumption of the drone depends on the flight time, the self-weight of the drone and the total weight of packages carried by the drone, which declines after each delivery throughout the drone flight. The MTSP-MD problem consists of three complicated sub-problems, namely (1) the drone flight problem with both a payload capacity constraint and an energy endurance constraint, (2) the traveling salesman problem with precedence constraints, and (3) the synchronization problem between the truck route and the drone schedules. The problem is first formulated into a mixed-integer linear program (MILP) model and we propose a multi-start tabu search (MSTS) algorithm with tailored neighborhood structure and a two-level solution evaluation method that incorporates a drone-level segment-based evaluation and a solution-level evaluation based on the critical path method (CPM). The experimental results demonstrate the accuracy and efficiency of our proposed algorithm on small-scale instances and show a significant cost reduction when considering multi-visits, multi-drones, and drones with higher payload capacity and higher battery capacity for medium and large-scale instances.