Two-indexed formulation of the traveling salesman problem with multiple drones performing sidekicks and loops

Aerial drone delivery has great potential to improve package delivery time, as drones can fly autonomously over obstacles at a possibly higher speed than trucks. The benefits of drones in delivery can even be increased in a truck-and-drone tandem where a truck carries multiple drones and releases them at advantageous places, i.e., the traveling salesman problem with multiple drones (TSPmD). We focus on a general version of this problem with makespan minimization, where the drones have two options after serving the customer: they can return to any node the truck visits at a later stage (sidekick) or return to the same node they were launched from (loop) - even at the depot. We introduce an efficient two-indexed mixed-integer linear program (MILP) for this TSPmD and show how to adapt the MILP to cover two problem variants, namely the multiple flying sidekick traveling salesman problem and the traveling salesman problem with drone. Our MILP formulation is an efficient formulation, as it outperforms eight state-of-the-art MILP formulations for these problem variants in nearly all larger instances. In a numerical study, we provide new optimal solutions with up to 28 nodes for benchmark purposes. Moreover, we analyze the impact of allowing drone loops on makespan minimization in general and at the depot. We find that loops mainly become relevant when drones travel faster than trucks, resulting in average makespan savings of up to 2.7%.