Pickup and delivery problem with multi-visit drones considering soft time windows

We extend the pickup and delivery problem with combined truck–drone operation by assuming that a fleet of trucks can stop at customer nodes and launch drones to perform multiple pickup and delivery services with soft time windows. Unlike other studied pickup and delivery problems, one-to-one pickup and delivery services (in which the pickup and delivery requests have a one-to-one relationship), which are typical in instant retail and food delivery, are considered. We mathematically model a mixed-integer nonlinear program and introduce strengthening strategies to capture this scenario, with the objective of minimising the total cost, including the penalty cost of time window violations. We approximate the nonlinear power of hovering drones with a piecewise linear method and propose an efficient metaheuristic approach, along with truck waiting time optimisation, to solve large-size problems. Finally, comprehensive computational experiments are conducted, which demonstrate the applicability of the algorithm and the impacts of different configurations. The numerical results indicate the efficiency of our proposed model and the solution approach, demonstrating the potential operational gain obtained by implementing the combined system. The cost savings rate compared to the truck-only mode is more than 40% on average, and our algorithm outperforms the benchmark algorithms in the literature by more than 10% in terms of solution quality.