The integration of trucks and drones in last-mile delivery has introduced new capabilities to the transportation industry. These two vehicles simultaneously offer unique features, which have improved performance and efficiency in the process of delivering products. This paper investigates a robust parallel drone scheduling traveling salesman problem with supporting drone, where drone travel times are uncertain and products gradually arrive at a depot over time. In this problem, a truck, a supporting drone, and service drones are located in the depot to deliver products with the goal of minimizing the total completion time. A mathematical model is proposed which is improved using the earliest release dates rule, followed by the development of an exact logic-based benders decomposition algorithm to solve the problem. In this algorithm, customers are initially assigned to the service drones or the truck in a master problem, and subsequent auxiliary problems are addressed utilizing the earliest release dates rule and a dynamic programming algorithm. Finally, various cuts are enhanced through strengthening techniques and sequentially added into the master problem. Numerical experiments demonstrate the efficiency of the improved mathematical model and the proposed algorithm. Furthermore, sensitivity analysis has provided several managerial recommendations for enhancing the delivery system performance.
