An exact algorithm for the two-echelon vehicle routing problem with drones

This paper studies a new variant of the vehicle routing problem with drones, i.e., the two-echelon vehicle routing problem with drones, where multiple vehicles and drones work collaboratively to serve customers. Drones can perform multiple back-and-forth trips when their paired vehicle stops at a customer node, forming a two-echelon network. Several practical constraints such as customers' delivery deadlines and drones' energy capacity are considered. Different from existing studies, we treat the number of drones taken by each vehicle as a deci-sion variable instead of a given parameter, which provides more flexibility for planning vehicle and drone routes. We first formulate this problem as a mixed-integer linear programming model, which is solvable by off-the-shelf commercial solvers. To tackle instances more efficiently, we next construct a set-partitioning model. To solve it, an exact branch-and-price algorithm is proposed, where a bidirectional labeling algorithm is used to solve the pricing problem. To speed up the algorithm, a tabu search algorithm is first applied before the exact labeling algorithm for finding desired columns in each iteration of the column generation process. Extensive numerical tests show that our algorithm can solve most instances within 25 customers to optimality in a short time frame and some instances of 35 customers to optimality within a three-hour time limit. Results also demonstrate that the allocation decisions of drones can help save the duration of all routes by 3.44% on average for 25-customer instances, compared to the case of fixing the number of paired drones on each vehicle. In addition, sensitivity analyses show that multiple strategies, e.g., adopting batteries of a higher energy density and developing faster drones, can be applied to further improve the delivery efficiency of a truck-drone system.

Automated summary

This study focuses on a new variant of the vehicle routing problem with drones, the two-echelon vehicle routing problem with drones, where multiple vehicles and drones collaborate to serve customers. The study considers practical constraints such as delivery deadlines and energy capacity of drones. Unlike previous research, the number of drones taken by each vehicle is treated as a decision variable, providing flexibility in planning vehicle and drone routes. The study proposes a mixed-integer linear programming model and an exact branch-and-price algorithm to solve the problem efficiently, and the results demonstrate the effectiveness of the algorithm in optimizing routes and the potential for further improvements.