Co-evolution with hierarchical decomposition for Vehicle Routing Problem with Drones

Abstract

Drone delivery technology has made rapid advances and has seen growing real-world adoption in recent years. To address the inherent limitations of drones in payload and endurance, the collaborative delivery system between trucks and drones, formulated as the Vehicle Routing Problem with Drones (VRP-D), has attracted increasing research interest. In this paper, we introduce a novel Co-evolution with Hierarchical Decomposition (CH-SaBO) framework for VRP-D. Firstly, we propose a co-evolutionary scheme to decompose the original VRP-D into multiple subproblems, each with one truck, effectively reducing the dimensionality of the problem. We then design a hierarchical decomposition strategy for problem decomposition. First, a top-down procedure progressively partitions the solution into finer elements—route, operation, subroute, path, and arc. Then, a recursive bottom-up construction of inter-layer correlation matrices captures the temporal and spatial dependencies among these elements. Finally, k-medoids clustering is applied to the correlation matrices to further group the elements. For subproblem boundaries, we design a simple yet effective overlapping strategy to enhance solution quality. Each subproblem is then independently optimized using a surrogate-assisted bi-level optimizer, enabling efficient search within smaller, well-structured solution spaces. Extensive experiments on 51 benchmark instances demonstrate that CH-SaBO significantly outperforms state-of-the-art algorithms in both solution quality and computational efficiency. Further analyses confirm the scalability of CH-SaBO, as well as the effectiveness of its hierarchical decomposition and overlapping strategies.