Rural first-mile pickup and last-mile delivery: A bus-assisted heterogeneous-drone model

With the rapid expansion of rural e-commerce, widely dispersed demand and limited road infrastructure have made conventional truck-based first-mile pickup and last-mile delivery increasingly unsustainable, creating an urgent need for alternative logistics models. We introduce a bus-assisted heterogeneous-drone scheme that treats fixed-route rural buses as mobile hubs while dispatching drones with complementary ranges and payloads for door-to-door service. A mixed-integer programming model captures bus schedules, drone heterogeneity, time-window constraints, and battery limits. To solve this model efficiently, we develop a two-stage framework-bus-stop clustering followed by an Improved Black-Kite Algorithm (IBKA). IBKA incorporates four enhancements: opposition-based learning, adaptive attack probability, random boundary shrinkage, and a Differential Evolution hybrid operator. Numerical experiments on adapted Solomon instances show the proposed method outperforms Gurobi, a standard Genetic Algorithm (GA), an Eel and Grouper Optimizer (EGO), and the original Black-Kite Algorithm (BKA) in terms of cost, stability, and convergence. On average, IBKA reduces total delivery cost by 5% relative to GA, 9% relative to EGO, and 13% relative to BKA, and enhances stability by 23%, 55%, and 23%, respectively. Sensitivity tests highlight the pivotal influence of drone payload and bus headway. A real-world study on the Xunyang-Tongqianguan line in Shaanxi Province further demonstrates substantial cost savings and operational advantages over both truck-only and homogeneous-drone delivery modes, underscoring the practical value of bus-drone collaboration for rural logistics.