Optimizing bidirectional delivery with multiple drones and trucks: a mixed-integer nonlinear model to addressing no-fly zone constraints

The rapid growth of e-commerce and emergency logistics has increased the demand for efficient and flexible delivery systems. However, traditional delivery models, constrained by traffic conditions, face significant challenges in meeting these requirements. This study addresses an integrated multi-truck and multi-drone delivery system as a mixed-integer nonlinear programming (MINLP) model. The system supports bidirectional logistics, multiple drone missions per sortie, service time windows, and spatio-temporal no-fly zones, with payload-dependent drone energy explicitly modeled. Drones are allowed to take off and land at various customer nodes and reroute around time-varying no-fly zones, enhancing operational flexibility. The nonlinear terms arising from detour-distance calculations are further linearized into an MILP model for efficient solution. To solve large-scale instances efficiently, an Adaptive Large Neighborhood Search (ALNS) algorithm is developed. Extensive experiments were conducted on sixteen test cases of varying scales based on real-world data from Seattle and Buffalo. Compared to traditional truck-only and non-detour settings, the proposed model significantly reduces cost and completion time. Sensitivity analyses on drone endurance, vehicle capacity, customer demand intensity, no-fly zone radius and duration further illustrate system adaptability and performance under varying constraints. The findings demonstrate the model’s effectiveness and scalability in complex urban scenarios, offering practical insights for future low-altitude logistics.