Multi-Modal Last-Mile Delivery Optimization With Smart Micro-Hubs: A Branch-and-Price Approach With Valid Inequalities

The rapid growth of e-commerce and urban logistics has intensified the challenges associated with last-mile delivery, necessitating the integration of autonomous delivery vehicles (ADVs), drones, and smart micro-hubs for cost-effective and sustainable operations. This study introduces a novel Multi-Depot Capacitated Vehicle Routing Problem with Time Windows and Multi-Modal Transportation (MD-CVRP-TW-MM), where deliveries are dynamically assigned to either ADVs or drones while optimizing hub locations and vehicle routing. The objective is to minimize total operational costs, including transportation, hub activation, and delivery mode selection, while ensuring service-level constraints such as time windows and capacity limits. To efficiently solve this complex combinatorial problem, we propose a mathematical formulation enhanced with valid inequalities for computational efficiency. Furthermore, a Branch-and-Price algorithm is developed, where column generation is employed to iteratively refine delivery routes while enforcing depot assignments and mode selection constraints. The solution approach integrates dual stabilization techniques and path-based heuristics to accelerate convergence. Experimental evaluations on benchmark datasets and real-world logistics scenarios demonstrate the effectiveness of the proposed method in reducing last-mile delivery costs, optimizing fleet utilization, and enhancing service reliability. The results highlight the significance of multi-modal last-mile delivery optimization in improving urban logistics and sustainability. This research contributes a new mathematical model, efficient optimization techniques, and practical insights for intelligent last-mile delivery planning in smart cities.