An exact method and a heuristic for last-mile delivery drones routing with centralized graph-based airspace control

The increasing demand for efficient last-mile delivery, driven by the rise of e-commerce, has intensified the need for innovative solutions to manage the complexities of urban logistics. Among the most pressing challenges are the Multi-Agent Pathfinding (MAPF) problem and collision avoidance, both of which are NP-hard and critical for the safe and efficient operation of drones. Collision avoidance is particularly challenging due to the expected high density of drones in future urban environments, making it a problem that remains largely unsolved. Traditional approaches often rely on heuristic and metaheuristic methods to manage these complexities, as large instances are beyond the reach of exact methods. Additionally, distributed relaxations to these problems can lead to suboptimal outcomes and highlights the need for a more centralized and controlled solution. This research adopts a graph-based representation of the delivery area, transforming the centralized Last-Mile Delivery Drones (LMDD) problem into a network flow optimization problem. We propose two graph-based novelty methods in LMDD, a purely exact, NP-hard Mixed Integer Linear Programming (MILP) solution that is evaluated against a heuristic. The complexity of the heuristic is bounded by $O\left(P^{1.5}K\right)$, where $P$ represents the number of permits and $K$ is the number of drones. In contrast, the complexity of the MILP model is approximated by $O\left(K^7{P}^{5.25}{2}^{K^{2}P\sqrt{P}}\right)$, making it intractable for larger instances. The findings from simulations indicate that the graph-based heuristic effectively balances computational efficiency and operational reliability, making it a viable solution for real-world LMDD applications, where large instances and practical execution times are required. This research significantly contributes to the fields of drone logistics and transportation by providing a scalable method for optimizing LMDD paths.