A truck and drones model for last-mile delivery: A mathematical model and heuristic approach

We present a mathematical formulation and a heuristic solution approach for the optimal planning of delivery routes in a multi-modal system combining truck and Unmanned Aerial Vehicle (UAV) operations. In this system, truck and UAV operations are synchronized, i.e., one or more UAVs travel on a truck, which serves as a mobile depot. Deliveries can be made by both UAVs and the truck. While the truck follows a multi-stop route, each UAV delivers a single shipment per dispatch. The presented optimization model minimizes the waiting time of customers in the system. The model determines the optimal allocation of customers to truck and UAVs, the optimal route sequence of the truck, and the optimal launch and reconvene locations of the UAVs along the truck route. We formulate the problem as a Mixed-Integer Linear Programming (MILP) model and conduct a bound analysis to gauge the maximum potential of the proposed system to reduce customer waiting time compared to a traditional truck-only delivery system. To be able to solve real-world problem size instances, we propose an efficient Truck and Drone Routing Algorithm (TDRA). The solution quality and computational performance of the mathematical model and the TDRA are compared together and with the truck-only model based on a variety of problem instances. Further, we apply the TDRA to a real-world case study for e-commerce delivery in São Paulo, Brazil. Our numerical results suggest significant reductions in customer waiting time to be gained from the proposed multi-modal delivery model.