Linearization Strategies for Energy-Aware Optimization of Single-Truck, Multiple-Drone Last-Mile Delivery Systems

The increasing demand for rapid and sustainable parcel delivery has motivated the exploration of innovative logistics systems that integrate drones with traditional ground vehicles. Among these, the single-truck, multiple-drone last-mile delivery configuration has attracted significant attention due to its potential to reduce both delivery time and environmental impact. However, optimizing such systems remains computationally challenging because of the nonlinear energy consumption behavior of drones, which depends on factors such as payload weight and travel time, among others. This study investigates the energy-aware optimization of truck–drone collaborative delivery systems, with a particular focus on the mathematical formulation as mixed-integer nonlinear problem (MINLP) formulations and linearization of drone energy consumption constraints. Building upon prior models proposed in the literature in the field, we analyze the MINLP computational complexity and introduce alternative linearization strategies that preserve model accuracy while improving performance solvability. The resulting linearized mixed-integer linear problem (MILP) formulations are solved using the PuLP software, a Python library solver, to evaluate the efficacy of linearization on computation time and solution quality across diverse problem instance sizes from a benchmark of instances in the literature. Thus, extensive computational results drawn from a standard dataset benchmark from the literature by running the solver in a cluster infrastructure demonstrated that the designed linearization methods can reduce optimization time of nonlinear solvers to several orders of magnitude without compromising energy estimation accuracy, enabling the model to handle larger problem instances effectively. This performance improvement opens the door to a real-time or near-real-time solution of the problem, allowing the delivery system to dynamically react to operational changes and uncertainties during delivery.