Physical internet drone logistics model: Optimizing cost and energy efficiency for urban delivery systems

Leading logistics service providers are increasingly adopting drone technologies to mitigate urban last-mile delivery challenges. However, many existing implementations remain siloed and lack integration with broader logistics infrastructures, limiting their operational scalability and energy efficiency. This study addresses this gap by proposing the Physical Internet Drone Logistics (PIDL) model, which embeds drones within the Physical Internet (PI) framework to optimize parcel routing and reduce energy consumption. Unlike traditional drone-only models, the PIDL framework explicitly incorporates open urban hubs and open recharge stations to enable dynamic energy and fleet management. The PIDL model is formulated as a mixed-integer linear programming problem to minimize total operational costs by optimizing delivery paths. To address computational intractability in large-scale instances, a tailored hybrid genetic algorithm (HGA) is developed. The HGA incorporates two novel components: the adaptive insertion and allocation heuristic for real-time coordination of drone routing and recharge logistics, and the hybrid heuristic crossover operator, which embeds domain-specific knowledge to improve solution quality and scalability. Computational experiments demonstrate the superior performance of the PIDL model across both small- and large-scale problem instances. A comparative analysis between the exact approach and the HGA-based metaheuristic highlights the trade-offs between computational efficiency and solution quality. The PIDL model yields up to 9.8% energy cost savings in large-scale tests and is further validated through a real-world setting. More importantly, the findings provide actionable implications for designing scalable, cost-efficient, and sustainable urban drone delivery systems aligned with the PI paradigm.