Collaborative truck multi-drone pollution routing problem with pickup and delivery under variable truck speeds: a cost minimisation approach

Collaborative truck-drone systems offer significant potential for reducing cost and emissions in last-mile logistics. However, prior studies often assume constant truck speeds, oversimplify fuel and drone energy use, and overlook differences between labor- and energy-dominant cost structures. This study introduces the Truck Multi-Drone Pollution Routing Problem with Pickup and Delivery under Variable Truck Speeds (TMD-PRP-PDVS), a generalisation of the PRP that integrates multi-drone coordination, segment-specific truck speed control, and realistic cost modelling. It is the first to unify variable truck speeds and coordinated truck-drone operations within a PRP framework for comparative analysis across distinct cost environments. A mixed-integer programming model is developed to minimise total cost, comprising truck fuel consumption, drone energy usage, and driver labor cost. For larger instances, an Adaptive Large Neighborhood Search heuristic is proposed with an embedded local search procedure that refines truck speeds at the route-segment level in polynomial time. Computational results show that drone deployment reduces total cost by about 40%, while speed optimisation yields an additional 7-11% on average and up to 18% in some cases. Case studies on real-world networks demonstrate applicability across contrasting cost regimes. The findings highlight the joint influence of truck speed regulation, drone deployment, and cost structures.