The selective multiple depot pickup and delivery problem with multiple time windows and paired demand

A recurring challenge for transportation companies is the inefficiency of returning (partially) empty vehicles, or backhauling, after delivering orders. To address this issue, companies search on freight exchange platforms for profitable pickup and delivery orders, aiming to reduce the costs associated with empty return trips. The increasing reliance on freight exchange platforms presents both an opportunity and a challenge: while they offer access to profitable loads, effectively selecting the right combination of orders to maximize returns is challenging. This paper addresses this challenge by introducing the Selective Multiple Depot Pickup and Delivery Problem with Multiple Time Windows and Paired Demand (SMDPDPMTWPD). We formulate the SMDPDPMTWP as a Mixed-Integer Linear Program (MILP) to maximize profit and optimize freight selection for return trips. In addition to the main model, three problem extensions are proposed: (i) profit maximization including CO2 costs, (ii) soft time windows, and (iii) soft time windows including CO2 costs. Given the complexity of the problem, we develop an Adaptive Large Neighborhood Search (ALNS) metaheuristic to solve large instances within reasonable computing times and compare it with a Simulated Annealing (SA) heuristic. Results show that ALNS outperforms SA and finds the same optimal solutions as the MILP formulation for small instances. Furthermore, ALNS achieves an average improvement of 308.17% over the initial solutions for the profit maximization variant. The model variant with CO2 costs shows a slight sensitivity of the routing schedules to the CO2 emissions costs, whereas we observe a significant change when allowing soft time windows. Finally, soft time windows significantly increase the profits earned compared to the hard time windows (179.54% on average), due to the additional flexibility created when late arrivals are possible.