A new branch-and-Benders-cut algorithm for the time-dependent vehicle routing problem

Daily traffic congestion poses significant challenges for companies operating in urban areas. By considering predicted travel times throughout the day, route planning systems can improve delivery schedules, thereby reducing costs associated with delays and congestion. Although the time-dependent vehicle routing problem presents a more realistic model for city logistics, it also introduces significant computational challenges due to the size of the network of the associated models. This paper presents the first exact method using logic- based Benders decomposition for the time-dependent vehicle routing problem. Two problem formulations based on two, and three-index vehicle routing models are proposed and solved using a branch-and-cut procedure. Moreover, a logic-based branch-and-Benders-cut algorithm is developed. Computational experiments on real- world instances demonstrate that the proposed algorithm achieves better solutions, especially when applied to the three-index formulation. Typical optimality gaps are lower than 5% even in the bigger instances. Moreover, the paper evaluates various design choices for the algorithm and their impact on the solutions of the problem. Finally, managerial insights highlight the importance of solving time-dependent vehicle routing problem by demonstrating that congestion adds significant delays and costs to transportation systems.