Robust vehicle routing and cross-dock scheduling with uncertain loading and unloading time

Cross-docking is a recently interesting logistics technique that consolidates incoming freights from suppliers and rapidly loads them on the outgoing vehicles to realize the economy of scale in the delivery process. Simultaneously considering the outgoing vehicles routing problem and the cross-dock scheduling is known as Vehicle Routing Problem with Cross-Docking (VRPCD) that is increasingly gaining ground for 3PL companies from the viewpoint of cost efficiency. This paper, minimizing the total cost (including transporting, cross-docking, inventory holding costs and the penalty cost of early and tardy deliveries), presents a rigorous MIP model for the VRPCD in which doors of the cross-dock are different in loading/unloading time, and a heterogeneous fleet of vehicles is used for delivery process. Considering the uncertainty of the loading/unloading times in dock-doors and unloading times in delivery nodes to retailers, a robust reformulation of the model is presented by applying interval-polyhedral (Bertsimas and Sim, 4002) approach. The robustness of the solutions and the effect of conservatism and variability levels on the solutions have been evaluated using simulation. The results of numerical analysis confirm the performance of the applied robust optimization approach and validate the robustness of the model.