An assignment-based decomposition approach for the vehicle routing problem with backhauls

In the context of the Vehicle Routing Problem with Backhauls, which involves delivering to linehaul and picking up from backhaul customers, we propose a novel mathematical model that can decompose the main problem into three sub-problems: two Open Vehicle Routing Problems and one Assignment Problem. In our proposed model, Open Vehicle Routing Problems optimize routes for homogeneous vehicles serving linehaul/backhaul customers, while the Assignment Problem matches linehaul and backhaul routes. We utilize Lagrangian decomposition approach and solve subproblems in parallel and sequential layouts. We measure the performance of the foregoing arrangements (in terms of solution quality and computational efficiency) by testing our model on two benchmark datasets, proposed by Goetschalckx and Jacobs-Blecha (1989) and Toth and Vigo (1997) and are known as GJ and TV datasets in the extant literature, respectively. Our model matches best known solutions in 35 % and 33 %, with most within 2 % deviation, for GJ and TV instances, respectively. We also showcase our model on a real-world transportation network containing 100, 250, and 500 customers and geographically located in Lansing, Michigan. To reduce the computational burden of solving the Vehicle Routing Problem with Backhauls on the Lansing dataset, we present a cluster first-route second algorithm and then analyze the impact of vehicle capacity on the solution quality of our proposed algorithm.