The Backhaul Profit Maximization Problem: Optimization Models and Solution Procedures

We present a compact mixed integer program (MIP) for the backhaul profit maximization problem in which a freight carrier seeks to generate profit from an empty delivery vehicle’s backhaul trip from its last scheduled delivery to its depot by allowing it to deviate from the least expensive (or fastest) route to accept pickup-and-delivery requests between various points on the route as allowed by its capacity and required return time. The MIP is inspired by a novel representation of multicommodity flow that significantly reduces the size of the constraint matrix compared with a formulation based on the classical node-arc representation. This, in turn, leads to faster solution times when using a state-of-the-art MIP solver. In an empirical study of both formulations, problem instances with 10 potential pickup/drop-off locations and up to 72 pickup-and-delivery requests were solved an average 1.44 times faster in real time with our formulation, whereas instances with 20 locations and up to 332 pickup-and-delivery requests were solved an average of 11.88 times faster. The largest instances in the comparative study had 60 locations and up to 3,267 pickup-and-delivery requests; these instances required an average of more than 54 hours of real time to solve with the node-arc–based formulation but were solved in an average of under two hours of real time using our compact formulation. We also present a heuristic algorithm based on our compact formulation that finds near optimal solutions to each of the 60-location instances within 22 minutes of real time and near optimal solutions to instances with up to 80 locations within four and a half hours of real time.