A Redesigned Benders Decomposition Approach for Large-Scale In-Transit Freight Consolidation Operations

The growth in online shopping and third-party logistics has caused a revival of interest in finding optimal solutions to the large-scale, in-transit freight consolidation problem. Given the shipment date, size, origin, destination, and due dates of multiple shipments distributed over space and time, the problem requires determining when to consolidate some of these shipments into one shipment at an intermediate consolidation point so as to minimize shipping costs while satisfying the due date constraints. In this article, the authors develop a mixed-integer programming formulation for a multi-period freight consolidation problem that involves multiple products, suppliers, and potential consolidation points. Benders decomposition is then used to replace a large number of integer freight-consolidation variables by a small number of continuous variables that reduce the size of the problem without impacting optimality. The results show that Benders decomposition provides a significant scale-up in the performance of the solver. The authors demonstrate their approach using a large-scale case with more than 27.5 million variables and 9.2 million constraints.