A Branch-and-Price Algorithm for the Two-Echelon Inventory-Routing Problem

The two-echelon inventory-routing problem (2E-IRP) addresses the coordination of inventory management and freight transportation throughout a two-echelon supply network. The latter consists of geographically widespread customers whose demand over a discrete planning horizon can be met from either their local inventory or intermediate facilities' inventory. Intermediate facilities are located in the city outskirt and are supplied from distant suppliers. The 2E-IRP aims to minimize transportation costs and inventory costs while meeting customers' demand. A route-based formulation is proposed and a branch-and-price algorithm is developed for solving the 2E-IRP. A labeling algorithm is used to solve several pricing subproblems associated with each period and intermediate facility. We generate 400 instances and obtain optimal solutions for 116 instances, and good upper bounds for 60 instances with a gap of less than 5% (with an average of 2.8%). Variations of the algorithm could solve 7 more instances to optimality. We provide comprehensive analyses to evaluate the performance of our solution approach.