Optimizing a multi-echelon location-inventory problem with joint replenishment: A Lipschitz ϵ-optimal approach using Lagrangian relaxation

This paper investigates a supply network design problem for cross-border e-commerce. The supply chain consists of suppliers, regional distribution centers, and distribution centers. Our goal is to determine simultaneously location decisions of regional distribution centers, and the inventory decisions for coordinated replenishment and delivery. We formulate this problem as a mixed-integer non-linear program, and develop a Lipschitz optimization algorithm and an iterative heuristic to solve it. Moreover, we propose a tight lower bound via constructing a dual problem. Computational results show that both the Lipschitz optimization method and the heuristic can produce a near-optimal solution with a gap below 1%. Finally, we analyze the influence of integrated decision making and capacity constraints. An interesting finding is that the capacity of regional distribution centers has a critical influence, while the capacity of distribution centers is trivial. Integrated decision making can achieve remarkable cost savings only if the capacity constraints of regional distribution centers are loose. Nevertheless, capacity constraints of distribution centers make almost no impact. This result can be a good guide for a company’s investment plan: one should give top priority to regional distribution centers, and give only low priority to distribution centers.