Joint Lagrangian Relaxation and Dynamic Programming Approach for a Multi-Period Multi-Product Location-Inventory Problem

We propose a multi-period multi-product location-inventory model considering an (R, s, S) periodic review policy and modular stochastic capacity constraints. The objective is to define which regional warehouses to open, maintain, operate, or close, which products to assign, and which customers should be supplied by the chosen distribution centers. Additionally, it involves setting order sizes and reorder points to minimize system costs while meeting service level conditions. This problem is structured as a non-convex mixed integer nonlinear programming model. We propose a Lagrangian relaxation algorithm and the subgradient approach as a solution method. We relax the customer allocation, distribution center variance, and demand constraints. Then, the relaxed problem is decomposed into location subproblems for each warehouse, in which inventory subproblems are embedded. So, each location subproblem is discomposed for each period and exactly solved through dynamic programming. Computational experiments prove that the presented approach gives solutions close to optimal and low gaps in a low computational time. Additionally, it exhibits meaningful incidences in the decisions on inventory control policy, facility location decisions, total costs, and risk pooling effects for different review intervals.