A Generalized Benders Decomposition based algorithm for an inventory location problem with stochastic inventory capacity constraints

This paper deals with an inventory location problem with order quantity and stochastic inventory capacity constraints, which aims to address strategic supply chain network design problems and is of a nonlinear, nonconvex mixed integer programming nature. The problem integrates strategic supply chain networks design decisions (i.e., warehouse location and customer assignment) with tactical inventory control decisions for each warehouse (i.e., order size and reorder point). A novel decomposition approach that deals with the nonconvex nature of the problem formulation is proposed and implemented, based on the Generalized Benders Decomposition. The proposed decomposition yields a Master Problem that addresses warehouses location and customer assignment decisions, and a set of underlying Subproblems (SPs) that deal with warehouse inventory control decisions. Based on this decomposition, nonlinearity of the original problem is captured by the SPs that are solved at optimality, while the Master Problem is a mixed integer linear programming problem. The master is solved using a commercial solver, the SPs are solved analytically by inspection, and cuts to be added into the Master Problem are obtained based on Lagrangian dual information. Optimal solutions were found for 160 instances in competitive times.