Integrated location-inventory modelling under forward and reverse product flows in the used merchandise retail sector: A multi-echelon formulation

This study presents a joint three-echelon location inventory model for a donation-demand driven industry in which the main warehouse (MW), distribution centers (DC), retail stores (RS) and donation-only centers (ADCs) exist. This unique inventory-location problem involves demand and supply uncertainties, coverage radius limitations, service level requirements, and multiple products consideration. Each retailer has two classes of products flowing from the assigned DC due to demands minus donations occurring in that retailer. The proposed model simultaneously determines the number of DCs to open, DC locations, and assignments of retailers to the open DCs for particular product types. The objective is to minimize the total annual cost including: facility location costs, transportation costs, inventory costs, and the lost sale costs. Due to the complexity of the problem, the proposed model structure allows for relaxing complicating constraints through recourse to Lagrangian relaxation. The use of robust branch-cut and price heuristics solves the mixed integer nonlinear problem to obtain a lower bound and a distance-based heuristic to get an upper bound. We formulate essential features of this novel problem, solve several numerical example problems and evaluate solution performance. We believe this is a novel problem environment, and that this initial study extends integrated location-inventory modeling to a new context.