A new Lagrangian-Benders approach for a concave cost supply chain network design problem

Abstract

This article presents an extended facility location model for multiproduct supply chain network design, that accounts for concave capacity, transportation, and inventory costs (induced by economies of scale, quantity discounts and risk pooling, respectively). The problem is formulated as a mixed-integer nonlinear program (MINLP) with linear constraints and a large number of separable concave terms in the objective function. We propose a solution approach that combines stabilized Lagrangian relaxation with a novel Benders decomposition. Lagrangian relaxation is used first to decompose the problem by potential warehouse location into low-rank concave minimization subproblems. Benders decomposition is then applied to solve the subproblems by shifting the concave terms to a low-dimensional master problem that is solved effectively through implicit enumeration. Finally, a feasible solution for the original problem is constructed from the partial subproblems solutions by solving a restricted set covering problem. The proposed approach is tested on several instances from the literature and compared against a state-of-the-art solver. Furthermore, a realistic case study is used to demonstrate the impact of concavities in the cost components with extensive sensitivity analyses.