A mixed-integer program and a Lagrangian-based decomposition algorithm for the supply chain network design with quantity discount and transportation modes

Abstract

We study the design of a two-echelon supply chain network in the presence of suppliers’ all-unit quantity discount and transportation mode selection. The problem involves determining the best location for plants and the allocation of customers to open plants. The problem additionally entails making decisions for the order quantity from each supplier for each plant and accordingly selecting the best transportation mode that can accommodate these order quantities among echelons. The objective is to minimize the total cost associated with fixed opening and operating costs of plants, fixed and variable costs of transportation modes, and purchasing costs of raw materials. To characterize and solve this problem, we develop a mixed-integer programming (MIP) model. We demonstrate that the MIP model has a special mathematical structure that makes it amenable to decomposition techniques. We, therefore, exploit this decomposable structure and develop an effective Lagrangian-based decomposition for solving the MIP. Our Lagrangian Relaxation (LR) method relaxes the complicating constraints associated with commodity flow conservation among echelons in the MIP, yielding more tractable subproblems, one for each echelon. Solutions obtained from the relaxed problem may be infeasible, e.g., the demand for some of the customers may not be satisfied. We remedy these subproblems’ infeasibilities using novel feasibility algorithms and appropriate Lagrangian multipliers that penalize constraints’ violations in the subproblems, leading the algorithm towards global feasibility/optimality. We appraise the performance of the MIP model and the LR algorithm on instances of varying sizes. We show that the MIP model solved via CPLEX finds integer feasible solutions for 42% of large problem instances and its average optimality gap for these solved instances is 64.56%. The LR algorithm significantly improves the solvability and optimality gap of the MIP model and finds integer feasible solutions for 100% of problem instances and achieves an average optimality gap of 1.78%. We investigate the robustness of our algorithm by conducting sensitivity analyses on the model parameters. We demonstrate that the LR technique remains robust and tractable with respect to various parameters’ values.