An efficient search algorithm for obtaining the optimal replenishment strategies in multi-stage just-in-time supply chain systems

In this study, we focus on the coordination of the replenishment activities among all firms in a multi-stage supply chain system in which Just-In-Time (JIT) philosophy applies. Similar to all JIT systems, kanban is used to control information flows and material flows between two adjacent manufacturing plants, which is called a kanban stage. In this supply chain, suppliers provide raw materials to the manufacturing plant in the first kanban stage and end-products are delivered from the plant in the last kanban stage to the warehouses or directly to the customers. Studies in the literature formulated this problem as a Mixed-Integer Non-Linear Program (MINLP). The objective of this study is to propose an efficient search algorithm that solves an optimal solution for this MINLP model. We conduct theoretical analysis on the optimality structure of the model and assert that the optimal cost curve is piece-wise convex. The characteristics of the optimal cost curve, especially the "junction points'', provide essential foundation to the design of our search algorithm. Also, in order to reinforce the effectiveness of the proposed search algorithm, we utilize a relaxation of the problem to obtain tight bounds on the search range. Using randomly generated instances in our numerical experiments, we demonstrated that the proposed search algorithm outperforms a previously published heuristic.