An Integrated Inventory Model for Three-tier Supply Chain Systems

--In this paper, we consider an integrated inventory model with one vendor and multiple retailers. In this three-tier supply chain system, the vendor purchases raw material produces into finished products, and delivers finished products to the retailers. We assume that the production (at the vendor) and the replenishment (at all the retailers) of the finished products share the common cycle time (T), but the replenishment cycle of raw material for the vendor is an integer multiple of T. The focus of this study is to solve the optimal common cycle to minimize the average joint total costs (AJTC) for the vendor in the whole supply chain system. To solve this problem, we derive the expression for the AJTC, and analyze the theoretical properties of the optimal AJTC curve. We show that the optimal AJTC curve is piece-wise convex with respect to T, and the junction points on the optimal AJTC curve can be easily located by a closed-form formula. By utilizing our theoretical properties, we propose an efficient search algorithm for solving this problem. Our random experiments demonstrate that our search algorithm effectively obtains the optimal solution, and interestingly, the vendor could gain more cost saving when more retailers join the three-tier supply chain system.