Single Item Multi-period Multi-retailer Inventory Replenishment Problem with Restricted Order Size

In this paper, we discuss the Multi-Period Multi-Retailer Inventory Replenishment Problem (MPMRIRP) denned by the following features: The planning horizon is T periods. There are n retailers denoted by J 1 ≤ J ≤ n and a central warehouse denoted by the index 0. Let pj,t, and Kj,t, denote respectively the purchasing price per unit and the fixed ordering cost of the item for warehouse/retailer j 0 ≤ j ≤ n, during period t 1 ≤ t ≤ T. Let Dj,t, be the demand of customer j, during period t. The warehouse replenishes its stock through bulk purchase and supplies to all the n retailers to satisfy their demands. The ordering cost is incurred whenever the order quantity is positive. Furthermore, a holding cost is incurred at the rate of hj,t, per unit of item held in inventory, at the end of period t, with the retailer j/warehouse, 0 ≤ j ≤ n. The order quantity is restricted to be less than or equal to Wj for retailer j/warehouse, 0 ≤ j ≤ n. We will assume that the initial and final physical inventory of the item at the stores (retail and warehouse) is zero. Stock out situations are not allowed at any of the retailers or warehouse. The problem is to determine the replenishment schedule for the single item for all the retailers and the warehouse so as to minimize the total cost incurred during the planning horizon of T periods.