A Queueing-inventory System with Defective Items and Poisson Demand

Abstract In this paper we consider a system consisting of a retailer, a supplier with a single processing unit, and a repair center. We assume that the retailer faces a Poisson demand, his ordering cost is zero or negligible, and for his inventory control he applies one-for-one ordering policy, that is, as soon as a demand arrives he orders one unit to the supplier. We also assume that when a demand arrives and the retailer is out of stock, this demand will be backordered. The retailer’s orders form a queue in the supplier processing unit. Further, we assume that a known fraction of the products produced by the supplier are defective. Defective items must be repaired in the repair center before going to the retailer. The processing time of supplier and service time of repair center are both exponentially distributed random variables with known means. In this paper we first derive the steady state probability of the number of outstanding orders of the retailer. Then we obtain the average on hand inventory and backorders of the retailer, and derive the long run unit total cost of the system. Finally, we investigate the convexity of the total system cost function and obtain the optimal value of the inventory position of the retailer which minimizes the total system cost.