A Queueing-Inventory System with Repair Center for Defective Items and One-for-One Ordering Policy

Document Type : Research Paper


1 Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California, USA

2 Department of Industrial Engineering,Sharif University of Technology, Tehran, Iran


In this paper we consider a system consisting of a supplier with a single processing unit, a repair center, and a retailer with Poisson demand. We assume that the retailer applies one-for-one ordering policy with backorders for his inventory control. The retailer’s orders form a queue in the supplier processing unit. We also assume that a certain fraction of the products produced by the supplier are defective and they must be repaired in the repair center before going to the retailer. Further, we assume that the processing time of each unit at the supplier and the service time of each defective item in the repair center are exponentially distributed random variables with known means. The purpose of this paper is to obtain the optimal value of the inventory position of the retailer which minimizes the total cost of the system. To achieve this purpose we consider two cases, Case (1) the ratio of the arrival rate to service rate, at the supplier and at the repair center are not equal and Case (2) these ratios are equal. For both cases, we first derive the long run probability distribution 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. We also investigate the convexity of this total system cost function and obtain the optimal value of the inventory position of the retailer and present a numerical example.


Main Subjects

[1] Berman O., Kim E. (2001), Dynamic order replenishment policy in internet-based supply chains;
Mathematics Methods of Operations Research. 53; 371–390.
[2] Hadley G., Whitin T.M. (1963), Analysis of Inventory Systems; Prentice Hall Inc.
[3] Haji R., Saffari M., Haji A. (2011), Queueing Inventory System in a Two-level Supply Chain with Onefor-
One Ordering Policy; Journal of Industrial and System Engineering, 5 (1); 52-62.
[4] He Q.M., Jewkes E.M., Buzacott J. (2002), Optimal and near-optimal inventory policies for a make to
order inventory-production system; European Journal of Operational Research 141; 113-132.
[5] Liu L., Liu X., Yao D.D. (2004), Analysis and Optimization of a Multistage Inventory-Queue System;
Management Science 50 (3); 365-380.
[6] Olsson R. J., Hill R.M. (2007), A two-echelon base-stock inventory model with Poisson demand and the
sequential processing of orders at the upper echelon; European Journal of Operational Research 177;
[7] Ross S. M. (2010), Introduction to Probability Modes; Academic Press 10th Edition.
[9] Saffari M., Haji R., Hassanzadeh F. (2011), A queueing system with inventory and mixed exponentially
distributed lead times; International Journal of Advance ManufacturingTechnology 53; 1231-1237
[8] Schwarz M., Daduna H. (2006), Queueing systems with inventory management with random lead times
and with backordering; Mathematical Methods of Operations Research 64; 383–414.
[9] Schwarz M., Sauer C., Daduna H., Kulik R., Szekli R. (2006), M/M/1 queueing systems with inventory;
Queueing System 54; 55–78.
[10] Wang Y., Cohen M.A., Zheng Y.S. (2000), A two echelon repairable inventory system with stockingcenter-dependent depot replenishment lead times; Management Science 46 (11); 1441-1453.
[11] Zhao N., Lian Z. (2011), Aqueuing-inventory system with two classes of customers; International
Journal of Production Economics 129; 225–231.