IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 8, NO. 1, JANUARY 2011 243 The Base Stock/Base Backlog Control Policy for a Make-to-Stock System With Impatient Customers Angelos A. Economopoulos, Vassilis S. Kouikoglou, Senior Member, IEEE, and Evangelos Grigoroudis Abstract—We study a single-stage production system that produces one product type. The system employs a base stock policy to maintain an inven- tory of finished items and cope with random demand. During stockout pe- riods, the system incurs three types of potential customer loss: (a) balking, i.e., arriving customers may be unwilling to place orders and leave im- mediately; (b) rejection, i.e., the system rejects new customer orders if its backlog has reached a certain limit, called the base backlog; (c) reneging, i.e., outstanding customers waiting in queue may become impatient and withdraw their orders. The objective is to determine the base stock and base backlog that maximize the mean profit rate of the system. This quan- tity is estimated analytically using a finite capacity M/M/1 queueing model, in which the arrival rate is a decreasing but otherwise arbitrary function of the backlog and customer reneging times have an arbitrary but known distribution. Certain properties are established which ensure that the op- timal control parameters can be determined in finite time by exhaustive search. The model is then extended to take into account a fixed order quan- tity policy for replenishing raw material. Numerical results show that man- aging inventories and backlog jointly achieves higher profit than other con- trol policies. Note to Practitioners—Throughput, inventory levels, sales management, and due-date performance are key factors of manufacturing businesses to achieve competitiveness and profitability. Because these factors represent conflicting criteria, they call for integrated production management approaches. These issues are investigated for a simple plant, in which decisions concerning safety stocks, raw material replenishment, and order acceptance are made by considering simultaneously profit from sales, customer impatience, and inventory costs. A policy that strikes a balance among these criteria is shown to outperform other commonly used control policies. This work is a start to studying complex systems facing customer impatience. Index Terms—Admission control, customer impatience, production/ inventory control, queueing analysis. I. INTRODUCTION Production control deals with the management of inventories in pro- duction systems. Control decisions specify when to produce and when to stop producing, how to replenish raw items, and whether an arriving order is accepted or rejected. The aim of control is to minimize the costs of inventory, lost sales, and delays in filling orders. These costs represent three conflicting criteria. 1) Low inventory levels lead to small inventory carrying costs but increase the other two cost components. During stockouts, each arriving order is either lost to competitors (balking of customers) or backlogged. An outstanding customer order is either withdrawn Manuscript received December 17, 2009; accepted June 06, 2010. Date of publication July 19, 2010; date of current version January 07, 2011. This paper was recommended for publication by Associate Editor S. Bhatnagar and Editor Y. Narahari upon evaluation of the reviewers’ comments. The work of A. A. Economopoulos was supported in part by a doctoral scholarship from the Alexander S. Onassis Public Benefit Foundation, Greece. The authors are with the Department of Production Engineering and Management, Technical University of Crete, 73100 Chania, Greece (e-mail: ubatou@hotmail.com; kouik@dpem.tuc.gr; vangelis@ergasya.tuc.gr). Digital Object Identifier 10.1109/TASE.2010.2052802 if the customer becomes impatient (reneging of customers) or sat- isfied after an elapsed time; in the latter case, the system incurs a backlog cost which is proportional to the customer delay (see, e.g., [1, p. 615] for an analysis of the backlog costs). On the other hand, maintaining high inventory levels minimizes the probability that an arriving order will become a lost or a delayed sale. In this case, however, the inventory holding cost is high and may diminish the profit from sales (various components of the holding cost are pre- sented in [2, p. 34]). 2) To avoid lost sales and reduce inventory costs, a firm may adopt a complete backordering (CB) policy, whereby all customer orders during a stockout period are accepted. Yet this policy results in increased backlogs and reneging rates. 3) Finally, a lost sales (LS) policy rejects all incoming orders during stockouts, avoids backlog costs and reneging, but it requires high safety stocks in anticipation of future demand to reduce the fre- quency of stockouts. The simplest inventory control policy is one that specifies a target value for the number of finished items, called the base stock (BS) [2]–[4]. When the inventory on hand reaches the base stock, the system stops producing to avoid excessive inventory costs. During a busy period, raw items are usually delivered to the production facility one at a time, to avoid unnecessary inventory costs. Therefore, the system has a work in process (WIP) of 1 if the production facility is busy or 0 if it is stopped. This policy is known as the base stock with a unit WIP cap. More sophisticated policies have been proposed in the literature, which can cope with delays and/or costs associated with the release raw items to a production stage. Examples of such policies are the kanban system and its extensions for multi stage systems (see, e.g., [3, Chap. 10]) as well as state-dependent, raw material replenishment policies for a simple inventory/production system [5]. Most applications in the literature of inventory control assume that when arriving orders cannot be filled immediately, they are all either rejected (LS) or accepted (CB). Moreover, under CB, customers are assumed to be patient enough so that no balking or reneging takes place during stockouts. An admission policy which generalizes LS and CB is one that rejects orders when the number of unsatisfied orders reaches a certain limit, called the base backlog (BB), and accepts them otherwise. This partial backordering policy was proposed in [6] for the control of admissions to a single-machine system of the M/M/1 type, i.e., with exponential customer interarrival times and exponential service times. When BB is applied to systems with zero base stock the policy is called make-to-order (MTO). Policies combining BB and inventory control have been studied recently for production systems with one [7]–[9] or several [10] machines. All these studies assume that cus- tomers have infinite patience. In this paper, we study the combined base-stock with unit WIP cap and base backlog policy, or BS-BB for short, for a single-stage produc- tion system with both types of customer impatience, namely, balking and reneging. The aim is to determine optimal values for BS and BB, denoted, respectively, and , so as to maximize the mean profit rate of the system. Profit includes revenue from sales diminished by the costs of purchasing raw items, inventory, order rejection, reneging and de- lays in filling customer orders. In Section II, we model the system as a Markov chain and use re- sults from [11] to derive expressions for various components of the mean profit rate of the system. In Section III, we show that the op- timal and belong to bounded sets and develop an algorithm which determines them in finite time. Section IV examines a combination of 1545-5955/$26.00 © 2010 IEEE