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  • Xiuli Chao (a1), Yifan Xu (a2) and Baimei Yang (a2)


One of the most fundamental results in inventory theory is the optimality of (s, S) policy for inventory systems with setup cost. This result is established under a key assumption of infinite ordering/production capacity. Several studies have shown that, when the ordering/production capacity is finite, the optimal policy for the inventory system with setup cost is very complicated and indeed, only partial characterization for the optimal policy is possible. In this paper, we consider a continuous review production/inventory system with finite capacity and setup cost. The demand follows a Poisson process and a demand that cannot be satisfied upon arrival is backlogged. We show that the optimal control policy has a very simple structure when the holding/shortage cost rate is quasi-convex. We also develop efficient algorithms to compute the optimal control parameters.



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1.Benjaafar, S., ElHafsi, M., & Vericourt, D. (2004). Demand allocation in multiple-product, multiple-facility, make-to-stock systems. Management Science 50: 14311448.
2.Chao, X. & Zhou, S.X. (2006). Joint inventory and pricing strategy for a stochastic continuous-review system, IIE Transactions 38: 401408.
3.Craven, B.D. (1988). Fractional programming. Berlin: Heldermann.
4.Dharmadhikari, S. & Joag-Dev, K.. (1988). Unimodality, convexity, and applications, San Diego, CA: Academic Press.
5.Federgruen, A. & Zipkin, P. (1986). An inventory model with limited production capacity and uncertain demands I. The average cost criterion. Mathematics of Operations Research 11: 193207.
6.Feng, Y. & Sun, J. (2001). Computing the optimal replenishment policy for inventory systems with random discount opportunities. Operations Research 49: 790795.
7.Feng, Y. & Xiao, B. (2002). Optimal threshold control in discrete failure-prone manufacturing systems, IEEE Transactions on Automatic Control 47: 11671174.
8.Gallego, G. (1998). New bounds and heuristics for (Q, r) policies. Management Science 44: 219233.
9.Gallego, G. & Scheller-Wolf, A. (2000). Capacitated inventory problems with fixed order costs: Some optimal policy structure. European Journal of Operational Research 126: 603613.
10.Gallego, G. & Toktay, L.B. (2004). All-or-nothing ordering under a capacity constraint. Operations Research 52: 10011002.
11.Gavish, B. & Graves, S. (1980). A one-product production/inventory problem under continuous review policy, Operations Research 28: 12281236.
12.Ha, A.Y. (1997). Optimal dynamic scheduling policy for a make-to-stock production system, Operations Research 45: 4253.
13.Heyman, D.P. (1968). Optimal operating policies for M/G/1 queuing systems, Operations Research 16: 362382.
14.Kapuscinski, R. & Tayur, S.R. (1990). Optimal policies and simulation based optimization for capacitated production inventory systems. In Tayur, S.R., Ganeshan, R., & Magazine, M. (eds.), Quantitative models for supply chain management. Kluwer: The Netherlands, pp. 740.
15.Kleinrock, L. (1975). Queueing systems, vol. 1: Theory. New York: John Wiley & Sons.
16.Li, L. (1988). A stochastic theory of the firm, Mathematics of Operations Research, 13: 447465.
17.Ohnishi, M. (1992). Policy iteration and Newton-Raphson methods for Markov decision processes under average cost criterion. Computers & Mathematics with Applications 24: 147155.
18.Puterman, M.L. (1994). Markov decision processes: discrete stochastic dynamic programming, New York: John Wiley & Sons.
19.Scarf, H. (1960). The optimality of (s,S) policies in the dynamic inventory problem. In Arrow, K., Karlin, S., & Suppes, P. (eds.), Mathematical methods in the social sciences. Stanford, CA, USA: Stanford University Press, pp. 196202.
20.Shaoxiang, C. (2004). The infinite horizon periodic review problem with setup costs and capacity. Operations Research 52: 409421
21.Shaoxiang, C. & Lambrecht. (1996). X–Y band and modified (s, S) policy. Operations Research 44: 10131019.
22.Sobel, M. (1982). The optimality of full-service policies. Operations Research 30: 636649.
23.Stidham, S. (2009). Optimal design of queueing systems. Boca Raton, FL: CRC Press.
24.Tijms, H.C. (2003). A first course in stochastic models. Chichester, UK: John Wiley & Sons, Ltd.
25.Veatch, M.H. & Wein, L.M. (1996). Scheduling a make-to-stock queue, Operations Resesarch 44: 643647.
26.Veinott, A. (1966). On the optimality of (s, S) invenotry polidies: New conditions and a new proof. SIAM Journal on Applied Mathematics 14: 10671083.
27.Wein, L.M. (1992). Dynamic scheduling of a make-to-stock queue. Operations Research 40: 724735.
28.Wijngaard, J. (1972). An inventory problem with constrained order capacity. TH-Report 72-WSK-63, Eindhoven University of Technology, Eindhoven, The Netherlands.
29.Xu, Y. & Chao, X. (2009). Dynamic pricing and inventory control for a production system with average profit criterion. Probability in Engineering and Informational Sciences 23: 489513.
30.Zheng, Y. (1992). On properties of stochastic inventory systems. Management Science 38: 87103.
31.Zheng, Y.S. & Zipkin, P. (1990). A queueing model to analyze the value of centralized inventory information, Operations Research 38: 296307.


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