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STOCHASTIC SETUP-COST INVENTORY MODEL WITH BACKORDERS AND QUASICONVEX COST FUNCTIONS

Published online by Cambridge University Press:  04 April 2019

Eugene A. Feinberg Open the ORCID record for Eugene A. Feinberg [Opens in a new window]  and
Yan Liang
Eugene A. Feinberg
Affiliation:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY11794, USA E-mail: eugene.feinberg@stonybrook.edu, yan.liang@stonybrook.edu
Yan Liang
Affiliation:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY11794, USA E-mail: eugene.feinberg@stonybrook.edu, yan.liang@stonybrook.edu
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Abstract

This paper studies a periodic-review single-commodity setup-cost inventory model with backorders and holding/backlog costs satisfying quasiconvexity assumptions. We show that the Markov decision process for this inventory model satisfies the assumptions that lead to the validity of optimality equations for discounted and average-cost problems and to the existence of optimal (s, S) policies. In particular, we prove the equicontinuity of the family of discounted value functions and the convergence of optimal discounted lower thresholds to the optimal average-cost lower threshold for some sequence of discount factors converging to 1. If an arbitrary nonnegative amount of inventory can be ordered, we establish stronger convergence properties: (i) the optimal discounted lower thresholds converge to optimal average-cost lower threshold; and (ii) the discounted relative value functions converge to average-cost relative value function. These convergence results previously were known only for subsequences of discount factors even for problems with convex holding/backlog costs. The results of this paper also hold for problems with fixed lead times.

Keywords

average-cost optimality equationsinventory controlrelative value functions(s, S) policies
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Special Issue 3: Analytic Methods in Health Care and in Clinical Trials In Honor and Memory of Lycourgos (Lee) Papayanopoulos , July 2020 , pp. 429 - 468
Copyright
Copyright © Cambridge University Press 2019

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References

1Bishop, C.J., Feinberg, E.A., & Zhang, J. (2014). Examples concerning Abel and Cesàro limits. Journal of Mathematical Analysis and Applications, 420(2): 1654–1661.CrossRefGoogle Scholar
2Beyer, D., Cheng, F., Sethi, S.P., & Taksar, M. (2010). Markovian demand inventory models. New York: Springer.CrossRefGoogle Scholar
3Beyer, D. & Sethi, S. (1999). The classical average-cost inventory models of Iglehart and Veinott—Wagner revisited. Journal of Optimization Theory and Applications, 101(3): 523–555.CrossRefGoogle Scholar
4Chen, X. & Simchi-Levi, D. (2004). Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case. Operations Research, 52(6): 887–896.CrossRefGoogle Scholar
5Chen, X. & Simchi-Levi, D. (2004). Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the infinite horizon case. Mathematics of Operations Research, 29(3): 698–723.CrossRefGoogle Scholar
6Feinberg, E.A. (2016). Optimality conditions for inventory control. In Gupta, A., & Capponi, A., (eds.), Tutorials in operations research. Optimization challenges in complex, networked, and risky systems, Cantonsville, MD: INFORMS, pp. 14–44.CrossRefGoogle Scholar
7Feinberg, E.A., Kasyanov, P.O., & Zadoianchuk, N.V. (2012). Average cost Markov decision processes with weakly continuous transition probability. Mathematics of Operations Research, 37(4): 591–607.CrossRefGoogle Scholar
8Feinberg, E.A., Kasyanov, P.O., & Zadoianchuk, N.V. (2013). Berge's theorem for noncompact image sets. Journal of Mathematical Analysis and Applications, 397(1): 255–259.CrossRefGoogle Scholar
9Feinberg, E.A., Kasyanov, P.O., & Zgurovsky, M.Z. (2016). Partially observable total-cost Markov decision processes with weakly continuous transition probabilities. Mathematics of Operations Research, 41(2): 656–681.CrossRefGoogle Scholar
10Feinberg, E.A. & Lewis, M.E. (2007). Optimality inequalities for average cost Markov decision processes and the stochastic cash balance problem. Mathematics of Operations Research, 32(4): 769–783.CrossRefGoogle Scholar
11Feinberg, E.A. & Lewis, M.E. (2017). On the convergence of optimal actions for Markov decision processes and the optimality of (s, S) inventory policies. Naval Research Logistic, DOI: 10.1002/nav.21750.Google Scholar
12Feinberg, E.A. & Liang, Y. (2017). Structure of optimal policies to periodic-review inventory models with convex costs and backorders for all values of discount factors. Annals of Operations Research, DOI: 10.1007/s10479-017-2548-6.CrossRefGoogle Scholar
13Feinberg, E.A. & Liang, Y. (2017). On the optimality equation for average cost Markov decision processes and its validity for inventory control. Annals of Operations Research, DOI: 10.1007/s10479-017-2561-9.CrossRefGoogle Scholar
14Goldberg, D.A., Katz-Rogozhnikov, D.A., Lu, Y., Sharma, M., & Squillante, M.S. (2016). Asymptotic optimality of constant-order policies for lost sales inventory models with large lead times. Mathematics of Operations Research, 41(3): 898–913.CrossRefGoogle Scholar
15Hernández-Lerma, O. (1989). Adaptive Markov Control Processes. New York: Springer-Verlag.CrossRefGoogle Scholar
16Hernández-Lerma, O. & Lasserre, J.B. (1996). Discrete-time Markov control processes: basic optimality creteria. New York: Springer-Verlag.CrossRefGoogle Scholar
17Huh, W.T., Janakiraman, G., & Nagarajan, M. (2011). Average cost single-stage inventory models: an analysis using a vanishing discount approach. Operations Research, 59(1): 143–155.CrossRefGoogle Scholar
18Iglehart, D.L. (1963). Dynamic programming and stationary analysis of inventory roblems. In Scarf, H., Gilford, D., & Shelly, M., (eds.), Multistage inventory control models and techniques Stanford, CA: Stanford University Press, pp. 1–31.Google Scholar
19Johnson, E.L. (1968). On (s, S) policies. Management Science, 15(1): 80–101.CrossRefGoogle Scholar
20Porteus, E. (2002). Foundations of stochastic inventory theory. Stanford, CA: Stanford University Press.Google Scholar
21Resnick, S.I. (1992). Adventures in stochastic processes. Boston: Birkhauser.Google Scholar
22Scarf, 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: Stanford University Press.Google Scholar
23Schäl, M. (1993). Average optimality in dynamic programming with general state space. Mathematics of Operations Research, 18(1): 163–172.CrossRefGoogle Scholar
24Simchi-Levi, D., Chen, X., & Bramel, J. (2005). The Logic of Logistics: Theory, Algorithms, and Applications for Logistics and Supply Chain Management. New York: Springer-Verlag.Google Scholar
25Veinott, A.F. (1966). On the optimality of (s, S) inventory policies: new condition and a new proof. SIAM Journal on Applied Mathematics, 14(5): 1067–1083.CrossRefGoogle Scholar
26Veinott, A.F. & Wagner, H.M. (1965). Computing optimal (s, S) policies. Management Science, 11(5): 525–552.CrossRefGoogle Scholar
27Zabel, E. (1962). A note on the optimality of (s, S) policies in inventory theory. Management Science, 9(1): 123–125.CrossRefGoogle Scholar
28Zheng, Y. (1991). A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems. Journal of Applied Probability, 28(4): 802–810.CrossRefGoogle Scholar
29Zipkin, P.H. (2000). Foundations of inventory management. New York: McGraw-Hill.Google Scholar