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On the optimality equation for average cost Markov decision processes and its validity for inventory control

Author

Listed:
  • Eugene A. Feinberg

    (Stony Brook University)

  • Yan Liang

    (Stony Brook University)

Abstract

As is well known, average-cost optimality inequalities imply the existence of stationary optimal policies for Markov decision processes with average costs per unit time, and these inequalities hold under broad natural conditions. This paper provides sufficient conditions for the validity of the average-cost optimality equation for an infinite state problem with weakly continuous transition probabilities and with possibly unbounded one-step costs and noncompact action sets. These conditions also imply the convergence of sequences of discounted relative value functions to average-cost relative value functions and the continuity of average-cost relative value functions. As shown in this paper, the classic periodic-review setup-cost inventory control problem with backorders and convex holding/backlog costs satisfies these conditions. Therefore, the optimality inequality holds in the form of an equality with a continuous average-cost relative value function for this problem. In addition, the K-convexity of discounted relative value functions and their convergence to average-cost relative value functions, when the discount factor increases to 1, imply the K-convexity of average-cost relative value functions. This implies that average-cost optimal (s, S) policies for the inventory control problem can be derived from the average-cost optimality equation.

Suggested Citation

  • Eugene A. Feinberg & Yan Liang, 2022. "On the optimality equation for average cost Markov decision processes and its validity for inventory control," Annals of Operations Research, Springer, vol. 317(2), pages 569-586, October.
    • Handle: RePEc:spr:annopr:v:317:y:2022:i:2:d:10.1007_s10479-017-2561-9
    DOI: 10.1007/s10479-017-2561-9
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    References listed on IDEAS

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    1. Arthur F. Veinott, Jr. & Harvey M. Wagner, 1965. "Computing Optimal (s, S) Inventory Policies," Management Science, INFORMS, vol. 11(5), pages 525-552, March.
    2. Michael Katehakis & Laurens Smit, 2012. "On computing optimal (Q,r) replenishment policies under quantity discounts," Annals of Operations Research, Springer, vol. 200(1), pages 279-298, November.
    3. Eugene A. Feinberg & Mark E. Lewis, 2007. "Optimality Inequalities for Average Cost Markov Decision Processes and the Stochastic Cash Balance Problem," Mathematics of Operations Research, INFORMS, vol. 32(4), pages 769-783, November.
    4. Xin Chen & David Simchi-Levi, 2004. "Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Infinite Horizon Case," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 698-723, August.
    5. Dirk Beyer & Feng Cheng & Suresh P. Sethi & Michael Taksar, 2010. "Markovian Demand Inventory Models," International Series in Operations Research and Management Science, Springer, number 978-0-387-71604-6, April.
    6. Eugene A. Feinberg & Pavlo O. Kasyanov & Nina V. Zadoianchuk, 2012. "Average Cost Markov Decision Processes with Weakly Continuous Transition Probabilities," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 591-607, November.
    7. Eugene A. Feinberg & Pavlo O. Kasyanov & Michael Z. Zgurovsky, 2016. "Partially Observable Total-Cost Markov Decision Processes with Weakly Continuous Transition Probabilities," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 656-681, May.
    8. D. Beyer & S. P. Sethi, 1999. "The Classical Average-Cost Inventory Models of Iglehart and Veinott–Wagner Revisited," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 523-555, June.
    9. Junmin Shi & Michael Katehakis & Benjamin Melamed, 2013. "Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands," Annals of Operations Research, Springer, vol. 208(1), pages 593-612, September.
    10. Xin Chen & David Simchi-Levi, 2004. "Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case," Operations Research, INFORMS, vol. 52(6), pages 887-896, December.
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