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Optimal Strategies in a Production Inventory Control Model

Author

Listed:
  • Pablo Azcue

    (Universidad Torcuato Di Tella)

  • Esther Frostig

    (Institute of Technology
    University of Haifa)

  • Nora Muler

    (Universidad Torcuato Di Tella)

Abstract

We consider a production-inventory control model with finite capacity and two different production rates, assuming that the cumulative process of customer demand is given by a compound Poisson process. It is possible at any time to switch over from the different production rates but it is mandatory to switch-off when the inventory process reaches the storage maximum capacity. We consider holding, production, shortage penalty and switching costs. This model was introduced by Doshi, Van Der Duyn Schouten and Talman in 1978. In their paper they found a formula for the long-run average expected cost per unit time as a function of two critical levels, in this paper we consider expected discounted cumulative costs instead. We seek to minimize this discounted cost over all admissible switching strategies. We show that the optimal cost functions for the different production rates satisfy the corresponding Hamilton-Jacobi-Bellman system of equations in a viscosity sense and prove a verification theorem. The way in which the optimal cost functions solve the different variational inequalities gives the switching regions of the optimal strategy, hence it is stationary in the sense that depends only on the current production rate and inventory level. We define the notion of finite band strategies and derive, using scale functions, the formulas for the different costs of the band strategies with one or two bands. We also show that there are examples where the switching strategy with two critical levels is not optimal.

Suggested Citation

  • Pablo Azcue & Esther Frostig & Nora Muler, 2023. "Optimal Strategies in a Production Inventory Control Model," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-43, March.
  • Handle: RePEc:spr:metcap:v:25:y:2023:i:1:d:10.1007_s11009-023-10024-3
    DOI: 10.1007/s11009-023-10024-3
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    References listed on IDEAS

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    1. (Ai-Chih) Chang, Jasmine & Lu, Haibing & (Junmin) Shi, Jim, 2019. "Stockout risk of production-inventory systems with compound Poisson demands," Omega, Elsevier, vol. 83(C), pages 181-198.
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    3. A. G. de Kok, 1985. "Approximations for a Lost-Sales Production/Inventory Control Model with Service Level Constraints," Management Science, INFORMS, vol. 31(6), pages 729-737, June.
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    5. Said Hamadène & Monique Jeanblanc, 2007. "On the Starting and Stopping Problem: Application in Reversible Investments," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 182-192, February.
    6. Erhan Bayraktar & Masahiko Egami, 2010. "On the One-Dimensional Optimal Switching Problem," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 140-159, February.
    7. Stephen C. Graves & Julian Keilson, 1981. "The Compensation Method Applied to a One-Product Production/Inventory Problem," Mathematics of Operations Research, INFORMS, vol. 6(2), pages 246-262, May.
    8. Yonit Barron & David Perry & Wolfgang Stadje, 2016. "A make-to-stock production/inventory model with MAP arrivals and phase-type demands," Annals of Operations Research, Springer, vol. 241(1), pages 373-409, June.
    9. B. T. Doshi & F. A. van der Duyn Schouten & J. J. Talman, 1978. "A Production-Inventory Control Model with a Mixture of Back-Orders and Lost-Sales," Management Science, INFORMS, vol. 24(10), pages 1078-1087, June.
    10. Jim (Junmin) Shi & Michael N. Katehakis & Benjamin Melamed & Yusen Xia, 2014. "Production-Inventory Systems with Lost Sales and Compound Poisson Demands," Operations Research, INFORMS, vol. 62(5), pages 1048-1063, October.
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    1. Brecht Verbeken & Marie-Anne Guerry, 2024. "Attainability for Markov and Semi-Markov Chains," Mathematics, MDPI, vol. 12(8), pages 1-14, April.

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