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Optimal Control of a Mean-Reverting Inventory

Author

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  • Abel Cadenillas

    (Department of Finance and Management Science, and Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada)

  • Peter Lakner

    (Department of Information, Operations, and Management Sciences, Stern School of Business, New York University, New York, New York 10012)

  • Michael Pinedo

    (Department of Information, Operations, and Management Sciences, Stern School of Business, New York University, New York, New York 10012)

Abstract

Motivated by empirical observations, we assume that the inventory level of a company follows a mean-reverting process. The objective of the management is to keep this inventory level as close as possible to a given target; there is a running cost associated with the difference between the actual inventory level and the target. If inventory deviates too much from the target, management may perform an intervention in the form of either a purchase or a sale of an amount of the goods. There are fixed and proportional costs associated with each intervention. The objective of this paper is to find the optimal inventory levels at which interventions should be performed as well as the magnitudes of the interventions to minimize the total cost. We solve this problem by applying the theory of stochastic impulse control. Our analysis yields the optimal policy, which at times exhibits a behavior that is not intuitive.

Suggested Citation

  • Abel Cadenillas & Peter Lakner & Michael Pinedo, 2010. "Optimal Control of a Mean-Reverting Inventory," Operations Research, INFORMS, vol. 58(6), pages 1697-1710, December.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:6:p:1697-1710
    DOI: 10.1287/opre.1100.0835
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    References listed on IDEAS

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    1. Abel Cadenillas & Tahir Choulli & Michael Taksar & Lei Zhang, 2006. "Classical And Impulse Stochastic Control For The Optimization Of The Dividend And Risk Policies Of An Insurance Firm," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 181-202, January.
    2. Abel Cadenillas & Sudipto Sarkar & Fernando Zapatero, 2007. "Optimal Dividend Policy With Mean‐Reverting Cash Reservoir," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 81-109, January.
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    5. Abel Cadenillas & Fernando Zapatero, 2000. "Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rates and Reserves," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 141-156, April.
    6. J. Michael Harrison & Thomas M. Sellke & Allison J. Taylor, 1983. "Impulse Control of Brownian Motion," Mathematics of Operations Research, INFORMS, vol. 8(3), pages 454-466, August.
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    8. Melda Ormeci & J. G. Dai & John Vande Vate, 2008. "Impulse Control of Brownian Motion: The Constrained Average Cost Case," Operations Research, INFORMS, vol. 56(3), pages 618-629, June.
    9. Manaster, Steven & Mann, Steven C, 1996. "Life in the Pits: Competitive Market Making and Inventory Control," The Review of Financial Studies, Society for Financial Studies, vol. 9(3), pages 953-975.
    10. Cadenillas, Abel & Zapatero, Fernando, 1999. "Optimal Central Bank Intervention in the Foreign Exchange Market," Journal of Economic Theory, Elsevier, vol. 87(1), pages 218-242, July.
    11. George M. Constantinides & Scott F. Richard, 1978. "Existence of Optimal Simple Policies for Discounted-Cost Inventory and Cash Management in Continuous Time," Operations Research, INFORMS, vol. 26(4), pages 620-636, August.
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    Cited by:

    1. Chi Seng Pun, 2022. "Robust classical-impulse stochastic control problems in an infinite horizon," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 291-312, October.
    2. René Aïd & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2020. "Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 205-232, February.
    3. Abel Cadenillas & Ricardo Huamán-Aguilar, 2016. "Explicit formula for the optimal government debt ceiling," Annals of Operations Research, Springer, vol. 247(2), pages 415-449, December.
    4. Yan, Tingjin & Park, Kyunghyun & Wong, Hoi Ying, 2022. "Irreversible reinsurance: A singular control approach," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 326-348.
    5. Abel Cadenillas & Ricardo Huamán-Aguilar, 2020. "The Optimal Control of Government Stabilization Funds," Mathematics, MDPI, vol. 8(11), pages 1-24, November.
    6. Abel Cadenillas & Peter Lakner & Michael Pinedo, 2013. "Optimal Production Management When Demand Depends on the Business Cycle," Operations Research, INFORMS, vol. 61(4), pages 1046-1062, August.
    7. Sandun Perera & Winston Buckley, 2017. "On the existence and uniqueness of the optimal central bank intervention policy in a forex market with jumps," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(8), pages 877-885, August.
    8. Matteo Basei, 2018. "Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates," Papers 1803.08166, arXiv.org, revised Mar 2019.
    9. Lu Xiao & Huacong Ding & Yu Zhong & Chaojie Wang, 2023. "Optimal Control of Industrial Pollution under Stochastic Differential Models," Sustainability, MDPI, vol. 15(6), pages 1-16, March.
    10. Lakdere Benkherouf & Michael Johnson, 2012. "Optimality of (s, S) policies for jump inventory models," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(3), pages 377-393, December.
    11. Weerasinghe, Ananda & Zhu, Chao, 2016. "Optimal inventory control with path-dependent cost criteria," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1585-1621.
    12. Shuangchi He & Dacheng Yao & Hanqin Zhang, 2017. "Optimal Ordering Policy for Inventory Systems with Quantity-Dependent Setup Costs," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 979-1006, November.
    13. Daniel Mitchell & Haolin Feng & Kumar Muthuraman, 2014. "Impulse Control of Interest Rates," Operations Research, INFORMS, vol. 62(3), pages 602-615, June.
    14. Jinbiao Wu, 2019. "Optimal exchange rates management using stochastic impulse control for geometric Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(2), pages 257-280, April.
    15. Perera, Sandun & Gupta, Varun & Buckley, Winston, 2020. "Management of online server congestion using optimal demand throttling," European Journal of Operational Research, Elsevier, vol. 285(1), pages 324-342.
    16. Ren'e Aid & Luciano Campi & Liangchen Li & Mike Ludkovski, 2020. "An Impulse-Regime Switching Game Model of Vertical Competition," Papers 2006.04382, arXiv.org.
    17. Gupta, Varun & Perera, Sandun, 2021. "Managing surges in online demand using bandwidth throttling: An optimal strategy amid the COVID-19 pandemic," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 151(C).
    18. Matteo Basei, 2019. "Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 355-383, June.
    19. René Aïd & Luciano Campi & Liangchen Li & Mike Ludkovski, 2021. "An Impulse-Regime Switching Game Model of Vertical Competition," Dynamic Games and Applications, Springer, vol. 11(4), pages 631-669, December.

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