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Technical Note—On the Optimality of Reflection Control

Author

Listed:
  • Jiankui Yang

    (School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China)

  • David D. Yao

    (Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027)

  • Heng-Qing Ye

    (Faculty of Business, Hong Kong Polytechnic University, Hong Kong, China)

Abstract

The goal of this paper is to illustrate the optimality of reflection control in three different settings, to bring out their connections and to contrast their distinctions. First, we study the control of a Brownian motion with a negative drift, so as to minimize a long-run average cost objective. We prove the optimality of the reflection control, which prevents the Brownian motion from dropping below a certain level by cancelling out from time to time part of the negative drift; and we show that the optimal reflection level can be derived as the fixed point that equates the long-run average cost to the holding cost. Second, we establish the asymptotic optimality of the reflection control when it is applied to a discrete production-inventory system driven by (delayed) renewal processes; and we do so via identifying the limiting regime of the system under diffusion scaling. Third, in the case of controlling a birth–death model, we establish the optimality of the reflection control directly via a linear programming–based approach. In all three cases, we allow an exponentially bounded holding cost function, which appears to be more general than what’s allowed in prior studies. This general cost function reveals some previously unknown technical fine points on the optimality of the reflection control, and extends significantly its domain of applications.

Suggested Citation

  • Jiankui Yang & David D. Yao & Heng-Qing Ye, 2020. "Technical Note—On the Optimality of Reflection Control," Operations Research, INFORMS, vol. 68(6), pages 1668-1677, November.
    • Handle: RePEc:inm:oropre:v:68:y:2020:i:6:p:1668-1677
    DOI: 10.1287/opre.2019.1935
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    References listed on IDEAS

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    1. J. Michael Harrison & Michael I. Taksar, 1983. "Instantaneous Control of Brownian Motion," Mathematics of Operations Research, INFORMS, vol. 8(3), pages 439-453, August.
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    3. Jingchen Wu & Xiuli Chao, 2014. "Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 163-189, February.
    4. Andrew Jack & Mihail Zervos, 2006. "A singular control problem with an expected and a pathwise ergodic performance criterion," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-19, June.
    5. 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.
    6. 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.
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    Cited by:

    1. Federico, Salvatore & Ferrari, Giorgio & Rodosthenous, Neofytos, 2021. "Two-Sided Singular Control of an Inventory with Unknown Demand Trend," Center for Mathematical Economics Working Papers 643, Center for Mathematical Economics, Bielefeld University.
    2. Salvatore Federico & Giorgio Ferrari & Neofytos Rodosthenous, 2021. "Two-sided Singular Control of an Inventory with Unknown Demand Trend (Extended Version)," Papers 2102.11555, arXiv.org, revised Nov 2022.

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