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Some Monotonicity Results for Partially Observed Markov Decision Processes

Author

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  • William S. Lovejoy

    (Stanford University, Stanford, California)

Abstract

This paper provides sufficient conditions for the optimal value in a discrete-time, finite, partially observed Markov decision process to be monotone on the space of state probability vectors ordered by likelihood ratios. The paper also presents sufficient conditions for the optimal policy to be monotone in a simple machine replacement problem, and, in the general case, for the optimal policy to be bounded from below by an easily calculated monotone function.

Suggested Citation

  • William S. Lovejoy, 1987. "Some Monotonicity Results for Partially Observed Markov Decision Processes," Operations Research, INFORMS, vol. 35(5), pages 736-743, October.
    • Handle: RePEc:inm:oropre:v:35:y:1987:i:5:p:736-743
    DOI: 10.1287/opre.35.5.736
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    Citations

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    Cited by:

    1. Wooseung Jang & J. George Shanthikumar, 2002. "Stochastic allocation of inspection capacity to competitive processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(1), pages 78-94, February.
    2. Hao Zhang, 2010. "Partially Observable Markov Decision Processes: A Geometric Technique and Analysis," Operations Research, INFORMS, vol. 58(1), pages 214-228, February.
    3. Sahin, Izzet & Zahedi, Fatemeh (Mariam), 2000. "Optimal policies under risk for changing software systems based on customer satisfaction," European Journal of Operational Research, Elsevier, vol. 123(1), pages 175-194, May.
    4. Chiel van Oosterom & Lisa M. Maillart & Jeffrey P. Kharoufeh, 2017. "Optimal maintenance policies for a safety‐critical system and its deteriorating sensor," Naval Research Logistics (NRL), John Wiley & Sons, vol. 64(5), pages 399-417, August.
    5. van Staden, Heletjé E. & Boute, Robert N., 2021. "The effect of multi-sensor data on condition-based maintenance policies," European Journal of Operational Research, Elsevier, vol. 290(2), pages 585-600.
    6. James T. Treharne & Charles R. Sox, 2002. "Adaptive Inventory Control for Nonstationary Demand and Partial Information," Management Science, INFORMS, vol. 48(5), pages 607-624, May.
    7. Jingyu Zhang & Brian T. Denton & Hari Balasubramanian & Nilay D. Shah & Brant A. Inman, 2012. "Optimization of Prostate Biopsy Referral Decisions," Manufacturing & Service Operations Management, INFORMS, vol. 14(4), pages 529-547, October.
    8. L M Maillart & T G Yeung & Z Gozde Icten, 2011. "Selecting test sensitivity and specificity parameters to optimally maintain a degrading system," Journal of Risk and Reliability, , vol. 225(2), pages 131-139, June.
    9. Oussama Habachi & Yezekael Hayel & Rachid El-Azouzi, 2018. "Optimal energy-delay tradeoff for opportunistic spectrum access in cognitive radio networks," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 67(4), pages 763-780, April.
    10. Vikram Krishnamurthy & Sujay Bhatt, 2015. "Sequential Detection of Market shocks using Risk-averse Agent Based Models," Papers 1511.01965, arXiv.org.
    11. Lu Jin & Undarmaa Bayarsaikhan & Kazuyuki Suzuki, 2016. "Optimal control limit policy for age-dependent deteriorating systems under incomplete observations," Journal of Risk and Reliability, , vol. 230(1), pages 34-43, February.
    12. Saghafian, Soroush, 2018. "Ambiguous partially observable Markov decision processes: Structural results and applications," Journal of Economic Theory, Elsevier, vol. 178(C), pages 1-35.
    13. Deep, Akash & Zhou, Shiyu & Veeramani, Dharmaraj & Chen, Yong, 2023. "Partially observable Markov decision process-based optimal maintenance planning with time-dependent observations," European Journal of Operational Research, Elsevier, vol. 311(2), pages 533-544.
    14. Sarang Deo & Seyed Iravani & Tingting Jiang & Karen Smilowitz & Stephen Samuelson, 2013. "Improving Health Outcomes Through Better Capacity Allocation in a Community-Based Chronic Care Model," Operations Research, INFORMS, vol. 61(6), pages 1277-1294, December.
    15. Miehling, Erik & Teneketzis, Demosthenis, 2020. "Monotonicity properties for two-action partially observable Markov decision processes on partially ordered spaces," European Journal of Operational Research, Elsevier, vol. 282(3), pages 936-944.
    16. Burhaneddin Sandıkçı & Lisa M. Maillart & Andrew J. Schaefer & Mark S. Roberts, 2013. "Alleviating the Patient's Price of Privacy Through a Partially Observable Waiting List," Management Science, INFORMS, vol. 59(8), pages 1836-1854, August.
    17. Hao Zhang & Weihua Zhang, 2023. "Analytical Solution to a Partially Observable Machine Maintenance Problem with Obvious Failures," Management Science, INFORMS, vol. 69(7), pages 3993-4015, July.
    18. Jue Wang & Chi-Guhn Lee, 2015. "Multistate Bayesian Control Chart Over a Finite Horizon," Operations Research, INFORMS, vol. 63(4), pages 949-964, August.
    19. Junbo Son & Yeongin Kim & Shiyu Zhou, 2022. "Alerting patients via health information system considering trust-dependent patient adherence," Information Technology and Management, Springer, vol. 23(4), pages 245-269, December.
    20. Lisa M. Maillart & Ludmila Zheltova, 2007. "Structured maintenance policies on interior sample paths," Naval Research Logistics (NRL), John Wiley & Sons, vol. 54(6), pages 645-655, September.
    21. Li, Weiyu & Denton, Brian T. & Morgan, Todd M., 2023. "Optimizing active surveillance for prostate cancer using partially observable Markov decision processes," European Journal of Operational Research, Elsevier, vol. 305(1), pages 386-399.
    22. Vikram Krishnamurthy & Udit Pareek, 2015. "Myopic Bounds for Optimal Policy of POMDPs: An Extension of Lovejoy’s Structural Results," Operations Research, INFORMS, vol. 63(2), pages 428-434, April.
    23. M. Reza Skandari & Steven M. Shechter, 2021. "Patient-Type Bayes-Adaptive Treatment Plans," Operations Research, INFORMS, vol. 69(2), pages 574-598, March.
    24. Vikram Krishnamurthy & Bo Wahlberg, 2009. "Partially Observed Markov Decision Process Multiarmed Bandits---Structural Results," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 287-302, May.
    25. James E. Smith & Canan Ulu, 2017. "Risk Aversion, Information Acquisition, and Technology Adoption," Operations Research, INFORMS, vol. 65(4), pages 1011-1028, August.

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