IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v289y2021i3p1127-1141.html
   My bibliography  Save this article

Time (in)consistency of multistage distributionally robust inventory models with moment constraints

Author

Listed:
  • Xin, Linwei
  • Goldberg, David A.

Abstract

Recently, there has been a growing interest in developing inventory control policies which are robust to model misspecification. One approach is to posit that nature selects a worst-case distribution for any relevant stochastic primitives from some pre-specified family. Several communities have observed that a subtle phenomena known as time inconsistency can arise in this framework. In particular, it becomes possible that a policy which is optimal at time zero may not be optimal for the associated optimization problem in which the decision-maker recomputes her policy at each point in time, which has implications for implementability. If there exists a policy which is optimal for both formulations, we say that the policy is time consistent, and the problem is weakly time consistent. If every optimal policy is time consistent, we say that the problem is strongly time consistent. We study these phenomena in the context of managing an inventory over time, when only the mean, variance, and support are known for the demand at each stage. We provide several illustrative examples showing that here the question of time consistency can be quite subtle. We complement these observations by providing simple sufficient conditions for weak and strong time consistency. Although a similar phenomena was previously identified by Shapiro for the setting in which only the mean and support of the demand are known, here our model is rich enough to exhibit a variety of additional interesting behaviors.

Suggested Citation

  • Xin, Linwei & Goldberg, David A., 2021. "Time (in)consistency of multistage distributionally robust inventory models with moment constraints," European Journal of Operational Research, Elsevier, vol. 289(3), pages 1127-1141.
  • Handle: RePEc:eee:ejores:v:289:y:2021:i:3:p:1127-1141
    DOI: 10.1016/j.ejor.2020.07.041
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221720306676
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.ejor.2020.07.041?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Homem-de-Mello, Tito & Pagnoncelli, Bernardo K., 2016. "Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective," European Journal of Operational Research, Elsevier, vol. 249(1), pages 188-199.
    2. Johanna Etner & Meglena Jeleva & Jean‐Marc Tallon, 2012. "Decision Theory Under Ambiguity," Journal of Economic Surveys, Wiley Blackwell, vol. 26(2), pages 234-270, April.
    3. Ioana Popescu, 2005. "A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 632-657, August.
    4. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 5, pages 145-154, World Scientific Publishing Co. Pte. Ltd..
    5. Awi Federgruen & Ziv Katalan, 1999. "The Impact of Adding a Make-to-Order Item to a Make-to-Stock Production System," Management Science, INFORMS, vol. 45(7), pages 980-994, July.
    6. William S. Lovejoy, 1992. "Stopped Myopic Policies in Some Inventory Models with Generalized Demand Processes," Management Science, INFORMS, vol. 38(5), pages 688-707, May.
    7. R. Jagannathan, 1978. "A Minimax Ordering Policy for the Infinite Stage Dynamic Inventory Problem," Management Science, INFORMS, vol. 24(11), pages 1138-1149, July.
    8. Philip Kaminsky & Onur Kaya, 2009. "Combined make-to-order/make-to-stock supply chains," IISE Transactions, Taylor & Francis Journals, vol. 41(2), pages 103-119.
    9. Dan A. Iancu & Nikolaos Trichakis, 2014. "Pareto Efficiency in Robust Optimization," Management Science, INFORMS, vol. 60(1), pages 130-147, January.
    10. R. H. Strotz, 1955. "Myopia and Inconsistency in Dynamic Utility Maximization," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 23(3), pages 165-180.
    11. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    12. Dimitris Bertsimas & Dan A. Iancu & Pablo A. Parrilo, 2010. "Optimality of Affine Policies in Multistage Robust Optimization," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 363-394, May.
    13. Retsef Levi & Georgia Perakis & Joline Uichanco, 2015. "The Data-Driven Newsvendor Problem: New Bounds and Insights," Operations Research, INFORMS, vol. 63(6), pages 1294-1306, December.
    14. Edward Ignall & Arthur F. Veinott, Jr., 1969. "Optimality of Myopic Inventory Policies for Several Substitute Products," Management Science, INFORMS, vol. 15(5), pages 284-304, January.
    15. Aharon Ben-Tal & Boaz Golany & Arkadi Nemirovski & Jean-Philippe Vial, 2005. "Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach," Manufacturing & Service Operations Management, INFORMS, vol. 7(3), pages 248-271, February.
    16. Tomasz R. Bielecki & Igor Cialenco & Marcin Pitera, 2018. "A Unified Approach to Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 204-221, February.
    17. Wolfram Wiesemann & Daniel Kuhn & Berç Rustem, 2013. "Robust Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 153-183, February.
    18. Herbert E. Scarf, 1960. "Some remarks on bayes solutions to the inventory problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 7(4), pages 591-596, December.
    19. Roorda, Berend & Schumacher, J.M., 2007. "Time consistency conditions for acceptability measures, with an application to Tail Value at Risk," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 209-230, March.
    20. Pierre Carpentier & Jean-Philippe Chancelier & Guy Cohen & Michel Lara & Pierre Girardeau, 2012. "Dynamic consistency for stochastic optimal control problems," Annals of Operations Research, Springer, vol. 200(1), pages 247-263, November.
    21. Dimitris Bertsimas & Aurélie Thiele, 2006. "A Robust Optimization Approach to Inventory Theory," Operations Research, INFORMS, vol. 54(1), pages 150-168, February.
    22. Dan A. Iancu & Marek Petrik & Dharmashankar Subramanian, 2015. "Tight Approximations of Dynamic Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 655-682, March.
    23. Kang Boda & Jerzy Filar, 2006. "Time Consistent Dynamic Risk Measures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 169-186, February.
    24. Ahmed, Shabbir & Cakmak, Ulas & Shapiro, Alexander, 2007. "Coherent risk measures in inventory problems," European Journal of Operational Research, Elsevier, vol. 182(1), pages 226-238, October.
    25. Shapiro, Alexander, 2012. "Minimax and risk averse multistage stochastic programming," European Journal of Operational Research, Elsevier, vol. 219(3), pages 719-726.
    26. Georgia Perakis & Guillaume Roels, 2008. "Regret in the Newsvendor Model with Partial Information," Operations Research, INFORMS, vol. 56(1), pages 188-203, February.
    27. Johanna Etner & Meglena Jeleva & Jean‐Marc Tallon, 2012. "Decision Theory Under Ambiguity," Journal of Economic Surveys, Wiley Blackwell, vol. 26(2), pages 234-270, April.
    28. Garud N. Iyengar, 2005. "Robust Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 257-280, May.
    29. Jinfeng Yue & Bintong Chen & Min-Chiang Wang, 2006. "Expected Value of Distribution Information for the Newsvendor Problem," Operations Research, INFORMS, vol. 54(6), pages 1128-1136, December.
    30. De Lara, Michel & Leclère, Vincent, 2016. "Building up time-consistency for risk measures and dynamic optimization," European Journal of Operational Research, Elsevier, vol. 249(1), pages 177-187.
    31. Wenqing Chen & Melvyn Sim, 2009. "Goal-Driven Optimization," Operations Research, INFORMS, vol. 57(2), pages 342-357, April.
    32. Henri G'erard & Michel de Lara & Jean-Philippe Chancelier, 2017. "Equivalence Between Time Consistency and Nested Formula," Papers 1711.08633, arXiv.org, revised May 2019.
    33. Guillermo Gallego, 1998. "New Bounds and Heuristics for (Q, r) Policies," Management Science, INFORMS, vol. 44(2), pages 219-233, February.
    34. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    35. repec:hal:pseose:halshs-00643580 is not listed on IDEAS
    36. S. Rajagopalan, 2002. "Make to Order or Make to Stock: Model and Application," Management Science, INFORMS, vol. 48(2), pages 241-256, February.
    37. Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
    38. Aharon, Ben-Tal & Boaz, Golany & Shimrit, Shtern, 2009. "Robust multi-echelon multi-period inventory control," European Journal of Operational Research, Elsevier, vol. 199(3), pages 922-935, December.
    39. Arthur F. Veinott, Jr., 1965. "Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem," Management Science, INFORMS, vol. 12(3), pages 206-222, November.
    40. Arnab Nilim & Laurent El Ghaoui, 2005. "Robust Control of Markov Decision Processes with Uncertain Transition Matrices," Operations Research, INFORMS, vol. 53(5), pages 780-798, October.
    41. Xin Chen & Melvyn Sim & David Simchi-Levi & Peng Sun, 2007. "Risk Aversion in Inventory Management," Operations Research, INFORMS, vol. 55(5), pages 828-842, October.
    42. Williams, T. M., 1984. "Special products and uncertainty in production/inventory systems," European Journal of Operational Research, Elsevier, vol. 15(1), pages 46-54, January.
    43. Chuen-Teck See & Melvyn Sim, 2010. "Robust Approximation to Multiperiod Inventory Management," Operations Research, INFORMS, vol. 58(3), pages 583-594, June.
    44. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath & Hyejin Ku, 2007. "Coherent multiperiod risk adjusted values and Bellman’s principle," Annals of Operations Research, Springer, vol. 152(1), pages 5-22, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Roos, Ernst & Brekelmans, Ruud & van Eekelen, Wouter & den Hertog, Dick & van Leeuwaarden, Johan S.H., 2022. "Tight tail probability bounds for distribution-free decision making," European Journal of Operational Research, Elsevier, vol. 299(3), pages 931-944.
    2. Adrián Esteban-Pérez & Juan M. Morales, 2022. "Partition-based distributionally robust optimization via optimal transport with order cone constraints," 4OR, Springer, vol. 20(3), pages 465-497, September.
    3. Wouter van Eekelen & Dick den Hertog & Johan S.H. van Leeuwaarden, 2022. "MAD Dispersion Measure Makes Extremal Queue Analysis Simple," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1681-1692, May.
    4. Arrigo, Adriano & Ordoudis, Christos & Kazempour, Jalal & De Grève, Zacharie & Toubeau, Jean-François & Vallée, François, 2022. "Wasserstein distributionally robust chance-constrained optimization for energy and reserve dispatch: An exact and physically-bounded formulation," European Journal of Operational Research, Elsevier, vol. 296(1), pages 304-322.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Andrew J. Keith & Darryl K. Ahner, 2021. "A survey of decision making and optimization under uncertainty," Annals of Operations Research, Springer, vol. 300(2), pages 319-353, May.
    2. Dan A. Iancu & Marek Petrik & Dharmashankar Subramanian, 2015. "Tight Approximations of Dynamic Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 655-682, March.
    3. Bakker, Hannah & Dunke, Fabian & Nickel, Stefan, 2020. "A structuring review on multi-stage optimization under uncertainty: Aligning concepts from theory and practice," Omega, Elsevier, vol. 96(C).
    4. Qiu, Ruozhen & Sun, Minghe & Lim, Yun Fong, 2017. "Optimizing (s, S) policies for multi-period inventory models with demand distribution uncertainty: Robust dynamic programing approaches," European Journal of Operational Research, Elsevier, vol. 261(3), pages 880-892.
    5. Qiu, Ruozhen & Shang, Jennifer & Huang, Xiaoyuan, 2014. "Robust inventory decision under distribution uncertainty: A CVaR-based optimization approach," International Journal of Production Economics, Elsevier, vol. 153(C), pages 13-23.
    6. Mengshi Lu & Zuo‐Jun Max Shen, 2021. "A Review of Robust Operations Management under Model Uncertainty," Production and Operations Management, Production and Operations Management Society, vol. 30(6), pages 1927-1943, June.
    7. Serrano, Breno & Minner, Stefan & Schiffer, Maximilian & Vidal, Thibaut, 2024. "Bilevel optimization for feature selection in the data-driven newsvendor problem," European Journal of Operational Research, Elsevier, vol. 315(2), pages 703-714.
    8. Metzker Soares, Paula & Thevenin, Simon & Adulyasak, Yossiri & Dolgui, Alexandre, 2024. "Adaptive robust optimization for lot-sizing under yield uncertainty," European Journal of Operational Research, Elsevier, vol. 313(2), pages 513-526.
    9. Xie, Chen & Wang, Liangquan & Yang, Chaolin, 2021. "Robust inventory management with multiple supply sources," European Journal of Operational Research, Elsevier, vol. 295(2), pages 463-474.
    10. Shin, Youngchul & Lee, Sangyoon & Moon, Ilkyeong, 2021. "Robust multiperiod inventory model with a new type of buy one get one promotion: “My Own Refrigerator”," Omega, Elsevier, vol. 99(C).
    11. Hamed Mamani & Shima Nassiri & Michael R. Wagner, 2017. "Closed-Form Solutions for Robust Inventory Management," Management Science, INFORMS, vol. 63(5), pages 1625-1643, May.
    12. Viktoryia Buhayenko & Dick den Hertog, 2017. "Adjustable Robust Optimisation approach to optimise discounts for multi-period supply chain coordination under demand uncertainty," International Journal of Production Research, Taylor & Francis Journals, vol. 55(22), pages 6801-6823, November.
    13. Saghafian, Soroush, 2018. "Ambiguous partially observable Markov decision processes: Structural results and applications," Journal of Economic Theory, Elsevier, vol. 178(C), pages 1-35.
    14. Henri G'erard & Michel de Lara & Jean-Philippe Chancelier, 2017. "Equivalence Between Time Consistency and Nested Formula," Papers 1711.08633, arXiv.org, revised May 2019.
    15. Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
    16. Yongchao Liu & Alois Pichler & Huifu Xu, 2019. "Discrete Approximation and Quantification in Distributionally Robust Optimization," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 19-37, February.
    17. Yanıkoğlu, İhsan & Gorissen, Bram L. & den Hertog, Dick, 2019. "A survey of adjustable robust optimization," European Journal of Operational Research, Elsevier, vol. 277(3), pages 799-813.
    18. Michal Melamed & Aharon Ben-Tal & Boaz Golany, 2016. "On the average performance of the adjustable RO and its use as an offline tool for multi-period production planning under uncertainty," Computational Management Science, Springer, vol. 13(2), pages 293-315, April.
    19. Henri Gérard & Michel Lara & Jean-Philippe Chancelier, 2020. "Equivalence between time consistency and nested formula," Annals of Operations Research, Springer, vol. 292(2), pages 627-647, September.
    20. Zhi Chen & Weijun Xie, 2021. "Regret in the Newsvendor Model with Demand and Yield Randomness," Production and Operations Management, Production and Operations Management Society, vol. 30(11), pages 4176-4197, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:289:y:2021:i:3:p:1127-1141. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.