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Dynamic Inventory Management with Learning About the Demand Distribution and Substitution Probability


  • Li Chen

    () (TrueDemand, Inc., Los Gatos, California 95032)

  • Erica L. Plambeck

    () (Graduate School of Business, Stanford University, Stanford, California 94305)


Awell-known result in the Bayesian inventory management literature is: If lost sales are not observed, the Bayesian optimal inventory level is larger than the myopic inventory level (one should "stock more" to learn about the demand distribution). This result has been proven in other studies under the assumption that inventory is perishable, so the myopic inventory level is equal to the Bayesian optimal inventory level with observed lost sales. We break that equivalence by considering nonperishable inventory. We prove that with nonperishable inventory, the famous "stock more" result is often reversed to "stock less," in that the Bayesian optimal inventory level with unobserved lost sales is lower than the myopic inventory level. We also prove that making lost sales unobservable increases the Bayesian optimal inventory level; in this specific sense, the famous "stock more" result of other studies generalizes to the case of nonperishable inventory. When the product is out of stock, a customer may accept a substitute or choose not to purchase. We incorporate learning about the probability of substitution. This reduces the Bayesian optimal inventory level in the case that lost sales are observed. Reducing the inventory level has two beneficial effects: to observe and learn more about customer substitution behavior and (for a nonperishable product) to reduce the probability of overstocking in subsequent periods. Finally, for a capacitated production-inventory system under continuous review, we derive maximum likelihood estimators (MLEs) of the demand rate and probability that customers will wait for the product. (Accepting a raincheck for delivery at some later time is a common type of substitution.) We investigate how the choice of base-stock level and production rate affect the convergence rate of these MLEs. The results reinforce those for the Bayesian, uncapacitated, periodic review system.

Suggested Citation

  • Li Chen & Erica L. Plambeck, 2008. "Dynamic Inventory Management with Learning About the Demand Distribution and Substitution Probability," Manufacturing & Service Operations Management, INFORMS, vol. 10(2), pages 236-256, May.
  • Handle: RePEc:inm:ormsom:v:10:y:2008:i:2:p:236-256

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    References listed on IDEAS

    1. William S. Lovejoy, 1990. "Myopic Policies for Some Inventory Models with Uncertain Demand Distributions," Management Science, INFORMS, vol. 36(6), pages 724-738, June.
    2. Katy S. Azoury, 1985. "Bayes Solution to Dynamic Inventory Models Under Unknown Demand Distribution," Management Science, INFORMS, vol. 31(9), pages 1150-1160, September.
    3. Martin A. Lariviere & Evan L. Porteus, 1999. "Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales," Management Science, INFORMS, vol. 45(3), pages 346-363, March.
    4. Wang, Qinan & Parlar, Mahmut, 1994. "A three-person game theory model arising in stochastic inventory control theory," European Journal of Operational Research, Elsevier, vol. 76(1), pages 83-97, July.
    5. Steven Nahmias & Stephen A. Smith, 1994. "Optimizing Inventory Levels in a Two-Echelon Retailer System with Partial Lost Sales," Management Science, INFORMS, vol. 40(5), pages 582-596, May.
    6. Rajaram, Kumar & Tang, Christopher S., 2001. "The impact of product substitution on retail merchandising," European Journal of Operational Research, Elsevier, vol. 135(3), pages 582-601, December.
    7. Ricardo Ernst & Panagiotis Kouvelis, 1999. "The Effects of Selling Packaged Goods on Inventory Decisions," Management Science, INFORMS, vol. 45(8), pages 1142-1155, August.
    8. Giora Harpaz & Wayne Y. Lee & Robert L. Winkler, 1982. "Learning, Experimentation, and the Optimal Output Decisions of a Competitive Firm," Management Science, INFORMS, vol. 28(6), pages 589-603, June.
    9. Mor Armony & Erica L. Plambeck, 2005. "The Impact of Duplicate Orders on Demand Estimation and Capacity Investment," Management Science, INFORMS, vol. 51(10), pages 1505-1518, October.
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    Cited by:

    1. Mor Armony & Erica Plambeck & Sridhar Seshadri, 2009. "Sensitivity of Optimal Capacity to Customer Impatience in an Unobservable M/M/S Queue (Why You Shouldn't Shout at the DMV)," Manufacturing & Service Operations Management, INFORMS, vol. 11(1), pages 19-32, June.
    2. Katy S. Azoury & Julia Miyaoka, 2009. "Optimal Policies and Approximations for a Bayesian Linear Regression Inventory Model," Management Science, INFORMS, vol. 55(5), pages 813-826, May.
    3. Zhang, Jian & Zhang, Juliang & Hua, Guowei, 2016. "Multi-period inventory games with information update," International Journal of Production Economics, Elsevier, vol. 174(C), pages 119-127.
    4. Marbán Sebastián & Rutten Cyriel & Vredeveld Tjark, 2010. "Tight performance in Bayesian Scheduling," Research Memorandum 052, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    5. Alp Akcay & Bahar Biller & Sridhar Tayur, 2011. "Improved Inventory Targets in the Presence of Limited Historical Demand Data," Manufacturing & Service Operations Management, INFORMS, vol. 13(3), pages 297-309, July.


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