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Estimation of Consumer Demand with Stock-Out Based Substitution: An Application to Vending Machine Products


  • Ravi Anupindi

    (J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, Illinois 60208)

  • Maqbool Dada

    (Krannert Graduate School of Management, Purdue University, West Lafayette, Indiana 47907)

  • Sachin Gupta

    (J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, Illinois 60208)


The occurrence of temporary stock-outs at retail is common in frequently purchased product categories. Available empirical evidence suggests that when faced with stock-outs, consumers are often willing to buy substitute items. An important implication of this consumer behavior is that observed sales of an item no longer provide a good measure of its core demand rate. Sales of items that stock-out are right-censored, while sales of other items are inflated because of substitutions. Knowledge of the true demand rates and substitution rates is important for the retailer for a variety of category management decisions such as the ideal assortment to carry, how much to stock of each item, and how often to replenish the stock. The estimated substitution rates can also be used to infer patterns of competition between items in the category. In this paper we propose methods to estimate demand rates and substitution rates in such contexts. We develop a model of customer arrivals and choice between goods that explicitly allows for possible product substitution and lost sales when a customer faces a stock-out. The model is developed in the context of retail vending, an industry that accounts for a sizable part of the retail sales of many consumer products. We consider the information set available from two kinds of inventory tracking systems. In the best case scenario of a perpetual inventory system in which times of stock-out occurrence and cumulative sales of all goods up to these times are observed, we derive Maximum Likelihood Estimates (MLEs) of the demand parameters and show that they are especially simple and intuitive. However, state-of-the-art inventory systems in retail vending provide only periodic data, i.e., data in which times of stock-out occurrence are unobserved or “missing.” For these data we show how the Expectation-Maximization (EM) algorithm can be employed to obtain the MLEs of the demand parameters by treating the stock-out times as missing data. We show an application of the model to daily sales and stocking data pooled across multiple beverage vending machines in a midwestern U.S. city. The vending machines in the application carry identical assortments of six brands. Since the number of parameters to be estimated is too large given the available data, we discuss possible restrictions of the consumer choice model to accomplish the estimation. Our results indicate that demand rates estimated naively by using observed sales rates are biased, even for items that have very few occurrences of stock-outs. We also find significant differences among the substitution rates of the six brands. The methods proposed in our paper can be modified to apply to many nonvending retail settings in which consumer choices are observed, not their preferences, and choices are constrained because of unavailability of items in the choice set. One such context is in-store grocery retailing, where similar issues of information availability arise. In this context an important issue that would need to be dealt with is changes in the retail environment caused by retail promotions.

Suggested Citation

  • Ravi Anupindi & Maqbool Dada & Sachin Gupta, 1998. "Estimation of Consumer Demand with Stock-Out Based Substitution: An Application to Vending Machine Products," Marketing Science, INFORMS, vol. 17(4), pages 406-423.
  • Handle: RePEc:inm:ormksc:v:17:y:1998:i:4:p:406-423

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    References listed on IDEAS

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    Cited by:

    1. Liu, Jun & Lee, Chi-Guhn, 2007. "Evaluation of inventory policies with unidirectional substitutions," European Journal of Operational Research, Elsevier, vol. 182(1), pages 145-163, October.
    2. Ioannis Ioannou & Julie Holland Mortimer & Richard Mortimer, 2011. "The Effects Of Capacity On Sales Under Alternative Vertical Contracts," Journal of Industrial Economics, Wiley Blackwell, vol. 59(1), pages 117-154, March.
    3. Aydinliyim, Tolga & Pangburn, Michael S. & Rabinovich, Elliot, 2017. "Inventory disclosure in online retailing," European Journal of Operational Research, Elsevier, vol. 261(1), pages 195-204.
    4. repec:eee:ejores:v:266:y:2018:i:1:p:99-111 is not listed on IDEAS
    5. Antonio Rosato, 2016. "Selling substitute goods to loss-averse consumers: limited availability, bargains, and rip-offs," RAND Journal of Economics, RAND Corporation, vol. 47(3), pages 709-733, August.
    6. repec:eee:proeco:v:193:y:2017:i:c:p:813-826 is not listed on IDEAS
    7. 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.
    8. Vishal Gaur & Young-Hoon Park, 2007. "Asymmetric Consumer Learning and Inventory Competition," Management Science, INFORMS, vol. 53(2), pages 227-240, February.
    9. Vaagen, Hajnalka & Wallace, Stein W. & Kaut, Michal, 2011. "Modelling consumer-directed substitution," International Journal of Production Economics, Elsevier, vol. 134(2), pages 388-397, December.
    10. Christopher T. Conlon & Julie Holland Mortimer, 2013. "Demand Estimation under Incomplete Product Availability," American Economic Journal: Microeconomics, American Economic Association, vol. 5(4), pages 1-30, November.
    11. Sandeep R. Chandukala & Yancy D. Edwards & Greg M. Allenby, 2011. "Identifying Unmet Demand," Marketing Science, INFORMS, vol. 30(1), pages 61-73, 01-02.
    12. 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.
    13. Tan, Baris & Karabati, Selcuk, 2013. "Retail inventory management with stock-out based dynamic demand substitution," International Journal of Production Economics, Elsevier, vol. 145(1), pages 78-87.
    14. David A. Matsa, 2011. "Running on Empty? Financial Leverage and Product Quality in the Supermarket Industry," American Economic Journal: Microeconomics, American Economic Association, vol. 3(1), pages 137-173, February.
    15. Shin, Hojung & Park, Soohoon & Lee, Euncheol & Benton, W.C., 2015. "A classification of the literature on the planning of substitutable products," European Journal of Operational Research, Elsevier, vol. 246(3), pages 686-699.
    16. Christopher T. Conlon & Julie Holland Mortimer, 2013. "Efficiency and Foreclosure Effects of Vertical Rebates: Empirical Evidence," NBER Working Papers 19709, National Bureau of Economic Research, Inc.
    17. Chiang, Wei-yu Kevin, 2010. "Product availability in competitive and cooperative dual-channel distribution with stock-out based substitution," European Journal of Operational Research, Elsevier, vol. 200(1), pages 111-126, January.
    18. Karakul, M. & Chan, L.M.A., 2008. "Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem," European Journal of Operational Research, Elsevier, vol. 190(1), pages 179-204, October.
    19. K. Sudhir & Nathan Yang, 2014. "Exploiting the Choice-Consumption Mismatch: A New Approach to Disentangle State Dependence and Heterogeneity," Cowles Foundation Discussion Papers 1941, Cowles Foundation for Research in Economics, Yale University.
    20. Andrés Musalem & Marcelo Olivares & Eric T. Bradlow & Christian Terwiesch & Daniel Corsten, 2010. "Structural Estimation of the Effect of Out-of-Stocks," Management Science, INFORMS, vol. 56(7), pages 1180-1197, July.
    21. Bayle-Tourtoulou, Anne-Sophie & Laurent, Gilles & Macé, Sandrine, 2006. "Assesing the frequency and clauses of out-of-stock events through store scanner data," Les Cahiers de Recherche 830, HEC Paris.
    22. Kirthi Kalyanam & Sharad Borle & Peter Boatwright, 2007. "Deconstructing Each Item's Category Contribution," Marketing Science, INFORMS, vol. 26(3), pages 327-341, 05-06.
    23. Yang, Hongsuk & Schrage, Linus, 2009. "Conditions that cause risk pooling to increase inventory," European Journal of Operational Research, Elsevier, vol. 192(3), pages 837-851, February.


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