Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales

Retailers are frequently uncertain about the underlying demand distribution of a new product. When taking the empirical Bayesian approach of Scarf (1959), they simultaneously stock the product over time and learn about the distribution. Assuming that unmet demand is lost and unobserved, this learning must be based on observing sales rather than demand, which differs from sales in the event of stockout. Using the framework and results of Braden and Freimer (1991), the cumulative learning about the underlying demand distribution is captured by two parameters, a shape parameter that reflects the precision with which the underlying demand distribution is known, and a scale parameter that reflects the predicted size of the underlying market. An important simplification result of Scarf (1960) and Azoury (1985), which allows the scale parameter to be removed from the optimization, is shown to extend to this setting. We present examples that reveal two interesting phenomena: (1) The retailer may prefer that customers not buy everything she has stocked, even though doing so would signal a stochastically larger underlying demand distribution, and (2) it can be optimal to drop a product after a period of successful sales. We also present specific conditions under which the following results hold: (1) Investment in excess stocks to enhance learning will occur in every dynamic problem, and (2) a product is never dropped after a period of poor sales. The unit stockout penalty, the number of periods left in the horizon, and the shape parameter all play key roles in understanding our results. The model is extended to multiple independent markets whose distributions depend proportionately on a single unknown parameter. We argue that smaller markets should be given better service as an effective means of acquiring information.