Closing the Gap: A Learning Algorithm for the Lost-Sales Inventory System with Lead Times

Introduction The periodic-review inventory control problem with lost-sales and positive lead times is one of the most fundamental yet notoriously difficult problems in the theory of inventory management (see Zipkin (2000)). The model assumes that unmet demand at the end of each period is lost, rather than being backlogged and carried over to the next period. For example, in many retail applications demand can be met by competing suppliers, making lost-sales a more appropriate modeling assumption (cf. Bijvank and Vis (2011)). There is a constant delivery lead time measured by the delay between placing an order and receiving it, which leads to an enlarged state-space in which the pipeline orders need to be tracked (cf. Zipkin (2008)). In this paper, contrary to the classical inventory setting, we assume that the firm does not know the demand distribution a priori but can only collect past sales data over time. Because the sales in a period are the minimum of the actual demand and the on-hand inventory level, the demand information is censored (cf. Huh et al. (2009a)). The firm wishes to minimize the long-run average holding and lost-sales penalty cost per period. As we have witnessed the recent progress for this fundamental class of problems, the incomplete information counterpart problem (under censored demand) remains relatively under-explored. In many practical scenarios (e.g., furniture retailing), the firm does not know the underlying demand distribution a priori and is forced to make replenishment decisions based on historical sales data. However, the sales data, as we discussed earlier, are in fact censored demand information. The joint learning and optimization problem in the underlying lost-sales system is therefore practically relevant and theoretically challenging. The only paper (and the closest to ours) in the literature is Huh et al. (2009a) who studied the exact same model and proposed an online learning algorithm whose regret against the full-information optimal base-stock policy is O(T ) over a T -period problem. The motivations and justifications for using the optimal base-stock policy as a valid benchmark for this incomplete information problem are two-fold. First, the class of base-stock policies is easily implemented and widely used (see e.g., Janakiraman and Roundy (2004)). Second,