We study stochastic inventory planning with lost sales and instantaneous replenishment, where contrary to the classical inventory theory, the knowledge of the demand distribution is not available. Furthermore, we observe only the sales quantity in each period, and lost sales are unobservable, that is, demand data are censored. The manager must make an ordering decision in each period based only on historical sales data. Excess inventory is either perishable or carried over to the next period. In this setting, we propose non-parametric adaptive policies that generate ordering decisions over time. We show that the T -period average expected cost of our policy differs from the benchmark newsvendor cost -the minimum expected cost that would have incurred if the manager had known the underlying demand distribution -by at most O(1/ √ T ).
IntroductionThe problem of inventory control and planning has received much interest from practitioners and academics from the early years of operations research. The early literature in this area modeled demand as deterministic and having known quantities, but it soon became apparent that deterministic modeling was often inadequate, and uncertainty needed to be incorporated in modeling future demand. As a result, a majority of the papers on inventory theory during the past fifty years employ stochastic demand models. In these models, future demand is given by a specific exogenous random variable, and the inventory decisions are made with full knowledge of the future demand distribution. In many applications, however, the demand distribution is not known a priori. Even when past data have been collected, the selection of the most appropriate distribution and its parameters remains ambiguous. In the case when excess demand is lost, the information available to the inventory manager is further limited since she does not observe the realized demand but only observes the sales quantity (often referred to as censored demand), which is the smaller of the stocking level and the realized demand. Motivated by these realistic constraints, we develop a non-parametric approach to stochastic inventory planning in the presence of lost sales and censored demand.