Confidence Intervals for Data-Driven Inventory Policies with Demand Censoring

We revisit the classical dynamic inventory management problem of Scarf [Scarf H (1959b) The optimality of (s, S) policies in the dynamic inventory problem. Arrow KJ, Karlin S, Suppes P, eds. Mathematical Methods in the Social Science (Stanford University Press, Stanford, CA), 196-202.] from the perspective of a decision maker who has n historical selling seasons of data and must make ordering decisions for the upcoming season. We develop a nonparametric estimation procedure for the (S, s) policy that is consistent and then characterize the finite sample properties of the estimated (S, s) levels by deriving their asymptotic confidence intervals. We also consider having at least some of the past selling seasons of data censored from the absence of backlogging and show that the intuitive procedure of first correcting for censoring in the demand data yields inconsistent estimates. We then show how to correctly use the censored data to obtain consistent estimates and derive asymptotic confidence intervals for this policy using Stein's method. We further show the confidence intervals can be used to effectively bound the difference between the expected total cost of an estimated policy and that of the optimal policy. We validate our results with extensive computations on simulated data. Our results extend to the repeated newsvendor problem and the base stock policy problem by appropriate parameter chokes.