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An Adaptive Algorithm for Finding the Optimal Base-Stock Policy in Lost Sales Inventory Systems with Censored Demand

Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
IOMS-OM Group, Stern School of Business, New York University, New York, New York 10012
School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853
School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853
huh{at}ieor.columbia.edu, http://www.columbia.edu/th2113/
gjanakir{at}stern.nyu.edu, http://pages.stern.nyu.edu/gjanakir/
jack{at}orie.cornell.edu, http://people.orie.cornell.edu/jack/
paatrus{at}cornell.edu, http://legacy.orie.cornell.edu/paatrus/

We consider a periodic-review, single-location, single-product inventory system with lost sales and positive replenishment lead times. It is well known that the optimal policy does not possess a simple structure. Motivated by recent results showing that base-stock policies perform well in these systems, we study the problem of finding the best base-stock policy in such a system. In contrast to the classical inventory literature, we assume that the manager does not know the demand distribution a priori but must make the replenishment decision in each period based only on the past sales (censored demand) data. We develop a nonparametric adaptive algorithm that generates a sequence of order-up-to levels whose running average of the inventory holding and lost sales penalty cost converges to the cost of the optimal base-stock policy, and we establish the cubic-root convergence rate of the algorithm. Our analysis is based on recent advances in stochastic online convex optimization and on the uniform ergodicity of Markov chains associated with bases-stock policies.

Key Words: base-stock policy; censored demand; lost sales inventory system; online optimization
History: Received: February 8, 2007; revision received: June 26, 2008;