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Asymptotic Optimality of Order-Up-To Policies in Lost Sales Inventory Systems

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
gjanakir{at}stern.nyu.edu
jack{at}orie.cornell.edu
paatrus{at}cornell.edu

We study a single-product single-location inventory system under periodic review, where excess demand is lost and the replenishment lead time is positive. The performance measure of interest is the long-run average holding cost and lost sales penalty cost. For a large class of demand distributions, we show that when the lost sales penalty becomes large compared to the holding cost, the relative difference between the cost of the optimal policy and the best order-up-to policy converges to zero. For any given cost parameters, we establish a bound on this relative difference. Numerical experiments show that the best order-up-to policy performs well, yielding an average cost that is within 1.5% of the optimal cost when the ratio between the lost sales penalty and the holding cost is 100. We also propose a heuristic order-up-to level using two newsvendor expressions; in our experiments, the cost of this order-up-to policy is 2.52% higher, on an average, than the best order-up-to policy.

Key Words: inventory; production; lost sales; backorders; finite horizon; infinite horizon; optimal costs
History: Received: December 4, 2006;

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