Operations Research Letters 35 (2007) 281 – 289 Operations Research Letters www.elsevier.com/locate/orl Inventory management under highly uncertain demand Guillermo Gallego a , ∗ , Kaan Katircioglu b , Bala Ramachandran b a Department of Industrial Engineering and Operations Research, Columbia University in the City of NewYork, S.W. Mudd Building, 500 West 120th Street, NY 10027, USA b IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA Received 1 February 2004; accepted 10 March 2006 Available online 24 May 2006 Abstract We show that base-stock levels first increase and then decrease as the standard deviation increases for a variety of non- negative random variables with a given mean and provide a distribution-free upper bound for optimal base-stock levels that grows linearly with the standard deviation and then remains constant. © 2006 Published by Elsevier B.V. Keywords: Inventory management; Base-stock policy; Risk management 1. Problem statement We are concerned with inventory management situ- ations where there is significant demand uncertainty as measured by the coefficient of variation, cv. In semi- conductor manufacturing, for example, it is common to find parts with cv > 2, and it is not unusual to see parts with cv > 3. Inventory managers often use base- stock levels of the form  + z = (1 + z ∗ cv), where  is the mean,  is the standard deviation and z is a safety factor based on the normal distribution. Typical values of z are 1.28, 1.64 and 2.33 corresponding to service levels 90%, 95% and 99%, respectively. When cv = 2, these safety factors lead to base-stock levels equal to 3.6, 4.3 and 5.7, respectively. Such excessive ∗ Corresponding author. E-mail address: ggallego@ieor.columbia.edu (G. Gallego). 0167-6377/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.orl.2006.03.012 orders often result in large financial losses. The use of the normal approximation may be even more widespread than intended as several commercial in- ventory management packages make the implicit assumption that demand is normal even when the coefficient of variation is large and decisions involve millions of dollars. Fig. 1 shows actual demand data for a semicon- ductor product with unit cost $5.00, unit selling price $10.00 and unit salvage value $3.00. Suppose, we are interested in maximizing expected profit. The empiri- cal distribution has a mean of 207 and a standard de- viation equal to 459 resulting in cv = 2.22. Although close to three quarters of the demand observations were for fewer than 100 units, there is a chance of receiving a demand for over 1000 units. The optimal order quantity under the normal demand assumption is 467 and it gives an expected loss of 291 when we