Scarf's min-max order formula for the distribution-free risk-neutral newsvendor problem is a classical result in the field of inventory management. The min-max order formula provides, in closed-form, the order quantity that maximizes the worst-case expected profit associated with the demand of a single product when only the mean and variance of the product's demand distribution, rather than the full distribution itself, is assumed to be known. It has been a long-standing question whether a similar closed-form order formula exists for the distribution-free risk-reward newsvendor problem; that is, for the order quantity that maximizes the worst-case risk-reward associated with the demand of a single product when only the mean and variance of the product's demand distribution is assumed to be known. The main contribution of this work is to extend Scarf's closed-form order formula to one of the most important risk-reward criteria, namely when the reward is defined by the expected profit, and the risk by the profit's standard deviation. Furthermore, we provide managerial insights associated with this result.
Working Paper Series #2012-001, University of New Brunswick, Canada