A distribution-free risk-reward newsvendor model: Extending Scarf’s min-max order formula

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.