On the Heavy-Tail Behavior of the Distributionally Robust Newsvendor

Abstract

Since the seminal work of Scarf (1958) [A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production, pages 201-209] on the newsvendor problem with ambiguity in the demand distribution, there has been a growing interest in the study of the distributionally robust newsvendor problem. The optimal order quantity is computed by accounting for the worst possible distribution from a set of demand distributions that is characterized by partial information, such as moments. The model is criticized at times for being overly conservative since the worst-case distribution is discrete with a few support points. However, it is the order quantity from the model that is typically of practical relevance. A simple observation shows that the optimal order quantity in Scarf’s model with known first and second moment is also optimal for a heavy-tailed censored student-t distribution with degrees of freedom 2. In this paper, we generalize this “heavy- tail optimality” property of the distributionally robust newsvendor to a more general ambiguity set where information on the first and the nth moment is known, for any real number n > 1. We provide a characterization of the optimal order quantity under this ambiguity set by showing that for high critical ratios, the order quantity is optimal for a regularly varying distribution with an approximate power law tail with tail index n. We illustrate the applicability of the model by calibrating the ambiguity set from data and comparing the performance of the order quantities computed via various methods in a dataset.