We consider stochastic finite-horizon inventory models with discrete distributions that are incompletely specified by selected moments, percentiles, or a combination of moments and percentiles. The objective is to determine an inventory policy that minimizes the maximum expected cost over the class of demand distributions satisfying the specifications described above. We show that many inventory models of this form can be solved by a sequence of linear programs.

1. Introduction

In this paper we consider discrete demand, finite-horizon inventory models where the demand distribution is unknown except for a finite number of parameters, such as its mean and variance. The problem is to determine an inventory policy that minimizes the maximum expected cost over all demand distributions with the given parameters. We make no further assumptions on the form of the demand distribution, allowing us to deal with situations in which the exact demand distribution is unknown or changes over time. For example, we may have sufficient data to accurately estimate the mean and variance of demand, but may be unable to confidently estimate higher moments or the shape of the distribution. As Gallego and Moon (1993) point out, this is frequently the case in the fashion and sporting goods industries.

To our knowledge, the first work on distribution free models is due to Scarf (1958), who considered the single period newsvendor problem. Scarf determines the order quantity that maximizes the minimum expected profit over all continuous demand distributions with a given mean and variance.

Kasugai and Kasegai (1960) present a dynamic programming approach to the distribution free, multi-period newsvendor problem when demand is assumed to belong to a known closed interval; no additional assumptions are made on the demand distribution. Kasugai and Kasegai (1961) also consider the minimax regret ordering principle for the distribution free newsvendor problem, and compare these results to the minimax policy. More recently, Gallego and Moon (1993) provide a concise derivation of Scarf's single period results and consider various extensions of the problem. Gallego (1992, 1996) considers a minimax approach to the infinite-horizon continuous review (Q, R) inventory model with incidence-oriented backorder costs (Gallego, 1992) and time-weighted backorder costs (Gallego, 1996). The thrust of this research has been to obtain distribution-free cost bounds that reveal the effects of randomness in the worst case, and distribution-free heuristics that are robust to the specification of demand. Moon and Galleg o (1994) apply similar techniques to the infinite-horizon continuous and periodic review models with backorders and lost sales, while Moon and Choi (1997) do so for Assemble-to-Order, Assemble-in-Advance, and composite policies. For a more detailed discussion of previous research on distribution-free procedures and other strategies that have been used to deal with inventory models in which the demand distribution is not specified, see Ryan (1997).