We use distributionally robust optimization (DRO) to model a general class of newsvendor problems with unknown demand distribution. The goal is to find an order quantity that minimizes the worst-case expected cost among an ambiguity set of distributions. The ambiguity set consists of those distributions that are not far—in the sense of the total variation distance—from a nominal distribution. The maximum distance allowed in the ambiguity set (called level of robustness) places the DRO between the risk-neutral stochastic programming and robust optimization models. An important problem a decision maker faces is how to determine the level of robustness—or, equivalently, how to find an appropriate level of riskaversion. We answer this question in two ways. Our first approach relates the level of robustness and risk to the regions of demand that are critical (in a precise sense we introduce) to the optimal cost. Our second approach establishes new quantitative relationships between the DRO model and the corresponding risk-neutral and classical robust optimization models. To achieve these goals, we first focus on a single-product setting and derive explicit formulas and properties of the optimal solution as a function of the level of robustness. Then, we demonstrate the practical and managerial relevance of our results by applying our findings to a healthcare problem to reserve operating room time for cardiovascular surgeries. Finally, we extend some of our results to the multi-product setting and illustrate them numerically

Introduction

The newsvendor problem is fundamental to many operations management models. It has been used, for instance, in production of influenza vaccines (Chick, Mamani, & Simchi-Levi, 2008), staffing problems (Harrison & Zeevi, 2005), reservation of operating room time (Olivares, Terwiesch, & Cassorla, 2008), and the classical seat allocation model in revenue management (Littlewood, 1972). The newsvendor decides on how many units of a product should be produced before the uncertain demand is revealed. Because the demand is uncertain, the newsvendor must balance the costs of under- and over-production to determine an optimal quantity. For a review on the newsvendor problem, we refer the readers to Qin, Wang, Vakharia, Chen, and Seref (2011). There are multiple ways of formulating a newsvendor problem. For instance, one could consider “classical” stochastic ming (SP) or robust optimization (RO) approaches1. In the classical SP-based newsvendor model, the decision maker (i) has complete knowledge of the underlying demand distribution, and (ii) is risk neutral—i.e., (s)he minimizes the expected cost with respect to that demand distribution. In the classical RO-based newsvendor problem, on the other hand, the decision maker (i) does not have any knowledge of the demand except for its range of possible values, and (ii) is very risk averse—i.e., (s)he minimizes the worst-case cost among all values of demand within that range of values.