Distributionally Robust Newsvendor Problems with Variation Distance

We use distributionally robust stochastic programs (DRSP) to model a general class of newsvendor problems where the underlying demand distribution is unknown, and so 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 so-called variation distance—from a nominal distribution. The maximum distance allowed in the ambiguity set (called level of robustness) places the DRSP between the “classical” stochastic programming and robust optimization models, which correspond to setting the level of robustness to zero and infinity, respectively. The structure of the newsvendor problem allows us to analyze the problem from multiple perspectives: First, we derive explicit formulas and properties of the optimal solution as a function of the level of robustness. Moreover, we determine the regions of demand that are critical (in a precise sense) to optimal cost from the viewpoint of a risk-averse decision maker. Finally, we establish quantitative relationships between the distributionally robust model and the corresponding risk-neutral and classical robust optimization models, which include the price of optimism/pessimism, and the nominal/worstcase regrets, among others. Our analyses can help the decision maker better understand the role of demand uncertainty in the problem and can guide him/her to choose an appropriate level of robustness. We illustrate our results with numerical experiments on a variety of newsvendor problems with different characteristics. ∗[email protected] †[email protected] ‡[email protected]; Corresponding author