Distributionally Robust Inventory Control When Demand Is a Martingale

Demand forecasting plays an important role in many inventory control problems. To mitigate the potential harms of model misspecification in this context, various forms of distributionally robust optimization have been applied. Although many of these methodologies suffer from the problem of time inconsistency, the work of Klabjan et al. established a general time-consistent framework for such problems by connecting to the literature on robust Markov decision processes. Motivated by the fact that many forecasting models exhibit very special structure as well as a desire to understand the impact of positing different dependency structures in distributionally robust multistage optimization, we formulate and solve a time-consistent distributionally robust multistage newsvendor model, which naturally robustifies some of the simplest inventory models with demand forecasting. In particular, in some of the simplest such models, demand evolves as a martingale (i.e., expected demand tomorrow equals realized demand today). We consider a robust variant of such models in which the sequence of future demands may be any martingale with given mean and support. Under such a model, past realizations of demand are naturally incorporated into the structure of the uncertainty set going forward. We explicitly compute the minimax optimal policy (and worst-case distribution) in closed form by combining ideas from convex analysis, probability, and dynamic programming. We prove that, at optimality, the worst-case demand distribution corresponds to the setting in which inventory may become obsolete at a random time. To gain further insight, we prove weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare with the analogous setting in which demand is independent across periods (analyzed previously by Shapiro) and identify several differences between these models in the spirit of the price of correlations studied by Agrawal et al. Finally, we complement our results by providing both numerical experiments that illustrate the potential benefits and limitations of our approach as well as additional theoretical analyses exploring what happens when our modeling assumptions do not hold.