Distributionally robust inventory control when demand is a martingale

Abstract

https://people.orie.cornell.edu/dag369/ 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, Simchi-Levi and Song [85] 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 this paper we formulate and solve a time-consistent distributionally robust multi-stage newsvendor model which naturally unifies and robustifies several inventory models with demand forecasting. In particular, many simple models of demand forecasting have the feature that 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 forwards.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, a scenario of practical interest. To gain further insight, we prove weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare to the analogous setting in which demand is independent across periods (analyzed previously in Shapiro [119]), and identify interesting differences between these models, in the spirit of the price of correlations studied in Agrawal et al. [2]. Finally, we complement our theoretical results by providing a targeted and concise numerical experiment further demonstrating the benefits of our model.

Distributionally Robust Optimization: A Review

“…They further relate time consistency to rectangularity of measures, see, e.g., Shapiro [288], and provide sufficient conditions for time consistency. Unlike Xin and Goldberg [326] that suppose the demand process is stage-wise independent, Xin and Goldberg [325] assume that the demand process is a martingale. They form the ambiguity set by all distributions with a known support and mean at each stage.…”

Optimization under Connected Uncertainty

“…Under similar conditions, Ang et al (2012) investigate storage assignment. For production problems, Xin and Goldberg (2015) consider connected uncertainties by incorporating stochastic processes, such as martingales, to describe the ambiguity sets within a DRO framework. These works have in common that they model connectedness in the context of a specific application to derive computational or theoretical insights.…”

“…In many practical settings, however, the realized uncertainty in a period can affect subsequent uncertainty realizations. Such connections have been addressed for unit commitment (Lorca and Sun 2015) and inventory control problems (Xin and Goldberg 2015).…”

Inventory Control with Modulated Demand and a Partially Observed Modulation Process

“…Sethi and Cheng (1997) extended the results of Song and Zipkin (1993) to a discrete time system and obtained analogous results. Xin and Goldberg (2015) dealt with martingale-demand under a robust optimization framework. Dvoretzky et al (1952), Scarf (1959), and Murray and Silver (1966) analyzed the case where the modulation process is static, partially observed by the demand process, completely unobserved by the AOD process, and represents unknown parameters of a single stationary distribution.…”