Closed-Form Solutions for Distributionally Robust Inventory Management: A Controlled Relaxation Method

Abstract

When only the moments (mean, variance or t-th moment) of the underline distribution are known, a variety of many max-min optimization models choose actions to maximize the firm’s expected profit against the most unfavorable distribution. We introduce relaxation scalars to reformulate the max-min model as a relaxed model and demonstrate that the closed form solutions (if they exist in the first place) can be quickly identified when we reduce the relaxation scalars to zero. To demonstrate the effectiveness of this new method, we provide closed-form solutions, hitherto unknown, for several distributionally robust inventory models, including the newsvendor problem with mean and t-th moment (for t > 1), the pricing model, the capacity planning model with multiple supply sources, and the two-product inventory system with common component.