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

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

Closed-Form Solutions for Distributionally Robust Inventory Management: Extended Reformulation using Zero-Sum Game

When only the moments (mean, variance or t-th moment) of the underline distribution are known, numerous max-min optimization models can be interpreted as a zero-sum game, in which the decision maker (DM) chooses actions to maximize her expected profit while Adverse Nature chooses a distribution subject to the moments conditions to minimize DM’s expected profit. We propose a new method to efficiently solve this class of zero-sum games under moment conditions. By applying the min-max inequality, our method reformulates the zero-sum game as a robust moral hazard model, in which Adverse Nature chooses both the distribution and actions to minimize DM’s expected profit subject to incentive compatibility (IC) constraints. Under quasi-concavity, these IC constraints are replaced by the first-order conditions, which give rise to extra moment constraints. Interestingly, these extra moment constraints drastically reduce the number of corner points to be considered in the corresponding semi-infinite programming models. We show that in the equilibrium, these moment constraints are binding but have zero Lagrangian multipliers and thus facilitate closed-form solutions in several application examples with different levels of complexity. The high efficiency of the method enables us to solve a large class of zero-sum games and the corresponding max-min robust optimization models