Time (in)consistency of multistage distributionally robust inventory models with moment constraints

Abstract: Recently, there has been a growing interest in developing inventory control policies which are robust to model misspecification. One approach is to posit that nature selects a worst-case distribution for any stochastic primitives from some pre-specified family. Several communities have observed that a subtle phenomena known as time inconsistency can arise in this framework. In particular, it becomes possible that a policy which is optimal at time zero may not be optimal for the associated optimization problem in which the decision-maker recomputes her policy at each point in time, which has implications for implementability. If there exists a policy which is optimal for both formulations, we say that the policy is time consistent, and the problem is weakly time consistent. If every optimal policy is time consistent, we say that the problem is strongly time consistent. We study these phenomena in the context of managing an inventory over time, when only the mean, variance, and support are known for the demand at each stage. We provide several illustrative examples showing that here the question of time consistency can be quite subtle, and complement these observations by providing simple sufficient conditions for weak and strong time consistency. Interestingly, our results show that time consistency may hold even when rectangularity does not. Although a similar phenomena was previously identified by Shapiro for the setting in which only the mean and support of the demand are known, there the problem was always weakly time consistent, with both formulations having the same optimal value. Here our model is rich enough to exhibit a variety of interesting behaviors, including lack of weak time consistency, strong time consistency even when both formulations have different optimal values, and non-existence of even a single optimal base-stock policy under the static formulation.