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Recently, there has been a growing interest in developing inventory controlpolicies which are robust to model misspecification. One approach is to positthat nature selects a worst-case distribution for any stochastic primitivesfrom some pre-specified family. Several communities have observed that a subtlephenomena known as time inconsistency can arise in this framework. Inparticular, it becomes possible that a policy which is optimal at time zero maynot be optimal for the associated optimization problem in which thedecision-maker recomputes her policy at each point in time, which hasimplications for implementability. If there exists a policy which is optimalfor both formulations, we say that the policy is time consistent, and theproblem is weakly time consistent. If every optimal policy is time consistent,we say that the problem is strongly time consistent. We study these phenomenain the context of managing an inventory over time, when only the mean,variance, and support are known for the demand at each stage. We provideseveral illustrative examples showing that here the question of timeconsistency can be quite subtle, and complement these observations by providingsimple sufficient conditions for weak and strong time consistency.Interestingly, our results show that time consistency may hold even whenrectangularity does not. Although a similar phenomena was previously identifiedby Shapiro for the setting in which only the mean and support of the demand areknown, there the problem was always weakly time consistent, with bothformulations having the same optimal value. Here our model is rich enough toexhibit a variety of interesting behaviors, including lack of weak timeconsistency, strong time consistency even when both formulations have differentoptimal values, and non-existence of even a single optimal base-stock policyunder the static formulation.

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