  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.
