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

Linwei Xin (linwei.xin***at***chicagobooth.edu)
David A. Goldberg (dag369***at***cornell.edu)

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 relevant 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 (i.e. solution to the multistage-static formulation) may not be optimal for the associated optimization problem in which the decision-maker recomputes her policy at each point in time (i.e. solution to the distributionally robust dynamic programming formulation), which has implications for implementability. If there exists a policy which is optimal for both formulations (w.p.1 under every joint distribution for demand belonging to the uncertainty set), we say that the policy is time consistent, and the problem is weakly time consistent. If every optimal policy for the multistage-static formulation 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. We complement these observations by providing simple sufficient conditions for weak and strong time consistency. We also relate our results to the well-studied notion of rectangularity of a family of measures. 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.

Keywords: inventory, news vendor, multistage distributionally robust optimization, rectangularity, moment constraints, time consistency, dynamic programming, base-stock policy

Category 1: Stochastic Programming


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Entry Submitted: 04/10/2013
Entry Accepted: 04/11/2013
Entry Last Modified: 08/20/2018

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