Closed-Form Solutions for Robust Inventory Management

We propose and analyze robust optimization models of an inventory management problem, where cumulative purchase, inventory, and shortage costs over n periods are minimized for correlated nonidentically distributed demand. We assume that the means and covariance matrix of stochastic demand are known; the distributions are not needed. We derive closed-form ordering quantities for both symmetric and asymmetric uncertainty sets, under capacitated inventory constraints, in both static and dynamic settings. The behaviors of our robust strategies differ qualitatively depending on the symmetry of the uncertainty set. For instance, considering our simplest static problem, (1) if the uncertainty set is symmetric, then there is positive ordering in all periods, whereas (2) if the set is asymmetric, then there is a set of periods in the middle of the planning horizon with zero orders. We also link the symmetry of the uncertainty set to the symmetry of the demand distribution. Finally, we present encouraging computational results where our solution compares favorably to previously studied, more complex robust solutions. This paper was accepted by Yinyu Ye, optimization.