  In this paper, we study a bivariate distributionally robust optimization
problem with mean-covariance ambiguity set and half-space support. Under a
conventional type of objective function widely adopted in inventory management,
option pricing, and portfolio selection, we obtain closed-form tight bounds of
the inner problem in six different cases. Through a primal-dual approach, we
identify the optimal distributions in each case. As an application in inventory
control, we first derive the optimal order quantity and the corresponding
worst-case distribution, extending the existing results in the literature.
Moreover, we show that under the distributionally robust setting, a centralized
inventory system does not necessarily reduce the optimal total inventory, which
contradicts conventional wisdom. Furthermore, we identify two effects, a
conventional pooling effect, and a novel shifting effect, the combination of
which determines the benefit of incorporating the covariance information in the
ambiguity set. Finally, we demonstrate through numerical experiments the
importance of keeping the covariance information in the ambiguity set instead
of compressing the information into one dimension.
