Distribution-free Inventory Risk Pooling in a Multi-location Newsvendor

We study a multi-location newsvendor network when the only information available on the joint distribution of demands are the values of its mean vector and covariance matrix. We adopt a distributionally robust model to find inventory levels that minimize the worst-case expected cost among the distributions consistent with this information. This problem is NP-hard. We find a closed-form tight bound on the expected cost when there are only two locations. This bound is tight under a family of joint demand distributions with six support points. This result extends the well-known Scarf (1958) bound for a single location. For the general case, we develop a computationally tractable upper bound on the worst-case expected cost if the costs of fulfilling demands have a nested structure. This upper bound is the optimal value of a semidefinite program whose dimensions are polynomial in the number of locations. We propose an algorithm that can approximate general fulfillment cost structures by nested structures, yielding a computationally tractable heuristic for distributionally robust inventory optimization on general newsvendor networks. We conduct experiments on networks resembling U.S. e-commerce distribution networks to show the value of a distributionally robust approach over a stochastic approach that assumes an incorrect demand distribution