Asymmetry and Ambiguity in Newsvendor Models

A basic assumption of the classical newsvendor model is that the probability distribution of the random demand is known. But in most realistic settings, only partial distribution information is available or reliably estimated. The distributionally robust newsvendor model is often used in this case where the worst-case expected profit is maximized over the set of distributions satisfying the known information, which is usually the mean and covariance of demands. However, covariance does not capture information on asymmetry of the demand distribution. In this paper, we introduce a measure of distribution asymmetry using second-order partitioned statistics. Semivariance is a special case with a single partition of the univariate demand. With mean, variance, and semivariance information, we show that a three-point distribution achieves the worst-case expected profit and derive a closed-form expression for the distributionally robust order quantity. For multivariate demand, the distributionally robust problem with partitioned statistics is hard to solve, but we develop a computationally tractable lower bound through the solution of a semidefinite program. We demonstrate in numerical experiments that asymmetry information significantly reduces expected profit loss particularly when the true distribution is heavy tailed. In computational experiments on automotive spare parts demand data, we provide evidence that the distributionally robust model that includes partitioned statistics outperforms the model that uses only covariance information.