Optimization problems in operations and finance often include a cost that is proportional to the expected amount by which a random variable exceeds some fixed quantity, known as the expected loss function. Representation of this function often leads to computational challenges, depending on the distribution of the random variable of interest. Moreover, in practice, a decision maker may possess limited information about this probability distribution, such as the mean and variance, but not the exact form of the associated probability density or distribution function. In such cases, a distributionally robust (DR) optimization approach seeks to minimize the maximum expected cost among all possible distributions that are consistent with the available information. Past research has recognized the overly conservative nature of this approach because it accounts for worst-case probability distributions that almost surely do not arise in practice. Motivated by this, we propose a DR approach that accounts for the worst-case performance with respect to a broad class of common continuous probability distributions, while producing solutions that are less conservative (and, therefore, less expensive, on average) than those produced by existing DR approaches in the literature. The methods we propose also permit approximation of the expected loss function for probability distributions under which exact representation of the function is difficult or impossible. Finally, we draw a connection between Scarf-type bounds from the literature, and mean-MAD (mean absolute deviation) bounds when MAD information is available in addition to variance.
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