We consider stochastic programs in which the probability distribution of uncertain parameters is unknown and partial information about it can only be captured from limited data. We use distributionally robust optimization (DRO) to model such problems. As opposed to the commonly used approach for DRO problems that suggests creating an ambiguity set by following a specific procedure, we propose to build it by adopting multiple procedures. Specifically, we design the ambiguity set by intersecting sets that are constructed by different discrepancy-based measures. The new ambiguity set excludes the probability distributions that reside in only one set, while preserving the common ones, which prevents the DRO problem from producing overly conservative solutions. We derive single-level convex programming reformulations for the resulting two-level DRO problems with various supports, namely, discrete known, univariate, and multivariate supports. We additionally design a three-level cutting-plane algorithm and a relaxation-based technique to tackle computationally challenging DRO problems with multivariate supports. To evaluate the quality of the solutions that our reformulations yield and the performance of these techniques, we conduct experiments on newsvendor and mean-risk portfolio allocation problems. Our results suggest that using multiple measures to build the ambiguity set decreases robustness and provides high-quality solutions, especially for small-size samples. Moreover, the outcomes illustrate that our proposed algorithm and the upper-bounding technique are indeed effective.
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