Two-stage distributionally robust optimization with a finite support

We consider two-stage distributionally robust mixed-integer problems where the uncertain parameters have discrete support. We propose an ambiguity set based on the feasible set of a transportation problem with a single knapsack constraint, extending the well-known Kantarovich ambiguity set in order to model a wider set of practical situations. The properties of this set are analysed. Based on different approaches to model the second-stage decisions and to impose the worst-case expected cost, three solution approaches are discussed: the Benders-like method proposed by Bansal, Huang, and Mehrotra (2018), a single-stage model obtained from the dualization of the transportation problem, and an epigraph formulation that enforces the expected cost through a series of optimality cuts which are generated dynamically. To evaluate the approaches a location-transportation problem is considered. Computational tests based on the three proposed approaches show that the best approach depends on the characteristics of the ambiguity set considered.