Distributionally robust optimization (DRO) aims at finding an optimal solution under the worst-case distribution within an ambiguity set, which is built from partial information about the true distribution. In this paper, we investigate a new class of risk-averse two-stage distributionally robust mixed-integer optimization problems where the ambiguity set is decision-dependent. Specifically, we consider distance-based ambiguity sets defined by \(\phi \) -divergence and Wasserstein metric, and these sets are influenced by the first-stage integer decisions. By adopting the Lagrangian dual theory and Slater’s condition, we reformulate the problem into tractable mixed-integer nonlinear programming problems. We develop a decomposition method to solve the resulting mixed-integer programming problems especially when the ambiguity set is defined using 1-Wasserstein metric. Furthermore, for cases where empirical data may be contaminated, we demonstrate the quantitative statistical robustness of the optimal value of decision-dependent distributionally robust optimization ( \(\hbox {D}^3\) RO) problem using the Fortet-Mourier metric. Finally, we conduct numerical experiments to exhibit variations in the optimal value and to illustrate the quantitative statistical robustness results.
