Stability of solutions for fuzzy set optimization problems with applications

In this paper, we introduce a new parametric fuzzy set optimization problem based on the ordering relations of fuzzy sets, where a parametric function and an objective function both take values on fuzzy sets. We show the continuity of solution mappings for such a problem under some mild conditions. Moreover, we give the Painlevé-Kuratowski upper convergence and lower convergence of solution sets for fuzzy set optimization problems. As applications, the stability results are applied to obtain the continuity of a minimal solution mapping and the Painlevé-Kuratowski convergence of minimal solution sets for continuous-time fuzzy set optimization problems.