Abstract：An optimization-based approach to determine the thresholds with Pythagorean fuzzy sets(PFSs) is proposed for threshold determination in three-way decisions(3WDs). Firstly, a pair of dual models from optimization angles are investigated, and it is proved that the dual models are equivalent to decision-theoretic rough sets models with the aid of the Karush-Kuhn-Tucker(KKT) condition. Next, the dual models are further generalized to the threshold determination of 3WDs with loss functions evaluated as PFSs, and a pair of nonlinear programming models are constructed based on nonlinear approaches for ranking PFSs. Meanwhile, the existence and the uniqueness of their optimal solution are proved and analyzed. Then, an optimization technique is exploited to solve these models, and a novel three-way decision approach under Pythagorean fuzzy evaluations is presented. Finally, an example and related comparison analysis indicate that the proposed method overcomes difficulties of the existing methods in determining the thresholds of Pythagorean fuzzy three-way decisions.
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2019, Vol. 32 
