基于勾股模糊集评价的三支决策方法

Three-Way Decisions Method Based on Evaluations of Pythagorean Fuzzy Sets

针对勾股模糊三支决策概率阈值难以确定的问题,文中提出基于优化表示的勾股模糊三支决策概率阈值确定方法.首先从优化的视角研究一对对偶模型,利用KKT条件证明该对偶模型与决策粗糙集模型的等价性.然后,在确定勾股模糊集评价的三支决策概率阈值时引入对偶模型,基于勾股模糊数非线性排序法建立一对非线性规划模型,证明模型最优解的存在性与唯一性.最后,采用优化技术搜索模型最优解,并提出基于勾股模糊集评价的三支决策方法.算例及对比分析表明文中方法能有效克服现有方法难以确定勾股模糊三支决策概率阈值的不足.

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.