Pythagorean Fuzzy Decision-making Theories And Methods Based On Isomorphic Archimedean T-norm And S-norm

The concept of fuzzy sets was proposed by scholar Zadeh in 1965.After more than 50 years of development,the related researches of fuzzy decision-making theories and methods have yielded fruitful results.The operational laws are the core of the development of fuzzy set theories,and the fuzzy weighted aggregation operator based on the operational laws is an important tool to solve the decision-making problems.The laws and operators are the research focus and difficulties for fuzzy decision-making theories.Archimedean t-norm and s-norm are widely used in various fuzzy sets as an important tool for the theory construction of various fuzzy sets.As a direct extension of intuitionistic fuzzy sets,the concept of Pythagorean fuzzy sets was proposed by scholar Yager in 2014.Compared with intuitionistic fuzzy sets,the advantage is that it can degenerate into intuitionistic fuzzy sets,and the value region is 1.57 times of that of intuitionistic fuzzy sets.At present,the research on the fuzzy theory of Pythagorean is in the early stage of development.This paper mainly studies the theory of the operation rules,the aggregation operators and the sort relations of the Pythagorean fuzzy sets.Based on the proposed theories,the multiple attribute decision making methods are studied with the Pythagorean fuzzy information.The main contents of this study are as follows:1.The Pythagorean fuzzy Archimedean operations and operators are studied.A special automorphism on the unit interval is introduced to propose a type of isomorphic Archimedean t-norm and s-norm,and it is used to define the Pythagorean fuzzy Archimedean operational laws.Then,the laws are used to develop the generalized Archimedean Pythagorean fuzzy weighted operators.Furthermore,two degenerate operators of the Archimedean operators are studied: the Pythagorean fuzzy Hamacher operators Frank operators with parameter.Firstly,the Pythagorean fuzzy Hamacher operational laws and Frank operational laws,and these laws are used to develop the Pythagorean fuzzy Hamacher operators and Frank operators,and the the monotonicity and the special cases of operators with respect to the parameter are analyzed.Then,some decision techniques were designed to deal with the Pythagorean fuzzy MADM problems by using these Hamacher operators Frank operators,and the practicability of the proposed decision-making methods is verified by solving the problem of airline service quality evaluation.2.The ranking method,generalized Archimedean operational laws and power operators of interval-valued Pythagorean fuzzy set are studied.In order to distinguish two different interval-valued Pythagorean fuzzy numbers,based on the score and the accuracy functions,the new order relation is defined by introducing the fluctuation function of the interval-valued Pythagorean fuzzy set.Based on the defined isomorphic Archimedean t-norm and s-norm,the Archimedean operational laws of the interval-valued Pythagorean fuzzy set are defined,and the Archimedean interval-valued Pythagorean fuzzy operators are developed.Then,the Archimedean interval-valued Pythagorean fuzzy power operators are proposed by combining the Archimedean interval-valued Pythagorean fuzzy operators with the power operators.Furthermore,two kinds of degenerate operators of the power operators are studied: the interval-valued Pythagorean fuzzy Hamacher power operators and Frank power operators.Firstly,the fuzzy Hamacher and Frank operation rules of interval-valued Pythagorean fuzzy set are defined,and then the fuzzy Hamacher operators and Frank operators are proposed,and the monotonicity and degeneracy of the operators with respect to the parameter are studied.Then,some fuzzy multi-attribute group decision-making methods are constructed by using the proposed operators and the feasibility of the proposed methods are demonstrated by solving the multi-attribute group decision-making problem of investment selection.3.The element repair approach,ranking method,Archimedean operational laws and aggregation operators of the hesitant Pythagorean fuzzy set are studied.Firstly,for two Pythagorean fuzzy numbers with different length,an effective and reasonable repair method is proposed.Then,based on the score function and the accuracy function,the distance measure is introduced to define the polymerization degree function,which can feed back the volatility of the hesitant Pythagorean fuzzy set,and then a new order relation is put forward.Then,the Archimedean operational laws are defined by using the isomorphic Archimedean t-norm and the s-norm.Based on these laws,the Archimedean hesitant Pythagorean fuzzy operators are proposed,and the Archimedean hesitant Pythagorean fuzzy power operators are proposed by combining Archimedean operator and power operators.Furthermore,we study the two classes of degenerate operators of the above power operators: Firstly,the fuzzy Hamacher and Frank operation rules of the hesitant Pythagorean fuzzy set are defined,and then the fuzzy Hamacher operators and the Frank operators of the hesitant Pythagorean fuzzy set are proposed,and the monotonicity and degeneracy of the operators on the parameter are investigated.Then,the fuzzy multi-attribute group decision-making approaches are developed by using the proposed operators,and the effectiveness of the proposed method is verified by solving the multi-attribute group decision problem of in joint research and PhD co-supervision.