The Research On Aggregation Operators Based On Interval-valued Pythagorean Fuzzy Number

Due to the uncertainty of people's cognition of things and the complexity of the real world,it is very difficult for decision makers to give decision information expressed by crisp numbers.So many scholars at home and abroad have been devoted to extending the concept of numbers,which is from the initial real number to the subsequent fuzzy number,and then to language variables and intuitionistic fuzzy numbers,etc.Because intuitionistic fuzzy number has more advantages in the expression of fuzzy information,it has been rapidly developed and become a hot topic.However,because of the restricted condition u~2+v~2?1 about membership degree and non-membership degree,intuitionistic fuzzy number can not express all fuzzy information.Therefore,Yager presented Pythagorean fuzzy number,which satisfies u~2+v~2?1.Pythagorean fuzzy number can express more fuzzy information.Garg proposed interval-valued Pythagorean fuzzy number by extending the membership degree and non-membership degree to interval numbers,which further enhanced its ability to represent fuzzy information.Interval-valued Pythagorean fuzzy number can describe the fuzzy nature of multi-attribute decision making more accurately.But the existing research results are only preliminary.So this paper will put forward three kinds of aggregation operators based on interval-valued Pythagorean fuzzy number.Then some multi-attribute decision making methods will be presented,which can be used as a guide to the real decision problems.The main work and achievements are as follows:(1)In this paper,based on the definition of interval-valued Pythagorean fuzzy number,we discuss the operational rule,operational property,the distance measure,the expectation and the ranking method of interval-valued Pythagorean fuzzy number.(2)We extend the Power averaging operator and Maclaurin symmetric mean operator to interval-valued Pythagorean fuzzy number.By using the operational rule of interval-valued Pythagorean fuzzy number,we propose the Power averaging operator and Maclaurin symmetric mean operator on the basis of interval-valued Pythagorean fuzzy number.The properties of these operators are discussed and proved,and some methods of multi-attribute group decision making based on these two kinds of operators are proposed.(3)Based on the characteristics of Power averaging operator and Maclaurin symmetric mean operator,we propose Power Maclaurin symmetric mean operator by combining Power averaging operator and Maclaurin symmetric mean operator,and extend it to interval-valued Pythagorean fuzzy number.In addition,we study the properties of the new operators,and establish a multi-attribute group decision making method based on the new operator.Finally,the validity of the method is verified by a numerical example and comparison with other method.