Some Induced Averaging Aggregation Operators Based on Pythagorean Fuzzy Numbers

: In this paper we present two new types aggregation operators such as, induced Pythagorean fuzzy ordered weighted averaging aggregation operator and induced Pythagorean fuzzy hybrid averaging aggregation operator. We also discuss of important properties of these proposed operators and construct some examples to develop these operators.


Introduction
Atanassov [1] introduced the concept of IFS characterized by a membership function and a non-membership function. It is more suitable for dealing with fuzziness and uncertainty than the ordinary fuzzy set developed by Zadeh [2] characterized by one membership function. Gau and Buehrer [3] proposed the notion of vague set. Chen and Tan [4] and Hong and Choi [5] presented some techniques for handling multi-criteria fuzzy decision-making problems based on vague sets. Bustine and Burillo [6] showed that the vague set is equivalent to IFS. In 1986, many scholars [7,8,9,10,11,12] have done works in the field of AIFS and its applications. Particularly, information aggregation is a very crucial research area in IFS theory that has been receiving more and more focus. Xu [13] developed some basic arithmetic aggregation operators, including IFWA operator, IFOWA operator and IFHA operator and applied them to group decision making. Xu and Yager [14] defined some basic geometric aggregation operators such as, IFWG operator, IFOWG operator and IFHG operator, and applied them to multiple attribute decision making (MADM) based on intuitionistic fuzzy information. Wang and Liu [15] introduced the notion of IFEWG operator and geometric IFEOWG operator and applied them to group decision making. In [16] Wang and Liu also introduced the concept of IFEWA operator and IFEOWA operator. Zhao and Wei [17] introduced the notion of two new types of hybrid aggregation operators such as, IFEHA operator and IFEHG operator. But there are many cases where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that their sum is greater than one. Therefore, Yager [18] introduced the concept of PFS. PFS is more powerful tool to solve uncertain problems. In 2013, Yager and Abbasov [19] introduced the notion of two new Pythagorean fuzzy aggregation operators such as PFWA operator and PFOWA operator. In [20,21,22,23] K. Rahman et al. introduced the concept of PFHA operator, PFWG operator, PFOWG operator and PFHG operator.
Thus keeping the advantages of the above mention aggregation operators in this article we introduce the notion of two new types aggregation operators based on PFNs, such as, induced Pythagorean fuzzy ordered weighted averaging (I-PFOWA) operator and induced Pythagorean fuzzy hybrid averaging (I-PFHA) operator. We also discuss some of their basic properties including idempotency, boundedness, commutativity and monotonicity. We also give some examples to develop these proposed operators.
The remainder paper can be constructed as. In Section 2, we present some basic definitions which will be used in our later sections. In Section 3, we introduce the notion of induced Pythagorean fuzzy ordered weighted averaging (I-PFOWA) operator and induced Pythagorean fuzzy hybrid averaging (I-PFHA) operator. In Section 4, we have conclusion.
The proof is complete Proof: Since From equation ( )

Conclusion
In this paper, we have familiarized the idea of induced Pythagorean fuzzy ordered weighted averaging (I-PFOWA) operator and induced Pythagorean fuzzy hybrid averaging (I-PFHA) operator and also discussed some of their basic properties.