Some aggregation operators based on Pythagorean fuzzy numbers and their applications in decision making problems

Abstract

The study of Multi-criteria decision making is that to identify and indicates the best option from all possible available options. Making a decision indicates that there are alternatives choice to be considered, and in such a case we want not only to identify as several of these alternatives as possible but to select that one which is more suitable for our aims, objectives, wants, values, and so on. Thus by decision making we have solve several problems in daily life. For this purpose Zadeh (Zadeh, 1965) presented the concept of fuzzy set theory. In fuzzy set theory he only discussed and conferred the membership function, called the degree of membership. After the presenting of FS theory, Zadeh also developed many applications of the fuzzy set theory in many fields, such as engineering, management science and computer science etc. After the positive and progressive applications and compensations of fuzzy set theory, Atanassov (Atanassov, 1986) utilized the theory of PF and industrialized the idea of a new set known as intuitionistic fuzzy set. IFS having each member in ordered pair form. The theory of IFS is more influential as compared to the theory of FS for the solution of problems. After the presenting and introducing of the theory of IFS, Atanassov and Gargov (Atanassov and Gergov, 1989) used the concep of IFS and presented the idea of another set, called interval-valued intuitionistic fuzzy set, having two elements such as membership and non-membership, whose values are intervals not real numbers. IFS and IVIFS become more popular and more attractive by introducing the various kinds of aggregation operators, information measures and employed them to solve the decision-making problems under the different environments. IFS and IVIFS become more popular and more attractive after the introducing. Several operators are developed using IFVs and IVIFVs. However, it has a shortcoming and limitation, the limitation is that this study is only valid for that situation where the sum of their degrees is less than or equal to one. But there are many problems, which cannot be solved by this study. In order to resolve this type of problems, which are accure in daily life and cannot be solving by IFS, Yager (Yager, 2013) develop and presented the notion of Pythagorean fuzzy set. PFS is the general form of IFS, because IFS is special case of PFS. Many operators have been introduced after the familiarizing and introducing of PFS. In PFS each member can be presented in ordered pair form, where sum of their square is less than are equal to one X. Peng, Y. Yang (Peng and Yang, 2015) presented the idea of interval-valued Pythagorean fuzzy set. This thesis contains eight chapters, which are discussed in the following in detail. In chapter one, we develop some basic and important definitions, which are directly related our work such as, fuzzy set (FS), intuitionistic fuzzy set (IFS), score function, accuracy function, interval-valued intuitionistic fuzzy set (IVIFS), Pythagorean fuzzy set (PFS), interval-valued Pythagorean fuzzy set (IVPFS), and several operators which are already developed. In chapter two, we explore the idea of Pythagorean fuzzy Einstein hybrid averaging (PFEHA) operator along with their properties, namely idempotency, boundedness and monotonicity. To develop the above method, we applied the proposed operator and method to multi-attribute group decision-making to show the validity, practicality and effectiveness of the new approach. In chapter three, we explore the idea of Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator, Pythagorean fuzzy Einstein hybrid geometric (PFEHG) operator along with their properties. In chapter four, we explore the idea of generalized Pythagorean fuzzy Einstein hybrid geometric (GPFEHG) operator, and generalized Pythagorean fuzzy Einstein hybrid averaging (GPFEHA) operator. To improve and develop the above concept, we present some of their basic properties such as, idempotency, boundedness and monotonicity. Finally, we give a numerical example to show the effectiveness and flexibility of the proposed method. All of the above chapters, we established many operators using PFVs, having a single valued for member and non-membership. But in chapter five, we explore the idea of interval-valued in which the member and nonmembership are not single valued but they accared in the form of closed interval. Thus we present the idea of IVPFEWA operator, IVPFEOWA operator and IVPFEHA operator. At the end of the chapter the above proposed operators have been applied to group decision-making problems to show their weight, practicality and efficiency. Like in chapter five, in chapter six, we introduce the notion of a series of geometric interval-valued operators, namely IVPFEWG operator, IVPFEOWG operator and IVPFEHG operator along with their properties namely, commutativity, idempotency, boundedness and monotonicity. In chapter seven, we introduce the notion three generalized operators using IVPVSs, such as GIVPFEWA operator, GIVPFEOWA operator and GIVPFEHA operator. Finally the proposed operators have been applied to decision making problems to show the validity, practicality and effectiveness of the new approach. Actually the operators proposed in chapter five, are the special cases of the new operators developed in chapter seven respectively. In chapter eight, we introduce the notion three generalized operators using IVPVSs, such as GIVPFEWG operator, GIVPFEOWG operator and GIVPFEHG operator. Actually the operators proposed in chapter six, are the special cases of the new operators and methods developed in chapter eight respectively. Finally the proposed operators have been applied to decision making problems to show the validity, practicality and effectiveness of the new approach.