Circu lar Pythagorean fuzzy sets and a pplications to multi -criteria decision making Mahmut Can Bozyiğit 1 Murat Olgun 2 Mehmet Ünver 2

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Circular Pythagorean fuzzy sets and applications to multi-criteria

decision making

Mahmut Can Bozyiğit, Murat Olgun, Mehmet Ünver

Ankara Yıldırım Beyazıt University, Faculty of Engineering and Natural Sciences,

Department of Mathematics, 06420 Ankara Türkiye

mcbozyigit@ybu.edu.tr

Ankara University, Faculty of Science, Department of Mathematics,

06100 Ankara Türkiye

olgun@ankara.edu.tr, munver@ankara.edu.tr

Abstract

In this paper, we introduce the concept of circular Pythagorean fuzzy set (value) (C-

PFS(V)) as a new generalization of both circular intuitionistic fuzzy sets (C-IFSs) proposed by

Atannassov and Pythagorean fuzzy sets (PFSs) proposed by Yager. A circular Pythagorean fuzzy

set is represented by a circle that represents the membership degree and the non-membership

degree and whose center consists of non-negative real numbers  and  with the condition

. A C-PFS models the fuzziness of the uncertain information more properly thanks to

its structure that allows modelling the information with points of a circle of a certain center and

a radius. Therefore, a C-PFS lets decision makers to evaluate objects in a larger and more flexible

region and thus more sensitive decisions can be made. After defining the concept of C-PFS we

define some fundamental set operations between C-PFSs and propose some algebraic operations

between C-PFVs via general -norms and -conorms. By utilizing these algebraic operations, we

introduce some weighted aggregation operators to transform input values represented by C-PFVs

to a single output value. Then to determine the degree of similarity between C-PFVs we define a

cosine similarity measure based on radius. Furthermore, we develop a method to transform a

collection of Pythagorean fuzzy values to a PFS. Finally, a method is given to solve multi-criteria

decision making problems in circular Pythagorean fuzzy environment and the proposed method

is practiced to a problem about selecting the best photovoltaic cell from the literature. We also

study the comparison analysis and time complexity of the proposed method.

Keywords: Circular Pythagorean fuzzy set, aggregation operators, multi-criteria decision making

1 Introduction

The concept of fuzzy set (FS) was developed by utilizing a function (called membership

function) assigning a value between zero and one as the membership degrees of the elements to

deal with ambiguity in real-life problems. Since the FS theory proposed by Zadeh [37] succeeded

to handle various types of uncertainty, it has been studied in detail by many researchers to model

uncertainty. Later the concept of intuitionistic fuzzy set (IFS), which is an extension of the concept

of FS, was proposed by Atanassov [1] via membership functions and non-membership functions.

The theory of IFS plays an important role in many research areas such as pattern recognition,

multi-criteria decision making (MCDM), data mining, classification, clustering and medical

diagnosis. Many aggregation operators, similarity measures, distance measures and entropy

measures have been developed for IFSs. Particularly, various generalizations of aggregation

operators for IFSs (see e.g. [6, 11]) have been defined via particular types of -norms and -

conorms.

The concept of Pythagorean fuzzy set (PFS) that is a ricing tool in MCDM (see, Figure 1)

was introduced by Yager [31, 32] to research in a wider environment to express uncertainty as a

generalization of the concept of IFS. A PFS is characterized via a membership function and a non-

membership function such that the sum of the squares of these non-negative functions are less

than . Moreover, a PFS has a quadratic form, which means a PFS expands the range of the

change of membership degree and non-membership degree to the unit circle and so is more

capable than an IFS in depicting uncertainty. Yager [32, 33] proposed some aggregation operators

for PFSs. After that, Peng et. al. [25] presented the axiomatic definitions of distance measure,

similarity measure and entropy measure for PFSs. Further studies on MCDM with fuzzy sets and

aggregation operators can be found in [5, 7, 10, 11, 12, 22, 23, 24, 28, 29, 35, 36].

Figure 1: Citation graph of the PFSs

Many types of fuzzy sets study with points, pairs of points or triples of points from the

closed interval  that makes the decision process more strict since they require (decision

makers) DMs to assign precise numbers. To overcome such a strict modelling Atanassov [2]

proposed the concept of circular intuitionistic fuzzy set (C-IFSs). A C-IFS is represented by a circle

standing for the uncertainty of the membership and non-membership functions. That is, the

membership and the non-membership of each element to a C-IFS are shown as a circle whose

center is a pair of non-negative real numbers with the condition that the sum of them is less than

. With the help of C-IFSs, the change of membership degree and non-membership degree can

be handled more sensitively to express uncertainty. Therefore, various types of MCDM methods

have been carried to circular intuitionistic fuzzy environment (see e.g. [3, 15, 16]). In this paper,

we carry the idea of representing membership degree and non-membership degree as circle to

the Pythagorean fuzzy environment by introducing the concept of circular Pythagorean fuzzy set

(C-PFS). In this new fuzzy set notion, the membership and non-membership degrees of an

element to a FS are represented by circles with center  instead of numbers and

with a more flexible condition 



. In this manner, we extend not only the

concept of the PFS, but also the concept of the C-IFS (see Figure 1). Thus the decision making

process become more sensitive since DMs can attain circles with certain properties instead of

precise numbers. Figure 1 illustrates the improvement of circular fuzzy sets.

Figure 2: The improvement of circular fuzzy set theory

Some main contributions of the present paper can be given as follows.

• This paper introduces the concepts of C-PFS and circular Pythagorean fuzzy value

(C-PFV).

• A method is developed to transform a collection of Pythagorean fuzzy values

(PFVs) to a C-PFS. In this way, multi-criteria group decision making (MCGDM) problem can be

relieved.

• The membership and non-membership of an element to a C-PFS are represented

by circles. Thanks to its structure a more sensitive modelling can be done in MCDM theory in

the continuous environment.

• Some algebraic operations are defined for C-PFVs via -norms and -conorms.

With the help of these operations some weighted arithmetic and geometric aggregation

operators are provided. These aggregation operators are used in MCDM and MCGDM.

The rest of this paper is organized as follows. In Section 2, we recall some basic concepts.

In Section 3, we introduce the concept of C-PFS(V) as new generalization of both C-IFSs and PFSs.

We also define some fundamental set theoretic operations for C-PFSs. Then we introduce some

algebraic operations for C-PFVs via continuous Archimedean - norms and - conorms. In

Section 4, we propose some weighted aggregation operators for C-PFVs by utilizing these

algebraic operations. In Section 5, motivating by a cosine similarity measure defined for PFVs in

[30], we define a cosine similarity measure for C-PFVs to determine the degree of similarity

between C-PFVs. Using the proposed similarity measure and the aggregation operators we

provide a MCDM method in circular Pythagorean fuzzy environment. We also apply the proposed

method to a MCDM problem from the literature [39] that deals with selecting the best

photovoltaic cell (also known as solar cell). We compare the results of the proposed method with

the existing result and calculate the time complexity of the MCDM method. In Section 6, we

conclude the paper.

2 Preliminaries

Atanassov [1] introduced the concept of IFS by taking into account the non-membership

functions with a membership functions of FSs. Throughout this section we assume that 

 is a finite set.

Definition 1 [1] An IFS  in  is defined by



where  are functions with the condition



that are called the membership function and the non-membership function, respectively.

The concept of PFS proposed by Yager [31, 32] which is a generalization of IFS.

Definition 2 [31, 32] A PFS  in  is defined by



where  are functions with the condition







that are called the membership function and the non-membership function, respectively. Let

 such that 



. Then the pair  is called a Pythagorean fuzzy

value (PFV).

Schweizer and Sklar [26] introduced the concepts of -norm and -conorm by motivating

the concept of probabilistic metric spaces proposed by Menger [21]. These concepts have

important roles in statistic and decision making. Algebraically, -norms and -conorms are

binary operations defined on the closed unit interval.

Definition 3 [17, 26] A -norm is a function  that satisfies the

following conditions:

(T1)  for all  (border condition),

(T2)  for all  (commutativity),

(T3)  for all  (associativity),

(T4)  whenever  and  for all 

(monotonicity).

Definition 4 [17, 26] A -conorm is a function  that satisfies the

following conditions:

(S1)  for all  (border condition),

(S2)  for all  (commutativity),

(S3)  for all  (associativity),

(S4)  whenever  and  for all 

(monotonicity).

Definition 5 [17, 18] A strictly decreasing function  with  is

called the additive generator of a -norm  if we have  for all

.

Next, we need the concept of fuzzy complement to find the additive generator of a dual

-conorm on .

Definition 6 [31, 32, 34] A fuzzy complement is a function  satisfying

the following conditions:

(N1)  and  (boundary conditions),

(N2)  whenever  for all  (monotonicity),

(N3) Continuity,

(N4)  for all  (involution).

The function  defined by  where  [31,

32] is a fuzzy complement. When ,  becomes the Pythagorean fuzzy complement

.

Definition 7 [20, 34] Let  be a -norm and let  be a -conorm on . If

 and , then  and  are called

dual with respect to a fuzzy complement .

Remark 1 Let  be a -norm on . Then the dual -conorm S with respect to the

Pythagorean fuzzy complement  is



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CircularPythagoreanfuzzysetsandapplicationstomulti-criteriadecisionmakingMahmutCanBozyiğit1,MuratOlgun2,MehmetÜnver2,∗1AnkaraYıldırımBeyazıtUniversity,FacultyofEngineeringandNaturalSciences,DepartmentofMathematics,06420AnkaraTürkiyemcbozyigit@ybu.edu.tr2AnkaraUniversity,FacultyofScience,Departmentof...

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Circu lar Pythagorean fuzzy sets and a pplications to multi -criteria decision making Mahmut Can Bozyiğit 1 Murat Olgun 2 Mehmet Ünver 2

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