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 centre consists of non-negative real numbers

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 (

The concept of Pythagorean fuzzy set (PFS), first introduced by Atanassov (

Citation graph of the PFSs.

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

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-PFV. 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

The rest of this paper is organized as follows. In Section

Atanassov (

An IFS

The concept of PFS proposed by Yager (

A PFS

Schweizer and Sklar (

A

A

A strictly decreasing function

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

A fuzzy complement is a function

Continuity,

The function

Let

Let

Note that

The notion of C-IFS was introduced by Atanassov (

Let

Since each IFS

Next, we introduce the concept of C-PFS that is a new extension of the concepts of C-IFS and PFS. C-PFSs allow decision makers to express uncertainty via membership and non-membership degrees represented by a circle in a more extended environment. Thus, more sensitive evaluations can be made in decision making process.

Let

Let

Let

A C-PFS can be considered as a collection of C-PFVs. Figure

Geometric representation of C-PFSs.

Comparison of the spaces of C-IFS and C-PFS.

Since each PFS

Now we can define some set operations among C-PFSs.

Let

The complement

The union of

The intersection of

Let

The following theorem shows that De Morgan’s rules are available for C-PFSs.

The proof is trivial from Definition

Now we develop a method to convert collections of PFVs to a C-PFV which is a useful method in group decision making.

We have

We present the following collections of PFVs:

The PFVs and C-PFVs in Example

Now we define some algebraic operations for C-PFVs.

Let

Algebraic operations among C-PFVs in Definition

Let

It is clear that with a particular choice of

We now show that the sum and the product of two C-PFVs are also C-PFVs with the following proposition.

We know that the dual

Klement

Let

The following proposition confirms that multiplication by constant and power of C-PFVs are also C-PFVs.

It is clear from Proposition

Let

The following theorem gives some basic properties of algebraic operations.

(i) and (ii) are trivial.

We obtain

It is obtained that

We get

It is clear that

We have

We have

Aggregation operators (see e.g. Beliakov

Let

It is seen from Proposition

Let

Let

It can be proved similar to Theorem

Let

In this section, we define a similarity measure for C-PFVs. Then using this similarity measure and the proposed aggregation operators we propose an MCDM method in circular Pythagorean fuzzy environment. Then we solve a real world decision problem from Zhang (

Similarity measures have an important role in the determination of the degree of similarity between two objects. Particularly, similarity measures for PFVs or PFSs have been investigated and developed by researchers since they are important tools for decision making, image processing, pattern recognition, classification and some other real life areas. Motivated by the cosine similarity measure for PFVs defined in Wei and Wei (

Let

The proof of (ii) and (iii) is trivial from the definition of

In this sub-section, an MCDM method is proposed in the circular Pythagorean fuzzy environment. The proposed method is applied to an MCDM problem adapted from Zhang (

Due to the scarcity of non-renewable energies and their harmful effects on the environment, the importance of renewable energy sources has increased gradually for supplying plentiful and clean energy. One of the current renewable energy sources is photovoltaic cell, which has almost no negative effects on the environment and is enormously productive. A photovoltaic cell, also known as a solar cell, is an energy generating device that converts solar energy into electricity because of the photovoltaic effect, which is a conversion discovered by Becquerel (

Photovoltaic cells with crystalline silicon (mono-crystalline and poly-crystalline),

Photovoltaic cells with inorganic thin layer (amorphous silicon),

Photovoltaic cells with inorganic thin layer (cadmium telluride/cadmium sulfide and copper indium gallium diselenide/cadmium sulfide),

Photovoltaic cells with advanced III–V thin layer with tracking systems for solar concentration, and

Photovoltaic cells with advanced, low cost, thin layers (organic and hybrid cells).

After viewing the photovoltaic cells determined as alternatives in the study, the criteria considered for the assessment of MCDM are the following:

Now let us consider this problem with the method developed in the present paper. Steps 1–2 are already conducted. Table

Pythagorean fuzzy group decision matrix.

Experts | Alternatives | |||||

Pythagorean fuzzy group normalized decision matrix.

Experts | Alternatives | |||||

Arithmetic average of Pythagorean fuzzy decision matrix.

Alternatives | |||||

Maximum radius lengths based on decision matrices.

Alternatives | |||||

0.16 | 0.07 | 0.09 | 0.19 | 0.24 | |

0.08 | 0.0 | 0.2 | 0.06 | 0.18 | |

0.18 | 0.3 | 0.33 | 0.37 | 0.16 | |

0.26 | 0.15 | 0.07 | 0.23 | 0.07 | |

0.23 | 0.18 | 0.12 | 0.1 | 0.32 |

Circular Pythagorean fuzzy decision matrix.

Aggregated circular Pythagorean fuzzy decision matrix.

Alternatives | ||||

The results of similarity measure between positive ideal alternative and alternatives.

0.325 | 0.493 | 0.465 | 0.411 | ||

0.377 | 0.548 | 0.475 | 0.425 | ||

0.301 | 0.429 | 0.328 | 0.468 | ||

0.311 | 0.439 | 0.343 | 0.488 |

Application of the proposed method to MCDM.

The best alternative remains same with the literature when the aggregation operators

The comparison of the other methods and proposed method.

Methods | Ranking order | Best Alternative |

Zhang ( |
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Biswas and Sarkar ( |
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Biswas and Sarkar ( |
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Proposed Method ( |
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Proposed Method ( |
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Proposed Method ( |
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Proposed Method ( |

The column chart comparison of other methods and the proposed method.

In this section, we investigate the time complexity of the MCDM method given in Section

Time complexity of the proposed MCDM method.

The main goal of this paper is to introduce the concept of C-PFS represented by a circle whose radius is

The authors are grateful to the reviewers for carefully reading the manuscript and for offering many suggestions which resulted in an improved presentation.