Circular Pythagorean fuzzy sets and applications to multi-criteria decision making

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 𝜇 and 𝜈 with the condition 𝜇 2 + 𝜈 2 ≤ 1 . 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 𝑡 -norms and 𝑡 -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


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 nonmembership function such that the sum of the squares of these non-negative functions are less than 1.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].Many types of fuzzy sets study with points, pairs of points or triples of points from the closed interval [0,1] 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 1.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   2 () +   2 () ≤ 1.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.• 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.

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  = { 1 , . . .,   } is a finite set.
The concept of PFS proposed by Yager [31,32] which is a generalization of IFS.Definition 2 [31,32] [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 5 [17,18] A strictly decreasing function : Next, we need the concept of fuzzy complement to find the additive generator of a dual -conorm on [0,1].

Remark 1
Let  be a -norm on [0,1].Then the dual -conorm S with respect to the Pythagorean fuzzy complement  is Note that  is an Archimedean -norm if and only if (, ) <  for all  ∈ (0,1) and  is an Archimedean -conorm if and only if (, ) >  for all  ∈ (0,1) [17,18].Klement and Mesiar [19] proved that continuous Archimedean -norms have representations via their additive generators in the following theorem.

Circular Pythagorean fuzzy sets
The notion of C-IFS was introduced by Atanassov [2] as an extension of the notion of IFS.Throughout this paper we assume that  = { 1 , . . .,   } is a finite set.

Remark 2 Since each IFS 𝐴 has the form
=  0 = {〈,   (),   (); 0〉:  ∈ } any IFS can be considered as a C-IFS.Hence, the notion of C-IFS is a generalization of the notion of 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 nonmembership degrees represented by a circle in a more extended environment.Thus more sensitive evaluations can be made in decision making process.A C-PFS can be considered as a collection of C-PFVs.Figure 3 shows some examples of C-PFVs and Figure 4 shows that the concept of C-PFS generalizes the concept of C-IFS.Proof.We have On the other hand it is clear that 0 ≤   ≤ 1 for each  .Therefore,   = {〈  , (  ), (  ); 〉:   ∈ } is a C-PFS.
We now show that the sum and the product of two C-PFVs are also C-PFVs with the following proposition.
Proposition 2 Let  and  be two C-PFVs.Assume that ,  are dual -norm and -conorm with respect to Pythagorean fuzzy complement () = √1 −  2 , respectively and  is a -norm or a -conorm.Then  ⊕   and  ⊗   are also C-PFVs.
Proof.We know that the dual  -conorm S with respect to the Pythagorean fuzzy = 1.Moreover, since the domain of  is the unit closed interval we conclude that  ⊕   is a C-PFV.Similarly, it can be shown that  ⊗   is a also C-PFV.Klement and Mesiar [19] showed that continuous Archimedean -norms and -conorms can be expressed with their additive generators.Thus, some algebraic operations among C-PFVs can be defined using additive generators of strict Archimedean -norms and -conorms.The following proposition confirms that multiplication by constant and power of C-PFVs are also C-PFVs.
Following theorem gives some basic properties of algebraic operations.

Aggregation Operators For C-PFVs
Aggregation operators (see e.g.[5,12,17]) have an important role while transforming input values represented by fuzzy values to a single output value.In this section, we introduce a weighted arithmetic aggregation operator and a weighted geometric aggregation operator for C-PFVs by using algebraic operations given in Section 3.

An Application of C-PFVs to a MCDM Problem
In this section, we define a similarity measure for C-PFVs.Then using this similarity measure and the proposed aggregation operators we propose a MCDM method in circular Pythagorean fuzzy environment.Then we solve a real world decision problem from the literature [39] that deals with selecting the best photovoltaic cell by utilizing the proposed method.

A Similarity Measure for C-PFVs
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.Motivating from the cosine similarity measure for PFVs defined in [30], we give the following similarity measure for C-PFVs.
Definition 17 Let  = 〈  ,   ;   〉 and  = 〈  ,   ;   〉 be two C-PFVs.The cosine similarity measure CSM is defined by Theorem 6 Let  = 〈  ,   ;   〉 and  = 〈  ,   ;   〉 be two C-PFVs.The cosine similarity measure  based on radius satisfies the following properties: On the other hand the expression is the cosine value of a certain angle in [0, The proof of (ii) and (iii) is trivial from the definition of .

A MCDM method
In this sub-section a MCDM method is proposed in the circular Pythagorean fuzzy environment.The proposed method is applied to a MCDM problem adapted from the literature [39] to show the efficiency of this method in next sub-section.We can present steps of the proposed method as follows: Step 1: Consider a set of  alternatives as  = { 1 , … ,   } evaluated by an expert with respect to a set of  criteria as  = { 1 , … ,   }.
Step 2: The expert expresses the evaluation results of alternatives as C-PFVs according to each criterion and determines the weight vector.
Step 3: If there exists a cost criterion, then the complement operation is taken to the values of this criterion.
Step 4: Using proposed weighted aggregation operators, evaluation results expressed as C-PFVs for each alternatives are transformed to a value expressed as C-PFVs.
Step 5: The cosine similarity measure  between aggregated value of each alternative and positive ideal alternative 〈1,0; 1〉 are calculated.
Step 6: Alternatives are ranked so that the maximum similarity value is the best alternative.

Evaluation of the problem of selecting photovoltaic cells
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 have 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 by the photovoltaic effect, which is a conversion discovered by Becquerel [4].Choosing the best photovoltaic cell has an important role to increase production, to reduce costs and to confer high maturity and reliability.There are many types of photovoltaic cells.The aim of this section is to solve a MCDM problem adapted from the literature [39] about selecting the best photovoltaic cell.In [27], the photovoltaic cells forms the alternatives of MCDM problem and these alternatives are the following: 1 : Photovoltaic cells with crystalline silicon (mono-crystalline and poly-crystalline),  2 : Photovoltaic cells with inorganic thin layer (amorphous silicon),  3 : Photovoltaic cells with inorganic thin layer (cadmium telluride/cadmium sulfide and copper indium gallium diselenide/cadmium sulfide),  4 : Photovoltaic cells with advanced IIIâ€"V thin layer with tracking systems for solar concentration, and  5 : 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: (1)  1 (manufacturing cost), (2)  2 (efficiency in energy conversion), (3)  3 (market share), (4)  4 (emissions of greenhouse gases generated during the manufacturing process), and (5)  5 (energy payback time).It is noted that the criteria  2 and  3 are the benefit criteria, and others are the cost criteria.According to these five criteria, three experts specializing in photovoltaic systems and technologies evaluate these five available photovoltaic cells.The weight vector of the criteria determined by experts is  = (0.2,0.4,0.1,0.1,0.2), and the weight vector of experts is fully unknown (see, [27]).Now let us consider this problem with the method developed in the present paper.Steps 1-2 are already conducted.Table 1 is the decision matrix taken from [27].Step 3: Since  1 ,  4 and  5 are the cost criteria, we take the complement of these values.Thus we obtain Pythagorean fuzzy group normalized decision matrix shown in Table 2.As this decision matrix consists of PFVs we need to convert these values to C-PFVs.For this purpose we use Proposition 1.For example, according to the  1 criterion of the  1 alternative, the evaluation results of the experts are 〈0.4,0.8〉, 〈0.3,0.9〉, 〈0.6,0.8〉, respectively.From Proposition 1, it is seen that the arithmetic average of the evaluation results is 〈0.45,0.83〉and the radius is 0.16.In this way, we attain aggregated Pythagorean fuzzy decision matrix given in Table 3 and maximum radius lengths based on decision matrix listed in Table 4.With C-PFVs the circular Pythagorean fuzzy decision matrix is shown in Table 5.      Step 5: The cosine similarity measure  defined in Definiton 17 is used to measure how each aggregated C-PFV and positive ideal alternative are related or closed to each other.The results of similarity measure between positive ideal alternative and alternatives is shown in Table 7. Step 6: With respect to the aggregation operators    and    we get the ranking  1 ≺  4 ≺  3 ≺  2 ≺  5 and with respect to the aggregation operators    and    we get the ranking  1 ≺  4 ≺  3 ≺  5 ≺  2 .The steps of the proposed method are visualized in Figure 6.

Comparative analysis
The best alternative remains same with the literature when the aggregation operators    and    are used.On the other hand the orders of best and second best alternative interchange when the aggregation operators    and    are used.The worst alternative is totally consistent with the literature.The comparison of the other methods proposed to solve this MCDM problem and the method we propose is shown in Table 8 and illustrated in Figure 7.

Time complexity of the proposed MCDM method
In this sub-section we investigate the time complexity of the MCDM method given in Sub-section 5.2.We assume that  experts assign PFVs to create the decision matrix as in the problem solved in Sub-section 5.3.Essentially the time complexity that depends on the number of times of multiplication, exponential, summation as in [9] and [13]

Conclusion
The main goal of this paper is to introduce the concept of C-PFS represented by a circle whose radius is  and whose center consists is a pair with the condition that sum of the square of the components is less than one.In such a fuzzy set the membership degree and the nonmembership degree are represented by a circle.Thus a C-PFS is a generalization of both C-IFSs and PFSs.C-PFSs allow decision makers or experts to evaluate objects in a larger and more flexible region compared to both C-IFSs and PFSs.Therefore, the change of membership degree and nonmembership degree can be handled to express uncertainty with the help of C-PFSs.In this way, more sensitive decisions can be made.In this paper, a method is developed to transform PFVs to a C-PFS.Also some fundamental set theoretic operations for C-PFSs are given and some algebraic operations for C-PFVs via continuous Archimedean  -norms and  -conorms are introduced.Then with the help of these algebraic operations some weighted aggregation operators for C-PFVs are presented.Inspired by a cosine similarity measure defined between PFVs, we give a cosine similarity measure based on radius to determine the degree of similarity between C-PFVs.Finally, by utilizing the concepts mentioned above we propose a MCDM method in circular Pythagorean fuzzy environment and we apply the proposed method to a MCDM problem from the literature about selecting the best photovoltaic cell (also known as solar cell).We compare the results of the proposed method with the existing results and calculate the time complexity of the MCDM method.In the future studies, different kind of aggregation operators and similarity measures can be investigated.Also while transforming PFVs to a C-PFS other aggregation tool as fuzzy integrals or aggregation operators can be used.Moreover, the proposed method can be used to solve MCDM problems such as classification, pattern recognition, data mining, clustering and medical diagnosis.

Figure 1 :
Figure 1: Citation graph of the PFSs

Figure 2 :
Figure 2: The improvement of circular fuzzy set theory

Figure 5 :
Figure 5: The PFVs and C-PFS in Example 3

𝜋 2 ]
specified by  and .So it is in the unit closed interval.Thus we have 0 ≤ (, ) ≤ 1.

Figure 6 :
Figure 6: Application of the proposed method to MCDM

Figure 8
visualizes the change of the time complexity with respect to the change in the numbers of the criteria, the alternatives and the experts for    ,    ,    and    .

Figure 8 :
Figure 8: Time complexity of the proposed MCDM method Then the pair  = 〈  ,   〉 is called a Pythagorean fuzzy value (PFV).
).Then we obtain Algebraic weighted arithmetic aggregation operators as particular cases of the aggregation operators given in Definition 15 as follows:

Table 1 :
Pythagorean fuzzy group decision matrix

Table 2 :
Pythagorean fuzzy group normalized decision matrix

Table 3 :
Arithmetic Average of Pythagorean fuzzy decision matrix

Table 4 :
Maximum radius lengths based on decision matrices

Table 6 :
Aggregated Circular Pythagorean fuzzy decision matrix

Table 8 :
The comparison of the other methods and proposed method