Generalized pythagorean fuzzy point operators and their application in multi-attributes decision making

1 Introduction

Many new means and theories have been brought forward since Zadeh [1] developed the fuzzy set. For example, Na Chen derived some correlation coefficient formulas for HFSs and applied them to clustering analysis under hesitant fuzzy environments [2]. DH Peng presented a generalized hesitant fuzzy synergetic weighted distance (GHFSWD) measure [3]. Based on this, intuitionistic fuzzy set was introduced by Atanassov [4] which led researchers investigating more meaningful conclusions and applied it to resolve issues especially in multicriteria decision making. In recent years, the definition of Pythagorean fuzzy set (PFS) has been discussed by Yager [5] and seen as an important expansion of the intuitionistic fuzzy sets. IFS and PFS both have membership degree and non-membership degree, what differs the two is that the limit of PFS is widen which only needs to meet the requirement that the square sum of the two degrees, rather than just the sum, is equal to or less than one. In a word, the PFS can hold more uncertain information and is able to perform better in practical decision making.

Since it appeared, many scholars have studied it and got a lot of achievements about PFS [6]. For example, Decui Liang introduced HFSs into PFSs and extend the existing research work of PFSs [7]. Yager [5] introduced some new fuzzy weighted average and geometric aggregated operators to treat Pythagorean fuzzy MADM issues. Zhang and Xu [8] proposed a method to discover the best alternative by using the ideal plan under the Pythagorean fuzzy environment. Vahid Mohagheghi [9] offers the newest procedure of a novel polymerization group decision-making, and it can be used to weigh and evaluate data. This method is very flexible and accurate when there is a big difference between judgement of makers. Ting-Yu Chen [10] introduced a method of novel remoteness index-based VIKOR, which can be applied to MCDA issues in PF range. Peijia Ren [11] introduced the TODIM method to deal with the MCDM problems. Harish Garg [12] investigated an interregional value Pythagorean fuzzy set along with two aggregation operators and developed an accuracy function under IVPFS.

Many studies focus on the interval-valued intuitionistic and pythagorean fuzzy set and aggregated operators to solve multicriteria decision making. However, few studies focus on point operators under Pythagorean fuzzy environment. So this paper proposes this theory for the purpose to decrease uncertain information and improve the accuracy of information. As far as we know, Xia [13] discussed many intuitionistic fuzzy point operators which the parameters can distribute membership degree, non-membership degree and hesitant degree for IFS in another way. Based on this, this paper investigates two point operators under pythagorean fuzzy environment and applies it to decision making.

In this paper, the definition of intuitionistic fuzzy sets and pythagorean fuzzy sets are first reviewed and some defined operations for PFS are introduced. Then some related concepts based on intuitionistic fuzzy point operators are given. Further, an in-depth study on the point operators is given, discover some meaningful results and put forward some new ideas. Also an example is introduced to verify these conclusions.

The structure of this paper is arranged as: The second part reviews some fundamental knowledge of IFS and PFS. Then section 3 investigates some new Pythagorean fuzzy point operators. Section 4 introduces Pythagorean fuzzy multi-attributes decision making based on analytic hierarchy procedure. Section 5 draws some concluding remarks.

2 Preliminaries

This section reviews some primary definitions and terms, describe the basic definitions corresponding to fuzzy sets and some principles involved in this paper are shown as follows:

Definition 1. [14] An intuitionistic fuzzy set (IFS) L on a fixed set.X. is delimited as L = { 〈 x , μ L ( x ) , ν L ( x ) 〉 | x ∈ X }(1) with the condition that 0 ≤ μ _L (x) + v _L (x) ≤1, μ _L (x) ≥0 and v _L (x) ≥0 the degree of hesitation is denoted by π _L (x) =1 - μ _L (x) - v _L (x).

For a given x, the pair 〈μ, ν〉 is called intuitionistic fuzzy number (IFN)where u ∈ [0, 1] v ∈ [0, 1] and u + v ∈ [0, 1].

Pythagorean fuzzy set (PFS), which was discussed by Yager [8], can be defined as:

Definition 2. [12] Assume X is a fixed set. We call the triad L as pythagorean fuzzy set (PFS): L = { 〈 x , μ L ( x ) , ν L ( x ) 〉 | x ∈ X }(2)

In which the function μ _L X → [0, 1] delimits the membership and ν _L : X → [0, 1] delimits the non-membership which based on the element x ∈ X of L. For every x ∈ X, it requires the condition that (μ _L (x)) ² + (ν _L (x)) ² ≤ 1. The degree of hesitant-ion is given by π L ( x ) = 1 - ( μ L ( x ) ) 2 - ( ν L ( x ) ) 2 .

In order to be more convenient, Zhang and Xu note 〈μ _L (x), ν _L (x)〉 as a Pythagorean fuzzy number (PFN) which can be simplified as L = 〈 μ _L , ν _L 〉. If it meets the condition that μ _L + ν _L ≤ 1, we can see that the PFN reduces to IFS. So the main difference between the two is their different restrictions, and PFN holds more space than IFN.

For the purpose to contrast with different PFNs, the score function for PFN was defined by Zhang and Xu [12]. Meanwhile, the score-based ranking method was also provided as:

Definition 3. [12] For arbitrary PFN L = 〈 μ _L , ν _L 〉 the score function is delimited as: S ( L ) = ( μ L ) 2 - ( ν L ) 2(3)

Definition 4. [15] For any two PFNs L₁, L₂ let S (L _i ) be the score value of L _i then

If S (L₁) ≺ S (L₂), then L₁ ≺ L₂

Where S (L) ∈ [- 1, 1]

The accuracy function of L can be delimited as H (L) = (μ _L ) ² + (ν _L ) ² in which H (L) ∈ [0, 1]

Definition 5. [16] For three PFNs γ = 〈 μ, ν 〉, γ₁ = 〈 μ₁, ν₁ 〉 and γ₂ =〈 μ₂, ν₂ 〉, λ ≻ 0, the operations defined as follows:

γ 1 ⊕ γ 2 = 〈 μ 1 2 + μ 2 2 - μ 1 2 μ 2 2 , ν 1 ν 2 〉

Since Yager introduced the OWA operator, which is used to aggregate fuzzy numbers, it has received more and more attention. Then he extended it and defined the GOWA operator as follows:

Definition 6. [17] Assume that m dimension of the GOWA operator can be cast light upon GOWA: H ^m → H and owns the shape below: GOWA [ γ 1 , γ 2 . . . γ m ] = ( ∑ j = 1 m ω j b j η ) 1 η(4)

Where η ∈[- ∞, + ∞], ω = (ω₁, ω₂, …, ω _m ) ^T is the relational weighting vector with ω _j ≥ 0, j = 1, 2, …, m, ∑ j = 1 m ω j = 1 , b _j is the jth biggest of γ _i (i = 1, 2, …, m),  H = [0, 1].

The GOWA operator is extended to GIFOWA in order to suit the situations where IFVs are the input parameters.

Definition 7. [18] A GIFOWA operator of dimension m can be seen as a mapping GIFOWA:V _m → V, and owns the form below: GIFOWA ω ( γ 1 , γ 2 , . . . , γ m ) = ( ω 1 γ σ ( 1 ) η ⊕ ω 2 γ σ ( 2 ) η ⊕ . . . ⊕ ω m γ σ ( m ) η ) 1 / - η(5)

Where η ≻ 0, ω = (ω₁, ω₂, …, ω _m ) ^T is the weight vector of (γ₁, γ₂, …, γ _m ), ω _j ≥ 0, j = 1, 2, …, m, ∑ j = 1 m ω j = 1 , and γ_σ(j) is the jth largest of γ _i (i = 1, 2, …, m).

However, the above aggregation operators only use the original information. In some situations, more information from the original one should be get. So point operators are developed to control the membership degree or non-membership degree of the IFVs using different parameters.

Definition 8. [19] Assume that there is an IFS A ={ 〈 x, μ _A (x), ν _A (x) 〉 |x ∈ X }, let κ, λ ∈ [0, 1], Atanassov presented the following operators: D κ ( A ) = { x , 〈 μ A ( x ) + κ π A ( x ) , ν A ( x ) + ( 1 - κ ) π A ( x ) 〉 | x ∈ X }(6)F κ , λ ( A ) = { x , 〈 μ A ( x ) + κ π A ( x ) , ν A ( x ) + λ π A ( x ) 〉 | x ∈ X }(7) where κ + λ ≤ 1.

Note IFS(X) as the set of all IFSs on X. For A ∈ IFS (x), an operator D _{κ x} (A) is defined for each x ∈ X: D κ x ( A ) = { x , 〈 μ A ( x ) + κ x π A ( x ) , ν A ( x ) + ( 1 - κ x ) π A ( x ) 〉 | x ∈ X }(8)

Where κ _x ∈ [0, 1]

Then, an IF point operator is defined for polymerization IFSs:

Definition 9. [20] Assume A ∈ IFS (x), for each x ∈ X, let κ _x , λ _x ∈ [0, 1] and κ _x + λ _x ≤ 1, then an IF point operator F_{κ x , λ x} (A) : IFS (X) → IFS (X) is noted as: F κ x , λ x ( A ) = { x , 〈 μ A ( x ) + κ x π A ( x ) , ν A ( x ) + λ x π A ( x ) 〉 | x ∈ X }(9)

The following are some new point operators delimited for polymerization IFVs:

Definition 10. [21] For an IFV α = (μ _α , ν _α ), let κ _α , λ _α ∈ [0, 1], some point operators: IFV ⟶ IFV are delimited as:

D _{κ α} (α) = (μ _α + κ _α π _α ,  ν _α + (1 - κ _α ) π _α )

where κ _α + λ _α ≤ 1. then we have the following theorem:

Theorem 1. [22] Assumeα = (μ _α , ν _α ) is an IFV, nis a positive integer, letκ _α , λ _α ∈ [0, 1], thenD κ α n ( α ) = ( μ α + κ α π α , ν α + ( 1 - κ α ) π α )(10)F κ α , λ α n ( α ) = ( μ α + κ α π α 1 - ( 1 - κ α - λ α ) n κ α + λ α , ν α + λ α π α 1 - ( 1 - κ α - λ α ) n κ α + λ α )(11) where κ _α + λ _α ≤ 1.

Definition 11. [22] Assume that α _j = (μ _j , ν _j ), j = (1, 2, …, n) is a set of PFNs. So the aggregated value is also a PFN then PFWA ω ( α 1 , α 2 , . . . , α n ) = 〈 1 - ∏ j = 1 n ( 1 - μ j 2 ) ω j , ∏ j = 1 n ν j ω j 〉(12) where ω _j is the weight of α _j (j = 1, 2, …, n), ω _j ∈ [0, 1] and ∑ j = 1 n ω j = 1 .

A new concept of point operators for PFN is developed in this section and some examples are given. Based on definition 8, the following point operators can be developed:

Definition 12. For a pythagorean fuzzy set A ={ 〈 x, μ _A (x), ν _A (x) 〉 |x ∈ X }, let κ, λ ∈ [0, 1], then D κ ( A ) = { x , 〈 μ A 2 ( x ) + κ π A 2 ( x ) , ν A 2 ( x ) + ( 1 − κ ) π A 2 ( x ) 〉 | x ∈ X }(13)F κ , λ ( A ) = { x , 〈 μ A 2 ( x ) + κ π A 2 ( x ) , ν A 2 ( x ) + λ π A 2 ( x ) 〉 | x ∈ X }(14) where κ + λ ≤ 1.

Proof.

Let x = μ A 2 ( x ) + κ π A 2 ( x ) , y = ν A 2 ( x ) + ( 1 - κ ) π A 2 ( x ) , x + y = μ A 2 ( x ) + ν A 2 ( x ) + π A 2 ( x ) = 1

Then 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, so we have x² + y² ≤ 1 satisfy the conditions.

Number (2) is similar to (1).

Definition 13. For a pythagorean fuzzy number α = (μ _α , ν _α ), let κ _α , λ _α ∈ [0, 1], some new point operators: PFV → PFV are defined as:

D κ α ( α ) = ( μ α 2 + κ α π α 2 , ν α 2 + ( 1 − κ α ) π α 2 )

Theorem 2.LetA ∈ PFS (∂), α ∈ ∂, κ _α , λ _α ∈ [0, 1] andκ _α + λ _α ≤ 1.

If μ A 2 ( α ) + ν A 2 ( α ) ≠ 0 for all α ∈ ∂ and κ α = μ A 2 ( α ) μ A 2 ( α ) + ν A 2 ( α ) and λ α = ν A 2 ( α ) μ A 2 ( α ) + ν A 2 ( α ) , Then F_{κ α , λ α} (A) ={ (α, κ _α , λ _α ) |α ∈ ∂ }.

Proof. F κ α , λ α = { ( α , μ A 2 ( α ) + μ A 2 ( α ) μ A 2 ( α ) + ν A 2 ( α ) π A 2 ( α ) , ν A 2 ( α ) + ν A 2 ( α ) μ A 2 ( α ) + ν A 2 ( α ) π A 2 ( α ) ) | α ∈ ∂ } = { α , μ A 2 ( α ) μ A 2 ( α ) + ν A 2 ( α ) , ν A 2 ( α ) μ A 2 ( α ) + ν A 2 ( α ) | α ∈ ∂ } = { ( α , κ α , λ α ) | α ∈ ∂ }(15)

Based on definition 13, assume that F κ α , λ α 0 ( A ) = D κ α 0 ( A ) = 0 , then we have the following theorem:

Theorem 3.Letα = (μ _α , ν _α ) be a PFV, n is noted as a positive integer, takingκ _α , λ _α ∈ [0, 1], then

Proof. π D κ α ( α ) = 1 - μ α 2 - κ α ( 1 - μ α 2 - ν α 2 ) - ν α 2 - ( 1 - κ α ) ( 1 - μ α 2 - ν α 2 ) = 0(16)

So we have π F κ α , λ α ( α ) = 1 - μ α 2 - κ α π α 2 - ν α 2 - λ α π α 2 = ( 1 - κ α - λ α ) π α 2 F κ α , λ α 2 ( α ) = ( μ α 2 + κ α π α 2 + κ α ( 1 - κ α - λ α ) π α 2 , ν α 2 + λ α π α 2 + λ α ( 1 - κ α - λ α ) π α 2 ) π F κ α , λ α ( α ) 2 = 1 - [ μ α 2 + κ α π α 2 + κ α ( 1 - κ α - λ α ) π α 2 ] - [ ν α 2 + λ α π α 2 + λ α ( 1 - κ α - λ α ) π α 2 ] = ( 1 - κ α - λ α ) 2 π α 2 F κ α , λ α 3 ( α ) = ( μ α 2 + κ α π α 2 + κ α ( 1 - κ α - λ α ) π α 2 + κ α ( 1 - κ α - λ α ) 2 π α 2 , ν α 2 + λ α π α 2 + λ α ( 1 - κ α - λ α ) π α 2 + λ α ( 1 - κ α - λ α ) 2 π α 2 ) so we can conclude that F κ α , λ α n ( α ) = ( μ α 2 + κ α π α 2 1 − ( 1 − κ α − λ α ) n κ α + λ α , ν α 2 + λ α π α 2 1 − ( 1 − κ α − λ α ) n κ α + λ α ) π F κ α , λ α ( α ) n = ( 1 - κ α - λ α ) n π α 2

From the above we can draw the following completion.

Theorem 4.LetM ∈ PFS (∂), nis a positive integer. Assumingκ _α , λ _α ∈ [0, 1] andκ _α + λ _α ≤ 1 for eachα ∈ ∂, we have: F k α , λ α n − 1 ( M ) ≤ F k α , λ α n ( M ) π F k α , λ α n ( M ) ( α ) ≤ π F k α , λ α n − 1 ( M ) ( α ) for any α ∈ ∂ .

Where for any M, N ∈ PFS (∂), M ≤ N only when μ _M (α) ≤ μ _N (α) and ν _M (α) ≤ ν _N (α) hold for all α ∈ ∂.

Definition 14. Let κ _α , λ _α ∈ [0, 1] and κ _α + λ _α ≤ 1 for each α ∈ ∂, A ∈ PFS (∂). We define the following limit: lim n → ∞ F k α , λ α n ( A ) = { ( α , lim n → ∞ μ F k α , λ α n ( A ) ( α ) , lim n → ∞ ν F k α , λ α n ( A ) ( α ) ) | α ∈ ∂ }(17)

Theorem 5.AssumeA ∈ PFS (∂), taking κ _α , λ _α ∈ [0, 1] andκ _α + λ _α ≤ 1 for allα ∈ ∂, then

lim n → ∞ F k α , λ α n ( A ) = D k α k α + λ α ( A )

Proof.

(2) is obvious. Prove (1) lim n → ∞ μ F k α , λ α n ( A ) ( α ) = lim n → ∞ [ μ A 2 ( α ) + κ α π A 2 ( α ) 1 - ( 1 - κ α - λ α ) n κ α + λ α ] = μ A 2 ( α ) + k α k α + λ α π A 2 ( α ) lim n → ∞ ν F k α , λ α n ( A ) ( α ) = lim n → ∞ [ ν A 2 ( α ) + λ α π A 2 ( α ) 1 - ( 1 - κ α - λ α ) n κ α + λ α ] = ν A 2 ( α ) + λ α k α + λ α π A 2 ( α )

So we have lim n → ∞ F k α , λ α n ( A ) = { ( α , μ A 2 ( α ) + k α k α + λ α π A 2 ( α ) , ν A 2 ( α ) + λ α k α + λ α π A 2 ( α ) ) } = D k α k α + λ α ( A )

By Definition 12, it is easy to get that the point operators convert an PFV into another form which has less uncertainty. Based on definition 6 and information introduced aboved, a new series of aggregation operators are introduced noted as PFWAD(F) and GPFWAD(F).

Definition 15. Assume that α _j = (μ _{α j} , ν _{α j} ) (j = 1, 2, …, n) is a set of PFVs. n is a positive integer, κ _{α j} , λ _{α j} ∈ [0, 1], ω = (ω₁, ω₂, …, ω _n ) ^T is a weight vector corresponding to PFWAD(F) operators, ω _j ≥ 0 and ∑ j = 1 n ω j = 1 , Then the PFWAD (F) aggregation value is also PFV, and P F W A D ( α 1 , α 2 , ... , α n ) = ω 1 ( D κ α 1 ( α 1 ) ) ⊕ ω 2 ( D κ α 2 ( α 2 ) ) ⊕ ... ⊕ ω n ( D κ α n ( α n ) ) = 〈 1 - ∏ j = 1 n ( 1 − ( μ α j 2 + κ α j π α j 2 ) 2 ) ω j , ∏ j = 1 n ( ν α j 2 + ( 1 − κ α j ) π α j 2 ) ω j 〉(18)P F W A F ( α 1 , α 2 , ... , α n ) = ω 1 ( F κ α 1 ( α 1 ) ) ⊕ ω 2 ( F κ α 2 ( α 2 ) ) ⊕ ... ⊕ ω n ( F κ α n ( α n ) ) = 〈 1 - ∏ j = 1 n ( 1 − ( μ α j 2 + κ α j π α j 2 ) 2 ) ω j , ∏ j = 1 n ( ν α j 2 + λ α j π α j 2 ) ω j 〉(19)GPFWAD ω n ( α 1 , α 2 , . . . , α m ) = ( ∑ i = 1 m ω i ( D κ α i n ( α i ) ) p ) 1 / p ( 1 - ∏ i = 1 m ( 1 - ( μ α i 2 + κ α i π α i 2 ) 2 p ) ω i ) 1 / p , 1 - [ 1 - ∏ i = 1 m ( 1 - ( 1 - ( ν α i 2 + ( 1 - κ α i ) π α i 2 ) 2 ) p ) ω i ] 1 / p 〉(20) GPFWAF ω n ( α 1 , α 2 , . . . , α m ) = ( ∑ i = 1 m ω i ( F κ α i , λ α i n ( α i ) ) p ) 1 / p = 〈 ( 1 - ∏ i = 1 m ( 1 - μ F κ α i , λ α i n ( α i ) 2 p ) ω i ) 1 / p , 1 - [ 1 - ∏ i = 1 m ( 1 - ( 1 - ν F κ α i , λ α i n 2 ( α i ) ) p ) ω i ] 1 / p 〉(21)

Which κ _{α i} + λ _{α i} ≤ 1, i = 1, 2, …, m and μ F κ α i , λ α i n ( α i ) = μ α i 2 + κ α i π α i 2 1 - ( 1 - κ α i - λ α i ) n κ α i + λ α i , ν F κ α i , λ α i n ( α i ) = ν α i 2 + λ α i π α i 2 1 - ( 1 - κ α i - λ α i ) n κ α i + λ α i

Theorem 6.Assumeα _j = (μ _{α j} , ν _{α j} ) (j = 1, 2, …, m) is a set of PFVs. n is noted as a positive integer, κ _{α j} , λ _{α j} ∈ [0, 1], ω = (ω₁, ω₂, …, ω _n ) ^T is the weighting vector corresponding to the GPFWA operators, ω _j ≥ 0 and ∑ j = 1 n ω j = 1 , then

α D n - ≤ GPFWAD ω n ( α 1 , α 2 , . . . , α m ) ≤ α D n + .

where κ _{α j} + λ _{α j} ≤ 1, j = 1, 2…, m. α D n - = ( min j ( μ D κ α j n ( α j ) ) , max j ( ν D κ α j n ( α j ) ) ) , α D n + = ( max j ( μ D κ α j n ( α j ) ) , min j ( ν D κ α j n ( α j ) ) ) α F n - = ( min j ( μ F κ α j , λ α j n ( α j ) ) , max j ( ν F κ α j , λ α j n ( α j ) ) ) , α F n + = ( max j ( μ F κ α j , λ α j n ( α j ) ) , min j ( ν F κ α j , λ α j n ( α j ) ) )

This section introduces Pythagorean analytic hierarchy process to cope with multi-criteria decision making issues. Analytic Hierarchy Process (AHP) takes the research object as a system and makes decisions according to the methods of decomposition, comparison and judgment. It can transfer multi-objective, multi-criteria decision-making problems into multi-level single-goal problem, through the comparison of each other to determine the relationship between elements of the same level and the previous level, and finally conduct a simple mathematical operation.

The main steps can be shown as follows:

Step 1. Establish an evaluation index system.

Step 2. Constructing Pythagorean Complementary Judgment Matrix.

Step 3. Multiplicative consistency checking on Pythagorean Complementary Judgment Matrix.

Step 4. Confirm the weight of the indicator.

Step 5. Comprehensive ability evaluation.