Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making

2.1 Pythagorean fuzzy set

Definition 1. [1] Let X be a fixed set. An IFS A is defined as: A = { 〈 x , ( μ A ( x ) , ν A ( x ) ) 〉 | x ∈ X } .(1) where the functions μ _A  (x) , ν _A  (x) : X → [0, 1] satisfy the condition: (μ _A  (x) + ν _A  (x)) ⊆ [0, 1] . μ _A  (x) , v _A  (x) denote, respectively, the degree of membership and the degree of non-membership of the element x ∈ X to the set A. The degree of indeterminacy π _A  (x) =1 - μ _A  (x) - ν _A  (x).

Definition 2. [27, 28]. Let X be a fixed set. A PFS P is defined as: P = { 〈 x , ( μ P ( x ) , ν P ( x ) ) 〉 | x ∈ X } .(2) where the functions μ _P  (x) , ν _P  (x) : X → [0, 1] satisfy the condition: 0 ≤ μ P 2 ( x ) + ν P 2 ( x ) ≤ 1 . μ _P  (x) , v _P  (x) denote, respectively, the degree of membership and the degree of non-membership of the element x ∈ X to the set P. The degree of indeterminacy π P ( x ) = 1 - μ P 2 ( x ) - ν P 2 ( x ) .

For convenience, Zhang and Xu [29] defined α = (μ _α , ν _α ) as the Pythagorean fuzzy number (PFN), where μ _α  ∈ [0, 1] , ν _α  ∈ [0, 1] and 0 ≤ μ α 2 + ν α 2 ≤ 1 . Some operational laws for PFNs were defined as follows:

Definition 3. [28, 29]. Let α = (μ _α , ν _α ) , α₁ = (μ _{α 1} , ν _{α 1} ) and α₂ = (μ _{α 2} , ν _{α 2} ) be three PFNs, then ( 1 ) α c = ( ν α , μ α ) ; ( 2 ) α 1 ⊕ α 2 = ( μ α 1 2 + μ α 2 2 - μ α 1 2 μ α 2 2 , ν α 1 ν α 2 ) ; ( 3 ) α 1 ⊗ α 2 = ( μ α 1 μ α 2 , ν α 1 2 + ν α 2 2 - ν α 1 2 ν α 2 2 ) ; ( 4 ) λ α = ( 1 - ( 1 - μ α 2 ) λ , ( ν α ) λ ) , λ > 0 ; ( 5 ) α λ = ( ( μ α ) λ , 1 - ( 1 - ν α 2 ) λ ) , λ > 0 .

Zhang and Xu [29] also defined the following laws of comparison.

Definition 4. Let α = (μ _α , ν _α ) be a PFN, s ( α ) = μ α 2 - ν α 2 and h ( α ) = μ α 2 + ν α 2 denote the score and the accuracy degree of α, respectively. For two PFNs α₁ = (μ _{α 1} , ν _{α 1} ) , α₂ = (μ _{α 2} , ν _{α 2} ),

If s (α₁) > s (α₂), then α₁ > α₂;

Definition 5. [42, 43]. Let X be a fixed set. An IVPFS P ˜ can be defined as: P ˜ = { 〈 x , ( μ P ˜ ( x ) , ν P ˜ ( x ) ) 〉 | x ∈ X } .(3) where the functions μ P ˜ ( x ) , ν P ˜ ( x ) ⊆ [ 0 , 1 ] satisfy μ P ˜ ( x ) = [ μ P ˜ L ( x ) , μ P ˜ U ( x ) ] , ν P ˜ ( x ) = [ ν P ˜ L ( x ) , ν P ˜ U ( x ) ] , ( μ P ˜ U ( x ) ) 2 + ( ν P ˜ U ( x ) ) 2 ≤ 1 . The degree of indeterminacy, π P ˜ ( x ) = [ π P ˜ L ( x ) , π P ˜ U ( x ) ] , where π P ˜ L ( x ) = 1 - ( μ P ˜ U ( x ) ) 2 - ( ν P ˜ U ( x ) ) 2 and π P ˜ U ( x ) = 1 - ( μ P ˜ L ( x ) ) 2 - ( ν P ˜ L ( x ) ) 2 . If μ P ˜ L ( x ) = μ P ˜ U ( x ) and ν P ˜ L ( x ) = ν P ˜ U ( x ) , then the IVPFS reduces to the PFS.

Zhang [42] defined p ˜ = ( μ p ˜ , ν p ˜ ) = ( [ μ p L , μ p U ] , [ ν p L , ν p U ] ) as the interval-valued Pythagorean fuzzy number (IVPFN), where ( μ p U ) 2 + ( ν p U ) 2 ≤ 1 . For convenience, the pair ( [ μ p L , μ p U ] , [ ν p L , ν p U ] ) is often denoted by ([a, b] , [c, d]), where [a, b] ⊆ [0, 1] , [c, d] ⊆ [0, 1], and 0 ≤ b² + d² ≤ 1.

Definition 6. [42] Let α ˜ = ( [ a , b ] , [ c , d ] ) , α ˜ 1 = ( [ a 1 , b 1 ] , [ c 1 , d 1 ] ) and α ˜ 2 = ( [ a 2 , b 2 ] , [ c 2 , d 2 ] ) be three IVPFNs, then: ( 1 ) α ˜ c = ( [ c , d ] , [ a , b ] ) ; ( 2 ) α ˜ 1 ⊕ α ˜ 2 = ( [ a 1 2 + a 2 2 - a 1 2 a 2 2 , b 1 2 + b 2 2 - b 1 2 b 2 2 ] , [ c 1 c 2 , d 1 d 2 ] ) ; ( 3 ) α ˜ 1 ⊗ α ˜ 2 = ( [ a 1 a 2 , b 1 b 2 ] , [ c 1 2 + c 2 2 - c 1 2 c 2 2 , d 1 2 + d 2 2 - d 1 2 d 2 2 ] ) ; ( 4 ) λ α ˜ = ( [ 1 - ( 1 - a 2 ) λ , 1 - ( 1 - b 2 ) λ ] , [ c λ , d λ ] ) , λ > 0 ; ( 5 ) α ˜ λ = ( [ a λ , b λ ] , [ 1 - ( 1 - c 2 ) λ , 1 - ( 1 - d 2 ) λ ] ) , λ > 0 . Definition 7. [42, 43] Let α ˜ = ( [ a , b ] , [ c , d ] ) be an IVPFN, S ( α ˜ ) = ( a 2 + b 2 - c 2 - d 2 ] ) / 2 and H ( α ˜ ) = ( a 2 + b 2 + c 2 + d 2 ] ) / 2 denote the score and the ccuracy degree of α ˜ , respectively. For two IVPFNs α ˜ 1 = ( [ a 1 , b 1 ] , [ c 1 , d 1 ] ) , α ˜ 2 = ( [ a 2 , b 2 ] , [ c 2 , d 2 ] ) , the following comparison laws can be given:

If S ( α ˜ 1 ) > S ( α ˜ 2 ) , then α ˜ 1 > α ˜ 2 ;

An improved score function for IVPFNs was developed by Garg [59] and Garg [60].

For simplicity, we denote that Θ is the set of all Pythagorean fuzzy values and that Ω is the set of all interval-valued Pythagorean fuzzy values.

Ordered weighted averaging (OWA) operator was introduced by Yager [50], which can be defined as follows:

Definition 8. An OWA operator of dimension n is a mapping OWA : R ⁿ  → R that has an associated weighting ω = (ω₁, ω₂, …, ω _n )  ^T with ω _j  ∈ [0, 1] and ∑ j = 1 n ω j = 1 such that OWA ( a 1 , a 2 , . . . , a n ) = ∑ j = 1 n ω j b j .(4) where b _j is jth largest of the arguments a _i and (a₁, a₂, . . . , a _n ) is a finite collection of argument.

The OWA operator is bounded, idempotent, commutative and monotonic [50].

Based on the operation of PFNs and the OWA operator, Zhang [37] introduced the Pythagorean fuzzy ordered weighted averaging (PFOWA) operator as follows:

Definition 9. [37] Let α _j = (μ _{α j} , ν _{α j} ) (j = 1, 2, ⋯, n) be a collection of PFNs, a PFOWA operator of dimension n is a mapping PFOWA : Θ ⁿ  → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω _n )  ^T with ω _j  ∈ [0, 1] and ∑ j = 1 n ω j = 1 such that ( α 1 , α 2 , … , α n ) = ⊕ j = 1 n ω j α σ ( j ) = ( 1 - ∏ j = 1 n ( 1 - μ α σ ( j ) 2 ) ω j , ∏ j = 1 n ν α σ ( j ) ω j ) .(5) where (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n) such that α_σ(j-1) ≥ α_σ(j) for all j.

A continuous ordered weighted averaging (C-OWA) operator was developed by Yager [52], which is the generalization of the OWA operator, the C-OWA operator can be defined as follows:

Definition 10. A C-OWA operator is a mapping φ : K → R⁺ associated with a BUM function Q such that φ Q ( [ a , b ] ) = ∫ 0 1 [ b + y ( b - a ) ] dQ ( y ) .(6) where [a, b] ∈ K, K is the set of all nonnegative interval numbers, and the basic unit interval monotonic (BUM) function Q satisfies Q (0) =0, Q (1) =1 and Q (x) ≥ Q (y) for any x ≥ y.

Based on the Definition 10, Liu et al. [55] developed continuous ordered weighted quadratic averaging(C-OWQA) operator, which can be defined as follows:

Definition 11. A continuous ordered weighted quadratic averaging(C-OWQA) operator f : K → R⁺ associated with a BUM function Q such that f Q ( [ a , b ] ) = ( ∫ 0 1 [ b 2 + y ( b 2 - a 2 ) ] dQ ( y ) ) 1 2 .(7) where the concepts K and Q are the same as in Definition 10.

If λ = ∫ 0 1 Q ( y ) dy is the attitudinal character of Q, then f _Q  ([a, b]) can be expressed as f Q ( [ a , b ] ) = ( 1 - λ ) a 2 + λ b 2 .(8)

For simplicity, f _Q is denoted by f _λ .

4.1 Weighted C-IVPFOWQA operator

Definition 15. Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ([a _j , b _j ] , [c _j , d _j ]) (j = 1, 2, …, n) be a collection of IVPFNs, a WC-IVPFOWQA operator of dimension n is a mapping WC - IVPFOWQA : Ω ⁿ  → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω _n )  ^T with ω _j  ∈ [0, 1] and ∑ j = 1 n ω j = 1 such that WC - IVPFOWQA ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) = ⊕ j = 1 n ω j g λ ( α ˜ j ) .(19) where λ is the attitudinal character of BUM function Q and g _λ is the C-IVPFOWQA operator.

Theorem 6. Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ( [ a j , b j ] , [ c j , d j ] ) (j = 1, 2, …, n) be a collection of IVPFNs, ω = (ω₁, ω₂, …, ω _n )  ^T be the weight vector of α ˜ j ( j = 1 , 2 , … , n ) with ω _j  ∈ [0, 1] and ∑ j = 1 n ω j = 1 , then the aggregated value deduced from WC-IVPFOWQA operator is a PFN, and WC - IVPFOWQA ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) = ( 1 - ∏ j = 1 n ( 1 - ( f λ ( μ α ˜ j ) ) 2 ) ω j , ∏ j = 1 n ( f 1 - λ ( ν α ˜ j ) ) ω j ) .(20)

Proof. Based on the Equation (10) and the results of Ref.37, we know that Theorem 6 holds.

Example 2. Let α ˜ 1 = ( [ 0.5 , 0.7 ] , [ 0.3 , 0.4 ] ) , α ˜ 2 = ( [ 0.3 , 0.6 ] , [ 0.2 , 0.6 ] ) , α ˜ 3 = ( [ 0.3 , 0.5 ] , [ 0.1 , 0.6 ] ) , and α ˜ 4 = ( [ 0.2 , 0.6 ] , [ 0.3 , 0.8 ] ) be four IVPFNs and ω = (0.2, 0.3, 0.1, 0.4)  ^T be the weight vector of a ˜ j ( j = 1 , 2 , 3 , 4 ) . Let the BUM function Q (y) = y², then λ = ∫ 0 1 y 2 dy = 1 3 . Here we use the WC-IVPFOWQA operator to aggregate these four IVPFNs, from Equation (10), we have g λ ( α ˜ 1 ) = ( ( 1 - 1 / 3 ) × 0 . 5 2 + ( 1 / 3 ) × 0 . 7 2 , ( 1 / 3 ) × 0 . 3 2 + ( 1 - 1 / 3 ) × 0 . 4 2 ) = ( 0 . 5745 , 0 . 3697 ) , g λ ( α ˜ 2 ) = ( ( 1 - 1 / 3 ) × 0 . 3 2 + ( 1 / 3 ) × 0 . 6 2 , ( 1 / 3 ) × 0 . 2 2 + ( 1 - 1 / 3 ) × 0 . 6 2 ) = ( 0 . 4243 , 0 . 5033 ) , g λ ( α ˜ 3 ) = ( ( 1 - 1 / 3 ) × 0 . 3 2 + ( 1 / 3 ) × 0 . 5 2 , ( 1 / 3 ) × 0 . 1 2 + ( 1 - 1 / 3 ) × 0 . 6 2 ) = ( 0 . 3786 , 0 . 4933 ) , g λ ( α ˜ 4 ) = ( ( 1 - 1 / 3 ) × 0 . 2 2 + ( 1 / 3 ) × 0 . 6 2 , ( 1 / 3 ) × 0 . 3 2 + ( 1 - 1 / 3 ) × 0 . 8 2 ) = ( 0 . 3830 , 0 . 6758 ) .

By Equation (20), it follows that WC - IVPFOWQA ( α ˜ 1 , α ˜ 2 , α ˜ 3 , α ˜ 4 ) = ( 1 - ∏ j = 1 4 ( 1 - ( f λ ( μ α ˜ j ) ) 2 ) ω j , ∏ j = 1 4 ( f 1 - λ ( ν α ˜ j ) ) ω j ) = ( 0 . 4431 , 0 . 5313 ) .

Based on the Theorem 6, the WC-IVPFOWQA operator satisfies the properties of the idempotency, boundedness and monotonicity, which can be demonstrated as the following theorems.

Theorem 7. (Idempotency) Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ( [ a j , b j ] , [ c j , d j ] ) (j = 1, 2, …, n) be a collection of IVPFNs and h _λ be the WC-IVPFOWQA operator. If α ˜ j = α ˜ = ( μ α ˜ , ν α ˜ ) = ( [ a , b ] , [ c , d ] ) ( j = 1 , 2 , … , n ) for all j, then h λ ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) = g λ ( α ˜ ) .(21) where λ is the attitudinal character of BUM function Q and g _λ is the C-IVPFOWQA operator.

Proof. If α ˜ i = α ˜ for all j, then based on Theorem 6, we obtain h λ ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) = ( 1 - ∏ j = 1 n ( 1 - ( f λ ( μ α ˜ j ) ) 2 ) ω j , ∏ j = 1 n ( f 1 - λ ( ν α ˜ j ) ) ω j ) = ( 1 - ∏ j = 1 n ( 1 - ( f λ ( μ α ˜ ) ) 2 ) ω j , ∏ j = 1 n ( f 1 - λ ( ν α ˜ ) ) ω j ) = ( 1 - ( 1 - ( f λ ( μ α ˜ ) ) 2 ) ∑ j = 1 n ω j , ( f 1 - λ ( ν α ˜ ) ) ∑ j = 1 n ω j ) = ( f λ ( μ α ˜ ) , f 1 - λ ( ν α ˜ ) ) = g λ ( α ˜ ) .

The proof is completed.

Theorem 8. (Boundedness) Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ( [ a j , b j ] , [ c j , d j ] ) (j = 1, 2, …, n) be a collection of IVPFNs and h _λ be the WC-IVPFOWQA operator, then α ˜ - ≤ h λ ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) ≤ α ˜ + . where α ˜ - = ( min j { f λ ( μ α ˜ j ) } , max j { f 1 - λ ( ν α ˜ j ) } ) , α ˜ + = ( max j { f λ ( μ α ˜ j ) } , min j { f 1 - λ ( ν α ˜ j ) } ) and λ is the attitudinal character of BUM function Q.

Theorem 9. (Monotonicity) Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ( [ a j , b j ] , [ c j , d j ] ) and β ˜ j = ( μ β ˜ j , ν β ˜ j ) = ( [ a β ˜ j , b β ˜ j ] , [ c β ˜ j , d β ˜ j ] ) (j = 1, 2, …, n) be two collections of IVPFNs, h _λ be the WC-IVPFOWQA operator. If f λ ( μ α ˜ j ) ≥ f λ ( μ β ˜ j ) and f 1 - λ ( ν α ˜ j ) ≤ f 1 - λ ( ν β ˜ j ) for all j, then h λ ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) ≥ h λ ( β ˜ 1 , β ˜ 2 , … , β ˜ n ) .

Ordered weighted C-IVPFOWQA (OWC-IVPFOWQA) operator is an extension of the WC-IVPFOWQA operator, which can be defined as follows:

Definition 16. Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ([a _j , b _j ], [c _j , d _j ]) (j = 1, 2, …, n) be a collection of IVPFNs, an OWC-IVPFOWQA operator of dimension n is a mapping OWC - IVPFOWQA : Ω ⁿ  → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω _n )  ^T with ω _j  ∈ [0, 1] and ∑ j = 1 n ω j = 1 such that OWC - IVPFOWQA ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) = ⊕ j = 1 n ω j g λ ( α ˜ σ ( j ) ) .(24) where λ is the attitudinal character of BUM function Q and g _λ is the C-IVPFOWQA operator, (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n) such that g λ ( α ˜ σ ( j - 1 ) ) ≥ g λ ( α ˜ σ ( j ) ) for all j = 1, 2, …, n.

Similarly to the Theorem 6, we have the following Theorem.

Theorem 10. Let α ˜ j = ( μ α ˜ j , ν α ˜ j ) = ([a _j , b _j ], [c _j , d _j ]) (j = 1, 2, …, n) be a collection of IVPFNs, ω = (ω₁, ω₂, . . . , ω _n )  ^T be the weight vector of α ˜ j ( j = 1 , 2 , … , n ) with ω _j  ∈ [0, 1] and ∑ j = 1 n ω j = 1 , then the aggregated value deduced from OWC-IVPFOWQA operator is a PFN, and OWC - IVPFOWQA ( α ˜ 1 , α ˜ 2 , … , α ˜ n ) = ( 1 - ∏ j = 1 n ( 1 - ( f λ ( μ α ˜ σ ( j ) ) ) 2 ) ω j , ∏ j = 1 n ( f 1 - λ ( ν α ˜ σ ( j ) ) ) ω j ) .(25)

Example 3. Let α ˜ 1 = ([0.2, 0.8], [0.4, 0.5]), α ˜ 2 = ([0.3, 0.7], [0.1, 0.6]), α ˜ 3 = ([0.1, 0.3], [0.2, 0.8]), and α ˜ 4 = ([0.4, 0.7], [0.2, 0.5]) be four IVPFNs and ω = (0.4, 0.2, 0.3, 0.1)  ^T be the weight vector of α ˜ j ( j = 1 , 2 , …, 4). Let the BUM function Q (y) = y², then λ = ∫ 0 1 y 2 dy = 1 3 . Here we use the OWC-IVPFOWQA operator to aggregate these four IVPFNs, by Equation (9), we get

g _λ ( α ˜ 1 ) = (0.4899, 0.4690), g _λ ( α ˜ 2 ) = (0.4726, 0.4933), g _λ ( α ˜ 3 ) = (0.1915, 0.6633), g _λ ( α ˜ 4 ) = (0.5196, 0.4243),

Based on Definition 13, we have

S _λ ( g λ ( α ˜ 1 ) ) = 0 . 0200, S _λ ( g λ ( α ˜ 2 ) ) = -0 . 0200, S _λ ( g λ ( α ˜ 3 ) ) = -0 . 4033, S _λ ( g λ ( α ˜ 4 ) ) = 0 . 0900, Since, S _λ ( g λ ( α ˜ 4 ) ) > S _λ ( g λ ( α ˜ 1 ) ) > S _λ ( g λ ( α ˜ 2 ) ) > S _λ (g _λ ( α ˜ 3 ) ) , then g _λ ( α ˜ 4 ) > g _λ ( α ˜ 1 ) > g _λ ( α ˜ 2 ) > g _λ ( α ˜ 3 ) and g _λ ( ( α ˜ ( 1 ) ) ) = g λ ( α ˜ 4 ) = (0.5196, 0.4243), g _λ ( ( α ˜ ( 2 ) ) ) = g λ ( α ˜ 1 ) = (0.4899, 0.4690), g _λ ( ( α ˜ ( 3 ) ) ) = g _λ ( α ˜ 2 ) = (0.4726, 0.4933) , g _λ ( ( α ˜ ( 4 ) ) ) = g _λ ( α ˜ 3 ) = (0.1915, 0.6633).

By Equation (25), it follows that OWC - IVPFOWQA ( α ˜ 1 , α ˜ 2 , α ˜ 3 , α ˜ 4 ) = ( 1 - ∏ j = 1 4 ( 1 - ( f λ ( μ α ˜ σ ( j ) ) ) 2 ) ω j , ∏ j = 1 4 ( f 1 - λ ( ν α ˜ σ ( j ) ) ) ω j ) = ( 0.4788 , 0.4736 ) .