Pythagorean Hesitant Fuzzy Hamacher Aggregation Operators in Multiple-Attribute Decision Making

2.1 Pythagorean Fuzzy Set

The basic concepts of PFSs [73, 74] are briefly reviewed in this section.

Definition 1 ([73, 74]). Let X be a fix set. A PFS is an object having the form

where the function μ_P : X→[0, 1] defines the degree of membership and the function ν_P : X→[0, 1] defines the degree of non-membership of the element x∈X to P, respectively, and, for every x∈X, it holds that

Definition 2 ([36]). Let p˜1=(μ1, ν1),p˜2=(μ2, ν2), and p˜=(μ, ν) be three PFNs, and some basic operations on them are defined as follows:

1. p˜1⊕⌢p˜2=((μ1)2+(μ2)2−(μ1)2(μ2)2, ν1ν2);

2. p˜1⊗⌢p˜2=(μ1μ2, (ν1)2+(ν2)2−(ν1)2(ν2)2);

3. λp˜=(1−(1−μ2)λ, νλ), λ>0;

4. (p˜)λ=(μλ, 1−(1−ν2)λ), λ>0;

5. p˜c=(ν, μ).

2.2 Pythagorean Hesitant Fuzzy Set

On the basis of the PFS [73, 74] and hesitant fuzzy set (HFS) [14, 27, 44, 47, 52, 62, 64, 65, 67, 68, 80, 81, 82, 84, 86], in the following, we propose the concept of PHFS, which permit the membership of an element to be a set of several possible PFNs. The motivation is that when defining the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error (as in PFSs) or some possibility distribution (as in type 2 fuzzy sets) on the possible values, but because we have several possible PFNs.

Definition 3. Given a fixed set X, then a PHFS on X is given in terms of a function that when applied to X returns a subset of Q. The PHFS can be expressed by the following mathematical symbol:

where h_P (x) is a set of some PFNs in Q, denoting the possible membership degree and non-membership degree of the element x∈X to the set P. For convenience, we call p=h_P (x) a PHFN and P the set of all PHFNs. If α∈p, then α is a PFN, and it can be denoted by α=(μ, ν) and μ²+ν²≤1.

For any α∈p, if α is a real number in [0, 1], then p reduces to a hesitant fuzzy number (HFN) [44]; if α is a closed subinterval of the unit interval, then p reduces to an interval-valued HFN (IVHFN) [58, 61]; if α is an intuitionistic fuzzy number [1], then p reduces to an intuitionistic HFN (IHFN). Therefore, HFNs, IVHFNs, and IHFNs are special cases of PHFNs.

To compare the PHFNs, we shall give the following comparison laws:

Definition 4. Let p_i =(μ_i , ν_i )(i=1, 2) be any two PHFNs, s(pi)=1#pi∑i=1#pi1+γi2−ηi22 is the score function of p_i = (μ_i , ν_i )(i=1, 2), and h(pi)=1#pi∑i=1#pi(γi2+ηi2) the accuracy function of p_i =(μ_i , ν_i )(i=1, 2), where #p_i is the numbers of the elements in p_i , (γ_i , η_i )∈p_i , i=1, 2; then

• If s(p₁)>s(p₂), then p₁ is superior to p₂, denoted by p₁≻p₂;

• If s(p₁)=s(p₂), then

  1. If h(p₁)=h(p₂), then p₁ is equivalent to p₂, denoted by p₁~p₂;

  2. If h(p₁)>h(p₂), then p₁ is superior to p₂, denoted by p₁≻p₂.

Then, we define some new operations on the PHFNs p₁, p₂ and p:

1. λp=∪(γ,η)∈(μ,ν){(1−(1−γ2)λ, ηλ)}, λ>0;

2. pλ=∪(γ,η)∈(μ,ν){(γλ, 1−(1−η2)λ)}, λ>0;

3. p1⊕p2=∪(γ1,η1)∈(μ1,ν1),(γ2,η2)∈(μ2,ν2){((γ1)2+(γ2)2−(γ1)2(γ2)2, η1η2)};

4. p1⊗p2=∪(γ1,η1)∈(μ1,ν1),(γ2,η2)∈(μ2,ν2){(γ1γ2, (η1)2+(η2)2−(η1)2(η2)2)};

5. pc=(ν, μ).

2.3 Hamacher Operations

T-norm and t-conorm are important notions in fuzzy set theory, which are used to define a generalized union and intersection of fuzzy sets [8]. Roychowdhury and Wang [40] gave the definition and conditions of t-norm and t-conorm. Based on a t-norm (T) and t-conorm (T^∗), a generalized union and a generalized intersection of IFSs were introduced by Deschrijver and Kerre [7]. Further, Hamacher [12] proposed a more generalized t-norm and t-conorm. The Hamacher operation [12] includes the Hamacher product and Hamacher sum, which are examples of t-norms and t-conorms, respectively. They are defined as follows:

The Hamacher product ⊗ is a t-norm and the Hamacher sum ⊕ is a t-conorm, where

Especially, when γ=1, then Hamacher t-norm and t-conorm will reduce to

which are the algebraic t-norm and t-conorm, respectively; when γ=2, then the Hamacher t-norm and t-conorm will reduce to

which are called the Einstein t-norm and t-conorm, respectively [45].

2.4 Hamacher Operations of Pythagorean Hesitant Fuzzy Set

Motivated by the arithmetic aggregation operators [29, 30, 31, 32, 33, 72], the Hamacher product ⊗ and the Hamacher sum ⊕, then the generalized intersection and union on two PHFNs p₁ and p₂ become the Hamacher product (denoted by p₁⊗p₂) and Hamacher sum (denoted by p₁⊕p₂) of two PHFNs p₁ and p₂, γ>0, respectively, as follows:

1. λp1=∪(γ1,η1)∈(μ1,ν1){((1+(γ−1)(γ1)2)λ−(1−(γ1)2)λ(1+(γ−1)(γ1)2)λ+(γ−1)(1−(γ1)2)λ, γ(η1)λ(1+(γ−1)(1−(η1)2))λ+(γ−1)(η1)2λ)}λ>0;

2.  (p1)λ=∪(γ1,η1)∈(μ1,ν1){(γ(γ1)λ(1+(γ−1)(1−(γ1)2))λ+(γ−1)(γ1)2λ, (1+(γ−1)(η1)2)λ−(1−(η1)2)λ(1+(γ−1)(η1)2)λ+(γ−1)(1−(η1)2)λ)}λ>0;

3. p1⊕p2=∪(γi,ηi)∈(μi,νi), i=1, 2{((γ1)2+(γ2)2−(γ1)2(γ2)2−(1−γ)(γ1)2(γ2)21−(1−γ)(γ1)2(γ2)2, η1η2γ+(1−γ)((η1)2+(η2)2−(η1)2(η2)2))};

4. p1⊗p2=∪(γi,ηi)∈(μi,νi), i=1, 2{(γ1γ2γ+(1−γ)((γ1)2+(γ2)2−(γ1)2(γ2)2), (η1)2+(η2)2−(η1)2(η2)2−(1−γ)(η1)2(η2)21−(1−γ)(η1)2(η2)2)}.

3.1 Pythagorean Hesitant Fuzzy Hamacher Arithmetic Aggregation Operators

In the following, we shall develop some Pythagorean hesitant fuzzy Hamacher arithmetic aggregation operator based on the operations of Pythagorean hesitant fuzzy Einstein and Hamacher operations.

Definition 5. Let p_j (j=1, 2, …, n) be a collection of PHFNs; then, we define the PHFHWA operator as follows:

where ω=(ω₁, ω₂, …, ω_n )^T be the weight vector of p_j (j=1, 2, …, n), and ω_j >0, ∑j=1nωj=1.

Based on Hamacher sum operations of the PHFNs described, we can drive Theorem 1.

Theorem 1.Let p_j (j=1, 2, …, n) be a collection of PHFNs; then, their aggregated value by using the PHFHWA operator is also a PHFN, and

whereω=(ω₁, ω₂, …, ω_n )^Tbe the weight vector ofp_j (j=1, 2, …, n), andω_j >0, ∑j=1nωj=1,γ>0.

Now, we can discuss some special cases of the PHFHWA operator with respect to the parameterγ.

• Whenγ=1, the PHFHWA operator reduces to the Pythagorean hesitant fuzzy weighted average (PHFWA) operator as follows:

  (12)PHFWAω(p1, p2, …, pn)=∪(γj,ηj)∈(hj,gj), j=1,2,…,n{(1−∏j=1n(1−(γj)2)ωj, ∏j=1n(ηj)ωj)}

• Whenγ=2, the PHFHWA operator reduces to the Pythagorean hesitant fuzzy Einstein weighted average (PHFEWA) operator as follows:

  (13)PHFEWAω(p1,p2, …, pn) =∪(γj,ηj)∈(hj,gj), j=1,2,…,n{(∏j=1n(1+(γj)2)ωj−∏j=1n(1−(γj)2)ωj∏j=1n(1+(γj)2)ωj+∏j=1n(1−(γj)2)ωj, 2∏j=1n(ηj)ωj∏j=1n(2−(ηj)2)ωj+∏j=1n(ηj)2ωj)}

Definition 6. Let p_j (j=1, 2, …, n) be a collection of PHFNs; then, we define the PHFHOWA operator as follows:

where (σ(1), σ(2), …, σ(n)) is a permutation of (1, 2, …, n), such that p_σ(j−1)≥p_σ(j) for all, j=2, …, n and w=(w₁, w₂, …, wn)^T is the aggregation-associated weight vector such that w_j ∈[0, 1] and ∑j=1nwj=1.

Based on Hamacher sum operations of the PHFNs described, we can drive Theorem 2.

Theorem 2.Letp_j (j=1, 2, …, n) be a collection of PHFNs, then their aggregated value by using the PHFHOWA operator is also a PHFN, and

where (σ(1), σ(2), …, σ(n)) is a permutation of (1, 2, …, n), such that p_σ(j−1)≥p_σ(j)for all j=2, …, n, and w=(w₁, w₂, …, w_n )^Tis the aggregation-associated weight vector such that w_j ∈[0, 1] and ∑j=1nwj=1, γ>0.

Now, we can discuss some special cases of the PHFHOWA operator with respect to the parameter γ.

• When γ=1, the PHFHOWA operator reduces to the Pythagorean hesitant fuzzy ordered weighted average (PHFOWA) operator as follows:

  (16)PHFOWAw(p1, p2, …, pn) =∪(γσ(j),ησ(j)) ∈(hσ(j),gσ(j)), j=1, 2, …, n{(1−∏j=1n(1−(γσ(j))2)wj, ∏j=1n(ησ(j))ωj)}.

• When γ=2, the PHFHOWA operator reduces to the Pythagorean hesitant fuzzy Einstein ordered weighted average (PHFEOWA) operator as follows:

  (17)PHFEOWAw(p1, p2, …, pn)=∪(γσ(j),ησ(j))∈(hσ(j),gσ(j)), j=1,2,…,n{(∏j=1n(1+(γσ(j))2)wj−∏j=1n(1−(γσ(j))2)wj∏j=1n(1+(γσ(j))2)wj+∏j=1n(1−(γσ(j))2)wj. 2∏j=1n(ησ(j))ωj∏j=1n(2−(ησ(j))2)ωj+∏j=1n(ησ(j))2ωj)}

From Definitions 5 and 6, we know that the PHFHWA operator weights the Pythagorean hesitant fuzzy argument itself, while the PHFHOWA operator weights the ordered positions of the Pythagorean hesitant fuzzy arguments instead of weighting the arguments themselves. Therefore, weights represent different aspects in both the PHFHWA and PHFHOWA operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose a PHFHHA operator.

Definition 7. A PHFHHA operator is defined as follows:

where w=(w₁, w₂, …, w_n ) is the associated weighting vector, with w_j ∈[0, 1], ∑j=1nwj=1, and p˙σ(j) is the j^th largest element of the Pythagorean hesitant fuzzy arguments p˙j(p˙j=(nωj)pj, j=1, 2, …, n),ω=(ω₁, ω₂, …, ω_n ) is the weighting vector of Pythagorean hesitant fuzzy arguments p_j (j=1, 2, …, n), with ω_j ∈[0, 1], ∑j=1nωj=1, and n is the balancing coefficient. Especially, if w=(1/n, 1/n, …, 1/n)^T , then PHFHHA is reduced to the PHFHWA operator; if ω=(1/n, 1/n, …, 1/n), then PHFHHA is reduced to the PHFHOWA operator.

Based on Hamacher sum operations of the PHFNs described, we can drive Theorem 3.

Theorem 3.Letpj(j=1, 2, …, n) be a collection of PHFNs; then, their aggregated value by using the PHFHHA operator is also a hesitant fuzzy Einstein, and

where w=(w₁, w₂, …, w_n ) is the associated weighting vector, with w_j ∈[0, 1], ∑j=1nwj=1,andp˙σ(j)is the j^thlargest element of the Pythagorean hesitant fuzzy argumentsp˙j(p˙j=(nωj)pj, j=1, 2, …, n),ω=(ω₁, ω₂, …, ω_n ) is the weighting vector of Pythagorean hesitant fuzzy argumentsp_j (j=1, 2, …, n), withω_j ∈[0, 1], ∑j=1nωj=1,andnis the balancing coefficient,γ>0.

Now, we can discuss some special cases of the PHFHHA operator with respect to the parameterγ.

• Whenγ=1, the PHFHHA operator reduces to the Pythagorean hesitant fuzzy hybrid average (PHFHA) operator as follows:

  (20)PHFHAω,w(p1, p2, …, pn)=⊕j=1n(wjp˙σ(j))=∪(γ˙σ(j),η˙σ(j))∈(h˙σ(j),g˙σ(j)),j=1,2,…,n{(1−∏j=1n(1−(γ˙σ(j))2)wj, ∏j=1n(η˙σ(j))wj)}.

• When γ=2, PHFHHA operator reduces to the Pythagorean hesitant fuzzy Einstein hybrid average (PHFEHA) operator as follows:

  (21)PHFEHAw,ω(p1, p2, …, pn)=⊕j=1n(wjp˙σ(j))=∪(γ˙σ(j),η˙σ(j))∈(h˙σ(j), g˙σ(j)), j=1,2,…,n{(∏j=1n(1+(γ˙σ(j))2)wj−∏j=1n(1−(γ˙σ(j))2)wj∏j=1n(1+(γ˙σ(j))2)wj+∏j=1n(1−(γ˙σ(j))2)wj, 2∏j=1n(η˙σ(j))ωj∏j=1n(2−(η˙σ(j))2)ωj+∏j=1n(η˙σ(j))2ωj)}

3.2 Pythagorean Hesitant Fuzzy Hamacher Geometric Aggregation Operators

Based on the Pythagorean hesitant fuzzy Hamacher arithmetic aggregation operators and the geometric mean [6, 18, 17, 57, 63, 70, 88], here we define some Pythagorean hesitant fuzzy Hamacher geometric aggregation operators:

Definition 8. Let p_j (j=1, 2, …, n) be a collection of PHFNs; then, we define the PHFHWG operator as follows:

where ω=(ω₁, ω₂, …, ω_n )^T be the weight vector of p_j (j=1, 2, …, n), and ω_j >0, ∑j=1nωj=1.

Based on Hamacher product operations of the PHFNs described, we can drive Theorem 4.

Theorem 4.Letp_j (j=1, 2, …, n) be a collection of PHFNs; then, their aggregated value by using the PHFHWG operator is also a PHFN, and

where ω=(ω₁, ω₂, …, ω_n )^Tbe the weight vector ofp_j (j=1, 2, …, n), andω_j >0, ∑j=1nωj=1,γ>0.

Now, we can discuss some special cases of the PHFHWG operator with respect to the parameterγ.

• Whenγ=1, the PHFHWG operator reduces to the Pythagorean hesitant fuzzy weighted geometric (PHFWG) operator as follows:

  (24)PHFWGω(p1, p2, …, pn)=∪(γj,ηj)∈(hj,gj),j=1,2,…,n{(∏j=1n(γj)ωj,1−∏j=1n(1−(ηj)2)ωj)}.

• When γ=2, the PHFHWG operator reduces to the Pythagorean hesitant fuzzy Einstein weighted geometric (PHFEWG) operator as follows:

  (25)PHFEWGω(p1, p2, …, pn)=∪(γj,ηj)∈(hj,gj),j=1,2,…,n{(2∏j=1n(γj)ωj∏j=1n(2−(γj)2)ωj+∏j=1n(γj)2ωj, ∏j=1n(1+(ηj)2)ωj−∏j=1n(1−(ηj)2)ωj∏j=1n(1+(ηj)2)ωj+∏j=1n(1−(ηj)2)ωj)}

Definition 9. Let p_j (j=1, 2, …, n) be a collection of PHFNs; then, we define the PHFHOWG operator as follows:

where (σ(1), σ(2), …, σ(n)) is a permutation of (1, 2, …, n), such that p_σ(j−1)≥p_σ(j) for all j=2, …, n, and w=(w₁, w₂, …, w_n )^T is the aggregation-associated weight vector such that w_j ∈[0, 1] and ∑j=1nwj=1.

Based on Hamacher product operations of the PHFNs described, we can drive Theorem 5.

Theorem 5.LetP_j (j=1, 2, …, n), be a collection of PHFNs; then, their aggregated value by using the PHFHHG operator is also a PHFN, and

where (σ(1), σ(2), …, σ(n)) is a permutation of (1, 2, …, n), such that P_σ(j−1)≥P_σ(j)for allj=2, …, n, andw=(w₁, w₂, …, w_n )^Tis the aggregation-associated weight vector such thatw_j ∈[0, 1], and∑j=1nwj=1,γ>0.

Now, we can discuss some special cases of the PHFHOWG operator with respect to the parameterγ.

• Whenγ=1, the PHFHOWG operator reduces to the Pythagorean hesitant fuzzy ordered weighted geometric (PHFOWG) operator as follows:

  (28)PHFOWGw(p1, p2, …, pn)=∪(γσ(j),ησ(j))∈(hσ(j),gσ(j)),j=1,2,…,n{(∏j=1n(γσ(j))wj, 1−∏j=1n(1−(ησ(j))2)wj)}.

• Whenγ=2, the PHFHOWG operator reduces to the Pythagorean hesitant fuzzy Einstein ordered weighted geometric (PHFEOWG) operator as follows:

  (29)PHFEOWGw(p1, p2, …, pn)=∪(γσ(j),ησ(j))∈(hσ(j),gσ(j)),j=1,2,…,n{(2∏j=1n(γσ(j))ωj∏j=1n(2−(γσ(j))2)ωj+∏j=1n(γσ(j))2ωj, ∏j=1n(1+(ησ(j))2)wj−∏j=1n(1−(ησ(j))2)wj∏j=1n(1+(ησ(j))2)wj+∏j=1n(1−(ησ(j))2)wj)}

From Definitions 8 and 9, we know that the PHFHWG operator weights the Pythagorean hesitant fuzzy argument itself, while the PHFHOWG operator weights the ordered positions of the Pythagorean hesitant fuzzy arguments instead of weighting the arguments themselves. Therefore, weights represent different aspects in both the PHFHWG and PHFHOWG operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose a PHFHHG operator.

Definition 10. A PHFHHG operator is defined as follows:

where w=(w₁, w₂, …, w_n ) is the associated weighting vector, with w_j ∈[0, 1], ∑j=1nwj=1, and p˙σ(j) is the j^th largest element of the Pythagorean hesitant fuzzy arguments p˙j(p˙j=(pj)nωj, j=1, 2, …, n),ω=(ω₁, ω₂, …, ω_n ) is the weighting vector of Pythagorean hesitant fuzzy arguments P_j (j=1, 2, …, n), with ω_j ∈[0, 1], ∑j=1nωj=1, and n is the balancing coefficient. Especially, if w=(1/n, 1/n, …, 1/n)^T , then PHFHHG is reduced to the PHFHWG operator; if ω=(1/n, 1/n, …, 1/n), then PHFHHA is reduced to the PHFHOWG operator.

Based on Hamacher product operations of the PHFNs described, we can drive Theorem 6.

Theorem 6.Let P_j (j=1, 2, …, n) be a collection of PHFNs, then their aggregated value by using the PHFHHG operator is also a PHFN, and