Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making

2.1 Hesitant Fuzzy Set

Definition 1 (see [7, 14]) Let X be a reference set, a hesitant fuzzy set (HFS) A on X is denoted by a function h_A(x) that returns a subset of [0, 1] when it is applied to X. In order to make it easier to understood, the HFS is expressed by a mathematical symbol

where h_A(x) is a set of some different values indicating all possible membership degrees of the element x∈X to the set A. For convenience, h_A(x) is called a hesitant fuzzy element (HFE), which is denoted by h_A(x) = γ|γ ∈ h_A(x).

Let h, h₁ and h₂ be three HFEs, then some operations are presented as follows^[14]:

1. h^c = {1 − γ|γ ∈ h};

2. h^λ = (γ)^λ|γ ∈ h;

3. λh = 1 − (1 − γ)²|γ ∈ h;

4. h₁ ⊕ h₂ = {γ₁ + γ₂ − γ₁γ₂|γ₁ ∈ h₁, γ₂ ∈ h₂};

5. h₁ ⊗ h₂ = {γ₁γ₂|γ₁ ∈ h₁, γ₂ ∈ h₂}.

Definition 2 (see [14]) For an HFE h={γ|γ∈h},s(h)=1lh∑y∈hγis called the score function of h, where l_h is the number of the values in h. For two HFEs h₁ and h₂ if s (h₁) > s(h₂) , then h₁> h₂; if s (h₁) = s (h₂) , then h₁ = h₂.

2.2 Hamacher Operations

T-norm and t-conorm are an important notion in fuzzy set theory, which are used to define a generalized union and intersection of fuzzy sets^[26]. Roychowdhury and Wang^[27] 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 intuitionistic fuzzy sets were introduced by Deschrijver and Kerre^[28]. Further, Hamacher^[29] proposed a more generalized t-norm and t-conorm. Hamacher operations^[29] include the Hamacher product and Hamacher sum, which are examples of t-norms and t-conorms, respectively. They are defined as follows:

Hamacher product ⊗ is a t-norm and 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 Hamacher t-norm and t-conorm will reduce to

which are called the Einstein t-norm and t-conorm, respectively^[30].

3.1 Trapezoid Fuzzy Numbers

In this section, we briefly describe some basic concepts and operational laws related to trapezoid fuzzy numbers, and define the degree of possibility of two trapezoid fuzzy numbers.

Definition 3 (see [25]) A trapezoid fuzzy numbers can be defined as [n₁, n₂, n₃, n₄]. The membership function μ(x) is defined as

For a trapezoidal fuzzy number n˜=[n1,n2,n3,n4], if n2=n3,then n˜is called a triangular fuzzy number.

Given any two trapezoidal fuzzy numbers, m˜=[m1,m2,m3,m4],n˜=[n1,n2,n3,n4],and λ > 0, some operations can be expressed as follows^[3]:

Motivated by the degree of possibility of two trapezoid fuzzy linguistic variables^[31], in the following, we introduce a formula for comparing trapezoid fuzzy numbers.

Definition 4 Let m˜=[m1,m2,m3,m4]and n˜=[n1,n2,n3,n4]be two trapezoid fuzzy numbers. Then, the possibility degree of m˜≥n˜is defined as follows:

Obviously, the possibility degree p( ˜m ≥ ˜n) satisfies the following properties:

1. 0≤p(m˜≥n˜)≤1,0≤p(n˜≥m˜)≤1;

2. p(m˜≥n˜)+p(n˜≥m˜)=1.Especially, p(m˜≥n˜)=p(n˜≥m˜)=0.5.

3.2 Hesitant Trapezoid Fuzzy Set

In many practical situations, it is relatively easy for decision makers to define the possible values rather than a precise number. Therefore, the trapezoid fuzzy number is usually more enough to describe real-life decision problems than crisp numbers. So, we propose the hesitant trapezoid fuzzy set based on HFS and trapezoid fuzzy numbers.

Definition 5 Let X be a reference set, a HTrFS A on X is denoted by a function h˜A(x)that returns a subset of [0, 1] when it is applied to X. In order to make it easier to understood, the HTrFS is expressed by a mathematical symbol.

where h˜A(x)is a set of some different trapezoid fuzzy numbers indicating all possible membership degrees of the element x ∈ X to the set A. For convenience,h˜A(x)is called a hesitant trapezoid fuzzy element (HTrFE), which is denoted byh˜A(x)=h˜={[a,b,c,d]}.

Given three HTrFEs, h˜={[a,b,c,d]},h˜1={[a1,b1,c1,d1]}and h˜2={[a2,b2,c2,d2]}and λ > 0, the operations are defined as follows.

1. h~1⊕h~2=⋃a1,b1,c1,d1∈h~1,a2,b2,c2,d2∈h~2a1+a2−a1a2,b1+b2−b1b2,c1+c2−c1c2,d1+d2−d1d2;

2. h˜1⊗h˜2=∪[a1,b1,c1,d1]∈h˜1,[a2,b2,c2,d2]∈h˜2{[a1a2,b1b2,c1c2,d1d2]};

3. h˜λ=∪[a,b,c,d]∈h˜{[aλ,bλ,cλ,dλ]};

4. λh~=∪[a,b,c,d]∈h~{[λa,λb,λc,λd]}.

Definition 6 For a HTrFE h~,sh~=1lh~∑γ∈h~γis called the score function of h˜,where lh˜and s(h˜)is the number of the trapezoid fuzzy values in h˜,is a trapezoid fuzzy number in the range of [0, 1]. For two HTrFEs h˜1 and h˜2,if s(h˜1)>s(h˜2),then h˜1>h˜2; if s(h˜1)=s(h˜2),then h˜1=h˜2.

So, we can utilize Equation (4) to compare two score functions and judge the magnitudes of two HTrFEs.

4.1 Hesitant Trapezoid Fuzzy Aggregation Operators

Based on the operational principle for HTrFEs, we shall develop the hesitant trapezoid fuzzy weighted average (HTrFWA) operator and the hesitant trapezoid fuzzy weighted geometric (HTrFWG) operator.

Definition 7 Let h˜j(j=1,2,⋯,n)be a collection of HTrFEs on X. AHTrFWA operator is a mapping Qⁿ → Q, and denoted by

where w=(w₁, w₂, · · · , w_n) is the weight vector of h˜j(j=1,2,⋯,n),wj≥0 and ∑j=1nwj=1.

Definition 8 Let h˜j(j=1,2,⋯,n)be a collection of HTrFEs on X. A HTrFWG operator is a mapping Qⁿ → Q, and denoted by

where w = (w₁, w₂, · · · , w_n) is the weight vector ofh˜j(j=1,2,⋯,n),wj≥0 and ∑j=1nwj=1.

Considering the weights of the ordered positions of hesitant trapezoid fuzzy arguments, the hesitant trapezoid fuzzy ordered weighted average (HTrFOWA) operator and the hesitant trapezoid fuzzy ordered weighted geometric (HTrFOWG) operator are defined as follows.

Definition 9 Let h˜j(j=1,2,⋯,n)be a collection of HTrFEs, then we define the HTrFOWA operator as follows

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that h˜σ(j−1)≥h˜σ(j)for all j=2,3,⋯,n,and w=(w1,w2,⋯,wn)Tis the aggregation-associated weight vector such that wj≥0,∑j=1nwj=1.

Definition 10 Let h˜j(j=1,2,⋯,n)be a collection of HTrFEs, then we define the HTrFOWG operator as follows

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that h˜σ(j−1)≥h˜σ(j)for all j=2,3,⋯,n, and w=(w1,w2,⋯,wn)Tis the aggregation-associated weight vector such that wj≥0,∑j=1nwj=1.

From the above-mentioned definitions, we know that the HTrFWA and HTrFWG operators weight the hesitant trapezoid fuzzy argument itself, while the HTrFOWA and HTrFOWG operators weight the ordered positions of hesitant trapezoid fuzzy arguments. However, neither of these operators can consider both the two aspects. To solve this drawback, in the following we shall propose the hesitant trapezoid fuzzy hybrid average (HTrFHA) operator and the hesitant trapezoid fuzzy hybrid geometric (HTrFHG) operator.

Definition 11 Let h˜j(j=1,2,⋯,n)be a collection of HTrFEs, then the HTrFHA operator is defined as follows

where w=(w1,w2,⋯,wn)Tis the associated weighting vector, with wj≥0,∑j=1nwj=1 ​​ ​​ and ​​ ​​ h˜σ(j)is the jth largest element of the hesitant trapezoid fuzzy argumentsh˜σ(j)=(nωj)h˜j(j=1,2,⋯,n),ω=(ω1,ω2,⋯,ωn)Tis the associated weighting vector, ωj≥0,∑j=1nωj=1, and n is the balancing coefficient.

Definition 12 Let h~j(j=1,2,⋯,n)be a collection of HTrFEs, then the HTrFHG operator is defined as follows

where w=(w1,w2,⋯,wn)Tis the associated weighting vector, with wj≥0,∑j=1nwj=1 and h˜σ(j)is the jth largest element of the hesitant trapezoid fuzzy argumentsh˜σ(j)=(nωj)h˜j(j=1,2,⋯,n),ω=(ω1,ω2,⋯,ωn)Tis the associated weighting vector, with ωj≥0,∑j=1nωj=1, and n is the balancing coefficient.

4.2 Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators

Motivated by Equations (2) and (3), let a t-norm T be the Hamacher product ⊗ and t-conorm T ^∗ be the Hamacher sum⊕, then the generalized intersection and union on two HTrFEs h˜1 and h˜2become the Hamacher product (denoted by h˜1⊗h˜2)and Hamacher sum (denoted by h˜1⊕h˜2)of two HTrFEs h˜1 and h˜2,respectively, as follows:

In the following, we shall develop some hesitant trapezoid fuzzy Hamacher aggregation operators, such as the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator.

Definition 13 Let h~j(j=1,2,⋯,n)be a collection of HTrFEs, then we define the HTrFHWA operator as follows:

where w = (w₁, w₂, · · · , w_n)^T be the weight vector of HTrFEs, and wj≥0, ∑j=1nwj=1.

Based on the Hamacher operations of hesitant trapezoid fuzzy elements described in Section 3, we can drive the following theorem.

Theorem 1 Leth~j(j=1,2,⋯,n)be a collection of HTrFEs, then their aggregated value by using the HTrFHWA operator is also a HTrFE, and

In what follows, we give some special cases of the HTrFHWA operator with respect to the parameter γ.

When γ = 1, the HTrFHWA operator reduces to the HTrFWA operator.

When γ = 2, the HTrFHWA operator reduces to the hesitant trapezoid fuzzy Einstein weighted average (HTrFEWA) operator as follows:

Definition 14 Let h~j(j=1,2,⋯,n)be a collection of HTrFEs, then we define the HTrFHWG operator as follows

where w = (w₁, w₂, · · · , w_n)^T be the weight vector of HTrFEs, and wj≥0, ∑j=1nwj=1.

Based on the Hamacher operations of hesitant trapezoid fuzzy elements described in Section 3, we can drive the following theorem.

Theorem 2 Leth~j(j=1,2,⋯,n)be a collection of HTrFEs, then their aggregated value by using the HTrFHWG operator is also a HTrFE, and

In the following, we can discuss some special cases of the HTrFHWG operator with respect to the parameter γ.

When γ = 1, the HTrFHWG operator reduces to the HTrFWG operator.

When γ = 2, the HTrFHWG operator reduces to the hesitant trapezoid fuzzy Einstein weigthed geometric (HTrFEWG) operator as follows: