Linguistic hesitant intuitionistic fuzzy linear assignment method based on Choquet integral

1 Preliminaries

Torra and Narukawa [1, 2] generalized fuzzy set to make each membership include several values and developed the hesitant fuzzy set (HFS).

Definition 1. Let X be a reference set, an HFS A on X is a function h that returns a subset of values in [0, 1] when it is applied to X: A = { < x , h A ( x ) > ∣ x ∈ X } ,

where h _A  (x) is a set of some different values in [0, 1], representing the possible membership degrees of the element x ∈ X to A. h _A  (x) is called a hesitant fuzzy element (HFE).

Suppose that S = {s _i | i = 1,. . . , g} is a finite and totally ordered discrete term set, where s _i represents a possible value for a linguistic variable. A set of nine terms S [54] can be expressed in the following S = {s₁ = extremelypoor,  s₂ = verypoor,  s₃ = poor, s₄ = slightly poor, s₅ = fair, s₆ = slightly good, s₇ = good, s₈ = verygood, s₉ = extremelygood} .

In order to preserve all information, the discrete linguistic term set S can be extended to a continuous one S ¯ = { s α ∣ s 0 ≤ s α ≤ s g , α ∈ [ 0 , g ] }.

Definition 2. [12]. Let X = {x₁, x₂,. . . , x _n } be a reference set. A linguistic hesitant intuitionistic fuzzy term set (LHIFS) A ˇ on X is defined as A ˇ = { < x i , h ˇ A ˇ ( x i ) > ∣ x i ∈ X } ,(1) where S ¯ = { s α ∣ s 0 ≤ s α ≤ s g } is a linguistic term set and H is the set of linguistic hesitant intuitionisic fuzzy elements. h ˇ A ˇ ( x i ) : X → H denotes all possible linguistic intuitionistic fuzzy evaluation values of element x _i  ∈ X. For convenience, we call h ˇ A ˇ ( x i ) a linguistic hesitant intuitionistic fuzzy element (LHIFE), which can be represented as h ˇ A ˇ ( x i ) = { ( s θ i , lh ( s θ i ) ) ∣ x i ∈ X } ,(2) where s θ i ∈ S ¯ is a linguistic argument and lh (s _{θ i} ) = {(α _i , β _i )} is the set of intuitionistic fuzzy membership values that s _{θ i} satisfies x _i . (s _{θ i} , lh (s _{θ i} )) is the linguistic intuitionistic fuzzy element (LIFE).

Definition 3. Let h ˇ, h ˇ 1 and h ˇ 2 be LHIFEs, λ > 0. a k = ( s θ ( k ) , lh ( s θ ( k ) ) ) ∈ h ˇ, a i = ( s θ ( i ) , lh ( s θ ( i ) ) ) ∈ h ˇ 1, a j = ( s θ ( j ) , lh ( s θ ( j ) ) ) ∈ h ˇ 2. (μ _k , ν _k ) ∈ lh (s_θ(k)) , (μ _i , ν _i ) ∈ lh (s_θ(i)) , (μ _j , ν _j ) ∈ lh (s_θ(j)). Some operation laws can be defined as follows

( 1 ) h ˇ 1 ⊕ h ˇ 2 = ⋃ a i ∈ h ˇ 1 , a j ∈ h ˇ 2 { ( s θ ( i ) + θ ( j ) , ⋃ {(μ _i  + μ _j  - μ _i μ _j , ν _i ν _j )})},

( 2 ) h ˇ 1 ⊗ h ˇ 2 = ⋃ a i ∈ h ˇ 1 , a j ∈ h ˇ 2 { ( s θ ( i ) θ ( j ) , ⋃ { ( μ i μ j , ν i +ν _j  - ν _i ν _j )})},

( 3 ) λ h ˇ = ⋃ a k ∈ h ˇ { ( s λ θ ( k ) , ⋃ ( μ k , ν k ) ∈ lh ( s θ ( k ) ) { ( 1 - ( 1 - μ k ) λ , ν k λ ) } ) },

( 4 ) ( h ˇ ) λ = ⋃ a k ∈ h ˇ { ( s θ ( k ) λ , ⋃ ( μ k , ν k ) ∈ lh ( s θ ( k ) ) { ( μ k λ , 1 - (1 - ν _k )  ^λ )})}.

Definition 4. Let a _i  = (s _{θ i} , lh (s _{θ i} )) be a LIFE, then the score function s (a _i ) of a _i can be defined as s ( a i ) = θ i g ∣ lh ( s θ i ) ∣ ( ∑ ( α i , β i ) ∈ lh ( s θ i ) ( α i - β i ) ) ,(3) the accuracy function h (a _i ) of a _i can be defined as h ( a i ) = θ i g ∣ lh ( s θ i ) ∣ ( ∑ ( α i , β i ) ∈ lh ( s θ i ) ( α i + β i ) ) ,(4) the deviation function d (a _i ) of a _i can be defined as

d ( a i ) = θ i g ∣ lh ( s θ i ) ∣ ( ∑ ( α i , β i ) ∈ lh ( s θ i ) ( ( α i + β i ) - s ( a i ) ) 2 ) ,(5) where g is the number of linguistic arguments in linguistic term set S and ∣lh (s _{θ i} )∣ is the number of intuitionistic fuzzy memberships in lh (s _{θ i} ). If lh (s _{θ i} ) = φ, let ∣lh (s _{θ i} ) ∣ =1.

The following ranking method for LIFEs can be presented based on the score function s (a _i ), the accuracy function h (a _i ) and the deviation function d (a _i ). Let a _i  = (s _{θ i} , lh (s _{θ i} )) and a _j  = (s _{θ j} , lh (s _{θ j} )) be two LIFEs, then

If s (a _i ) < s (a _i ), then a _i is smaller than a _j , denoted by a _i  < a _j ;

If s (a _i ) = s (a _i ), then

If h (a _i ) < h (a _j ), then a _i is smaller than a _j , denoted by a _i  < a _j ;

Definition 5. Let h ˇ = { ( s θ i , lh (s _{θ i} )) ∣ (s _{θ i} , lh ( s θ i ) ) ∈ h ˇ } be LHIFE, the score function S ( h ˇ ) can be defined as

S ( h ˇ ) = 1 l h ˇ ( ∑ θ i g ∣ lh ( s θ i ) ∣ ( ∑ ( α i , β i ) ∈ lh ( s θ i ) ( α i - β i ) ) ) ,(6) the accuracy function A ( h ˇ ) can be defined as

A ( h ˇ ) = 1 l h ˇ ( ∑ θ i g ∣ lh ( s θ i ) ∣ ( ∑ ( α i , β i ) ∈ lh ( s θ i ) ( α i + β i ) ) ) ,(7) the deviation function D ( h ˇ ) can be defined as

D ( h ˇ ) = 1 l h ˇ ( ∑ θ i g ∣ lh ( s θ i ) ∣ ( ∑ ( α i , β i ) ∈ lh ( s θ i ) ( α i + β i - S ( h ˇ ) ) 2 ) ) ,(8)

where l h ˇ is the number of LIFEs in h ˇ and g is the number of linguistic arguments in linguistic term set S, ∣lh (s _{θ i} )∣ is the count of intuitionistic fuzzy memberships in lh (s _{θ i} ).

Based on the score function and accuracy function, we present the following method to compare LHIFEs. Let h ˇ i and h ˇ j be two LHIFEs,

If S ( h ˇ i ) < S ( h ˇ j ) , then h ˇ i is smaller than h ˇ j, denoted by h ˇ i < h ˇ j;

If S ( h ˇ i ) = S ( h ˇ j ) , then

If A ( h ˇ i ) < A ( h ˇ j ), then h ˇ i is smaller than h ˇ j, denoted by h ˇ i < h ˇ j;

Different LHIFEs may have different number of LIFEs and different LIFEs may have different number of intuitionistic fuzzy memberships in most cases. In order to define the distance between two LHIFEs h ˇ 1 , h ˇ 2 more accurately, they should be extended according to the risk attitude of the decision maker until they have the same number of LIFEs and each IFLE should have the same number of intuitionistic fuzzy memberships. If decision maker is risk-seeking, the largest LIFE and the largest intuitionistic fuzzy membership can be added; if decision maker is risk-averse, the smallest LIFE and the smallest intuitionistic fuzzy membership can be added; if decision maker is risk-neutral, the average LIFE and average intuitionistic fuzzy membership can be added.

Definition 6. Let h ˇ 1 = { a i ( 1 ) } = { ( s θ i ( 1 ) , lh ( s θ i ( 1 ) ) ) } , h ˇ 2 = { a i ( 2 ) } = { ( s θ i ( 2 ) , lh ( s θ i ( 2 ) ) ) } . a σ ( i ) ( k ) ≥ a σ ( j ) ( k ) for i ≥ j, k = 1, 2. ( α i k ( t ) , β i k ( t ) ) ∈ lh ( s θ i ( t ) ). l is the number of LIFEs in LHIFE, g is the number of linguistic variables in linguistic term set and ∣lh (s _{θ i} )∣ is the number of intuitionistic fuzzy memberships in lh (s _{θ i} ). Based on the operation laws of LHIFEs and the extension principle, we can define some distance measures between LHIFEs.

The hesitant Euclidean distance d ( h ˇ 1 , h ˇ 2 ) for LHIFEs h ˇ 1 , h ˇ 2 can be defined as:

d ( h ˇ 1 , h ˇ 2 ) = ( 1 3 l ∑ i = 1 l ( ( θ ( σ ( i ) ) ( 1 ) - θ ( σ ( i ) ) ( 2 ) ) 2 / g 2 + 1 ∣ lh ( s θ i ) ∣ ∑ k = 1 ∣ lh ( s θ i ) ∣ ( ( α σ ( i ) k ( 1 ) - α σ ( i ) k ( 2 ) ) 2 + ( β σ ( i ) k ( 1 ) - β σ ( i ) k ( 2 ) ) 2 ) ) ) 1 / 2 .(9)

The linguistic hesitant intuitionistic fuzzy Hamming distance can be defined as follows:

d ( h ˇ 1 , h ˇ 2 ) = 1 3 l ∑ i = 1 l ( | θ ( σ ( i ) ) ( 1 ) - θ ( σ ( i ) ) ( 2 ) | / g + 1 ∣ lh ( s θ i ) ∣ ∑ k = 1 ∣ lh ( s θ i ) ∣ ( | α σ ( i ) k ( 1 ) - α σ ( i ) k ( 2 ) | + | β σ ( i ) k ( 1 ) - β σ ( i ) k ( 2 ) | ) ) .(10)

In this section, we consider some linguistic hesitant intuitionistic fuzzy correlated aggregation operators based on Choquet integral.

Definition 7. [55]. A fuzzy measure μ on the set X is a set function m :  P (X) →[0, 1] satisfying the following axioms:

m (φ) =0,  m (X) =1 ;

In the above definition, if ρ = 0, then the third condition reduces to the axiom of the additive measure:

m ( B ∪ C ) = m ( B ) + m ( C ) for all B , C ⊆ X and B ∩ C = φ .(11)

If the elements of B in X are independent, we have m ( B ) = ∑ x i ∈ B m ( x i ) , for all B ⊆ X .(12)

Let X = {x₁, x₂,. . . , x _n } be a finite set. Sugeno [56] gave the following equation to determine fuzzy measure on X avoiding the computational complexity. m ( X ) = { 1 ρ ( Π i = 1 n ( 1 + ρ m ( x i ) ) - 1 ) , ρ ≠ 0 , ∑ i = 1 n m ( x i ) , ρ = 0 ,(13) the value ρ can be uniquely determined from m (X) =1 by the following equation ρ = Π i = 1 n ( 1 + ρ m ( x i ) ) - 1 .(14)

Especially, for every subset A ⊂ X, we have m ( X ) = { 1 ρ ( Π x i ∈ A ( 1 + ρ m ( x i ) ) - 1 ) , ρ ≠ 0 , ∑ x i ∈ A m ( x i ) , ρ = 0 .(15)

We define linguistic hesitant intuitionistic fuzzy correlated averaging (LHIFCA) operator based on the Choquet integral as follows.

Definition 8. Let X = {x₁, x₂,. . . , x _n } be a reference set, m be the fuzzy measure [55] on X and { h ˇ 1 , h ˇ 2 , . . . , h ˇ n } be a collection of LHIFEs on X. The LHIFCA operator can be defined as follows:

LHIFCA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊕ j = 1 n ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) h ˇ σ ( j ) ,(16) where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that h ˇ σ ( 1 ) ≥ h ˇ σ ( 2 ) ≥ . . . ≥ h ˇ σ ( n ).

Theorem 1. Let m be a fuzzy measure on X = {x₁, x₂,. . . , x _n } and h ˇ j ( j = 1 , 2 , . . . , n ) be a collection of LHIFEs on X. The aggregated value using LHIFCA operator is also LHIFE and

LHIFCA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊕ j = 1 n w j h ˇ σ ( j ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∑ j = 1 n ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( 1 - ∏ j = 1 n ( 1 - μ σ ( j ) k it ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) , ∏ j = 1 n ( ν σ ( j ) k it ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ) ) } , where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that h ˇ σ ( 1 ) ≥ h ˇ σ ( 2 ) ≥ . . . ≥ h ˇ σ ( n ). h ˇ j = { a j i | a j i ∈ h ˇ j , i = 1 , 2 , . . . , l j } ( j = 1 , 2 , . . . , n ), a j i = { ( s θ j it , lh ( s θ j it ) ) | ( s θ j it , lh ( s θ j it ) ) ∈ a j i } , t = 1 , 2 , . . . , l ij, ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ) , k = 1 , 2 , . . . , l ijt.

Proof. We can prove the theorem by using mathematical induction method. Let w _i  = m (A_σ(j)) - m (A_σ(j-1)). Then if n = 2, w 1 h ˇ σ ( 1 ) ⊕ w 2 h ˇ σ ( 2 ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∑ j = 1 2 w j θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ k it ) { ( 1 - ∏ j = 1 2 ( 1 - μ σ ( j ) k it ) w j , ∏ j = 1 2 ( ν σ ( j ) k it ) w j ) } ) } . If ⊕ j = 1 m w j h ˇ σ ( j ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∑ j = 1 m w j ( θ σ ( j ) it ) , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ k it ) { ( ( 1 - ∏ j = 1 m ( 1 - μ σ ( j ) k it ) w j , ∏ j = 1 m ( ν σ ( j ) k it ) w j ) } ) } holds for n = m, then for n = m + 1, we have ⊕ j = 1 m + 1 w j h ˇ σ ( j ) = ⊕ j = 1 m w j h ˇ σ ( j ) + w m + 1 h ˇ σ ( m + 1 ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∑ j = 1 m w j θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ k it ) { ( 1 - ∏ j = 1 m ( 1 - μ σ ( j ) k it ) w j , ∏ j = 1 m ν σ ( j ) k it ) w j ) } ) } ⊕ ⋃ a σ ( m + 1 ) i ∈ h ˇ σ ( m + 1 ) { ( s w m + 1 θ σ ( m + 1 ) it , ⋃ ( μ σ ( m + 1 ) j it , ν σ ( m + 1 ) j it ) ∈ lh ( s θ σ ( m + 1 ) it ) { ( 1 - ( 1 - μ σ ( m + 1 ) k it ) w m + 1 , ( ν σ ( m + 1 ) k it ) w m + 1 ) } ) } = ⋃ a σ ( i ) ∈ h ˇ σ ( i ) { ( s ∑ j = 1 m + 1 w j θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ k it ) { ( 1 - ∏ j = 1 m + 1 ( 1 - ( μ σ ( j ) k it ) λ ) w j , ∏ j = 1 m + 1 ( 1 - ( 1 - ν σ ( j ) k it ) λ ) w j ) } ) } , then ⊕ j = 1 m + 1 w j h ˇ σ ( j ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∑ j = 1 m + 1 w j θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ k it ) { ( 1 - ∏ j = 1 m + 1 ( 1 - μ σ ( j ) k it ) w j , ∏ j = 1 m + 1 ( ν σ ( j ) k it ) w j ) } ) } . Hence, the value of LHIFCA operator is a LHIFE.

Definition 9. Let X = {x₁, x₂,. . . , x _n } be a reference set, m be the fuzzy measure on X and { h ˇ 1 , h ˇ 2 , . . . , h ˇ n } be a collection of LHIFEs on X. The linguistic hesitant intuitionistic fuzzy correlated geometric averaging (LHIFCGA) operator can be defined as follows:

LHIFCGA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊗ j = 1 n ( h ˇ σ ( j ) ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ,(17) where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that h ˇ σ ( 1 ) ≥ h ˇ σ ( 2 ) ≥ . . . ≥ h ˇ σ ( n ).

Theorem 2. Let m be a fuzzy measure on X = {x₁, x₂,. . . , x _n } and h ˇ j ( j = 1 , 2 , . . . , n ) be a collection of LHIFEs on X. The aggregated value using LHIFCGA operator is also LHIFE and LHIFCGA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊗ j = 1 n ( h ˇ σ ( j ) ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∏ j = 1 n ( θ σ ( j ) it ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( ∏ j = 1 n ( μ σ ( j ) k it ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) , 1 - ∏ j = 1 n ( 1 - ν σ ( j ) k it ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ) ) } , where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that h ˇ σ ( 1 ) ≥ h ˇ σ ( 2 ) ≥ . . . ≥ h ˇ σ ( n ). h ˇ j = { a j i | a j i ∈ h ˇ j , i = 1 , 2 , . . . , l j } ( j = 1 , 2 , . . . , n ), a j i = { ( s θ j it , lh ( s θ j it ) ) | ( s θ j it , lh ( s θ j it ) ) ∈ a j i } , t = 1 , 2 , . . . , l ij , ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) , k = 1 , 2 , . . . , l ijt.

Definition 10. Let X = {x₁, x₂,. . . , x _n } be a reference set, m be the fuzzy measure on X and { h ˇ 1 , h ˇ 2 , . . . , h ˇ n } be a collection of LHIFEs on X. The generalized linguistic hesitant intuitionistic fuzzy correlated averaging (GLHIFCA) operator can be defined as follows:

GLHIFCA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ( ⊕ j = 1 n ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ( h ˇ σ ( j ) ) λ ) 1 / λ ,(18) where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that h ˇ σ ( 1 ) ≥ h ˇ σ ( 2 ) ≥ . . . ≥ h ˇ σ ( n ).

Theorem 3. Let m be a fuzzy measure on X = {x₁, x₂,. . . , x _n } and h ˇ j ( j = 1 , 2 , . . . , n ) be a collection of LHIFEs on X. The aggregated value using GLHIFCA operator is also LHIFE and

GLHIFCA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ( ∑ j = 1 n ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ( θ σ ( j ) it ) λ ) 1 / λ , ⋃ ( μ kj it , ν kj it ) ∈ lh ( s θ k it ) { ( ( 1 - ∏ j = 1 n ( 1 - ( μ σ ( j ) k it ) λ ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ) 1 / λ , 1 - ( 1 - ∏ j = 1 n ( 1 - ( 1 - ν σ ( j ) k it ) λ ) ( m ( A σ ( j ) ) - m ( A σ ( j - 1 ) ) ) ) 1 / λ ) } ) } .

where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that h ˇ σ ( 1 ) ≥ h ˇ σ ( 2 ) ≥ . . . ≥ h ˇ σ ( n ). h ˇ j = { a j i | a j i ∈ h ˇ j , i = 1 , 2 , . . . , l j } ( j = 1 , 2 , . . . , n ), a j i = { ( s θ j it , lh ( s θ j it ) ) | ( s θ j it , lh ( s θ j it ) ) ∈ a j i } , t = 1 , 2 , . . . , l ij,( μ jk it , ν jk it ) ∈ lh ( s θ j it ) , k = 1 , 2 , . . . , l ijt.

We consider some special cases of the new aggregation operators in the following.

Remark 1. For any A, B ∈ P (X) such that ∣A ∣ = ∣B∣, where ∣A∣ is the number of elements in the set A. If m (A) = m (B) and m ( A σ ( i ) ) = i n , 1 ≤ i ≤ n. Then the LHIFCA operator becomes the linguistic hesitant intuitionistic fuzzy averaging (LHIFA) operator as follows LHIFA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊕ j = 1 n 1 n ( h ˇ σ ( j ) ) = ⊕ j = 1 n 1 n ( h ˇ j ) = ⋃ a j i ∈ h ˇ j { ( s ∑ j = 1 n 1 n θ j it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( 1 - ∏ j = 1 n ( 1 - μ jk it ) 1 n , ∏ j = 1 n ν jk it ) 1 n ) ) } .

The LHIFCGA operator becomes the linguistic hesitant intuitionistic fuzzy geometric averaging (LHIFGA) operator as follows LHIFGA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊗ j = 1 n ( h ˇ σ ( j ) ) 1 n = ⊗ j = 1 n ( h ˇ j ) 1 n = ⋃ a j i ∈ h ˇ j { ( s ∏ j = 1 n 1 n θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( ∏ j = 1 n ( μ jk it ) 1 n , 1 - ∏ j = 1 n ( 1 - ν jk it ) 1 n ) ) } .

The GLHIFCA operator becomes the generalized linguistic hesitant intuitionistic fuzzy averaging (GLHIFA) operator as follows GLHIFA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ( ⊕ j = 1 n 1 n ( h ˇ σ ( j ) ) λ ) 1 / λ = ( ⊕ j = 1 n 1 n ( h ˇ j ) λ ) 1 / λ = ⋃ a j i ∈ h ˇ j { ( s ( ∑ j = 1 n 1 n ( θ j it ) λ ) 1 / λ , ⋃ ( μ kj it , ν kj it ) ∈ lh ( s θ k it ) { ( ( 1 - ∏ j = 1 n ( 1 - ( μ jk it ) λ ) 1 n ) 1 / λ , 1 - ( 1 - ∏ j = 1 n ( 1 - ( 1 - ν jk it ) λ ) 1 n ) 1 / λ ) } ) } .

Remark 2. If m (A) = ∑_{x i ∈A}m ({x _i }) holds for all A ⊆ X with ∑_{x i ∈A}m ({x _i }) =1, then m ({x_σ(i)}) = m (A_σ(i)) - m (A_σ(i-1)) , i = 1, 2,. . . , n . Then the LHIFCA operator becomes the linguistic hesitant intuitionistic fuzzy weighted averaging (LHIFWA) operator as follows LHIFWA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊕ j = 1 n m ( x σ ( j ) ) h ˇ σ ( j ) = ⊕ j = 1 n m ( x j ) h ˇ j = ⋃ α j k ∈ h ˇ j { ( s ∑ j = 1 n m ( x j ) θ j it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( 1 - ∏ j = 1 n ( 1 - μ jk it ) m ( x j ) , ∏ j = 1 n ( ν jk it ) m ( x j ) ) ) } .

Then the LHIFCGA operator becomes the linguistic hesitant intuitionistic fuzzy geometric weighted averaging (LHIFGWA) operator as follows LHIFGWA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊗ j = 1 n ( h ˇ σ ( j ) ) m ( x σ ( j ) ) = ⊗ j = 1 n ( h ˇ j ) m ( x j ) = ⋃ α j k ∈ h ˇ j { ( s ∏ j = 1 n m ( x j ) θ j it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( ∏ j = 1 n ( μ jk it ) m ( x j ) , 1 - ∏ j = 1 n ( 1 - ν jk it ) m ( x j ) ) ) } .

The GLHIFCA operator becomes the generalized linguistic hesitant intuitionistic fuzzy weighted averaging (GLHIFWA) operator as follows GLHIFWA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ( ⊕ j = 1 n m ( x σ ( j ) ) ( h ˇ σ ( j ) ) λ ) 1 / λ = ( ⊕ j = 1 n m ( x j ) ( h ˇ j ) λ ) 1 / λ = ⋃ a j i ∈ h ˇ j { ( s ( ∑ j = 1 n m ( x j ) ( θ j it ) λ ) 1 / λ , ⋃ ( μ kj it , ν kj it ) ∈ lh ( s θ k it ) { ( ( 1 - ∏ j = 1 n ( 1 - ( μ jk it ) λ ) m ( x j ) ) 1 / λ , 1 - ( 1 - ∏ j = 1 n ( 1 - ( 1 - ν jk it ) λ ) m ( x j ) ) 1 / λ ) } ) } .

Remark 3. If m ( { x i } ) = ∑ i = 1 | A | w i holds for all A ⊆ X with ∑_{x i ∈A}m ({x _i }) =1, then w _i  = m (A_σ(i)) - m (A_σ(i-1)) , i = 1, 2,. . . , n, and ∑ i = 1 n w i = 1. Then the LHIFCA operator becomes the linguistic hesitant intuitionistic fuzzy ordered weighted averaging (LHIFOWA) operator LHIFOWA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊕ j = 1 n w j h ˇ σ ( j ) = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∑ j = 1 n w j θ σ ( j ) it , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( 1 - ∏ j = 1 n ( 1 - μ σ ( j ) k it ) w j , ∏ j = 1 n ( ν σ ( j ) k it ) w j ) ) } .

The LHIFCGA operator becomes the linguistic hesitant intuitionistic fuzzy ordered weighted geometric averaging (LHIFOWGA) operator as follows LHIFOWGA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ⊗ j = 1 n ( h ˇ σ ( j ) ) w j = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ∏ j = 1 n ( θ σ ( j ) it ) w j , ⋃ ( μ jk it , ν jk it ) ∈ lh ( s θ j it ) ( ∏ j = 1 n ( μ σ ( j ) k it ) w j , 1 - ∏ j = 1 n ( 1 - ν σ ( j ) k it ) w j ) ) } .

The GLHIFCA operator becomes the generalized linguistic hesitant intuitionistic fuzzy ordered weighted averaging (GLHIFOWA) operator as follows GLHIFOWA ( h ˇ 1 , h ˇ 2 , . . . , h ˇ n ) = ( ⊕ j = 1 n w j ( h ˇ σ ( j ) ) λ ) 1 / λ = ⋃ a σ ( j ) i ∈ h ˇ σ ( j ) { ( s ( ∑ j = 1 n w j ( θ σ ( j ) it ) λ ) 1 / λ , ⋃ ( μ kj it , ν kj it ) ∈ lh ( s θ k it ) { ( ( 1 - ∏ j = 1 n ( 1 - ( μ σ ( j ) k it ) λ ) w j ) 1 / λ , 1 - ( 1 - ∏ j = 1 n ( 1 - ( 1 - ν σ ( j ) k it ) λ ) w j ) 1 / λ ) } ) } .