Trapezoidal cubic hesitant fuzzy aggregation operators and their application in group decision-making

1.1 Trapezoidal cubic hesitant fuzzy sets

Definition 2.1.1. Let b ˜ be the trapezoidal cubic hesitant fuzzy number on the set of real numbers, its IVTrHFS is defined as:

and its trapezoidal hesitant fuzzy set is

Γ b ˜ ( h ) = { s θ ( h ) , [ ( h - r ) ( s - r ) ] [ η b ˜ ] r ≤ h s [ η b ˜ ] s ≤ h t s θ ( h ) , ( t - h ) ( d - s ) [ η b ˜ ] t ≤ h u 0 h ( r , u ) or h ( r , u ) where 0 ≤ λ b ˜ ( h ) ≤ 1 , 0 ≤ Γ b ˜ ( h ) ≤ 1 and r, s, t, u are real numbers. Then the TrCHFN b ˜ basically denoted by b = { [ r , s , t , u ] , 〈 [ μ b - , μ b + ] , η b ˜ 〉 } . Further, the TrCHFN reduced to a TCHFN. If the TrCHFN b ˜ is called a normal TCHFN denoted as b ˜ = { [ r , s , t , u ] ; 〈 [ 1 , 1 ] , 0 〉 } . Therefore, the TrCHFN considered now can be regarded as generalized TrHCFN. Then b ˜ is called trapezoidal cubic hesitant fuzzy number.

Definition 2.1.2. Let, h = { [ ɛ , ξ , ψ , υ ] , 〈 [ Ω - , Ω + ] , Ω 〉 } , h 1 = { [ ɛ 1 , ξ 1 , ψ 1 , υ 1 ] , 〈 [ Ω 1 - , Ω 1 + ] , Ω 1 〉 } and h 2 = { [ ɛ 2 , ξ 2 , ψ 2 , υ 2 ] , 〈 [ Ω 2 - , Ω 2 + ] , Ω 2 〉 } be any three TrCHFNs. Then,

h ̈ c = { α c | α ∈ h ̈ } = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω [ a , b , c , d ] , 〈 [ μ b - , μ b + ] , η b ˜ 〉 ; h ̈ 1 ∪ h ̈ 2 = ⋃ a 1 ∈ ɛ 1 , a 2 ∈ ɛ 2 , b 1 ∈ ξ 1 , b 2 ∈ ξ 2 , c 1 ∈ ψ 1 , c 2 ∈ ψ 2 , d 1 ∈ υ 1 , d 2 ∈ υ 2 , μ 1 - ∈ Ω 1 - , μ 2 - ∈ Ω 2 - , μ 1 + ∈ Ω 1 + , μ 2 + ∈ Ω 2 + , η 1 ∈ Ω 1 , η 2 ∈ Ω 2 { [ a 1 ∪ a 2 , b 1 ∪ b 2 , c 1 ∪ c 2 , d 1 ∪ d 2 ] , 〈 [ μ h ̈ 1 - ∪ μ h ̈ 2 - , μ h ̈ 1 + ∪ μ h ̈ 2 + ] , η h ̈ 1 ∩ η h ̈ 2 〉 } ; h ̈ 1 ∩ h ̈ 2 = ⋃ a 1 ∈ ɛ 1 , a 2 ∈ ɛ 2 , b 1 ∈ ξ 1 , b 2 ∈ ξ 2 , c 1 ∈ ψ 1 , c 2 ∈ ψ 2 , d 1 ∈ υ 1 , d 2 ∈ υ 2 , μ 1 - ∈ Ω 1 - , μ 2 - ∈ Ω 2 - , μ 1 + ∈ Ω 1 + , μ 2 + ∈ Ω 2 + , η 1 ∈ Ω 1 , η 2 ∈ Ω 2 { [ a 1 ∩ a 2 , b 1 ∩ b 2 , c 1 ∩ c 2 , d 1 ∩ d 2 ] , 〈 [ μ h ̈ 1 - ∩ μ h ̈ 2 - , μ h ̈ 1 + ∩ μ h ̈ 2 + ] , η b ˜ 1 ∪ η h ̈ 2 〉 } ;

h ̈ 1 ⊕ h ̈ 2 = ⋃ a 1 ∈ ɛ 1 , a 2 ∈ ɛ 2 , b 1 ∈ ξ 1 , b 2 ∈ ξ 2 , c 1 ∈ ψ 1 , c 2 ∈ ψ 2 , d 1 ∈ υ 1 , d 2 ∈ υ 2 , μ 1 - ∈ Ω 1 - , μ 2 - ∈ Ω 2 - , μ 1 + ∈ Ω 1 + , μ 2 + ∈ Ω 2 + , η 1 ∈ Ω 1 , η 2 ∈ Ω 2

{ [ a 1 + a 2 - a 1 a 2 , b 1 + b 2 - b 1 b 2 , c 1 + c 2 - c 1 c 2 , d 1 + d 2 - d 1 d 2 ] , 〈 [ μ h ̈ 1 - + μ h ̈ 2 - - μ h ̈ 1 - μ h ̈ 2 - , μ h ̈ 1 + + μ h ̈ 2 + - μ h ̈ 1 + μ h ̈ 2 + ] , η b ˜ 1 η b ˜ 2 〉 } ;

λ h ̈ = { λ h | a ∈ h ̈ } = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω

{ [ 1 - ( 1 - a ) λ ] , [ 1 - ( 1 - b ) λ ] , [ 1 - ( 1 - c ) λ ] , [ 1 - ( 1 - d ) λ ] ; 〈 [ 1 - ( 1 - μ h ̈ - ) λ ] , [ 1 - ( 1 - μ h ̈ + ) λ ] , ( η h ̈ ) λ 〉 } ;

h ̈ λ = { α λ | a ∈ h ̈ } = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω

{ [ ( a ) λ , ( b ) λ , ( c ) λ , ( d ) λ ] , 〈 [ ( μ h ̈ - ) λ , ( μ h ̈ + ) λ ] , 1 - ( 1 - η h ̈ ) λ 〉 } .

Example 2.1.3. Let α = { [ 0.4 , 0.6 , 0.8 , 0.10 ] , 〈 [ 0.2 , 0.4 ] , 0.3 〉 } α 1 = { [ 0.2 , 0.4 , 0.6 , 0.8 ] , 〈 [ 0.4 , 0.6 ] , 0.5 〉 } and α 2 = { [ 0.9 , 0.11 , 0.13 , 0.15 ] , 〈 [ 0.11 , 0.13 ] , 0.10 〉 } be any three TrCHFNs. Then,

α ^c = 〈[0.4, 0.6, 0.8, 0.10], [0.2, 0.4], 0.3〉,

α 1 ∪ α 2 = { [ 0.2 ∪ 0.9 , 0.4 ∪ 0.11 , 0.6 ∪ 0.13 , 0.8 ∪ 0.15 ] , 〈 [ 0.4 ∪ 0.11 , 0.6 ∪ 0.13 ] , 0.5 ∩ 0.10 〉 } ;

α 1 ∩ α 2 = { [ 0.2 ∩ 0.9 , 0.4 ∩ 0.11 , 0.6 ∩ 0.13 , 0.8 ∩ 0.15 , 〈 [ 0.4 ∩ 0.11 , 0.6 ∩ 0.13 ] , 0.5 ∪ 0.10 〉 } ; α 1 ⊕ α 2 = { [ 0.2 + 0.9 - ( 0.2 ) ( 0.9 ) , 0.4 + 0.11 - ( 0.4 ) ( 0.11 ) , 0.6 + 0.13 - ( 0.6 ) ( 0.13 ) , 0.8 + 0.15 - ( 0.8 ) ( 0.15 ) ] , 〈 [ 0.4 + 0.11 - ( 0.4 ) 0 ( 0.11 ) , 0.6 + 0.13 - ( 0.6 ) ( 0.13 ) ] , ( 0.5 ) ( 0.10 ) 〉 } ; α 1 ⊗ α 2 = { [ ( 0.2 ) ( 0.9 ) , ( 0.4 ) ( 0.11 ) , ( 0.6 ) ( 0.13 ) , ( 0.8 ) ( 0.15 ) ] , [ ( 0.4 ) ( 0.11 ) , ( 0.6 ) ( 0.13 ) ] , 0.5 + 0.10 - ( 0.5 ) ( 0.10 ) 〉 } ; λ = ( 0.25 , 0.25 , 0.25 , 0.25 ) T λ α = { S 3 × 0.25 , [ 1 - ( 1 - 0.4 ) 0.25 , 1 - ( 1 - 0.6 ) 0.25 , 1 - ( 1 - 0.8 ) 0.25 , 1 - ( 1 - 0.10 ) 0.25 ] , 〈 [ 1 - ( 1 - 0.2 ) 0.25 , 1 - ( 1 - 0.4 ) 0.25 ] , ( 0.3 ) 0.25 〉 } ; λ α = { [ ( 0.4 ) 0.25 , ( 0.6 ) 0.25 , ( 0.8 ) 0.25 , ( 0.10 ) 0.25 ] , 〈 [ ( 0.2 ) 0.25 , ( 0.4 ) 0.25 ] , 1 - ( 1 - 0.3 ) 0.25 〉 }

Theorem 2.1.4. Let h = { [ ɛ , ξ , ψ , υ ] , 〈 [ Ω - , Ω + ] , Ω 〉 } , h 1 = { [ ɛ 1 , ξ 1 , ψ 1 , υ 1 ] , 〈 [ Ω 1 - , Ω 1 + ] , Ω 1 〉 } and h 2 = { [ ɛ 2 , ξ 2 , ψ 2 , υ 2 ] , 〈 [ Ω 2 - , Ω 2 + ] , Ω 2 〉 } be any three TrCHFNs. Then, the score function S ( α ) = 〈 { [ a + b + c + d ] + [ μ - + μ + ] - η } 12 〉 , accuracy function h ̈ ( α ) = 〈 { [ a + b + c + d ] + [ μ - + μ + ] + η } 12 gave an order relation between two TrCHFNs α ₁ and α ₂ .

If s (α ₁) > s (α ₂), then s (α ₁) > s (α ₂).

Example 2.1.5. Let h ̈ 1 = { [ 0.4 , 0.5 , 0.6 , 0.7 ] , 〈 [ 0.1 , 0.3 ] , 0.2 〉 } , h ̈ 2 = { [ 0.8 , 0.9 , 0.10 , 0.11 ] , 〈 [ 0.14 , 0.16 ] , 0.15 〉 } and h ̈ 2 = { [ 0.2 , 0.4 , 0.6 , 0.8 ] , 〈 [ 0.10 , 0.12 ] , 0.11 〉 } be three TrCHFNs. Then, the score function S ( h 1 ) = 2.4 12 = 0.2 , S ( h 1 ) = 2.06 12 = 0.1716 , S ( h 3 ) = 2.11 12 = 0.1758 .

Accuracy function H ( h 1 ) = 2.8 12 = 0.2333 , H ( h 2 ) = 2.36 12 = 0.1966 , H ( h 3 ) = 2.33 12 = 0.1941 .

Theorem 2.1.6. Let h = { [ ɛ , ξ , ψ , υ ] , 〈 [ Ω - , Ω + ] , Ω 〉 } , h 1 = { [ ɛ 1 , ξ 1 , ψ 1 , υ 1 ] , 〈 [ Ω 1 - , Ω 1 + ] , Ω 1 〉 } and h 2 = { [ ɛ 2 , ξ 2 , ψ 2 , υ 2 ] , 〈 [ Ω 2 - , Ω 2 + ] , Ω 2 〉 } are three TrCHFEs and, then we have that

h ̈ 1 ∪ h ̈ 2 = ( h ̈ 1 ∩ h ̈ 2 ) c ,

Definition 2.1.7. For a TrCHFE h ̈ , s ( h ̈ ) = ∑ Ω ˜ ∈ h ̈ Ω ˜ / # h ̈ is called the score function of h ̈ , where # h ̈ is the number of the elements in h ̈ . For two TrCHFEs 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 .

3.1 The TrCHFWA and TrCHFWG Operators

Definition 3.1.1. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } are the collections of TrCHFEs and let τ = (τ ₁, τ ₂, τ _n ) ^T be the weight vector of TrCHFEs h ̈ i ( i = 1 , 2 , … , n ) where τ _i ∈ [0, 1], ∑ i = 1 n τ i = 1 . The trapezoidal cubic hesitant fuzzy weighted averaging operator is a mapping h ̈ n → h ̈ such that TrCHFWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊕ i = 1 n ( τ i h ̈ i ) . If τ = ( 1 n , 1 n , … , 1 n ) T , then the TrCHFWA operator reduces to the trapezoidal cubic hesitant fuzzy averaging operator: { TrCHFA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊕ i = 1 n ( 1 n h ̈ i ) } .

Theorem 3.1.2. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFWA operator is also a TrCHFE and

{ TrCHFWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊕ i = 1 n ( τ i h ̈ i ) = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω [ ∏ i = 1 n ( a i ) τ i , ∏ i = 1 n ( b i ) τ i , ∏ i = 1 n ( c i ) τ i , ∏ i = 1 n ( d i ) τ i ] , 〈 [ [ 1 - ∏ i = 1 n ( 1 - μ i - ) τ i ] , [ 1 - ∏ i = 1 n ( 1 - μ i + ) τ i ] ] , ∏ i = 1 n ( η i ) τ i 〉 } .

Definition 3.1.3. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs and let Ψ ̈ = ( Ψ ̈ 1 , Ψ ̈ 2 , … , Ψ ̈ n ) T be the weight vector of TrCHFEs h ̈ i (i = 1, 2, …, n), where Ψ ̈ i ∈ [ 0 , 1 ] , ∑ i = 1 n Ψ ̈ i = 1 . The trapezoidal cubic hesitant fuzzy weighted geometric operator is a mapping h ̈ n → h ̈ such that TrCHFWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊗ i = 1 n ( h ̈ i Ψ ̈ i ) . If Ψ ̈ = ( 1 n , 1 n , … , 1 n ) T , then the TrCHFWG operator reduces to the trapezoidal cubic hesitant fuzzy averaging operator: TrCHFG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊗ i = 1 n ( h ̈ i 1 n ) .

Theorem 3.1.4. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFWG operator is also a TrCHFE and

{ TrCHFWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω [ ∏ i = 1 n ( a i ) Ψ ̈ i , ∏ i = 1 n ( b i ) Ψ ̈ i , ∏ i = 1 n ( c i ) Ψ ̈ i , ∏ i = 1 n ( d i ) Ψ ̈ i ] 〈 [ ∏ i = 1 n ( μ - ) Ψ ̈ i , ∏ i = 1 n ( μ + ) Ψ ̈ i ] , ∏ i = 1 n ( η ) Ψ ̈ i 〉 } .

Corollary 3.1.5. Suppose that h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } is a collection of TrCHFEs and Ψ ̈ = ( Ψ ̈ 1 , Ψ ̈ 2 , … , Ψ ̈ n ) T is their weight vector with Ψ ̈ i ∈ [ 0 , 1 ] and ∑ i = 1 n Ψ ̈ i = 1 , then one has TrCHFWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) ⩽ TrCHFWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) .

Theorem 3.1.6. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs having the weight vector Ψ ̈ = ( Ψ ̈ 1 , Ψ ̈ 2 , … , Ψ ̈ n ) T such that Ψ ̈ i ∈ [ 0 , 1 ] and ∑ i = 1 n Ψ ̈ i = 1 , λ> 0; then TrCHFWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) ⩽ TrCHFWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) .

Example 3.1.7. Suppose that h ̈ 1 = { 〈 { [ 0.2 , 0.4 , 0.6 , 0.8 ] , [ 0.5 , 0.8 ] , 0.6 } , 〉 } , h ̈ 2 = { 〈 { [ 0.4 , 0.8 , 0.10 , 0.12 ] , [ 0.7 , 0.9 ] , 0.8 } , 〉 } and h ̈ 3 = { 〈 { [ 0.1 , 0.2 , 0.3 , 0.4 ] , [ 0.8 , 0.12 ] , 0.10 } , 〉 } , are three TrCLHFEs and τ = {0.18, 0.35, 0.47} is their weight vector. Then, by Definition above, we can obtain TrCHFWA λ ( h ̈ 1 , h ̈ 2 , h ̈ 3 ) = { [ 0.2354 , 0.5423 , 0.3088 , 0.43702 ] , 〈 [ 0.7281 , 0.2278 ] , 0.4595 〉 } ; TrCHFWA ( h ̈ 1 , h ̈ 2 , h ̈ 3 ) = { [ [ 0.7646 , 0.4677 , 0.6912 , 0.5629 ] , 〈 [ 0.2719 , 0.7722 ] , 0.5405 〉 } . score function S ( h 1 ) = 2.7 12 = 0.225 , S ( h 2 ) = 2.23 12 = 0.1858 , s ( h 3 ) = 1.82 12 = 0.1516 .

3.2 The TrCHFOWA and TrCHFOWG operators

Definition 3.2.1. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs. The trapezoidal cubic hesitant fuzzy ordered weighted averaging operator is denoted and defined by the mapping h ̈ n → h ̈ , such that TrCHFOWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊕ i = 1 n ( τ i h ̈ σ ( i ) ) , where τ = (τ ₁, τ ₂, …, τ _n ) ^T is the aggregation-associated vector of TrCHFEs, such that τ _i ∈ [0, 1] and ∑ i = 1 n τ i = 1 .

Theorem 3.2.2. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFOWA operator is also a TrCHFE and { TrCHFOWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω [ 1 - ∏ i = 1 n ( 1 - a α σ ( i ) ) τ i , 1 - ∏ i = 1 n ( 1 - b α σ ( i ) ) τ i , 1 - ∏ i = 1 n ( 1 - c α σ ( i ) ) τ i , 1 - ∏ i = 1 n ( 1 - d α σ ( i ) ) τ i ] , [ ∏ i = 1 n ( μ α σ ( i ) - ) τ i , ∏ i = 1 n ( μ α σ ( i ) + ) τ i ] , ∏ i = 1 n ( η α σ ( i ) ) τ i | α σ ( 1 ) , α σ ( 2 ) , … , α σ ( n ) ∈ h ̈ α σ ( n ) } .

Definition 3.2.3. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs. The trapezoidal cubic hesitant fuzzy ordered weighted geometric operator is denoted and defined by the mapping h ̈ n → h ̈ , such that TrCHFOWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊗ i = 1 n ( h ̈ σ ( i ) τ i ) . Where τ = (τ ₁, τ ₂, …, τ _n ) ^T is the aggregation-associated vector of TrCHFEs, such that τ _i ∈ [0, 1] and ∑ i = 1 n τ i = 1 .

Theorem 3.2.4. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be a collection of TrCHFEs. Then the aggregated value is calculated by using the TrCHFOWG operator is also a TrCHFE and { TrCHFOWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⋃ a ∈ ɛ , b ∈ ξ , c ∈ ψ , d ∈ υ , μ - ∈ Ω - , μ + ∈ Ω + , η ∈ Ω [ ∏ i = 1 n ( a α σ ( i ) ) τ i , ∏ i = 1 n ( b α σ ( i ) ) τ i , ∏ i = 1 n ( c α σ ( i ) ) τ i , ∏ i = 1 n ( d α σ ( i ) ) τ i ] , [ ∏ i = 1 n ( μ α σ ( i ) - ) τ i , ∏ i = 1 n ( μ α σ ( i ) + ) τ i ] , ( 1 - ∏ i = 1 n ( 1 - η α σ ( i ) ) τ i ) | α σ ( 1 ) , α σ ( 2 ) , . . . , α σ ( n ) ∈ h ̈ α σ ( n ) } .

Theorem 3.2.5. Assume h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } are the collections of TrCHFEs, λ > 0 and let τ = (τ ₁, τ ₂, …, τ _n ) ^T be the aggregation associated vector such that τ _i ∈ [0, 1] and ∑ i = 1 n τ i = 1 then TrCHFOWG ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) ≤ TrCHFOWA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) .

Example 3.2.6. Suppose that h ̈ 1 = { [ 0.5 , 0.7 , 0.9 , 0.11 ] , 〈 [ 0.15 , 0.17 ] , 0.16 〉 } , h ̈ 2 = { [ 0.8 , 0.10 , 0.12 , 0.14 ] , 〈 [ 0.22 , 0.24 ] , 0.23 〉 } and h ̈ 3 = { [ 0.26 , 0.28 , 0.30 , 0.32 ] , 〈 [ 0.5 , 0.7 ] , 0.9 〉 } are three TrCHFEs. Then, we can define score function s ( h ̈ 1 ) = { [ 0.5 + 0.7 + 0.9 + 0.11 ] + 〈 [ 0.15 + 0.17 ] - 0.16 〉 } 12 = 2.37 12 = 0.1976 ; S ( h ̈ 2 ) = { [ 0.8 + 0.10 + 0.12 + 0.14 ] + 〈 [ 0.22 + 0.24 ] - 0.23 〉 } 12 = { [ 1.16 ] + 〈 [ 0.46 ] - 0.23 〉 } 12 = 1.39 12 = 0.1159 ; S ( h ̈ 3 ) = { [ 0.26 + 0.28 + 0.30 + 0.32 ] + 〈 [ 0.5 + 0.7 ] - 0.9 〉 } 12 = { [ 1.16 ] + 〈 [ 1.2 ] - 0.9 〉 } 12 = 1.46 12 = 0.1216 .

3.3 The TrCHFHA and TrCHFHG operators

Definition 3.3.1. Let h ̈ i = { [ ɛ i , ξ i , ψ i , υ i ] , 〈 [ Ω i - , Ω i + ] , Ω i 〉 } be the gathering of TrCHFEs. The trapezoidal cubic hesitant fuzzy hybrid averaging operator is defined by the mappin h ̈ n → h ̈ such that, TrCHFHA ( h ̈ 1 , h ̈ 2 , … , h ̈ n ) = ⊕ i = 1 n ( τ i h ̈ σ ( i ) ) , where h ̈ σ ( i ) is the largest i th of h ̈ k = n τ i h ̈ σ ( i ) (k = 1, 2, …, n) and where τ = (τ ₁, τ ₂, …, τ _n ) ^T is the weight vector of TrCHFEs, with τ _i ∈ [0, 1] and ∑ i = 1 n τ i = 1 , and n is the balancing coefficient which plays a role of balance. Then we define the following aggregation operators, which are all based on the mapping h ̈ n → h ̈ with an aggregation-associated vector τ = (τ ₁, τ ₂, …, τ _n ) ^T such that τ _i ∈ [0, 1] and ∑ i = 1 n τ i = 1 .