Cubic Fuzzy multi-attribute group decision-making with an application to plant location selected based on a new extended Vikor method

3.1 The crisp weighted possibility means of TrCFNs

Definition 3.1.1. For a TrCFNs b ˜ = [ r , s , t , u ] ; 〈 [ ω b - , ω b + ] , η b ˜ 〉 , the (α, β) –cut set, the α - cut set and the β - cut set are define as b ˜ α , β = { h | λ b ˜ - ( h ) ⩾ α , λ b ˜ + ( h ) ⩾ α , Γ b ˜ ( h ) ⩽ β } , b ˜ α - = { h | λ b ˜ - ( h ) ⩾ α } , b ˜ α = { h | λ b ˜ + (h) ⩾ α} and b ˜ β = { h | Γ b ˜ ( h ) ⩽ β } b ˜ α = 〈 [ { L b ˜ - ( α ) , R b ˜ - ( α ) } , { L b ˜ + ( α ) , R b ˜ + ( α ) } ] 〉 b ˜ α = 〈 [ { r + α ( s - r ) ω b - , α ( t - u ) ω b - + u } , { r + α ( s - r ) ω b + , α ( t - u ) ω b + + u } ] 〉(1) b ˜ β = [ L a ˜ ( β ) , L a ˜ ( β ) ] b ˜ β = 〈 [ { r + α ( s - r ) η b ˜ , α ( t - u ) η b ˜ + u } ] 〉(2)

Definition 3.1.2. Let b ˜ α = 〈 [ { L b - ( α ) , R b - ( α ) } , {L _{b ⁺}  (α) , R _{b ⁺}  (α)}] 〉 be the a^-, a⁺ -cut set of a TrCFN b ˜ = [ r , s , t , u ] , 〈 [ ω b - , ω b + ] , η b ˜ 〉 . The φ weighted lower and upper possibility means of the membership function λ^-, λ⁺ for the TrCFN b ˜ are defined as N λ - ( b ˜ ) = ∫ 0 ω b - φ ( pos [ b ˜ ⩽ L b - ( α ) ] ) L b - ( α ) d α -(3) Q λ - ( b ˜ ) = ∫ 0 ω b - φ ( pos [ b ˜ ⩾ R b - ( α ) ] ) R b - ( α ) d α -(4) N λ + ( b ˜ ) = ∫ 0 ω b + φ ( pos [ b ˜ ⩽ L b + ( α ) ] ) L b + ( α ) d α +(5) Q λ + ( b ˜ ) = ∫ 0 ω b + φ ( pos [ b ˜ ⩾ R b + ( α ) ] ) R b + ( α ) d α +(6) respectively, where φ   : [0, ω _{b ^-} ] → R^- and φ : [0,ω _{b ⁺} ] → R⁺ are a non-negative, monotone increasing weighted function satisfying ∫ 0 ω b - φ ( α ) d α - = ω b - and φ (0) =0, ∫ 0 ω b + φ ( α ) d α + = ω b + and φ (0) =0. Here, Pos means possibility which are defined as Pos [ b ˜ ⩽ L a - ( α ) ] ) = ∨ h ⩽ L b - ( α ) λ a - ( h ) = α -(7) Pos [ b ˜ ⩽ L a + ( α ) ] = ∨ h ⩽ L b + ( α ) λ a + ( h ) = α +(8) Pos [ b ˜ ⩾ R a - ( α ) ] = ∨ h ⩽ R b + ( α ) λ a - ( h ) = α -(9) Pos [ b ˜ ⩾ R a + ( α ) ] = ∨ x ⩽ R a + ( α ) λ a + ( x ) = α +(10) using Equations (7), (8) and (9), (10), we get N λ - ( b ˜ ) = ∫ 0 ω b - φ ( pos [ b ˜ ⩽ L b - ( α ) ] ) L b - ( α ) d α - = ∫ 0 ω b - φ ( α ) L b - ( α ) d α - Q λ - ( b ˜ ) = ∫ 0 ω b - φ ( pos [ b ˜ ⩾ R b - ( α ) ] ) R b - ( α ) d α - = ∫ 0 ω b - φ ( α ) R b - ( α ) d α - N λ + ( b ˜ ) = ∫ 0 ω b + φ ( pos [ b ˜ ⩽ L b + ( α ) ] ) L b + ( α ) d α + = ∫ 0 ω b + φ ( α ) L b + ( α ) d α + Q λ + ( b ˜ ) = ∫ 0 ω b + φ ( pos [ b ˜ ⩾ R b + ( α ) ] ) R b + ( α ) d α + = ∫ 0 ω b + φ ( α ) R b + ( α ) d α +

Evidently, N λ - ( b ˜ ) and Q λ - ( b ˜ ) shows the φ weighted lower possibility weighted average of the minimum of the α -cut set and hence called the φ weighted lower possibility mean of interval values. N λ + ( b ˜ ) and Q λ + ( b ˜ ) shows the weighted upper possibility weighted average of the maximum of the α -cut set and thus called the φ weighted upper possibility medium of interval values.

Definition 3.1.3. b ˜ β = [ L b ( β ) , L b ( β ) ] be the β -cut set of a TrCFN b ˜ = [ r , s , t , u ] ; 〈 [ ω b - , ω b + ] , η b ˜ 〉 . The φ weighted lower and upper possibility means of the non-membership function Γ for the TrCFN b ˜ are defined as N Γ ( b ˜ ) = ∫ η b ˜ 1 ψ ( pos [ b ˜ ⩽ L b ˜ ( β ) ] ) L b ˜ ( β ) d β ,(11) and Q Γ ( b ˜ ) = ∫ η b ˜ 1 ψ ( pos [ b ˜ ⩾ R b ( β ) ] ) R b ( β ) d β(12) respectively, where ψ : [ η b ˜ , 1 ] → R is a non-negative, monotone decreasing weighting function satisfying ∫ η b ˜ 1 ψ ( β ) ] ) d β = 1 - η b ˜ and ψ ( 1 ) = 1 , and Pos [ b ˜ ⩽ L b ( β ) ] ) = ∨ h ⩾ L b ˜ ( β ) Γ b ˜ ( h ) = β(13) Pos [ b ˜ ⩾ R b ( β ) ] = ∨ x ⩾ R a ˜ ( β ) Γ b ˜ ( h ) = β(14)

It follows from Equations (13) and (14) N Γ ( b ˜ ) = ∫ η b ˜ 1 ψ ( pos [ b ˜ ⩽ L b ˜ ( β ) ] ) L b ˜ ( β ) d β = ∫ η b ˜ 1 ψ ( β ) L b ˜ ( β ) d β

Most likely, N Γ ( b ˜ ) reflects the ψ weighted lower possibility weighted average of the minimum of the β -cut set, hence the name ψ weighted down possibility mean of non-membership function. Likewise, Q Γ ( b ˜ ) reflects the ψ weighted upper possibility weighted average of the maximum of the β -cut set and is called ψ weighted upper possibility mean of non-membership function.

Definition 3.1.4. For a TrCFN b ˜ = [ r , s , t , u ] ; 〈 [ ω b - , ω b + ] , η b ˜ 〉 the φ weighted possibility mean of interval value trapezoidal fuzzy number λ is defined as M λ - ( b ˜ ) = 1 2 [ N Γ - ( b ) + Q μ - ( b ) ] ,(15) M λ + ( b ˜ ) = 1 2 [ N Γ + ( b ) + Q Γ + ( b ) ](16) and the ψ weighted possibility mean of trapezoidal fuzzy number Γ is defined as M Γ ( b ˜ ) = 1 2 [ N Γ ( b ) + Q Γ ( b ) ](17)

Example 3.1.5. If we selected and respectively of the following shape φ ( α - ) = 2 α ω a - , α - ∈ [ 0 , ω a - ] , φ ( α + ) = 2 α ω a + , α + ∈ [ 0 , ω a + ] and ψ ( β ) = 2 ( 1 - β ) 1 - η a ˜ , β ∈ [ η a ˜ , 1 ] then according to Equations (3), (4), (5) and (6), we have N λ - ( b ˜ ) = [ ω b - ( r + 2 s ) 3 ] , Q λ - ( b ˜ ) = [ ω b - ( 2 t + u ) 3 ] and N λ + ( b ˜ ) = [ ω b + ( r + 2 s ) 3 ] , Q λ + ( b ˜ ) = [ ω b + ( 2 t + u ) 3 ] according to Equations (11) and (12) we have N Γ ( b ˜ ) = [ ( 1 - η b ) ( r + 2 s ) 3 ] and Q Γ ( b ˜ ) = [ ( 1 - η b ) ( 2 t + u ) 3 ]

Further, from Equations (15), (16) and (17), it follows M λ - ( b ˜ ) = ( r + 2 s + 2 t + u ) ( ω b - ) 6 ,(18) M λ + ( b ˜ ) = ( r + 2 s + 2 t + u ) ( ω b + ) 6 ,(19) and M Γ ( b ˜ ) = ( r + 2 s + 2 t + u ) ( 1 - η b ) 6(20)

Thus, for a TrCFN M λ - ( b ˜ ) , M λ + ( b ˜ ) the weighted possibility mean interval can be create from M λ - ( b ˜ ) , M λ + ( b ˜ ) and M Γ ( b ˜ ) as follows: { [ M λ - ( b ˜ ) , M λ + ( b ˜ ) ] , M Γ ( b ˜ ) = [ ( r + 2 s + 2 t - u ) ( ω b - ) 6 , ( r + 2 s + 2 t + u ) ( ω b + ) 6 ] ] , ( ( r + 2 s + 2 t + u ) ( 1 - η b ) 6 ) }

Definition 3.2.1. For TrCFNs b ˜ = [ r , s , t , u ] , 〈 [ ω b - , ω b + ] , η b ˜ 〉 , the weighted average operator TrCFN-Wa: φ ⁿ  → φ is defined as TrCF-Wa [ { ( b ˜ 1 - , b ˜ 2 - ) , ( b ˜ 1 + , b ˜ 2 + ) , ( a ˜ 1 , a ˜ 2 ) ] ... [ ( b ˜ n - , b ˜ n + ) , b ˜ n } ] = 〈 { ∑ j = 1 n w j b j - , ∑ j = 1 n w j b j + } , ∑ j = 1 n w j b j 〉 where the set of all TrCFNs is denoted by φ, W = ( w 1 , w 2 , . . . . w n ) T is the weight vector of b ˜ j ( j = 1 , 2 , . . . . , n ) and w j ∈ [ 0 , 1 ] , ∑ j = 1 n w j = 1 The TrCFN-WA operator is called the trapezoidal cubic fuzzy weighted average operator.

Theorem 3.2.2. The aggregated value of TrCFNs b ˜ j ( j = 1 , 2 , . . . . , n ) under the TrCFN-Wa operator is also a TrCFN given by TrCF - Wa ( b ˜ 1 , b ˜ 2 , . . . . , b ˜ n ) = [ ∑ j = 1 n w j ( r j ) , ∑ j = 1 n w j ( s j ) , ∑ j = 1 n w j ( t j ) , ∑ j = 1 n w j ( u j ) ] ; 〈 ∑ j = 1 n ( w b j - , w b j + ) ] } ; ∑ j = 1 n ( η b j ) ] 〉(23)

Proof Let b ˜ j = [ r , s , t , u ] ; 〈 ω b - , ω b + ] ; η b ˜ 〉 , be the TrCFNs.

Now, we use mathematical induction for n to prove the theorem. For n = 1, using Definition 3,i.e.,TrCFN-Wa TrCFN - Wa b ˜ j = [ r 1 , s 1 , t 1 , u 1 ] ; 〈 [ ω b 1 - , ω b 1 + ] , η b ˜ 1 〉 ,

Now, we prove it for n = 2. Using, the case of n = 1, we have w₁b₁ + w₂b₂ as TrCFNs. Now, we obtain the aggregation of these numbers as w 1 b 1 + w 2 b 2 = [ ∑ j = 1 2 w j ( r j ) , ∑ j = 1 2 w j ( s j ) , ∑ j = 1 2 w j ( t j ) , ∑ j = 1 2 w j ( u j ) ] ; 〈 ∑ j = 1 2 w j { w a j - , w a j + } ] , ∑ j = 1 2 w j { η a ˜ j } 〉(24) which is also a TrCFN using Definition 3. We now assume that it is true for n = k, i.e.,TrCFN-Wa TrCFN - Wa ( { ( b ˜ 1 - , b ˜ 1 + ) , b ˜ 1 } , { ( b ˜ 2 - , b ˜ 2 + ) , b ˜ 2 } . . . { ( b ˜ n - , b ˜ n + ) , b ˜ n } ) = [ ∑ j = 1 k w j r j , ∑ j = 1 k w j s j , ∑ j = 1 k w j t j , ∑ j = 1 k w j u j ] ; 〈 ∑ j = 1 k w j { w b j - , w b j + } ] , ∑ j = 1 k w j { η a ˜ j } 〉(25) is a TrCFN. We prove it for n = k + 1, we have TrCFN-Wa [ { ( b 1 - , b 2 - , . . . b k + 1 - ) , ( b 1 + , b 2 + , . . . . , b k + 1 + ) ] , ( b ˜ 1 , b ˜ 2 , . . . . , b ˜ k + 1 } ] = TrCFN - Wa [ { ( b 1 - , b 2 - , . . . b k + 1 - ) , ( b 1 + , b 2 + , . . . . , b k + 1 + ) ] , ( , b ˜ 1 , b ˜ 2 , . . . . , b ˜ k + 1 } = ∑ j = 1 k w j b ˜ j + w k + 1 b ˜ k + 1 which is also a TrCFN using Equations (24) and (25).

Definition 3.3.1. Consider the Banach space X, the hausdorff distance between any two subsets E and F can be defined as d ˜ ( E , F ) = max { Sup e ∈ E inf e ∈ F | e - f | , sup f ∈ F inf e ∈ E | e - f | } . If X = R and E and F are closed intervals [ e 1 , e 2 ] and [ f 1 , f 2 ] , respectively, then the above distance becomes to d ˜ ( E , F ) = Max { | e 1 - f 1 | , | e 2 - f 2 | } .

Definition 3.3.2. Let b₁ = [r₁, s₁, t₁, u₁]; 〈 [ ω b 1 - , ω b 1 + ] , η _{b 1} 〉 and b₂ = [r₂, s₂, t₂, u₂]; 〈 [ ω b 2 - , ω b 2 + ] , η b 2 〉 be two TrCFNs. The hammimg distance between b ˜ 1 and b ˜ 2 is defined as follows: d H ( b ˜ 1 , b ˜ 2 ) = 1 12 〈 [ | r 1 - r 2 | + | s 1 - s 2 | + | t 1 - t 2 | + | u 1 - u 2 | ] + | ω b 1 - - ω b 2 - | + | ω b 1 + - ω b 2 + | + | η b 1 - η b 2 | ] 〉(26)

4.1 MAGDM problems using TrCFNs

Assume that there exists l non-inferior alternatives {B₁, B₂, . . . . B _l } to be evaluated against n attributes {b₁, b₂,. . . . b _n } with the associated weighting vector W = (w₁, w₂,. . . . w _n )  ^T , satisfying w _j  ∈ [0, 1] (j = 1, 2,. . . , n) and ∑ j = 1 q w j = 1 . Further, suppose there exist q decision-makers {DM₁, DM₂,. . . . . DM _n } in the experts group with the associated weighing vector as ρ = (ρ₁, ρ₂,. . . . ρ _q )  ^T satisfying ρ ∈ [0, 1] (k = 1, 2,. . . q) and ∑ k = 1 q ρ k = 1 . Assume that each decision-maker provides the characteristics of the alternatives B _j with respect to each attribute b _j using TrCFNs as b ij k = [ r ij k , s ij k , t ij k , u ij k ] ; 〈 [ ω b ij - k , ω b ij + k ] , η b ij k 〉 .

Next, we determine the weights W = (w₁, w₂,. . . . w _n )  ^T of the attributes using Shannon entropy theory and the weights ρ = (ρ₁, ρ₂,. . . . ρ _q )  ^T of the decision-makers using the Evidence theory with Bayes approximation.

4.2 Weights of the attributes

To determine the attribute weights, which are unknown using Equation (23), the fuzzy assessment data provided by the decision-maker for each alternative with respect to each attribute is defuzzified into the crisp weighted possibility mean values V k = ( v ij k ) n × 1 (k = 1, 2,. . . . q), where v ij k = 1 6 s [ r ij k + 2 s ij k + 2 t ij k + u ij k ) ] , 〈 [ ( ω b ij - k ) ( s ) , ( ω b ij + k ) ( s ) ] , ( 1 - η b ij k ) ] 〉(27)

In general, attributes classified into two types: benefit attributes and cost attributes, i.e., the set F of attributes can be divided into two sets F _c and F _d , where F _c denotes the set of benefit attributes and F _d denotes the set of cost attributes, F _c  ∩ F _d  = φ and F _c  ∪ F _d  = F. So , Let y ij k = { v ij k v j k + , j ∈ F b v j k - v ij k , j ∈ F c(28) where v j k + = max { v ij k | i = 1 , 2 , . . . l } and v j k - = min { v ij k | i = 1 , 2 , . . . l } . Since, by using Shannon entropy measure, we objectively determine the weights of the attributes, the crisp weighted possibilities mean values V ^k need to be normalized unless all the attributes b _j  (j = 1, 2,. . . , n) are of the same type. we have e ij = 〈 [ 1 - y ij - , 1 - y ij + ] , 1 - y ij k 〉 .(29) j = 1 , 2 , . . . , n , k = 1 , 2 . . . , q .

The entropy measure g j k of the attribute b _j with respect DM _k is computed as follows: g j k = { [ ( - 1 ln l ∑ i = 1 n e ij - k ln e ij - k ) , ( - 1 ln l ∑ i = 1 n e ij + k ln e ij + k ) ] , ( - 1 ln l ∑ i = 1 n e ij k ln e ij k ) }(30)

Thus corresponding to attribute b _j , the individual weights ω j k with respect DM _k can be obtained as follows: ω j k = { [ ( 1 - g j - k ) ( 1 - g j + k ) ] ( 1 - g j k ) } { ( ∑ j = 1 n ( 1 - g j - k ) ) ( ∑ j = 1 n ( 1 - g j + k ) ) ( ∑ j = 1 n ( 1 - g j k ) ) }(31) ω j = ∑ k = 1 q ρ k ω j k(32)

Note that to have the composite attribute weight vector W = (ω₁, ω₂,. . . . . , ω _n )  ^T , we need the weight vector ρ = (ρ₁, ρ₂,. . . ρ _q )  ^T of the decision-makers.

Let there be a fixed set of l mutually exclusive and exhaustive propositions called the frame of discernment which is indicated by Ω = {B₁, B₂,. . . B _l }. A mass function over Ω (also known as basic probability assignment (BPA)) is a function M (.) :   2 ^Ω  → [0, 1] such that M (φ) =0 and ∑_{a⊆2 ^Ω} M (b) =1 where 2 ^Ω represents the power set of Ω and M (b) corresponds to the measure of belief that is committed exactly to the proposition B. Voorbraak [24] defined Bayes approximation of belief functions, which is computational less involving as it distributes mass M to a subset of Ω over its elements rather than assigning the mass M on the power set of Ω as in the case of Dempster-Shafer theory thus making the amount of computation required for the combination of evidence as exponentially increasing. The Bayesian approximation M of belief function convinced by the BPA M is computed as M ˜ ( b ) = { ∑ a ⊆ B M ( B ) ∑ C ⊆ Ω [ M ( c ) . | c | ] , if b is a singleton 0 , otherwise ,(33)

Where | · | denotes the cardinality function. Considering the alternatives {b₁, b₂,. . . . b _n } as the propositions of the frame of discernment of the MAGDM problems, let 〈 [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) 〉 be the evidence body of the proposition b _i on attribute b _j provided by DM _k . It follows 〈 [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) 〉 = e ij k , i = 1 , 2 , . . . l , j = 1 , 2 , . . . n , k = 1 , 2 , . . . q ,(34) where the assessment of alternative b _i on attribute b _j , e ij k are considered as evidence bodies. The weighted attribute evidence body of alternative b _i on attribute b _j by DM _k is computed as { [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) } = { ω j k ( M j - k ( b i ) ) , ω j k ( M j + k ( b i ) ) ] , ω j k ( M j k ( b i ) ) }(35) M ( Ω ) = 1 - ∏ i = 1 l { [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) } = 1 - ω j k ∏ i = 1 l { [ M j - k ( b i ) , M j + k ( b i ) ] , ω j k ∏ i = 1 l M j k ( b i ) }(36)

Incorporating combinational rule with Bayesian approximation, then n Bayes approximations M ˜ k ( b i ) (j = 1, 2,. . . n) corresponding to the mass function { [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) } could be aggregated into the respective composite evidence body { [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) } of alternative b _i with respect to DM _k as follows: [ [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) ] = { 0 , if b i = φ ∏ j = 1 m [ M j - k ( Y i ) , M j + k ( Y i ) ] , ∏ j = 1 m M j k ( Y i ) ∑ ∩ Y ≠ φ ∏ j = 1 m [ M j - k ( Y i ) , M j + k ( Y i ) ] , M j k ( Y i ) ] , if b i ≠ φ(37) where Y _i is a singleton and ∑_∩Y≠φ ∏ j = 1 m [ M j - k ( Y i ) , M j + k ( Y i ) ] , M j k ( Y i ) ] represents the extent of fracas between bodies of evidence. Let the bodies of evidence { [ M j - k ( b i ) , M j + k ( b i ) ] , M j k ( b i ) } . The distance between any two bodies of evidence M ˜ λ and M ˜ Γ denoted as dis ( M ˜ λ , M ˜ Γ ) can be defined as follows: dis H ( M ˜ λ , M ˜ Γ ) = 1 2 ( < m λ , m λ > + 2 < m Γ , m Γ > - < m λ , m Γ > )(38) sim ( M ˜ λ , M ˜ Γ ) = 1 - dis ( M ˜ λ , M ˜ Γ ) , ( u , v = 1 , 2 , . . . q )(39) which is formulated into the similarity matrix S = ( sim ( M ˜ λ , M ˜ Γ ) ) q × q . Obviously, similarity would be enhanced if the distance reduces. Let the support degree, denoted by s M ˜ k , the function of similarity which represents the degree to which an evidence body M ˜ k is supported by other bodies of evidence, is expressed as follows: s ( M ˜ k ) = ∑ Γ = 1 , Γ ≠ k q sim ( M ˜ k , M ˜ Γ ) , ( k = 1 , 2 , . . . q ) .(40)

The credibility degree, denoted by c ( M ˜ k ) of the body of evidence M ˜ k represents the relative importance of the evidence M ˜ k given by DM _k and hence measures the weight of DM _k , defined as c ( M ˜ k ) = s ( M ˜ k ) ∑ Γ = 1 q s ( M ˜ Γ ) , k = 1 , 2 , . . . q .(41)

Of course ∑ Γ = 1 q s ( M ˜ Γ ) = 1 . Thus, the weight vector of DM _s , ρ = (ρ₁, ρ₂,. . . . ρ _q )  ^T is obtained as follows: ρ k = c ( M ˜ k ) , k = 1 , 2 , . . . , q .(42)

Here, we extend the typical VIKOR method created on the beyond analysis. The steps are defined as below:

Step 1: Calculate the crisp weighted possibility mean values V k = ( v ij k ) n × 1 of decision-makers byEquation (27) from the TrCFN assessment date b ij k ( i = 1 , 2 , . . . , l ; j = 1 , 2 , . . . , n ; k = 1 , 2 , . . . , q ) given by each DM _k based on attribute b ij k with respect to alternative B _i .

Step 2: Forecast the individual weights W k = ( ω 1 k , ω 2 k , . . . . , ω n k ) T for each attribute b _j with respect to each DM _k using Equations (28)–(31).

Step 3: Using Equations (33)–(41), determine the weights of each decision-maker ρ = (ρ₁, ρ₂,. . . . ρ _q )  ^T .

Step 4: Compute the Combined weight vector of attribute W = (ω₁, ω₂,. . . ω _n )  ^T using Equation (32)

Step 5: Using the TrCFN-WA operator (i.e., Equation (23)) TrCFN - WA [ b ˜ 1 , b ˜ 2 , . . . . , b ˜ n } ] = ∑ j = 1 n w j ( λ ij ) = [ ∑ j = 1 n w j ( r j ) , ∑ j = 1 n w j ( s j ) , ∑ j = 1 n w j ( t j ) , ∑ j = 1 n w j ( u j ) ] ; 〈 ∑ j = 1 n w j [ w b j - , w b j + ] , ∑ j = 1 n w j ( η b j ) ] 〉 compute the TrCFN group decision values by aggregating the fuzzy assessment data given by each DM k H = ( h ij ) n × l , where h ˜ ij = [ h 1 ( b j ) , h 2 ( b j ) , h 3 ( b j ) , h 4 ( b j ) ] ; 〈 [ ω h ij - , ω h ij + } ] ; η h ij 〉 = TrCFN - WA ( b ˜ ij 1 , b ˜ ij 2 , . . . . . , b ˜ ij q ) = [ ∑ j = 1 n ρ k ( b j ) ] = [ ∑ j = 1 n ρ k ( r j ) , ∑ j = 1 n ρ k ( s j ) , ∑ j = 1 n ρ k ( t j ) , ∑ j = 1 n ρ k ( u j ) ] , 〈 ∑ j = 1 n ρ k [ w b j - , w b j + ] , ∑ j = 1 n ρ k ( η b j ) ] 〉(43)