Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 105

n

v α i w i

v l 1 , n = v α i )) λ + (γ

2

( 1 + (γ − 1 )( 1 − − 1 ) (1.265)

i = 1

n

(

v α i w i

v 1 , n =

α i )) λ −

1 + (γ − 1 )( 1 − v (1.266)

i = 1

then

l
1

r
1

v 1 , n
v l 1 , n + (γ −

μ

n − μ

γ

k
i = 1 β i

, , n

=

n ,

(1.267)

l
1

r
1

v 1 , n μ

n + (γ −

1 )μ 1 )

, ,

and

k
⊕

1 β i

1
λ

i =

⎛
⎝

1
λ

l 1 , n − μ 1 , n

μ

γ

l
1 , n + (γ −

r
1 , n

μ 1 )μ

=

λ ,

1 1

1
λ

1

l 1 1 l 1 1

μ

n − μ

μ

n − μ , , n , , n

+ (γ − 1 ) − + (γ − 1 )

l 1 , n + (γ − l 1 , n + (γ −

1 , n 1 , n

μ 1 )μ μ 1 )μ

⎞
⎠

1 1

1
λ

1
λ

v 1 , n
v l 1 , n + (γ −

v 1 , n
v l 1 , n + (γ −

γ γ

+ (γ − 1 ) − −

v 1 , n v 1 , n 1 ) 1 )

1 1

1
λ

1
λ

v 1 , n
v l 1 , n + (γ −

v 1 , n
v l 1 , n + (γ −

γ γ

+ (γ − 1 ) + (γ − 1 ) −

v 1 , n v 1 , n

1 ) 1 )

⎛
⎝

n

1
λ

l
1

1

γ μ

n − μ , ,

=

λ ,

1 , n 1 , n

1
λ

1

l 2 r l r

μ

1 , n + (γ

− 1 )μ + (γ − 1 ) μ

1 , n − μ

⎞

v l 1 , n + (γ

v 1 , n

v l 1 , n −

v l 1 , n

1
λ

1
λ

2

− 1 ) −

⎠

(1.268)

v l 1 , n + (γ

v l 1 , n

v l 1 , n −

v l 1 , n

1
λ

1
λ

2

− 1 ) + (γ − 1 )

which completes the proof.

Then in what follows, we introduce some desirable properties of the GHIFWA
operator (Xia and Xu 2011):

Theorem 1.39 If all α i ( i = 1 , 2 ,..., n ) are equal, i.e., α i = α = (μ α , v α ) , for all

i , then