Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 109

which is called a generalized Einstein intuitionistic fuzzy weighted averaging
(GEIFWA) operator.
Case 5 If

γ = 2 and λ = 1, then Eq. ( 1.243 ) is transformed as:

EIFWA (α 1 ,α 2 ,...,α n )

i = 1 (

− i = 1 ( 2 i = 1 v w i

w i w i

1 + μ α i ) 1 − μ α i )

α i

= i = 1 (

+ i = 1 (

w i ,

i = 1 (

+ i = 1 v w i

1 + μ α i )

w i

1 − μ α i )

w i

2 − v α i )) α i

(1.283)

which is the EIFWA operator (Xia et al. 2012c).
Combining the GHIFWA operator and the geometric mean, then we introduce the
following:

T

Definition 1.25 (Xia and Xu 2011) Let w = ( w 1 , w 2 ,..., w n ) be the weight

vector of the IFVs α i ( i = 1 , 2 ,..., n ) , where w i indicates the importance degree

and i = 1 w i of α i , satisfying w i > 0( i = 1 , 2 ,..., n ) = 1, if

n
⊗

1
λ

w i
i

GAIFWG (α 1 ,α 2 ,...,α n ) =

1 λα

(1.284)

i =

then GAIFWG is called a generalized Archimedean intuitionistic fuzzy geometric
(GHIFWG) operator.

Similarly, the following theorem can be obtained:

Theorem 1.42 (Xia and Xu 2011) The aggregated value by using the GHIFWG
operator is also an IFV, and

⎛
⎝

μ

n n

1
λ

1
λ

l
1

2 r

1

l
1

l
1

n + (γ

− 1 )μ − μ

n − μ , , , ,

GHIFWG (α 1 ,α 2 ,...,α n ) = ,

n n

1
λ

1
λ

l
1

l
1

l
1

l
1

2

μ

n + (γ

− 1 )μ + (γ − 1 ) μ

n − μ , , , ,

⎞
⎠

v l 1 , n − v 1 , n

1
λ

γ

v l 1 , n + (γ

− 1 ) v 1 , n

v l 1 , n − v 1 , n

1
λ

1
λ

2

+ (γ − 1 )

(1.285)

where

)μ α i w i

n

l
1

− μ α i )) λ + (γ

2

μ

n =

( 1 + (γ − 1 )( 1 − 1 (1.286)

,

i = 1

n

(

− μ α i )) λ − μ α i w i

r
1

μ

n =

1 + (γ − 1 )( 1 (1.287)

,

i = 1