Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 108

(α 1 ,α 2 ,...,α n )

HIFWA

i = 1 (

− i = 1 (

w i w i

1 + (γ − 1 )μ α i ) 1 − μ α i )

= i = 1 (

) i = 1 (

w i ,

1 + (γ − 1 )μ α i )

w i

+ (γ − 1 1 − μ α i )

γ i = 1 v w i

α i

i = 1 (

) i = 1 v w i

(1.277)

w i

1 + (γ − 1 )( 1 − v )) + (γ − 1

α α i i

which is the Hamacher intuitionistic fuzzy averaging (HIFWA) operator (Xia et al.
2012c).
Case 2 If

λ = γ =

1 and 1, then Eq. ( 1.243 ) becomes the IFWA operator (Xu

2007):

1

n n

(

− μ α i w i

v w i IFWA (α 1 ,α 2 ,...,α n ) = − 1 , (1.278)

α i

i = 1 i = 1

1, then Eq. ( 1.243 ) becomes the generalized intuitionistic fuzzy
weighted averaging (GIFWA) operator (Zhao et al. 2010):

Case 3 If γ =

GIFW A (α 1 ,α 2 ,...,α n )

⎛

λ ⎞
⎠

1

− μ α i w i

1

v α i ) λ w i

1
λ

1

n n

( (

⎝

= − 1 , 1 − − 1 − ( 1 −

i = 1 i = 1

(1.279)

Case 4 If γ = 2, then Eq. ( 1.243 ) is written as:

GEIFWA (α 1 ,α 2 ,...,α n )

⎛
⎝

⎞
⎠

2

1 , n

v l 1 , n +

3 v 1 , n

v l 1 , n −

v 1 , n

1
λ

1
λ

1
λ

l r

μ

1 , n − μ

−

=

λ ,

n n

v l 1 , n +

3 v l 1 , n

v l 1 , n −

v 1 , n

1
λ

1 1

λ

1
λ

l
1

r
1

l
1

r
1

μ

n +

3 μ + μ

n − μ

+

, , , ,

(1.280)

where

n n

(

μ α i w i

(

− μ α i )) λ − μ α i w i

l
1

− μ α i )) λ +

r
1

μ

n =

1 + ( 1 3 ,μ

n =

1 + ( 1

, ,

i = 1 i = 1

(1.281)

n n

(

3 v α i w i

(

v α i w i

v l 1 , n =

α i )) λ +

v 1 , n =

α i )) λ −

1 + ( 1 − v , 1 + ( 1 − v

i = 1 i = 1

(1.282)