Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 100

Theorem 1.38 (Xia and Xu 2011) The aggregated value by using the GHIFWA
operator is also an IFV, and

⎛
⎝

1 , n

1
λ

l r

γ μ

1 , n − μ

GHIFWA (α 1 ,α 2 ,...,α n ) = ,

n n

1
λ

1
λ

l
1

r
1

l
1

r
1

2

μ

n + (γ

− 1 )μ + (γ − 1 ) μ

n − μ , , , ,

⎞
⎠

v l 1 , n + (γ

− 1 ) v 1 , n

v l 1 , n − v l 1 , n

1
λ

1
λ

2

−

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 )

(1.243)

where

n

)μ α i w i

l
1

− μ α i )) λ + (γ

2

μ

n =

( 1 + (γ − 1 )( 1 − 1 (1.244)

,

i = 1

n

(

− μ α i )) λ − μ α i w i

1

μ = + (γ − )(

1 1 1 (1.245)

, n

i = 1

v α i w i

n

v l 1 , n =

α i )) λ + (γ

2

( 1 + (γ − 1 )( 1 − v − 1 ) (1.246)

i = 1

n

(

v α i w i

v 1 , n = v α i )) λ − 1 + (γ − 1 )( 1 − (1.247)

i = 1

w i α i

Proof Let β i = , then Eq. ( 1.242 ) can be written as:

n
⊕ w i α i

n
⊕

1 β i

1
λ

1
λ

GHIFWA (α 1 ,α 2 ,...,α n ) = = (1.248)

i = 1 i =

and we first prove the following equation by using mathematical induction on n :

i = 1 (

)μ β i ) − i = 1 (

1 + (γ − 1 1 − μ β i )

n
⊕

1 β i

= i = 1 (

) i = 1 ( − μ β i ) , 1 + (γ − 1 )μ β i ) + (γ − 1 1

i =

γ i = 1 v

β i

i = 1 (

) i = 1 v

(1.249)

1 + (γ − 1 )( 1 − v

β i )) + (γ −

1

β i

For n = 2, Eq. ( 1.249 ) holds obviously. Suppose that Eq. ( 1.249 ) holds for n = k ,

that is,