Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 147

GIFPOWAQ w (α 1 ,α 2 ,...,α m ) (9)

w 1 Q n

α σ ( 1 ) ρ

w 2 Q n

α σ ( 2 ) ρ

= ⊕ ⊕···

κ α σ ( 1 ) ,λ α σ ( 1 ) κ α σ ( 2 ) ,λ α σ ( 2 )

w m Q n

α σ ( m ) ρ

1
ρ

⊕

κ α σ ( m ) ,λ α σ ( m )

α σ ( j ) is the j th largest

m , Q n
κ α σ ( j ) ,λ α σ ( j )

where κ α σ ( j ) + λ α σ ( j ) ≤ 1, j = 1 , 2 ,...,

of Q n

κ α i ,λ α i (α i ) (

i = 1 , 2 ,..., m ) .

Then the functions GIFPOWAD w , GIFPOWAF w , GIFPOWAG w , GIFPOWAH w ,
GIFPOWAH ∗ , n
w

, GIFPOWAJ w , GIFPOWAJ ∗ , n , GIFPOWAP w and GIFPOWAQ w are

w

called the GIFPOWA operators.
The GIFPOWA operators have some properties similar to those of the GIFPWA
operators.

Theorem 1.49 (Xia and Xu 2010) The aggregated value by using the GIFPOWA
operators are also IFVs, and

GIFPOWAD w (α 1 ,α 2 ,...,α m ) (1)

⎛
⎝

⎛

− μ α σ ( j ) + κ α σ ( j ) π α σ ( j ) ρ w j ⎞

1
ρ

m

1

⎝ 1 ⎠

= − ,

j = 1

ρ ⎞
⎠

⎛

− κ α σ ( j ) π α σ ( j ) ρ w j ⎞

1

m

1

− 1 v α σ ( j ) − 1

⎝ 1 ⎠

1 − − −

j = 1

GIFPOWAF w (α 1 ,α 2 ,...,α m ) (2)

⎛

⎛

w j ⎞
⎠

1

1
ρ

m

⎝

− μ F n

⎝ 1

= − ,

κ α σ ( j ) ,λ α σ ( j ) ( α σ ( j ) )

j = 1

ρ ⎞
⎠

⎛

ρ w j ⎞
⎠

1

1 1

m

⎝ 1

1 − − − − v F n

κ α σ ( j ) ,λ α σ ( j ) ( α σ ( j ) )

j = 1

where κ α σ ( j ) + λ α σ ( j ) ≤ 1, j = 1 , 2 ,..., m , and