Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 158

GIFPHAP w ,ω (α 1 ,α 2 ,...,α m )

( )

8

⎛
⎝

⎛

− max κ α σ ( j ) ,μ α σ ( j ) ρ w j ⎞

1
ρ

m

1

⎝ 1 ⎠

= − ,

j = 1

ρ ⎞
⎠

⎛

v α σ ( j ) ρ w j ⎞

1

m

1

− 1 min λ α σ ( j ) ,

⎝ 1 ⎠

1 − − −

j = 1

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

GIFPHAQ w ,ω (α 1 ,α 2 ,...,α m ) ( 9 )

⎛
⎝

⎛

− min κ α σ ( j ) ,μ α σ ( j ) ρ w j ⎞

1
ρ

m

1

⎝ 1 ⎠

= − ,

j = 1

ρ ⎞
⎠

⎛

v α σ ( j ) ρ w j ⎞

1

m

1

− 1 max λ α σ ( j ) ,

⎝ 1 ⎠

1 − − −

j = 1

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

Theorem 1.57 (Xia and Xu 2010)
(1) If w

T and n

0, then the GIFPHA operators reduce to
the GIFWA operators (Zhao et al. 2010).
(2) If

= ( 1 / n , 1 / n ,..., 1 / n ) =

T and n

0, then the GIFPHA operators reduce to
the GIFOWA operator (Zhao et al. 2010).
(3) If

ω = ( 1 / n , 1 / n ,..., 1 / n ) =

η = 1 and n = 0, then the GIFPHA operators reduce to the following:

⎛ ⎞

m m

v w j
α σ( j )

⎝ 1

w j

⎠ (1.417)

IFHA w ,ω (α 1 ,α 2 ,...,α m ) = −

1 (

1 − μ α σ( j ) ) ,

j = j = 1

which is called an intuitionistic fuzzy hybrid averaging operator (Xu 2007).