Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 153

Theorem 1.54 (Xia and Xu 2010)
(1) If n 0, then the GIFPOWA operators reduce to the generalized intuitionistic

fuzzy ordered weighted average operator (Zhao et al. 2010).
(2) If

=

0, then the GIFPOWA operators reduce to the intuitionistic
fuzzy ordered weighted average operator (Xu 2007).
(3) If

ρ = 1 and n =

0, then the GIFPOWA operators reduce to the intuitionistic
fuzzy ordered weighted geometric operator (Xu and Yager 2006).
(4) If

ρ → 0 and n =

0, then the GIFPOWA operators reduce to the intuitionistic
fuzzy maximum operator (Chen and Tan 1994).
(5) If w

ρ →+∞ and n =

T , n

1, then theGIFPOWAoperators
reduce to the intuitionistic fuzzy average operator (Xu 2007).
(6) If w

= ( 1 / m , 1 / m ,..., 1 / m ) = 0, and ρ =

T , n

= ( 1 / m , 1 / m ,..., 1 / m ) = 0, and ρ → 0, then the GIFPOWA opera-

tors reduce to the IFGM (Xu and Yager 2006).
(7) If w

T

0, then the GIFPOWA operators reduce to the
intuitionistic fuzzy maximum operator (Chen and Tan 1994).
(8) If w

= ( 1 , 0 ,..., 0 ) and n =

T

0, then the GIFPOWA operators reduce to the
intuitionistic fuzzy minimum operator (Chen and Tan 1994).

= ( 0 , 0 ,..., 1 ) and n =

The GIFPWA operators weight only the IFVs, while the GIFPOWA operators
weight only the ordered positions of the IFVs instead of weighting the IFVs them-
selves. To overcome this limitation, motivated by the idea of combining the weighted
averaging operator and the OWA operators (Torra 1997; Xu and Da 2003), Xia and
Xu (2010) developed a generalized intuitionistic fuzzy point hybrid aggregation
(GIFPHA) operator, which weights each given IFV and its ordered positions.

Theorem 1.55 (Xia and Xu 2010) The GIFPHA operators of dimension m is a map-
ping GIFPHA: V m

T ,

→ V , which has an associated vector w = ( w 1 , w 2 ,..., w m )

m , j = 1 w j with w j ≥ 0, j = 1 , 2 ,..., = 1, such that

(1) GIFPHAD w ,ω (α 1 ,α 2 ,...,α m )

w 1 D n

α σ ( 1 ) ρ

w 2 D n

α σ ( 2 ) ρ

= ⊕

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

w m D n

α σ ( m ) ρ

1
ρ

⊕···⊕

κ α σ ( m ) ,λ α σ ( m )

α σ ( j ) is the j th largest of m

where D n
κ α σ ( j ) ,λ α σ ( j )

ω i D n

κ α i ,λ α i (α i ) (

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

(2) GIFPHAF w ,ω (α 1 ,α 2 ,...,α m )

w 1 F n

α σ ( 1 ) ρ

w 2 F n

α σ ( 2 ) ρ

= ⊕

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

w m F n

α σ ( m ) ρ

1
ρ

⊕···⊕

κ α σ ( m ) ,λ α σ ( m )