Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 130

− μ α ) 1 − (

p p

= μ α + ( − κ α ) − κ α ( − κ α )

1 1 1

⎛ ⎛ ⎞ ⎞

p − 1

⎝ ⎝ ⎠ + λ ⎠

p − 1 − t t + 1 p

α

−

v α κ α λ α

0 λ

− κ α )

( 1

α

t =

p

t

1

1 p + p − t

= μ α + ( 1 − μ α ) − ( 1 − κ α ) − v α κ α λ α

0 λ

( 1 − κ α )

α

t =

(1.329)

p
α

p + 1

v J ∗ , p + 1

κ α ,λ α (α) = κκ

v

α = λ

v (1.330)

α

α

= +

and hence, (7) holds for n 1. Therefore, (7) holds for all n .

In addition, by Definition 1.27, we can easily get that intuitionistic fuzzy point
operators translate one IFV to another IFV. In the following subsection, we introduce
some generalized intuitionistic fuzzy point averaging operators (Xia and Xu 2010)
combining the developed point operators with Zhao et al. (2010)'s operators.

p

1.9 Generalized Point Operators for Aggregating IFVs

Definition 1.28 (Xia andXu 2010) Let V be the set of all IFVs, α j = (μ α j , v

α j )(

j =

1 , 2 ,..., m ) a collection of IFVs, and n a positive integer, taking κ α j ,λ α j ∈ [0 , 1],

0, and let GIFPWA: V m j = 1 , 2 ,..., m , ρ> → V ,if

GIFPWAD w (α 1 ,α 2 ,...,α m )

( 1 )

w 1 D n ρ

w 2 D n

ρ

w m D n

ρ

1
ρ

=

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕ ···⊕

κ α m ,λ α m (α m )

.

GIFPWAF w (α 1 ,α 2 ,...,α m ) ( 2 )

w 1 F n ρ

w 2 F n

ρ

w m F n

ρ

1
ρ

=

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕ ···⊕

κ α m ,λ α m (α m )

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

GIFPWAG w (α 1 ,α 2 ,...,α m ) ( 3 )

w 1 G n ρ

w 2 G n

ρ

w m G n

ρ

1
ρ

=

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕···⊕

κ α m ,λ α m (α m )

.

GIFPWAH w (α ( 4 ) ,α ,...,α ) 1 2 m

w 1 H n ρ

w 2 H n

ρ

w m H n

ρ

1
ρ

=

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕···⊕

κ α m ,λ α m (α m )

.

GIFPWAH ∗ , n
w

( 5 ) (α ,α ,...,α ) 1 2 m

w 1 H ∗ , n ρ

w 2 H ∗ , n

ρ

w m H ∗ , n

ρ

1
ρ

=

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕···⊕

κ α m ,λ α m (α m )

.