Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 20

1.2.3 Power Aggregation Operators for IFVs

Let α i = (μ α i , v α i ,π α i )( i = 1 , 2 ,..., n ) be a collection of IFVs, and

T

w = ( w 1 , w 2 ,..., w n ) the weight vector of α i ( i = 1 , 2 ,..., n ) , where w i ≥

n , and i = 1 w i 0 1, then Xu (2011) defined an intuitionistic fuzzy

power weighted average (IFPWA) operator as follows:

, i = 1 , 2 ,..., =

IFPWA (α 1 ,α 2 ,...,α n )

= (

w 1 ( 1 + T (α 1 ))α 1 ) ⊕ ( w 2 ( 1 + T (α 2 ))α 2 ) ⊕···⊕ ( w n ( 1 + T (α n ))α n )

i = 1 w i (

1 + T (α i ))

(1.25)

By Definition 1.3, Eq. ( 1.25 ) can be transformed into the following form by using
mathematical induction on n :

IFPWA (α 1 ,α 2 ,...,α n )

⎛

n n

w j ( 1 + T (α j ))
i = 1 w i ( 1 + T (α i ))

w j ( 1 + T (α j ))
i = 1 w i ( 1 + T (α i ))

⎝ 1

= −

1 (

1 − μ α j ) ,

1 (

v

α j )

,

j = j =

⎞

n n

w j ( 1 + T (α j ))
n
i = 1 w i ( 1 + T (α i ))

w j ( 1 + T (α j ))

n
i = 1 w i ( 1 + T (α i ))

⎠

1 (

1 − μ α j ) −

1 (

v

α j )

(1.26)

j = j =

where

n

T (α i ) = w j Sup (α i ,α j ) (1.27)

j = 1

j = i

and Sup (α i ,α j ) is the support for α i from α j , with the following conditions:

(1) Sup (α i ,α j ) ∈[ 0 , 1 ] .

(α i ,α j ) = (α j ,α i )

(2) Sup Sup .

(α i ,α j ) ≥ (α s ,α t ) (α i ,α j )< (α s ,α t )

(3) Sup , where d is a distance

measure, such as the normalized Hamming distance or the normalized Euclidean
distance (Szmidt and Kacprzyk 2000; Narukawa and Torra 2006; Xu and Yager
2008), where

Sup ,if d d

(a) The normalized Hamming distance for IFVs:

2 μ α i − μ α j + v

j + π α i − π α j

1

d H (α i ,α j ) = − v (1.28)

α α i

(b) The normalized Euclidean distance for IFVs:

1

2 (μ α i − μ α j )

2

2 2

d E (α i ,α j ) = + ( v − v ) + (π α i − π α j ) (1.29)

α α i j