Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 37

(a) The normalized Hamming distance for IVIFVs:

μ α i − μ α j + μ α i − μ α j +

v α j + v α j

1
4

v α i − v α i −

d H ( α i , α j ) =

π α i − π α j + π α i − π α j

+ (1.73)

(b) The normalized Euclidean distance for IVIFVs:

d E ( α i , α j )

1

4 (μ α i

2
(1.74)

− μ α j ) + (μ α i − μ α j ) + ( v α i − v α j ) + ( v α i − v α j ) + (π α i − π α j ) + (π α i − π α j ) =

2 2 2 2 2

T , then the IVIFPWA operator ( 1.71 )
reduces to an interval-valued intuitionistic fuzzy power average (IVIFPA) operator:

= ( / , / ,..., / )

Especially, if w 1 n 1 n 1 n

IVIFPA ( α 1 , α 2 ,..., α n )

(

( α 1 )) α 1 ⊕ ( ( α 2 )) α 2 ⊕···⊕ ( ( α n )) α n

1 + T 1 + T 1 + T

= i = 1 (

1 + T ( α i ))

⎛ ⎡ ⎤

( 1 + T ( α j ))

n

n

n

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

⎝ ⎣ ⎦ ,

1 ( 1 + T ( α i ))

− μ α j ) − μ α j ) = −

1 (

, −

1 (

1 1 1 1

i =

j = j =

⎡ ⎤

n n

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

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

⎣

v α j ) v α j )

⎦ ,

1 (

,

1 ( j = j =

⎡

n n

( 1 + T ( α j )) ( 1 + T ( α j ))

n
i = 1 ( 1 + T ( α i ))

n
i = 1 ( 1 + T ( α i ))

⎣

− μ α j ) v α j )

1 (

1 −

1 (

,

j = j =

⎤ ⎞

n

( 1 + T ( α j ))

n

( 1 + T ( α j ))

i i

− μ α j ) v α j )

⎦ ⎠

1 (

+ T ( α i ))

1 (

+ T ( α i ))

1 (

1

1

−

1 (

1

(1.75)

= =

j = j =

where

n

1
n

T ( α i ) = Sup ( α i , α j ) (1.76)

j = 1

j = i

Based on the IVIFPWA operator ( 1.71 ) and the geometric mean, Xu (2011) intro-
duced an interval-valued intuitionistic fuzzy power weighted geometric (IVIFPWG)
operator: