Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 38

( α 1 , α 2 ,..., α n )

IVIFPWG

1 1

w 1 ( 1 + T ( α 1 )) w 2 ( 1 + T ( α 2 ))

1 1

− −

i i

n − 1 n − 1

1 w i (

1 + T ( α i ))

1 w i (

1 + T ( α i ))

= ( α 1 ) ⊗ ( α 2 )

= =

1

1
n − 1

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

−

⊗···⊗ ( α n ) (1.77)

which can be transformed into Eq. ( 1.78 ) by using mathematical induction on n :

IVIFPWG ( α 1 , α 2 ,..., α n )

⎛ ⎡

⎤
⎦ ,

1 − 1 −

n n

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

n
i = 1 w i ( 1 + T ( α i ))

1
n − 1

1
n − 1

n
i = 1 w i ( 1 + T ( α i ))

⎝ ⎣

1 (μ α j ) 1 (μ α j )

= ,

j = j =

⎡

1 1

n

w j ( 1 + T ( α j ))

1

−

i

⎣ 1

v α j )

n −

1 w i (

1 + T ( α i ))

−

1 (

1 − ,

=

j =

⎤
⎦ ,

1

n

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

j = 1 (

1
n − 1

−

v α j ) 1 − 1 −

⎡ 1 − 1 −

n n

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

n
i = 1 w i ( 1 + T ( α i ))

1
n − 1

1
n − 1

n
i = 1 w i ( 1 + T ( α i )) ⎣

v α j )

1 (μ α j )

1 (

1 − − ,

j = j =

⎤
⎦

⎞

1 1

n

w j ( 1 + T ( α j ))

n

w j ( 1 + T ( α j ))

1 1

− −

i i

v α j )

n − 1

1 (μ α j )

n − 1

⎠ 1 w i (

1 + T ( α i ))

1 w i (

1 + T ( α i ))

1 (

1 − −

= =

j = j =

(1.78)

with the condition ( 1.72 ).
Especially, if w

T , then the IVIFPWG operator ( 1.78 )
reduces to an interval-valued intuitionistic fuzzy power geometric (IVIFPG) operator:

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

IVIFPG ( α 1 , α 2 ,..., α n )

1 1

1 + T ( α 1 )
i = 1 ( 1 + T ( α i ))

1 + T ( α 2 )
i = 1 ( 1 + T ( α i ))

1
n − 1

1
n − 1

− −

= ( α 1 ) ⊗ ( α 2 )

1 1

1 + T ( α n )

1
n −

−

i

1 (

1 + T ( α i ))

⊗···⊗ ( α n )

=

⎛ ⎡

⎤
⎦ ,

1 1

n

( 1 + T ( α j ))

n

( 1 + T ( α j ))

1 1

− −

i i

⎝ ⎣

1 (μ α j )

n − 1

1 (μ α j )

n − 1

1 (

1 + T ( α i ))

1 (

1 + T ( α i ))

= ,

= =

j = j =

⎡

1

n

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

1
n − 1

−

⎣ 1

v α j ) −

1 (

1 − ,

j =