Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 22

(α 1 ,α 2 ,...,α n )

IFPWG

1 1 1 1

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

1 1

− −

i i

n − n −

1 w i (

1 + T (α i ))

1 w i (

1 + T (α i ))

= (α 1 ) ⊗ (α 2 ) ⊗···

= =

1

w n ( 1 + T (α n ))

1

−

i

n − 1

1 w i (

1 + T (α i ))

⊗ (α n ) (1.37)

=

which can be transformed into the following form by using mathematical induction
on n :

IFPWG (α 1 ,α 2 ,...,α n )

⎛ 1 1

n

w j ( 1 + T (α j ))
i = 1 w i (

n

w j ( 1 + T (α j ))
i = 1 w i (

1 1

− −

⎝

n − 1 n − 1

1 + T (α i )) 1 + T (α i ))

=

1 (μ α j )

, 1 −

1 (

1 − v

α j )

,

⎞
⎠

j = j =

1 1

n

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

n

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

1
n − 1

1
n − 1

− −

1 (

1 − v ) −

1 (μ α j )

α

j

j = j =

(1.38)

with the condition ( 1.27 ).
Especially, if w

T , then the IFPWGoperator ( 1.37 ) reduces
to an intuitionistic fuzzy power geometric (IFPG) operator:

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

IFPG (α 1 ,α 2 ,...,α n )

1 1

(α 1 )
i = 1 ( 1 + T (α i ))

1 + T (α 2 )

i = 1 ( 1 + T (α i ))

1 + T

1 1

− −

n − 1 n − 1

= (α 1 ) ⊗ (α 2 ) ⊗···

1 1

1 + T (α n )

1

−

i

n −

1 (

1 + T (α i ))

⊗ (α n )

=

⎛

1 1

n

1 + T (α j )

1

−

i

⎝

n −

1 (

1 + T (α i ))

=

1 (μ α j )

,

=

j =

1 1

n

1 + T (α j )

n

1 + T (α j )

1
n − 1

1
n − 1

− −

i i

1 (

1 + T (α i ))

1 (

1 + T (α i ))

−

1 (

−

v α j )

,

1 (

−

v α j )

1 1 1

= =

j = j =

⎞
⎠

1

n

1 + T (α j )

1

−

i

n − 1

1 (

1 + T (α i ))

−

1 (μ α j )

(1.39)

=

j =

with the condition ( 1.31 ).
Similar to the IFPWA operator, the IFPWG operator has the following three prop-
erties (Xu 2011):

(α 1 ,α 2 ,...,α n ) Theorem 1.7 (Commutativity) Let be any permutation of (α 1 ,α 2 ,

...,α n ) , then

(α 1 ,α 2 ,...,α n )

IFPWG (α 1 ,α 2 ,...,α n ) = IFPWG (1.40)