Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 91

then

ATS − IFWA (α 1 ⊕ β 1 ,α 2 ⊕ β 2 ,...,α n ⊕ β n )

h − 1 n

, g − 1 n h − 1 w i g(g − 1 = w i h ( ( h (μ α i ) + h (μ β i ))) (g( v α i ) + g( v β i )))

i = 1 i = 1

h − 1 n

, g − 1 n

(1.227)

= w i ( h (μ α i ) + h (μ β i )) w i (g( v α i ) + g( v β i ))

i = 1 i = 1

and

ATS − IFWA (α 1 ,α 2 ,...,α n ) ⊕ ATS − IFWA (β 1 ,β 2 ,...,β n )

h − 1 n

, g − 1 n

= w i h (μ α i ) w i g( v α i )

i = 1 i = 1

h − 1 n

, g − 1 n

⊕ w i h (μ β i ) w i g( v β i )

i = 1 i = 1

h − 1 h h − 1 n

+ h h − 1 n = w i h (μ α i ) w i h (μ β i ) ,

i = 1 i = 1

g − 1 g − 1 n g − 1 n

g w i g( v α i ) + g w i g( v β i )

i = 1 i = 1

h − 1 n

, g − 1 n

n n

= w i h (μ α i ) + w i h (μ β i ) w i g( v α i ) + w i g( v β i )

i = 1 i = 1 i = 1 i = 1

(1.228)

which completes the proof.

If the additive generator g is assigned different forms, then some specific intu-
itionistic fuzzy aggregation operators can be obtained (Xia et al. 2012c):
Case 1 If g(

t ) =− log ( t ) , then the ATS-IFWA operator reduces to the following:

1

n n

w i

v w i
α i

IFWA (α 1 ,α 2 ,...,α n ) = −

1 (

1 − μ α i ) , (1.229)

i = i = 1

which is the IFWA operator defined by Xu (2007).
Case 2 If g( log 2 − t , then the ATS-IFWAoperator reduces to the following:

t ) =

EIFWA (α 1 ,α 2 ,...,α n )

i = 1 (

− i = 1 ( 2 i = 1 v w i

w i w i

1 + μ α i ) 1 − μ α i )

α i

= i = 1 (

+ i = 1 (

w i ,

i = 1 (

+ i = 1 v w i + μ α i )

w i

− μ α i )

w i

1 1 2 − v α i )

α i

(1.230)

which is called anEinstein intuitionistic fuzzyweighted averaging (EIFWA) operator.