Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 86

1.6.2 Intuitionistic Fuzzy Aggregation Operators Based
on Archimedean t-conorm and t-norm

The operational laws defined in Sect. 1.6.1 can be used to aggregate the intuitionistic
fuzzy information, which is the focus of this subsection.

T be the weight vector

Definition 1.22 (Xia et al. 2012c) Let w = ( w 1 , w 2 ,..., w n )

of the IFVs α i = (μ α i , v )( i = 1 , 2 ,..., n ) , where w i indicates the importance

α

and i = 1 w i

i

degree of α i , satisfying w i > 0 ( i = 1 , 2 ,..., n ) = 1, if

n
⊕ ATS − IFWA (α 1 ,α 2 ,...,α n ) =

1 (

w i α i ) (1.202)

i =

then ATS-IFWA is called an Archimedean t-conorm and t-norm based intuitionistic
fuzzy weighted averaging (ATS-IFWA) operator.

Theorem 1.29 (Xia et al. 2012c) The aggregated value by using the ATS-IFWA
operator is also an IFV, and

n
⊕

ATS − IFWA (α 1 ,α 2 ,...,α n ) = w i α i

i = 1

h − 1 n

, g − 1 n = w i h (μ α i ) w i g( v

α i )

i = 1 i = 1

(1.203)

which has been investigated by Beliakov et al. (2011), Xu and Yager (2009), Xu and
Cai (2010a), and next we give a further study:

Proof By using mathematical induction on n :For n = 2, we have

ATS − IFWA (α 1 ,α 2 )

2
⊕ = w i α i = w 1 α 1 ⊕ w 2 α 2

i = 1

h − 1 h

h − 1 h − 1 = ( ( w 1 h (μ α 1 ))) + h ( ( w 2 h (μ α 2 ))) ,

g − 1 g(g − 1 v α 1 ))) + g(g − 1 ( w 1 g( ( w 2 g( v α 2 )))

α 2 )

g − 1 w 1 g(μ α 1 ) + w 2 g(μ α 2 ) , h − 1 w 1 h

= ( v

α 1 ) +

w 2 h ( v (1.204)

Suppose that Eq. ( 1.203 ) holds for n = k , that is,