Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 244

b
a

f

A , B (

x ) dx

c 10 ( A

, B

˙ ) = b

a

(2.184)

· b
a

g

A (

x ) dx g

B (

x ) dx

For convenience, we introduce the concept of interval-valued intuitionistic fuzzy
association matrix:

A j

Definition 2.33 (Xu et al. 2008) Let ( j = 1 , 2 ,..., m ) be m IVIFSs, then

C

( A i , A j ) is the interval-

valued intuitionistic fuzzy association coefficient of A i and A j , which has the follow-
ing properties: (1) 0

= ( ˙ c ij ) m × m is called an association matrix, where ˙ c ij =˙ c

≤˙ c ij ≤ 1 for all i , j = 1 , 2 ,..., m ;(2) c ij ˙ = 1 if and only if

A i = A j ; and (3)

c ij =˙ ˙ c ji , for all i , j = 1 , 2 ,..., m .

Based on the definition above, in what follows, we introduce an algorithm for
clustering IVIFSs (Zhao et al. 2012b):

Algorithm 2.13

Step 1 Use Eqs. ( 2.168 )or( 2.172 ) (if the weights of the attributes are the same,
we use Eq. ( 2.168 ); otherwise, we use Eq. ( 2.172 )) to calculate the association coef-
ficients of the IVIFSs A j ( j = 1 , 2 ,..., m ) , and then construct an association matrix

C

c 7 ( A i , A j ) c 8 ( A i , A j ) = ( ˙ c ij ) m × m , where ˙ c ij =˙ or c ij =˙ ˙ , i , j = 1 , 2 ,..., m .

λ = λ ˙

c ij m × m of

-cutting matrix C

C by using Eq. ( 2.87 ).

Step 2 Construct a λ

Step 3 See Algorithm 2.12.
Step 4 See Algorithm 2.12.
Step 5 End.

Example 2.11 (Zhao et al. 2012b) Suppose that there are six samples

y i ( i = 1 , 2 ,..., 6 ) to be classified. According to the attributes G i ( i = 1 , 2 ) ,

their attribute values are expressed by IVIFSs as follows:

y 1 ={ G 1 , [0 . 60 , 0 . 80] , [0 . 10 , 0 . 20] , G 2 , [0 . 50 , 0 . 70] , [0 . 10 , 0 . 30] }

y 2 ={ G 1 ,

. , . , . , . ,

G 2 ,

. , . , . , . }

[0 30 0 50] [0 25 0 45] [0 70 0 85] [0 00 0 15]

y 3 ={ G 1 ,

. , . , . , . ,

G 2 ,

. , . , . , . }

[0 45 0 65] [0 15 0 35] [0 60 0 80] [0 05 0 20]

y 4 ={ G 1 ,

[ 0

. 34 , 0 . 54 ] ,

[ 0

. 25 , 0 . 45 ] , G 2 ,

[ 0

. 50 , 0 . 70 ] ,

[ 0

. 10 , 0 . 30 ] }

y 5 ={ G 1 , [0 . 40 , 0 . 60] , [0 . 25 , 0 . 40] , G 2 , [0 . 65 , 0 . 80] , [0 . 10 , 0 . 20] }

y 6 ={ G 1 , [0 . 45 , 0 . 65] , [0 . 15 , 0 . 35] , G 2 , [0 . 47 , 0 . 67] , [0 . 05 , 0 . 25] }

Suppose that the weights of the attributes G j ( j = 1 , 2 ) are equal, now we utilize

Algorithm 2.13 to group these samples y i ( i = 1 , 2 ,..., 6 ) :

Step 1 Use Eq. ( 2.168 ) to compute the association coefficients of the IFSs

y i ( i = 1 , 2 ,..., 6 ) , and then construct an association matrix C = ( c ij ) 6 × 6 , where

c ij =˙ c 7 ( y i , y j ), i , j = 1 , 2 ,..., 6: