Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 233

From Theorem 2.18, we can see that if the association matrix is equivalent, then

λ

its -cutting matrix is an equivalent Boole matrix, and then we can use the equivalent

Boole matrix to do clustering directly. But if the association matrix doesn't satisfy the
transitivity, then we know that the

-cutting matrix of C is just only a similar Boole
matrix, and thus, we cannot do clustering. In this situation, we can transform the
similar Boole matrix into an equivalent matrix for clustering. Let's see the following
theorem:

λ

Theorem 2.19 (Lei 1979) Let Bo be a similar Boole matrix over a discrete universe
of discourse X

={ x 1 , x 2 ,..., x n } , then Bo is transitive if and only if Bo has not the

following special sub-matrices:
11
10

11
01

10
11

01
11

, , ,

(2.167)

no matter how the matrix Bo is arranged.
We can judge from Theorem 2.19 whether or not a similar Boole matrix is an
equivalent one.
Based on Theorems 2.18 and 2.19, Zhao et al. (2012b) developed an intuitionistic
fuzzy clustering algorithmbased onBoolematrix and associationmeasure as follows:

Algorithm 2.12

Step 1 Use Eq. ( 2.159 )or( 2.164 ) (if the weights of the attributes are the same,
we use Eq. ( 2.159 ); otherwise, we use Eq. ( 2.164 )) to compute the association coef-
ficients of the IFSs A j ( j = 1 , 2 ,..., m ) , and then construct an association matrix

C = ( c ij ) m × m , where c ij = c 3 ( A i , A j ) or c ij = c 4 ( A i , A j ) , i , j = 1 , 2 ,..., m .

-cutting matrix C λ = λ c ij m × m of C by using Eq. ( 2.87 ).
Step 3 If C λ is an equivalent Boole matrix, then we can cluster the m samples
as follows: If all the elements of the i th column are the same as the corresponding
elements of the j th column in C λ , then the IFSs A i and A j are in the same cluster. By
this principle, we can cluster all these m samples A j (

Step 2 Construct a λ

.
If C λ is not an equivalent Boole matrix, then by Theorem 2.19, we know that
no matter how the matrix C λ is arranged, it must have some of the special sub-
matrixes in Eq. ( 2.167 ). In such cases, we can transform the elements 0 into 1 in such
special sub-matrices until C λ

j = 1 , 2 ,..., m )

has not any special sub-matrix, and thus, we get a new

equivalent matrix C λ

.
Step 4 Employ the equivalent matrix C λ to classify all the given IFSs A j

( j = 1 , 2 ,..., m ) by the procedure in Step 3.

Step 5 End.

: Based on the idea of constructing the association
matrix whose elements are association coefficients between every two alternatives
(samples) in this paper, we balance the similarity degree between two alternatives
mainly through the association coefficient (that is, the confidence level) of them. We

The principal of choosing λ