Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 250

n n n

1
n

1
n

1
n

μ ik +˙ v ik = 1 −

1 |

v ij − v kj |−

1 | π ij − π kj |+ 1 |

v ij − v kj |

j = j = j =

n

1
n

= 1 −

1 | π ij − π kj |≤

1 (2.195)

j =

then we have 0 ≤ μ ij +˙ v kj ≤ 1, with μ ik +˙ v ik = 1 if and only if π ij = π kj ,for

all j = 1 , 2 ,..., n , and μ ik +˙ v ik = 0, if and only if π ij = 1 and π kj = 0, for all

j = 1 , 2 ,..., n ,or π ij = 0 and π kj = 1, for all j = 1 , 2 ,..., n .

(2) Since Z ( y i , y i ) = ( 1 , 0 ) , then Z is reflexive.

(3) Since | v ij − v kj |=| v kj − v ij | and | π ij − π kj |=| π kj − π ij | , then Z ( y i , y k ) =

Z ( y k , y i ) , i.e., Z is symmetrical. Thus, Z ( A , B ) is an intuitionistic fuzzy similarity

relation.
From Eq. ( 2.193 ), we can see that if all the differences of the non-membership
degrees and the differences of the uncertainty degrees of two alternatives y i and y k
with respect to the attributes G j (

j = 1 , 2 ,..., n ) get smaller, then the two alternatives

are more similar to each other.

In the following section, we shall use Eq. ( 2.193 ) to introduce a clustering method.

2.8.2 A Netting Clustering Method

The so called netting means a simple process: Firstly, for an intuitionistic fuzzy
similarity matrix Z , we should choose a confidence level λ ∈[ 0 , 1 ] , and then get a

λ with the symbol of

the alternatives. Under the diagonal, we replace '1' with the nodal point '*' and ignore
all the '0' in Z

-cutting matrix Z and change the elements on the diagonal of Z

λ λ

. From the node '*', we draw the vertical line and the horizontal line
to the diagonal and the corresponding alternatives on the diagonal will be clustered
into one type (He 1983).
Netting method was first used to cluster data in the field of fuzzy mathematics (He
1983). With this method, we can get the clustering results by 'netting' the elements of
similaritymatrix directly. Wang et al. (2011) proposed a nettingmethod for clustering
the objects with intuitionistic fuzzy information:

λ

Step 1 For a multi-attribute decision making problem, Let Y ={ y 1 , y 2 ,..., y m }

and G be defined previously, and assume that the charac-

teristics of the alternatives y i

={ G 1 , G 2 ,..., G n }

( i = 1 , 2 ,..., m ) with respect to the attributes

G j (

= , ,..., )

are represented as in Eq. ( 2.185 ).
Step 2 Construct the intuitionistic fuzzy similarity matrix Z

j 1 2 n

= (

z ij ) m × m by using

Eq. ( 2.193 ), where z ij is an IFV, and z ij = (μ ij , m .

Step 3 Delete all the elements above the diagonal and replace the elements on the
diagonal with the symbol of the alternatives.

v ij ) = Z ( y i , y j ) , i , j = 1 , 2 ,...,