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d wH ( A 2 , A 4 ) =
d wH ( A 4 , A 2 ) =
d wH ( A 3 , A 4 ) =
d wH ( A 4 , A 3 ) =
.
,
.
0
4237
0
2288
then the IVIFSs A j (
j
=
1
,
2
,
3
,
4
)
can be clustered into the following three clusters
at the second stage:
{ A 1 } ,
{ A 2 } ,
{ A 3 , A 4 }
Step 3 Calculate the center of each cluster by using Eq. ( 2.109 ):
{ A 1 }=
{ A 2 }= A 2
c
A 1 ,
c
{ A 3 , A 4 }=
( A 3 , A 4 )
={
c
f
x 1 , [
0
.
43
,
0
.
51
] , [
0
.
35
,
0
.
40
] ,
x 2 , [
0
.
67
,
0
.
78
] , [
0
.
00
,
0
.
20
] ,
x 3 , [
0
.
15
,
0
.
23
] , [
0
.
67
,
0
.
75
] ,
x 4 , [
0
.
18
,
0
.
25
] , [
0
.
59
,
0
.
67
] ,
x 5 , [
0
.
00
,
0
.
10
] , [
0
.
77
,
0
.
85
] ,
x 6 , [
0
.
55
,
0
.
65
] , [
0
.
20
,
0
.
27
]}
and then compare each cluster with all the other two clusters by using the weighted
Hamming distance ( 2.111 ):
{ A 1 } ,
{ A 2 } ) =
{ A 2 } ,
{ A 1 } ) =
d wH (
c
c
d wH (
c
c
0
.
4600
{ A 1 } ,
{ A 3 , A 4 } ) =
{ A 3 , A 4 } ,
{ A 1 } ) =
d wH (
c
c
d wH (
c
c
0
.
3211
{ A 2 } ,
{ A 3 , A 4 } ) =
{ A 3 , A 4 } ,
{ A 2 } ) =
d wH (
d wH (
.
c
c
c
c
0
3871
A j
As a result, the IVIFSs
(
j
=
1
,
2
,
3
,
4
)
can be clustered into the following two
clusters at the third stage:
{ A 2 } , { A 1 , A 3 , A 4 }
In the final stage, the above clusters can be further clustered into a unique cluster:
{ A 1 , A 2 , A 3 , A 4 }
All the above processes can be shown as in Fig. 2.2 (Xu 2009).
2.4 Intuitionistic Fuzzy Orthogonal Clustering Algorithm
We first introduce some basic concepts:
Definition 2.16 (Bustince 2000) Let X and Y be two non-empty sets. Then
={ (
,
), μ R (
,
),
v R (
,
) |
,
}
R
x
y
x
y
x
y
x
X
y
Y
(2.113)
is called an intuitionistic fuzzy relation, where
 
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