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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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