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and considering only two clusters can be jointed in each stage, the IFSs
A
j
(
=
,
,
,
,
)
j
1
2
3
4
5
can be clustered into the following three clusters at the second
stage:
{
A
1
}
,
{
A
2
,
A
3
}
,
{
A
4
,
A
5
}
Step 3
Calculate the center of each cluster by using Eq. (
2.103
):
c
{
A
1
}=
A
1
c
{
A
2
,
A
3
}=
f
(
A
2
,
A
3
)
={
x
1
,
0
.
48
,
0
.
37
,
x
2
,
0
.
60
,
0
.
21
,
x
3
,
1
.
00
,
0
.
00
,
x
4
,
0
.
13
,
0
.
72
,
x
5
,
0
.
11
,
0
.
75
,
x
6
,
0
.
65
,
0
.
17
,
x
7
,
0
.
35
,
0
.
49
,
x
8
,
0
.
47
,
0
.
36
}
c
{
A
4
,
A
5
}=
f
(
A
4
,
A
5
)
={
x
1
,
1
.
00
,
0
.
00
,
x
2
,
1
.
00
,
0
.
00
,
x
3
,
0
.
83
,
0
.
10
,
x
4
,
0
.
73
,
0
.
17
,
x
5
,
0
.
37
,
0
.
35
,
x
6
,
0
.
23
,
0
.
74
,
x
7
,
0
.
13
,
0
.
72
,
x
8
,
0
.
35
,
0
.
46
}
and then compare each cluster with all the other two clusters by using the weighted
Hamming distance (
2.111
):
d
wH
(
c
{
A
1
}
,
c
{
A
2
,
A
3
}
)
=
d
wH
(
c
{
A
2
,
A
3
}
,
c
{
A
1
}
)
=
0
.
4921
d
wH
(
c
{
A
1
}
,
c
{
A
4
,
A
5
}
)
=
d
wH
(
c
{
A
4
,
A
5
}
,
c
{
A
1
}
)
=
0
.
4007
d
wH
(
{
A
2
,
A
3
}
,
{
A
4
,
A
5
}
)
=
d
wH
(
{
A
4
,
A
5
}
,
{
A
2
,
A
3
}
)
=
.
c
c
c
c
0
3879
Hence, the IFSs
A
j
(
j
=
1
,
2
,
3
,
4
,
5
)
can be clustered into the following two
clusters at the third stage:
{
A
1
}
,
{
A
2
,
A
3
,
A
4
,
A
5
}
Finally, the above two clusters can be further clustered into a unique cluster:
{
A
1
,
A
2
,
A
3
,
A
4
,
A
5
}
All the above processes can be shown as in Fig.
2.1
(Xu 2009).
In the process of clustering, the number of clusters can be determined according
to practical applications.
A
j
Example 2.4
(Xu 2009) Given four enterprises, represented by the IVIFSs
(
, where (1)
x
1
:The
ability of sale; (2)
x
2
: The ability of management; (3)
x
3
: The ability of pro-
duction; (4)
x
4
: The ability of technology; (5)
x
5
: The ability of financing; and
(6)
x
6
: The ability of risk bearing (the weight vector of
x
i
j
=
1
,
2
,
3
,
4
)
in the attribute set
X
={
x
1
,
x
2
,...,
x
6
}
(
i
=
1
,
2
,...,
6
)
is
T
. The given data are listed as follows:
=
(
.
,
.
,
.
,
.
,
.
,
.
)
w
0
25
0
20
0
15
0
10
0
15
0
15
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