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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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