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Fig. 2.7
The fuzzy graph
Fig. 2.8
The MST of the
fuzzy graph
Step 3
Choose a threshold
λ
and cut down all the edges of the MST with weights
greater than
λ
so that we could arrive at a certain number of sub-trees (clusters)
automatically.
(1) If
λ
=
d
16
=
0
.
1115, then we get
{
y
1
,
y
2
,
y
3
,
y
4
,
y
5
,
y
6
}
.
(2) If
λ
=
d
25
=
d
35
=
0
.
1, then we get
{
y
1
}
,
{
y
2
,
y
3
,
y
4
,
y
5
,
y
6
}
.
(3) If
λ
=
d
46
=
0
.
088, then we get
{
y
1
}
,
{
y
2
}
,
{
y
5
}
,
{
y
3
,
y
4
,
y
6
}
.
(4) If
λ
=
d
36
=
0
.
0715, then we get
{
y
1
}
,
{
y
2
}
,
{
y
4
}
,
{
y
5
}
,
{
y
3
,
y
6
}
.
(5) If
λ
=
0, then we get
{
y
1
}
,
{
y
2
}
,
{
y
3
}
,
{
y
4
}
,
{
y
5
}
,
{
y
6
}
.
From the results of Algorithms 2.9 and 2.10, we have found that they coincide
with each other on the whole.
Sometimes, it is not suitable to assume that the membership degrees and the non-
membership degrees for certain elements are exactly real numbers, but fuzzy ranges
can be given. As a result, Zhao et al. (2012a) defined the concept of interval-valued
intuitionistic fuzzy distance matrix:
=
,
,...,
Definition 2.30
(Zhao et al. 2012a) Let
y
j
(
j
1
2
n
)
be
m
IVIFSs. Then
=
d
ij
n
×
n
D
is called an interval-valued intuitionistic fuzzy distance matrix, where
d
y
i
,
y
j
is the distance between
y
i
and
y
j
, which has the following properties:
d
ij
=
(1) 0
≤
d
ij
≤
1, for all
i
,
j
=
1
,
2
,...,
n
.
(2)
d
ij
=
0 if and only if
y
i
=
y
j
.
(3)
d
ij
=
d
ji
, for all
i
,
j
=
1
,
2
,...,
n
.
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