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Algorithm 2.10
Step 1
Compute the intuitionistic fuzzy distance matrix and draw the fuzzy graph:
d
A
i
,
A
j
by Eq. (
2.111
)or(
2.112
) and get the
(1) Calculate the distance
d
ij
=
=
d
ij
n
×
n
which is actually a fuzzy similarity
intuitionistic fuzzy distance matrix
D
relation.
(2) Draw the fuzzy graph
V
D
. Although the
n
nodes associated to the samples
,
A
i
(
to be clustered are still expressed by IFSs, the weight
d
ij
of every
edge between
A
i
and
A
j
changes into a real number which comes from the second
kind of intuitionistic fuzzy distance matrix
D
i
=
1
,
2
,...,
n
)
d
ij
)
n
×
n
(the graph here is really a
fuzzy graph and is quite different from the one in Algorithm 2.9).
Step 2
Compute the minimum spanning tree (MST) of the fuzzy graph
V
=
(
D
,
,
which is similar to Step 2 of Algorithm 2.9.
Step 3
See Step 3 of Algorithm 2.9.
In the following, we use an example to illustrate Algorithms 2.9 and 2.10:
Example 2.7
In an operational mission (adapted from Zhang et al. (2007), there are
six operational plans
y
i
. In order to group these operational plans
with respect to their comprehensive functions, a military committee has been set up
to provide assessment information on the operational plans. The attributes which are
considered here in assessment of
y
i
(
i
=
1
,
2
,...,
6
)
are: (1)
G
1
: The effectiveness of
operational organization; and (2)
G
2
: The effectiveness of operational command. The
military committee evaluates the performance of all the operational plans according
to the attributes
G
j
(
i
=
1
,
2
,...,
6
)
(
j
=
1
,
2
)
, and gives the data as follows:
y
1
={
G
1
,
0
.
70
,
0
.
15
,
G
2
,
0
.
60
,
0
.
20
}
y
2
={
G
1
,
0
.
40
,
0
.
35
,
G
2
,
0
.
80
,
0
.
10
}
y
3
={
G
1
,
0
.
55
,
0
.
25
,
G
2
,
0
.
70
,
0
.
15
}
y
4
={
G
1
,
0
.
44
,
0
.
35
,
G
2
,
0
.
60
,
0
.
20
}
y
5
={
G
1
,
0
.
50
,
0
.
35
,
G
2
,
0
.
75
,
0
.
20
}
y
6
={
G
1
,
0
.
55
,
0
.
25
,
G
2
,
0
.
57
,
0
.
15
}
T
.We
Let the weight vector of the attributes
G
j
(
j
=
1
,
2
)
be
w
=
(
0
.
45
,
0
.
55
)
first utilize Algorithm 2.9 to group these operational plans
y
j
(
j
=
1
,
2
,...,
6
)
:
Step 1
Construct the intuitionistic fuzzy distance matrix and the intuitionistic
fuzzy graph:
d
y
i
,
y
j
(see Definition 2.28), and let
(1) Calculate the distance
d
ij
=
λ
=
2
,
α
=
β
=
γ
=
1
/
3. Then
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