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situations. Based on the distance measures (
2.105
) and (
2.106
), and the intuitionistic
fuzzy aggregation operator (
2.103
), Xu (2009) extended the traditional hierarchical
clustering algorithm to the IFS theory:
Algorithm 2.4
Given a collection of
m
IFSs
A
j
(
j
=
1
,
2
,...,
m
)
, in the first stage each of
the
m
IFSs
A
j
(
j
=
1
,
2
,...,
m
)
is considered as a unique cluster. The IFSs
A
j
(
j
=
1
,
2
,...,
m
)
are then compared among themselves by using the weighted
Hamming distance:
d
wH
(
A
1
,
A
2
)
n
1
2
=
w
i
(
|
μ
A
i
(
x
i
)
−
μ
A
j
(
x
i
)
|+|
v
A
i
(
x
i
)
−
v
A
j
(
x
i
)
|+|
π
A
i
(
x
i
)
−
π
A
j
(
x
i
)
|
)
i
=
1
(2.111)
or the weighted Euclidean distance:
d
wE
(
A
1
,
A
2
)
1
2
1
/
2
n
2
2
2
=
w
i
((μ
A
i
(
x
i
)
−
μ
A
j
(
x
i
))
+
(
v
A
i
(
x
i
)
−
v
A
j
(
x
i
))
+
(π
A
i
(
x
i
)
−
π
A
j
(
x
i
))
)
i
=
1
(2.112)
The two clusters with smaller distance are jointed. The procedure is then repeated
time after time until the desirable number of clusters is achieved. Only two clusters
can be jointed in each stage and they cannot be separated after they are jointed. In
each stage the center of each cluster is recalculated by using the average (derived
from Eq. (
2.103
)) of the IFSs assigned to the cluster, and the distance between two
clusters is defined as the distance between the centers of each clusters.
If the collected data information is expressed as IVIFSs, then based on the distance
measures (
2.105
) and (
2.107
), and the interval-valued intuitionistic fuzzy aggregation
operator (
2.110
), Xu (2009) gave an interval-valued intuitionistic fuzzy hierarchical
algorithm for clustering IVIFSs:
Algorithm 2.5
A
j
Given a collection of
m
IVIFSs
(
j
=
1
,
2
,...,
m
)
, in the first stage each of
A
j
(
=
,
,...,
)
the
m
IVIFSs
j
1
2
m
is considered as a unique cluster. The IVIFSs
A
j
(
are then compared among themselves by using the weighted
Hamming distance (
2.105
) or the weighted Euclidean distance (
2.107
). The two
clusters with smaller distance are jointed. The procedure is then repeated time after
time until the desirable number of clusters is achieved. Only two clusters can be
jointed in each stage and they cannot be separated after they are jointed. In each
stage the center of each cluster is recalculated by using the average (derived by
j
=
1
,
2
,...,
m
)
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