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(2) The weighted Euclidean distance:
d
wE
(
A
1
,
A
2
)
1
4
n
w
i
((μ
A
1
(
x
i
)
−
μ
A
2
(
+
(μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
v
A
2
(
2
2
2
=
x
i
))
x
i
))
+
(
x
i
)
−
x
i
))
i
=
1
1
2
+
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
))
2
+
(π
A
1
(
x
i
)
−
π
A
2
(
x
i
))
2
+
(π
A
1
(
x
i
)
−
π
A
2
(
x
i
))
2
(2.107)
)
T
, then Eq. (
2.107
) reduces to the normalized
Especially, if
w
=
(
1
/
n
,
1
/
n
,...,
1
/
n
)
Euclidean distance:
(
A
1
,
A
2
d
NE
)
1
4
n
n
i
=
1
((μ
A
1
(
)
−
μ
A
2
(
2
+
(μ
A
1
(
)
−
μ
A
2
(
2
v
A
1
(
v
A
2
(
2
=
x
i
x
i
))
x
i
x
i
))
+
(
x
i
)
−
x
i
))
2
v
A
1
(
v
A
2
(
2
+
(π
A
1
(
x
i
)
−
π
A
2
(
2
+
(π
A
1
(
x
i
)
−
π
A
2
(
2
+
(
x
i
)
−
x
i
))
x
i
))
x
i
))
)
(2.108)
A
j
x
i
)
=[
μ
A
j
(
Moreover, let
={
x
i
, μ
A
j
(
x
i
),
˜
v
A
j
(
x
i
)
|
x
i
∈
X
}
, where
μ
A
j
(
x
i
),
μ
A
j
(
v
A
j
(
v
A
j
(
. Then,
based on the operations of IVIFSs, Xu (2009) defined the average of a collection of
m
IVIFSs
A
j
(
x
i
)
]⊂[
0
,
1
]
and
v
˜
A
j
(
x
i
)
=[
x
i
),
x
i
)
]⊂[
0
,
1
]
(
j
=
1
,
2
,...,
m
)
=
,
,...,
)
j
1
2
m
as:
1
m
(
A
1
⊕
A
2
⊕···⊕
A
m
)
(
A
1
,
A
2
,...,
A
m
)
=
f
(2.109)
which can be further transformed into the following:
(
A
1
,
A
2
,...,
A
m
)
f
⎧
⎨
⎩
⎡
⎤
m
m
1
m
1
m
⎣
1
−
μ
A
j
(
−
μ
A
j
(
⎦
,
=
x
i
,
−
1
(
1
x
i
))
,
1
−
1
(
1
x
i
))
j
=
j
=
⎫
⎬
⎡
⎣
⎤
⎦
|
m
m
1
m
1
m
v
A
j
(
v
A
j
(
1
(
x
i
))
,
1
(
x
i
))
x
i
∈
X
(2.110)
⎭
j
=
j
=
The traditional hierarchical clustering algorithm (Anderberg 1972) is generally
used to cluster numerical information. However, in many fields including medical
informatics, information retrieval and bio-informatics, where the data information
sometimes may be imprecise or uncertain, and is suitable to be expressed in IFSs or
IVIFSs, the traditional hierarchical clustering algorithm fails in dealing with these
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