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≤˙
c
ij
≤
,
,
=
,
,...,
(1) 0
1
i
j
1
2
m
.
1 if and only if
A
i
=
A
j
.
(2)
c
ij
=
˙
(3)
c
ij
=˙
˙
c
ji
,
i
,
j
=
1
,
2
,...,
m
.
Based on the association matrix of the IVIFSs, in what follows, we introduce an
algorithm for clustering IVIFSs (Xu et al. 2008):
Algorithm 2.3
Step
1
Let
X
={
x
1
,
x
2
,...,
x
n
}
be a discrete universe of discourse,
T
w
=
(
w
1
,
w
2
,...,
w
n
)
the weight vector of the elements
x
i
(
i
=
1
,
2
,...,
n
)
,
n
, and
i
=
1
w
i
=
1, and let
A
j
(
with
w
i
∈[
0
,
1
]
,
i
=
1
,
2
,...,
j
=
1
,
2
,...,
m
)
be
a collection of IVIFSs:
A
j
={
x
i
, μ
A
j
(
x
i
),
˜
v
A
j
(
x
i
)
|
x
i
∈
X
}
(2.98)
where
x
i
)
=[
μ
A
j
(
x
i
), μ
A
j
(
v
A
j
(
v
A
j
(
μ
A
j
(
x
i
)
]⊂[
,
]
,
v
A
j
(
˜
x
i
)
=[
x
i
),
x
i
)
]⊂[
,
]
,
0
1
0
1
μ
A
j
(
v
A
j
(
x
i
)
+
x
i
)
≤
1
,
x
i
∈
X
(2.99)
x
i
)
=[
π
A
j
(
x
i
), π
A
j
(
]
,π
A
j
(
−
μ
A
j
(
Additionally,
π
A
j
(
x
i
)
]⊂[
0
,
1
x
i
)
=
1
x
i
)
−
1
−
v
A
j
(
x
i
), π
A
j
(
−
μ
A
j
(
v
A
j
(
.
Step 2
Utilize the interval-valued intuitionistic fuzzy association measures:
x
i
)
=
1
x
i
)
−
1
−
x
i
)
k
=
1
w
k
μ
A
i
(
x
k
)
·
μ
A
j
(
x
k
)
+
μ
A
i
(
x
k
)
·
μ
A
j
(
x
k
)
v
A
i
(
v
A
j
(
v
A
i
(
v
A
j
(
+
x
k
)
·
x
k
)
+
x
k
)
·
x
k
)
+
π
A
i
(
x
k
)
·
π
A
j
(
x
k
)
+
π
A
i
(
x
k
)
·
π
A
j
(
x
k
)
(
A
i
,
A
j
)
=
c
(2.100)
max
k
=
1
w
k
2
2
v
A
i
(
2
μ
A
i
(
μ
A
i
(
x
k
)
+
x
k
)
+
x
k
)
v
A
i
(
2
2
2
π
A
i
(
π
A
i
(
+
x
k
)
+
x
k
)
+
x
k
)
,
k
=
1
w
k
2
2
v
A
j
(
2
μ
A
j
(
μ
A
j
(
x
k
)
+
x
k
)
+
x
k
)
v
A
j
(
2
2
2
π
A
j
(
π
A
j
(
+
x
k
)
+
x
k
)
+
x
k
)
to calculate the association coefficients of the IVIFSs
A
i
and
A
j
(
i
,
j
=
1
,
2
,...,
m
)
,
(
A
i
,
A
j
),
C
and then construct an association
=
(
˙
c
ij
)
m
×
m
, where
˙
c
ij
=˙
c
i
,
j
m
.
Step 3
If the association matrix
=
1
,
2
,...,
C
=
(
˙
c
ij
)
m
×
m
is an equivalent association
C
λ
=
(
λ
c
ij
)
m
×
m
of
C
by using
λ
matrix, then we construct a
-cutting matrix
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