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the IIFCM algorithm, then the objective function of the IIFCM algorithm can be
formulated as follows:
p
c
,
V
u
ij
d
NE
(
A
j
,
V
i
)
min
J
m
(
U
)
=
(2.148)
j
=
1
i
=
1
Subject to
c
u
ij
=
1
,
1
≤
j
≤
p
i
=
1
u
ij
≥
0
,
1
≤
i
≤
c
,
1
≤
j
≤
p
p
u
ij
>
0
,
1
≤
i
≤
c
j
=
1
A
j
where
(
j
=
1
,
2
,...,
p
)
are
p
IVIFSs each with
n
elements,
c
is the number of
V
i
clusters (1
<
c
<
p
)
, and
(
i
=
1
,
2
,...,
c
)
are the prototypical IVIFSs of the
clusters. The parameter
m
is the fuzzy factor (
m
>
1
)
,
u
ij
is the membership degree
of the
j
th sample
A
j
to the
i
th cluster,
U
p
.
To solve the optimization problem in Eq. (
2.148
), we also employ the Lagrange
multiplier method. Let
=
(
u
ij
)
c
×
p
isamatrixof
c
×
p
p
c
c
u
ij
d
wE
(
A
j
,
V
i
)
−
=
1
ς
j
(
u
ij
−
)
L
1
(2.149)
j
=
1
i
=
1
j
=
i
=
1
where
d
wE
(
A
j
,
V
i
)
w
l
n
1
4
((μ
A
j
(
x
l
))
−
μ
V
i
(
x
l
))
+
(μ
A
j
(
x
l
)
−
μ
V
i
(
x
l
))
+
(
v
A
j
(
x
l
)
−
v
V
i
(
x
l
))
2
2
2
=
l
=
1
2
v
A
j
(
v
V
i
(
2
+
(π
A
j
(
x
l
)
−
π
V
i
(
2
+
(π
A
j
(
x
l
)
−
π
V
i
(
+
(
x
l
)
−
x
l
))
x
l
))
x
l
))
(2.150)
Similar to Algorithm 2.7, we can establish the system of partial differential func-
tions of
L
as follows:
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