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μ
B
x
j
2
v
B
x
j
2
π
B
x
j
2
μ
B
x
j
2
B
x
j
=
g
+
+
+
v
B
x
j
2
π
B
x
j
2
+
+
(2.170)
,
B
x
j
=
μ
A
x
j
μ
B
x
j
+
v
A
x
j
v
B
x
j
f
A
+
π
A
x
j
π
B
x
j
+
μ
A
x
j
μ
B
x
j
+
v
A
x
j
v
B
x
j
+
π
A
x
j
π
B
x
j
(2.171)
If we need to consider the weights of the element
x
i
∈
X
, then Eq. (
2.166
) can be
extended to its weighted counterpart:
j
=
1
w
j
f
,
B
x
j
A
c
8
(
A
,
B
˙
)
=
j
=
1
w
j
g
(2.172)
A
x
j
·
j
=
1
w
j
g
B
x
j
T
where
w
=
(
w
1
,
w
2
,...,
w
n
)
is the weight vector of
x
i
(
i
=
1
,
2
,...,
n
)
, with
n
and
j
=
1
w
j
=
w
j
≥
0
,
i
=
1
,
2
,...,
1. If
w
1
=
w
2
= ··· =
w
n
=
1
/
n
, then
Eq. (
2.172
) reduces to Eq. (
2.168
).
In the following, we prove that Eq. (
2.172
) satisfies all the conditions of Definition
2.32:
Proof
Since
A
,
B
∈
IVIFS
(
X
)
, then
≤
μ
A
(
x
j
)
≤
μ
A
(
v
A
(
v
A
(
≤
π
A
(
x
j
)
≤
π
A
(
0
x
j
)
≤
1
,
0
≤
x
j
)
≤
x
j
)
≤
1
,
0
x
j
)
≤
1
,
for all
x
j
∈
X
(2.173)
≤
μ
B
(
x
j
)
≤
μ
B
(
v
B
(
v
B
(
≤
π
B
(
x
j
)
≤
π
B
(
0
x
j
)
≤
1
,
0
≤
x
j
)
≤
x
j
)
≤
1
,
0
x
j
)
≤
1
,
for all
x
j
∈
X
(2.174)
c
8
(
A
,
B
and thus, by Eq. (
2.172
), we get
˙
)
≥
0. According to the famous Cauchy-
Schwarz inequality Eq. (
2.160
), we have
⎛
⎞
⎛
⎞
n
n
n
A
x
j
B
x
j
⎝
⎠
⎝
⎠
w
j
f
,
B
≤
w
j
g
w
j
g
(2.175)
A
j
=
1
j
=
1
j
=
1
c
8
(
A
,
B
and thus,
˙
)
≤
1 with equality if and only if there exists a nonzero real number
λ
, such that
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