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⎛
α
k
w
i
w
j
w
k
⎞
1
p
+
q
+
r
1
n
⎝
1
p
q
r
⎠
−
−
μ
α
i
μ
α
j
μ
i
,
j
,
k
=
1
⎛
r
w
i
w
j
w
k
⎞
1
p
+
q
+
r
n
1
⎝
1
p
q
⎠
+
1
−
−
−
(
1
−
v
)
(
1
−
v
)
(
1
−
v
)
α
α
α
i
j
k
i
,
j
,
k
=
1
⎛
r
w
i
w
j
w
k
⎞
1
p
+
q
+
r
n
1
⎝
1
p
q
⎠
≤
1
+
−
−
(
1
−
v
)
(
1
−
v
)
(
1
−
v
)
α
α
α
i
j
k
i
,
j
,
k
=
1
⎛
r
w
i
w
j
w
k
⎞
1
p
+
q
+
r
n
1
⎝
1
p
q
⎠
−
−
−
(
1
−
v
α
i
)
(
1
−
v
α
j
)
(
1
−
v
α
k
)
=
1
i
,
j
,
k
=
1
(1.164)
which completes the proof of the theorem.
Moreover, the GIFWBM also has the following properties (Xia et al. 2012b):
Theorem 1.17
If all
α
i
(
i
=
1
,
2
,...,
n
)
are equal, i.e.,
α
i
=
α
, for all
i
, then
GIFWBM
p
,
q
,
r
GIFWBM
p
,
q
,
r
(α
1
,α
2
,...,α
n
)
=
(α,α,...,α)
=
α
(1.165)
Theorem 1.18
Let
β
i
=
(μ
β
i
,
v
β
i
,π
β
i
)(
i
=
1
,
2
,...,
n
)
be a collection of IFVs,
if
μ
α
i
≤
μ
β
i
and
v
≥
v
β
i
, for all
i
, then
α
i
GIFWBM
p
,
q
,
r
GIFWBM
p
,
q
,
r
(α
1
,α
2
,...,α
n
)
≤
(β
1
,β
2
,...,β
n
)
(1.166)
Proof
Since
μ
α
i
≤
μ
β
i
and
v
≥
v
β
i
, for all
i
, then
α
i
p
β
i
μ
q
β
j
μ
p
q
r
α
k
r
β
k
μ
α
i
μ
α
j
μ
≤
μ
n
1
k
w
i
w
j
w
k
n
1
k
w
i
w
j
w
k
p
β
i
μ
q
β
j
μ
p
α
q
α
r
α
r
β
⇒
−
μ
μ
μ
≥
−
μ
i
j
i
,
j
,
k
=
1
i
,
j
,
k
=
1
n
1
k
w
i
w
j
w
k
n
1
k
w
i
w
j
w
k
p
β
i
μ
q
β
j
μ
p
α
q
α
r
α
r
β
⇒
1
−
−
μ
μ
μ
≤
1
−
−
μ
i
j
i
,
j
,
k
=
1
i
,
j
,
k
=
1
⎛
α
k
w
i
w
j
w
k
⎞
1
1
p
+
q
+
r
n
⎝
1
p
q
r
⎠
⇒
−
−
μ
α
i
μ
α
j
μ
i
,
j
,
k
=
1
⎛
β
k
w
i
w
j
w
k
⎞
1
1
p
+
q
+
r
n
p
β
q
β
⎝
1
r
⎠
≤
−
−
μ
i
μ
j
μ
(1.167)
i
,
j
,
k
=
1
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