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=
0, then the GBM reduces to the Bonferroni mean. However, it
is noted that both the Bonferroni mean and the GBM do not consider the situation
that
i
Especially, if
r
k
, and the weight vector of the aggregated arguments
is not also considered. To overcome this drawback, Xia et al. (2012b) defined the
weighted version of the GBM:
=
j
or
j
=
k
or
i
=
T
be the weight vector
Definition 1.11
(Xia et al. 2012b) Let
w
=
(
w
1
,
w
2
,...,
w
n
)
n
and
i
=
1
w
i
of
a
i
(
i
=
1
,
2
,...,
n
)
such that
w
i
>
0,
i
=
1
,
2
,...,
=
1. If
⎛
⎞
1
p
+
q
+
r
n
⎝
w
i
w
j
w
k
a
i
a
j
a
k
⎠
GWBM
p
,
q
,
r
a
1
,
a
2
,...,
a
n
)
=
(
(1.146)
i
,
j
,
k
=
1
then
GWBM
p
,
q
,
r
is called a generalized weighed Bonferroni mean (GWBM).
Especially, if
w
T
, then the GWBM reduces to the follow-
=
(
1
/
n
,
1
/
n
,...,
1
/
n
)
ing:
⎛
⎝
⎞
⎠
1
p
+
q
+
r
n
1
n
3
a
i
a
j
a
k
RBM
p
,
q
,
r
(
a
1
,
a
2
,...,
a
n
)
=
(1.147)
i
,
j
,
k
=
1
which is called the revised Bonferroni mean (RBM).
Moreover, the GWBM has the following properties:
Theorem 1.15
(Xia et al. 2012b)
(1)
GWBM
p
,
q
,
r
(
0
,
0
,...,
0
)
=
0.
(2)
GWBM
p
,
q
,
r
(
a
,
a
,...,
a
)
=
a
,if
a
i
=
a
, for all
i
.
(3)
GWBM
p
,
q
,
r
GWBM
p
,
q
,
r
, i.e.,
GWBM
p
,
q
,
r
(
a
1
,
a
2
,...,
a
n
)
≥
(
b
1
,
b
2
,...,
b
n
)
is monotonic, if
a
i
≥
b
i
, for all
i
.
GWBM
p
,
q
,
r
(4) min
{
a
i
}≤
(
a
1
,
a
2
,...,
a
n
)
≤
max
{
a
i
}
.
Some special cases can be obtained as the change of the parameters (Xia et al.
2012b):
(1) If
r
=
0, then the GWBM reduces to the following:
⎛
⎝
⎞
⎠
1
p
+
q
n
w
i
w
j
w
k
a
i
a
j
GWBM
p
,
q
,
0
(
a
1
,
a
2
,...,
a
n
)
=
i
,
j
,
k
=
1
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
1
p
+
q
1
p
+
q
n
n
n
w
i
w
j
a
i
a
j
w
i
w
j
a
i
a
j
(1.148)
=
w
k
=
i
,
j
=
1
k
=
1
i
,
j
=
1
which we call a weighted Bonferroni mean (WBM).
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