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which gives more importance to the individual opinions. In this subsection, we first
introduce the generalized weighted Bonferroni geometric mean, based on which, the
generalized intuitionistic fuzzy weighted Bonferroni geometric mean is given.
Let
p
,
q
,
r
≥
0, and
a
i
(
i
=
1
,
2
,...,
n
)
be a collection of nonnegative numbers,
then
T
be the weight vector
Definition 1.15
(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
n
1
GWBGM
p
,
q
,
r
w
i
w
j
w
k
(
a
1
,
a
2
,...,
a
n
)
=
1
(
pa
i
+
qa
j
+
ra
k
)
p
+
q
+
r
i
,
j
,
k
=
(1.178)
then
GWBGM
p
,
q
,
r
is called a generalized weighted Bonferroni geometric mean
(GWBGM), which has the following properties:
Theorem 1.21
(Xia et al. 2012b)
(1)
GWBGM
p
,
q
,
r
(
0
,
0
,...,
0
)
=
0.
(2)
GWBGM
p
,
q
,
r
(
a
,
a
,...,
a
)
=
a
,if
a
i
=
a
, for all
i
.
(3)
GWBGM
p
,
q
,
r
GWBGM
p
,
q
,
r
, i.e.,
GWBGM
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
.
GWBGM
p
,
q
,
r
(4) min
{
a
i
}≤
(
a
1
,
a
2
,...,
a
n
)
≤
max
{
a
i
}
.
In addition, some special cases can be obtained as the change of the parameters:
(1) If
r
=
0, then the GWBGM reduces to:
n
1
GWBGM
p
,
q
,
0
w
i
w
j
w
k
(
a
1
,
a
2
,...,
a
n
)
=
1
(
pa
i
+
qa
j
)
p
+
q
i
,
j
,
k
=
n
1
w
i
w
j
=
1
(
pa
i
+
qa
j
)
(1.179)
p
+
q
i
,
j
=
which is called a weighted Bonferroni geometric mean (WBGM).
(2) If
q
=
0 and
r
=
0, then
n
n
1
p
a
w
i
i
GWBGM
p
,
0
,
0
w
i
w
j
w
k
(
a
1
,
a
2
,...,
a
n
)
=
1
(
pa
i
)
=
(1.180)
i
,
j
,
k
=
i
=
1
which is the usual geometric mean.
Let
p
,
q
,
r
>
0 and
α
i
=
(μ
α
i
,
v
α
i
,π
α
i
)(
i
=
1
,
2
,...,
n
)
be a collection of
IFVs.
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