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=
0, then by Eq. (
1.113
), the Bonferroni mean reduces to the
generalized mean operator (Dyckhoff and Pedrycz 1984) as follows:
Especially, if
q
⎛
⎛
⎞
⎞
1
p
+
0
1
n
1
p
n
n
n
⎝
⎝
⎠
⎠
1
n
1
a
p
i
a
p
i
BM
p
,
0
a
j
(
a
1
,
a
2
,...,
a
n
)
=
=
(
n
−
1
)
i
=
1
j
=
1
i
=
1
j
=
i
(1.114)
If
p
=
1 and
q
=
0, then Eq. (
1.113
) reduces to the well-known arithmetic
average:
n
1
n
BM
1
,
0
(
a
1
,
a
2
,...,
a
n
)
=
a
i
(1.115)
i
=
1
Based on the usual geometric mean and the Bonferroni mean, Xia et al. (2012a)
proposed a geometric Bonferroni mean as follows:
,
>
(
=
,
,...,
)
Definition 1.7
(Xia et al. 2012a) Let
p
q
0, and
a
i
i
1
2
n
be a
collection of non-negative numbers. If
n
1
1
GBM
p
,
q
(
a
1
,
a
2
,...,
a
n
)
=
(
pa
i
+
qa
j
)
(1.116)
n
(
n
−
1
)
p
+
q
i
,
j
=
1
i
=
j
then we call
GBM
p
,
q
a geometric Bonferroni mean (GBM).
Obviously, the GBM has the following properties (Xia et al. 2012a):
(1)
GBM
p
,
q
(
0
,
0
,...,
0
)
=
0.
(2)
GBM
p
,
q
(
a
,
a
,...,
a
)
=
a
,if
a
i
=
a
,
for all
i
.
(3)
GBM
p
,
q
GBM
p
,
q
, i.e.,
GBM
p
,
q
(
a
1
,
a
2
,...,
a
n
)
≥
(
b
1
,
b
2
,...,
b
n
)
is mono-
tonic, if
a
i
≥
b
i
, for all
i
.
GBM
p
,
q
(4) mi
i
{
a
i
}≤
(
a
1
,
a
2
,...,
a
n
)
≤
max
i
{
a
i
}
.
Furthermore, if
q
=
0, then by Eq. (
1.116
), the GBM reduces to the geometric
mean:
n
n
1
p
1
1
n
GBM
p
,
0
(
a
1
,
a
2
,...,
a
n
)
=
(
pa
i
)
=
1
(
a
i
)
(1.117)
n
(
n
−
1
)
i
,
j
=
1
i
=
i
=
j
1.4.2 Intuitionistic Fuzzy Geometric Bonferroni Mean
Let
be a collection of IFVs, based on
Eq. (
1.116
), Xia et al. (2012a) defined a geometric Bonferroni mean for IFVs:
α
i
=
(μ
α
i
,
v
α
i
,π
α
i
)(
i
=
1
,
2
,...,
n
)
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