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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
= α i ,
v α i α i )(
i
=
1
,
2
,...,
n
)
 
 
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