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T
in Definition 1.13, then
the IFWBM cannot reduce to the IFBM given in Definition 1.12. Moreover, both the
IFBM and the IFWBM can only deal with the situations where there are correlations
between any two aggregated arguments, but not the situations where there exist con-
nections among any three aggregated arguments. To solve this issue, and motivated
by Definition 1.11, we define the following:
=
(
/
,
/
,...,
/
)
However, it is noted that if
w
1
n
1
n
1
n
Definition 1.14
(Xia et al. 2012b) For any
p
,
q
,
r
>
0, if
n
⊕
i
,
j
,
k
=
1
k
w
i
w
j
w
k
1
p
+
q
+
r
p
i
q
j
GIFWBM
p
,
q
,
r
r
(α
1
,α
2
,...,α
n
)
=
α
⊗
α
⊗
α
(1.152)
then
GIFWBM
p
,
q
,
r
is called a generalized intuitionistic fuzzy weighted Bonferroni
mean (GIFWBM).
→
Especially, if
r
0, then the GIFWBM reduces to:
0
GIFWBM
p
,
q
,
r
lim
r
(α
1
,α
2
,...,α
n
)
→
1
p
+
q
n
⊕
p
i
q
j
)
=
w
i
w
j
w
k
(α
⊗
α
i
,
j
,
k
=
1
n
n
⊕
i
,
j
=
1
p
+
q
n
⊕
i
,
j
=
1
p
+
q
p
i
q
j
)
p
i
q
j
)
=
w
k
w
i
w
j
(α
⊗
α
=
w
i
w
j
(α
⊗
α
1
1
k
=
1
(1.153)
which is called an intuitionistic fuzzy weighted Bonferroni mean (IFWBM).
Especially, if
q
→
0 and
r
→
0, then the GIFWBM reduces to:
GIFWBM
p
,
q
,
r
lim
r
→
0
q
(α
1
,α
2
,...,α
n
)
→
0
n
⊕
i
,
j
,
k
=
p
p
i
=
w
i
w
j
w
k
α
1
⎛
⎞
p
n
⊕
n
n
1
p
n
⊕
p
i
p
i
⎝
⎠
=
w
j
w
k
w
i
α
=
w
i
α
(1.154)
i
=
1
i
=
1
j
=
1
k
=
1
which is the generalized intuitionistic fuzzy weighted mean (GIFWM) (Zhao et al.
2010).
Based on the operational laws of IFVs, we can derive the following theorem:
Theorem 1.16
(Xia et al. 2012b) Let
p
,
q
,
r
>
0, then the aggregated value by
using the GIFWBM is also an IFV, and
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