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s
1
−
ω
(
k
)
s
1
−
ω
(
k
)
ij
s
−
1
ij
s
−
1
v
+
(
index
(
k
))
ij
1
(μ
+
(
index
(
k
))
1
(
1
−
)
−
)
,
ij
k
=
k
=
s
−
ω
(
k
)
ij
s
−
ω
(
k
)
ij
1
1
v
−
(
index
(
k
))
ij
1
(μ
−
(
index
(
k
))
1
(
1
−
)
−
)
(1.112)
s
−
1
s
−
1
ij
k
=
k
=
to aggregate all the individual interval-valued intuitionistic fuzzy decision matrices
R
(
k
)
=
(
˜
r
(
k
)
ij
)
m
×
n
(
k
=
1
,
2
,...,
s
)
into the collective interval-valued intuitionistic
R
fuzzy decision matrix
r
ij
)
m
×
n
.
Step 4
See Approach 1.3.
Step 5
See Approach 1.3.
Clearly, Approaches 1.3 and 1.4 are the extensions of Approaches 1.1 and 1.2 in
interval-valued intuitionistic fuzzy environments, and thus, they have similar char-
acteristics.
In the case where the evaluation values given by the experts are expressed with
IVIFVs in the example of Sect.
1.2.5
, we can utilize Approach 1.3 to solve the
problem, here omitted for brevity.
=
(
˜
1.4 Intuitionistic Fuzzy Geometric Bonferroni Means
The Bonferroni mean was originally introduced by Bonferroni (1950) and then gen-
eralized by Yager (2009). The desirable characteristic of the Bonferroni mean is its
capability to capture the interrelationship between input arguments. Xu and Yager
(2011) further applied the Bonferroni mean to intuitionstic fuzzy environments and
introduced the intuitionistic fuzzy Bonferroni mean and the weighted Bonferroni
mean. Xia et al. (2012a) developed a geometric Bonferroni mean based on the Bon-
ferroni mean and the geometric mean and further extended it to intuitionistic fuzzy
environments.
1.4.1 Geometric Bonferroni Mean
The Bonferroni mean, introduced by Bonferroni (1950), can be defined as follows:
Definition 1.6
(Bonferroni 1950) Let
a
i
(
i
=
1
,
2
,...,
n
)
be a collection of crisp
data, where
a
i
≥
0, for all
i
, and
p
,
q
≥
0, then
⎛
⎞
1
p
+
q
n
⎝
⎠
1
a
p
i
a
q
j
BM
p
,
q
(
a
1
,
a
2
,...,
a
n
)
=
(1.113)
n
(
n
−
1
)
i
,
j
=
1
i
=
j
is called a Bonferroni mean.
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