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In-Depth Information
˜
=
(
˜
r
1
j
,
˜
r
2
j
,...,
˜
r
mj
)
r
j
IVIFWA
1
w
i
m
w
i
m
m
m
−
μ
ij
)
−
μ
ij
)
v
ij
)
v
ij
)
w
i
w
i
=
−
1
(
,
−
1
(
,
1
(
,
1
(
,
1
1
1
i
=
i
=
i
=
i
=
m
w
i
m
m
m
−
μ
ij
)
v
ij
)
−
μ
ij
)
v
ij
)
w
i
w
i
w
i
1
(
−
1
(
,
1
(
−
1
(
,
1
1
i
=
i
=
i
=
i
=
j
=
1
,
2
,...,
n
(1.106)
or the interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operator
(Xu and Cai 2009):
r
j
˜
=
IVIFWG
(
˜
r
1
j
,
˜
r
2
j
,...,
˜
r
mj
)
m
1
1
1
m
m
m
1
−
w
i
1
−
w
i
1
−
w
i
1
−
w
i
1
(μ
ij
)
1
(μ
ij
)
v
ij
)
v
ij
)
=
,
,
−
1
(
1
−
,
1
−
1
(
1
−
,
m
−
1
m
−
m
−
1
m
−
i
=
i
=
i
=
i
=
m
m
m
m
1
−
w
i
m
−
1
1
−
w
i
m
−
1
1
−
w
i
m
−
1
1
−
w
i
m
−
1
v
ij
)
1
(μ
ij
)
v
ij
)
1
(μ
ij
)
1
(
1
−
−
,
1
(
1
−
−
,
i
=
i
=
i
=
i
=
(1.107)
j
=
1
,
2
,...,
n
Step 5
Rank
r
j
˜
(
j
=
1
,
2
,...,
n
)
in descending order by using the ranking
method described in Definition 1.4.
Step 6
Rank all the alternatives
y
j
(
j
=
1
,
2
,...,
n
)
and select the best one in
˜
(
=
,
,...,
)
accordance with the ranking of
.
In the case where the information about the weights of experts is unknown, we
can utilize the IVIFPOWA (or IVIFPWG) operator to develop an approach to multi-
attribute group decisionmakingwith interval-valued intuitionistic fuzzy information,
which can be described as follows (Xu 2011):
r
j
j
1
2
n
Approach 1.4
Step 1
Calculate
Sup
r
index
(
k
)
ij
r
index
(
l
)
ij
˜
,
˜
d
r
index
(
k
)
ij
r
index
(
l
)
ij
=
1
−
˜
,
˜
μ
−
(
index
(
k
))
ij
+
μ
+
(
index
(
k
))
ij
1
4
−
μ
−
(
index
(
l
))
ij
−
μ
+
(
index
(
l
))
ij
=
1
−
+
v
−
(
index
(
k
))
ij
v
−
(
index
(
l
))
ij
v
+
(
index
(
k
))
ij
v
+
(
index
(
l
))
ij
+
−
−
π
−
(
index
(
k
))
+
π
+
(
index
(
k
))
(1.108)
−
π
−
(
index
(
l
))
ij
−
π
+
(
index
(
l
))
ij
+
ij
ij
r
index
(
l
)
ij
which indicates the support of the
l
th largest IVIFV
˜
for the
k
th largest IVIFV
r
index
(
k
)
ij
r
(
t
)
ij
˜
˜
(
=
,
,...,
)
of
t
1
2
s
.
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