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In-Depth Information
r
(
1
)
ij
r
(
2
)
ij
r
(
s
)
ij
r
ij
=
IFPWG
(
,
,...,
)
⎛
1
1
)
ij
))
k
=
1
η
k
(
1
+
T
(
r
(
k
)
r
(
k
)
ij
))
k
=
1
η
k
(
1
+
T
(
r
(
k
)
r
(
k
η
k
(
1
+
T
(
η
k
(
1
+
T
(
s
1
s
−
s
1
s
−
−
−
⎝
1
1
1
(μ
(
k
)
v
(
k
)
ij
))
))
=
)
,
1
−
1
(
1
−
)
,
ij
ij
ij
k
=
j
=
⎞
⎠
1
1
η
k
(
1
+
T
(
r
(
k
)
ij
η
k
(
1
+
T
(
r
(
k
)
ij
))
))
s
1
s
−
1
s
1
s
−
1
−
−
k
k
r
(
k
)
ij
r
(
k
)
ij
v
(
k
)
ij
1
(μ
(
k
)
1
η
k
(
1
+
T
(
))
1
η
k
(
1
+
T
(
))
1
(
1
−
)
−
)
=
=
ij
k
=
k
=
(1.56)
r
(
k
)
ij
to aggregate all the individual intuitionistic fuzzy decisionmatrices
R
(
k
)
=
(
)
m
×
n
(
k
=
1
,
2
,...,
s
)
into the collective intuitionistic fuzzy decision matrix
R
=
(
r
ij
)
m
×
n
, where
r
ij
=
(μ
ij
,
v
ij
,π
ij
),
i
=
1
,
2
,...,
m
;
j
=
1
,
2
,...,
n
.
Step 4
To get the overall preference value
r
j
corresponding to the alternative
y
j
,
we aggregate all the preference values
r
ij
(
in the
j
th column of
R
by using the intuitionistic fuzzy weighted average (IFWA) operator (Xu 2007):
i
=
1
,
2
,...,
m
)
r
j
=
IFWA
(
r
1
j
,
r
2
j
,...,
r
mj
),
1
w
i
m
m
m
m
w
i
w
i
w
i
=
−
1
(
1
−
μ
ij
)
,
1
(
v
ij
)
,
1
(
1
−
μ
ij
)
−
1
(
v
ij
)
,
i
=
i
=
i
=
i
=
j
=
1
,
2
,...,
n
(1.57)
or the following aggregation operator:
r
j
=
IFWG
(
r
1
j
,
r
2
j
,...,
r
mj
)
m
m
m
m
1
−
w
i
m
−
1
1
−
w
i
m
−
1
1
−
w
i
m
−
1
1
−
w
i
m
−
1
=
1
(μ
ij
)
,
1
−
1
(
1
−
v
ij
)
,
1
(
1
−
v
ij
)
−
1
(μ
ij
)
,
i
=
i
=
i
=
i
=
=
,
,...,
j
1
2
n
(1.58)
which is defined based on the intuitionistic fuzzy weighted geometric (IFWG) oper-
ator (Xu and Yager 2006).
Step 5
Rank
r
j
(
j
=
1
,
2
,...,
n
)
in descending order by using the ranking
method described in Sect.
1.1.2
.
Step 6
Rank all the alternatives
y
j
(
j
=
1
,
2
,...,
n
)
and select the best one in
accordance with the ranking of
r
j
.
If the information about the weights of experts is unknown, then we utilize the
IFPOWA (or IFPWG) operator to develop an approach to multi-attribute group deci-
sion making with intuitionistic fuzzy information, which can be described as follows
(Xu 2011):
(
j
=
1
,
2
,...,
n
)
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