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r
ij
=
(
t
ij
,
f
ij
,π
ij
)
b
ij
,
for benefit attribute G
j
=
,
i
=
1
,
2
,...,
n
;
j
=
1
,
2
,...,
m
b
ij
,
for cost attribute G
j
(1.143)
where
b
ij
is the complement of
b
ij
, such that
b
ij
=
(
v
ij
,μ
ij
,π
ij
)
.
of the
i
th line,
and get the overall performance value
r
i
corresponding to the alternative
y
i
by the
WIFGBM:
Step 2
Aggregate all the performance values
r
ij
(
j
=
1
,
2
,...,
m
)
IFGBM
p
,
q
w
r
i
=
(
t
i
,
f
i
,π
i
)
=
(
r
i
1
,
r
i
2
,...,
r
im
)
⎛
⎛
⎝
⎞
⎠
1
p
+
q
⎝
1
1
ij
p
ik
q
m
1
1
w
j
w
k
m
(
m
−
1
)
=
1
−
1
−
−
−
μ
−
μ
,
j
,
k
=
1
j
=
k
⎛
⎝
⎞
⎠
1
p
+
q
m
1
−
1
w
j
p
1
w
k
q
1
1
−
−
(
1
−
v
ij
)
−
(
1
−
v
ik
)
m
(
m
−
1
)
,
j
,
k
=
1
j
=
k
⎛
⎝
⎞
⎠
1
p
+
q
1
1
ij
p
ik
q
m
1
1
w
j
w
k
m
(
m
−
1
)
1
−
−
−
μ
−
μ
j
,
k
=
1
j
=
k
⎛
⎞
1
p
+
q
m
⎝
1
−
1
w
j
p
1
w
k
q
⎠
1
−
1
−
−
(
1
−
v
ij
)
−
(
1
−
v
ik
)
m
(
m
−
1
)
j
,
k
=
1
j
=
k
(1.144)
where
p
0.
Step 3
Rank the overall performance values
r
i
,
q
>
according to
Xu and Yager (2006)'s ranking method, and obtain the priority of the alternatives
y
i
(
i
=
1
,
2
,...,
n
)
(
.
Next, we give an example to illustrate the proposed method:
i
=
1
,
2
,...,
n
)
according to
r
i
(
i
=
1
,
2
,...,
n
)
Example 1.5
(Xia et al. 2012a) A city is planning to build a municipal library. One
of the problems facing the city development commissioner is to determine what
kind of air-conditioning system should be installed in the library (adapted from
Yoon (1989)). The contractor offers five feasible alternatives
y
i
,
which might be adapted to the physical structure of the library. Suppose that three
attributes: (1)
G
1
: economic; (2)
G
2
: functional; and (3)
G
3
: operational, are taken
(
i
=
1
,
2
,
3
,
4
,
5
)
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