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t
(
k
)
ij
=[
t
−
(
k
)
ij
,
t
+
(
k
)
ij
f
(
k
)
ij
=[
f
−
(
k
)
ij
,
f
+
(
k
)
ij
t
+
(
k
)
ij
f
+
(
k
)
ij
]⊆[
0
,
1
]
,
]⊆[
0
,
1
]
,
+
≤
1
,
π
−
(
k
)
ij
=
1
−
t
+
(
k
)
ij
f
+
(
k
)
ij
,π
+
(
k
)
ij
=
1
−
t
−
(
k
)
ij
f
−
(
k
)
ij
−
−
,
(1.96)
i
=
1
,
2
,...,
m
;
j
=
1
,
2
,...,
n
In general, there are benefit attributes and cost attributes inmulti-attribute decision
making, and all the attribute values need to be transformed into the same type.
Without loss of generality, here, we transform the attribute values of cost type into
the attribute values of benefit type, i.e., transform
B
(
k
)
=
(
b
(
k
)
ij
)
m
×
n
into the interval-
r
(
k
)
ij
R
(
k
)
=
(
˜
valued intuitionistic fuzzy decision matrix
)
m
×
n
, where
⎧
⎨
b
(
k
)
ij
,
for benefit attribute G
i
ij
c
r
(
k
)
ij
=
( μ
(
k
)
ij
v
(
k
)
ij
, π
(
k
)
ij
˜
,
˜
)
=
for cost attribute G
i
,
j
=
1
,
2
,...,
n
b
(
k
)
⎩
,
(1.97)
where
b
(
k
)
ij
c
, such that
b
(
k
)
ij
c
b
(
k
)
ij
=
(
f
(
k
)
ij
,
t
(
k
)
ij
, π
(
k
)
ij
)
is the complement of
,
where
π
(
k
)
ij
=[
π
−
(
k
)
ij
,π
+
(
k
)
ij
]
,π
−
(
k
)
ij
t
+
(
k
)
ij
f
+
(
k
)
ij
−
μ
+
(
k
)
ij
v
+
(
k
)
ij
=
1
−
−
=
1
−
,
π
+
(
k
)
ij
t
−
(
k
)
ij
f
−
(
k
)
ij
−
μ
−
(
k
)
ij
v
−
(
k
)
ij
=
1
−
−
=
1
−
(1.98)
Then, we can utilize the IVIFPWA (or IVIFPWG) operator to develop an approach
to multi-attribute group decision making with interval-valued intuitionistic fuzzy
information, which involves the following steps (Xu 2011):
Approach 1.3
Step 1
Calculate the supports:
Sup
ij
d
ij
r
(
k
)
ij
r
(
l
)
r
(
k
)
ij
r
(
l
)
˜
,
˜
=
1
−
˜
,
˜
,
l
=
1
,
2
,...,
s
(1.99)
which satisfy the support conditions (1)-(3) in Sect.
1.3.2
. Without loss of generality,
here we let
d
ij
μ
−
(
k
)
+
μ
+
(
k
)
+
1
4
r
(
k
)
ij
r
(
l
)
−
μ
−
(
l
)
ij
−
μ
+
(
l
)
ij
v
−
(
k
)
ij
v
−
(
l
)
ij
˜
,
˜
=
−
ij
ij
+
π
−
(
k
)
ij
+
π
+
(
k
)
ij
v
+
(
k
)
ij
v
+
(
l
)
ij
−
π
−
(
l
)
ij
−
π
+
(
l
)
ij
+
−
,
l
=
1
,
2
,...,
s
(1.100)
Step 2
Utilize the weights
η
k
(
k
=
1
,
2
,...,
s
)
of the experts
e
k
(
k
=
1
,
2
,...,
q
)
r
(
k
)
ij
r
(
k
)
ij
to calculate the weighted support
T
(
˜
)
of the IVIFV
˜
by the other IVIFVs
r
(
l
)
ij
˜
(
=
,
,...,
=
)
l
1
2
s
and
l
k
:
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