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α
i
=
( μ
α
i
,
˜
v
α
i
, π
α
i
)(
=
,
,...,
)
Let
i
1
2
n
be a collection of IVIFVs, and
T
=
(
w
1
,
w
2
,...,
w
n
)
α
i
(
=
,
,...,
)
≥
,
w
the weight vector of
i
1
2
n
, where
w
i
0
n
, and
i
=
1
w
i
i
1, then Xu (2011) defined an interval-valued
intuitionistic fuzzy power weighted average (IVIFPWA) operator:
=
1
,
2
,...,
=
I V IFPWA
( α
1
, α
2
,..., α
n
)
=
(
w
1
(
1
+
T
( α
1
)) α
1
)
⊕
(
w
2
(
1
+
T
( α
2
)) α
2
)
⊕···⊕
(
w
n
(
1
+
T
( α
n
)) α
n
)
i
=
1
w
i
(
1
+
T
( α
i
))
(1.70)
which can be transformed into the following form:
( α
1
, α
2
,..., α
n
)
IVIFPWA
⎛
⎝
⎡
⎣
1
⎤
⎦
,
n
n
w
j
(
1
+
T
( α
j
))
i
=
1
w
i
(
w
j
(
1
+
T
( α
j
))
i
=
1
w
i
(
−
μ
α
j
)
−
μ
α
j
)
=
−
1
(
1
+
T
( α
i
))
,
−
1
(
1
+
T
( α
i
))
1
1
1
j
=
j
=
⎡
⎤
n
n
w
j
(
1
+
T
( α
j
))
i
=
1
w
i
(
1
+
T
( α
i
))
w
j
(
1
+
T
( α
j
))
i
=
1
w
i
(
1
+
T
( α
i
))
⎣
v
α
j
)
v
α
j
)
⎦
,
1
(
,
1
(
j
=
j
=
⎡
n
n
w
j
(
1
+
T
( α
j
))
i
=
1
w
i
(
1
+
T
( α
i
))
w
j
(
1
+
T
( α
j
))
i
=
1
w
i
(
1
+
T
( α
i
))
⎣
−
μ
α
j
)
v
α
j
)
1
(
1
−
1
(
,
j
=
j
=
⎤
⎞
n
w
j
(
1
+
T
( α
j
))
n
w
j
(
1
+
T
( α
j
))
n
i
=
n
i
=
−
μ
α
j
)
v
α
j
)
⎦
⎠
1
(
1
1
w
i
(
1
+
T
( α
i
))
−
1
(
1
w
i
(
1
+
T
( α
i
))
(1.71)
j
=
j
=
where
n
T
( α
i
)
=
w
j
Sup
( α
i
, α
j
)
(1.72)
j
=
1
j
=
i
and
Sup
( α
i
, α
j
)
is the support for
α
i
from
α
j
, with the following conditions:
(1)
Sup
( α
i
, α
j
)
∈[
0
,
1
]
.
(2)
Sup
( α
i
, α
j
)
=
Sup
( α
j
, α
i
)
.
(3)
Sup
, where
d
is a distance
measure, such as the normalized Hamming distance or the normalized Euclidean
distance, where
( α
i
, α
j
)
≥
Sup
( α
s
, α
t
)
,if
d
( α
i
, α
j
)<
d
( α
s
, α
t
)
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