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In-Depth Information
1
⎤
⎦
,
n
(
1
+
T
( α
j
))
i
=
1
(
1
+
T
( α
i
))
1
n
−
1
−
v
α
j
)
1
−
1
(
1
−
j
=
⎡
1
1
n
n
(
1
+
T
( α
j
))
i
=
1
(
1
+
T
( α
i
))
(
1
+
T
( α
j
))
i
=
1
(
1
+
T
( α
i
))
1
n
−
1
1
n
−
1
−
−
⎣
v
α
j
)
1
(μ
α
j
)
1
(
1
−
−
,
j
=
j
=
⎤
⎦
⎞
1
−
1
−
n
n
(
1
+
T
( α
j
))
(
1
+
T
( α
j
))
1
n
−
1
1
n
−
1
n
i
=
1
(
1
+
T
( α
i
))
n
i
=
1
(
1
+
T
( α
i
))
v
α
j
)
1
(μ
α
j
)
⎠
1
(
1
−
−
j
=
j
=
(1.79)
with the condition (
1.76
).
Similar to the IFPOWA operator (
1.43
), Xu (2011) defined an interval-valued
intuitionistic fuzzy power ordered weighted average (IVIFPOWA) operator as fol-
lows:
IVIFPOWA
( α
1
, α
2
,..., α
n
)
=
ω
1
α
index
(
1
)
⊕
ω
2
α
index
(
2
)
⊕···⊕
ω
n
α
index
(
n
)
(1.80)
which can be further expressed as:
IVIFPOWA
( α
1
, α
2
,..., α
n
)
⎛
⎡
−
μ
α
index
(
j
)
)
ω
j
⎤
n
n
⎝
⎣
1
−
μ
α
index
(
j
)
)
ω
j
⎦
,
=
−
1
(
1
,
1
−
1
(
1
j
=
j
=
⎡
v
α
index
(
j
)
)
ω
j
⎤
⎞
n
n
j
=
1
(
j
=
1
(
⎣
v
α
index
(
j
)
)
ω
j
⎦
⎠
,
(1.81)
where
α
j
=
( μ
α
j
,
˜
v
, π
α
j
)(
j
=
1
,
2
...,
n
)
are a collection of IVIFVs, and
α
index
(
i
)
α
j
is the
i
th largest of the IVIFVs
α
j
(
j
=
1
,
2
,...,
n
)
.
ω
i
(
i
=
1
,
2
,...,
n
)
are a
collection of weights such that
D
i
TV
D
i
−
1
TV
i
n
ω
i
=
g
−
g
,
D
i
=
V
index
(
j
)
,
TV
=
V
index
(
i
)
j
=
1
i
=
1
V
index
(
j
)
=
1
+
T
( α
index
(
j
)
)
(1.82)
( α
index
(
j
)
)
α
index
(
j
)
and
T
denotes the support of the
j
th largest IVIFV
by all the other
IVIFVs, i.e.,
n
Sup
α
index
(
j
)
, α
index
(
i
)
T
( α
index
(
j
)
)
=
(1.83)
i
=
1
i
=
j
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