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T
, then the IFPWAoperator (
1.25
) reduces
to an intuitionistic fuzzy power average (IFPA) operator:
=
(
/
,
/
,...,
/
)
Especially, if
w
1
n
1
n
1
n
IFPA
(α
1
,α
2
,...,α
n
)
=
((
1
+
T
(α
1
))α
1
)
⊕
((
1
+
T
(α
2
))α
2
)
⊕···⊕
((
1
+
T
(α
n
))α
n
)
i
=
1
(
1
+
T
(α
i
))
⎛
n
n
(
1
+
T
(α
j
))
i
=
1
(
1
+
T
(α
i
))
(
1
+
T
(α
j
))
i
=
1
(
1
+
T
(α
i
))
⎝
1
=
−
1
(
1
−
μ
α
j
)
,
1
(
v
α
j
)
,
j
=
j
=
⎞
n
n
(
1
+
T
(α
j
))
(
1
+
T
(α
j
))
n
i
=
1
(
1
+
T
(α
i
))
n
i
=
1
(
1
+
T
(α
i
))
⎠
1
(
1
−
μ
α
j
)
−
1
(
v
α
j
)
(1.30)
j
=
j
=
where
n
1
n
T
(α
i
)
=
Sup
(α
i
,α
j
)
(1.31)
j
=
1
j
=
i
be a vector of
n
IFVs, then it can be easily proven that the
IFPWA operator has the following desirable properties (Xu 2011):
Let
(α
1
,α
2
,...,α
n
)
(α
1
,α
2
,...,α
n
)
Theorem 1.4
(Commutativity) Let
be any permutation of
(α
1
,α
2
,...,α
n
)
, then
(α
1
,α
2
,...,α
n
)
IFPWA
(α
1
,α
2
,...,α
n
)
=
IFPWA
(1.32)
Theorem 1.5
(Idempotency) If
α
j
=
α
, for all
j
, then
IFPWA
(α
1
,α
2
,...,α
n
)
=
α
(1.33)
Theorem 1.6
(Boundedness)
α
−
≤
(α
1
,α
2
,...,α
n
)
≤
α
+
IFPWA
(1.34)
where
mi
j
{
μ
α
j
}
,
α
−
=
ma
j
{
v
α
j
}
,
1
−
mi
j
{
μ
α
j
}−
ma
j
{
v
α
j
}
(1.35)
ma
j
{
μ
α
j
}
,
α
+
=
mi
j
{
v
α
j
}
,
1
−
ma
j
{
μ
α
j
}−
mi
j
{
v
α
j
}
(1.36)
Based on the IFPWA operator (
1.25
) and the geometric mean, Xu (2011)
defined an intuitionistic fuzzy power weighted geometric (IFPWG) operator:
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