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n
Sup α index ( j ) index ( i )
T
index ( j ) ) =
(1.46)
i
=
1
i
=
j
where Sup α index ( j ) index ( i ) indicates the support of i th largest IFV
α index ( i )
for
the j th largest IFV
α index ( j )
, and g :[
0
,
1
]→[
0
,
1
]
is a basic unit-interval monotonic
(BUM) function, having the properties:
(1)
g(
0
) =
0.
(2)
g(
1
) =
1.
(3)
g(
x
) g(
y
)
,if x
>
y .
Especially, if g(
x
) =
x , then the IFPOWA operator ( 1.43 ) reduces to the IFPA
operator ( 1.30 ).
Furthermore, based on the IFPOWA operator ( 1.44 ) and the geometric mean,
Xu (2011) defined an intuitionistic fuzzy power ordered weighted geometric
(IFPOWG) operator:
IFPOWG
1 2 ,...,α n )
= index ( 1 ) )
1 ω 1
n 1
1 ω 2
n 1
1 ω n
n 1
index ( 2 ) )
⊗···⊗ index ( n ) )
(1.47)
which can be further expressed as:
n
n
1
ω j
n
1
ω j
n
IFPOWG
1 2 ,...,α n ) =
1 α index ( j ) )
,
1
1 (
1
v
α index ( j ) )
,
1
1
j
=
j
=
n
n
1 ω j
n
1 ω j
n
1 (
1
v
α index ( j ) )
1 α index ( j ) )
1
1
j
=
j
=
(1.48)
where
are a collection of weights satisfying the conditions
( 1.45 ) and ( 1.46 ). Especially, if g(
ω i
(
i
=
1
,
2
,...,
n
)
x
) =
x , then the IFPOWG operator ( 1.47 ) reduces
to the IFPG operator ( 1.39 ).
Clearly, the weighting vectors of both the IFPWA and IFPWG operators not only
depend upon the input arguments and allow the values being aggregated to support
and reinforce each other, but also emphasize the ordered positions of all the given
arguments. Furthermore, the IFPWA and IFPWG operators have also the properties:
commutativity, idempotency and boundedness.
 
 
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