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In-Depth Information
1.8 Point Operators for Aggregating IFVs
For an IFS A
={
x
A (
x
),
v A (
x
) |
x
X
}
,let
κ, λ
[0
,
1], Atanassov (1995) gave
the following operators:
( 1 )
D κ ( A ) = { x , μ A ( x ) + κπ A ( x ) , v A ( x ) + ( 1 κ) π A ( x ) | x X } .
(
2
)
F κ,λ (
A
) = { x
, μ A (
x
) + κπ A (
x
) ,
v A (
x
) + λπ A (
x
) | x
X } ,
where
κ + λ
1
.
(
3
)
G κ,λ (
A
) = {
x
, κμ A (
x
) ,λ
v A (
x
) |
x
X
} .
H κ,λ ( A ) = { x , κμ A ( x ) , v A ( x ) + λπ A ( x ) | x X } .
( 4 )
H κ,λ ( A ) = { x , κμ A ( x ) , v A ( x ) + λ ( 1 κμ A ( x ) v A ( x )) | x X } .
( 5 )
(
6
)
J κ,λ (
A
) = {
x
, μ A (
x
) + κπ A (
x
) ,λ
v A (
x
) |
x
X
} .
J κ,λ ( A ) = { x , μ A ( x ) + κ (
(
7
)
1
μ A ( x ) λ v A ( x )) ,λ v A ( x ) | x X } .
( 8 )
P κ,λ ( A ) = { x , max (κ, μ A ( x )) , min (λ, v A ( x )) | x X } , where κ + λ 1 .
(
9
)
Q κ,λ (
A
) = { x
, min
(κ, μ A (
x
)) ,
max
(λ,
v A (
x
)) | x
X } ,
where
κ + λ
1
.
Let IFS
(
X
)
be the set of all IFSs on X .For A
IFS
(
X
)
, Burillo and Bustince
(1996) defined an operator D
κ x (
A
)
for each point x
X :
D
κ x (
A
) = {
x
, μ A (
x
) + κ x π A (
x
) ,
v A (
x
) + (
1
κ x ) π A (
x
) |
x
X
}
(1.300)
where
1].
Then, Liu andWang (2007) defined an intuitionistic fuzzy point operator for IFSs:
κ x
[0
,
Definition 1.26 (Liu and Wang 2007) Let A
IFS
(
X
)
, for each point x
X ,
taking
κ x x
[0
,
1] and
κ x + λ x
1, then an intuitionistic fuzzy point operator
F κ x x (
A
)
:IFS
(
X
)
IFS
(
X
)
is as follows:
F κ x x (
A
) = {
x
, μ A (
x
) + κ x π A (
x
) ,
v A (
x
) + λ x π A (
x
) |
x
X
}
(1.301)
and if let F 0
κ x x (
A
) =
A , then
x
n
1
(
1
κ x λ x )
F n
κ x x (
A
) =
,
μ A (
x
) + κ x π A (
x
)
,
κ x + λ x
X
n
1
(
1
κ x λ x )
v A (
x
) + λ x π A (
x
)
x
(1.302)
κ x + λ x
Xia and Xu (2010) defined a series of point operators for aggregating IFVs:
Definition 1.27 (Xia and Xu 2010) For an IFV
α = α ,
v
α )
,let
κ α α
[0
,
1],
we define some point operators as follows:
(1)
D
κ α α (α) = α + κ α π α ,
v
α + (
1
κ α ) π α )
.
(2)
F κ α α (α) = α + κ α π α ,
v α + λ α π α )
, where
κ α + λ α
1.
 
 
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