Information Technology Reference
In-Depth Information
1.8 Point Operators for Aggregating IFVs
For an IFS
A
={
x
,μ
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
,λ
x
∈
[0
,
1] and
κ
x
+
λ
x
≤
1, then an intuitionistic fuzzy point operator
F
κ
x
,λ
x
(
A
)
:IFS
(
X
)
→
IFS
(
X
)
is as follows:
F
κ
x
,λ
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
,λ
x
(
A
)
=
A
, then
x
n
1
−
(
1
−
κ
x
−
λ
x
)
F
n
κ
x
,λ
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
=
(μ
α
,
v
α
)
,let
κ
α
,λ
α
∈
[0
,
1],
we define some point operators as follows:
(1)
D
κ
α
,λ
α
(α)
=
(μ
α
+
κ
α
π
α
,
v
α
+
(
1
−
κ
α
) π
α
)
.
(2)
F
κ
α
,λ
α
(α)
=
(μ
α
+
κ
α
π
α
,
v
α
+
λ
α
π
α
)
, where
κ
α
+
λ
α
≤
1.
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