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a
i
)
=
j
=
1
j
(
(
a
i
,
a
j
)
(
a
i
,
a
j
)
where
T
Sup
, and
Sup
is the support for
a
i
from
a
j
,
=
i
which satisfies the following properties:
(1)
Sup
(
a
i
,
a
j
)
∈[
0
,
1
]
.
(2)
Sup
(
a
i
,
a
j
)
=
Sup
(
a
j
,
a
i
)
.
(3)
Sup
(
a
i
,
a
j
)
≥
Sup
(
a
s
,
a
t
)
,if
|
a
i
−
a
j
|
<
|
a
s
−
a
t
|
.
Based on the PA operator and the geometric mean, Xu and Yager (2010) further
defined a power geometric (PG) operator:
1
+
T
(
a
i
)
n
i
1
(
1
+
T
(
a
i
))
PG
(
a
1
,
a
2
,...,
a
n
)
=
a
=
(1.18)
i
i
=
1
Obviously, the PA and PG operators are a nonlinear weighted aggregation tool,
whose weighting vectors depend upon the input data and allow the values being
aggregated to support and reinforce each other, that is, the closer two values
a
i
and
a
j
, the more similar they are, and the more they support each other.
1.2.2 Some Operational Laws of IFVs
Xu and Yager (2006), and Xu (2007) introduced some operational laws of IFVs as
follows:
Definition 1.3
(Xu and Yager 2006; Xu 2007) Let
α
i
=
(μ
α
i
,
v
α
i
,π
α
i
)(
i
=
1
,
2
)
be any two IFVs, then
(1)
α
1
⊕
α
2
=
(μ
α
1
+
μ
α
2
−
μ
α
1
μ
α
2
,
v
α
1
v
α
2
,(
1
−
μ
α
1
)(
1
−
μ
α
2
)
−
v
α
1
v
α
2
)
.
(2)
α
1
⊗
α
2
=
(μ
α
1
μ
α
2
,
v
α
1
+
v
α
2
−
v
α
1
v
α
2
,(
1
−
v
α
1
)(
1
−
v
α
2
)
−
μ
α
1
μ
α
2
)
.
−
μ
α
1
)
λ
,
v
α
1
,(
−
μ
α
1
)
λ
−
v
α
1
), λ >
λα
1
=
(
−
(
(3)
1
1
1
0.
α
1
=
(μ
α
1
,
)
λ
,(
)
λ
−
μ
α
1
), λ >
(4)
1
−
(
1
−
v
1
−
v
0.
α
α
1
1
All the results of the above operations are also IFVs and the following are all
right.
Theorem 1.2
(Xu 2011)
1
−
λ
1
1
−
λ
2
λ
1
>λ
2
, then
λ
1
α
≥
λ
2
α, α
≥
α
,
<λ
1
,λ
2
≤
(1) If
0
1.
λα
1
≥
λα
2
,α
1
≥
α
2
,
(2) If
μ
α
1
≥
μ
α
2
,
v
α
1
≤
v
α
2
, then
0
<λ
≤
1.
(3) If
μ
α
1
≥
μ
α
3
,μ
α
2
≥
μ
α
4
,
v
≤
v
,
v
≤
v
α
4
, then
α
1
⊕
α
3
≥
α
2
⊕
α
4
,α
1
⊗
α
1
α
3
α
2
α
3
≥
α
2
⊗
α
4
.
Proof
(1) If
λ
1
>λ
2
, then
−
μ
α
)
λ
1
−
μ
α
)
λ
2
v
λ
1
v
λ
2
α
1
−
(
1
≥
1
−
(
1
,
α
≤
(1.19)
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