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and get Frank t-conorm and t-norm (Beliakov et al. 2007):
1
1
−
x
1
−
y
+
(γ
−
1
)(γ
−
1
)
s
F
(
˙
x
,
y
)
=
1
−
log
,γ
∈
(
1
,
+∞
)
(1.197)
γ
γ
−
1
1
x
y
+
(γ
−
1
)(γ
−
1
)
τ
F
(
x
,
y
)
=
log
,γ
∈
(
1
,
+∞
)
(1.198)
γ
γ
−
1
Especially, if
γ
→
1, then
log
γ
−
log
1
t
1
lim
γ
→
1
g(
t
)
=
lim
γ
→
=
lim
γ
→
=−
log
t
(1.199)
t
t
−
1
γ
−
1
γ
−
1
1
1
which indicates that lim
γ
→
1
˙
s
F
(
x
,
y
)
=˙
s
A
(
x
,
y
)
and lim
γ
→
1
τ
F
(
x
,
y
)
=
τ
A
(
x
,
y
)
.
Considering the relationships among all the three components:
π
α
i
=
1
−
μ
α
i
−
v
i
,
α
we usually denote an IFV
α
=
(μ
α
,
v
α
,π
α
)
only by its two former components
α
=
(μ
α
,
for brevity. Based on Archimedean t-norm and t-conorm (Klir and
Yuan 1995), Beliakov et al. (2011) defined the sum operation on two IFVs
v
α
)
α
i
=
(μ
α
i
,
v
α
i
)(
i
=
1
,
2
)
:
α
1
⊕
α
2
=
˙
v
α
2
)
s
(μ
α
1
,μ
α
2
), τ (
v
α
1
,
(1.200)
which can be expressed by the following:
α
1
⊕
α
2
=
˙
v
α
2
)
(μ
α
1
,μ
α
2
), τ (
v
α
1
,
s
h
−
1
(μ
α
2
)), g
−
1
=
(
h
(μ
α
1
)
+
h
(g(
v
α
1
)
+
g(
v
α
2
))
(1.201)
Beliakov et al. (2011) also mentioned that for an IFV
α
=
(μ
α
,
v
α
)
,let
λα
=
(μ
λα
,
v
λα
)
, then
g(
v
λα
)
=
λg(
v
α
)
and
h
(μ
λα
)
=
λ
h
(μ
α
)
.
be three IFVs, then with the
above analysis, the operations about these IFVs based on Archimedean t-norm and
Archimedean t-conorm (Klir and Yuan 1995) can be also expressed as below:
Let
α
i
=
(μ
α
i
,
v
α
i
)(
i
=
1
,
2
)
and
α
=
(μ
α
,
v
α
)
Definition 1.21
(Xia et al. 2012b)
)α
1
⊕
α
2
=
˙
α
2
)
(
1
s
(μ
α
1
,μ
α
2
), τ (
v
α
1
,
v
h
−
1
(μ
α
2
)), g
−
1
=
(
h
(μ
α
1
)
+
h
(g(
v
α
1
)
+
g(
v
α
2
))
.
)α
1
⊗
α
2
=
τ(μ
α
1
,μ
α
2
),
˙
α
2
)
(
2
s
(
v
α
1
,
v
g
−
1
h
−
1
=
(g(μ
α
1
)
+
g(μ
α
2
)),
(
h
(
v
α
1
)
+
h
(
v
α
2
))
.
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