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Therefore,
y
6
y
4
y
5
y
2
y
3
y
1
From the above analysis, the results obtained by Xia et al. (2012b)'s approach are
very similar to the ones obtained by Wu and Chen (2011)'s approach, but the former
is much simpler. In addition, Xia et al. (2012b)'s approach is more flexible than Wu
and Chen (2011)'s one, because it can provide the decision makers (or experts) more
choices as the parameters are assigned different values.
1.6 Intuitionistic Fuzzy Aggregation Operators Based
on Archimedean t-conorm and t-norm
1.6.1 Intuitionistic Fuzzy Operational Laws Based on t-conorm
and t-norm
Definition 1.17
(Klir and Yuan 1995) A function
τ
:
[
0
,
1
]×[
0
,
1
]→[
0
,
1
]
is called
a t-norm if it satisfies the following four conditions:
(1)
τ(
1
,
x
)
=
x
, for all
x
.
(2)
τ(
x
,
y
)
=
τ(
y
,
x
)
, for all
x
and
y
.
τ(
,τ(
,
))
=
τ(τ(
,
),
)
(3)
x
y
z
x
y
z
, for all
x
,
y
and
z
.
x
and
y
y
, then
x
,
y
)
≤
≤
τ(
,
)
≤
τ(
(4) If
x
x
y
.
Definition 1.18
(Klir and Yuan 1995) A function
s
:
˙
[
0
,
1
]×[
0
,
1
]→[
0
,
1
]
is called
a t-conorm if it satisfies the following four conditions:
(1)
s
˙
(
0
,
x
)
=
x
, for all
x
.
(2)
s
˙
(
x
,
y
)
=˙
s
(
y
,
x
)
, for all
x
and
y
.
(3)
s
˙
(
x
,
˙
s
(
y
,
z
))
=˙
s
(
˙
s
(
x
,
y
),
z
)
, for all
x
,
y
and
z
.
x
and
y
y
, then
x
,
y
)
(4) If
x
≤
≤
˙
s
(
x
,
y
)
≤˙
s
(
.
Definition 1.19
(Klir and Yuan 1995) A t-norm function
τ(
x
,
y
)
is called
Archimedean t-norm if it is continuous and
.An
Archimedean t-norm is called strict Archimedean t-norm if it is strictly increasing
in each variable for
x
τ(
x
,
x
)<
x
for all
x
∈
(
0
,
1
)
,
y
∈
(
0
,
1
)
.
Definition 1.20
(Klir and Yuan 1995) A t-conorm function
s
˙
(
x
,
y
)
is called
˙
(
,
)>
∈
(
,
)
Archimedean t-conorm if it is continuous and
.An
Archimedean t-conorm is called strict Archimedean t-conorm if it is strictly increas-
ing in each variable for
x
s
x
x
x
for all
x
0
1
,
y
∈
(
0
,
1
)
.
It is well known (Klement and Mesiar 2005) that a strict Archimedean t-norm is
expressed via its additive generator
h
as
h
−
1
s
˙
(
x
,
y
)
=
(
h
(
x
)
+
h
(
y
))
, and similarly,
)
=
g
−
1
applied to its dual t-conorm
T
(
x
,
y
(g(
x
)
+
g(
y
))
with
h
(
t
)
=
g(
1
−
t
)
.
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