Civil Engineering Reference
In-Depth Information
number is immensely important. In
Ma et al.
(
2000
) first studied symmetric trian-
gular approximations of fuzzy numbers. Consequently, in
Grzegorzewski
(
2002
)
proposed interval approximations, in
Abbasbandy and Asady
(
2004
) proposed
trapezoidal approximations, in
Zeng and Li
(
2007
) proposed weighted triangular
approximations which were improved by
Ye h
(
2008
,
2009
), and in
Nasibov and
Peker
(
2008
) proposed the nearest parametric approximations which were improved
by
Ban
(
2008
,
2009
)and
Ye h
(
2011
), independently. In addition, during the last
years, approximations of fuzzy numbers preserving some attributes were studied
too. For example, trapezoidal approximations preserving the expected interval were
proposed by
Grzegorzewski and Mrowka
(
2005
,
2008
) and improved by
Ban
(
2008
)
and
Ye h
(
2007
,
2008
) independently, trapezoidal approximations preserving cores
of fuzzy numbers were proposed by Grzegorzewski and Stefanini in 2009 and
further studied by
Abbasbandy and Hajjari
(
2010
), and trapezoidal approximations
preserving the value and ambiguity were proposed by
Ban et al.
(
2011
). In this paper,
we study more general approximations without preserving any attribute, named
general f-trapezoidal approximations and general f-triangular approximations. In
Sect.
2
, we present several preliminaries and state our main problem. In Sect.
3
,
the formulas for computing general f-trapezoidal approximations and general f-
triangular approximations are provided. In Sect.
4
, we study an efficient algorithm
and illustrated by an example. The conclusions are drawn in Sect.
5
.
2
Problem Statement
A fuzzy number
A
is a subset of the real line R with membership function
μ
A
:
→
[
0
,
1
]
such that (
Dubois and Prade 1978
):
A
is normal, i.e., there is an
x
0
∈
1.
R
with
μ
(
x
0
)=
1
.
A
A
is fuzzy convex, i.e.,
2.
μ
(
rx
+(
1
−
r
)
y
)
≤
min
{
μ
(
x
)
,
μ
(
y
)
}
for all
x
,
y
∈
[
0
,
1
]
.
A
A
A
A
is upper semicontinuous, i.e.,
μ
−
1
A
3.
([
α
,
1
])
is closed for all
α
∈
[
0
,
1
]
.
4. The support of
μ
A
is bounded, i.e., the closure of
{
x
∈
R
:
μ
>
0
}
is bounded.
A
Recall that
A
can be also represented by using its
α
−
cuts
[
A
L
(
α
)
,
A
U
(
α
)]
,
α
∈
[
0
,
1
]
(an ordered pair of left continuous functions)
which satisfy the following conditions:
A
L
is increasing on [0,1].
1.
A
U
is decreasing on [0,1].
2.
A
L
(
α
)
≤
A
U
(
α
)
3.
,forall
α
∈
[
0
,
1
]
.
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
be a left continuous and decreasing function such that
A
is called general f-trapezoidal if its
f
(
0
)=
1
,
and f
(
1
)=
0. A fuzzy number
α
-
cuts
are of the form
[
x
2
−
(
x
2
−
x
1
)
f
(
α
)
,
x
3
+(
x
4
−
x
3
)
f
(
α
)]
,
α
∈
[
0
,
1
]
Search WWH ::
Custom Search