Civil Engineering Reference
In-Depth Information
2
ab
bc
−
1
A
A
∈
Γ
(
)=[
+
(
)
,
]
(
,
)=(
+
,
l
f
)
If
4
,thenT
x
yf
a
x
,where
x
y
l
u
.
A
A
Theorem 3.
Let T
(
)
∈
F
(
R
)
and let
Δ
(
)
be its general f-triangular approxi-
1
Γ
A
A
mation. Then,
Δ
(
)
can be computed in the following cases: If
∈
Γ
2
,then
A
T
(
)=[
t
1
+
t
2
f
(
a
)
,
t
1
+
t
3
f
(
a
)]
2
ab
bc
−
1
A
A
If
∈
Γ
3
,thenT
(
)=[
x
,
x
+
yf
(
a
)]
,where
(
x
,
y
)=(
l
+
u
,
u
f
)
.
2
ab
bc
−
1
A
A
If
∈
Γ
4
,thenT
(
)=[
x
+
yf
(
a
)
,
x
]
,where
(
x
,
y
)=(
l
+
u
,
l
f
)
.
4
Algorithm and Examples
In the previous section, we have presented formulas for computing the general f-
trapezoidal approximation
T
A
A
)
of any fuzzy number
A
. In the process of applying Theorems
2
and
3
, we need to
determine which one subset
(
)
and the general f-triangular approximation
Δ
(
4, the given fuzzy number
A
belongs to.
In the following algorithm, we straightforwardly compute
T
Γ
(
i
)
,
1
≤
i
≤
A
(
)
. It is really more
efficient.
4.1
Algorithm 4
A
A
Let
=[
A
L
(
α
)
,
A
U
(
α
)]
be a fuzzy number and
T
(
)
be general f-trapezoidal
approximation of
A
.
Step 1.
Compute the following objectives: a,b,c and
l
,
l
f
,
u
,
u
f
.
φ
−
1
,and
u
f
)
φ
−
1
.If
s
1
≤
Step 2.
Compute
φ
,
(
s
1
,
s
2
,
s
3
,
s
4
)=(
l
,
l
f
,
u
,
s
3
,then
A
T
(
)=[
s
1
+
s
2
f
(
a
)
,
s
3
+
s
4
f
(
a
)]
.
ψ
−
1
,and
u
f
)
ψ
−
1
.
Step 3.
Otherwise, compute
ψ
,
(
t
1
,
t
2
,
t
3
)=(
l
+
u
,
l
f
,
A
Step 4.
If
t
2
≤
0and
t
3
≥
0, then
T
(
)=[
t
1
+
t
2
f
(
a
)
,
t
1
+
t
3
f
(
a
)]
.
2
ab
bc
−
1
A
Step 5.
If
t
2
>
0, then
T
(
)=[
x
,
x
+
yf
(
a
)]
,where
(
x
,
y
)=(
l
+
u
,
u
f
)
.
2
ab
bc
−
1
A
Step 6.
If
t
3
<
0, then
T
(
)=[
x
+
yf
(
a
)
,
x
]
,where
(
x
,
y
)=(
l
+
u
,
l
f
)
.
Note that, to obtain an algorithm for computing the general f-triangular approxima-
tion
A
A
, it only drops Step 2 from the above Algorithm 4.
Δ
(
)
of
Search WWH ::
Custom Search