Information Technology Reference
In-Depth Information
FN
~
with the following membership function
⎧
⎩
⎨
⎧
a
−
x
⎭
⎬
⎫
a
−
x
L
1
,
0
<
1
≤
1
a
>
0
⎪
L
a
a
⎪
L
L
⎪
⎛
x
−
a
⎞
x
−
a
()
μ
x
=
R
⎜
⎜
⎝
2
⎟
⎟
⎠
,
0
<
2
≤
1
a
>
0
⎨
~
A
R
a
a
⎪
(1.6)
R
R
⎪
1
a
≤
x
≤
a
;
1
2
⎪
0
x
<
a
−
a
or
x
>
a
+
a
.
⎩
1
L
2
R
is referred to as tolerance
-number.
FN
~
is symbolically written in the form
(
)
L
−
R
~
(
)
A
≡
a
,
a
,
a
L
a
,
1
2
R
() (
)
μ
x
≡
a
,
a
,
a
,
a
a
,
a
,
a
L
a
,
], where
are parameters of tolerance
[or
~
1
2
L
R
A
1
2
R
-number
~
;
a segment
[
]
(
)
a
1
,
a
a
,
a
are left
is a tolerance interval; and
L
−
R
2
and right coefficients of fuzziness, accordingly;
⎛
⎞
⎛
⎞
a
L
x
x
−
R
a
⎜
⎜
⎝
⎟
⎟
⎠
⎜
⎜
⎝
⎟
⎟
⎠
L
1
and
R
2
a
a
(
)
are left and right boundaries of membership function of tolerance
-number:
L
−
R
with
a
=
0
⎛
a
−
L
x
⎞
L
⎜
⎜
⎝
1
⎟
⎟
⎠
=
0
a
with
a
=
0
R
⎛
x
−
R
a
⎞
R
⎜
⎜
⎝
2
⎟
⎟
⎠
=
0
.
a
The unimodal
~
-
(
)
(
)
number has membership function of tolerance
-
L
−
R
L
−
R
(
)
number under the condition of
a
=
a
. A unimodal
number is written
L
−
R
1
2
~
~
~
(
)
(
)
(
)
symbolically as
A
≡
a
,
a
L
a
,
. If
A
≡
a
,
a
,
a
L
a
,
,
B
≡
b
,
b
,
b
L
b
,
,
1
R
1
2
R
1
2
R
then:
~
~
(
)
1.
A
⊕
B
≡
a
+
b
,
a
+
b
,
a
+
b
,
a
+
b
;
1
1
2
2
L
L
R
R
~
(
)
2.
β
A
≡
β
a
,
β
a
,
β
a
,
β
a
on
β
≥
0
1
2
L
R
~
(
)
β
A
≡
β
a
,
β
a
,
β
a
,
β
a
3.
on
β
<
0
2
1
R
L
