Information Technology Reference
In-Depth Information
essential role for issues of processing and analysis of fuzzy information. There are
different methods of defuzzification of fuzzy numbers, in particular, a gravity
method, a minimax method, a method of a maximum of membership function, a
method of the weighed point etc. [40. 46. 83—93] which have the certain merits
and demerits described in [87]. Most frequently applied of these methods is the
gravity method, according to which pointwise value defined under the formula:
1
()
∫
x
μ
x
dx
E
=
0
.
(1.8)
1
()
∫
μ
x
dx
0
()
μ
x
is assigned to fuzzy number with membership function
.
() (
)
()
()
μ
x
≡
a
,
a
,
a
L
a
,
If
, and
R
x
in (1.6) are linear functions, and
,
L
x
1
2
R
then (1-8) looks like:
[
]
1
() ()
() ()
2
2
2
2
a
−
a
+
a
a
+
a
a
+
a
−
a
(1.9)
2
1
1
L
2
R
R
L
3
E
=
.
(
)
2
a
−
a
+
a
+
a
The method of the maximum of membership function is applied to unimodal fuzzy
numbers; the abscissa of a point of a maximum of its membership function is to be
taken as an integral index of fuzzy number. In the minimax method a minimum
value of abscissas of membership function's maximum points is to be selected as
an integral index of fuzzy number. The essence of the method of the weighed
point [87] is that generation of an integral index of a normal triangular number is
carried out with accounting for weighs of its
2
1
L
R
-level sets. If the normal triangular
α
()
() (
)
number
~
with membership function
μ
x
≡
b
,
b
,
b
μ
x
looks like
, the
~
~
L
R
B
corresponding weighed point is defined by the formula:
1
1
[
(
)
(
)
]
()
∫
B
=
b
−
1
−
α
b
+
b
+
1
−
α
b
p
α
d
α
.
L
R
2
0
()
Function
-levels' importance distribution
density function [87]. This function shall be nonnegative and satisfy a
normalization condition:
p
α
at
0
≤ α
≤
1
is referred to as
α
1
()
∫
p
α
d
α
=
1
.
A weighed point for a normal triangular number with membership function
() (
0
)
()
μ
x
≡
b
,
b
,
b
and
is defined in [84] by the formula
p
α
=
2
α
~
L
R
B
