Information Technology Reference
In-Depth Information
6.2 A Method of Defuzzification of Fuzzy Numbers on the Basis
of the Weighed Sets
6.2 A Metho d of Defuzzif ication of Fuzzy Numbers
Known methods of defuzzification of fuzzy numbers allow obtaining only
pointwise aggregating indexes for these numbers, thereby, making indiscernible,
for example, unimodal numbers with different coefficients of fuzziness or
unimodal numbers with tolerance numbers. Thus, these methods essentially lose
informational features of fuzzy numbers.
The development of defuzzification method is aimed in keeping, whenever
possible, informational features of fuzzy numbers, and to use them for building of
regression models.
In [84] for a normal triangular number
~
(
)
definition of the weighed
B
≡
b
,
b
L
b
,
R
point is provided
1
1
[
(
)
(
)
]
(
)
=
∫
B
b
−
1
−
α
b
+
b
+
1
−
α
b
α
d
α
=
b
+
b
−
b
.
L
R
R
L
6
(6.2)
0
(
)
Let us consider symmetric
triangular numbers with a typical point
b
.
Computing by the formula (6.2) weighed points for such fuzzy numbers, we
obtain that irrespective of values of fuzziness coefficients
b
=
b
L
R
, the weighed
b
,
L
b
R
points are equal to the same number
b
.
To have a possibility to determine various indexes for similar numbers, the
authors have introduced the new concept — the weighed set.
As known,
{
()
}
[
]
=
1
,
2
A
=
x
∈
R
:
μ
x
≥
α
=
A
A
α
A
α
α
()
()
[]
[
]
−
1
−
1
=
a
−
L
α
a
,
a
+
R
α
a
,
α
∈
0
.
(6.3)
1
L
2
R
~
(
)
is referred to as set of
.
Let us extend definition of the weighed point (6.2) of normal triangular number
onto
-level of
-tolerance number
A
≡
a
,
a
,
a
L
a
,
Λ
α
1
2
R
-unimodal numbers [93, 221].
The number
Λ
1
[
]
()
()
=
∫
−
1
−
1
B
2
b
−
L
α
b
+
R
α
b
α
d
α
=
b
+
rb
−
lb
;
L
R
R
L
0
1
1
()
()
∫
−
∫
−
l
=
L
1
α
α
d
α
,
r
=
R
1
α
α
d
α
.
(6.4)
0
0
~
(
)
is referred to as weighed point of
Λ
-unimodal number
B
≡
b
,
b
L
b
,
.
R
As described above, when operating with
-tolerance numbers (sometimes
Λ
also with
-unimodal numbers) pointwise representation of fuzzy numbers is not
always informative enough. For reasons given, definition of the weighed set for
Λ
Λ
-tolerance numbers is introduced.
