Information Technology Reference
In-Depth Information
Chapter 2
Methods of Expert Information Formalization
Based on Complete Orthogonal Semantic
Spaces
2.1 Fuzzy Numbers Used for Formalization of Linguistic
Values of Characteristics
Let us consider tolerance and unimodal
(
)
-numbers with membership functions
L
−
R
⎧
⎛
⎞
a
−
x
a
−
x
⎜
⎜
⎝
1
⎟
⎟
⎠
1
L
,
0
≤
≤
1
a
>
0
⎪
L
a
a
⎪
L
L
⎪
⎛
x
−
a
⎞
x
−
a
R
⎜
⎜
⎝
2
⎟
⎟
⎠
,
0
≤
2
≤
1
a
>
0
⎪
⎪
R
a
a
()
μ
x
=
⎨
R
R
~
A
a
−
x
x
−
a
⎪
1
1
<
0
∩
2
<
0
⎪
a
a
L
R
⎪
a
−
x
x
−
a
⎪
1
2
0
>
1
∪
>
1
⎪
a
a
⎩
L
R
and following conditions for functions
L
and
R
:
1.
( )
( )
()
()
L
0
=
R
0
=
1
L
1
=
R
1
=
0
()
()
R
x
2.
and
are monotonically decreasing functions over set [0,1].
L
x
Let us denote
Λ
for a population of all tolerance and unimodal numbers with
conditions 1 and 2.
Let us call elements of the population
Λ
as
Λ
-numbers which are in turn
subdivided into
Λ
-tolerance and
Λ
-unimodal numbers.
As
L
and
R
are monotonically decreasing functions, the set of α -level of
~
(
)
Λ
-tolerance number
A
≡
a
,
a
,
a
L
a
,
will look like:
1
2
R
}
[
]
=
{
()
A
x
∈
R
:
μ
x
≥
α
=
A
1
,
A
2
~
α
α
α
A
[
]
[]
()
()
−
1
−
1
=
a
−
L
α
a
;
a
+
R
α
a
,
α
∈
0
1
.
(2.1)
1
L
2
R
