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In-Depth Information
[
]
[
]
()
()
1
2
−
1
−
1
B
=
B
,
B
=
b
−
L
α
b
,
b
+
R
α
b
.
α
α
α
1
2
L
2
2
R
2
2
-level of
~
looks like
The set of
α
[
]
(
)
(
)
1
1
1
2
2
1
2
2
1
1
1
2
2
1
2
2
D
=
min
A
B
,
A
B
,
A
B
,
A
B
,
max
A
B
,
A
B
,
A
B
,
A
B
.
(2.2)
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
bb
(
~
— a positive number,
~
— a negative number). The proof of the proposition for other relations between
numbers
a
−
a
>
0
+
<
0
Let us prove the proposition for
,
1
L
R
~
~
−
>
0
is carried out similarly. In a case
a
a
,
b
+
b
<
0
for (2.2)
A
,
B
1
L
R
we'll obtain
[
]
=
1
,
2
D
=
D
D
α
α
α
[
]
[
]
[
]
[
]
{
}
()
()
()
()
−
1
−
1
−
1
−
1
=
a
+
R
α
a
b
−
L
α
b
,
a
−
L
α
a
b
+
R
α
b
.
2
1
R
1
2
L
1
1
L
2
2
R
1
2
1
2
a
−
a
>
0
Functions
L
,
L
,
R
,
R
are monotonically decreasing, therefore at
,
1
L
α
>
α
b
+
b
<
0
and
R
2
1
()
()
−
1
−
1
0
<
a
−
L
α
a
<
a
−
L
α
a
;
1
1
1
L
1
1
2
L
1
1
()
()
−
1
−
1
0
<
a
+
R
α
a
<
a
+
R
α
a
;
2
1
2
R
2
1
1
R
1
1
()
()
−
1
−
1
b
−
L
α
b
<
b
−
L
α
b
<
0
1
2
1
L
1
2
2
L
2
2
()
()
b
+
R
−
1
α
b
<
b
+
R
−
1
α
b
<
0
.
2
2
2
R
2
2
1
R
2
2
α
>
α
Thus, with
, and hence, the left boundary of
membership function
~
monotonically increases, and the right monotonically
decreases, that ensures a membership of
~
to
,
D
2
>
D
2
D
1
>
D
1
2
1
α
2
α
1
α
2
α
1
. The proposition 2.1 is proved.
Similarly, it is possible to prove that product of
Λ
-unimodal numbers is
Λ
-unimodal number, and product of
-tolerance and
-unimodal numbers is
-
Λ
Λ
Λ
Λ
tolerance number.
Further paragraphs of the Chapter are devoted to methods of formalization of
expert evaluations of qualitative characteristics and description of values of
quantitative characteristics in linguistic terms based on COSS. The constructed
population of fuzzy numbers of
Λ
is offered to be used for formalization of
linguistic values of these characteristics or for building of membership functions
of COSS terms.
2.2 For mali zatio n of Ex pert Eval uatio ns of Q ualitative C haracteristics
2.2 Formalization of Expert Evaluations of Qualitative
Characteristics in a Verbal Scale
2.2 For mali zatio n of Ex pert Eval uatio ns of Q ualitative C haracteristics
As the input information the a posteriori information resulted from expert
evaluations of a qualitative characteristic
X
for a population of objects
[121—122] is considered.
