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1
a
≥
a
,
a
≥
a
;
⎧
l
3
il
3
l
4
il
4
⎪
⎨
1
δ
=
−
1
a
≤
a
,
a
≤
a
;
il
l
3
il
3
l
4
il
4
⎪
⎩
0
a
>
a
,
a
<
a
or
a
<
a
,
a
>
a
;
l
3
il
3
l
4
il
4
l
3
il
3
l
4
il
4
1
a
<
a
,
a
>
a
;
⎧
l
3
il
3
l
4
il
4
⎪
⎨
2
δ
=
−
1
a
>
a
,
a
<
a
;
il
l
3
il
3
l
4
il
4
⎪
⎩
0
a
≥
a
,
a
≥
a
or
a
≤
a
,
a
≤
a
;
l
3
il
3
l
4
il
4
l
3
il
3
l
4
il
4
a
a
Unknown parameters
,
,
l
=
l
,
m
−
1
are solutions of optimization problem
l
3
l
4
[150]
(
)
(
)
⎡
⎛
2
2
⎞
⎤
m
−
1
k
1
a
−
a
+
a
−
a
(
)
∑∑
1
2
⎜
⎜
⎝
⎟
⎟
⎠
σ
=
δ
a
−
a
+
a
−
a
+
δ
l
3
il
3
l
4
il
4
→
min
.
⎢
⎣
⎥
⎦
il
l
3
il
3
l
4
il
4
il
2
k
a
−
a
+
a
−
a
⎢
⎥
l
11
i
==
l
4
il
4
il
3
l
3
Solutions meet limits of known methods (152].
The generalized model of expert evaluations of qualitative characteristic or
expert description of quantitative property values in linguistic terms, being
constructed within the scope of set
elements, maintains a maxima of the
information included in the set elements. However, unlike the model constructed
in §4.2, it generally does not satisfy to Pareto condition. In particular practical
problems, the fulfillment of Pareto condition for the generalized model
constructed on the basis of a information loss minimum, is directly verified.
Besides, the method of determination of generalized model described in §4.2 can
be applied within the limits of any membership function of fuzzy numbers from
Λ
Ξ
k
group, and the method of the current paragraph can be applied only in case of
T
-numbers or normal triangular numbers.
Thus, for practical problems, while determining the generalized models within
the limits of set
elements, we propose to define models according to methods
described in §4.2 and §4.4. Then, it is necessary to check Pareto optimality of the
generalized model constructed on the basis of information loss minimum. After
that, on the basis of two characteristics, namely, fuzziness degrees of models and
information loss occurred while constructing the models, we can define the
optimum generalized model.
4.5 Buil di ng of t he G eneralize d For mali zed Result of Ex pert Eval uatio ns
Ξ
k
4.5 Building of the Generalized Formalized Result of Expert
Evaluations of the Qualitative Characteristic Based on the
Least Squares Method
4.5 Buil di ng of t he G eneralize d For mali zed Result of Ex pert Eval uatio ns
Let
{
}
n
i
M
= μ
,
n
=
1
N
Θ
k
(the formalized results of
expert evaluations of qualitative characteristic of an object group),
(
,
i
=
1
k
be elements of set
i
)
n
i
in
in
in
L
in
R
μ
≡
a
,
a
,
a
,
a
,
n
=
1
N
. Let us construct the generalized formalized
1
2
