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membership functions of the term-sets of the set
k
elements; i.e. if the fuzzy
Ξ
numbers corresponding to the term-sets of the set
elements are tolerance or
Ξ
k
unimodal numbers of the group
described in §2.1, then fuzzy numbers
corresponding to a term-set of the generalized model are also identified as
tolerance or unimodal numbers of the group
Λ
Λ
. Let us suppose that weight
ω
coefficients
of of set
Ξ
k
elements are revealed. Let us define parameters of
i
(
)
l
l
l
L
l
R
membership functions
,
21
of a term-set of the generalized model of
expert qualitative characteristic evaluations or expert description of quantitative
characteristic values in linguistic terms, from the condition
a
a
,
a
,
a
[
]
m
k
(
)
(
)
(
)
(
)
2
2
2
2
∑∑
==
F
=
ω
a
il
−
a
l
+
a
il
−
a
l
+
a
il
L
−
a
l
L
+
a
il
R
−
a
l
R
→
min
.
(4.2)
i
1
1
2
2
l
11
i
l
=
1
m
Unknown parameters are determined from system of normal equations at
∂
F
⎡
k
⎤
∂
F
⎡
k
⎤
∑
=
∑
=
il
l
il
l
=
2
ω
a
−
a
=
0
=
2
ω
a
−
a
=
0
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
i
1
1
i
2
2
l
l
∂
a
∂
a
i
1
i
1
1
2
k
k
∂
F
⎡
⎤
∂
F
⎡
⎤
∑
=
∑
=
il
L
l
L
il
R
l
R
=
2
ω
a
−
a
=
0
=
2
ω
a
−
a
=
0
(4.3)
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
l
L
i
l
R
i
∂
a
∂
a
i
1
i
1
We obtain solutions
k
k
k
k
1
∑
=
2
∑
=
∑
=
l
il
∑
=
a
=
ω
a
;
a
l
=
ω
a
il
;
l
L
il
L
a
l
R
=
ω
a
il
R
.
a
=
ω
a
;
(4.4)
i
1
i
2
i
i
i
1
i
1
i
1
i
1
According to optimization problem solutions, the generalized model is a linear
combination of the set
elements (see §3.1). Coefficients of the linear
combination are weight coefficients of the characteristic expert evaluations
models which are used to construct the generalized model. Thus, we obtain
k
Ξ
⎩
⎨
⎧
k
⎭
⎬
⎫
{
()
}
()
()
=
∑
=
X
=
f
x
=
f
x
=
ω
μ
x
l
l
i
il
i
1
⎧
⎫
⎛
k
k
k
k
⎞
()
∑
∑
∑
∑
il
il
il
L
il
R
=
f
x
≡
⎜
⎝
ω
a
,
ω
a
,
ω
a
,
ω
a
⎟
⎠
.
⎩
⎨
⎭
⎬
l
i
1
i
2
i
i
i
=
1
i
=
1
i
=
1
i
=
1
(4.5)
Let us denote the constructed generalized model of expert qualitative
characteristic evaluations or expert description, in linguistic terms, of physical
values of quantitative characteristic as optimum by noisiness of the set
k
Ξ
element parameters.
Let us prove the fulfillment of the condition (4.1) ensuring Pareto optimality of
generalized model of an expert evaluation or description of a characteristic. As
