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Thus, the generalized formalized result of an expert evaluation of qualitative
characteristic of an object group constructed within the limits of set
k
Θ
elements
is a linear combination of these elements. Linear combination coefficients are
weight coefficients of set
k
Θ
elements.
Thus,
{
}
⎩
⎨
⎧
k
⎭
⎬
⎫
()
()
∑
=
n
n
i
M
=
μ
x
,
n
=
1
N
=
ω
μ
x
,
n
=
1
N
=
i
i
1
⎩
⎨
⎧
⎭
⎬
⎫
⎛
k
k
k
k
⎞
()
∑
∑
∑
∑
(4.30)
=
μ
n
x
≡
⎜
⎝
ω
a
in
,
ω
a
in
,
ω
a
in
L
,
ω
a
in
R
⎟
⎠
,
n
=
1
N
.
i
1
i
2
i
i
i
=
1
i
=
1
i
=
1
i
=
1
Let us prove fulfillment of the condition (4.26), which ensures a Pareto optimality
of generalized formalized result of expert evaluations of qualitative characteristic
of an object group. Since
[
]
=
()
()
()
n
n
n
k
m
in
μ
x
,
μ
x
,...,
μ
x
1
2
i
k
=
1
{
}
k
[
]
k
()
()
()
()
=
∑
∑
n
n
n
k
n
i
ω
min
μ
x
,
μ
x
,...,
μ
x
≤
ω
μ
x
i
1
2
i
i
=
1
k
i
=
1
i
=
1
k
[
]
⎩
⎨
⎧
⎭
⎬
⎫
()
()
()
()
∑
=
n
n
n
n
k
=
μ
x
≤
ω
max
μ
x
,
μ
x
,...,
μ
x
=
i
1
2
i
=
1
k
i
1
[
]
[]
()
()
()
=
max
μ
n
x
,
μ
n
x
,...,
μ
n
k
x
;
∀
x
∈
0
,
n
=
1
N
,
(4.31)
1
2
i
=
1
k
Then we obtain
[
]
()
()
()
()
≤
n
n
n
k
n
min
μ
x
,
μ
x
,...,
μ
x
≤
μ
x
1
2
i
=
1
k
[
]
[]
()
()
()
n
n
n
k
μμ
Thus, the generalized result of expert evaluations of qualitative characteristic of an
object group constructed within the limits of set
≤
max
x
,
x
,...,
μ
x
;
∀
x
∈
0
,
n
=
1
N
.
1
2
i
=
1
k
k
Θ
elements is Pareto optimal
one.
In order to identify fuzzy evaluations of appearance of q
ualit
ative characteristic
for real objects with one of linguistic values
X
,
l
=
1
m
of an
esti
mated
()
n
attribute, it is necessary to compar
e m
embership functions
μ
x
,
n
=
1
N
with
()
f
l
x
X
,
membership functions
,
l
=
1
m
of terms
l
=
1
m
of the generalized
model constructed withi
n th
e limits of set
k
elements. To compare me
mbe
rship
Ξ
()
()
n
μ
x
f
l
x
n
=
1
N
l
=
1
m
functions
,
with membership functions
,
, the
indexes can be used, without limiting generality,
