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⎛
m
m
m
m
⎞
()
()
()
()
~
∑
∑
∑
∑
m
n
i
≡
μ
m
n
i
a
l
,
μ
m
n
i
a
l
,
μ
m
n
i
a
l
L
,
μ
m
n
i
a
l
R
.
⎜
⎝
⎟
⎠
(5.4)
l
1
l
2
l
l
The vector of evaluations of
n
-th object (5.1) is replaced with a new vector of
fuzzy evaluations:
l
=
1
l
=
1
l
=
1
l
=
1
(
)
~
~
~
n
n
n
k
=
(5.5)
Each evaluation shall contribute to a total rating point with some weight coeffi-
cient which is determined based on importance of a stage related to the evaluation.
With the system of preferences not available, sta
ges
and related evaluations are
considered equivalent with weights
m
,
m
,...,
m
,
n
1
N
.
1
2
i
1= . With the system of prefer-
ences is available, we range stages, each of them aimed at measuring of one sub-
characteristic, in the order of decrease of their contribution to the total evaluation.
Let us use Fishburn scale and determine weight coefficients of sub-characteristics
as follows:
ω
=
1
/
k
k
,
i
(
)
2
k
−
i
+
1
ω
=
,
i
=
1
k
.
(
)
i
k
k
+
1
Other approaches to determination of weight coefficients are described in [175].
Let us determine fuzzy rating of characteristic manifestation
X
for
n
-th object
by the formula
~
~
~
n
n
k
A
=
ω
⊗
m
⊕
...
⊕
ω
⊗
m
,
n
=
1
N
(5.6)
n
1
1
k
or
~
⎡
k
m
k
m
()
()
∑∑
∑∑
n
i
l
n
i
l
A
≡
ω
μ
m
a
,
ω
μ
m
a
;
⎢
⎣
n
i
l
1
i
l
2
i
=
1
l
=
1
i
=
1
l
=
1
k
m
k
m
⎤
()
()
(
)
∑∑
∑∑
n
i
l
L
n
i
l
R
n
n
n
L
n
R
ω
μ
m
a
,
ω
μ
m
a
≡
Δ
,
Δ
,
Δ
,
Δ
.
⎥
⎦
i
l
i
l
1
2
i
=
1
l
=
1
i
=
1
l
=
1
x
, which characterizes
Let us define a confidential interval for definite rating
characteristic manifestation
X
of
n
-th object,
. From definition of level
sets for fuzzy number it follows that for confidence level
n
=
1
N
()
α
η
n
x
≥
0
< α
<
1
,
n
x
of characteristic manifestation
X
for
n
-th object,
rating
n
=
1
N
is within an
interval
()
()
n
n
L
−
1
n
n
R
−
1
Δ
−
Δ
L
α
≤
x
≤
Δ
+
Δ
R
α
,
(5.7)
1
n
2
~
η
where
is membership function of fuzzy number
,
n
=
1
N
.
n
