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Characteristics measured within verbal scales.
Let us consider group
of
N
objects, the intensity of manifestation of qualitative characteristics
X
,
of
j
=
1
k
j
which is estimated. Let
,
l
=
1
m
be levels of the verbal scales applied to
X
j
ij
estimate characteristics
, accordingly. Levels are arranged in ascend-
ing order of intensity of manifestation of these characteristics.
Let us denote with
,
X
j
=
1
k
j
j
a
relative numbers of objects of the considered group,
i
which are referred to level
while estimating characteristic
.
X
X
lj
j
Based on these data, let us construct
k
COSS's with names
, and term-sets
X
j
~
X
. Let us denote membership function of fuzzy number
corresponding to
lj
lj
~
()
μ
x
l
-th object of
j
-th COSS with
. Let us refer fuzzy numbers
or their
lj
lj
()
μ
as object evaluations. Let us denote an evaluation of
n
-th object within the limits of characteristic
x
membership function
lj
~
with
n
j
and
X
j
~
()
(
)
n
j
n
j
n
j
n
jL
n
jR
μ
x
≡
a
,
a
,
a
,
a
n
j
n
=
1
N
,
,
j
=
1
k
. Fuzzy number
with membership
1
2
~
()
μ
n
j
x
function
is equal to one of fuzzy numbers
. Let us denote weight
lj
coefficients of estimated characteristics with
n
∑
=
ω
,
ω
=
1
j
j
The fuzzy rating of
n
-th object within the limits of characteristic
j
1
X
is defined
j
in the form of fuzzy number
~
~
~
A
=
ω
⊗
X
n
⊕
...
⊕
ω
⊗
X
n
k
(5.19)
n
1
1
k
with membership function
⎛
⎞
k
k
k
k
()
∑
∑
∑
∑
⎜
⎜
⎝
n
j
n
j
n
jL
n
jR
⎟
⎟
⎠
μ
x
≡
ω
a
,
ω
a
,
ω
a
,
ω
a
.
(5.20)
n
j
1
j
2
j
j
j
=
1
j
=
1
j
=
1
j
=
1
y
characterizing manifes-
Let us define a confidential interval for definite rating
()
α
μ
n
y
≥
for
n
-th object. With confidence level
tations of characteristics
X
,
n
j
y
of manifestation of characteristics
for
n
-th object lies
0
< α
<
1
rating
X
j
within the interval
k
k
k
k
∑
()
∑
∑
()
∑
n
j
−
1
n
jL
n
j
−
1
n
jR
ω
a
−
L
α
ω
a
≤
x
≤
ω
a
+
R
α
ω
a
.
(5.21)
j
1
j
n
j
2
j
j
=
1
j
=
1
j
=
1
j
=
1