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In-Depth Information
()
()
()
⎛
μ
s
n
μ
s
n
...
μ
s
n
k
⎞
⎜
1
1
1
2
1
⎟
()
()
()
μ
s
n
μ
s
n
...
μ
s
n
k
⎜
⎟
2
1
2
2
2
M
=
.
⎜
⎟
n
(5.11)
.
.
...
.
⎜
⎜
⎟
⎟
()
()
()
μ
s
n
μ
s
n
...
μ
s
n
k
⎝
⎠
m
1
m
2
m
(
~
~
~
)
Let
be a vector the co-ordinates of which be fuzzy numbers.
Result of the generalized multiplication of this vector by a matrix (5.11) is the
vector
Р
=
X
,
X
,...,
X
1
2
m
~
~
~
(
)
n
n
n
k
H
=
P
⊗
M
,
H
=
A
,
A
,...,
A
,
with co-ordinates in the form
n
=
1
N
n
1
2
n
n
of fuzzy numbers
()
~
~
~
()
n
j
n
j
n
j
= μμ
(5.12)
These fuzzy numbers
ar
e fuzzy evaluations of manifestation of qualitative
characteristics
Y
,
A
s
⊗
X
⊕
...
⊕
s
⊗
X
,
j
=
1
k
,
n
=
1
N
.
1
1
m
m
for
n
-th object. Let us assign corresponding weight to
j
=
1
k
each characteristic
k
∑
=
ω
,
j
=
1
k
,
ω
=
1
j
j
Let us define the f
uzz
y rating corresponding to manifestation of estimated
characteristic
Y
,
j
1
for
n
-th object as follows:
j
=
1
k
~
~
~
A
=
ω
⊗
A
n
⊕
...
⊕
ω
⊗
A
n
k
.
(5.13)
n
1
1
k
From (5.12), (5.13) it follows, that
⎡
~
k
m
()
k
m
()
∑∑
∑∑
n
j
l
n
j
l
A
≡
ω
μ
s
a
,
ω
μ
s
a
,
⎢
⎣
n
j
l
1
j
l
2
j
=
1
l
=
1
j
=
1
l
=
1
⎤
k
m
k
m
()
()
∑∑
∑∑
ω
μ
s
n
j
a
l
L
,
ω
μ
s
n
j
a
l
R
.
⎥
⎦
(5.14)
j
l
j
l
j
=
1
l
=
1
j
=
1
l
=
1
~
Let us denote parameters of fuzzy number
,
n
=
1
N
in (5.14) with
()
η
x
n
n
n
L
n
R
δ
,
δ
,
δ
,
δ
and membership function with
, accordingly. Let us determine
n
1
2
x
which characterize manifestations
a confidential interval for the definite rating
()
α
for
n
-th object. With confidence level
η
n
x
≥
of characteristics
Y
,
j
=
1
k
,
n
x
of manifestations of characteristics
Y
,
for
n
-th
0
< α
<
1
, rating
j
=
1
k
object lies within the interval
()
()
n
n
L
−
1
n
n
R
−
1
δ
−
δ
L
α
≤
x
≤
δ
+
δ
R
α
.
(5.15)
1
n
2
