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Additive and multiplicative indexes, accordingly
1
[
]
()
()
()
n
n
n
k
min
μ
x
,
μ
x
,...,
μ
x
dx
1
2
1
N
k
=
0
;
ij
1
N
[
]
()
()
()
n
=
1
max
μ
n
x
,
μ
n
x
,...,
μ
n
k
x
dx
1
2
(3.9)
0
1
(
)
n
n
n
k
dx
min
μ
,
μ
,...,
μ
1
2
~
N
x
k
=
0
.
N
1
(
)
n
=
1
n
n
n
k
max
μ
,
μ
,...,
μ
1
2
x
(3.10)
All indexes vary from zero to unity. If values of consistency indexes are close to
zero, it means incompetence of at least several experts or it means a fuzzy
proposition of evaluation procedure.
3.4 The Fuzzy Cl uster Ana lysi s of Set of the For mal i zed Res ults
0
3.4 The Fuzzy Cluster Analysis of Set of the Formalized Results
of an Evaluation of Qualitative Characteristic of a
Population of Objects
3.4 The Fuzzy Cl uster Ana lysi s of Set of the For mal i zed Res ults
Building of fuzzy binary relations of similarity and conformity over a set of
formalized results of expert evaluations allows to carry out fuzzy cluster analysis
of this set and, by that, to study its structural composition.
The proposition 3.2. [135] Fuzzy sets
with membership functions
R
1 , R
2
~
(
)
(
)
(
)
μ
M
,
M
=
k
μ
M
,
M
=
k
i
=
1
k
;
j
=
1
k
,
, accordingly, define fuzzy
R
i
j
ij
R
i
j
ij
2
1
relations of similarity over set
.
Θ
k
R and
R possess characteristics of reflexivity and
The proof . Let us prove that
symmetry
(
1
~
1
N
)
(
)
()
()
=
n
i
n
i
μ
M
,
M
=
k
=
1
d
M
,
M
1
μ
x
μ
x
dx
=
1
R
i
i
ii
i
i
N
1
n
1
0
1
[
]
()
()
n
i
n
i
min
μ
x
,
μ
x
dx
1
N
(
)
μ
M
M
k
0
,
=
=
=
1
.
R
i
i
ij
1
N
2
[
]
()
()
n
=
1
max
μ
n
i
x
,
μ
n
i
x
dx
0
R are reflective. Let us prove their symmetry
Thus,
R and
1
1
N
(
)
(
)
()
()
=
n
i
n
j
μ
M
,
M
=
1
d
M
,
M
=
1
μ
x
μ
x
dx
=
R
i
j
i
j
N
1
n
1
0
 
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