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be fuzzy relations of similarity which are transitive closures of fuzzy
relations of similarity
Let
R
p
R
, they are defined by matrixes
(
)
(
)
1
μ
X
,
X
.
.
.
μ
X
,
X
⎛
⎞
1
2
1
k
⎜
R
R
⎟
p
p
(
)
(
)
μ
X
,
X
1
.
.
.
μ
X
,
X
⎜
⎟
k
R
1
2
R
2
=
R
.
p
p
⎜
⎟
p
.
.
.
.
.
.
⎜
⎜
⎟
⎟
(
)
(
)
μ
X
,
X
μ
X
,
X
.
.
.
1
⎝
⎠
1
k
2
k
R
R
p
p
k
Thus,
the fuzzy binary relations of similarity, and,
accordingly, hierarchy of partitioning of set of expert indicator evaluation models
by equivalence classes.
That is, according to the decomposition theorem for similarity relations
R
define within
Ξ
p
R
ˆ
it is
p
possible to decompose:
⎧
1
δ
.
.
.
δ
⎫
⎛
⎞
12
1
k
⎜
⎟
⎪
⎪
⎨
⎪
⎪
⎬
δ
1
.
.
.
δ
⎜
⎟
12
2
k
R
=
∪
α
,
⎜
⎟
(3.6)
p
.
.
.
.
.
.
α
⎪
⎩
⎪
⎭
⎜
⎜
⎟
⎟
⎪
⎪
δ
δ
.
.
.
1
⎝
⎠
1
k
2
k
0
⎩
⎨
⎧
onto equivalence relations, where
δ
=
i
=
1
k
,
j
=
1
k
.
ij
1
Populations of units form matrix partitions on minors corresponding to clusters.
Thus, depending on
-levels of fuzzy relations of similarity, the set
k
can be
Ξ
α
divided into clusters of elements similar among themselves with
α
levels of
confidence.
An application of the method of fuzzy cluster analysis of set
is check and
rejection of erroneous information. Under conditions of unsatisfactory consistency
indexes of elements of this set (value 0.5 can be a threshold of satisfactory
consistency), it is possible to select falling out elements (models of expert evaluations
of qualitative characteristics or expert description in linguistic terms of quantitative
characteristics). The invalidity of information of these models is confirmed by raised
concordance indexes of elements of set
Ξ
k
k
Ξ
when being rejected.
3.3 Buil di ng of C omparative I ndexes and I ndexes of Consistency
3.3 Building of Comparative Indexes and Indexes of
Consistency of Formalized Outcomes of an Qualitative
Characteristic Evaluation for a Population of Objects
3.3 Buil di ng of C omparative I ndexes and I ndexes of Consistency
Let us assume that
k
experts estimate appearance of qualitative characteristic at a
populatio
n o
f objects. According to the methods stated in §2.2 and 2.3,
k
COSS
X
;
can be constructed with membership functions of term-sets
i
=
1
k
