Information Technology Reference
In-Depth Information
(
)
(
)
1
,
.
.
.
,
⎛
μ
M
M
μ
M
M
⎞
1
2
1
k
⎜
R
R
⎟
p
p
(
)
(
)
μ
M
,
M
1
.
.
.
μ
M
,
M
⎜
⎟
1
2
2
k
R
R
R
=
,
p
=
1
2
p
p
⎜
⎟
p
.
.
.
.
.
.
⎜
⎜
⎟
⎟
(
)
(
)
μ
M
,
M
μ
2
M
,
M
.
.
.
1
⎝
1
k
2
k
⎠
R
R
(3.11)
p
p
which define over
fuzzy binary relations of similarity and, accordingly,
hierarchy of partitioning of set of the formalized results of an qualitative index
expert evaluation for a population of objects by equivalence classes.
According to (3.12), it is possible to decompose matrixes of relations of
similarity
k
Θ
R
onto equivalence relations:
p
⎧
1
δ
.
.
.
δ
⎫
⎛
⎞
12
1
k
⎜
⎟
⎪
⎪
⎨
⎪
⎪
⎬
δ
1
.
.
.
δ
⎜
⎟
12
2
k
R
=
∪
α
,
⎜
⎟
p
.
.
.
.
.
.
α
⎪
⎩
⎪
⎭
⎜
⎜
⎟
⎟
⎪
⎪
δ
δ
.
.
.
1
⎝
⎠
1
k
2
k
(3.12)
0
⎩
⎨
⎧
where
δ
=
ij
1
Populations of objects form partitions of the matrix onto minors corresponding
to clusters.
Thus, depending on
k
α
-levels of fuzzy similarity relations, the set
Θ
can be
divided into clusters of similar elements with levels of confidence
α
.
3.5 Examples of Practical Application of the Fuzzy Cluster
Analysis Methods
Example 3.1.
[146]
Assignment of boards of examiners.
Quality of checks of
entrants' exam papers directly depends on to what extent accurate and coordinated
knowledge evaluation criteria of the examiners appointed to the subject boards
are. To solve a problem of assigning the boards of examiners the method based on
the fuzzy cluster analysis is offered. Let us consider pupils' knowledge on
mathematics estimated results provided by five examiners within the limits of a
scale “E”,"C”, "B", “A", summarized in Table 3.1
Table 3.1
Estimated results of knowledge on mathematics provided by examiners
No. of an examiner
A (“5”)
B (“4”)
C (“3”)
E (“2”)
1
2
3
4
5
48
50
50
48
42
110
87
99
100
99
90
120
92
97
93
36
27
43
39
50
