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set (set of
formalized expert evaluation results of a qualitative characteristic of an object
group) can be divided into clusters of similar elements.
Let us define weight coefficients of elements of sets
Thus, depending on confidence level
, set
k
(COSS set) or
k
α
Ξ
Θ
k
and
Θ
k
depending on
Ξ
confidence level within the limits of following criteria:
I.
Weight coefficient of the element which has entered a larger cluster exceeds
weight coefficient of the element which has entered a smaller cluster.
II.
If elements have entered the same cluster, their weight coefficients are equal.
Let us consider various cases:
α
<
1
1.
With confidence level
all elements enter the same cluster.
Based on the criterion II, their weight coefficients
k
∑
=
ω
,
i
=
1
k
,
ω
=
1
i
i
i
1
are considered equal, i.e.
ω
=
1
/
k
,
i
=
1
k
.
i
α
<
1
2.
With confidence level
two clusters of potency (number of elements):
k
−
1
and 1 occur.
i
=
1
k
−
1
Let us assume that elements with indexes
have entered a cluster of
ω
potency
are considered equal,
and based on the criterion I, the weight coefficient of the element with the index
k
has smaller value than values of weight coefficients
k
−
1
. Based on the criterion II, weight indexes
i
k
∑
=
ω
,
ω
=
1
.
i
i
Let us use Fishburn scale [151] according to which weight coefficients of units
ranked in decrease of their importance order (within the limits of certain criterion)
are determined under the formula:
i
1
(
)
2
k
−
i
+
1
(4.7)
ω
=
,
i
=
1
k
.
(
)
i
k
k
+
1
Since the weight coefficient of an element with
k
index is the least one, it is
determined by substitution
i
=
k
in the formula (4.7), hence we obtain
2
+
ω
(4.8)
Let
us c
ompute the sum of weight coefficients of elements with indexes
1
=
.
(
)
k
k
k
1
using the formula (4.7):
(
i
=
1
k
−
)
k
−
1
k
−
1
2
k
−
i
+
1
⎡
2
k
+
4
⎤
⎡
k
+
2
⎤
(
)
(
)
∑
∑
ω
=
=
k
−
1
=
k
−
1
.
(4.9)
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
(
)
(
)
(
)
i
k
k
+
1
2
k
k
+
1
k
k
+
1
i
=
1
i
=
1
