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2.3 Formalization of Expert Evaluations of Qualitative
Characteristics in a Mark Scale
2.3 For mali zatio n of Ex pert Eval uatio ns of Q ualitative C haracteristics
Let us assume that appearances (demonstrations) of qualitative characteristics are
estimated by experts using a mark scale.
Let us construct COSS on the basis of a posteriori information resulted from an
evaluation of appearances of qualitative characteristic
X
for a population of
objects
. The minimum quantity of points which allows evaluation of
characteristic appearance equals zero, and maximum quantity of points is
M
.
Besides it is assumed that
the
verbal scale is developed for the evaluation
purposes, with levels
Y
,
2
Y
,...,
Y
1
N
X
,
arranged in ascending intensity order.
l
=
1
m
[]
=
represent relative evaluations of appearance of
characteristic
X
, which were obtained by dividing mark scale-based evaluations
by the maximum estimate
M
. For example, in terms of educational process
evaluation they can be obtained as a result of testing within the scope of a certain
school subject, and those evaluation values are equal to the ratio of number of
properly performed tasks to the total quantity of all tasks performed.
Let us construct COSS
X
with terms
Let
y
∈
0
,
j
1
N
j
X
(according to levels of a verbal scale)
()
and membership functions
of
T
-numbers or normal triangular numbers.
The additional expert information consisting of results of preliminary allocation
of outcomes obtained to one of levels of a verbal scale and comparison of
outcomes between each other is required for model-building process. In this case,
the standard approach of paired comparisons of outcomes of qualitative
characteristics availability, which is applied to construct membership functions of
fuzzy sets [49], is used.
Let us start constructing with term
X
which corresponds to maximum
characteristic
X
appearance intensity degree. Let the point
μ
x
l
()
x
=
1
, i.e.
μ
1
=
1
be
m
considered as the typical one for the membership function of this term.
Let
Y
,...,
1
Y
;
are the objects allocated by the expert to level
. Let us
j
≥
3
X
j
m
y
;
arrange them in decreasing order of evaluations
. We obtain a
i
=
1
j
conditional ordered series
Y
,
Y
,..,
Y
,
()
( )
( )
1
2
j
to which a numerical ordered series corresponds
y
,
y
,..,
y
.
()
()
()
1
2
j
Let us perform paired comparisons
of
object
s o
f that conditional ordered series
using Saati scale [123]. Let
be Saati scale evaluations of
characteristic
X
superiority availability at an object
a
=
1
9
,
k
=
1
j
ik
versus an object
. If
Y
X
()
k
()
i
objects
Y
and
Y
have approximately equal characteristic
X
appearances, then
()
()
i
k
