Rough Sets Modeling - Probabilistic Composition of Preferences, Theory and Applications - page 104

proportion of those not belonging to Cl t does not exceed 1-l. Similarly, for an
upper class, the lower approximation of consistency level l is:

P l Cl t ¼

Dp þ ðÞ\ Cl t

x 2 Cl t

l

:

j Dp þ ðÞ j

fixed at l = 1, the result of application of VC-
DRSA will be the same as that of DRSA, but, for l < 1, more elements may belong
to the approximations and the boundaries may become smaller.
A dif

So, if the consistency level is

culty with this rule is that it has the effect of, for instance, leading some
alternative x to be accepted in the lower approximation of a lower class while other
alternative, dominated by x, is not accepted because it dominates less alternatives.
This is not reasonable, as being dominated by x and not dominating other alter-
natives should be a stronger reason for this second alternative to belong to the lower
approximation of the lower class.
Changing the condition of being dominated by that of dominating, identical
inconsistencies may occur in the higher approximations of the upper classes.
Several alternatives (Blaszczynski et al. 2006 , 2009 ; Inuiguchi and Yoshioka 2006 ;
Inuiguchi et al. 2009 ; Deng et al. 2011 ) have been proposed to avoid this contra-
diction, but all preserving the principle of establishing the level of consistency in
terms of proportion of inconsistencies, disregarding the distance involved in the
dominance.

10.4 Aggregation of Classes

In Sant Anna and Moreira Filho ( 2013 ) is proposed the strategy of changing the

original data by augmenting the roughness in the measurements to avoid con-
tradictions that may increase the number of alternatives in the boundaries. The basic
idea of this strategy is to move alternatives in small classes with extreme values to
neighboring classes, by replacing such extreme values by approximations.
Increasing roughness in this way identifies alternatives with close values in the
decision attribute. With the values of the condition attributes unchanged this will
result necessarily in the possibility of reducing the number of contradictions,
increasing the quality of approximation and reducing the size of the reducts.
In fact, the quality of the approximation is increased whenever the cardinality of
the boundaries decreases and what reduces the boundaries is reducing, for each
alternative x, the number of alternatives indiscernible from x and located outside the
class in which the decision attribute locates x. If the values of the condition attri-
butes are not changed, the indiscernible objects remain the same. Thus, the increase
in the size of the classes determined by the decision attribute is forcefully associated
to reduction of the number of indistinguishable alternatives outside.
The shrinking of the reducts can be proved in two steps. First, it is easy to see
that no reduction is lost when joining the classes determined by the decision

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