Rough Sets Modeling - Probabilistic Composition of Preferences, Theory and Applications - page 103

For dominance relations, it is interesting to consider the set of alternatives
dominating a given alternative x:

D P ðÞ¼ y 2 U : yD P x f g;

as well as the set of alternatives dominated by x:

D P ðÞ¼ y 2 U : xD P y f g:

ned the classes and the entire list of the P-dominating and P-dominated
sets, it is possible to de

Once de

ne the upper and lower approximations. Regarding the
definitions above, the difference is that now, dominance relationships are used
instead of indiscernibility.
The lower and upper approximations can be set respectively for the upper
cumulative classes by:

PCl t ¼

x 2 U : D P ðÞ Cl t

and

PCl t ¼

:

x 2 U : D P ðÞ\ Cl t 6¼ [

Analogously for the lower cumulative classes:
PCl t ¼ x 2 U : D P ðÞ Cl t

and

PCl t ¼

:

x 2 U : D P ðÞ\ Cl t 6¼ [

ned as before,
being enough consider in the computation of the index of quality of approximation
the boundaries of cumulative classes. Likewise, it is enough formulating the
decision rules in terms of pertinence to cumulative classes.

Boundaries and the index of quality of approximation can be de

10.3 VC-DRSA

VC-DRSA is based on a relaxation of the conditions for entrance in the approxi-
mations, controlled by a level of consistency l, which can vary from zero to one. In
the case of a lower class Cl t , for an alternative x be included in the lower
approximation, the requirement of no alternative outside Cl t being dominated by x
is relaxed and some alternatives outside Cl t may be dominated by x, as long as the