Rough Sets Modeling - Probabilistic Composition of Preferences, Theory and Applications - page 102

The boundary of X according to P is de ned by

F P ðÞ¼ P ðÞ P ðÞ;

Pawlak ( 1982 )de nes a set X as rough if its boundary is not empty and as crisp if it

is empty.
Pawlak ( 1991 )de by the set of

condition attributes P for the set of decision attributes D as the proportion of
alternatives whose classi

nes an index of quality of approximation

γ

cation according to D is not contradicted by indiscern-

ibilities according to P. can be determined complementarily by the proportion of

alternatives in boundaries of the sets of the partition

γ

f X i g i ¼ 1 ; ... ; n of U determined by

D. Formally,

X

j F P X ðÞ j

c P ðÞ¼

1 :

j U j

i

The goal of RST is to establish decision rules in terms of values of the condition
attributes to inform on the relevant classes determined by the decision attributes. By
a principle of parsimony, rules employing less condition attributes are preferable.
If a set of condition attributes P has the same index of quality of approximation
of a larger one, i.e., if

Q, this means that there is no
preference between two alternatives x and y according to attributes in Q\P that
might erase any indiscernibility increasing a boundary according to P. If that
happens, decision rules employing the attributes in Q can be replaced by rules
employing only the attributes in P.
Pawlak ( 1991 )de

c P ðÞ¼ c Q ðÞ

for P

⊆

nes as a reduct a set of condition attributes P such that any
subset of P has a smaller quality of approximation than P. The smaller the number
of attributes in the reducts, the simpler the decision rules.
If the attributes determine dominance relationships, i.e., if each of them deter-
mines an order relation in U, it is easier to measure the quality of approximation.
The adaptation of RST for the case of dominance as set forth below was developed
by Greco et al. ( 2001 ).
Suppose the classes of the partition

f Cl t g t ¼ 1 ; ... ; n determined by the decision
attributes ordered in such way that, if r [ s, the alternatives in Cl r are preferable to
those in Cl s . Then, it is enough to deal with the cumulative classes

Cl t or Cl t ¼ s t Cl s ¼ s t Cl s ; t ¼ 1 ; ... ; n :

For the condition attributes, x dominates y with respect to P, or “ x P-dominates y ” ,

or y is , if the preference for x is higher or equal than

the preference for y according to all the elements of P.

“ P-dominated by x ” or “ xD P y ”