Dominance-based Rough Set Approach - Example

Example

Consider the following problem of high school students evaluations:

Table 1: Example—High School Evaluations

object (student)
	medium	medium	bad		bad
	good	medium	bad		medium
	medium	good	bad		medium
	bad	medium	good		bad
	bad	bad	medium		bad
	bad	medium	medium		medium
	good	good	bad		good
	good	medium	medium		medium
	medium	medium	good		good
	good	medium	good		good

Each object (student) is described by three criteria, related to the levels in Mathematics, Physics and Literature, respectively. According to the decision attribute, the students are divided into three preference-ordered classes:, and . Thus, the following unions of classes were approximated:

• i.e. the class of (at most) bad students,

• i.e. the class of at most medium students,

• i.e. the class of at least medium students,

• i.e. the class of (at least) good students.

Notice that evaluations of objects and are inconsistent, because has better evaluations on all three criteria than but worse global score.

Therefore, lower approximations of class unions consist of the following objects:

Thus, only classes and cannot be approximated precisely. Their upper approximations are as follows:

while their boundary regions are:

Of course, since and are approximated precisely, we have, and

The following minimal set of 10 rules can be induced from the decision table:

1. if then
2. if and and then
3. if then
4. if and then
5. if and then
6. if and then
7. if and then
8. if then
9. if then
10. if and then

The last rule is approximate, while the rest are certain.