Dealing with Missing Data in Rough Set Analysis of Multi-Attribute and Multi-Criteria Decision Problems

Rough sets methodology is a useful tool for analysis of decision problems concerning a set of objects described in a data table by a set of condition attributes and by a set of decision attributes. In practical applications, however, the data table is often not complete because some data are missing. To deal with this case, we propose an extension of the rough set methodology to the analysis of incomplete data tables. The adaptation concerns both the classical rough set approach based on the use of indiscernibility relations and the new rough set approach based on the use of dominance relations. While the first approach deals with the multi-attribute classification problem, the second approach deals with the multi-criteria sorting problem. In the latter, condition attributes have preference-ordered scales, and thus are called criteria, and the classes defined by the decision attributes are also preference-ordered. The adapted relations of indiscernibility or dominance between a pair of objects are considered as directional statements where a subject is compared to a referent object. We require that the referent object has no missing data The two adapted rough set approaches boil down to the original approaches when there are no missing data. The rules induced from the newly defined rough approximations defined are either exact or approximate, depending whether they are supported by consistent objects or not, and they are robust in a sense that each rule is supported by at least one object with no missing data on the condition attributes or criteria represented in the rule.