Consistency measure, inclusion degree and fuzzy measure in decision tables

Classical consistency degree has some limitations for measuring the consistency of a decision table, in which the lower approximation of a target decision is only taken into consideration. In this paper, we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory. For three types of decision tables (complete, incomplete and maximal consistent blocks), the membership functions of an object are defined through using the equivalence class, tolerance class and maximal consistent blocks including itself, respectively. Based on these membership functions, we introduce consistency measures to assess the consistencies of a target set and a decision table, and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables. In addition, the relationships among the consistency, inclusion degree and fuzzy measure are established as well. These results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues.