Knowledge Reduction In Several Ordered Decision Information Systems

The rough set theory proposed by Pawlak (1982) is established on the classification ofuniverse determined by equivalent relation and has been proved to be an excellentmathematical tool for dealing with uncertain, imprecise, and incomplete information systems.Pawlak rough set theory is defined using the indiscernibility relation, which implies that allattributes are nominal. However, in the real world, we may face cases when some attributevalues are ordinal. Therefore, Pawlak rough set theory is inapplicable in dealing withknowledge reduction problem of the ordered information systems. To overcome thisinsufficiency, Greco, Matarazzo and Slowinski proposed the dominance-based rough setapproach to take the ordering properties of attributes into account.Knowledge reduction is one of the major advantages of rough set analysis, and includesattribute reduction and optimal decision rules acquisition. In this paper, for several differentordered decision information systems, we discussed the knowledge reduction based on thedominance-based rough set approach and its extended models. This paper is organized asfollows:In Chapter2, some notation and basic concepts for the ordered decision informationsystems and the dominance-based rough set approach such as incomplete,lower(upper)approximation and knowledge reduction are introduced.In Chapter3, we discussed the minimal decision rules acquisition by using thediscernibility function in ordered decision information systems. For the lower approximation,we construct the set of relatively minimal elementary objects, from which we can obtain theminimal elementary decision rules. To obtain the optimal decision rules and make thecondition part of the decision rules more simply, we give the reduct of the minimalelementary decision rules and the corresponding discernibility function. Finally, we define therule-vector to describe the decision rule. By computing minimum rule-vector, we caneliminate the implication between the optimal elementary decision rules. Then, we obtain theminimal decision rules.In Chapter4, credible decision rules acquisition in incomplete ordered informationsystem is discussed. In incomplete ordered information system, regular objects are defined to describe the objects which do not contain any unknown attribute value. In order to derive theoptimal decision rules, firstly, we define the dominance relation and dominating class in theset of regular objects. Secondly, the concept of compatible dominating class is proposed byusing the regular objects and non-regular objects, and we give the corresponding descriptor.Finally, the concept of the relative reduct of compatible dominating class and its discernibilityfunction computing approach are proposed, and then we can compute the credible decisionrules.In Chapter5, approximate decision rules acquisition and its optimization based onconsistent objects in ordered decision information systems with fuzzy decision are discussed.In an ordered decision information system (ODIS), upper (lower) consistent objects areconsidered, and used to cope with inconsistency of the system and draw “at least”(“at most”)decision rules. The upper (lower) consistent objects are described by generalized decisions ofthe objects, and thereby generalized conveniently to the ordered decision information systemwith fuzzy decisions (FODIS). This new innovation can be used to draw decision rules fromthe FODIS. Furthermore, upper (lower) approximate consistent objects, which can inducedecision rules more efficiently in the FODIS, are discussed. At last, upper (lower)approximate consistent reducts, which preserve all the upper (lower) approximate consistentobjects, are defined. By constructing discernibility function, the upper (lower) approximateconsistent reducts can be computed by Boolean reasoning techniques.In Chapter6, we conclude our work and point out the future work.