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In this paper we discuss the basic relationship between the ordinal fuzzy set theory and the rough set theory. A new generalized definition of lower and upper approximations and a method for detecting the ordinal definable is put forward.

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Double-quantitative decision-theoretic rough sets (Dq-DTRS) provide more comprehensive description methods for rough approximations of concepts, which lay foundations for the development of attribute reduction and rule extraction of rough sets. Existing researches on concept approximations of Dq-DTRS pay more attention to the equivalence class of each object in approximating a concept, and calculate concept approximations from the whole data set in a batch. This makes the calculation of approximations time consuming in dynamic data sets. In this paper, we first analyze the variations of equivalence classes, decision classes, conditional probability, internal grade and external grade in dynamic data sets while objects vary sequentially or simultaneously over time. Then we propose the updating mechanisms for the concept approximations of two types of Dq-DTRS models from incremental perspective in dynamic decision information systems with the sequential and batch variations of objects. Meanwhile, we design incremental sequential insertion, sequential deletion, batch insertion, batch deletion algorithms for two Dq-DTRS models. Finally, we present experimental comparisons showing the feasibility and efficiency of the proposed incremental approaches in calculating approximations and the stability of the incremental updating algorithms from the perspective of the runtime under different inserting and deleting ratios and parameter values.

This paper proposes new definitions of lower and upper
approximations, which are basic concepts of the rough set theory. These
definitions follow naturally from the concept of ambiguity introduced in
this paper. The new definitions are compared to the classical
definitions and are shown to be more general, in the sense that they are
the only ones which can be used for any type of indiscernibility or
similarity relation

We investigate in this paper approximate operations on sets, approximate equality of sets, and approximate inclusion of sets. The presented approach may be considered as an alternative to fuzzy sets theory and tolerance theory. Some applications are outlined.

The original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation. It operates on a data table composed of a set U of objects (actions) described by a set Q of attributes. Its basic notions are: indiscernibility relation on U, lower and upper approximation of either a subset or a partition of U, dependence and reduction of attributes from Q, and decision rules derived from lower approximations and boundaries of subsets identified with decision classes. The original rough set idea is failing, however, when preference-orders of attribute domains (criteria) are to be taken into account. Precisely, it cannot handle inconsistencies following from violation of the dominance principle. This inconsistency is characteristic for preferential information used in multicriteria decision analysis (MCDA) problems, like sorting, choice or ranking. In order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary. The main change is the substitution of the indiscernibility relation by a dominance relation, which permits approximation of ordered sets in multicriteria sorting. To approximate preference relations in multicriteria choice and ranking problems, another change is necessary: substitution of the data table by a pairwise comparison table, where each row corresponds to a pair of objects described by binary relations on particular criteria. In all those MCDA problems, the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model. It is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax. In order to workout a recommendation in one of the MCDA problems, we propose exploitation procedures of the set of decision rules. Finally, some other recently obtained results are given: rough approximations by means of similarity relations, rough set handling of missing data, comparison of the rough set model with Sugeno and Choquet integrals, and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive.

Rough set theory is a new approach to decision making in the presence of uncertainty and vagueness. Basic concepts of rough set theory will be outlined and its possible application will be briefly discussed. Further research problems will conclude the paper.

We consider a sorting (classification) problem in the presence of multiple attributes and criteria, called the MA&C sorting problem. It consists in assignment of some actions to some pre-defined and preference-ordered decision classes. The actions are described by a finite set of attributes and criteria. Both attributes and criteria take values from their domains; however, the domains of attributes are not preference-ordered, while the domains of criteria (scales) are totally ordered by preference relations. Among the attributes we distinguish between qualitative attributes and quantitative attributes. In order to construct a comprehensive preference model that could be used to support the sorting task, we consider preferential information of the decision maker (DM) in the form of assignment examples, i.e. exemplary assignments of some reference actions to the decision classes. The preference model inferred from these examples is a set of ''if ... , then ... '' decision rules. The rules are derived from rough approximations of decision classes made up of reference actions. They satisfy conditions of completeness and dominance, and manage with possible ambiguity (inconsistencies) in the set of examples. Our idea of rough approximations involves three relations together: indiscernibility, similarity and dominance defined on qualitative and quantitative attributes, and on criteria, respec- tively. The usefulness of this approach is illustrated by an example. � 2002 Published by Elsevier Science B.V.

Fuzzy set theory has been used as a framework for interpreting imprecise linguistic expressions. In general, a linguistic term is described by the compatibility ordering induced in some universe of discourse (UoD). A membership function in fuzzy set theory serves to reflect this ordering by assignment of values in [0, 1] for objects in UoD. When we compute the meaning of a linguistic expression such as "young and tall" using fuzzy membership functions, two implicit assumptions are made. First, we assume the membership values have quantitative meaning so that they can be quantitatively manipulated, for example, by adding or subtracting (the extensive scale assumption). Second, we assume that the scales of the membership values used in describing the different linguistic terms are comparable and the same (the common scale assumption). In many cases, these assumptions cannot be justified. Some proposals have been made to address the first issue by using ordinal scale in defining fuzzy membership functions. However, the second issue has not been properly investigated. In this paper, we propose a framework that does not depend on both of these assumptions. Such framework will facilitate our understanding and investigation of qualitative reasoning without the extensive scale and common scale assumptions.