Benedetto Matarazzo

Rough sets theory for multicriteria decision analysis

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 approximation of a preference relation by dominance relations

An original methodology for using rough sets to preference modeling in multi-criteria decision problems is presented. This methodology operates on a pairwise comparison table (PCT), including pairs of actions described by graded preference relations on particular criteria and by a comprehensive preference relation. It builds up a rough approximation of a preference relation by graded dominance relations. Decision rules derived from the rough approximation of a preference relation can be used to obtain a recommendation in multi-criteria choice and ranking problems. The methodology is illustrated by an example of multi-criteria programming of water supply systems.

Rough approximation by dominance relations

In this article we are considering a multicriteria classification that differs from usual classification problems since it takes into account preference orders in the description of objects by condition and decision attributes. To deal with multicriteria classification we propose to use a dominance‐based rough set approach (DRSA). This approach is different from the classic rough set approach (CRSA) because it takes into account preference orders in the domains of attributes and in the set of decision classes. Given a set of objects partitioned into pre‐defined and preference‐ordered classes, the new rough set approach is able to approximate this partition by means of dominance relations (instead of indiscernibility relations used in the CRSA). The rough approximation of this partition is a starting point for induction of if‐then decision rules. The syntax of these rules is adapted to represent preference orders. The DRSA keeps the best properties of the CRSA: it analyses only facts present in data, and possible inconsistencies are not corrected. Moreover, the new approach does not need any prior discretization of continuous‐valued attributes. In this article we characterize the DRSA as well as decision rules induced from these approximations. The usefulness of the DRSA and its advantages over the CRSA are presented in a real study of evaluation of the risk of business failure.

Fuzzy Similarity Relation as a Basis for Rough Approximations

The rough sets theory proposed by Pawlak [8,9] was originally founded on the idea of approximating a given set by means of indiscernibility binary relation, which was assumed to be an equivalence relation (reflexive, symmetric and transitive). With respect to this basic idea, two main theoretical developments have been proposed: some extensions to a fuzzy context (e.g. Dubois and Prade, [1,2], Slowinski and Stefanowski, [13,14,15], Yao, [19]) and some extensions of the indiscernibility relation by means of more general binary relations (e.g. Nieminen, [7], Lin, [5], Marcus, [6], Polkowski, Skowron and Zytkow, [10], Skowron and Stepaniuk, [11], Slowinski, [12], Slowinski and Vanderpooten, [16,17,18], Yao and Wong, [20]). In the latter extensions, we wish to point out the proposal of Slowinski and Vanderpooten( [16,17,18]) who introduced and characterized a general definition of rough approximations using a similarity relation which is a reflexive binary relation, relaxing the assumption of symmetry and transitivity.

The Use of Rough Sets and Fuzzy Sets in MCDM

The rough sets theory has been proposed by Z. Pawlak in the early 80’s to deal with inconsistency problems following from information granulation. It operates on an information 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 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 sets idea has proved to be particularly useful in the analysis of multiattribute classification problems; however, it was failing when preferential ordering of attributes (criteria) had to be taken into account In order to deal with problems of multicriteria decision making (MCDM), like sorting, choice or ranking, a number of methodological changes to the original rough sets theory were necessary. The main change is the substitution of the indiscernibility relation by a dominance relation (crisp or fuzzy), which permits approximation of ordered sets in multicriteria sorting In order to approximate preference relations in multicriteria choice and ranking problems, another change is necessary: substitution of the information 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 MCDM 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 classic 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 MCDM 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 (crisp or fuzzy) and the equivalence of a decision rule preference model with a conjoint measurement model which is neither additive nor transitive.

A New Rough Set Approach to Evaluation of Bankruptcy Risk

We present a new rough set method for evaluation of bankruptcy risk. This approach is based on approximations of a given partition of a set of firms into pre-defined and ordered categories of risk by means of dominance relations, instead of indiscernibility relations. This type of approximations enables us to take into account the ordinal properties of considered evaluation criteria. The new approach maintains the best properties of the original rough set analysis: it analyses only facts hidden in data, without requiring any additional information, and possible inconsistencies are not corrected. Moreover, the results obtained in terms of sorting rules are more understandable for the user than the rules obtained by the original approach, due to the possibility of dealing with ordered domains of criteria instead of non-ordered domains of attributes. The rules based on dominance are also better adapted to sort new actions than the rules based on indiscernibility. One real application illustrates the new approach and shows its advantages with respect to the original rough set analysis.

Decision Rule Approach

In this chapter we present the methodology of Multiple-Criteria Decision Aiding (MCDA) based on preference modelling in terms of “if…, then …” decision rules. The basic assumption of the decision rule approach is that the decision maker (DM) accepts to give preference information in terms of examples of decisions and looks for simple rules justifying her decisions. An important advantage of this approach is the possibility of handling inconsistencies in the preference information, resulting from hesitations of the DM. The proposed methodology is based on the elementary, natural and rational principle of dominance. It says that if action x is at least as good as action y on each criterion from a considered family, then x is also comprehensively at least as good as y. The set of decision rules constituting the preference model is induced from the preference information using a knowledge discovery technique properly adapted in order to handle the dominance principle. The mathematical basis of the decision rule approach to MCDA is the Dominance-based Rough Set Approach (DRSA) developed by the authors. We present some basic applications of this approach, starting from multiple-criteria classification problems, and then going through decision under uncertainty, hierarchical decision making, classification problems with partially missing information, problems with imprecise information modelled by fuzzy sets, until multiple-criteria choice and ranking problems, and some classical problems of operations research. All these applications are illustrated by didactic examples whose aim is to show in an easy way how DRSA can be used in various contexts of MCDA.

Assessing non-additive utility for multicriteria decision aid

Abstract In the framework of Multi-Attribute Utility Theory (MAUT) several methods have been proposed to build a Decision-Makers (DM) utility function representing his/her preferences. Among such methods, the UTA method infers an additive utility function from a set of exemplary decisions using linear programming. However, the UTA method does not guarantee to find a utility function which is coherent with the available information. This drawback is due to the underlying utility model of UTA, viz. the additive one, which does not allow to include additional information such as an interaction among criteria. In this paper we present a methodology for building a non-additive utility function, in the framework of the so called fuzzy integrals, which permits to model preference structures with interaction between criteria. Like in the UTA method, we aim at searching a utility function representing the DMs preferences, but unlike UTA, the functional form is a specific fuzzy integral (Choquet integral). As a result, we obtain weights which can be interpreted as the “importance” of coalitions of criteria, exploiting the potential interaction between criteria, as already proposed by other authors. However, within the same framework, we obtain also the marginal utility functions relative to each one of the considered criteria, that are evaluated on a common scale, as a consequence of the implemented methodology. Finally, we illustrate our approach with an example.

Rough-Set-Based Decision Support

In this chapter, we are concerned with discovering knowledge from data. The aim is to find concise classification patterns that agree with situations that are described by the data. Such patterns are useful for explanation of the data and for the prediction of future situations. They are particularly useful in such decision problems as technical diagnostics, performance evaluation and risk assessment. The situations are described by a set of attributes, which we might also call properties, features, characteristics, etc. Such attributes may be concerned with either the input or output of a situation. These situations may refer to states, examples, etc. Within this chapter, we will refer to them as objects. The goal of the chapter is to present a knowledge discovery paradigm for multi-attribute and multicriteria decision making, which is based upon the concept of rough sets. Rough set theory was introduced by (Pawlak 1982, Pawlak 1991). Since then, it has often proved to be an excellent mathematical tool for the analysis of a vague description of objects. The adjective vague (referring to the quality of information) is concerned with inconsistency or ambiguity. The rough set philosophy is based on the assumption that with every object of the universe U there is associated a certain amount of information (data, knowledge). This information can be expressed by means of a number of attributes. The attributes describe the object. Objects which have the same description are said to be indiscernible (similar) with respect to the available information.

Variable Consistency Model of Dominance-Based Rough Sets Approach

Consideration of preference-orders requires the use of an extended rough set model called Dominance-based Rough Set Approach (DRSA). The rough approximations defined within DRSA are based on consistency in the sense of dominance principle. It requires that objects having not-worse evaluation with respect to a set of considered criteria than a referent object cannot be assigned to a worse class than the referent object. However, some inconsistencies may decrease the cardinality of lower approximations to such an extent that it is impossible to discover strong patterns in the data, particularly when data sets are large. Thus, a relaxation of the strict dominance principle is worthwhile. The relaxation introduced in this paper to the DRSA model admits some inconsistent objects to the lower approximations; the range of this relaxation is controlled by an index called consistency level. The resulting model is called variable-consistency model (VC-DRSA). We concentrate on the new definitions of rough approximations and their properties, and we propose a new syntax of decision rules characterized by a confidence degree not less than the consistency level. The use of VC-DRSA is illustrated by an example of customer satisfaction analysis referring to an airline company.