Salvatore Greco

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.

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTAGMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives ARÂ [subset, double equals]Â A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTAGMS method is intended to be used interactively, with an increasing subset AR and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTAGMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTAGMS software. Some extensions of the method are also presented.

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.

Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method

We present a method called Generalized Regression with Intensities of Preference (GRIP) for ranking a finite set of actions evaluated on multiple criteria. GRIP builds a set of additive value functions compatible with preference information composed of a partial preorder and required intensities of preference on a subset of actions, called reference actions. It constructs not only the preference relation in the considered set of actions, but it also gives information about intensities of preference for pairs of actions from this set for a given decision maker (DM). Distinguishing necessary and possible consequences of preference information on the considered set of actions, GRIP answers questions of robustness analysis. The proposed methodology can be seen as an extension of the UTA method based on ordinal regression. GRIP can also be compared to the AHP method, which requires pairwise comparison of all actions and criteria, and yields a priority ranking of actions. As for the preference information being used, GRIP can be compared, moreover, to the MACBETH method which also takes into account a preference order of actions and intensity of preference for pairs of actions. The preference information used in GRIP does not need, however, to be complete: the DM is asked to provide comparisons of only those pairs of reference actions on particular criteria for which his/her judgment is sufficiently certain. This is an important advantage comparing to methods which, instead, require comparison of all possible pairs of actions on all the considered criteria. Moreover, GRIP works with a set of general additive value functions compatible with the preference information, while other methods use a single and less general value function, such as the weighted-sum.

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.

Can bayesian confirmation measures be useful for rough set decision rules

Bayesian confirmation theory considers a variety of non-equivalent confirmation measures which say in what degree a piece of evidence confirms a hypothesis. In this paper, we apply some well-known confirmation measures within the rough set approach to discovering relationships in data in terms of decision rules. Moreover, we discuss some interesting properties of these confirmation measures and we propose a new property of monotonicity that is particularly relevant within rough set approach. The main result of this paper states that only two from among confirmation measures considered in the literature have the desirable properties from the viewpoint of the rough set approach. Moreover, we clarify relationships between logical implications and decision rules, and we compare the confirmation measures to several related measures, like independence (dependence) of logical formulas, interestingness measures in data mining and Bayesian solutions of ravens paradox.