Rough sets theory for multicriteria decision analysis

Abstract

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.

Introduction

The rough sets theory introduced by Pawlak, 1982, Pawlak, 1991 has often proved to be an excellent mathematical tool for the analysis of a vague description of objects (called actions in decision problems). The adjective vague, referring to the quality of information, means inconsistency or ambiguity which follows from information granulation. The rough sets philosophy is based on the assumption that with every object of the universe there is associated a certain amount of information (data, knowledge), expressed by means of some attributes used for object description. Objects having the same description are indiscernible (similar) with respect to the available information. The indiscernibility relation thus generated constitutes a mathematical basis of the rough sets theory; it induces a partition of the universe into blocks of indiscernible objects, called elementary sets, that can be used to build knowledge about a real or abstract world. The use of the indiscernibility relation results in information granulation.

Any subset X of the universe may be expressed in terms of these blocks either precisely (as a union of elementary sets) or approximately only. In the latter case, the subset X may be characterized by two ordinary sets, called lower and upper approximations. A rough set is defined by means of these two approximations, which coincide in the case of an ordinary set. The lower approximation of X is composed of all the elementary sets included in X (whose elements, therefore, certainly belong to X), while the upper approximation of X consists of all the elementary sets which have a non-empty intersection with X (whose elements, therefore, may belong to X). Obviously, the difference between the upper and lower approximation constitutes the boundary region of the rough set, whose elements cannot be characterized with certainty as belonging or not to X, using the available information. The information about objects from the boundary region is, therefore, inconsistent or ambiguous. The cardinality of the boundary region states, moreover, to what extent it is possible to express X in exact terms, on the basis of the available information. For this reason, this cardinality may be used as a measure of vagueness of the information about X.

The rough sets theory, dealing with representation and processing of vague information, presents a series of intersections and complements with respect to many other theories and mathematical techniques handling imperfect information, like probability theory, evidence theory of Dempster–Shafer, fuzzy sets theory, discriminant analysis and mereology (see Dubois and Prade, 1990, Dubois and Prade, 1992, Krusinska et al., 1992, Pawlak, 1985a, Pawlak, 1985b, Polkowski and Skowron, 1994, Skowron and Grzymala-Busse, 1994, Slowinski, 1995).

Some important characteristics of the rough set approach make of this a particularly interesting tool in a number of problems and concrete applications. With respect to the input information, it is possible to deal with both quantitative and qualitative data, and inconsistencies need not to be removed prior to the analysis. With reference to the output information, it is possible to acquire a posteriori information regarding the relevance of particular attributes and their subsets to the quality of approximation considered in the problem at hand, without any additional inter-attribute preference information. Moreover, the final result in the form of “if..., then...” decision rules, using the most relevant attributes, is easy to interpret.

Several attempts have already been made to use the rough sets theory to decision support (Pawlak and Slowinski, 1994; Slowinski, 1993b). The original rough set approach is not able, however, to deal with preference-ordered attribute domains and decision classes. Solving this problem was crucial for application of the rough set approach to multicriteria decision analysis (MCDA). Why this application seems so important? The answer is connected with the nature of the input preferential information available in MCDA and of the output of the analysis. As to the input, the rough set approach requires a set of examples which is also convenient for acquisition of preferential information from decision makers (DMs). Very often in MCDA, this information has to be given in terms of preference model parameters, like importance weights, substitution ratios and various thresholds. Giving such information requires a great cognitive effort of the DM. It is generally acknowledged that people prefer to make exemplary decisions than to explain them in terms of specific parameters. For this reason, the idea of inferring preference models from exemplary decisions provided by the DM is very attractive. Furthermore, the exemplary decisions may be inconsistent because of limited discriminatory power of criteria and because of hesitation of the DM (see, e.g., Roy, 1989). These inconsistencies cannot be considered as a simple error or noise. They can convey important information that should be taken into account in the construction of the DMs preference model. The rough set approach is intended to deal with inconsistency and this is another argument for its application to MCDA. Finally, the output of the analysis, i.e. the model of preferences in terms of decision rules seems very convenient for decision support because it is transparent and speaks the same language as the DM.

Let us explain shortly why the original rough set approach is not able to deal with inconsistencies coming from consideration of criteria, i.e. attributes with preference-ordered domains (scales), like product quality, market share, debt ratio. Consider, for example, two firms, A and B, evaluated for assessment of bankruptcy risk by a set of criteria including the “debt ratio” (total debt/total assets). If firm A has a low value while firm B has a high value of the debt ratio, and evaluations of these firms on other attributes are equal, then, from bankruptcy risk point of view, firm A dominates firm B. Suppose, however, that firm A has been assigned by a DM to a class of higher risk than firm B. This is obviously inconsistent with the dominance principle. Within the original rough set approach, the two firms will be considered as just discernible and no inconsistency will be stated.

For this reason, Greco et al., 1995, Greco et al., 1997a, Greco et al., 1998, Greco et al., 1999c, Greco et al., 1999l have proposed an extension of the rough sets theory that is able to deal with inconsistencies typical to exemplary decisions in MCDA problems. This innovation is mainly based on substitution of the indiscernibility relation by a dominance relation in the rough approximation of decision classes. An important consequence of this fact is a possibility of inferring from exemplary decisions the preference model in terms of decision rules being logical statements of the type “if..., then...” The separation of certain and doubtful knowledge about the DM's preferences is done by distinction of different kinds of decision rules, depending whether they are induced from lower approximations of decision classes or from the boundaries of these classes composed of inconsistent examples that do not observe the dominance principle. Such preference model is more general than the classical functional models considered within Multi-Attribute Utility Theory (MAUT) or relational models considered, for example, in outranking methods.

The paper is organized as follows. In Section 2, a general view of the rough set approach is given. In Section 3, two extensions of the classical rough set approach based on generalizations of the basic concept of indiscernibility are presented: the first is the similarity relation being only reflexive and not necessarily symmetric and transitive; the second is a specific indiscernibility relation handling missing values in objects' description – it is transitive but neither reflexive nor symmetric. In Section 4, we introduce a distinction between classification and sorting problems. The sorting problem involves preference-orders on domains of considered attributes (criteria) and among decision classes. To deal with multicriteria sorting problems rough set approximation based on dominance is proposed in this section. Furthermore, in order to handle missing values in multicriteria sorting problems a specific dominance relation is proposed. In Section 5, choice and ranking problems are considered. They are based on pairwise comparisons of objects, so the rough set approach concerns in this case approximation of a preference binary relation by specific dominance relations. These dominance relations can be multigraded, when the preferences with respect to considered criteria are cardinal, or without any degree of preference, when the preferences with respect to criteria are ordinal. Section 6 presents some results about equivalence between preference models of conjoint measurement and preference models expressed in terms of decision rules induced from rough approximations. Section 7 groups conclusions.