Computational Intelligence and Inform. ManagementDominance-based rough fuzzy set approach and its application to rule induction
Introduction
It is widely accepted that imprecise and vague information pervades many aspects of our daily lives especially in decision making contexts. Zadeh (1965) established the fuzzy set theory to deal with this kind of uncertainty under such circumstances. The membership function associated with a fuzzy set maps every object of the universe to a specific value in the unit interval [0, 1] extended originally from {0, 1}. The higher the degree of membership, the greater the belongingness of that element to the corresponding set. The primary union and intersection of fuzzy sets are pointwise defined as max and min operators, respectively. Since its introduction, fuzzy set theory has advanced in a variety of ways and in many disciplines (Klir, Yuan, 1995, Zimmermann, 2001). Usefulness and versatility of this theory have amply been demonstrated by successful applications in a variety of domains, such as artificial intelligence, control engineering, operational research, decision theory, just to name a few.
Another excellent extension of the classical set theory is the theory of rough sets initially developed by Pawlak (1982). It serves as a powerful mathematical tool for modeling and processing insufficient and inconsistent information in intelligent systems Greco, Matarazzo, Slowinski, Zanakis, 2011, Pawlak, 1991, Yao, Zhou, 2016. The basic idea of this theory comes from the perception that objects characterized by the same description are indiscernible in view of the available information about them. The equivalence classes or elementary building blocks which are induced from an indiscernibility relation form a partition of the universe of discourse, and constitute the basic granules of knowledge. Lower and upper approximations are constructed by these equivalence classes, from which the knowledge hidden in information systems can be unraveled and expressed in the form of decision rules.
In a decision system, the set of all objects that correspond to the same decision value(s) is called a concept, the knowledge to be approximated in Pawlak’s rough set theory. While in reality, the concept, for example a linguistic category, is always fuzzy rather than crisp due to the uncertainty in classification. In an information system with fuzzy decisions, the grade of membership represents the credibility that the object gets that decision. To address this issue, Dubois, Prade, 1990, Dubois, Prade, 1992 introduced the lower and upper approximations of a fuzzy concept in a Pawlak’s approximation space, generalizing rough sets to fuzzy environment and obtaining the rough fuzzy sets. It is worth mentioning that the discussion of the fuzzy rough set, another combination of fuzzy and rough sets, is beyond the scope of this paper.
Until now, the research on rough fuzzy sets has mainly been concerned with three themes: fundamentals, extensions and applications. Two types of roughness measure for fuzzy sets were presented in Banerjee and Pal (1996) and Huynh and Nakamori (2005) with and without parameters. Sarkar (2002) defined the rough-fuzzy membership function for a fuzzy output class in pattern classification tasks. To improve the performance of related algorithms, Cheng (2011) put forward two incremental methods for fast computing the rough fuzzy approximations. Some extensions are needed to make the rough fuzzy sets more widely used in practical applications. First, the equivalence relation is generalized to an arbitrary binary relation. Wu, Leung, and Zhang (2006) proposed a general framework for the study of rough fuzzy sets with respect to (w.r.t.) a binary relation by using both constructive and axiomatic approaches. Li and Zhang (2008) investigated the rough fuzzy sets by means of a binary relation between two universes of discourse. Second, the target fuzzy set is generalized to its extensions. For example, approximations of interval-valued fuzzy sets, bipolar-valued fuzzy sets and interval type-2 fuzzy sets were discussed in Gong, Sun, and Chen (2008), Han, Shi, and Chen (2015) and Zhang (2012). Third, the crisp approximation space can be generalized as well. In a probabilistic approximation space, for instance, Sun, Ma, and Zhao (2014) examined the probabilistic rough fuzzy set by defining the conditional probability of a fuzzy event. The use of rough fuzzy sets has found their way into numerous fields of application, such as spatial data mining (Bai et al., 2014), image compression (Petrosino & Ferone, 2009), text information retrieval (Singh & Dey, 2005).
On the other hand, the Pawlak’s rough set theory is not able to discover and process inconsistencies coming from consideration of criteria, that is, attributes with preference-ordered domains (scales), such as student performance evaluation, hotel rating assessment, and investment risk analysis. To alleviate this problem, Greco, Matarazzo, Slowinski, 2001, Greco, Matarazzo, Slowinski, 2002a proposed an extension of the rough set theory, which is called the dominance-based rough set approach (DRSA) to take into account the ordering properties of criteria. This innovation is mainly based on the substitution of the equivalence relation by a dominance relation, which permits approximations of ordered sets within multi-criteria decision making and multi-criteria sorting problems (Błaszczyński, Greco, and Słowiński, 2007; Chakhar, Ishizaka, Labib, and Saad, 2016; Dembczyński, Greco, and Słowiński, 2009; Fan, Liu, and Tzeng, 2007; Fan, Liau, and Liu, 2011b; Greco, Matarazzo, Slowinski, 2002b, Greco, Matarazzo, Slowinski, 2010). In DRSA, where condition attributes are criteria and decision classes are preference-ordered, the knowledge used for approximation is a collection of upward and downward unions of decision classes and the dominance classes are sets of objects defined by using a dominance relation (Du, Hu, 2014, Du, Hu, 2016a, Du, Hu, 2016b, Li, Liao, Zhao, 2009, Susmaga, Słowiński, 2015).
Pioneering work on the employment of DRSA for rough approximations of fuzzy sets was also done by Greco et al., a parallel idea to the integration of rough sets and fuzzy sets. Based on the ordinal properties of fuzzy membership degrees only, the concepts of fuzzy lower and upper approximations were proposed, creating a base for induction of fuzzy decision rules having syntax and semantics of gradual rules (Greco, Inuiguchi, & Slowinski, 2006). Moreover, it was demonstrated in (Greco, Matarazzo, & Słowiński, 2008) that the classical rough approximation is a particular case of the rough approximation within a fuzzy information system. Nevertheless, the issue of attribute reduction has not been considered yet in the framework of dominance-based rough fuzzy set approach (DRFSA) which introduces dominance-based rough approximations of cumulated fuzzy decisions. One objective of this paper is to investigate this problem with minor modifications of the upward and downward cumulated fuzzy sets. It is noteworthy that according to Greco et al., each ordered target concept should be a fuzzy set of the universe. However, it is also suggested that only one fuzzy set describing the confidence of each object being assigned to the best evaluation is sufficient enough. For example, Huang (2011) analyzed attribute reduction and rule extraction in ordered information systems with fuzzy decisions. Later on, the result was further generalized to the situations of intuitionistic fuzzy decisions and interval-valued intuitionistic fuzzy decisions (Huang, Li, Wei, 2012, Huang, Wei, Li, Zhuang, 2013). Along this line of research, recently the condition criteria were abstracted to the lattice-valued case (Zhang, Wei, & Xu, 2016), in a natural way, equipped with an order structure.
From the logical point of view, a decision system may be seen as a series of decision rules. The lower and upper approximations support certain and possible decision rules respectively, each of which gives, in general, a synthetic representation of knowledge contained in the given decision system. As pointed out by Greco et al. (2001), the rules derived by DRSA are more understandable and more applicable for final users because of the more general syntax of the rules. Four types of decision rules induced in the context of DRFSA correspond to dominance-based rough fuzzy approximations which provide a generalized description of objects. In addition, the way to simplify these decision rules is also discussed by utilizing the reduct with respect to the type of decision rules. This is another contribution of our work.
The remainder of this paper is built up as follows. We give some preliminaries on dominance-based rough sets in Section 2. In Section 3, lower and upper dominance-based rough fuzzy approximations of cumulated fuzzy sets are proposed and their properties examined. In Section 4, attribute reduction in ordered fuzzy decision systems is in discussion by using the discernibility matrix method and the reduct construction technique, while the rule acquisition method is studied in Section 5. Moreover, in Section 6, the feasibility of the proposed methods is demonstrated through a case study of bankruptcy risk evaluation in Greece. Section 7 concludes with a summary of the present work and a suggestion for further research.
Section snippets
Preliminaries
In this section, we recall several basic concepts including dominance cones, lower and upper approximation operators in dominance-based rough set approach.
Definition 2.1 An information system is a quadruple where U is a finite nonempty set of objects, AT is a finite nonempty set of attributes, and Va is the domain of attribute a, and f: U × AT → V is a total function such that f(x, a) ∈ Va for every a ∈ AT, x ∈ U, called an information function.(Pawlak, 1982)
In practical decision-making analysis,
Dominance-based rough fuzzy sets
In this section, the dominance-based rough fuzzy approximation operators are introduced in ordered fuzzy decision systems, and their properties are also investigated.
Attribute reduction in ordered fuzzy decision systems
A reduct is a minimal subset of all attributes which has the same power as the full set of condition attributes in the given information system. In this section, the concepts of upward/downward lower and upper reducts are proposed in OFDSs, and approaches to calculate these reducts are presented.
Definition 4.1 Let be an OFDS and A ⊆ C. If ∀x ∈ U, then A is referred to as an l- (u-) consistent set relative to of S. If A is an l- (u-)
Rule induction in ordered fuzzy decision systems
Induction of decision rules is an important application of the traditional rough set theory and its extensions. In this section, this problem is discussed within the DRFSA framework.
The decision rules are derived from the lower and upper dominance-based rough fuzzy sets with the syntax as follows:
- (Type 1)
If then with credibility at least .
- (Type 2)
If
Case study: application in bankruptcy risk analysis
The data collected in Table 6 are adapted from the experience of a Greek industrial development bank that finances industrial and commercial firms. The universe is a sample of 39 firms, the condition criteria is a set of 12 criteria used to assess the firms – all are to be maximized according to preference, and three pre-defined classes Cl1, Cl2, Cl3 represent unacceptable, uncertain and acceptable firms, respectively. The detailed information about the criteria
Conclusions
The hybrid notion of rough fuzzy sets comes from the combination of two models of uncertainty (coarseness for rough sets and vagueness for fuzzy sets). The DRSA is considered as an important branch of rough set theory because it takes users’ preferences into consideration as many real life applications need. In this paper, the dominance-based rough approximations of upward and downward cumulated fuzzy sets are considered in ordered fuzzy decision systems. Then, lower and upper reducts relative
Acknowledgments
The authors would like to thank the three anonymous reviewers for their insightful comments and constructive suggestions that helped improve this paper. This research was supported by the National Natural Science Foundation of China (Grant nos. 11571010, 61179038) and the Fundamental Research Funds for the Central Universities (Grant no. 2015201020201).
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