Overlap function-based variable precision fuzzy rough sets with application to multi-attribute decision-making

The fuzzy rough set model and interval-valued fuzzy rough set model have been introduced to handle databases with real values and interval values, respectively. Variable precision rough set was advanced by Ziarko to overcome the shortcomings of misclassification and/or perturbation in Pawlak rough sets. By combining fuzzy rough set and variable precision rough set, a variety of fuzzy variable precision rough sets were studied, which cannot only handle numerical data, but are also less sensitive to misclassification. However, fuzzy variable precision rough sets cannot effectively handle interval-valued data-sets. Research into interval-valued fuzzy rough sets for interval-valued fuzzy data-sets has commenced; however, variable precision problems have not been considered in interval-valued fuzzy rough sets and generalized interval-valued fuzzy rough sets based on fuzzy logical operators nor have interval-valued fuzzy sets been considered in variable precision rough sets and fuzzy variable precision rough sets. These current models are incapable of wide application, especially on misclassification and/or perturbation and on interval-valued fuzzy data-sets. In this paper, these models are generalized to a more integrative approach that not only considers interval-valued fuzzy sets, but also variable precision. First, we review generalized interval-valued fuzzy rough sets based on two fuzzy logical operators: interval-valued fuzzy triangular norms and interval-valued fuzzy residual implicators. Second, we propose generalized interval-valued fuzzy variable precision rough sets based on the above two fuzzy logical operators. Finally, we confirm that some existing models, including rough sets, fuzzy variable precision rough sets, interval-valued fuzzy rough sets, generalized fuzzy rough sets and generalized interval-valued fuzzy variable precision rough sets based on fuzzy logical operators, are special cases of the proposed models.

One limitation of the fuzzy rough sets is its sensitivity to the perturbation of original numerical data. In this paper we construct a model of fuzzy variable precision rough sets (FVPRS) by combining the fuzzy rough sets and variable precision rough sets, which is non-sensitive to the perturbation of the original numerical data. First, the fuzzy lower and upper approximations of FVPRS model are defined, and their properties are described. Second, the concepts of attributes reduction of FVPRS model, such as attributes reduct, core and positive region, etc, are defined. Third, a discernibility matrix is adopted to develop an algorithm to obtain all the attributes reduction of FVPRS. By the strict mathematical reasoning, we prove that the results obtained by the algorithm based on the discernibility matrix are the exact attributes reducts of FVPRS. Finally, the experimental results demonstrate that the model of FVPRS is feasible and effective in the real problems.

Rough set has been shown to be a valuable approach to mine rules from a remote monitoring manufacturing process. In this research, an application of the fuzzy set theory with the fuzzy variable precision rough set approach for mining the causal relationship rules from the database of a remote monitoring manufacturing process is presented. The membership function in the fuzzy set theory is used to transfer the data entries into fuzzy sets, and the fuzzy variable precision rough set approach is applied to extract rules from the fuzzy sets. It is found that the induced rules are identical to the practical knowledge and fault diagnosis thinking of human operators. The induced rules are then compared with the rules induced by the original rough set approach. The comparison shows that the rules induced by the fuzzy rough set are expressed in linguistic forms, and are evaluated by plausibility and future effectiveness measures. The fuzzy rough set approach, being less sensitive to noisy data, induces better rules than the original rough set approach.

In order to effectively handle the real-valued data sets in practice, it is valuable from theoretical and practical aspects to combine fuzzy rough set (FRS) and variable precision rough set (VPRS) so that a powerful tool can be developed. That is, the model of fuzzy variable precision rough set (FVPRS), which not only can handle numerical data but also is less sensitive to misclassification and perturbation,In this paper, we propose a new variable precision rough fuzzy set by introducing the variable precision parameter to generalized rough fuzzy set, i.e., the variable precision rough fuzzy set based on general relation. We define the lower and upper approximations of any fuzzy set with variable precision parameter by constructive approach. Then we establish the relation between the model proposed in this paper and the existing models. Finally, we present the properties of the proposed model in detail. The results enrich the theory and also extend the application fields for rough set theory.

In order to effectively handle the real-valued data sets in practice, it is valuable from theoretical and practical aspects to combine fuzzy rough set and variable precision rough set so that a powerful tool can be developed. That is, the model of fuzzy variable precision rough set, which not only can handle numerical data but also is less sensitive to misclassification and perturbation,In this paper, we propose a new variable precision rough fuzzy set by introducing the variable precision parameter to generalized rough fuzzy set, i.e., the variable precision rough fuzzy set based on general relation. We, respectively, define the variable precision rough lower and upper approximations of any fuzzy set and it level set with variable precision parameter by constructive approach. Also, we present the properties of the proposed model in detail. Meanwhile, we establish the relationship between the variable precision rough approximation of a fuzzy set and the rough approximation of the level set for a fuzzy set. Furthermore, we give a new approach to uncertainty measure for variable precision rough fuzzy set established in this paper in order to overcome the limitations of the traditional methods. Finally, some numerical example are used to illuminate the validity of the conclusions given in this paper.

Granular variable precision fuzzy rough sets with general fuzzy relations

There are many ways to generalize rough sets. Researching fuzzy rough sets under the framework of residuated lattices is one of the generalizations that broadens the truth values to lattices. This paper studies fuzzy variable precision approximation operators based on residuated lattices and presents the concept of -fuzzy variable precision rough sets. After the presentation of this concept, we investigate the corresponding properties and some special classes of -fuzzy variable precision rough sets based on different -relations. Finally, we study the topological structures of -fuzzy variable precision rough sets based on -lower sets and -upper sets, and conclude that the family of all -lower exact -sets and the family of all -upper exact -sets are -topologies, for each .

New results on granular variable precision fuzzy rough sets based on fuzzy (co)implications

Zhan and Jiang defined covering-based compact and loose variable precision fuzzy rough set models. Soon after, they proposed a reflexive fuzzy β-neighborhood operator and defined a covering-based generalized variable precision fuzzy rough set. Based on them, in this paper, we use the reflexive fuzzy β-neighborhood operator to establish two covering-based generalized variable precision fuzzy rough set models, which are called covering-based generalized compact and loose variable precision fuzzy rough set models, respectively. Then, we investigate the important properties of the two rough set models and their relationship to the original models. Finally, we apply the covering-based generalized compact variable precision fuzzy rough set model to decision-making problems. A simple example is given to verify the validity of the model and compare the results with other models.

Generalized fuzzy neighborhood system-based multigranulation variable precision fuzzy rough sets with double TOPSIS method to MADM

The model of covering-based fuzzy rough sets (CFRSs) can be regarded as a hybrid one by combining covering-based rough sets with fuzzy sets. In this paper, based on fuzzy neighborhoods, we propose two types of covering-based variable precision fuzzy rough sets (CVPFRSs) via fuzzy logical operators, i.e., type-I CVPFRSs and type-II CVPFRSs. Then, several basic properties of the two types of CVPFRSs are discussed. In addition, by virtue of the idea of PROMETHEE II methods, we construct a novel method to multi-attribute decision-making (MADM) in the context of medical diagnosis based on the proposed rough approximation operators. Finally, a test example for illustrating the proposed method is given. Meanwhile, a comparative analysis and an experimental evaluation are further discussed to interpret and evaluate the effectiveness and superiority of the proposed method. The proposed rough set model not only extends the theory of CFRSs, but also provides a new perspective for MADM with fuzzy evaluation information.

This paper presents an approach which combines unified variable precision fuzzy rough set model together with the concept of fuzzy linguistic labels. A real world application of the standard fuzzy rough sets can be problematic, especially in the case of large universes and noisy data. Due to relaxation of strict inclusion requirement in determining approximations of sets, a more tolerant variable precision fuzzy rough set model is better suited to be useful in analysis of this kind of data. Furthermore, a crucial issue at the initial stage of the fuzzy rough set approach consists in generating a fuzzy partition of a universe, with respect to condition and decision attributes. It requires comparing of elements by using a suitable fuzzy similarity relation. We simplify this process by applying the concept of fuzzy linguistic labels for determining the family of fuzzy similarity classes. This is done by performing a comparison of elements of the universe to a subset of representative elements which are described with the help of dominating linguistic values of attributes. The notions of the variable precision fuzzy rough set model, which is expressed in a unified parameterized form, can be used to determine the quality of the considered information system by evaluating its consistency, and to obtain a system of fuzzy decision rules.

This paper proposes an approach to representation and analysis of information systems with fuzzy attributes, which combines the variable precision fuzzy rough set (VPFRS) model with the fuzzy flow graph method. An idea of parameterized approximation of crisp and fuzzy sets is presented. A single Ɛ -approximation, which is based on the notion of fuzzy rough inclusion function, can be used to express the crisp approximations in the rough set and variable precision rough set (VPRS) model. A unified form of the Ɛ -approximation is particularly important for defining a consistent VPFRS model. The introduced fuzzy flow graph method enables alternative description of decision tables with fuzzy attributes. The generalized VPFRS model and fuzzy flow graphs, taken together, can be applied to determining a system of fuzzy decision rules from process data.

Traditional rough sets only could handle the datasets with discrete attributes, and have difficulty in handling real-valued attributes. The fuzzy rough set model which could deal with real-valued datasets has been introduced. However, fuzzy rough sets are sensitive to misclassification and perturbation. The variable precision fuzzy rough set model was introduced to handle datasets with misclassification and perturbation, but it could not effectively handle highly uncertain data. Interval type-2 fuzzy rough set model is a powerful tool to handle highly uncertain data. However, interval type-2 fuzzy rough set model is sensitive to misclassification. In this paper, the concept of variable precision interval type-2 fuzzy rough sets (VPI2FRS) by combining variable precision fuzzy rough sets and interval type-2 fuzzy rough sets is introduced. Furthermore, a new attribute reduction approach within VPI2FRS framework is developed. In the end, we by experiments demonstrate the feasibility and effectiveness of the proposed reduction algorithm.

Axiomatization is a lively research direction in fuzzy rough set theory. Fuzzy variable precision rough set (FVPRS) incorporates fault-tolerant factors to fuzzy rough set, so its axiomatic description becomes more complicated and difficult to realize. In this paper, we present an axiomatic approach to FVPRSs based on residuated lattice (L-fuzzy variable precision rough set (LFVPRS)). First, a pair of mappings with three axioms is utilized to characterize the upper (resp., lower) approximation operator of LFVPRS. This is distinct from the characterization on upper (resp., lower) approximation operator of fuzzy rough set, which consists of one mapping with two axioms. Second, utilizing the notion of correlation degree (resp., subset degree) of fuzzy sets, three characteristic axioms are grouped into a single axiom. At last, various special LFVPRS generated by reflexive, symmetric and transitive L-fuzzy relation and their composition are also characterized by axiomatic set and single axiom, respectively.