An Interval-Valued Intuitionistic Fuzzy Rough Set Model

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak's rough approximations have been made in the literature over these years. This paper proposes a general framework for the study of intuitionistic fuzzy rough approximation operators based on intuitionistic fuzzy triangle norm. In the constructive approach, a pair of lower and upper induced from intuitionistic fuzzy relation are defined. Basic properties of intuitionsitic fuzzy rough approximation operators are then examined. By introducing intuitionistic fuzzy residual implication, Further properties of intuitionistic fuzzy rough approximation operators are then investigated. we propose that classical rough set and fuzzy rough are special types of intuitionistic fuzzy rough set based on intuitionistic fuzzy triangle norm.

Axiomatic characterizations of approximation operators are important in the study of rough set theory. In this paper, axiomatic characterizations of relation-based intuitionistic fuzzy rough approximation operators determined by an intuitionistic fuzzy implication operator \(\mathcal{I}\) are investigated. We present a set of axioms of lower/upper \(\mathcal{I}\)-intuitionistic fuzzy set-theoretic operator which is necessary and sufficient for the existence of an intuitionistic fuzzy relation producing the same operator. We show that the lower and upper \(\mathcal{I}\)-intuitionistic fuzzy rough approximation operators generated by an arbitrary intuitionistic fuzzy relation can be described by single axioms. Moreover, the \(\mathcal{I}\)-intuitionistic fuzzy rough approximation operators generated by reflexive and \(\mathcal{T}\)-transitive intuitionistic fuzzy relations can also be characterized by single axioms.

In this paper, dual intuitionistic fuzzy rough approximation operators determined by an intuitionistic fuzzy implication operator ${\mathcal I}$ in infinite universes of discourse are investigated. Lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space in infinite universes of discourse are first introduced. Properties of ${\mathcal I}$ -intuitionistic fuzzy rough approximation operators are then examined. Relationships between special types of intuitionistic fuzzy relations and properties of ${\mathcal I}$ -intuitionistic fuzzy rough approximation operators are further established.

Interval-valued intuitionistic fuzzy soft sets and their properties

Representations of L-fuzzy rough approximation operators

The variable precision rough set (VPRS) model generalizes the Pawlak rough set model with variable parameters. The Bayesian rough set (BRS) model improves the VPRS model with non-parametric modification by using the prior probability as a reference. This paper presents two research results related to rough set model and rough vague sets. One is a modification to the Bayesian rough set model and the other is a modification to rough vague sets. First, the Bayesian rough set model is analyzed and discussed. Second, a modification to this model is proposed and verified. Finally, a modification to rough vague sets is presented and its related properties are discussed..

The paper aims at the problem of multi-targets threat degree being hard to be evaluated accurately in complex air defense battlefield environments. Combined with multi-sensors information fusion and interval-valued intuitionistic fuzzy sets (IVIFS) theories, the target priority determination is studied. The score and accuracy functions of IVIFS are improved with thinking about the hesitating information in order to increase the rationality. Then, the influence factors of target priority and the nonlinear relationship between the influence factors and target priority are analyzed. Next, the algorithms for calculating the factor weights and sensor weights are given. Based on the theory of IVIFS and technique for order preference by similarity to an ideal solution (TOPSIS), two methods of target priority determination based on the IVIFS and TOPSIS are proposed. At last, an application example verifies the effectiveness and flexibility of the proposed algorithms.

Rough set theory for the interval-valued fuzzy information systems

Precision and grade are two important quantitative indexes. This paper is to combine precision and grade, and explore new extended rough set model. Based on logical OR requirement of precision and grade, this paper proposes logical OR model of variable precision upper approximation operator and grade lower approximation operator. In the new model, basic structure and properties are obtained by transformation formulas between variable precision approximations and grade approximations, furthermore regular algorithm and structural algorithm are proposed and analyzed. Finally a practical example is given to illustrate the model and the algorithms. The new model has extended variable precision rough set model, graded rough set model and classical rough set model, and so these models have obtained corresponding properties of approximation operators.

This paper presents a general framework for the study of covering-based fuzzy approximation operators in which a fuzzy set can be approximated by some elements in a crisp or a fuzzy covering of the universe of discourse. Two types of approximation operators, crisp-covering-based rough fuzzy approximation operators and fuzzy-covering-based fuzzy rough approximation operators, are defined, their properties are examined in detail. Finally, the comparison of these new approximation operators is done, a sufficient and necessary condition is given under which some operators are equivalent, and approximation operator characterization of fuzzy partitions of the universe is obtained.KeywordsCrisp coveringsfuzzy coveringsfuzzy-covering-based fuzzy rough approximation operatorsfuzzy partitionsfuzzy sets

Hesitant fuzzy set is a generalization of the classical fuzzy set by returning a family of the membership degrees for each object in the universe. Since how to use the rough set model to solve fuzzy problems plays a crucial role in the development of the rough set theory, the fusion of hesitant fuzzy set and rough set is then firstly explored in this paper. Both constructive and axiomatic approaches are considered for this study. In constructive approach, the model of the hesitant fuzzy rough set is presented to approximate a hesitant fuzzy target through a hesitant fuzzy relation. In axiomatic approach, an operators-oriented characterization of the hesitant fuzzy rough set is presented, that is, hesitant fuzzy rough approximation operators are defined by axioms and then, different axiom sets of lower and upper hesitant fuzzy set-theoretic operators guarantee the existence of different types of hesitant fuzzy relations producing the same operators.

Rough set theory and vague set theory are powerful tools for managing uncertain, incomplete and imprecise information. This paper extends the rough vague set model based on equivalence relations and the rough fuzzy set model based on fuzzy relations to vague sets. We mainly focus on the lower and upper approximation operators of vague sets based on vague relations, and investigate the basic properties of approximation operators on vague sets. Specially, we give some essential characterizations of the lower and upper approximation operators generated by reflexive, symmetric, and transitive vague relations. Finally, we structure a parameterized roughness measure of vague sets and similarity measure methods between two rough vague sets, and obtain some properties of the roughness measure and similarity measures. We also give some valuable counterexamples and point out some false properties of the roughness measure in the paper of Wang et al.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfy many interesting properties. Many types of generalized rough set models have been developed in the literature. This paper presents a general framework for the study of rough intuitionistic fuzzy sets in which intuitionistic fuzzy sets are approximated with respect to a crisp approximation space. A pair of lower and upper rough intuitionistic fuzzy approximation operators are first defined. Properties of rough intuitionistic fuzzy approximations are then examined. The relationships between special types of crisp binary relations and properties of rough intuitionistic fuzzy approximation operators are further established.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.