Covering-based Rough Sets Research Articles

Neighborhood-related q-rung orthopair fuzzy covering-based rough set models and their applications for multi-attribute decision making

Neighborhood-related q-rung orthopair fuzzy covering-based rough set models and their applications for multi-attribute decision making

The novel WASPAS method for roughness of bipolar fuzzy sets based bipolar fuzzy covering

The research employs Covering-based Rough Set and Fuzzy Set theories to handle uncertainty in data analysis. However, when dealing with data uncertainty and bipolarity in various scenarios, the Bipolar Fuzzy Set (BFS) theory proves advantageous by simultaneously managing positive and negative information. Other sets, such as traditional fuzzy sets, often fail to capture this duality, leading to less comprehensive data analysis. This study pioneers a new methodology called the Theory of Roughness of Bipolar Fuzzy Sets, integrating Fuzzy Covering, Monotone Fuzzy Covering, and Bipolar Fuzzy Covering to propose an innovative decision-making approach. This novel concept undergoes comprehensive structural analysis. By incorporating the bipolar fuzzy covering based bipolar fuzzy rough set ( BFCBFRS ) model into conventional decision-making methods like the WASPAS technique, the research introduces a fresh perspective to address Multi-Criteria Decision-Making ( MCDM ) challenges. The efficacy of this extended method is evaluated by applying it to agricultural diagnosis, demonstrating its superiority over existing approaches through comparative analysis.

Intuitionistic Fuzzy Granular Matrix: Novel Calculation Approaches for Intuitionistic Fuzzy Covering-Based Rough Sets

Intuitionistic fuzzy (IF) β-minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF β-covering approximation space (i.e., an IF environment) with a high cardinality, it would be tedious and complicated to use IF set representations to calculate them. Therefore, it is necessary to find a quick method to obtain them. In this paper, we present the notion of IF β-maximal description based on the definition of IF β-minimal description, along with the concepts of IF granular matrix and IF reduction. Moreover, we propose matrix calculation methods for IF covering-based rough sets, such as IF β-minimal descriptions, IF β-maximal descriptions, and IF reductions. Firstly, the notion of an IF granular matrix is presented, which is used to calculate IF β-minimal description. Secondly, inspired by IF β-minimal description, we give the notion of IF β-maximal description. Furthermore, the matrix representations of IF β-maximal descriptions are presented. Next, two types of reductions for IF β-covering approximation spaces via IF β-minimal and fuzzy β-minimal descriptions are presented, along with their matrix representations. Finally, the new calculation methods are compared with corresponding set representations by carrying out several experiments.

Covering-Based Intuitionistic Hesitant Fuzzy Rough Set Models and Their Application to Decision-Making Problems

In this paper, we present four categories of covering-based intuitionistic hesitant fuzzy rough set (CIHFRS) models using intuitionistic hesitant fuzzy β-neighborhoods (IHF β-neighborhoods) and intuitionistic hesitant fuzzy complementary β-neighborhoods (IHFC β-neighborhoods. Through theoretical analysis of covering-based IHFRS models, we propose the intuitionistic hesitant fuzzy TOPSIS (IHF-TOPSIS) technique for order of preference by similarity to an ideal solution, addressing multicriteria decision-making (MCDM) challenges concerning the assessment of IHF data. A compelling example aptly showcases the suggested approach. Furthermore, we address MCDM problems regarding the assessment of IHF information based on CIHFRS models. Through comparison and analysis, it is evident that addressing MCDM problems by assessing IHF data using CIHFRS models proves more effective than utilizing intuitionistic fuzzy data with CIFRS models or hesitant fuzzy information with CHFRS models. IHFS emerges as a unique and superior tool for addressing real-world challenges. Additionally, covering-based rough sets (CRSs) have been successfully applied to decision problems due to their robust capability in handling unclear data. In this study, by combining CRSs with IHFS, four classes of CIFRS versions are established using IHF β-neighborhoods and IHFC β-neighborhoods. A corresponding approximation axiomatic system is developed for each. The roughness and precision degrees of CBIHFRS models are specifically talked about. The relationship among these four types of IHFRS versions and existing related versions is presented based on theoretical investigations. A method for MCDM problems through IHF information, namely, IHF-TOPSIS, is introduced to further demonstrate its effectiveness and applicability. By conducting a comparative study, the effectiveness of the suggested approach is evaluated.

Retraction Note: Covering-based rough set classification system

Retraction Note: Covering-based rough set classification system

Three-way decisions with dual hesitant fuzzy covering-based rough set and their applications in medical diagnosis

As an extension of partition, fuzzy β-covering can provide a more realistic and accurate description of incomplete information. In this paper, we mainly integrate the idea of fuzzy β-covering with dual hesitant fuzzy (DHF) information and construct some novel three-way decision (3WD) models with DHF covering-based rough set. Firstly, we propose the notions of DHF β-covering, DHF β-minimal description and DHF β-maximal description, and then construct three types of DHF neighborhood operators. Meanwhile, we introduce DHF conditional probability by using DHF neighborhood operator. Secondly, in terms of DHF conditional probability, DHF covering-based probabilistic rough set, DHF covering-based decision-theoretic rough set and their 3WD models are established. In light of DHF neighborhood operator, we propose two pairs of optimistic and pessimistic DHF decision evaluation functions and build two 3WD models through them. Lastly, a numerical example is employed to elaborate the application of the above models, which is effective and credible to medicine screening for Alzheimer’s disease.

L-valued covering-based rough sets and corresponding decision-making applications

Considering L to be a complete residuated lattice, by introducing the notion of L-valued covering on an L-set (as a universe), and then an L-valued neighborhood based on it, we present the concept of L-valued covering-based rough sets. We mainly address the following issues in this paper: Firstly, we present four types of L-valued neighborhood operators and study some of their respective properties. Secondly, we construct L-valued lower (resp., upper) approximation operators and then discuss some of their properties. Finally, we try to propose a kind of MADM problem based on an L-valued covering-based rough set model (for L=[0,1]).

Different types of approximation operators on Gn-CAS via ideals

A mathematical approach to dealing with the problems of ambiguity and indeterminacy in knowledge is called a rough set theory. It begins by using an equivalence relation to divide the universe into parts. Numerous generalized rough set models have been developed and investigated to increase their adaptability and extend their range of applications. In this context, we introduce new generalized rough set models that are inspired by covering-based rough sets and ideals. In this paper, lower and upper approximations of new types of covering rough sets based on j-neighborhoods, complementary j-neighborhoods, and j-adhesions are defined via ideals. The main features of these approximations are examined. The relationships among them are given by various examples and propositions. Some comparisons between our methods and others? methods such as Abd El-Monsef et al.?s method [2] and Nawar et al.?s method [22] are given. A practical example is given to illustrate one of our methods is more precise.

Fermatean fuzzy covering-based rough set and their applications in multi-attribute decision-making

Rough set theory (RST) is a common and effective tool for data processing. In order to extend the range of data that RST and its related theories can handle, scholars have proposed many innovations on the basis of Pawlak’s model. This paper focuses on solving the multi-attribute decision-making (MADM) issue in Fermatean fuzzy environment and interval-valued Fermatean fuzzy environment by using Fermatean fuzzy β-covering rough set (CFFRS) model and interval-valued Fermatean fuzzy β-covering set (CIVFFS) model, respectively. Firstly, we give the definitions of Fermatean fuzzy β-neighborhood and interval-valued Fermatean fuzzy β-neighborhood, and then we construct CFFRS model and CIVFFRS model, respectively. Furthermore, the related properties and the complementary concepts of these two models are also discussed. Next, in order to generalize MADM approaches in Fermatean fuzzy and interval-valued Fermatean fuzzy environments, from the point of view of comparison among the advantages of the alternatives, we establish two TOPSIS methodologies based on CFFRS and CIVFFRS, respectively. Moreover, we discuss the effect of value of precision parameter β for decision-making results by two examples and conclude that the construction of fuzzy β-neighborhood is the most important influence factor in decision-making results. And then, we give a principle for the selection of the precision parameter β, and suggest the treatment when there are alternatives in decision that cannot be compared. To demonstrate the decision-making processes of novel methodologies, aiming at an electric vehicle charging stations selection issue in an India city, including Raniganj, Jamuria, Kulti, and Burnpur, the decision-making results based on CFFRS and CIVFFRS models show that Jamuria appears to be the best location to build an electric vehicle charging station. Finally, we compare the decision-making results to other existing methods, through the Spearman ranking correlation coefficient and Pearson correlation coefficient methods, we verify the effectiveness of novel approaches.

Lattices arising from fuzzy coverings

A fuzzy covering is a natural generalization of coverings. This paper shows that the families of all definable fuzzy sets induced by a fuzzy covering-based rough set have lattice structures. There may be some superfluous elements in the universal set and the fuzzy covering for the families of all definable sets. We define class reduction, element reduction, and bireduction to remove these elements respectively. For the family of all lower definable fuzzy sets, it is shown that the class reduction coincides with covering reduction, and the element reduction can be transformed into a reduction in information systems. Bireduction considers class reduction and element reduction, and it can be achieved by class and element reductions. Moreover, our results also hold for coverings because a covering is a special fuzzy covering.

Two new fuzzy covering-based rough approximation operators via inclusion degree

Approximate accuracy is an important concept in rough set theory, which is defined by upper and lower approximations. Generally speaking, the higher precision means the better application performance. The approximation accuracy can be improved by minimizing the upper approximation and maximizing the lower approximation. Recently, Zhou [52] introduced two types of fuzzy-covering based rough set models by using inclusion relation between fuzzy sets. In this paper, by replacing inclusion relation with implication degree, we investigate two new fuzzy covering-based rough set models. Compared with inclusion relationship, the inclusion degree can describe the contained relation between fuzzy sets in more detail. This makes our upper approximation smaller than Zhou’s upper approximation, while the lower approximation is larger than Zhou’s. Therefore, the approximate accuracy of our model is higher than that of Zhou. Furthermore, we apply the new model to the study of multi-attribute decision-making (MADM). Combined with the car buying problem, we demonstrate the effectiveness of our model and compare it with other methods. The results show that we can get the same optimal choice as other methods. However, according to Zhou’s model, we cannot get the optimal choice.

Three-way decisions with fuzzy probabilistic covering-based rough sets and their applications in credit evaluation

As a generalization of equivalent class, fuzzy covering makes descriptions of the objective world more realistic, practical and accurate in some cases. In this paper, we establish four kinds of fuzzy probabilistic covering-based rough set, which combine the fuzzy covering-based neighborhood operator and the fuzzy probabilistic rough set, and then obtain the result of their three-way decision. The differences among the above models lie in the selection of thresholds, which further leads to different classification results. In addition, in order to interpret the thresholds rationally under different models, Bayesian decision procedure is adopted to construct the fuzzy probabilistic covering-based upper and lower approximation operators. Finally, a numerical example of the credit evaluation is applied to illustrate the validity of the proposed model.

Single-Valued Neutrosophic Covering-Based Rough Set Model Over Two Universes and Its Application in MCDM

Single-Valued Neutrosophic Covering-Based Rough Set Model Over Two Universes and Its Application in MCDM

Some neighborhood-related fuzzy covering-based rough set models and their applications for decision making

Fuzzy rough set (FRS) has a great effect on data mining processes and the fuzzy logical operators play a key role in the development of FRS theory. In order to generalize the FRS theory to more complicated data environments, we firstly propose four types of fuzzy neighborhood operators based on fuzzy covering by overlap functions and their implicators in this paper. Meanwhile, the derived fuzzy coverings from an original fuzzy covering are defined and the equalities among overlap function-based fuzzy neighborhood operators on a finite fuzzy covering are also investigated. Secondly, we prove that new operators can be divided into seventeen groups according to equivalence relations, and the partial order relations among these seventeen classes of operators are discussed, as well. And then, two types of neighborhood-related fuzzy covering-based rough set models are proposed via the overlap function-based fuzzy neighborhood operators and the t-norm-based fuzzy neighborhood operators. Furthermore, the groups and partially order relations are also discussed. Finally, a novel fuzzy TOPSIS methodology is put forward to solve a biosynthetic nanomaterials select issue, and the rationality and enforceability of our new approach is verified by comparing its results with nine different methods.

Boundary region-based variable precision covering rough set models

As an essential way of knowledge representation and knowledge acquisition in uncertainty theory, variable precision rough set (VPRS) has been widely considered by researchers. Firstly, the corresponding CVPRS model via boundary region is investigated based on an existing covering-based rough set model. Its algebraic structures and properties are systematically studied. This model mainly depicts a pair of boundary operators and a pair of approximation operators with a certain threshold from the perspective of single knowledge granularity. And it not only relaxes the complete inclusion relation but also retains the basic properties of the classical rough set. Secondly, an attribute reduction approach is given for a covering-based decision information systemusing the proposed CVPRS model, which keeps the lower or upper approximation unchanged. The different performances of boundary operators and their related three indices in these reduction methods are compared. The results show that boundary operators and distribution functions have outstanding advantages in these applications. Finally, an extraction method of necessity rules and possibility rules corresponding to decision classes are established. The validity and security of the above methods are verified theoretically. Numerical experiments further show that the proposed models and methods have a certain application value.

Noise-Tolerant Fuzzy-$\beta$-Covering-Based Multigranulation Rough Sets and Feature Subset Selection

As a novel fuzzy covering, fuzzy <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> covering has attracted considerable attention. However, the traditional fuzzy-<inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula>-covering-based rough set and most of its extended models cannot well fit the distribution of samples in real data, which limits their application in classification learning and decision making. First, the upper and lower approximations of these models have no inclusion relation, so they cannot characterize a given objective concept accurately. Moreover, most of these models are hard to resist the influence of noise data, resulting in poor robustness in feature learning. For these reasons, a robust rough set model is set forth by combining fuzzy rough sets, covering-based rough sets, and multigranulation rough sets. To this end, the optimistic and pessimistic lower and upper approximations of a target concept are reconstructed by means of the fuzzy <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> neighborhood related to a family of fuzzy coverings, and a new multigranulation fuzzy rough set model is presented. Furthermore, a fuzzy dependence function is introduced to evaluate the classification ability of a family of fuzzy <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> coverings at different granularity levels. The dimensionality reduction of a given fuzzy covering decision table is carried out from the perspective of maintaining the discrimination power, and a forward algorithm for feature selection is developed by using the optimistic significance of candidate features as heuristic information. Three groups of numerical experiments on 16 different types of datasets demonstrate that the proposed model exhibits good robustness on datasets contaminated with noise and outperforms some state-of-the-art feature learning algorithms in terms of classification accuracy and the size of the selected feature subset.

Attribute reduction based on observational consistency in intuitionistic fuzzy multi-covering decision systems

Attribute reduction is an important issue in data mining, machine learning and other applications of big data processing. Covering-based rough set and intuitionistic fuzzy (IF) set models are both the effective theoretical tools of uncertainty or imprecise computation, and thus IF covering rough set model has been acknowledged as a positive approach to attribute reduction. Based on IF covering rough set model, this study explores a kind of parameterized IF observational consistency in IF multi-covering decision system, and proposes an attribute reduction method. This article firstly defines the concepts of regular IF β-covering, parameterized IF observational sets on the regular IF β-covering approximation space. Secondly, the parameterized IF observational consistency is defined to be the principal of attribute reduction in the IF multi-covering decision system, and the related IF discernibility matrix is developed to provide a way of attribute reduction. For multi-observational consistency corresponding to an observational parameters set, an unified multi-observational discernibility matrix is constructed, which avoids the disadvantage of needing to construct multiple corresponding discernibility matrices separately. Furthermore, an attribute reduction algorithm based on iterative dissolving of unified multi-observational discernibility matrix is proposed, and the experiment to demonstrate effectiveness of algorithm is presented. Experiments with UCI datasets shows that, the proposed method is a good way for improving both the rates of attribute-reduced and the classification accuracy of reduced datasets.

Grained matrix and complementary matrix: Novel methods for computing information descriptions in covering approximation spaces

As two important tools of information descriptions, minimal description (MinD) and maximal description (MaxD) are used to eliminate redundant information when describing an object in covering-based rough sets (CbRSs). Hence, there are many key issues (such as reductions, neighborhoods and approximations) have to do with them in CbRSs. However, it is complicated and time-consuming to use set approaches to compute them in a large cardinal covering approximation space. So the matrix approaches have been used to calculate them by which computers can implement the corresponding calculations easily. Based on the previous research work and inspired by the demand of knowledge discovery under large scale covering information systems, we propose two new matrix methods to calculate MinD and MaxD in CbRSs in this paper. Firstly, a note on the existing methods (matrix approach-1 and approach-2) is presented. Secondly, a new matrix, called grained matrix, is presented. After combining grained matrices and the mean of “sum”, the new matrix approach-1 is presented to calculate MinD and MaxD. Thirdly, three new matrix operations and the notion of complementary matrix are given which can simplify the calculation of computers. Based on new operations, we present the new matrix approach-2 to compute MinD and MaxD. Finally, the computational efficiency of the presented approaches are shown and compared with other methods by experiments on several UCI data sets.

Fuzzy covering-based rough set on two different universes and its application

In this paper, we propose a new type of fuzzy covering-based rough set model over two different universes by using Zadeh’s extension principle. We mainly address the following issues in this paper. First, we present the definition of fuzzy \(\beta\)-neighborhood, which can be seen as a fuzzy mapping from a universe to the set of fuzzy sets on another universe and study its properties. Then we define a new type of fuzzy covering-based rough set model on two different universes and investigate the properties of this model. Meanwhile, we give a necessary and sufficient condition under which two fuzzy \(\beta\)-coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation. Moreover, matrix representations of the fuzzy covering lower and fuzzy covering upper approximation operators are investigated. Finally, we propose a new approach to a kind of multiple criteria decision making problem based on fuzzy covering-based rough set model over two universes. The proposed models not only enrich the theory of fuzzy covering-based rough set but also provide a new perspective for multiple criteria decision making with uncertainty.

Decision self-information based on parameterized fuzzy β neighborhood and its application in three-way multi-attribute group decision-making

&lt;abstract&gt;&lt;p&gt;As a special kind of entropy, decision self-information effectively considers the uncertainty information of both the lower and upper approximations. However, it is limited to rough binary relations, which limits its application to complex problems. In addition, parameterized fuzzy β covering, as an extension of the covering-based rough set model, can effectively characterize the similarity between samples. We combine decision self-information with a parameterized fuzzy β neighborhood to propose decision self-information in fuzzy environments, and we study its important properties. On this basis, a three-way multi-attribute group decision-making algorithm is established, and a practical problem is solved. The effectiveness of the proposed method is verified by experimental analysis.&lt;/p&gt;&lt;/abstract&gt;