Multi-granulation Fuzzy Rough Sets Research Articles

New covering techniques and applications utilizing multigranulation fuzzy rough sets

In order to conduct an in-depth study of Zhan’s methodology pertaining to the covering of multigranulation fuzzy rough sets (CMG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {C}_{{MG}}$$\\end{document}FRSs), we build two families: the family of fuzzy β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-minimum descriptions and the family of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-maximum descriptions. Subsequently, utilizing these notions, we proceed to develop two variations of covering via optimistic (pessimistic) multigranuation rough set samples (CO(P)MG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {CO(P)}_{{MG}}$$\\end{document}FRS). The axiomatic properties are examined. In this study, we examine four models of covering using variable precision multigranulation fuzzy rough sets (CVPMG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {CVP}_{{MG}}$$\\end{document}FRSs). We proceed with analyzing the features of these models. Interconnections between these planned plans are also elucidated. This study explores algorithms that aim to identify innovative strategies for addressing multiattribute group decision-making problems (MAGDM) and multicriteria group decision-making problems (MCGDM). The test examples have been elucidated to provide an inclusive grasp of the efficacy of the offered samples. Ultimately, the distinctions between our methodologies and the preexisting research have been demonstrated.

Uncertainty measures and feature selection based on composite entropy for generalized multigranulation fuzzy neighborhood rough set

With the continuous advancement of information technology, the information and data covered by various information systems become increasingly complex and diverse, it is essential to perform knowledge mining from multiple perspectives to extract valuable insights. Fuzzy neighborhood multigranulation rough set, as an excellent feature selection model, is capable of handling heterogeneous datasets more effectively, significantly improving learning efficiency. In this study, we investigate a feature selection method based on a generalized multigranulation fuzzy rough set (GMFNRS) in fuzzy decision systems. First, the concepts of fuzzy neighborhood rough sets and generalized multigranulation rough sets are introduced. Subsequently, the GMFNRS model is established to enable data mining and rule extraction from various perspectives. Secondly, from an informational perspective, the study investigates uncertainty measurement methods through fuzzy neighborhood joint entropy. Furthermore, a novel fuzzy neighborhood generalized composite entropy is proposed by integrating the GMFNRS model with uncertainty measures. Finally, a forward greedy feature selection algorithm is considered to extract essential information from complex datasets. Experimental results on 15 public datasets demonstrate that the proposed model effectively selects important features in fuzzy systems and exhibits excellent classification performance.

Matrix methods for some new covering-based multigranulation fuzzy rough set models under fuzzy complementary β-neighborhoods

Fuzzy complementary β-neighborhoods (FCNs) are used to find information relevant to the target in data mining. Based on FCNs, there are six types of covering-based multigranulation fuzzy rough set (CMFRS) models have been constructed, which can be used to deal with the problem of multi-criteria information systems. These CMFRS models are calculated by set representations. However, it is time-consuming and error-prone when set representations are used to compute these CMFRS models in a large multi-criteria information system. Hence, it is important to present a novel method to compute them quickly, which is our motivation for this paper. In this paper, we present the matrix representations of six types of CMFRS models on FCNs. Firstly, some new matrices and matrix operations are given in a multi-criteria information system. Then, matrix representations of three types of optimistic CMFRSs on FCNs are proposed. Moreover, matrix approaches are also used for computing three types of pessimistic CMFRSs on FCNs. Finally, some experiments are presented to show the effectiveness of our approaches.

Matrix representation of z-fuzzy soft β-covering based multigranulation fuzzy rough sets

Matrix representation of z-fuzzy soft β-covering based multigranulation fuzzy rough sets

Kernelized multi-granulation fuzzy rough set over hybrid attribute decision system and application to stroke risk prediction

Kernelized multi-granulation fuzzy rough set over hybrid attribute decision system and application to stroke risk prediction

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

Multigranulation fuzzy rough sets (MFRSs), variable precision fuzzy rough sets (VPFRSs) and their combinations with traditional decision-making methods are fundamental in the research of multi-attribute decision-making (MADM) problems. However, the existing theories and methods have two deficiencies: (1) many existing MFRSs are based on fuzzy relations or fuzzy coverings, which may not effectively solve some MADM problems; (2) VPFRSs usually do not meet the inclusion property, which hinders their further development in theory and application. Noted that both fuzzy relations and fuzzy coverings can generate fuzzy neighborhood systems in a natural way, so the fuzzy neighborhood system provides a broader multigranulation framework. Inspired by this, by using the generalized fuzzy neighborhood system, we propose three multigranulation variable precision fuzzy rough set models, namely optimistic, pessimistic and compromise, respectively. As expected, these models contain many existing models as special cases, thus broadening the framework of rough set. Additionally, we verify that these models fulfill many basic algebra properties, particularly the generalized inclusion properties, which overcome the limitation of VPFRSs. Finally, by combining the proposed fuzzy rough set models with the TOPSIS method, we develop a new method to MADM. A series of experiments show that our method has some obvious advantages.

A Novel Fuzzy Covering Rough Set Model Based on Generalized Overlap Functions and Its Application in MCDM

As nonassociative fuzzy logic connectives, it is important to study fuzzy rough set models using overlap functions that replace the role of t-norms. Overlap functions and t-norms are logical operators with symmetry. Recently, intuitionistic fuzzy rough set and multi-granulation fuzzy rough set models have been proposed based on overlap functions. However, some results (that contain five propositions, two definitions, six examples and a proof) must be improved. In this work, we improved the existing results. Moreover, to extend the existing fuzzy rough sets, a new fuzzy covering rough set model was constructed by using the generalized overlap function, and it was applied to the diagnosis of medical diseases. First, we improve some existing results. Then, in order to overcome the limitations of the fuzzy covering rough set model based on overlap functions, a fuzzy β-covering rough set model based on generalized overlap functions was established. Third, some properties of the fuzzy β-covering rough set model based on generalized overlap functions are discussed. Finally, a multi-criteria decision-making (MCDM) method of the fuzzy β-covering rough set based on generalized overlap functions was proposed. Taking medical disease diagnosis as an example, the comparison with other methods shows that the proposed method is feasible and effective.

Matrix Approaches for Covering-Based Multigranulation Fuzzy Rough Set Models

Multigranulation rough set theory is one of the most effective tools for data analysis and mining in multicriteria information systems. Six types of covering-based multigranulation fuzzy rough set (CMFRS) models have been constructed through fuzzy β -neighborhoods or multigranulation fuzzy measures. However, it is often time-consuming to compute these CMFRS models with a large fuzzy covering using set representation approaches. Hence, presenting novel methods to compute them quickly is our motivation for this paper. In this article, we study the matrix representations of CMFRS models to save time in data processing. Firstly, some new matrices and matrix operations are proposed. Then, matrix representations of optimistic CMFRSs are presented. Moreover, matrix approaches for computing pessimistic CMFRSs are also proposed. Finally, some experiments are proposed to illustrate the effectiveness of our approaches.

Multi-granulation fuzzy rough sets based on overlap functions with a new approach to MAGDM

A common approach to constructing fuzzy rough sets (FRSs) is using t-norms. Furthermore, establishing multi-granulation fuzzy rough sets (MGFRSs) is also usually undertaken by means of t-norms. However, most of these sets cannot satisfy the property that upper approximations contain lower approximations under fuzzy binary relations, which is imperative for rough set models. The overlap function is a type of aggregate function that is widely used in multi-attribute decision-making (MADM), image processing and other fields. In this paper, two novel types of MGFRS models under n-dimensional overlap functions are established to overcome the shortcomings of existing models, which are then applied to the multi-attribute group decision-making (MAGDM) problem. First, idempotent n-dimensional overlap functions are used to establish optimistic and pessimistic MGFRSs. These new models fully preserve the important properties of the traditional MGFRS model. Second, through the theoretical analysis of MGFRS based on overlap functions (OMGFRS) and in combination with the TOPSIS method, a solution mechanism for the MAGDM method is proposed. Finally, to fully illustrate this decision-making method, an effective example is developed. A comparison with available methods indicates that this approach is more suitable and has wider adaptability, and that is can be used to handle decision-making problems.

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.

Measurement, modeling, reduction of decision-theoretic multigranulation fuzzy rough sets based on three-way decisions

Variable precision multigranulation fuzzy rough sets (VP-MFRSs) use two direct integrations: the multigranulation maximum and minimum. Their optimistic and pessimistic models facilitate uncertainty informatization but also imply a potential limitation of extremes. This study improves VP-MFRSs for extension and balance, and thus, decision-theoretic multigranulation fuzzy rough sets (DT-MFRSs) are proposed by systematically fusing the multigranulation maximum and minimum. For DT-MFRSs, their tri-level analysis of measurement, modeling, and reduction is deeply acquired via three-way decisions. First, maximum and minimum membership degrees are linearly combined, and the weight parameter guides a generalized multigranulation membership degree. This adjustable measure motivates DT-MFRSs with positive, negative, and boundary regions, while attitude-preference values of 1, 0, and 0.5 respectively produce optimistic, pessimistic, and compromised models. Then, nonmonotonicity and uncertainty of membership degrees and model regions are determined, and these fundamental characteristics induce new reduction criteria of region preservations. Also, three-way attribute reducts are proposed by preserving positive, negative, and positive–negative regions, and their systematic relationships are obtained. Finally, tri-level results of measures, models, and reducts are validated by table examples and data experiments. In this study, DT-MFRSs extend and improve VP-MFRSs via systematic fusion of membership measurement, and contain optimistic, pessimistic, compromised models, etc., thereby exhibiting extended diversity and applied robustness. Their three-way attribute reduction also perfects uncertainty optimization.

Some New Covering-Based Multigranulation Fuzzy Rough Sets and Corresponding Application in Multicriteria Decision Making

Multigranulation rough set theory is an important tool to deal with the problem of multicriteria information system. The notion of fuzzy β -neighborhood has been used to construct some covering-based multigranulation fuzzy rough set (CMFRS) models through multigranulation fuzzy measure. But the β -neighborhood has not been used in these models, which can be seen as the bridge of fuzzy covering-based rough sets and covering-based rough sets. In this paper, the new concept of multigranulation fuzzy neighborhood measure and some types of covering-based multigranulation fuzzy rough set (CMFRS) models based on it are proposed. They can be seen as the further combination of fuzzy sets: covering-based rough sets and multigranulation rough sets. Moreover, they are used to solve the problem of multicriteria decision making. Firstly, the definition of multigranulation fuzzy neighborhood measure is given based on the concept of β -neighborhood. Moreover, four types of CMFRS models are constructed, as well as their characteristics and relationships. Then, novel matrix representations of them are investigated, which can satisfy the need of knowledge discovery from large-scale covering information systems. The matrix representations can be more easily implemented than set representations by computers. Finally, we apply them to manage the problem of multicriteria group decision making (MCGDM) and compare them with other methods.

Risk assessment for PPP waste-to-energy incineration plant projects in china based on hybrid weight methods and weighted multigranulation fuzzy rough sets

Implementing effective risk assessment for public–private partnership (PPP) waste-to-energy (WTE) incineration plant projects promotes the sustainable development of WTE incineration plants. However, some studies fail to consider multi-individuals participation and handle mutual compensation among risk factors, which affects the reliability of evaluation results and increases decision risks. Thus, we propose a methodology based on weighted multigranulation fuzzy rough sets (MGFRSs) over two universes to perform risk evaluation for PPP WTE incineration plant projects. Firstly, alternative risk factors are detected from different dimensions to construct a risk evaluation index system. Secondly, the BWM (best-worst method) is utilized to determined subjective expert weights and an improved DEMATEL (Decision Making Trial and Evaluation Laboratory) method is developed to derive objective expert weights. Combining subjective and objective expert weights, an optimization model is built to form the collective expert weights. Thirdly, to avoid mutual compensation among risk factors, multi-source heterogeneous information system-based weighted MGFRSs over two universes are proposed to evaluate PPP WTE plant projects and select a promising one. Finally, a case study is performed to illustrate the feasibility of the proposed methodology. Sensitivity analysis and comparative analysis are implemented to demonstrate the robustness, effectiveness, and superiority of the proposed methodology.

Fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications

In this paper, with the aid of fuzzy soft $$\beta $$ -neighborhoods, we introduce fuzzy soft covering-based multi-granulation fuzzy rough set models. We examine some of the relevant properties of fuzzy soft covering based on optimistic, pessimistic, and variable precision multi-granulation fuzzy rough set models. Then, we give fuzzy soft coverings based on $$\psi $$ -optimistic and $${\mathscr {D}}$$ -optimistic ( $$\psi $$ -pessimistic and $${\mathscr {D}}$$ -pessimistic) multi-granulation fuzzy rough sets from fuzzy soft measures. It also discusses the interactions between these forms of fuzzy soft coverings based on multi-granulation fuzzy rough sets. Eventually, we apply the proposed models for solving MAGDM problems. The effectiveness and feasibility of our approach are noted from the introduced comparisons between our method and some methods given in the previous studies.

Some extensions of covering-based multigranulation fuzzy rough sets from new perspectives

Some extensions of covering-based multigranulation fuzzy rough sets from new perspectives

Generalized multigranulation fuzzy rough sets based on upward additive consistency

In this paper, we point out that the transfer function for computing the fuzzy preference degree of Pan et al. (Fuzzy Sets Syst 312:87–108, 2017) for the construction of upward/downward fuzzy relations is not additive consistent. Appropriate counterexample is given. Further their modified versions are presented. Meanwhile, we construct upward consistency matrices of experts which satisfy the upward additive consistency and the upward order consistency simultaneously. After that, by introducing some new fuzzy upward $$\beta $$ -coverings, fuzzy upward $$\beta $$ -neighborhoods and fuzzy upward complement $$\beta $$ -neighborhoods are proposed and related properties are studied. Furthermore, we propose multigranulation optimistic/pessimistic $$\left( {\mathcal {I}},{\mathcal {T}}\right) $$ -fuzzy upward rough set based on fuzzy upward $$\beta $$ -covering and investigate some of their properties. Finally we construct a new approach to multiple attribute decision making problem based on multigranulation optimistic/pessimistic $$\left( {\mathcal {I}}, {\mathcal {T}}\right) $$ -fuzzy upward rough set. The decision making procedure, methodology and the algorithm of the proposed technique are given. The detailed comparison of the present work with other methods to multiple attribute decision making problem illustrates the advantages of the this work and limitations of other studies.

Type-2 fuzzy multigranulation rough sets

Aiming at expanding the application range of the multigranulation rough set (MGRS) theory, a type-2 fuzzy multigranulation rough set (T2FMGRS) model is proposed by combining type-2 fuzzy sets with multigranulation rough sets (MGRSs) in this paper. At first, definitions and properties of optimistic and pessimistic type-2 fuzzy multigranulation rough sets (T2FMGRSs) are introduced. Then, the rough measure is proposed to measure the uncertainty of T2FMGRSs. Finally, T2FMGRSs over two universes are put forward and some examples of decision-making are given to illustrate the applicability of the newly proposed model. In summary, the establishment of T2FMGRSs is a meaningful generalization of the MGRS theory from both theoretical and practical aspects.

Diversified multiple attribute group decision-making based on multigranulation soft fuzzy rough set and TODIM method

This paper investigates the diversified multi-attribute group decision-making problem which means that the different decision-makers have different evaluation attribute sets for all candidate alternatives. We analyze the diversified multi-attribute group decision-making problem from the perspective of granular computing models and creatively combine soft sets with multigranulation fuzzy rough sets to construct the multigranulation soft fuzzy rough set model. Subsequently, we define the optimistic upper and lower approximations and pessimistic upper and lower approximations with respect to multigranulation soft fuzzy rough set model. Meanwhile, we discuss some important mathematical conclusions and properties based on the established model. Then, we propose a novel approach to the diversified multi-attribute group decision-making problem based on multigranulation soft fuzzy rough set and TODIM method, which fully consider the reference dependence and loss aversion of decision-makers in the decision-making process. Finally, we use a numerical example of evaluation for the policy selection group decision-making problem to describe and explain the calculation principle and process and compare the results with existing related work, and our approach has great stability and rationality. And the contributions of this paper include the following two points: (1) construct the multigranulation soft fuzzy rough set model; (2) combine the established model with TODIM method to provide a novel perspective to solve the diversified multi-attribute group decision-making problem.

An Intuitionistic Fuzzy Set Approach for Multi-attribute Information Classification and Decision-Making

This article introduced a new multi-attribute information classification method by employing intuitionistic fuzzy set (IFS) approach. The proposed method was referred as four-way intuitionistic decision space (4WIDS). In the 4WIDS, IFS theory was used to model the inherent uncertainty of multi-attribute information. For generating more precise level of decision-rules, granular computing (GrC) approach was employed. The proposed 4WIDS method was appropriate for the classification of the multi-attribute information into four different regions as positive IFS, negative IFS, uncertain IFS and gray IFS regions. Detail methodology of the 4WIDS was explained by presenting its representation in a precise way. This study also presented various definitions, properties and theorems in the support of the 4WIDS method. The 4WIDS was applied in benchmark datasets that included Pima Indians diabetes, Thyroid disease, Fisher’s Iris and Spambase datasets. Experimental results including statistical analysis indicated that the proposed 4WIDS outperformed existing classification methods, such as Naive Bayes, Decision tree, PART, J48, logistic model trees (LMT), rough set (RS), gray multi-granulation rough set (GMGRS) and multi-granulation fuzzy rough set (MGFRS).

Novel classes of coverings based multigranulation fuzzy rough sets and corresponding applications to multiple attribute group decision-making

The notion of covering based multigranulation fuzzy rough set (CMGFRS) models is a generalization of both granular computing and covering based fuzzy rough sets. Therefore it has become a powerful tool for coping with vague and multigranular information in cognition. In this paper we introduce three kinds of CMGFRS models by means of fuzzy β-neighborhoods and fuzzy complementary β-neighborhoods, and we investigate their axiomatic properties. We investigate three respective types of coverings based CMGFRS models, namely, optimistic, pessimistic and variable precision setups. In particular, by using multigranulation fuzzy measure degrees and multigranulation fuzzy complementary measure degrees, we derive three types of coverings based γ-optimistic (γ-pessimistic) CMGFRSs and E (F, G)-optimistic and E (F, G)-pessimistic CMGFRSs, respectively. We discuss the interrelationships among these three types of CMGFRS models and covering based Zhan-CMGFRS models. In view of the theoretical analysis for these three types of CMGFRS models, we put forward a novel methodology to multiple attribute group decision-making problem with evaluation of fuzzy information. An effective example is fully developed, hence concluding the applicability of the proposed methodology.