Multigranulation Rough Set Research Articles

Covering based multigranulation fuzzy rough sets and corresponding applications

By combining covering based rough sets, fuzzy rough sets, and multigranulation rough sets, we introduce covering based multigranulation fuzzy rough set models by means of fuzzy $$\beta $$-neighborhoods. We investigate axiomatic characterizations of covering based optimistic, pessimistic and variable precision multigranulation fuzzy rough set models. We propose coverings based $$\alpha $$-optimistic (pessimistic) multigranulation fuzzy rough sets and D-optimistic (pessimistic) multigranulation fuzzy rough sets from fuzzy measures. We examine the relationships among these kinds of coverings based fuzzy rough sets. Finally, we apply the proposed models to solve problems for multi-criteria group decision-making.

Multigranulation rough set model based on granulation of attributes and granulation of attribute values

Existing multigranulation rough set models often suppose the family of equivalence relations is existed and do not give the methods to construct the multigranulation space. That is, the multigranulation space is given. So in multigranulation rough set models, how to derive the family of equivalence relations and construct the multigranulation space is an important problem to be resolved. In this paper, considering there are different granulation of attributes and attribute values, I propose multigranulation rough set model based on granulation of attributes and granulation of attribute values (MRAAV), which provides a method to construct the multigranulation space in multigranulation rough set models. Based on MRAAV, multigranulation problem solving space is constructed. The search strategy for problem solving in multigranulation problem solving space is presented, which make the information processing process of rough set is structured and hierarchical. Finally, the rule extraction algorithm based on MRAAV is given, which can not only extract rules from multiple abstract concept levels, but also with low costs. The example and experimental results demonstrate the effectiveness of the proposed model and algorithms.

Multi-granulation Pythagorean fuzzy decision-theoretic rough sets based on inclusion measure and their application in incomplete multi-source information systems

Multi-granulation rough sets (MGRSs) and decision-theoretic rough sets (DTRSs) are two important and popular generalizations of classical rough sets. The combination of two generalized rough sets have been investigated by numerous researchers in different extensions of fuzzy settings such as interval-valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs), bipolar-valued fuzzy sets (BVFSs), etc. Pythagorean fuzzy (PF) set is another extension of fuzzy set, which is more capable in comparison to IFS handle vagueness in real world. However, few studies have focused on the combination of the two rough sets in PF settings. In this study, we combine the two generalized rough sets in PF settings. First, we introduce a type of PF subset (of subset of the given universe) of the PF Set (of the given universe). Then we establish two basic models of multi-granulation PF DTRS (MG-PF-DTRS) of PF subset of the PF set based on PF inclusion measure within the framework of multi-granulation PF approximation space. One model is based on a combination of PF relations (PFRs) and the construction of approximations with respect to the combined PFR. By combining PFRs through intersection and union, respectively, we construct two models. The other model is based on the construction of approximations from PFRs and a combination of the approximations. By using intersection and union to combine the approximations, respectively, we again get two models. As a result, we have total four models. Further for different constraints on parameters, we obtain three kinds of each model of the MG-PF-DTRSs. Then, their principal structure, basic properties and uncertainty measure methods are investigated as well. Second, we give a way to compute PF similarity degrees between two objects and also give a way to compute PF decision-making objects from incomplete multi-source information systems (IMSISs). Then we design an algorithm for decision-making to IMSISs using MG-PFDTRSs and their uncertainty measure methods. Finally, an example about the mutual funds investment is included to show the feasibility and potential of the theoretic results obtained.

Matrix-Based Approaches for Updating Approximations in Multigranulation Rough Set While Adding and Deleting Attributes

With advanced technology in medicine and biology, data sets containing information could be huge and complex that sometimes are difficult to handle. Dynamic computing is an efficient approach to solve some problems. Since multigranulation rough sets were proposed, many algorithms have been designed for updating approximations in multigranulation rough sets, but they are not efficient enough in terms of computational time. The purpose of this study is to further reduce the computational time of updating approximations in multigranulation rough sets. First, searching regions in data sets for updating approximations in multigranulation rough sets are shrunk. Second, matrix-based approaches for updating approximations in multigranulation rough set are proposed. The incremental algorithms for updating approximations in multigranulation rough sets are then designed. Finally, the efficiency and validity of the designed algorithms are verified by experiments.

A Decision-Making Approach Based on a Multi Q-Hesitant Fuzzy Soft Multi-Granulation Rough Model

In this paper, we propose a new hybrid model, multi Q-hesitant fuzzy soft multi-granulation rough set model, by combining a multi Q-hesitant fuzzy soft set and multi-granulation rough set. We demonstrate some useful properties of these multi Q-hesitant fuzzy soft multi-granulation rough sets. Furthermore, we define multi Q-hesitant fuzzy soft ( M k Q H F S ) rough approximation operators in terms of M k Q H F S relations and M k Q H F S multi-granulation rough approximation operators in terms of M k Q H F S relations. We study the main properties of lower and upper M k Q H F S rough approximation operators and lower and upper M k Q H F S multi-granulation rough approximation operators. Moreover, we develop a general framework for dealing with uncertainty in decision-making by using the multi Q-hesitant fuzzy soft multi-granulation rough sets. We analyze the photovoltaic systems fault detection to show the proposed decision methodology.

Novel Three-Way Decisions Models with Multi-Granulation Rough Intuitionistic Fuzzy Sets

The existing construction methods of granularity importance degree only consider the direct influence of single granularity on decision-making; however, they ignore the joint impact from other granularities when carrying out granularity selection. In this regard, we have the following improvements. First of all, we define a more reasonable granularity importance degree calculating method among multiple granularities to deal with the above problem and give a granularity reduction algorithm based on this method. Besides, this paper combines the reduction sets of optimistic and pessimistic multi-granulation rough sets with intuitionistic fuzzy sets, respectively, and their related properties are shown synchronously. Based on this, to further reduce the redundant objects in each granularity of reduction sets, four novel kinds of three-way decisions models with multi-granulation rough intuitionistic fuzzy sets are developed. Moreover, a series of concrete examples can demonstrate that these joint models not only can remove the redundant objects inside each granularity of the reduction sets, but also can generate much suitable granularity selection results using the designed comprehensive score function and comprehensive accuracy function of granularities.

Granulation selection and decision making with multigranulation rough set over two universes

Multigranulation rough set over two universes provides a new perspective to combine multiple granulation knowledge in a multigranulation space in practical reality. Note that there are always non-essential neighborhood granulations, which would affect the efficiency and quality of decision making. Therefore, selecting valuable granulations and reducing worthless ones are necessary for the application of multigranulation rough set in decision process. In this paper, we first define several measurements to compare the granularity of neighborhood granulations, using which the granulation selection with multigranulation rough set is characterized. Then, the selection algorithms in the multigranulation space are developed. Third, we generate “OR” and “AND” decision rules based on multigranulation fusion strategies. As an application, these decision rules are employed to make decisions in the presence of disease diagnosis problems. In the end, the effectiveness and efficiency of the proposed algorithms are examined with numerical experiments on selective data sets.

Vector‐based approaches for computing approximations in multigranulation rough set

Vector‐based approaches for computing approximations in multigranulation rough set

A Novel Decision-Making Approach to Fund Investments Based on Multigranulation Rough Set

Fund investment is a hot issue in today’s society. How to choose a project for investment is affected by many factors. In view of this problem, this paper starts from the granular computing point of view and combines the multigranulation rough set decision-making method to construct a fund investment decision information system; then, the fund investment decision information system is reduced under different thresholds, and the decision rules are extracted through reduction. And from the aspects of decision accuracy and rule accuracy, the rules are analyzed. Finally, decision rules are used to give the decision of the fund investment project. This study provides a new approach to fund management.

Incremental updating knowledge in neighborhood multigranulation rough sets under dynamic granular structures

Neighborhood Multigranulation Rough Sets (NMGRS), constructed by a family of neighborhood relations, can effectively obtain required knowledge from Neighborhood Information Systems (NIS). In various practical situations, NIS may alter dynamically with time. Incremental learning is an alternative manner for maintaining knowledge by utilizing previous computational results under dynamic data contexts. In dynamic NIS with numerical data, potential useful knowledge, e.g., the positive, boundary and negative regions in NMGRS, needs to be updated for various applications. To address this issue, we present matrix-based incremental approaches to update knowledge in NMGRS with the addition or deletion of granular structures. First, a matrix-based representation of neighborhood multigranulation rough approximations is developed. Then, the matrix-based dynamic strategies are proposed to update the positive, boundary, and negative regions in the optimistic and pessimistic NMGRS under the variation of a granular structure. In accordance with the proposed updating strategies, the matrix-based dynamic algorithms are explored for maintaining the positive, boundary and negative regions while adding or deleting granular structures. Finally, comparative experiments are conducted to show that the matrix-based dynamic algorithms are feasible and effective.

Operation properties and algebraic properties of multi-covering rough sets

The multi-covering rough sets (MCRSs) are a popular aspect of rough sets. It is easy to see that classical rough sets, covering rough sets (CRSs) and multi-granulation rough sets (MGRSs) are all the special cases of the MCRSs. Recently, the algebraic theory of these rough set models mentioned above have been researched in detail. However, the algebraic theory of MCRSs has not been studied until now. It is necessary for researchers to explore the algebraic theory of MCRSs. In this paper, we focus on the operation and algebraic theories of two types of MCRS models. First, the properties of the two types of multi-covering set approximations are discussed. Especially, the properties of multi-covering approximation operators based on the unary coverings are deeply researched. Second, the operation properties with respect to intersection and union of MCRSs are researched. Meanwhile, to compute the intersection and union of MCRSs, several algorithms are constructed. Finally, on the basis of the operation properties of MCRSs, many meaningful algebraic properties of MCRSs are deeply studied.

Variable precision multigranulation rough fuzzy set approach to multiple attribute group decision-making based on λ-similarity relation

Abstract This study proposes a fuzzy multigranulation rough set approach to the problem of multiple attribute group decision-making with uncertainty. Based on the classical Pawlak rough set theory, we define the λ - similarity ( 0 ⩽ λ ≤ 1 ) relation classes over the universe of discourse by introducing a distance measure to all alternatives with respect to attribute set. Subsequently, we present the α ( 0.5 α ≤ 1 ) rough approximation of a crisp decision-making object and a fuzzy decision-making object under the framework of multigranulation rough set theory, respectively. That is, we establish the variable precision multigranulation rough set model and variable precision multigranulation rough fuzzy set model based on λ - similarity relation, respectively. Meanwhile, we discuss the interrelationship between the proposed multigranulation rough fuzzy set model and the existing generalized rough set models. After that, we construct a new approach to multiple attribute group decision-making problems based on variable precision multigranulation rough fuzzy set theory. The decision-making procedure and the methodology as well as the algorithm of the proposed method are given and a detailed comparison of the traditional methods to multiple attribute group decision-making problems illustrates the advantages and limitations. Finally, an example of handling multiple criteria group decision-making problem of evaluation of emergency plans for unconventional emergency events illustrates this approach. The main contribution of this paper is twofold. One is to provide a new way to construct multigranulation rough set model with the fuzzy environment. Another is to try making a new way to handle multiple criteria group decision-making problems based on generalized rough set theory and methodologies.

Granularity reduction method based on positive decision holding for multi‐granulation decision‐theoretic rough set

Multi-granulation decision-theoretic rough set (MG-DTRS) is a generalisation of the multi-granulation rough set (MGRS) model through integrating the related properties of MGRSs and decision-theoretic rough sets (DTRSs), and it can realise decision analysis and processing of information systems from multiple perspectives and multi-level. Taking the optimistic MG-DTRS and the pessimistic MG-DTRS model as an example, MG-DTRS model and its related properties were discussed according to the DTRSs which construct positive, boundary and negative regions on the basis of two thresholds given by experts. Especially, the positive decision and its properties in MG-DTRS model were analysed. At the same time, it was found and proved that the positive decision is monotonic with the change of granularity in MG-DTRS model. Further, the granular structure selection problem was investigated under the MG-DTRS model, the concept of α-decision granularity importance was introduced and a granularity reduction algorithm based on the positive decision holding for MG-DTRS model was designed and constructed. Finally, an example was given to prove that the algorithm is effective, and it has a small time complexity.

New types of dominance based multi-granulation rough sets and their applications in conflict analysis problems

New types of dominance based multi-granulation rough sets and their applications in conflict analysis problems

Parallel Attribute Reduction Algorithm for Complex Heterogeneous Data Using MapReduce

Parallel attribute reduction is one of the most important topics in current research on rough set theory. Although some parallel algorithms were well documented, most of them are still faced with some challenges for effectively dealing with the complex heterogeneous data including categorical and numerical attributes. Aiming at this problem, a novel attribute reduction algorithm based on neighborhood multigranulation rough sets was developed to process the massive heterogeneous data in the parallel way. The MapReduce-based parallelization method for attribute reduction was proposed in the framework of neighborhood multigranulation rough sets. To improve the reduction efficiency, the hashing Map/Reduce functions were designed to speed up the positive region calculation. Thereafter, a quick parallel attribute reduction algorithm using MapReduce was developed. The effectiveness and superiority of this parallel algorithm were demonstrated by theoretical analysis and comparison experiments.

Attributes set reduction in multigranulation approximation space of a multi-source decision information system

The multiple granular structures in multigranulation approximation space can be integrated by different approximation operators. These operators are induced by multigranulation rough set or others. In this paper, numerical algorithms of attributes set reduction are developed by using evidence theory or minimal elements of discernibility matrix, respectively. Firstly, a pair of multi-source rough approximation operators and their corresponding multigranulation rough approximation operators are defined based on a bijective and new transitional domain. Then, we propose attributes set reduction with respect to multi-source rough approximation. It is based on the relationship between multi-source rough approximations and evidence theory. Therefore, a heuristic algorithm of attributes set reduction for multi-source lower rough approximation is given. Secondly, we define a multigranulation variable precision rough set (MVRS) by considering weight of each attributes set relative to all of attributes sets. Finally, we investigate attributes set reduction with MVRS by combing the minimal elements of discernibility matrix and distribution discernibility function. There are some illustrative examples to elaborate the operation mechanism of above conclusions.

Multigranulation vague rough set over two universes and its application to group decision making

This article studies the rough approximation of a vague concept based on multiple granularity over two universes. With the description of the background of risk decision making problems in reality, we first present the optimistic multigranulation rough vague set, pessimistic multigranulation rough vague set and variable precision multigranulation rough vague set on the multigranulation approximation space over two universes. Then, under the multigranulation fuzzy approximate space over two universes, we establish the corresponding three types of multigranulation vague rough set over two universes. Subsequently, the interesting properties and results as well as the relationship between the multigranulation rough vague set models and multigranulation vague rough set model over two universes are investigated in detail. The results show that multigranulation vague rough set models are extensions of the existing generalized rough set models under the framework of two universes. At the same time, we construct a new approach to group decision making under uncertainty by using the theory of multigranulation vague rough set over two universes. The basic principal and the detailed steps of the decision making model given in this paper are presented in detail. Meanwhile, an example of handling a medical diagnosis group decision making problem illustrates this approach. The main contribution of this paper is twofold. One is to provide the theoretical model of multigranulation vague rough set over two universes. Another is to try making a new way to handle group decision making problems under uncertainty based on multigranulation vague rough set theory and methodologies over two universes.

Inclusion measure-based multi-granulation decision-theoretic rough sets in multi-scale intuitionistic fuzzy information tables

Multi-granulation rough sets (MGRSs) and decision-theoretic rough sets (DTRSs) are important and popular extended types of Pawlak’s classical rough set model. Multi-granulation DTRSs (MG-DTRSs), which combine these two generalized rough set models, have been investigated in depth in recent years to handle noisy distributed data. However, this combination cannot be used to acquire knowledge from multi-scale information systems, in which an object may take on different values under the same attribute depending on the scale used to measure it. Two novel types of MG-DTRSs in multi-scale intuitionistic fuzzy (IF) information tables have been developed on the basis of IF inclusion measures in this study to solve this problem. First, we introduce a type of inclusion measure between two IF sets and present the concept of inclusion measure-based DTRSs in multi-scale IF information tables. Second, we present the inclusion measure-based optimistic and pessimistic MG-DTRSs in multi-scale IF information tables, examine their properties, and analyze the three-way decision method based on the presented models. Third, we define the optimal scale selection and present the two optimal scale selection algorithms based on MG-DTRSs in multi-scale IF information tables. Fourth, we provide the reducts of the optimal scales based on MG-DTRSs in multi-scale IF information tables, examine the discernibility function reduction method, and devise two algorithms for computing an optimal approximation scale reduct. Finally, we discuss several possible generalizations related to MG-DTRSs in multi-scale IF information tables. This study provides an MG-DTRS method for acquiring knowledge from multi-scale IF information tables.

Two types of coverings based multigranulation rough fuzzy sets and applications to decision making

Covering based multigranulation rough fuzzy set, as a generalization of granular computing and covering based rough fuzzy set theory, is a vital tool for dealing with the vagueness and multigranularity in artificial intelligence and management sciences. By means of neighborhoods, we introduce two types of coverings based (optimistic, pessimistic and variable precision) multigranulation rough fuzzy set models, respectively. Some axiomatic systems are also obtained. The relationships between two types of coverings based (optimistic, pessimistic and variable precision) multigranulation rough fuzzy set models are established. Based on the theoretical discussion for the covering based multigranulation rough fuzzy set models, we present an approach to multiple criteria group decision making problem. These two types of basic models and the procedure of decision making methods as well as the algorithm for the new approach are given in detail. By comparative analysis, the ranking results based on two different models have a highly consensus. Although there exist some different ranking results on these two methods, the optimal selected alternative is the same.

Decision-making Method for an Ordered Information System Based on Multi-granulation Rough Sets

Decision-making Method for an Ordered Information System Based on Multi-granulation Rough Sets