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Multi-granulation decision-theoretic rough set effectively combines Bayesian decision approaches with multi-granulation rough set theory, and provides an important theoretical framework for studying rough set. In this paper, we explore several extensional models of multi-granulation decision-theoretic rough sets under the normal distribution of the decision loss function. Using the 3σ rule of normal distribution, we transform the decision loss of the multi-granulation decision-theoretic rough set into a set of interval values. We construct the upper and lower approximations from the optimistic, weakly optimistic, pessimistic, weakly pessimistic, optimistic-pessimistic, weakly optimistic-pessimistic, pessimistic-optimistic, and weakly pessimistic-optimistic viewpoints, and provide the decision rules of the proposed rough set models. The work in this paper brings the decision behavior based on a multi-granulation decision-theoretic rough set closer to the actual situation. [ABSTRACT FROM AUTHOR]

Neighborhood granulation is a classical granulation method. Although it is adequate for clustering and classification tasks, its granules are more complex, and the data representation is binary. This paper proposes a new granulation method based on the neighborhood granulation. Firstly, a detailed definition of the granular form is given with fuzzy rough set theory. Then, a modified fuzzy rough discriminant function is proposed based on neighborhood systems. The samples are globally granulated on single features to construct granules and on multiple features to construct granular vectors. Also, a feature selection technique based on the Chi-square, which strikingly reduces the complexity of the fuzzy rough granular vectors, is introduced to address the disadvantage of the fuzzy rough granular vectors. An ensemble model structure is also proposed in the paper for the mixed nature of fuzzy rough granular vectors. The paper makes a detailed comparison between the fuzzy rough granulation and the neighborhood granulation. The results show that fuzzy rough granulation has higher computational efficiency and classification performance. Finally, a detailed comparison is made between the fuzzy rough granular ensemble model and various classical ensemble algorithms. The final results show that the fuzzy rough granular ensemble model has better robustness and generalization. [ABSTRACT FROM AUTHOR]

Neighborhood granulation is a classical granulation method. Although it is adequate for clustering and classification tasks, its granules are more complex, and the data representation is binary. This paper proposes a new granulation method based on the neighborhood granulation. Firstly, a detailed definition of the granular form is given with fuzzy rough set theory. Then, a modified fuzzy rough discriminant function is proposed based on neighborhood systems. The samples are globally granulated on single features to construct granules and on multiple features to construct granular vectors. Also, a feature selection technique based on the Chi-square, which strikingly reduces the complexity of the fuzzy rough granular vectors, is introduced to address the disadvantage of the fuzzy rough granular vectors. An ensemble model structure is also proposed in the paper for the mixed nature of fuzzy rough granular vectors. The paper makes a detailed comparison between the fuzzy rough granulation and the neighborhood granulation. The results show that fuzzy rough granulation has higher computational efficiency and classification performance. Finally, a detailed comparison is made between the fuzzy rough granular ensemble model and various classical ensemble algorithms. The final results show that the fuzzy rough granular ensemble model has better robustness and generalization. [ABSTRACT FROM AUTHOR]

The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques. [ABSTRACT FROM AUTHOR]

Water quality inspection (WQI) is one of the primary ways to ensure the safe utilization of water resources, and complicated data modeling, fusion and analysis play a significant role in seeking the resource with the best water quality. Nevertheless, the challenges of missing data, relatively large differences in decision results and bounded rationality owned by decision-makers (DMs) in terms of WQI still exist nowadays. Thus, from the aspect of stable and behavioral decision-making in multi-granularity incomplete intuitionistic fuzzy information systems (MG-IIFISs), the paper investigates a comprehensive multi-attribute group decision-making (MAGDM) approach for the application of WQI. First, the concept of MG-IIFISs is built by modeling MAGDM problems with intuitionistic fuzzy numbers (IFNs), then a new transformation scheme is constructed for transforming MG-IIFISs into multi-granularity intuitionistic fuzzy information systems (MG-IFISs) based on the similarity principle. Second, three types of multigranulation intuitionistic fuzzy probabilistic rough sets (MG IF PRSs) are developed by referring to the MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus the full MULTIplicative form) method. Afterwards, attribute weights are objectively calculated based on the best-worst method (BWM), and a new stable and behavioral MAGDM approach is constructed by means of the TODIM (an acronym in Portuguese for interactive multi-criteria decision-making) method. At last, a case study in the setting of WQI is conducted with the support of a UCI data set, and sensitivity analysis, comparative analysis and experimental analysis are investigated to display the validity of the proposed approach. In general, the proposed approach improves the stability of decision results via MULTIMOORA and BWM, and also fully considers the bounded rationality of DMs' psychological behaviors from the aspect of the TODIM method, which has certain advantages in the community of MAGDM studies. [ABSTRACT FROM AUTHOR]

Attribute reduction based on rough sets is an effective approach of data learning in intelligent systems, and it has two basic types. Traditional classification-based attribute reducts mainly complete the classification task, while recent class-specific reducts directly realize the class-pattern recognition. Neighborhood rough sets have the covering-structure extension and data-diversity applicability, but their attribute reducts concern only the neighborhood classification-based reducts. This paper proposes class-specific attribute reducts based on neighborhood rough sets, so as to promote the optimal identification and robust processing of specific classes. At first, neighborhood class-specific reducts are defined, and their basic properties and heuristic algorithms are acquired by granulation monotonicity. Then, hierarchical relationships between the neighborhood classification-based and class-specific reducts are analyzed, and mutual derivation algorithms are designed. Finally, the theoretical constructions and mutual relationships are effectively verified by both decision table examples and data set experiments. The neighborhood class-specific reducts robustly extend the existing class-specific reducts, and they also provide a hierarchical mechanism for the neighborhood classification-based reducts, thus facilitating wide applications of class-pattern processing. [ABSTRACT FROM AUTHOR]

Granular computing is a relatively new platform for constructing, describing and processing information or knowledge. For crisp information granulation, the universe is decomposed into granules by binary relations on the universe, say, preorder, tolerance and equivalence relations. A knowledge structure is composed of all information granules induced by a relation that corresponds to the granulation. This paper establishes a novel theoretical framework for the measurement of information granularity of knowledge structures. First, two new relations between knowledge structures are introduced through the use of their respective Boolean relation matrices, where the granular equality relation is defined based on an orthogonal transformation with the transformation matrix being a permutation matrix, and the granularly finer relation is presented by combining the classical finer relation and the orthogonal transformation. Then, it is demonstrated that the simplified knowledge structure base with the granularly finer relation is a partially ordered set, which can be represented by a Hasse diagram. Subsequently, an axiomatic definition of information granularity is proposed to satisfy the constraints regarding these two relations. Moreover, a general form of the information granularity is given, and some existing measures are proved to be its special cases. Finally, as an application of the proposed measure, the attribute significance measure is developed based on the information granularity. [ABSTRACT FROM AUTHOR]

Multi-granulation decision-theoretic rough sets uses the granular structures induced by multiple binary relations to approximate the target concept, which can get a more accurate description of the approximate space. However, Multi-granulation decision-theoretic rough sets is very time-consuming to calculate the approximate value of the target set. Local rough sets not only inherits the advantages of classical rough set in dealing with imprecise, fuzzy and uncertain data, but also breaks through the limitation that classical rough set needs a lot of labeled data. In this paper, in order to make full use of the advantage of computational efficiency of local rough sets and the ability of more accurate approximation space description of multi-granulation decision-theoretic rough sets, we propose to combine the local rough sets and the multigranulation decision-theoretic rough sets in the covering approximation space to obtain the local multigranulation covering decision-theoretic rough sets model. This provides an effective tool for discovering knowledge and making decisions in relation to large data sets. We first propose four types of local multigranulation covering decision-theoretic rough sets models in covering approximation space, where a target concept is approximated by employing the maximal or minimal descriptors of objects. Moreover, some important properties and decision rules are studied. Meanwhile, we explore the reduction among the four types of models. Furthermore, we discuss the relationships of the proposed models and other representative models. Finally, illustrative case of medical diagnosis is given to explain and evaluate the advantage of local multigranulation covering decision-theoretic rough sets model. [ABSTRACT FROM AUTHOR]

The original rough set model, developed by Pawlak depends on a single equivalence relation. Qian et al, extended this model and defined multigranulation rough sets by using finite number of equivalence relations. This model provide new direction to the research. Recently, Shabir et al. proposed a rough set model which depends on a soft relation from an universe V to an universe W. In this paper we are present multigranulation roughness based on soft relations. Firstly we approximate a non-empty subset with respect to aftersets and foresets of finite number of soft binary relations. In this way we get two sets of soft sets called the lower approximation and upper approximation with respect to aftersets and with respect to foresets. Then we investigate some properties of lower and upper approximations of the new multigranulation rough set model. It can be found that the Pawlak rough set model, Qian et al. multigranulation rough set model, Shabir et al. rough set model are special cases of this new multigranulation rough set model. Finally, we added two examples to illustrate this multigranulation rough set model. [ABSTRACT FROM AUTHOR]

Multigranulation rough set (MGRS) theory provides an effective manner for the problem solving by making use of multiple equivalence relations. As the information systems always dynamically change over time due to the addition or deletion of multiple objects, how to efficiently update the approximations in multigranulation spaces by making fully utilize the previous results becomes a crucial challenge. Incremental learning provides an efficient manner because of the incorporation of both the current information and previously obtained knowledge. In spite of the success of incremental learning, well-studied findings performed to update approximations in multigranulation spaces have relatively been scarce. To address this issue, in this paper, we propose matrix-based incremental approaches for updating approximations from the perspective of multigranulation when multiple objects vary over time. Based on the matrix characterization of multigranulation approximations, the incremental mechanisms for relevant matrices are systematically investigated while adding or deleting multiple objects. Subsequently, in accordance with the incremental mechanisms, the corresponding incremental algorithms for maintaining multigranulation approximations are developed to reduce the redundant computations. Finally, extensive experiments on eight datasets available from the University of California at Irvine (UCI) are conducted to verify the effectiveness and efficiency of the proposed incremental algorithms in comparison with the existing non-incremental algorithm. [ABSTRACT FROM AUTHOR]

This study proposes the multi attribute group decision making in the presence of incomplete multi attribute and incomplete multi decision while making a decision with preferences in an incomplete information system. We then consider resolving the problem in an incomplete information system by using two different approximation strategies, that is seeking the common reserving difference and common rejecting difference, four kinds of soft dominance based multi-granulation rough sets namely soft dominance based optimestic multi-granulation rough sets and soft dominance based pessimistic multi-granulation rough sets are presented. Another worth mentioning contibution of this paper is to disclose the ideas of two kinds of approximate precision, rough degree, approximate quality, maximal and minimal rough member ships and their mutual relationships. Finally the validity of these concepts are proved by constructing two algorithms and applying them in solving incomplete multi-agent conflict analysis problem. [ABSTRACT FROM AUTHOR]

Three-way decisions have become a representative of the models dealing with decision-making problems with uncertainty and fuzziness. However, most of the current models are single granular structures that cannot meet the needs of complex fuzzy environmental decision-making. Multi-granulation rough sets can better deal with fuzzy problems of multiple granularity structures. Therefore, three-way decisions will be a more reasonable decision-making model to address uncertain decision problems in the context of multiple granularity structures. In this paper, firstly we propose the four different conditional probabilities based on support intuitionistic fuzzy sets, which are referred to as support intuitionistic fuzzy probability. Then, a multi-granulation support intuitionistic fuzzy probabilistic approximation space is defined. Secondly, we calculate the thresholds α and β by the Bayesian theory, and construct four different types of multi-granulation support intuitionistic fuzzy probabilistic rough sets models in multi-granulation support intuitionistic fuzzy probabilistic approximation space. Moreover, some properties of lower and upper approximation operators of these models are discussed. Thirdly, by combining these proposed models with three-way decision theory, the corresponding three-way decision models are constructed and three-way decision rules are derived. Finally, an example of person-job fit procedure is given to prove and compare the validity of these proposed models. [ABSTRACT FROM AUTHOR]

Z-soft rough covering models introduced by zhan et al are important generalizations of classical rough set theory to deal with more complex problems of real world. So far, the existing studies mainly focus on constructing various forms of approximation operators and their related properties by means of neighborhoods. In this paper, we introduce different kinds of uncertainty measures related to Z-soft rough covering sets and discuss their limitations. An axiomatic definition of knowledge granulation for soft covering approximations space is introduced. Some main theoretical results are obtained and investigated with the help of examples. Finally, a fully developed example describing the application of the proposed theory in multicriteria decision making is constructed. [ABSTRACT FROM AUTHOR]

As a new and meaningful extension of the Pawlak rough set, multi-granulation rough sets (MGRSs) have attracted much attention and fruitful achievements have been reported in different aspects. By combining with fuzzy rough set, the paper introduces multi-granulation fuzzy rough sets in the covering approximation space, namely, covering-based multi-granulation fuzzy rough sets (CMFRS), which form the extension of fuzzy rough sets. We first investigate several important properties of lower and upper approximations of concepts in covering-based multi-granulation fuzzy rough sets and elaborate on the differences between the proposed models and the existing ones in literature. By employing the notions of reduct and exclusion of a covering, the paper studies the necessary and sufficient conditions for two CMFRS to generate identical lower and upper approximations of a target concept in the given covering approximation space. Finally, the relationships between the new models are explored in the paper. [ABSTRACT FROM AUTHOR]

This paper proposes a new approach to uncertainty multiple attribute group decision problem with linguistic preference relation based on multigranulation probabilistic rough set and the Multimoora method. According to the classical Pawlak rough set and the neighborhood rough set, we present a multigranulation probabilistic fuzzy rough set based on neighborhood relation with linguistic preference information. We investigate the rough approximation of a crisp decision-making object and a fuzzy decision-making object under the framework of multigranulation rough set theory with linguistic preference inforamtion, respectively. That is, a multigranulation probabilistic rough set model and a multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation are established, respectively. Meanwhile, the proposed multigranulation probabilistic fuzzy rough set model is compared with the existing model and illustrate the superiority of the new model established. Furthermore, by combining the multigranulation probabilistic fuzzy rough set and the Multimoora method, a new method for multiple attribute group decision making with linguistic preference information is proposed. The decision-making procedure and the algorithm of the proposed method are also given. Finally, the effectiveness and validity of the proposed method is verified by investigating a multiple attribute group decision problem with the selection of suppliers in the context of e-commerce. [ABSTRACT FROM AUTHOR]

Rough set theory proposed by professor Pawlak in 1982 is an important tool to solve uncertain problems. In order to make rough sets deal with uncertain problems better, by considering the stringent notions of mathematical equality and inclusion, the definitions of rough equality and rough inclusion were introduced by Pawlak. But the researches of rough equality and rough inclusion are on a certain granularity space. In this paper, some change rules will be given with respect to the relations of rough equality and rough inclusion between two uncertain target sets in multi-granulation spaces, and the similarity degree between two roughly equal sets is proposed to describe the similarity of two uncertain sets in multi-granulation spaces. In order to describe two uncertain sets which are rough equality by an approximation set at the same time, the definitions of optimistic λ-approximation set and pessimistic λ-approximation set will be defined from the points of optimism and pessimism, and some properties of them are discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper we propose fuzzy multi-agent recommender system for candidates in state elections that is intended to serve to voters as a filter for excess of information on Internet and other media on their election candidates. The recommender system we propose should guide voters to the candidates that suit their preferences, and help them make a choice of a representative by accessing relevant information. For this system we have enrolled computing with words (CW) methodology which main concepts are graduation (linguistic variables) and granulation (a fuzzy set of points drawn together by similarity). Our proposed recommender system is appropriate for application in developing countries, because of its simplicity and its low cost. It can also serve as foundation for future development and sophistication, taking into consideration both technical and political aspects. [ABSTRACT FROM AUTHOR]

We analyze the properties and characteristics of the information structure in incomplete lattice-valued information system (ILIS), we redefine the information structure and the dependence and information distance between the two information structures. In addition, in order to evaluate the uncertainty of ILIS, the concepts of granular measure and entropy measure are expounded, including information granulation, information quantity, rough entropy and information entropy. Finally, we carry out numerical experiments to verify the feasibility of the method, and carry out effective statistical analysis. These results are conducive to the establishment of granular computing framework in ILIS. [ABSTRACT FROM AUTHOR]

As an extension of the classical Pawlak rough sets, covering multigranulation trapezoidal fuzzy decision-theoretic rough set models have been researched, in which objects approximated are crisp sets or accurate concepts of the universe of discourse, and there are only two states, which are disjoint and opposite each other for an accurate concept of the universe of discourse for a decision-making problem. However, the objects of many decision-making problems can have more than two states in practice. Moreover, the states of the decision object are not necessarily disjoint and opposite each other, but these decision states are fuzzy descriptions of the states of the object on the universe. In order to deal with decision-making problems in which decision states are fuzzy sets, covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models are proposed in this paper. In addition, the mean, optimistic and pessimistic covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets are investigated and characterized. Furthermore, the relationships between covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and the existing models are discussed. Finally, an example is presented to illustrate the practicality and generality of the proposed models. [ABSTRACT FROM AUTHOR]

Atanassov's intuitionistic fuzzy 𝒯 equivalence information systems are natural extensions of fuzzy 𝒯 equivalence information systems. The aim of this paper is to investigate the uncertainty measures of knowledge in Atanassov's intuitionistic fuzzy 𝒯 equivalence information systems. At the first, we introduce the concepts of knowledge granulation, knowledge entropy and knowledge uncertainty measure in Atanassov's intuitionistic fuzzy 𝒯 equivalence information systems, and some important properties of them are studied. From these properties, it can be shown that these measures provide important approaches to measuring the discernibility ability of different knowledge in Atanassov's intuitionistic fuzzy 𝒯 equivalence information systems. And relationships among knowledge granulation, knowledge entropy and knowledge uncertainty measure are considered. Furthermore, we introduce the definition of rough entropy of rough sets in Atanassov's intuitionistic fuzzy 𝒯 equivalence information systems. By an example, it is shown that the rough entropy of rough set is more accurate than natural extension of classical rough degree to measure the roughness of rough set in Atanassov's intuitionistic fuzzy 𝒯 equivalence information systems. [ABSTRACT FROM AUTHOR]

Intrusion detection systems have become one of the important tools for network security due to the frequent attacks brought about by the explosive growth of network traffic. Autoencoder is an unsupervised learning model with a neural network structure. It has a powerful feature learning capability and is effective in intrusion detection. However, its network construction suffers from overfitting and gradient disappearance problems. Traditional granular computing methods have advantages in solving such problems, but the process is relatively complex, the granularity dimension is high, and the computational cost is large, which is not suitable for application in intrusion detection systems. To address these problems, we propose a novel autoencoder: Granular AutoEncoders (GAE). The granulation reference set is constructed by random sampling. The granulation of training samples is based on single-feature similarity in a reference set to form granules. The granulation of multiple features results in granular vectors. Some operations of granules are defined. Furthermore, we propose some granular measures, including granular norms and granular loss functions. The GAE is further applied to the field of intrusion detection by designing an anomaly detection algorithm based on the GAE. The algorithm determines whether the network flows are anomalous by comparing the difference between an input granular vector and its output granular vector that is reconstructed by the GAE. Finally, some experiments are conducted using an intrusion detection dataset, comparing multiple metrics in terms of precision, recall, and F1-Score. The experimental results validate the correctness and effectiveness of the intrusion detection method based on GAE. And contrast experiments show that the proposed method has stronger ability for detecting anomalies than the correlation algorithms. [ABSTRACT FROM AUTHOR]

Learning Vector Quantization (LVQ) is a clustering method with supervised information, simple structures, and powerful functions. LVQ assumes that the data samples are labeled, and the learning process uses labels to assist clustering. However, the LVQ is sensitive to initial values, resulting in a poor clustering effect. To overcome these shortcomings, a granular LVQ clustering algorithm is proposed by adopting the neighborhood granulation technology and the LVQ. Firstly, the neighborhood granulation is carried out on some features of a sample of the data set, then a neighborhood granular vector is formed. Furthermore, the size and operations of neighborhood granular vectors are defined, and the relative and absolute granular distances between granular vectors are proposed. Finally, these granular distances are proved to be metrics, and a granular LVQ clustering algorithm is designed. Some experiments are tested on several UCI data sets, and the results show that the granular LVQ clustering is better than the traditional LVQ clustering under suitable neighborhood parameters and distance measurement. [ABSTRACT FROM AUTHOR]

In this study, we develop a novel clustering with double fuzzy factors to enhance the performance of the granulation-degranulation mechanism, with which a fuzzy rule-based model is designed and demonstrated to be an enhanced one. The essence of the developed scheme is to optimize the construction of the information granules so as to eventually improve the performance of the fuzzy rule-based models. In the design process, a prototype matrix is defined to express the Fuzzy C-Means based granulation-degranulation mechanism in a clear manner. We assume that the dataset degranulated from the formed information granules is equal to the original numerical dataset. Then, a clustering method with double fuzzy factors is derived. We also present a detailed mathematical proof for the proposed approach. Subsequently, on the basis of the enhanced version of the granulation-degranulation mechanism, we design a granular fuzzy model. The whole design is mainly focused on an efficient application of the fuzzy clustering to build information granules used in fuzzy rule-based models. Comprehensive experimental studies demonstrate the performance of the proposed scheme. [ABSTRACT FROM AUTHOR]

The multi-granulation rough sets serve as important hierarchical models for intelligent systems. However, their mainstream optimistic and pessimistic models are respectively too loose and strict, and this defect becomes especially serious in hierarchical processing on an attribute-expansion sequence. Aiming at the attribute-addition chain, compromised multi-granulation rough set models are proposed to systematically complement and balance the optimistic and pessimistic models. According to the knowledge refinement and measure order induced by the attribute-enlargement sequence, the basic measurement positioning and corresponding pointer labeling based on equilibrium statistics are used, and thus we construct four types of compromised models at three levels of knowledge, approximation, and accuracy. At the knowledge level, the median positioning of ordered granulations derives Compromised-Model 1; at the approximation level, the average positioning of approximation cardinalities is performed, and thus the separation and integration of dual approximations respectively generate Compromised-Models 2 and 3; at the accuracy level, the average positioning of applied accuracies yields Compromised-Model 4. Compromised-Models 1–4 adopt distinctive cognitive levels and statistical perspectives to improve and perfect the multi-granulation rough sets, and their properties and effectiveness are finally verified by information systems and data experiments. [ABSTRACT FROM AUTHOR]

An extraction of granular structures using graphs is a powerful mathematical framework in human reasoning and problem solving. The visual representation of a graph and the merits of multilevel or multiview of granular structures suggest the more effective and advantageous techniques of problem solving. In this research study, we apply the combinative theories of rough fuzzy sets and rough fuzzy digraphs to extract granular structures. We discuss the accuracy measures of rough fuzzy approximations and measure the distance between lower and upper approximations. Moreover, we consider the adjacency matrix of a rough fuzzy digraph as an information table and determine certain indiscernible relations. We also discuss some general geometric properties of these indiscernible relations. Further, we discuss the granulation of certain social network models using rough fuzzy digraphs. Finally, we develop and implement some algorithms of our proposed models to granulate these social networks. [ABSTRACT FROM AUTHOR]

In this research, we succeeded in introducing a new reasoning procedure which applies interval type-2 fuzzy sets into a rule induction process. Our proposal allows information granulation which resulted in achieving good experimental results. We introduced decision tables with elements assumed as interval type-2 fuzzy sets which greatly generalize information. Next, by applying corresponding rule induction procedure, we introduced the possibility to generate directly from a benchmark data fuzzy rulebases for type-2 fuzzy inference models. We strongly believe that our reasoning approach will be a proper solution for different research issues such as classification or ranking procedures as well as determining knowledge for fuzzy inference models. The method proposed was tested in a classification problem verified by using medical benchmark data. [ABSTRACT FROM AUTHOR]

Through a combination of hesitant fuzzy sets with rough sets, this study develops a single-granulation hesitant fuzzy rough set model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough set model, called the optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets. In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the single-granulation hesitant fuzzy rough sets, optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets are also investigated. Finally, we develop an approximation reduction approach of multi-granulation hesitant fuzzy rough sets to eliminate redundant hesitant fuzzy granulations with a detailed example. [ABSTRACT FROM AUTHOR]

Hesitant Pythagorean fuzzy set (HPFS) is a generalization of fuzzy set (FS). By integrating HPFS with rough set (RS), the notion of single granulation hesitant Pythagorean fuzzy rough sets (SGHPFRSs) is firstly put forward. Then, in view of the multigranulation framework, two types of multigranulation hesitant Pythagorean fuzzy rough sets (MGHPFRSs) are proposed, which are called the optimistic multigranulation hesitant Pythagorean fuzzy rough sets (OMGHPFRSs) and pessimistic multigranulation hesitant Pythagorean fuzzy rough sets (PMGHPFRSs). The connections between serial hesitant Pythagorean fuzzy relations and hesitant Pythagorean fuzzy approximation operators are established. The relationship between the MGHPFRSs and SGHPFRSs will be explored. Moreover, the relationship between the OMGHPFRSs and PMGHPFRSs will be established subsequently. Ultimately, an illustrative example will be given to expound the availability about the MGHPFRSs in multi-attribute decision making. [ABSTRACT FROM AUTHOR]

Similarity measure is an important uncertainty measurement in intuitionistic fuzzy set (IFS) theory. In this study, a novel similarity measure is presented by the combination of the information carried by hesitancy degree and the endpoint distance of membership and nonmembership, respectively. Moreover, a numerical example is used to verify the reasonable of the proposed similarity measure. After that, the similarity measure is applied to construct the IF decision-theoretic rough set (IF-DTRS) model and multigranulation IF decision-theoretic rough set (MG-IF-DTRS) model. Some properties of IF-DTRS and MG-IF-DTRS are also investigated. Thirdly, based on granular significance, a novel approach of optimal granulation selection is formulated. Finally, a heuristic algorithm is designed and the effectiveness of this algorithm is demonstrated by an illustrative example. [ABSTRACT FROM AUTHOR]