Single Granulation Research Articles

Soft ordered double quantitative approximations based three-way decisions and their applications

The classical theory of rough set was established by Pawlak, which mainly focusses on the approximation of sets characterized by a single equivalence relation over the universe. However, most of the current single granulation structure models cannot meet the user demand or the target of solving problems. Multigranulation rough sets approach can better deal with the problems, where data might be spread over various locations. In this article, we present the idea of soft preference and soft dominance relation for the development of soft dominance rough set in an incomplete information system. Subsequently, several important structural properties and results of the proposed model are carefully analyzed. After employing soft dominance based rough set approach to it for any times, we can only get six different sets at most in an incomplete information system. 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. Based on soft dominance relation, we introduce logical disjunction/conjunction soft dominance optimistic/pessimistic multigranulation decision theoretic rough approximations in an incomplete information. Meanwhile, to measure the uncertainty of soft dominance optimistic/pessimistic multigranulation decision theoretic rough approximation and some of their interesting properties are examined. Thereafter, a novel multi attribute with multi decision making problem approach based on logical disjunction/conjunction soft dominance optimistic/pessimistic multigranulation decision theoretic rough sets approach are developed to solve the selection of medicine to treat the coronavirus disease (COVID-19). The basic principle and the detailed steps of the decision making model (algorithms) are presented in detail. To demonstrate the applicability and potentiality of the proposed model, we present a practical example of a medical diagnosis is given to validate the practicality of the technique.

Uncertainty Optimization-Based Rough Set for Incomplete Information Systems

Often the information in the surrounding world is incomplete, and such incomplete information gives rise to uncertainties. Pawlak's rough set model is an approach to approximation under uncertainty. It uses a tolerance relation to obtain single granulation of the incomplete information system for approximation. In this work, we extend the single granulation rough set for the incomplete information system to an uncertainty optimization-based rough set (UOBRS). The proposed approach is used to minimize the uncertainty using multiple tolerance relations. We list properties of the UOBRS for incomplete information systems. We compare UOBRS with the classical single granulation rough set (SGRS) and multi-granular rough set (MGRS). We list the basic properties of the UOBMGRS. We introduce the application of the UOBRS for attribute subset selection in case of incomplete information system. We use the measure of approximation quality to assess the uncertainties of the attributes. We compare the approximation quality of the attributes using UOBRS with the approximation quality using SGRS and MGRS. We get higher approximation quality with the less number of attributes using UOBRS as compared to SGRS and MGRS. The proposed method is a novel approach to dealing with incomplete information systems for more effective dataset analysis.

Multigranulation Modified Rough Bipolar Soft Sets and Their Applications in Decision-Making

The classical theory of rough sets (RSs) established by Pawlak, mainly focused on the approximation of sets characterized by a single equivalence relation (ER) over the universe. However, most of the current single granulation structure models cannot meet the user demand or the target of solving problems. Multi-granulation rough sets (MGRS) can better deal with the problems where data might be spread over various locations. In this article, based on modified rough bipolar soft sets (MRBSs), the concept of multi-granulation MRBSs (MGMRBSs) is introduced. A finite collection of bipolar soft sets (BSs) has been used for this purpose. Several important structural properties and results of the suggested model are carefully analyzed. Meanwhile, to measure the uncertainty of MGMRBSs, some important measures associated with MGMRBSs are presented in MGMRBS-approximation space, and some of their interesting properties are examined. In the framework of multi-granulation, we developed optimistic MGMRBSs (OMGMRBSs) and pessimistic MGMRBSs (PMGMRBSs). The relationships among the MGMRBSs, OMGMRBSs, and PMGMRBSs are also established. After that, a novel multi-criteria group decision-making (MCGDM) approach based on OMGMRBSs is developed to solve some problems in decision-making (DM). The basic principles and the detailed steps of the DM model are presented in detail. To demonstrate the applicability and potentiality of the developed model, we give a practical example of a medical diagnosis. Finally, we conduct a comparative study of the proposed MCGDM approach with some existing techniques to endorse the advantages of the proposed model.

A Study on Soft Multi-Granulation Rough Sets and Their Applications

Rough set (RS) and soft set (SS) theories are two successful mathematical approaches to dealing with uncertainty in data analysis. The classical soft rough set (SRS) theory proposed by Feng et al. [8] offers a formal theoretical framework for solving the uncertainty under a single granulation environment. However, it is essential to note that the SRS theory cannot be applied in the context of multi-granulation in the real world. To address this issue, in this paper, we introduce the idea of soft multi-granulation RS (SMGRS) model based on two soft binary relations (S-BRs). Axiomatic operations, lower soft rough approximation space (lower SRA-space) and upper soft rough approximation space (upper SRA-space), are defined through after sets of soft relations. After that, the concept of SMGRSs is applied to a significant part of commutative algebra, group theory. In this respect, the primitive notions of SRA-spaces are defined with the help of two normal soft groups (NSGs). In groups, several important structural properties related to SMGRS are investigated in detail with illustrative examples. It is shown that SMGRS in groups may be influential in decision-making (DM) by some numerical examples. To demonstrate the flexibility, superiority, and effectiveness of the suggested technique, some comparative examples are given with some existing methods.

Anaerobic granulation of single culture Clostridium beijerinckii

This study utilizes Clostridium beijerinckii P260, an anaerobic bacterium capable of acetone-butanol-ethanol (ABE) production, as model bacteria to test the hypothesis that single culture anaerobic granulation can be achieved in accordance with the biogranulation theory developed from the mixed culture studies. After four months of cultivation, C. beijerinckii granules formed under the selection pressure of a selective settling velocity at 9.0 m h−1 in a sequential batch reactor (SBR). These C. beijerinckii granules possess excellent settleability and exceptionally high cell and enzyme concentrations, which demonstrate promising potential in the application for high-rate fermentation and separation of value-added products. However, C. beijerinckii granules only accounted for 25% of the cell mass in the SBR, leaving the rest 75% still in the form of planktonic cells, which indicates an incomplete anaerobic granulation. The lack of ideal feast and famine conditions in the cyclic SBR operation was regarded as the reason. Future research should focus on the optimization of the bioreactor operation to create optimal conditions for inducing the phenotype switch from planktonic to granulated growth mode.

(α, β)-Multi-granulation bipolar fuzzified rough sets and their applications to multi criteria group decision making

Pawlak’s rough set theory based on single granulation has been extended to multi-granulation rough set structure in recent years. Multi-granulation rough set theory has become a flouring research direction in rough set theory. In this paper, we propose the notion of (α, β)-multi-granulation bipolar fuzzified rough set ((α, β)-MGBFRSs). For this purpose, a collection of bipolar fuzzy tolerance relations has been used. In the framework of multi-granulation, we proposed two types of (α, β)-multi-granulation bipolar fuzzified rough sets model. One is called the optimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) o-MGBFRSs) and the other is called the pessimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) p-MGBFRSs). Subsequently, a number of important structural properties and results of proposed models are investigated in detail. The relationships among the (α, β)-MGBFRSs, (α, β) o-MGBFRSs and (α, β) p-MGBFRSs are also established. In order to illustrate our proposed models, some examples are considered, which are helpful for applying this theory in practical issues. Moreover, several important measures associated with (α, β)-multi-granulation bipolar fuzzified rough set like the measure of accuracy, the measure of precision, and accuracy of approximation are presented. Finally, we construct a new approach to multi-criteria group decision-making method based on (α, β)-MGBFRSs, and the validity of this technique is illustrated by a practical application. Compared with the existing results, we also expound its advantages.

A Novel Concept of Uncertainty Optimization Based Multi-Granular Rough Set and Its Application

Data is generating at an exponential pace with the advancement in information technology. Such data highly contain uncertain and vague information. The rough set approximation is a way to find information in the data-set under uncertainty and to classify objects of the dataset. This work presents a mathematical approach to evaluate the data-sets uncertainties and their application to data reduction. In this work, we have extended the multi-granulation variable precision rough set in the context of uncertainty optimization. We develop an uncertainty optimization-based multi-granular rough set (UOMGRS) to minimize the uncertainties in the data set more effectively. Using UOMGRS, we find the most informative attribute in the feature space. It is desirable to minimize the rough set boundary region using the attribute having the highest approximation quality. Thus we group the attributes whose relative quality of approximation is the maximum to maximize the positive region and to minimize the uncertain region. We compare the UOMGRS with the single granulation rough set (SGRS) and the multi-granular rough set (MGRS). By our proposed method, we require only an average of 62% attributes for approximation whereas, SGRS and MGRS need an average of at least 72% of attributes in the data set for approximation of the concepts in the data-set. Our proposed method requires less amount of data for the classification of objects in the dataset. The method helps minimize the uncertainties in the dataset in a more efficient way.

A fuzzy granular sparse learning model for identifying antigenic variants of influenza viruses

Sparse learning has significant applications in statistics, big data, bioinformatics and machine learning. In big data systems, a large amount of redundant, missing and noisy data cause sparsity, and the rapid changes of information result in uncertainty. Since the traditional sparse learning model is difficult to deal with uncertain data, we propose a Fuzzy Granular Sparse Learning (FGSL) model for identifying antigenic variants of influenza viruses. Firstly, a fuzzy set theory is introduced to measure and granulate the influenza viruses. Some fuzzy granules are induced by a single feature fuzzy granulation. Then, a fuzzy granular vector is constructed from these fuzzy granules, and the fuzzy granular regression is presented. Some constraint norms for granules and granular vectors are proposed, which are two granule norms and four granular vector norms. Therefore, the FGSL model is constructed based on granular regression and constraint norms. The FGSL model includes granular ridge and lasso regressions under different constraint norms. Furthermore, we prove the derivative forms of two granular regression functions, guaranteeing the convergence of the FGSL model. The optimization problem of the FGSL model is discussed and two gradient descent algorithms of the FGSL model are designed. Finally, we employ the FGSL model to serologic data and hemagglutinin sequences for learning antigenicity-associated mutations and inferring antigenic variants. The experimental results confirm some advantages of the FGSL model with fast convergence, low RMSE and strong feature selection ability. We successfully identify antigenic variants of influenza viruses by the FGSL model.

Multiple Granulation Rough Set Approach to Interval-Valued Intuitionistic Fuzzy Ordered Information Systems

As a further extension of the fuzzy set and the intuitive fuzzy set, the interval-valued intuitive fuzzy set (IIFS) is a more effective tool to deal with uncertain problems. However, the classical rough set is based on the equivalence relation, which do not apply to the IIFS. In this paper, we combine the IIFS with the ordered information system to obtain the interval-valued intuitive fuzzy ordered information system (IIFOIS). On this basis, three types of multiple granulation rough set models based on the dominance relation are established to effectively overcome the limitation mentioned above, which belongs to the interdisciplinary subject of information theory in mathematics and pattern recognition. First, for an IIFOIS, we put forward a multiple granulation rough set (MGRS) model from two completely symmetry positions, which are optimistic and pessimistic, respectively. Furthermore, we discuss the approximation representation and a few essential characteristics for the target concept, besides several significant rough measures about two kinds of MGRS symmetry models are discussed. Furthermore, a more general MGRS model named the generalized MGRS (GMGRS) model is proposed in an IIFOIS, and some important properties and rough measures are also investigated. Finally, the relationships and differences between the single granulation rough set and the three types of MGRS are discussed carefully by comparing the rough measures between them in an IIFOIS. In order to better utilize the theory to realistic problems, an actual case shows the methods of MGRS models in an IIFOIS is given in this paper.

Granulation in Rough Set Theory: A novel perspective

Considering data from different perspectives (views or granulations) is very common in several applications nowadays. Unfortunately, Rough Sets lack of effective tools for handling multiple granulation in mixed and incomplete information systems. This paper introduces a novel approach for dealing with such information systems: The Parameterized Granulation. The demonstration of some of the properties of such granulation approach, as well as the theoretical and experimental analysis carried out, show the capabilities of this novel granulation, and result in great utility in the study of mixed and incomplete information systems, not only in the context of a single granulation, but in the context of multiple granulations.

Powder bed packing and API content homogeneity of granules in single drop granule formation

Single drop granule formation on a static powder bed of pharmaceutical mixtures was studied to investigate the effects of hydrophobicity and primary particle size distribution on the powder bed packing structure and the content homogeneity of active pharmaceutical ingredient (API) in granules formed. The granule formation mechanisms, drop penetration time, granule morphology and internal structure have been previously investigated in a mixture of coarse microcrystalline cellulose (MCC) and fine acetaminophen (APAP). When the APAP amount increased (decreasing particle size and increasing hydrophobicity), drop penetration time increased, formation mechanisms transitioned from Spreading to Tunneling, the granules became smaller in size, and the internal porosity of the granules decreased (Gao et al., 2018). In the current study, single drop granulation on mixtures of MCC and APAP with different particle sizes were investigated for formation mechanisms and granule morphology. Additionally, the powder bed packing structure was characterized by X-ray micro-CT and the API content uniformity was measured by UV–vis spectrometry. It was found that in the mixture made from coarse MCC and fine APAP, the internal structure became heterogeneous and there were dense aggregate regions in both the powder bed and granules from 25% APAP proportion, where the transition from Spreading to Tunneling occurs. The content uniformities of granules from fine powder beds are much more compromised (indicating a discrepancy between the actual value and theoretical value) than those from coarse powder beds. This content discrepancy becomes much larger when the APAP proportion in the powder bed is higher (above 50%). This was previously observed by other researchers (Nguyen et al., 2010) and was attributed to the preferential wetting of the ingredients. It is believed that the primary particle size of the powder bed is more significant than the hydrophobicity in affecting the formation mechanism, granule internal structure, and content uniformity.

Granular neural networks: The development of granular input spaces and parameters spaces through a hierarchical allocation of information granularity

The issue of granular output optimization of neural networks with fixed connections within a given input space is explored. The numeric output optimization is a highly nonlinear problem if nonlinear activation functions are used; the granular output optimization becomes an even more challenging task. We solve the problem by developing an optimal hierarchical allocation of information granularity, proposing a new objective function which considers both specificity and evidence, and engaging here efficient techniques of evolutionary optimization. In contrast to the existing techniques, the hierarchical one builds a three-level hierarchy to allocate information granularity to the input space and the architecture (parameters) of the network. Granulating both the input features and the architecture at the same time return a different result with the single factor granulation. The constructed granular neural network emphasizes the abstract nature of data and the granular nature of nonlinear mapping of the architecture. Experimental studies completed for synthetic data and publicly available data sets are used to realize the algorithm.

New Multigranulation Neutrosophic Rough Set with Applications

After the neutrosophic set (NS) was proposed, NS was used in many uncertainty problems. The single-valued neutrosophic set (SVNS) is a special case of NS that can be used to solve real-word problems. This paper mainly studies multigranulation neutrosophic rough sets (MNRSs) and their applications in multi-attribute group decision-making. Firstly, the existing definition of neutrosophic rough set (we call it type-I neutrosophic rough set (NRSI) in this paper) is analyzed, and then the definition of type-II neutrosophic rough set (NRSII), which is similar to NRSI, is given and its properties are studied. Secondly, a type-III neutrosophic rough set (NRSIII) is proposed and its differences from NRSI and NRSII are provided. Thirdly, single granulation NRSs are extended to multigranulation NRSs, and the type-I multigranulation neutrosophic rough set (MNRSI) is studied. The type-II multigranulation neutrosophic rough set (MNRSII) and type-III multigranulation neutrosophic rough set (MNRSIII) are proposed and their different properties are outlined. We found that the three kinds of MNRSs generate tcorresponding NRSs when all the NRs are the same. Finally, MNRSIII in two universes is proposed and an algorithm for decision-making based on MNRSIII is provided. A car ranking example is studied to explain the application of the proposed model.

Drum granulation of single super phosphate fertilizer: Effect of process variables and optimization

In this work we investigated the effects of the drum rotation speed, liquid-phase ratio, granulation time, and drum hold-up on the single super-phosphate granulation performance in a rotary drum. Regression techniques were used to quantify the effects of those variables on the granulation efficiency and product properties. Equations to predict these responses as functions of the studied variables were proposed. Sequential Quadratic Programming (SQP) algorithm was used for the optimization study. The results showed that the levels of the process variables that led to the highest efficiency were not the same for the highest granule hardness. From the results of this study it is possible to identify conditions that improve the granulation efficiency and the physical properties of the final product.

A Comparative Study of MGRSs and their Uncertainty Measures

Multi-granulation rough set(MGRS), as a kind of fusion mechanism of different information or data, is an useful development of Pawlak rough set theory. Firstly, this paper gives an introduction for various types of MGRS, their properties and axiomatization characterizations are studied. We show that, except for the optimistic one, each of the existing MGRS means a single granulation rough set. Then, we made a comparative analysis on the different uncertainty measures among the various multi-granulation approximation spaces. At the basis of investigating for the existing uncertainty measures, we discuss their limitations via some examples, and propose a total ordered relation among approximation spaces, even in the more general covering ones. It will be better than the original partial relation in revealing uncertainty, which conceal in the approximation space or covering one. Finally, based on the total ordered relation, we present improved information entropy, rough entropy, knowledge granulation and axiomatic definition of the knowledge granulation measures. It is proved that they are more reasonable than the original ones. Then, some novel uncertainty measures and improved fusion uncertainty measures about various granulations are also proposed. By employing these measures, granulation measures of various MGRSs are defined and studied.

Variable precision multigranulation decision-theoretic fuzzy rough sets

This paper studies variable precision multigranulation fuzzy decision-theoretic rough sets in an information system. We firstly review definitions and properties of multigranulation fuzzy rough sets. A novel membership degree based on single granulation rough sets is proposed. Then two operators based on this membership degree are defined. By employing these operators, two types of variable precision multigranulation fuzzy rough sets in an information system are proposed. Finally, inspired by three-way decisions, we propose Type-1 variable precision multigranulation decision-theoretic fuzzy rough sets.

Study of Covering Based Multi Granular Rough Sets and Their Topological Properties

The notions of basic rough sets introduced by Pawlak as a model of uncertainty, which depends upon a single equivalence relation has been extended in many directions. Over the years, several extensions to this rough set model have been proposed to improve its modeling capabilities. From the granular computing point of view these models are single granulations only. This single granulation model has been extended to multi-granulation set up by taking more than one equivalence relations simultaneously. This led to the notions of optimistic and pessimistic multi-granulation. One direction of extension of the basic rough set model is dependent upon covers of universes instead of partitions and has better modeling power as in many real life scenario objects cannot be grouped into partitions but into covers, which are general notions of partitions. So, multigranulations basing on covers called covering based multi-granulation rough sets (CBMGRS) were introduced. In the literature four types of CBMGRSs have been introduced. The first two types of CBMGRS are based on minimal descriptor and the other two are based on maximal descriptor. In this paper all these four types of CBMGRS are studied from their topological characterizations point of view. It is well known that there are four kinds of basic rough sets from the topological characterisation point of view. We introduce similar characterisation for CBMGRSs and obtained the kinds of the complement, union, and intersection of such sets. These results along with the accuracy measures of CBMGRSs are supposed to be applicable in real life situations. We provide proofs and counter examples as per the necessity of the situations to establish our claims.

Optimistic Multi-granulation Rough Set Based Classification for Medical Diagnosis

Medical analysis has been approached by several machine learning methods for many years.Pattern recognition algorithms are capable of improving the quality of prediction, early diagnosis of diseases and disease classification. The classification complications in medical area are solved based on the outcome of medical analysis or report of medical treatment by the medical specialist. This research focuses on applying Rough set based data mining techniques for medical data to discover locally frequent diseases. This work applies Optimistic Multi-granulation rough set model (OMGRS) for medical data classification. Multi-granulation rough set provides efficient results than single granulation rough set model and soft rough set based classifier model. The results of applying the OMGRS methodology to medical diagnosis based upon selected information. The performance of the proposed optimistic multi granulation Rough set based classification is compared with other rough set based (RS), Kth Nearest Neighbor (KNN) and Back propagation neural network (BPN) approaches using various classification Measures.

Algebraic Properties of Rough Set on Two Universal Sets based on Multigranulation

The rough set philosophy is based on the concept that there is some information associated with each object of the universe. The set of all objects of the universe under consideration for particular discussion is considered as a universal set. So, there is a need to classify objects of the universe based on the indiscernibility relation (equivalence relation) among them. In the view of granular computing, rough set model is researched by single granulation. The granulation in general is carried out based on the equivalence relation defined over a universal set. It has been extended to multi-granular rough set model in which the set approximations are defined by using multiple equivalence relations on the universe simultaneously. But, in many real life scenarios, an information system establishes the relation with different universes. This gave the extension of multi-granulation rough set on single universal set to multi-granulation rough set on two universal sets. In this paper, we define multi-granulation rough set for two universal sets U and V. We study the algebraic properties that are interesting in the theory of multi-granular rough sets. This helps in describing and solving real life problems more accurately.

Multigranulations Rough Set Method of Attribute Reduction in Information Systems Based on Evidence Theory

Attribute reduction is one of the most important problems in rough set theory. However, from the granular computing point of view, the classical rough set theory is based on a single granulation. It is necessary to study the issue of attribute reduction based on multigranulations rough set. To acquire brief decision rules from information systems, this paper firstly investigates attribute reductions by combining the multigranulations rough set together with evidence theory. Concepts of belief and plausibility consistent set are proposed, and some important properties are addressed by the view of the optimistic and pessimistic multigranulations rough set. What is more, the multigranulations method of the belief and plausibility reductions is constructed in the paper. It is proved that a set is an optimistic (pessimistic) belief reduction if and only if it is an optimistic (pessimistic) lower approximation reduction, and a set is an optimistic (pessimistic) plausibility reduction if and only if it is an optimistic (pessimistic) upper approximation reduction.