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Multi-tier sustainable supply chain relationships (MSSCR) are complex with involvement of multiple organisations, practices, objectives, and interconnectivities. These complexities make it difficult to effectively study MSSCR using traditional empirical statistical correlative research. In MSSCR the relationship between the focal firm and sub-supplier may be affected by the relationship between the focal firm and direct-supplier, and the relationship between the direct-supplier and the sub-supplier. This paper introduces a research methodology, based on K-Means clustering, rough set theory, and cluster membership to investigate complex relationships across multi-tier sustainable supply chain triads. Collaborative advantage variables including knowledge-sharing routines, relation-specific investments, complementary capabilities, effective governance mechanisms, and long-term commitment are conceptually presented to evaluate relationships among MSSCR practices and sustainability performance. Insights for practitioners and researchers are provided. Research insights can be gained by analytically providing alternative explanatory and equifinality relationships amongst variables and multi-pair organisations. The methodology can mitigate the endogeneity problem, such as omitted factors (variables), measurement error, even in small sample data situations. Directions for future research are discussed. [ABSTRACT FROM AUTHOR]

Rough sets (RSs) and fuzzy sets (FSs) are designed to tackle the uncertainty in the data. By taking into account the control or reference parameters, the linear Diophantine fuzzy set (LD-FS) is a novel approach to decision making (DM), broadens the previously dominant theories of the intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS), and allows for a more flexible representation of uncertain data. A promising avenue for RS theory is to investigate RSs within the context of LD-FS, where LD-FSs are approximated by an intuitionistic fuzzy relation (IFR). The major goal of this article is to create a novel method of roughness for LD-FSs employing an IFR over dual universes. The notions of lower and upper approximations of an LD-FS are established by using an IFR, and some axiomatic systems are carefully investigated in detail. Moreover, a link between LD-FRSs and linear Diophantine fuzzy topology (LDF-topology) has been established. Eventually, based on lower and upper approximations of an LD-FS, several similarity relations are investigated. Meanwhile, we apply the recommended model of LD-FRSs over dual universes for solving the DM problem. Furthermore, a real-life case study is given to demonstrate the practicality and feasibility of our designed approach. Finally, we conduct a detailed comparative analysis with certain existing methods to explore the effectiveness and superiority of the established technique. [ABSTRACT FROM AUTHOR]

Nano topology is a modern type of topology based on rough sets, introducing nano open sets and their properties. This study explores the new properties of nano-β-open sets and nano-closures in nano topological space, providing some equivalent definitions, theorems, and lemmas for illustration. The effectiveness of incorporating nano β-open sets in medical decision-making is demonstrated in diagnosing breast cancer in two cases for women. Results indicated that utilizing nano open sets enhanced the accuracy, sensitivity, and specificity of cancer diagnosis. Additionally, an algorithm has been developed to aid clinicians in diagnosing breast cancer infections. The overarching objective is to provide physicians with a valuable tool to make informed decisions and reach optimal conclusions in patient care. [ABSTRACT FROM AUTHOR]

In this article, we study the zero-divisor graphsΓ(Zn) of rings of integers modulo η as information systems L(Γ(Zn)) using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set P of all prime divisors of η and (ii) the set consisting of P/pi, prime powers of P/pi in the prime factorization of η and the elements of the form P/pi pj, where pj ε P/pi . Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of L(Γ(Zn)) either consist of two prime divisors of η or one prime divisor P/pi combined with prime powers of P/pi . Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function forΓ(Zn). Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets ofΓ(Zn). Based on these results, we determine information-granularity measures corresponding to the notable partitions of Γ(Zn) and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory. [ABSTRACT FROM AUTHOR]

Worldwide, the healthcare system is greatly impacted by the changing requirements of the people. Diabetes is a long-lasting condition that can lead to serious complications if not controlled correctly. It is divided into Type 1 (TID) and Type 2 (T2D) diabetes. Research shows that almost 90% of Diabetes cases are T2D, with TID making up around 10% of all Diabetes cases. This paper suggests a Rough-Neuro classification model for identifying Type 2 Diabetes, which includes a two-stage process. The approach includes utilising Rough sets JohnsonReducer to eliminate unnecessary features or characteristics and multi-layer perceptron for illness categorization. The suggested technique seeks to reduce the amount of input characteristics, which results in a reduction in the time needed to train the neural network and the storage space required. The findings show that decreasing the amount of input characteristics results in a lower neural network training time, enhances model performance, and reduces storage needs by 63%. It is worth mentioning that a smaller neural network with only seven hidden layers, trained for 1000 epochs with a learning rate of 0.01, attained the best performance, but time and storage were much decreased. [ABSTRACT FROM AUTHOR]

Type-2 fuzzy information systems (TFISs-2), one interesting form of complex data, are extensions of fuzzy information systems. Uncertainty measurement and attribute reduction of TFISs-2 are key issues when describing and analyzing complex fuzzy data. However, existing researches mainly focus on the traditional fuzzy data, and seldom consider type-2 fuzzy data. This paper focuses on finding an effective indicator to measure the uncertainty of TFISs-2, and carries out the research of attribute reduction based on this indicator. Firstly, the weighted membership distance and the weighted membership similarity are defined, which are used to describe the relation between two objects in TIFSs-2, and σ similarity relations are introduced. Secondly, a fuzzy neighborhood structure is defined based on σ similarity relations. The classical rough set model is extended on TFISs-2, and the definitions of upper and lower approximation, accuracy and roughness are given. Then, a novel indicator called σ self-information is used to describe the uncertainty of TFISs-2, and a heuristic greedy selection attribute reduction algorithm is proposed. Finally, the experimental results show that the constructed uncertainty measurement indicator is suitable for TFISs-2, and the proposed attribute reduction method is effective. This study is beneficial to promote the understanding of complex fuzzy data, and contributes to the data analysis and processing of TFISs-2. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct the lattice structure for Z-soft covering based rough set. First, we define an equivalence relation R' on a universal set to obtain the equivalence classes induced by Z-soft covering-based rough set. Also, we define a relation RS on the family of Z-soft covering-based rough set (TS) to show that the relation RS is a poset on TS. Second, we define two operations join ∨ and meet ∧ on TS. Using these two operations, we prove that every pair of elements of RS has a least upper bound and a greatest lower bound and as a result, TS is a lattice. Finally, we develop a novel Multiple Attribute Group Decision Making (MAGDM) model using Z-soft covering based rough set in medical diagnosis to determine the patients at high risk of chronic kidney disease using the collected data from the UCI Machine Learning Repository. [ABSTRACT FROM AUTHOR]

In this article, we study the zero-divisor graphsΓ(Zn) of rings of integers modulo η as information systems L(Γ(Zn)) using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set P of all prime divisors of η and (ii) the set consisting of P/pi, prime powers of P/pi in the prime factorization of η and the elements of the form P/pi pj, where pj ε P/pi . Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of L(Γ(Zn)) either consist of two prime divisors of η or one prime divisor P/pi combined with prime powers of P/pi . Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function forΓ(Zn). Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets ofΓ(Zn). Based on these results, we determine information-granularity measures corresponding to the notable partitions of Γ(Zn) and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory. [ABSTRACT FROM AUTHOR]

The main weakness of complex and sizeable fuzzy rule systems is the complexity of data interpretation in terms of classification. Classification interpretation can be affected by reducing rules and removing important rules for several reasons. Based on the results of experiments using the fuzzy grid partition (FGP) approach for high-dimensional data, the difficulty in generating many fuzzy rules still increases exponentially as the number of characteristics increases. The solution to this problem is a hybrid method that combines the advantages of the rough set method and the FGP method, which is called the fuzzy grid partition rough set (FGPRS) method. In the Irish data, the rough set approach reduces the number of characteristics and objects so that data with excessive values can be minimized, and the fuzzy rules produced using the FGP method are more concise. The number of fuzzy rules produced using the FGPRS method at K=2 is 50%; at K=K+1, it is reduced by 66.7% and at K=2 K, it is reduced by 75%. Based on the findings of the data collection classification test, the FGPRS method has a classification accuracy rate of 83.33%, and all data can be classified. [ABSTRACT FROM AUTHOR]

The rough set (RS) and multi-granulation rough set (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, bipolarity refers to the explicit handling of positive and negative aspects of data. In this paper, with the help of bipolar fuzzy preference relation (BFPR) and bipolar fuzzy preference δ -covering (BFP δ C), we put forward the idea of BFP δ C based optimistic multi-granulation bipolar fuzzy rough set (BFP δ C-OMG-BFRS) model and BFP δ C based pessimistic multi-granulation bipolar fuzzy rough set (BFP δ C-PMG-BFRS) model. We examine several significant structural properties of BFP δ C-OMG-BFRS and BFP δ C-PMG-BFRS models in detail. Moreover, we discuss the relationship between BFP δ C-OMG-BFRS and BFP δ C-PMG-BFRS models. Eventually, we apply the BFP δ C-OMG-BFRS model for solving multi-criteria decision-making (MCDM). Furthermore, we demonstrate the effectiveness and feasibility of our designed approach by solving a numerical example. We further conduct a detailed comparison with certain existing methods. Last but not least, theoretical studies and practical examples reveals that our suggested approach dramatically enriches the MGRS theory and offers a novel strategy for knowledge discovery, which is practical in real-world circumstances. [ABSTRACT FROM AUTHOR]

Machine learning algorithms investigate relationships in data to deliver useful outputs. However, past models required complete datasets as a prerequisite. In this study, rough set-based machine learning was applied using real-world incomplete datasets to generate a prediction model of biochar's adsorption capacity based on key attributes. The predictive model consists of if–then rules classifying properties by fulfilling certain conditions. The rules generated from both complete and incomplete datasets exhibit high certainty and coverage, along with scientific coherence. Based on the complete dataset model, optimal pyrolysis conditions, biomass characteristics and adsorption conditions were identified to maximize tetracycline adsorption capacity (> 200 mg/g) by biochar. This study demonstrates the capabilities of rough set-based machine learning using incomplete practical real-world data without compromising key features. The approach can generate valid predictive models even with missing values in datasets. Overall, the preliminary results show promise for applying rough set machine learning to real-world, incomplete data for generating biomass and biochar predictive models. However, further refinement and testing are warranted before practical implementation. Highlights: • It is the first explainable AI-based rough set model to study the tetracycline adsorption capacity of biochar. • Usage of an incomplete Practical dataset through RSML evaded the biasness due to imputations. • Higher accuracy and precision of incomplete Practical datasets revealed the uniqueness of the model. [ABSTRACT FROM AUTHOR]

Rough Set (RS) theory is a useful mathematical strategy to handle uncertainty. In 1982, Pawlak presented the idea, and numerous authors have undertaken in-depth studies on RS in both ordinary cases and fuzzy situations. In terms of both the theoretical investigations and the practical applications, progress in this field of RS theory has yielded favorable outcomes over the last three decades and it is typically considered to be an extension of classical sets. A universe in RS is separated by two subsets known as lower and upper approximations. Upper approximations are nonempty intersections of equivalence classes, whereas lower approximations are subsets of the set. In this study, rough sets are examined when the universe set has a ring structure. The main contribution of this study is to focus on rough fuzzy prime and semi-prime ideals of the gamma ring structure and explain certain respects of its upper and lower approximations. The goal of this research is to investigate some of the characterizations of prime and semi-prime ideals and prove some related results. Moreover, the study discusses rough fuzzy ideals (RFI) in Γ Residue class and gives some findings. [ABSTRACT FROM AUTHOR]

Rough Set (RS) is a relatively new technique for making imprecise and uncertain decisions. It is an emerging area of ambiguous mathematics associated with fuzzy set theory. In comparison with fuzzy sets and other theories, RS has more advantages. Unlike statistics, evidence theory, and fuzzy set theory, RS theory does not require any additional or preliminary information about data. RS theory deals with approximating arbitrary subsets of the universe using two subsets namely lower and upper approximation. The advantage of RS is that rough approximations can be expanded into fuzzy environments which aids in solving a wide variety of realtime problems. The goal of this study is to focus on Rough Fuzzy Prime Ideals (RFPI) in ring structure and explain specific aspects of its upper and lower approximations. Furthermore, Rough Fuzzy Ideal (RFI) and RFPI notions and characteristics are described. [ABSTRACT FROM AUTHOR]

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The main goal of informal computing is to overcome the limitations of hypersensitivity to defects and uncertainty while maintaining a balance between high accuracy, accessibility, and cost-effectiveness. This paper investigates the potential applications of intuitionistic fuzzy sets (IFS) with rough sets in the context of sparse data. When it comes to capture uncertain information emanating from both upper and lower approximations, these intuitionistic fuzzy rough numbers (IFRNs) are superior to intuitionistic fuzzy sets and pythagorean fuzzy sets, respectively. We use rough sets in conjunction with IFSs to develop several fairly aggregation operators and analyze their underlying properties. We present numerous impartial laws that incorporate the idea of proportionate dispersion in order to ensure that the membership and non-membership activities of IFRNs are treated equally within these principles. These operations lead to the development of the intuitionistic fuzzy rough weighted fairly aggregation operator (IFRWFA) and intuitionistic fuzzy rough ordered weighted fairly aggregation operator (IFRFOWA). These operators successfully adjust to membership and non-membership categories with fairness and subtlety. We highlight the unique qualities of these suggested aggregation operators and investigate their use in the multi-attribute decision-making field. We use the intuitionistic fuzzy rough environment's architecture to create a novel strategy in situation involving several decision-makers and non-weighted data. Additionally, we developed a novel technique by combining the IFSs with quaternion numbers. We establish a unique connection between alternatives and qualities by using intuitionistic fuzzy quaternion numbers (IFQNs). With the help of this framework, we can simulate uncertainty in real-world situations and address a number of decision-making problems. Using the examples we have released, we offer a sophisticated and systematically constructed illustrative scenario that is intricately woven with the complexity of medical evaluation in order to thoroughly assess the relevance and efficacy of the suggested methodology. [ABSTRACT FROM AUTHOR]

Accurately and effectively detecting the growth position and contour size of apple fruits is crucial for achieving intelligent picking and yield predictions. Thus, an effective fruit edge detection algorithm is necessary. In this study, a fusion edge detection model (RED) based on a convolutional neural network and rough sets was proposed. The Faster-RCNN was used to segment multiple apple images into a single apple image for edge detection, greatly reducing the surrounding noise of the target. Moreover, the K-means clustering algorithm was used to segment the target of a single apple image for further noise reduction. Considering the influence of illumination, complex backgrounds and dense occlusions, rough set was applied to obtain the edge image of the target for the upper and lower approximation images, and the results were compared with those of relevant algorithms in this field. The experimental results showed that the RED model in this paper had high accuracy and robustness, and its detection accuracy and stability were significantly improved compared to those of traditional operators, especially under the influence of illumination and complex backgrounds. The RED model is expected to provide a promising basis for intelligent fruit picking and yield prediction. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The significance of studying voting behaviour is underscored by its ability to gauge the continuity or divergence of electoral politics from historical trends, elucidating the real impact of the transformative ballot box, and contributing to the examination of democracy as a value among both masses and elites. Additionally, it aids in comprehending the intricate process of political socialization. Methods: An inherent strength of the rough set lies in its reliance solely on raw data, devoid of external inputs. The decision-theoretic rough set framework, an evolution of the rough set, has garnered widespread application across diverse domains, serving as a proficient tool for acquiring knowledge, particularly in navigating situations marked by vagueness and uncertainty. Despite the proliferation of mathematical models designed to discern people's voting behavior, a decision-based rough set recommendation remains noticeably absent in existing literature. This paper introduces an innovative three-way decision approach grounded in linguistic information for identifying voting behavior. The proposed approach is based on a hybrid probabilistic rough fuzzy model incorporating linguistic information and providing insights into voting patterns. Results: The three-way decision hybrid models are tested on people and a highly satisfactory result was achieved for identifying their voting behaviours. The justification of results was validated through the mathematical process. Conclusion: A practical illustration is provided to highlight the importance of this hybrid model and to confirm its usefulness in identifying and forecasting voting behaviour. [ABSTRACT FROM AUTHOR]

In this paper rudiments of the theory will be outlined, and basic concepts of the theory will be illustrated by a simple tutorial example, concerning churn modeling in telecommunications. Real life applications require more advanced extensions of the theory but we will not discuss these extensions here. Rough set theory has an overlap with many other theories dealing with imperfect knowledge, e.g., evidence theory, fuzzy sets, Bayesian inference and others. Nevertheless, the theory can be regarded as an independent, complementary, not competing, discipline in its own rights. [ABSTRACT FROM AUTHOR]

Rough set theory is an advanced uncertainty tool that is capable of processing sophisticated real-world data satisfactorily. Rough approximation operators are used to determine the confirmed and possible data that can be obtained by using subsets. Numerous rough approximation models, inspired by neighborhood systems, have been proposed in earlier studies for satisfying axioms of Pawlak approximation spaces (P-approximation spaces) and improving the accuracy measures. This work provides a formulation a novel type of generalized approximation spaces (G-approximation spaces) based on new neighborhood systems inspired by Ij-neighborhoods and ideal structures. The originated G-approximation spaces are offered to fulfill the axiomatic requirements of P-approximation spaces and give more information based on the data subsets under study. That is, they are real simulations of the P-approximation spaces and provide more accurate decisions than the previous models. Several examples are provided to compare the suggested G-approximation spaces with existing ones. To illustrate the application potentiality and efficiency of the provided approach, a numerical example for Chikungunya disease is presented. Ultimately, we conclude our study with a summary and direction for further research. [ABSTRACT FROM AUTHOR]

Fuzzy rough entropy established in the notion of fuzzy rough set theory, which has been effectively and efficiently applied for feature selection to handle the uncertainty in real-valued datasets. Further, Fuzzy rough mutual information has been presented by integrating information entropy with fuzzy rough set to measure the importance of features. However, none of the methods till date can handle noise, uncertainty and vagueness simultaneously due to both judgement and identification, which lead to degrade the overall performances of the learning algorithms with the increment in the number of mixed valued conditional features. In the current study, these issues are tackled by presenting a novel intuitionistic fuzzy (IF) assisted mutual information concept along with IF granular structure. Initially, a hybrid IF similarity relation is introduced. Based on this relation, an IF granular structure is introduced. Then, IF rough conditional and joint entropies are established. Further, mutual information based on these concepts are discussed. Next, mathematical theorems are proved to demonstrate the validity of the given notions. Thereafter, significance of the features subset is computed by using this mutual information, and corresponding feature selection is suggested to delete the irrelevant and redundant features. The current approach effectively handles noise and subsequent uncertainty in both nominal and mixed data (including both nominal and category variables). Moreover, comprehensive experimental performances are evaluated on real-valued benchmark datasets to demonstrate the practical validation and effectiveness of the addressed technique. Finally, an application of the proposed method is exhibited to improve the prediction of phospholipidosis positive molecules. RF(h2o) produces the most effective results till date based on our proposed methodology with sensitivity, accuracy, specificity, MCC, and AUC of 86.7%, 90.1%, 93.0% , 0.808, and 0.922 respectively. [ABSTRACT FROM AUTHOR]

Rough set theory is a mathematical tool widely employed in various fields to handle uncertainty. Feature selection, as an essential and independent research area within rough set theory, aims to identify a small subset of important features by eliminating irrelevant, redundant, or noisy ones. In human life, data characteristics constantly change over time and other factors, resulting in ordered datasets with varying features. However, existing feature extraction methods are not suitable for handling such datasets since they do not consider previous reduction results when features change and need to be recomputed, leading to significant time consumption. To address this issue, the incremental attribute reduction algorithm utilizes prior reduction results effectively reducing computation time. Motivated by this approach, this paper investigates incremental feature selection algorithms for ordered datasets with changing features. Firstly, we discuss the dominant matrix and the dominance conditional entropy while introducing update principles for the new dominant matrix and dominance diagonal matrix when features change. Subsequently, we propose two incremental feature selection algorithms for adding (IFS-A) or deleting (IFS-D) features in ordered data set. Additionally, nine UCI datasets are utilized to evaluate the performance of our proposed algorithm. The experimental results validate that the average classification accuracy of IFS-A and IFS-D under four classifiers on twelve datasets is 82.05% and 80.75%, which increases by 5.48% and 3.68% respectively compared with the original data. [ABSTRACT FROM AUTHOR]

In recent years, advanced persistent threat (APT) attacks have become a significant network security threat due to their concealment and persistence. Correlation analysis of APT groups is vital for understanding the global network security landscape and accurately attributing threats. Current studies on threat attribution rely on experts or advanced technology to identify evidence linking attack incidents to known APT groups. However, there is a lack of research focused on automatically discovering potential correlations between APT groups. This paper proposes a method using attack behavior patterns and rough set theory to quantify APT group relevance. It extracts two types of features from threat intelligence: APT attack objects and behavior features. To address the issues of inconsistency and limitations in threat intelligence, this method uses rough set theory to model APT group behavior and designs a link prediction method to infer correlations among APT groups. Experimental results on publicly available APT analysis reports show a correlation precision of 90.90%. The similarity coefficient accurately reflects the correlation strength, validating the method's efficacy and accuracy. [ABSTRACT FROM AUTHOR]

This paper constructs a two-layer road data asset revenue allocation model based on a modified Shapley value approach. The first layer allocates revenue to three roles in the data value realization process: the original data collectors, the data processors, and the data product producers. It fully considers and appropriately adjusts the revenue allocation to each role based on data risk factors. The second layer determines the correction factors for different roles to distribute revenue among the participants within those roles. Finally, the revenue values of the participants within each role are synthesized to obtain a consolidated revenue distribution for each participant. Compared to the traditional Shapley value method, this model establishes a revenue allocation evaluation index system, uses entropy weighting and rough set theory to determine the weights, and adopts a fuzzy comprehensive evaluation and numerical analysis to assess the degree of contribution of participants. It fully accounts for differences in both the qualitative and quantitative contributions of participants, enabling a fairer and more reasonable distribution of revenues. This study provides new perspectives and methodologies for the benefit distribution mechanism in road data assets, which aid in promoting the market-based use of road data assets, and it serves as an important reference for the application of data assetization in the road transportation industry. [ABSTRACT FROM AUTHOR]

Based on equivalence relation R on X, equivalence class [ x ] of a point and equivalence class [ A ] of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space (X , R) depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology τ R on X is equivalent to the generated topologies associated with metric d, proximity δ and uniformity U on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology τ R on X. [ABSTRACT FROM AUTHOR]

This paper mainly studies and proves the roughness bound of disturbation fuzzy sets. Firstly, based on the theory of determining self-increment and uncertain self-decrement operators, the problem that the execution subsets are not equal sets is effectively solved, which hinders the quantitative study of disturbed fuzzy sets and lays a foundation for the quantitative study of the related properties of disturbed fuzzy sets in the future. The boundary of roughness measure of disturbing fuzzy set is further studied and proved. The new territories proposed in this paper can effectively avoid the unnecessary calculation space outside the boundary in the calculation process, so as to improve the work efficiency. [ABSTRACT FROM AUTHOR]

Numerous studies have extensively examined the correlation between convex structures and covering rough set models. However, limited attention has been devoted to investigating the relationship between convex structures and rough set models based on relations. In this paper, we aim to integrate convex geometry with rough sets based on relations. Firstly, a novel class of binary relation known as quasi-atomic relation is introduced. We show that each ordinal convex geometry can be reformulated by rough sets based on quasi-atomic relations. Subsequently, we illustrate that any convex geometry can be embedded in an ordinal convex geometry using the rough set approach. Secondly, we develop a new model called the multi-valued rough set model which serves as a valuable tool for investigating convex geometry. A novel representation of any convex geometry is provided by multi-valued rough sets based on the family of quasi-atomic relations. Furthermore, we demonstrate that any convex geometry can be decomposed into a union of ordinal convex geometries. Lastly, We develop a method to determine whether an element belongs to a convex closure of a given set. In addition, we incorporate terminologies from rough set theory into the study of convex geometry by defining concepts such as rough convex sets and degree of convexity which are utilized to analyze the structure of convex geometry. In summary, this paper offers an efficient method to integrate rough sets with convex geometry, which extends the theory and application of rough sets. [ABSTRACT FROM AUTHOR]

Reducing doctor-patient conflict is an important part of coordinating doctor-patient disputes and easing doctor-patient relationship, which is conducive to building a harmonious medical environment and promoting the healthy development of medical undertakings. This paper constructs a multi-decision-maker mixed conflict model based on rough set theory, puts forward the matrix operation expression of the conflict degree theory in the Pawlak model, and gives a more objective and scientific evaluation function. Combined with hot issues of doctor-patient conflict, the proposed multi-decision-maker mixed conflict model is applied to doctor-patient conflict, examines the doctor-patient relationship in the medical institution system from multiple internal perspectives, and calculates feasible solutions in the conflict system. The results show that high medical quality, high standardize medication, high institutional efficiency, high staff efficiency, high hospital benefits, high hospital revenue, medium employee development, medium equipment development, or high medical quality, high standardize medication, high institutional efficiency, medium staff efficiency, medium hospital benefits, high hospital revenue, high employee development, and high equipment development are important conditions for building a harmonious medical environment and reducing doctor-patient conflicts. [ABSTRACT FROM AUTHOR]

As an important extension of the rough set theory, interval-set rough sets provide an effective method to solve the problem that the objective sets cannot be accurately expressed in the rough approximation process and to induce classification rules from incomplete information systems. In practical information systems, the value of an object's decision attribute may be set-valued due to missing decision information or multiple decision values. However, this situation is not considered by any of the existing rough set models. Therefore, to properly deal with the information system where decision attributes are set-valued, we construct the interval-set rough set model on the information system and study attribute reductions of interval-set rough sets. First, we define the information system where decision attributes are set-valued as an extended set-valued decision information system (ESVDIS) and construct the interval-set rough set model on the ESVDIS. Second, the accuracy and the roughness are extended to the interval-set approximation accuracy and the interval-set approximation roughness, respectively. The two extended measures can be used to measure the uncertainty caused by rough approximations. In addition, the dependency degree of interval-set rough sets called the interval-set dependency degree is defined to estimate the significance of attribute subsets. By adopting the interval-set dependency degree, we propose the definition of attribute reduction based on interval-set rough sets and design a heuristic attribute reduction algorithm. Finally, the effectiveness of the proposed attribute reduction algorithm is demonstrated using 12 public data sets. Experimental results show that the model is applicable to some fields, such as missing decision information and group decision-making. [ABSTRACT FROM AUTHOR]

In order to address the problem of low algorithm efficiency caused by repeated calculation of the relative importance of attributes in the attribute reduction algorithm based on positive region, this paper proposed a fast attribute reduction algorithm based on knowledge rough entropy from the perspectives of attribute measurement and search strategy. Firstly, it introduced knowledge rough entropy into decision information systems by incorporating knowledge distance to measure the degree of knowledge roughness. Next, it employed knowledge rough entropy as the criterion for evaluating the significance of attributes, assessing the importance of individual attributes. Finally, it ranked all attributes based on attribute importance, and eliminated redundant attributes through dependency, so as to achieve rapid attribute reduction. The proposed algorithm was compared with other three algorithms in terms of running efficiency and classification accuracy on six publicly available datasets. The results demonstrate that the proposed algorithm improves running efficiency by 83.24%,28.77%, and 59.92% respectively compared to other three algorithms. Among the three classifiers, the classification accuracy increases on average by 0.83%,0.63%, and 1.37% respectively. Therefore, the proposed algorithm is able to achieve attribute reduction more quickly while ensuring classification performance. [ABSTRACT FROM AUTHOR]

Fuzzy rough set models are useful tools for dealing with fuzzy and real-valued data. They have been used in many real-world applications. In this paper, we investigate the fuzzy rough set model based on triangular norms and fuzzy implications. First, we extend some results in the published literature by removing the condition that is the continuity of triangular norms, and obtain more general conclusion about fuzzy upper approximation operators. Then, for the fuzzy neighborhood and the fuzzy lower approximation operator based on fuzzy implications, we investigate their characterization with each other. Finally, we establish the relationships between fuzzy rough sets and fuzzy topology. In this work, researches on the properties of fuzzy rough sets based on triangular norms which need not be continuous provide generalization results for fuzzy rough set theory from viewpoint of mathematics. [ABSTRACT FROM AUTHOR]

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras using only a single axiom. [ABSTRACT FROM AUTHOR]

Market analyzers use different parameters as features in the market data to analyze the market trends. The feature's values act as a signal to market fluctuations. Many studies have examined these features to predict market movement more effectively. However, the method to minimize the uncertainties associated with the features is not available in the literature. This exploratory study introduces the uncertainty optimization based feature selection method for stock marketing. We introduce a notion of certainty region of the feature as the set of feature values, which signify particular happening with certainty. We use rough set theory to find the feature's certainty region and uncertainty region and measure each feature's significance. The feature whose certainty region is the maximum is the most significant in the feature space. Hence we group the features by minimizing the uncertainty region of the most informative features to get feature subsets for feature selection. We propose an algorithm based on uncertainty optimization to find subsets of the feature set for effectiveness and performance enhancement in the feature selection. We obtain the decision rules with comprehensive coverage and excellent support using the selected features. The accuracy of classification using the chosen parameters is up to 85.91%, which is higher than 79.54% of the complete feature set. The study provides an uncertainty optimization model for more efficient market movement prediction. [ABSTRACT FROM AUTHOR]

The note concerns two-universe rough sets. First, we discuss the name of two-universe model. Second, we outline two lines of the development of two-universe rough sets and affirm that three-way decision unifies models of two-universe rough sets. [ABSTRACT FROM AUTHOR]

Liver diseases have emerged as a growing concern in modern times, encompassing various disorders that can harm the liver and impede its function. The employment of data acquisition instruments introduces novel obstacles to the identification and diagnosis of these diseases, such as ambiguity in information modeling, potential risks in decision-making when interpreting diagnostic data, and the demand for comprehensible tools to facilitate informed choices. Therefore, the goal of this paper is to investigate a three-way group decision (3WGD) scheme with evidential reasoning (ER) in incomplete hesitant fuzzy information systems (I-HF-ISs) for liver disease diagnosis. Specifically, the form of multigranulation (MG) I-HF-ISs is established to describe realistic incomplete, imprecise and hesitant information existed in liver disease diagnosis. By utilizing the HF similarity principle and MG three-way decisions (3WD), an adjustable MG HF probability rough set (PRS) concept is developed. The ER method is then employed to determine the optimal threshold. Subsequently, an HF MAGDM approach is established for multi-attribute group decision-making (MAGDM) using adjustable MG HF PRSs and the ER method. Ultimately, the rationality of the proposed methodology is demonstrated through a real-life example using two UCI data sets for liver diseases. The approach's effectiveness is substantiated through experimental analyses. [ABSTRACT FROM AUTHOR]

In this paper, we use the notions of lower set LR(A) and the upper set UR(A) to define the interior operator intAR and the closure operator clAR associated with a set A in an approximation space (X,R). These operators generate an approximation topological space different from the generated Nano topological space in (X,R). Ideal approximation spaces (X,R, ℓ) based on an ideal ℓ joined to the approximation space (X,R) are introduced as well. The approximation continuity and the ideal approximation continuity are defined. The lower separation axioms Ti, i = 0, 1, 2 are introduced in the approximation spaces and also in the ideal approximation spaces. Examples are given to explain the definitions. Connectedness in approximation spaces and ideal connectedness are introduced and the differences between them are explained. The interior and the closure operators are deduced using a grill G defined on (X,R), yielding the same results. [ABSTRACT FROM AUTHOR]

Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-Rs and complete cng-Rs. [ABSTRACT FROM AUTHOR]

Point cloud data generated by LiDAR sensors play a critical role in 3D sensing systems, with applications encompassing object classification, part segmentation, and point cloud recognition. Leveraging the global learning capacity of dot product attention, transformers have recently exhibited outstanding performance in point cloud learning tasks. Nevertheless, existing transformer models inadequately address the challenges posed by uncertainty features in point clouds, which can introduce errors in the dot product attention mechanism. In response to this, our study introduces a novel global guidance approach to tolerate uncertainty and provide a more reliable guidance. We redefine the granulation and lower-approximation operators based on neighborhood rough set theory. Furthermore, we introduce a rough set-based attention mechanism tailored for point cloud data and present the rough set transformer (RST) network. Our approach utilizes granulation concepts derived from token clusters, enabling us to explore relationships between concepts from an approximation perspective, rather than relying on specific dot product functions. Empirically, our work represents the pioneering fusion of rough set theory and transformer networks for point cloud learning. Our experimental results, including point cloud classification and segmentation tasks, demonstrate the superior performance of our method. Our method establishes concepts based on granulation generated from clusters of tokens. Subsequently, relationships between concepts can be explored from an approximation perspective, instead of relying on specific dot product or addition functions. Empirically, our work represents the pioneering fusion of rough set theory and transformer networks for point cloud learning. Our experimental results, including point cloud classification and segmentation tasks, demonstrate the superior performance of our method. [ABSTRACT FROM AUTHOR]

Searching for quantitative scientific prediction models and methods under uncertain hybrid information environment is helpful to the scientific decision-making of practical management problems. In this paper, we discuss the uncertain decision making problem with hybrid attributes and fuzzy decision objects. Considering the characteristic of hybrid attribute information, we introduce kernel functions to abstract the similarities from different types of attributes. On this basis, the kernel-based upper and lower approximations of arbitrary fuzzy decision objects over hybrid attribute information system are given under a multi-granularity framework, i.e., kernelized multi-granulation fuzzy rough sets, called KMGFRS in this paper. Then, the attribute reduction on hybrid attribute decision system based on KMGFRS is discussed. In addition, we combine KMGFRS and fuzzy k-nearest neighbors (FkNN) to propose a new prediction method without prior information and apply it to stroke risk assessment and prediction in clinical decision-making. Finally, we use real clinical data and UCI datasets for experiment analysis, and the results verify the applicability and validity of the model. [ABSTRACT FROM AUTHOR]

The establishment of the correspondence between the important features of the topside weld pool and the penetration state is a key link in penetration control. This article presents a scheme of variable gap welding penetration control based on rough-fuzzy modeling. By utilizing the basic concepts of variable precision rough sets and conditional information entropy, relevant algorithms for attribute reduction and rule extraction in decision information systems are addressed. Raw data for modeling are obtained through current–gap combination experiments, and then rough-set-based knowledge acquisition algorithms are used to generate classification rules for the penetration state. Based on the correspondence between the important features of the topside pool and the penetration state, a fuzzy control model is established. Welding experiments under variable gaps are conducted to examine the proposed control scheme, and the results indicate that the weld has good penetration and meets the requirements of welding specifications. [ABSTRACT FROM AUTHOR]

The rough set (RS) and multi-granulation RS (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, the bipolar fuzzy sets (BFSs) are effective tools for handling bipolarity and fuzziness of the data. In this study, with the description of the background of risk decision-making problems in reality, we present (α β)-optimistic multi-granulation bipolar fuzzified preference rough sets ((α β)o-MG-BFPRSs) and (α β)-pessimistic multi-granulation bipolar fuzzified preference rough sets ((α β)p-MG-BFPRSs) using bipolar fuzzy preference relation (BFPR). Subsequently, the relevant properties and results of both (α β)o-MG-BFPRSs and (α β)p-MG-BFPRSs are investigated in detail. At the same time, a relationship among the (α β)-BFPRSs, (α β)o-MGBFPRSs and (α β)p-MG-BFPRSs is given. [ABSTRACT FROM AUTHOR]

In this paper, two new uncertainty modeling concepts, namely strait soft set and strait rough set, between the structures of rough sets and soft sets, which will bring new perspectives to both theoretical and practical aspects, are presented. A reduction method of alternatives is given using strait soft sets. Another convenience arising from the structure of strait soft sets is that they allow the parameters to be combined. Thus, the fusion of parameters is defined and its use in soft set operations is demonstrated. Strait rough sets naturally contain the characteristics of the rough sets and also allow the parameters to be characterized. The strait soft set and strait rough set are supported by many examples and comparisons. In addition, a new decision-making approach based on the strait soft set and strait rough set is proposed and then followed by real-life applications to illustrate the computational processes. [ABSTRACT FROM AUTHOR]

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Preference analysis is an essential component of the decision-making (DM) process for identifying the optimal object. The rough set (RS) theory has been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, bipolarity refers to the explicit handling of positive and negative aspects of data. In this article, we first established the idea of bipolar fuzzy preference relation (BFPR) from the bipolar fuzzy information system and studied some of its basic properties. Then, based on BFPR, a novel idea of (α , β) -bipolar fuzzified preference RSs ( (α , β) -BFPRSs) is offered. Several significant structural properties of (α , β) -BFPRSs are analyzed in detail. Moreover, some significant uncertainty measures associated with (α , β) -BFPRSs are proposed. Meanwhile, we put forward the idea of bipolar fuzzy preference δ -covering ( B F P δ C ), bipolar fuzzy preference δ -neighborhood (BFP δ -nghd), and bipolar preference δ -neighborhood (BP δ -nghd). Moreover, we establish a new type of bipolar fuzzy RS (BFRS) model in the context of BFP δ -nghd, and several relevant properties are explored. Using the theory of the B F P δ C -based BFRS model, we develop a novel approach for multi-criteria decision-making (MCDM) problem. A real-life example of a smartphone selection is provided to show the applicability of our suggested approach. We further investigate the effectiveness of the developed technique using a comparative analysis. Last but not least, theoretical investigations and practical examples reveal that our suggested approach dramatically enriches the RS theory and offers a novel knowledge discovery strategy that is applicable in real-world circumstances. [ABSTRACT FROM AUTHOR]

The relationship between Rough Set (RS) and algebraic systems has been long studied by mathematicians. RS is a growing research area that encourages studies into both real-world applications and the theory itself. In RS, a universe subset is characterized by a pair of ordinary sets called lower and upper approximations. In this study, we look attentively at the use of rough sets when the universe set has a ring structure. The main contribution of the paper is to concentrate on the study of rough fuzzy ideals concerning the gamma ring and to describe some properties of its lower and upper approximations. This paper deals with the connection between Rough Fuzzy Sets (RFS) and ring theory. The goal of this paper is to present the notion of Left Operator Rings (LOR) and Right Operator Rings (ROR) in the gamma ring structure. We introduce some basic concepts of rough fuzzy left and right operator rings. Furthermore, we investigate some characterizations of left and right operator rings and prove some theorems based on these results. [ABSTRACT FROM AUTHOR]

In this paper, a new uncertainty modelling concept called strait fuzzy set is introduced, which brings new perspectives to both theoretical and practical advances in fuzzy mathematics. This set type allows objects/points to be graded with fuzzy membership intervals that are partitions of [0,1] (so that the union of these partitions is [0,1] and their intersection is empty) instead of fuzzy membership degrees represented as exact values in [0,1]. Moreover, some basic operations and properties of strait fuzzy sets are studied in detail. The concept of strait fuzzy rough set is put forward and its theoretical aspects are discussed. In addition, two different similarity approaches are proposed for strait fuzzy sets and strait fuzzy rough sets, and applied to measure the similarity rates of vaccines against influenza viruses. [ABSTRACT FROM AUTHOR]

This study aims to identify significant risks and their relationship to the successful operation of the Trans-Sumatra toll road in Indonesia. The research utilizes the Delphi and DEMATEL methods, along with rough set analysis, to identify and evaluate 28 risks associated with toll road operations in Sumatra. The research identifies 13 dominant risks, including policy changes, government intervention, inflation, financial distress, fluctuation of interest rate, fluctuation of currency, high cost of maintenance, low volume of traffic, competitive routes, overloading vehicles, many infrastructure defects, and natural disasters. In this case, natural disasters, inflation, and vehicles with excessive loads are the most dominant cause factors because those risks have the highest cause value based on the cause-effect diagram. Furthermore, the prominence diagram reveals that income risk, policy changes, and financial distress have notable implications for operational activities. The study presents a MCDM risk assessment approach that incorporates rough set analysis, providing a comprehensive understanding of the critical risk relationship factors for toll road operations. By integrating rough set analysis, this research contributes to the field of toll road operations and risk assessment. The identified risks and their relationships serve as a foundation for developing effective strategies for toll road operational management. [ABSTRACT FROM AUTHOR]

In fuzzy set theory, the concept of a non-membership function and the hesitation margin were not considered while these two concepts have been included along with the membership function for intuitionistic fuzzy sets. It is also to be noted that the intuitionistic fuzzy set is re ected as an extension of the fuzzy set accommodating both membership and non-membership functions together with a hesitation margin. In the intuitionistic fuzzy set theory, the sum of the membership function and the non-membership function is a value between 0 and 1. In recent times, intuitionistic fuzzy rough set theory has emerged as a powerful tool for dealing with imprecision and uncertain information in relational database theory. Measures of similarity between fuzzy rough sets as well as intuitionistic fuzzy rough sets provide wide applications in real-life problems and that is why many researchers paid more attention to this concept. Intuitionistic fuzzy rough set theory behaves like an excellent tool to tackle impreciseness or uncertainties. In this paper, we propose a new approach of similarity measure on an intuitionistic fuzzy rough set based on a set-theoretic approach. The proposed measure is able to give an exact result. In the application part, we consider a real-life problem for selecting a fair play award-winning team in a cricket tournament and describe the algorithm. [ABSTRACT FROM AUTHOR]

In multi-label learning, the classification obtained by attribute set is usually inconsistent with that based on label set. This makes multi-label learning to possess fuzziness and uncertainty, which leads to the poor performance of learning algorithms. To address this fuzziness and uncertainty, this paper first combines multiple labels into a granulation multi-label decision function by using the classification obtained by attribute set. Then we give three kinds of formulations to construct the granulation multi-label decision functions, which are divided three levels, that is, macroscopic level, mesoscopic level and microscopic level. Moreover, by using the granulation multi-label decision function, this paper provides the method that transforms a multi-label decision table into a granulation multi-label decision table. Then three-level granulation attribute reductions of a multi-label decision table are defined and investigated. Furthermore, this paper shows that the existing multi-label attribute reduction, complementary decision reduction, can be viewed as the coarsest granularity reduction proposed by this paper, that is, a macroscopic-level granulation attribute reduction. In summary, this paper establishes a relatively systematic theoretical framework for attribute reduction of multi-label data based on rough set theory. Finally, by several comparative analysis, the reasonability, feasibility and effectiveness of these granulation attribute reductions are demonstrated. [ABSTRACT FROM AUTHOR]

Issues in daily life, where making the best decisions is crucial, are frequently encountered. But, in the majority of these situations, the best course of action is uncertain. We must take into account a number of parameters in order to find the best possible solution to these difficulties. The best mathematical instrument for this is fuzzy soft set F S S theory in decision making. Nutrition is the process of supplying cells and organisms with the nutrients they need to grow and thrive and to sustain life. A healthy diet has the potential to prevent or mitigate numerous prevalent health issues. The purpose of this paper is to select a burning problem for the nutrition of students and successfully apply the F S S theory in decision making. We aim to prove that the approach to decision-making problems with imprecise data via F S S s is more accurate than other types of approaches, and we present a new approach to the F S S model and its applications in decision-making problems. [ABSTRACT FROM AUTHOR]

In March 2020, a cohort of 26 is treated critically ill hospitalized SARS-CoV-2 infected patients who received EEGs to assess unexplained altered mental status, loss of consciousness, or poor arousal and responsiveness. The objective of the present work is to develop a method that is able to automatically determine mental status of vigilance, i.e., a person's state of alertness. Such a task is relevant to diverse domains, where a person is expected or required to be in a particular state of mind. Aiming at the EEG feature selection and classification model in the identification of fatigue driving, the discretization algorithm using rough set theory is proposed to select the channel and EEG signal feature quantities. The support vector machine (SVM) is selected as the fatigue driving recognition model, and the risk of fatigue misjudgment is taken as SVM model parameters for model optimization. The experimental results of subjects show that compared with the principal component method, the rough set discretization algorithm selects fewer features, and the compatibility threshold 0.8. The number of features selected among the candidate features is 208. The features selected by different subjects are different and have an impact on the establishment of the support vector machine recognition model. Fatigue misjudgment risk control parameters can adjust the support vector machine recognition model error judgment risk. Even if the present approach is costly in computation time, it allows constructing a decision rule that provides an accurate and fast prediction of the alertness state of an unseen individual. [ABSTRACT FROM AUTHOR]