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Logistics is a sophisticated system involving third-party logistics (3PL) providers, freight forwarders, warehousing companies, and transport service producers in various transport modes, such as road, rail, air, maritime, and multimodal transportation. Furthermore, it is possible to add customs clearance agencies, insurance companies, banks, and relevant institutions and organizations to the system. Effective logistics systems must sustainably provide customers with quality and satisfactory logistics services using data shared over advanced technologies. Nevertheless, data and information on logistics are usually challenging to collect, process, and understand, as they are primarily unstructured, unreachable, and unstandardized. Blockchain is a new and advanced technology promising to eliminate or mitigate the adverse effects of these difficulties. However, blockchain technology practices in the logistics industry are extraordinarily scarce; and a few blockchain platforms have attempted to produce solutions for a few large-scale and global logistics firms. Hence, the digital transformation process involving blockchain technology for a logistics company encounters the challenge of selecting an appropriate blockchain platform for the logistics industry's needs. Although studies have been carried out in the relevant literature to select a suitable blockchain platform for various industries, few of these studies dealt with selecting the best blockchain platform for the logistics industry. Hence, experimental studies on choosing the proper blockchain platform in the logistics industry that tries to manage outstandingly complicated relations and linkages among stakeholders are currently inadequate. The current study presents a novel, robust, practical, and powerful decision-making tool that can also overcome highly complex uncertainties to identify the most feasible blockchain technology for the logistics industry. The robustness of the study's findings is validated with comprehensive sensitivity and comparative analyses. [ABSTRACT FROM AUTHOR]

• Propose an IoMT security defense method based on blockchain and fuzzy set theory. • Propose DFCM algorithm to solve the clustering problem of balanced data. • Use FSVM model to classify data points far from the sample center. • Use the combination of DFCM and FSVM in IoT attack detection. This work aims to improve the stability and efficiency of the Internet of Medical Things (IoMT), strengthen the security and privacy of personal information, and create a healthy and safe network environment. First, art therapy in the current medical environment is discussed, and the security defenses of the IoMT supporting art therapy are studied. Then, a security defense method for IoMT is proposed by combining basic Blockchain knowledge with the Fuzzy Sets Theory. Specifically, a Density-awarded Fuzzy C-means (DFCM) algorithm is developed to solve the clustering issue of unbalanced data sets. Besides, the Fuzzy Support Vector Machine (FSVM) model is used to classify data points far from the sample center to increase classification accuracy. In the simulation experiment, the DFCM algorithm achieves a pleasing clustering effect on the four data sets, involving the data sets with density changes and imbalanced sample points within the cluster. Moreover, the combination of DFCM and FSVM (DFCM-FSVM) for attack detection in the Internet of Things (IoT) realizes the optimal accuracy in the detection of different behaviors compared with similar algorithms; the detection accuracy of normal behavior, scheduled attack, gray hole attack, and black hole attack is 99.8%, 89.8%, 99.4%, and 91.5%, respectively. This work creatively proposes a hybrid intrusion detection method based on fuzzy theory. As the second line of defense in security protection, it plays a critical role and is suitable for the network environment of the perception layer. In conclusion, the attack detection method based on Blockchain combined with the Fuzzy Sets Theory can precisely identify potential attacks in abnormal data, which has important practical value for maintaining data security and the regular operation of the IoMT. [ABSTRACT FROM AUTHOR]

The first step of implementing computing with words using a perceptual computer is to establish an encoder to transform words into interval type-2 fuzzy sets. In this paper, a novel approach named the retained region approach is introduced for encoding words into interval type-2 fuzzy sets. In the retained region approach, the data part is redesigned relative to the existing approaches, and a new fuzzy set part is established in accordance with the principle of justifiable granularity. The different means and standard deviations of the embedded type-1 fuzzy sets associated with a word are recognized as the origins of its inter-uncertainty, and the relations between them are taken into account in the construction of interval type-2 fuzzy sets. Importantly, the retained region approach is a versatile approach that does not impose redundant constraints on the selection of embedded type-1 fuzzy sets, and the capacity of the retained region can be flexibly adjusted along the two dimensions of the mean and the standard deviation. Finally, the performance of the retained region approach is illustrated by several simulations and comparisons with existing approaches. [ABSTRACT FROM AUTHOR]

Factorization of classical, two-valued Boolean data became a widely studied topic in the past decade due to its role in analyzing relational data as well as its significance for other fields. Recently, various extensions to factorization of ordinal data, or data with graded (fuzzy) attributes, have been proposed. We identify and describe a fundamental problem regarding quality of factors, which is non-existent in the Boolean case, but naturally appears in the more general setting of ordinal data. As we demonstrate, the problem gets more significant with growing size of the factorized data. We analyze the problem, propose a method to alleviate it, and evaluate experimentally our solution to the problem. We also provide a discussion regarding ramifications of our findings for the concept of cardinality of fuzzy sets. [ABSTRACT FROM AUTHOR]

Variable precision (fuzzy) rough sets are interesting generalizations of Pawlak rough sets and can handle uncertain and imprecise information well due to their error tolerance capability. The comparable property (CP for short), i.e., the lower approximation is included in the upper approximation, is fundamental in Pawlak rough sets since many theories and applications rely on this property. However, the CP is lost in many existing variable precision (fuzzy) rough sets. Therefore, a novel variable precision fuzzy rough set model with CP is proposed and an associated three-way decision model is developed. Firstly, a pair of variable precision fuzzy upper and lower approximation operators are defined and studied through fuzzy implication and co-implication operators. We show that this pair of approximation operators satisfy various properties (including CP) that the existing variable precision fuzzy approximation operators do not possess. Secondly, new methods for constructing loss functions and conditional probabilities are given by using CP, and then a new three-way decision model with three strategies (optimistic, pessimistic, and compromise strategies) is established. Finally, an explanatory example is provided to examine validity, stability and sensitivity of the proposed three-way decision model. [ABSTRACT FROM AUTHOR]

Inconsistency in classification and regression problems occurs when instances that relate in a certain way on the condition attributes, do not follow the same relation on the decision attribute. It typically appears as a result of perturbation in data caused by incomplete knowledge (missing attributes) or by random effects that occur during data generation (instability in the assessment of decision attribute values). Inconsistencies with respect to a crisp preorder relation (expressing either dominance or indiscernibility between instances) can be handled with set-theoretic approaches like rough sets and by using statistical/machine learning approaches that involve optimization methods. In particular, the Kotłowski-Słowiński (KS) approach relabels the objects from a dataset such that inconsistencies are removed, and such that the new class labels are as close as possible to the original ones in terms of a given loss function. In this paper, we generalize the KS approach to handle inconsistency determined by a fuzzy preorder relation rather than a crisp one. The method produces a consistent fuzzy relabeling of the instances and may be used as a preprocessing tool with algorithms for binary classification and regression. As the obtained fuzzy sets can be represented as unions of meaningful simple fuzzy sets or granules, we call them granular approximations. We provide statistical foundations for our method, develop appropriate optimization procedures, provide didactic examples, and prove several important properties. • Proposing the definition of granular approximations; a novel way of integrating machine learning, fuzzy sets, and rough sets. • Inconsistencies in data, defined according to the rough set framework, are removed with optimization methods. • The new method generalizes Kotłowski and Słowiński's approach defined for crisp relations. • The fuzzy rough set lower and upper approximations are extreme cases of the granular approximation. [ABSTRACT FROM AUTHOR]

• A novel hesitant fuzzy set with SVM modeling scheme is proposed for FTSF. • Length-based discretization (LBD) method is proposed to compute number of interval. • Two set of UODs are used to handle the non-determinism in time series data. • FLRs are established using normalized crisp value with its mean of aggregated HFEs. • FLRs are modeled using SVM. Over the last three decades, several researchers have been putting their efforts into developing non-deterministic fuzzy time series (FTS) models using the traditional fuzzy set. However, considering a set of membership values to each element of the time series, the hesitant fuzzy set glorifies the chances to capture the fuzziness and uncertainty due to randomness better than the traditional fuzzy set. Motivated by this, the present study proposes a novel hesitant FTS forecasting model employing a support vector machine (HFTSF-SVM). In this model, to diminish the computational complexity and increase the forecasting accuracy, we have addressed three key issues by proposing three new methods such as (a) a length-based discretization technique is developed to compute the number of intervals, (b) two sets of universe of discourse are determined for the same time series to compute the non-membership and membership value for each observation (c) each observation along with its mean aggregated membership values are adopted to construct the fuzzy logical relationships (FLR). To assess the competency and reliability of the proposed HFTSF-SVM model, six promising FTS models from the recent literature and four promising time series forecasting (TSF) models are considered for comparison using sixteen time series datasets. Extensive statistical analyses of obtained results confirm the supremacy of the proposed HFTSF-SVM model. [ABSTRACT FROM AUTHOR]

The sustainable supplier selection is treated as a type of complex decision-making problem due to involving lots of uncertain information, and the appropriate linguistic terms are usually used to handle such uncertainties. However, the descriptions of linguistic terms in the form of traditional fuzzy sets are not perfectly supported in the sustainable supplier selection process. Luckily, the intuitionistic fuzzy sets and interval-valued fuzzy sets can effectively deal with this uncertain and incomplete data information when selecting evaluation criteria and facing with the ambiguity of human thinking, whereas providing us with a helpful tool to resolve fuzzy multi-criteria decision-making (MCDM) problems in a more flexible way. Moreover, the cumulative prospect theory can be introduced to well depict decision-makers' risk preferences and psychological behaviors. Therefore, this study develops a novel fuzzy MCDM approach based on the combination of intuitionistic fuzzy sets, interval-valued fuzzy sets and cumulative prospect theory for selecting the most sustainable supplier. A case study from the sustainable recycling supplier selection of e-bike sharing is studied to illustrate applicability of the proposed method. By comparing with other methods, the rationality and feasibility of the proposed method are verified. This can provide an insightful implication for decision-makers with different risk preferences to select the most sustainable supplier under uncertain environment. [ABSTRACT FROM AUTHOR]

• We apply fuzzy rough nearest neighbour methods to classify text based on emotions. • Feature engineering and ensembles can gain accuracy similar to deep learning methods. • Our approach provides a more interpretable alternative than black box methods. Due to the ever-expanding volumes of information available on social media, the need for reliable and efficient automated text understanding mechanisms becomes evident. Unfortunately, most current approaches rely on black-box solutions rooted in deep learning technologies. In order to provide a more transparent and interpretable framework for extracting intrinsic text characteristics like emotions, hate speech and irony, we propose to integrate fuzzy rough set techniques and text embeddings. We apply our methods to different classification problems originating from Semantic Evaluation (SemEval) competitions, and demonstrate that their accuracy is on par with leading deep learning solutions. [ABSTRACT FROM AUTHOR]

Overlap functions, as a class of not necessarily associative novel aggregation functions, have played an important role in the relevant theory and applications involving fuzzy sets and systems. Recently, extension study has become a hot topic in the theoretical research of overlap functions, especially for the so-called quasi-overlap functions. In this paper, we investigate the set-based extended quasi-overlap functions, which are one kind of set-based extended aggregation functions proposed by Mesiar et al. lately. Specifically, after introducing the definition of set-based extended quasi-overlap functions, we study some of their elementary properties. Furthermore, we give two equivalent characterizations of set-based extended quasi-overlap functions by means of mappings from the set composed by all finite subsets of unit closed interval with no repetition of elements to unit closed interval and 0,1-aggregation functions recently proposed by the author, respectively. In particular, it is shown that the family of set-based extended quasi-overlap functions is closed to convex combination and that the repetition of values in the input vectors to be aggregated and adding any value between its minimum and maximum values have no effect on the output value under the aggregation of set-based extended quasi-overlap functions. The obtained results afford theoretical basis for more potential applications of quasi-overlap functions in real problems, especially when there are more than two classes or objects in fuzzy context or fuzzy sets for which we need to measure the degree of overlapping or generate overlap indices. [ABSTRACT FROM AUTHOR]

The probabilistic hesitant fuzzy sets (PHFSs), which have received a lot of attention and have been widely used in many domains, are effective in describing hesitant evaluations. Despite significant progress, entropy and cross-entropy, as the famous uncertainty measures of probabilistic hesitant fuzzy information, have not yet been thoroughly investigated. However, these uncertainty measures are quite important and useful in the context of multi-attribute decision-making (MADM). In this contribution, we revisit the concept of uncertainty measures for probabilistic hesitant fuzzy information by comprehensively considering their fuzziness and hesitancy, and proposed some novel entropy and cross-entropy measures for them. Specifically, the hybrid entropy measures are designed as convex combinations of the fuzzy entropy factor and the hesitant entropy factor, and in the same line of thought, the hybrid cross-entropy measures are also designed as convex combinations of the fuzzy cross-entropy factor and the hesitant cross-entropy factor. It is shown that the hybrid entropy measures are simple in structure, clear in physical meaning, and strong in discriminating capability, while the hybrid cross-entropy measures can avoid the design flaws of existing cross-entropy measures and employ natural symmetry. It is further shown that the hybrid entropy and cross-entropy measures can be designed in pairs and are mutually supportive, which suggests that they are internally unified as the uncertainty measures of probabilistic hesitant fuzzy information. In the end, based on the proposed uncertainty measures, an MADM model is developed within the TOPSIS framework to evaluate a company's strategic initiatives under probabilistic hesitant fuzzy environment. Numerical results show that these uncertainty measures can support group decision-making effectively and efficiently with probabilistic hesitant fuzzy information. [ABSTRACT FROM AUTHOR]

Data classification is a core component of many intelligent systems, and it has been almost monopolizing the interest of the scientific community for decades. However, developing classifiers capable of explaining how a classification result was derived, in a way that is compatible to the human perception, is still a challenge. Steps towards this direction have been made via interpretable fuzzy classifiers capable of extracting linguistically expressible rules from data. Motivated by the need for interpretable data classification, this study proposes a novel interpretable fuzzy classification framework based on Fuzzy Similarity Phrases (FSPs), aiming to provide an improved tradeoff between interpretability and classification performance. This framework incorporates the following novel components: a) a generic vocabulary-based feature extraction scheme which considers similarity features and fuzzy sets to construct FSPs; b) a rule generation methodology based on FSPs; c) an embedded feature selection methodology targeting to keep both the number of extracted rules and their length low; and an interpretable class prediction mechanism based on the interpretation of the FSPs. The experimental evaluation of the proposed framework on various publicly available datasets, in comparison with well-known and state-of-the-art classifiers, validate that it can provide a better average classification performance with fewer rules. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new entropy measure of Pythagorean fuzzy sets (PFSs). The proposed entropy measure of PFSs can conquer the shortcomings of the existing entropy measure of PFSs. We also propose the Pythagorean fuzzy weighted arithmetic mean (PFWAM) aggregation operator (AO) of Pythagorean fuzzy numbers (PFNs). The proposed PFWAM AO of PFNs can conquer the shortcomings of the existing sine trignometry Pythagorean fuzzy weighted averaging (ST-PFWA) AO and the existing sine trignometry Pythagorean fuzzy weighted geometric (ST-PFWG) AO of PFNs. Based on the proposed entropy measure of PFSs and the proposed PFWAM AO of PFNs, we propose a new group decision making (GDM) approach in the environment of PFNs. The proposed GDM approach can conquer the shortcomings of existing GDM approaches, where they cannot distinguish the ranking orders (ROs) of alternatives in some conditions. It offers us a very useful approach to deal with GDM problems in the environment of PFNs. [ABSTRACT FROM AUTHOR]

• New non-iterative reasoning algorithm for Fuzzy Cognitive Maps (FCMs) is developed. • Weights are generated based on experts surveys and modelled using Type 2 Fuzzy Sets. • Uncertainties in experts' opinions are aggregated using Interval Agreement Approach. • Sensitivity analysis provides insight into uncertainty propagation in FCMs. A Fuzzy Cognitive Map (FCM) is a causal knowledge graph connecting concepts using directional and weighted connections making it an effective approach for reasoning and decision making. However, the modelling and reasoning capabilities of a conventional FCM for real world problems in the presence of uncertain data is limited as it relies on Type 1 Fuzzy Sets (T1FSs). In this work, we extend the capability of FCMs for capturing greater uncertainties in the interrelations of the modelled concepts by introducing a new reasoning algorithm that uses Type 2 Fuzzy Sets based on z slices (zT2FSs) for the modelling of uncertain weights connecting FCM's concepts. These Type 2 Fuzzy Sets are generated using interval valued data from surveyed participants and aggregated using the Interval Agreement Approach method. Our algorithm performs late defuzzification of the FCM's values at the end of the reasoning process, preserving the uncertainty in values for as long as possible. The proposed algorithm is applied to the evaluation of the performance of modules of an undergraduate Mathematical programme. The results obtained show a greater correlation to domain experts' subjective knowledge on the modules' performance than both a corresponding FCM with weights modelled using T1FS and a statistical method currently used for evaluating the modules' performance. Sensitivity analysis conducted demonstrates that the new algorithm effectively preserves the propagation of uncertainty captured from input data. [ABSTRACT FROM AUTHOR]

• Electric vehicles (EVs) based multiple fuel supply system modelling are developed. • A novel hybrid fuzzy MCDM model for solving FSSMA problems is proposed. • PPH–FWZIC is used to determine the weights of the sustainable fuel consideration. • MARCOS method are applied to rank the FSSMA alternatives. Various companies have developed electric vehicle (EV)-based multiple fuel supply system modeling approaches (FSSMAs). Nonetheless, no superior approach concurrently satisfies all essential criteria, including 'sustainability' and 'fuel consideration' criteria. Furthermore, benchmarking the FSSMA alternatives to determine the most sustainable ones does not come without issues. The five main most common concerns are the use of various evaluation criteria, effecting the weights of the criteria with sublayers, criteria prioritization, trade-offs among the criteria, and data variations. Thus, this study proposes a novel FSSMA for EV benchmarking based on two methods—the Pythagorean probabilistic hesitant fuzzy sets and fuzzy weighted zero inconsistency (PPH–FWZIC) and the measurement of alternatives and ranking according to the compromise solution (MARCOS)—which are integrated as a single method. The PPM–FWZIC method was developed to solve the criteria prioritization issue, while the MARCOS method was developed to solve the various evaluation criteria, trade-offs among the criteria, and data variation issues to benchmark the FSSMA for EV alternatives. The integrated multicriteria decision-making (MCDM) method allows the system to perform a backward scoring process (BSP) and derive a scoring decision matrix from the formulated decision matrices that are performed based on the feed-forward data presentation (FFDP) procedure to solve the multiple criteria layers that affect the proper assessment of the impact of a certain criterion and its subcriteria in the weighting purpose issues. Subsequently, the FSSMAs for EVs are benchmarked, and the most sustainable approach is selected. The results were tested via sensitivity analysis and the Spearman correlation coefficient. The present study is also compared with a benchmark study based on a benchmarking checklist. [ABSTRACT FROM AUTHOR]

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

In the real word, due to the existence of uncertainty in decision-making information, it is often difficult to accurately evaluate utility values of alternatives. Recently, a series of discrete choice models based on fuzzy subjective utility have been proposed. Specifically, utility evaluation of each alternative has been expressed by fuzzy membership grades. However, traditional fuzzy membership function has only considered supporting degree, but has not considered the degrees of opposition and indeterminacy involved in decision makers' evaluations. In this paper, a series of novel discrete choice models based on Atanassov-type intuitionistic fuzzy set are proposed, in which membership degrees, non-membership degrees, and hesitation degrees are simultaneously taken into account to jointly express the utility evaluations. Moreover, the attitudinal character of decision makers and relative weights of attributes are also considered to further extend the proposed models. Compared with existing discrete choice models, these new models are more flexible and general, which increase the applicability of discrete choice models for decision-making under an uncertain and imprecise environment. Two real-world case studies are given to illustrate the effectiveness and rationality of the new models. [ABSTRACT FROM AUTHOR]

Feature selection attempts to capture the more discriminative features and plays a significant role in multilabel learning. As an efficient mathematical tool to handle incomplete and uncertain information, fuzzy rough set theory has been widely used in feature selection. However, there are two pivotal issues to be addressed when adopting fuzzy rough sets for multilabel data analysis. One is how to effectively characterize the discriminative ability of the features to multilabel samples. Another one is how to alleviate the effect of data noise on feature selection. For these reasons, this study mainly studies the noise-tolerant fuzzy neighborhood rough set model for multilabel learning and its feature selection strategy. Firstly, a parameterized hybrid fuzzy similarity relation is introduced to granulate multilabel data, and the parameterized fuzzy decision is extended to multilabel learning. Then, a noise-resistant multilabel fuzzy neighborhood rough set model using inclusion relation is constructed for describing the discriminative ability of the features to multilabel samples. Furthermore, a noise-resistant heuristic feature selection algorithm titled NRFSFN is designed. Besides, motivated by local sampling and ensemble strategy, an efficient extended version of NRFSFN is designed from the perspective of data distribution, named ENFSFN. Finally, a series of comparison experiments on sixteen multilabel datasets demonstrate the effectiveness of the proposed two algorithms. [ABSTRACT FROM AUTHOR]

Fuzzy rough set (FRS) has a great effect on data mining processes and the fuzzy logical operators play a key role in the development of FRS theory. In order to generalize the FRS theory to more complicated data environments, we firstly propose four types of fuzzy neighborhood operators based on fuzzy covering by overlap functions and their implicators in this paper. Meanwhile, the derived fuzzy coverings from an original fuzzy covering are defined and the equalities among overlap function-based fuzzy neighborhood operators on a finite fuzzy covering are also investigated. Secondly, we prove that new operators can be divided into seventeen groups according to equivalence relations, and the partial order relations among these seventeen classes of operators are discussed, as well. And then, two types of neighborhood-related fuzzy covering-based rough set models are proposed via the overlap function-based fuzzy neighborhood operators and the t -norm-based fuzzy neighborhood operators. Furthermore, the groups and partially order relations are also discussed. Finally, a novel fuzzy TOPSIS methodology is put forward to solve a biosynthetic nanomaterials select issue, and the rationality and enforceability of our new approach is verified by comparing its results with nine different methods. [ABSTRACT FROM AUTHOR]

Bipolar complex fuzzy information is a new idea that can easily depict awkward and vague information in real life problems. The main theory of the bipolar complex fuzzy set is the fusion of two different theories, called bipolar fuzzy set and complex fuzzy set. The bipolar complex fuzzy set contained two different grades, called positive membership grade which is considered a positive aspect of the problem, and negative membership grade which is considered a negative aspect of the problem, both contain in a unit square of complex plane. The major analysis of this theory is to diagnose the interrelationship among any number of attributes. For this, we defined various Aczel-Alsina operational laws and proved their influential results. Further, by using the Aczel-Alsina operational laws and bipolar complex fuzzy information, we diagnosed the theory of bipolar complex fuzzy Aczel-Alsina weighted averaging (BCFAAWA), bipolar complex fuzzy Aczel-Alsina ordered weighted averaging (BCFAAOWA), and bipolar complex fuzzy Aczel-Alsina hybrid averaging (BCFAAHA) operators. To modify the influence of the explored work, we discussed various important results and some properties (Idempotency, Monotonicity, and Boundedness) of the proposed work. Additionally, we computed a multi-attribute decision-making rule under the presence of the initiated approaches and tried to justify it with the help of some examples. Finally, to evaluate the influence and dominancy of the diagnosed work, we compared our deliberated work with various prevailing approaches and also described their geometrical representations. [ABSTRACT FROM AUTHOR]

Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called I O -implications) in this paper. More specifically, we first give the definition of the 1 st type of FRSs and study some essential properties. Subsequently, the definition of the 2 nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1 st type of FRSs and the 2 nd type of FRSs is described. That is, in a fuzzy approximation space, the 2 nd type of lower approximation is contained in the 1 st type of lower approximation, while the 1 st type of upper approximation is included in the 2 nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses. [ABSTRACT FROM AUTHOR]

Frequent pattern mining (FM) has a wide range of applications in the real world. But FM sometimes discovers many uninteresting patterns at the same time. Constraint-based FM, especially periodic constraint FM, becomes of interest, and it reduces redundant patterns. Most of the state-of-the-art algorithms for mining periodic frequent patterns (PFPs) are designed to find PFPs only in binary temporal databases. However, vast information in real situations is more suitable to be modelized as quantitative databases, in which most existing methods are inapplicable. Patterns obtained by those methods may change due to a tiny perturbation in the database, which is not expected in many situations. In this paper, we propose an algorithm (FP2M) to mine fuzzy-driven periodic frequent patterns and an algorithm (SFP2M) to find stable fuzzy-driven periodic frequent patterns in quantitative temporal databases. Fuzzy sets are used in our algorithms to deal with quantitative data since they provide a relaxed interval segmentation and are easy to implement. Novel pruning strategies and a new structure called Estimated Period Co-occurrence Structure (EPCS) are designed to speed up the mining process and improve efficiency. An improved method (SFP2M) is also presented by using the lability measurement to find stable patterns. Experimental evaluations on both real and synthetic datasets show good performance of the designed algorithms. [ABSTRACT FROM AUTHOR]

• Bibliometric techniques identify the productivity and influence on research. • Fuzzy systems are an area of study with potential for further research. • Zadeh is known as the most influential author in fuzzy research in North American. • The United States is the most productive and influential country in fuzzy research. The objective of this paper is to apply bibliometric techniques to analyze the evolution of fuzzy systems research (FSR) in the United States of America and Canada. The aim is to visualize the progression of FSR since Zadeh's first publication of fuzzy sets and the regional impact that these ideas have in the scientific community. This work obtains data retrieved from the Web of Science (WoS) scientific database. Using a hybrid methodology designed to maximize the obtention of representative results, we analyze 20,317 documents published between 1965 and 2020. The main findings show that Zadeh is the most influential author and that Pedrycz is the most productive author at FSR. Journals such as Fuzzy Sets and Systems (FSS) and IEEE Transactions on Fuzzy Systems (TFS) publish most of the FSR articles. The results show a greater contribution by the United States of America compared to Canada in FSR. This analysis contributes to visualization and evolution in the field of study of FSR in diverse regions. [ABSTRACT FROM AUTHOR]

The generalized three-way decision (G3WD) theory is featured by a trisecting-acting-outcome paradigm. Experiments in psychology and economics have indicated that risky decision-making problems should take into account the psychological characteristic of agents. Moreover, the loss of information acts as a serious challenge in G3WD. Thus, exploring valid G3WD approaches that consider the mental state of decision-makers in incomplete information systems (IISs) is imperative. This paper explores a new three-way decision (3WD) approach that combines the prospect theory with the regret theory for IISs with the support of fuzzy set pair dominance degrees. First, the notion of fuzzy set pair dominance degrees to process evaluation information in IISs is put forward. Then, an approach that obtains objective reference points and determines the weighting function in the prospect theory is given by combining 3WD with the behavioral decision theory. In light of the prospect theory, a prospect value function is obtained, which is introduced into the regret theory to obtain two utility perception functions. As a consequence, a G3WD approach based on fuzzy set pair dominance degrees and the behavioral decision theory is constructed for IISs. By using two medical cases in the KEEL and UCI datasets, the superiority, effectiveness and stability of the constructed approach are verified via corresponding comparative and experimental analyses. [ABSTRACT FROM AUTHOR]

Fuzzy relation systems are important knowledge representation systems in fuzzy rough sets. In this paper, we explore attribute reduction from a topological perspective for fuzzy relation systems. First, using general fuzzy relations as a foundation, we provide an improved definition of topological reduction for fuzzy relation systems. Next, we propose the notion of topological reduction for fuzzy relation decision systems. Moreover, we create corresponding reduction algorithms to identify all topological reducts based on discernibility matrices. Because of the feature redundancy issue with the term frequency-inverse document frequency (TF-IDF) algorithm in text categorization, we suggest a hybrid feature selection algorithm that combines the TF-IDF algorithm and topological reduction algorithm. Comparative analyses of Chinese and English text datasets demonstrated the effectiveness and practicality of our proposed hybrid feature selection approach. [ABSTRACT FROM AUTHOR]

Gaussian random fuzzy numbers are random fuzzy sets generalizing Gaussian random variables and possibility distributions. They define belief functions on the real line that can be conveniently combined by the product-intersection rule under the independence assumption. In this paper, we introduce various extensions of this rule to account for dependence and partial reliability of the pieces of evidence. We first provide formulas for the combination of an arbitrary number of Gaussian random fuzzy numbers whose dependence is described by a correlation matrix, and we introduce a minimum-conflict combination operation. To account for partially reliable evidence, we then introduce two discounting operations called possibilistic and evidential discounting, as well as several combination operators based on different assumptions, each one parameterized by a correlation matrix and a vector of discounting coefficients. We demonstrate the application of these operators to the combination of predictions with different sets of inputs in machine learning, and show that performance can be enhanced by optimizing the parameters of the combination operators. • Gaussian random fuzzy numbers (GRFNs) represent evidence about real variables. • We address the combination of dependent GRFNs with an arbitrary correlation matrix. • We propose two discounting mechanisms for GRFNs. • We introduce combination rules for combining dependent and partially reliable GRFNs. • We demonstrate an application of these rules in machine learning. [ABSTRACT FROM AUTHOR]

Image segmentation allows us to separate an image into distinct, non-overlapping parts by utilizing specific features such as hue, texture, and shape. The technique is prevalent in different domains, including target detection, medical imaging, and pattern recognition owing to its importance in analyzing the image. The fuzzy C-means (FCM) algorithm is a popular method for image segmentation and pattern recognition. However, uncertainty and unknown noise in the data impair the effectiveness of the algorithm. Alternatively, uncertainty in real world can be addressed by the intuitionistic fuzzy set (IFS). This article presents a new approach to image representation using IFS and local information about the image. We introduce the concept of filtering into the intuitionistic fuzzy set and utilize a specially designed exponential distance for IFS. We propose the intuitionistic fuzzy local information C-means (IFLICM) algorithm. The goal of IFLICM is to increase the tolerance to noise and the maintain the details in image better than existing FCM variants. We test the performance of our algorithm on a public dataset and compare it with existing FCM methods and Double Deep-Image-Prior (Double-DIP). The experimental results demonstrate that IFLICM is highly effective in image segmentation and outperforms existing methods. • We use an IFS to represent uncertainty, combine local information propose a novel image representation method. • We propose an exponential distance measure for IFSs, and show its superiority by illustrating examples. • We develop a novel image segmentation algorithm, intuitionistic fuzzy local information C-means (IFLICM). • We complete performance evaluation, the results show great improvement and robustness of proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a method for fuzzy model reference adaptive control system design for a partially known multi-input multi-output (MIMO) linear dynamic plant. Differential or convolutional equations can describe the plant to be controlled, and the fuzzy controller is assumed to be from the MIMO sector-bounded nonstationary nonlinearities class. The task is to obtain robust adaptive stabilization. Some absolute stability theorems and integral evaluations of the state and control vectors are applied in the MIMO fuzzy adaptive controller design procedure for the first time. A method for automatically obtaining a near-optimal nonlinear dynamic reference model is described. A practical step-by-step procedure of the MIMO adaptive fuzzy controller design is proposed, resulting in highly interpretable MIMO fuzzy control rules based on triangular fuzzy sets and strong triangular fuzzy partition. An example of the fuzzy controller design according to the proposed procedure is given. • A MIMO plant represented by differential or convolutional equations. • Systematic procedure for designing a fuzzy MIMO controller. • Highly interpretable fuzzy control rules obtained through adaptation. • The controller function placed in the stable and quasi-optimal sector. [ABSTRACT FROM AUTHOR]

• Novel implementation technique of T1 and IT2 rule-based fuzzy systems. • Decomposition based on first-order rule partitions for singleton and non-singleton fuzzifiers. • Proposed technique outperforms the traditional method regarding computational and memory efficiency. • Case of identical partitions shows that execution time and memory costs remain unaffected by fuzzy sets increase. Rule-based fuzzy systems are widely adopted in various fields for their effectiveness in reducing uncertainties. However, large rule bases present significant computational and memory challenges, especially for real-time applications. To overcome this limitation, this paper introduces a novel first-order rule partitions-based decomposition technique (FORPs-DT) for type-1 (T1) and interval type-2 (IT2) fuzzy logic systems (FLSs). This method involves decomposing the universe of discourse, membership functions (MFs), and fuzzy sets (FSs) by defining conditions for non-zero firing levels (intervals). Significantly, this approach allows the construction of subfuzzy systems with separate knowledge bases while maintaining the input–output relationship. The paper provides detailed explanations and visual representations of FORPs-DT and proposes a case of identical FORPs to illustrate its simplicity and resource optimization benefits. Extensive experiments demonstrate the superiority of this technique, showcasing not only reduced execution time and memory usage but also enhanced performance through identical partitions, which allows an increase in the number of FSs without additional computational or memory overhead. [ABSTRACT FROM AUTHOR]

Railway cold chain service network design (RCC-SND) aims to optimize the utilization of stations and lines as well as train allocations in a manner that minimizes costs while satisfying the service requirements of shippers. Furthermore, the uncertainties associated with freight demand, transportation costs, quality loss, station handling capacity, and arc capacity make the RCC-SND a complex decision-making problem. To tackle this challenge, we first formulate a Mixed-Integer Nonlinear Programming (MINLP) model to determine hub locations, freight wagon flows, and service frequency. To cope with uncertain parameters with varying degrees of uncertainty incorporated in the model, we extend the problem using fuzzy programming and further convert it to its crisp counterpart. A real-world cases study in Southwest China is performed to validate the proposed model, whose results provide different strategies for decision-makers with varying preferences. There are some main findings: As the number of hubs increases from 5 to 6, a maximum total cost savings of 1.99% can be achieved. Railway operators may opt for different decision preferences, for decisions prioritizing economic efficiency, the cost can decrease by 2.69% compared to deterministic optimization. [ABSTRACT FROM AUTHOR]

Pedrycz shadowed sets are three-way approximations of fuzzy sets by transforming the infinite levels of fuzzy set membership grades in the unit interval [ 0 , 1 ] into three levels. The three levels represent qualitatively the sets of the white, grey, and black members of a shadowed set. In this paper, we generalize the notion of shadowed sets to the case of L-fuzzy sets by making three new contributions. First, we consider two representations of a shadowed set. One is a three-valued L-fuzzy set and the other is three pairwise disjoint sets. Second, we introduce two methods for constructing a shadowed set. One divides a finite lattice based on the notion of a pair of a set of designated core membership grades and a set of designated null membership grades. The other uses a pair of threshold sets, which generalizes the method that uses a pair of thresholds. We study formal properties of the two methods and show that they are equivalent. Finally, based on a distance function on a lattice, we present a simple method to build the sets of designated core and null membership grades. [ABSTRACT FROM AUTHOR]

Although many methods have been proposed for tackling the class-imbalance problem, they still suffer from the insufficient feature representative capability and the overfitting problem. This paper proposes a new deep im balance learning approach based on the F uzzy set T heory and the P rototypical learning mechanism, abbreviated as imFTP for short, which consists of an adaptive smooth sampling module, a self-learnable prototypical learning module, and a fuzzy transition module. The adaptive smooth sampling module adaptively adjusts the sampling frequency of different classes to ensure their adequate opportunity to participate in the training process, which can mitigate the overfitting problem. The self-learnable prototypical learning module devises a clustering distance based Softmax cross-entropy loss and an intra-class clustering loss to improve the feature representation and discrimination capability of the model. The fuzzy transition module utilizes the fuzzy set theory to transform sample features effectively, which further enhances the feature representation capability of the model, meanwhile alleviates the overfitting problem. Experimental results on 15 benchmark datasets demonstrate that our method outperforms the best competitor by more than 3% in terms of the Macro-F1 metric, which is very significant. [ABSTRACT FROM AUTHOR]

Most of the existing anomaly detection methods are based on a single and fine granularity input pattern, which is susceptible to noisy data and inefficient for detecting anomalies. Granular-ball computing, as a novel multi-granularity representation and computation method, can effectively compensate for these shortcomings. We utilize the fuzzy rough sets to mine the potential uncertainty information in the data efficiently. The combination of granular-ball computing and fuzzy rough sets takes into account the benefits of both methods, providing great application and research value. However, this novel combination still needs to be explored, especially for unsupervised anomaly detection. In this study, we first propose the granular-ball fuzzy rough set model, and the relevant definitions in the model are given. Subsequently, we pioneeringly present an unsupervised anomaly detection method based on granular-ball fuzzy rough sets called granular-ball fuzzy rough sets-based anomaly detection (GBFRD). Our method introduces the granular-ball fuzzy rough granules-based outlier factor to characterize the outlier degree of an object effectively. The experimental results demonstrate that GBFRD exhibits superior performance compared to the state-of-the-art methods. The code is publicly available at https://github.com/Mxeron/GBFRD. [ABSTRACT FROM AUTHOR]

Label distribution learning (LDL) is a novel framework for handling label ambiguity problems and has been used widely in practice. However, dealing with high-dimensional data or data with redundant features in the LDL context is still an open problem. Existing feature selection algorithms cannot be directly applied to LDL due to the unique challenges caused by the label uncertainty nature. In this paper, we propose a novel LDL feature selection algorithm based on neighborhood rough sets. Specifically, we first introduce dual-similarity that is used to measure sample similarity in both the feature and the label spaces. Second, we invent a novel neighborhood fuzzy entropy as a feature evaluation metric, with which neighborhood rough sets can be applied to deal with LDL problems. Lastly, we complete a feature selection model that inherits the spirit of neighborhood rough sets and neighborhood fuzzy entropy. Extensive experiments have been conducted on twelve real-world LDL datasets, and the results demonstrate the superiority of our proposed model against to other six state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

• The fact that objects of interval-valued data could be noticed as interval-valued fuzzy sets (IVFS) is emphasized. • New similarity measures for IVFS based on average embedding functions are proposed. • Hierarchical clustering algorithm is applied to real intervalar data using new similarity measures for IVFS. • The similarity measures proposed are applied to clustering real data of the Spanish weather in 2021. Clustering algorithms create groups of objects based on their similarity. As objects are usually defined by data points, this similarity is commonly measured by a distance function. When the objects are defined by variables that are intervals, it is more difficult to determine how to measure the similarity between the objects of the dataset. In this work, we propose some similarity measures between intervals based on average embedding functions. Using these, new similarity measures between interval-based objects are proposed. All the proposed similarities are based on measuring the similarity between the objects variable by variable and then averaging the obtained results to get a single value. By its definition, the objects can be considered as interval-valued fuzzy sets (IVFS), so the similarities introduced are proved to be valid similarities for IVFS. The measures proposed are used in a hierarchical clustering algorithm with the aim of grouping the objects of the dataset into different clusters based on their similarity to interval-valued data. The described process is applied to real data regarding the Spanish weather in order to cluster the provinces of Spain based on the interval temperature of each month in 2021, showing different results that the ones obtained using non-interval-valued data. [ABSTRACT FROM AUTHOR]

Decision making is undertaken by individuals whose perceptions for the objectives are vague due to the dynamics of decision-making objectives, the complexity of decision-making situation and the limitations of individuals' education backgrounds, experiences, etc. Therefore, decision making which uses fuzzy set (FS) and its variants to describe the individuals' vague perceptions has become more and more popular. Furthermore, in real-life decision-making situations, individuals are always bounded rationality, which is clearly described by prospect theory (PT). Thus, PT has been integrated into fuzzy decision making (FDM). Since then, FDM with PT have been rapidly developed. In this paper, we systematically analyze the existing FDM with prospect framework. Firstly, a detailed analysis about the existing research on this topic is given separately, including how to determine the reference point, how to express the value function and weighting function in PT under vague conditions. With this in mind, the applications and challenges of the existing methods are described and some possible directions for future research are provided. According to our review, it is worth mentioning that the FDM with psychological factors of decision makers will surely receive much more attention from researchers and practitioners in the future. [ABSTRACT FROM AUTHOR]

Linguistic intuitionistic fuzzy numbers (LIFNs) are useful to express the uncertainty of qualitative aspects of information, which have received many attentions in recent years. In this paper, we propose the distance measure of linguistic intuitionistic fuzzy sets (LIFSs), where the membership grade and the non-membership grade of each element in the universe of discourse belonging to a LIFS are represented by LIFNs. We also provide the proofs of the validity and some desirable properties of the proposed distance measure of LIFSs. Moreover, we also propose the weighted distance measure of LIFSs. Based on the proposed weighted distance measure of LIFSs and the "Technique for Order Preference by Similarity to Ideal Solution" (TOPSIS) method, we propose a new group decision making (GDM) approach in the environments of LIFNs. We also use some examples to illustrate the practicability and the feasibility of the proposed GDM approach. The proposed GDM approach can overcome the drawbacks of the existing GDM approaches, where they have the drawbacks that they cannot distinguish the ranking orders of the alternatives in some situations. It provides us a very useful method for dealing with GDM problems in the environments of LIFNs. [ABSTRACT FROM AUTHOR]

[Display omitted] • Generalized preference information is used to reduce subjective randomness. • Imputation of missing values occurs systematically under different scenarios. • Weights of experts and criteria are systematically determined. • Waste treatment options are ranked rationally. • Practicality is validated by using a case study from Tamil Nadu. A generalized orthopair fuzzy set can express uncertain information easier than most other processes, which gives us more space for decision-making. In recent times, the selection of healthcare waste treatment (HCWT) technology can be considered as multi-criteria decision-making (MCDM) problem due to the involvement of multiple conflicting criteria. To reduce the unhealthy impact on the ecosystem and promote sustainable disposal, researchers have investigated MCDM problems to select apt HCWTs. In a bid to the minimize negative environmental impact of healthcare waste, researchers adopted MCDM, and faced challenges such as: (i) handling uncertainty/subjective randomness; (ii) imputation of missing values; and (iii) consideration to experts' attitude and interdependencies during MCDM process. To remedy these, this paper has attempted to establish a novel decision model with generalized orthopair fuzzy information (GOFI). Initially, missing values are systematically imputed. Experts' preferences were fused to obtain an aggregated matrix by considering the interdependencies among experts. Also, criteria weights were computed utilizing attitude-based entropy measures, and HCWTs were ranked using the GOFI-evaluation based on distance from average solution (GOFI-EDAS) approach. Lastly, the methodology's superiority was validated using an illustrative example, followed by a comparison with extant models. The results confirm that the developed framework is more efficient than and consistent with earlier methods under uncertainty. [ABSTRACT FROM AUTHOR]

Fuzzy rough set theory can tackle feature redundancy in data and select more informative features for machine learning tasks. Gaussian kernel is often coupled with fuzzy rough set theory to measure fuzzy relation between data instances. However, Gaussian kernel has a serious long-tail phenomenon, which would perform poorly in modeling the fuzzy relation for high-dimensional data. Moreover, a robust feature evaluation function is also nontrivial in a fuzzy rough set model because a naive model may select those non-optimal feature subsets due to the perturbations from redundant features. This paper delves into Student- t kernel and fuzzy divergence to address these challenges for fuzzy rough feature selection. This paper proposes a new Student- t Kernelized Fuzzy Rough Set (SKFRS) model. The new model uses fuzzy divergence to evaluate uncertain information in the data. It also explores a newly-defined feature evaluation function on the biases of the dynamic relation between the relevance and indispensability of features in feature selection process. A novel forward greedy search algorithm is then presented to solve the final objective function. The selected features are subsequently evaluated on downstream classification tasks. Experimental results using real-world datasets demonstrate the effectiveness of the proposed model and its superiority against the baseline methods. [ABSTRACT FROM AUTHOR]

Multi-attribute decision making (MADM) plays an important role in decision analysis. Based on a new fuzzy rough set theory (FRS), this paper presents a multi-attribute prediction analysis model in fuzzy information systems (FISs). In this prediction model, the proposed FRS is based on an attribute-oriented fuzzy similarity relation; the cosine deviation of a candidate to optimistic decision and pessimistic decision is proposed to construct a trend function, for further predicting the development trend of the candidate. It deals with MADM problems from the perspective of multi-attribute predictive analysis, and is a valuable continuation of fuzzy decision making research. To evaluate the validity and feasibility of the proposed prediction model, experiments with real data sets and comparative experimental analysis are conducted. The experimental results prove that the prediction performance of the proposed model is well-pleasing in terms of both prediction accuracy and AUC results. [ABSTRACT FROM AUTHOR]

The TOPSIS method follows a basic principle, that is, the better the alternative, the farther the alternative is to the pessimistic ideal solution, and the closer the alternative is to the optimistic ideal solution. However, in the decision-making process, there may be some alternatives whose ranking cannot be directly judged from this perspective. Therefore, researchers have proposed some sorting formulas, which can accomplish the overall sorting of alternatives. But, these formulas can only be used to deal with the sorting of alternatives. To expand the application scope and decision prospect of the TOPSIS method, we introduce decision-theoretic rough fuzzy sets, and design a novel TOPSIS method that simultaneously deals with the sorting and classification of alternatives. Using the basic principle of the TOPSIS method to acquire the optimistic and pessimistic ideal distances, we regard these two distances as two fuzzy concepts, here their features are opposite, and utilize the newly defined decision-theoretic rough fuzzy set models to obtain the corresponding decision areas. Then, according to these two distances, we analyze connection between the ordering of alternatives and the total loss of alternatives and summarize the sorting rules when alternatives are classified into the same decision area and the sorting rules when alternatives are classified into different decision areas. Further, the concept of joint decision area is defined. Meanwhile, we summarize the sorting rules when alternatives are classified into the same joint decision area and the sorting rules when alternatives are classified into different joint decision areas. Using these rules, we introduce the detailed process of the novel TOPSIS method and analyze the rationality of the method. Subsequently, we test the feasibility of our method with project investment decision example and illustrate its superiority and stability through comparative analysis and parameter test. Besides, we conduct simulation experiment to further verify the similarities and differences of the sorting function of the traditional TOPSIS method and our method. [ABSTRACT FROM AUTHOR]

In this study, the fusion of probabilistic hesitant fuzzy sets (PHFSs) and rough sets is explored and applied to uncertain multi-criteria decision-making (MCDM). First, to reduce the huge amount of computation of probabilistic hesitant fuzzy operations in the MCDM process, we introduce an advanced method to obtain normalized PHFSs (NPHFSs), in which each probabilistic hesitant fuzzy element (PHFE) has the same length and probability distribution. Second, based on NPHFSs, a novel probabilistic hesitant fuzzy rough set (PHFRS) model is established to solve the information aggregation problem in MCDM. Subsequently, we propose the concept of fuzziness of a PHFRS and develop a fuzziness-based objective weight determination method. Further, we construct an extended PROMETHEE II method based on a PHFRS for MCDM problems. Finally, we analyze the effectiveness of our proposed method through two MCDM problems along with experimental comparisons. The results illustrate that the proposed method is useful in numerous uncertain application fields. [ABSTRACT FROM AUTHOR]

In recent years, fuzzy β -covering, as a natural extension of fuzzy coverings, has attracted considerable attention. However, existing fuzzy β -neighborhood operators cannot accurately describe the relationship between objects, which greatly restricts the application of fuzzy β -covering. For this reason, we first construct four new fuzzy β -neighborhood operators by using the existing fuzzy β -neighborhood operator and generalized fuzzy logic operators, and investigate their properties. To better portray the similarity between samples, inspired by the definition of fuzzy similarity relation, we define the concept of fuzzy β -covering relation. On this basis, we develop a new framework of fuzzy β -covering rough set models. We further propose an attribute reduction method by employing the new fuzzy β -covering relation, and design a heuristic attribute reduction algorithm with reference to an uncertainty measure called attribute significance. Finally, experimental results show the superiority of our proposed method through a series of experimental analyses. [ABSTRACT FROM AUTHOR]

In this paper we show a methodology for designing operators on spaces of lattice-valued mappings. More precisely, from a family of operators on a bounded lattice L and mappings from a set X to itself, we may construct an operator, that we call the induced operator, on the lattice of set mappings from X to L. Furthermore, if X is also a bounded lattice, under suitable conditions preserving the orders on L and X , the induced operator belongs to the lattice of monotone mappings from X to L. The procedure is quite simple, versatile and allows to obtain plenty of different examples in a wide range of lattices. In particular, by appropriate choices of X and L , it can be applied to the most important types of fuzzy sets. The relation with some properties associated to popular types of operators is studied. Hence, we show that, under certain conditions, aggregation operators, implications, negations, overlap functions and others are preserved by the induction process. [ABSTRACT FROM AUTHOR]

Shadowed set divides a fuzzy set into three regions through fuzzy-rough transformation to denote acceptance, rejection and uncertain decision. Based on the tri-partition property, shadowed sets are utilized to implement the machine learning methods for uncertain data analysis. The extant uncertain machine learning methods with shadowed sets include the unsupervised clustering on only unlabeled data and the supervised classification on only labeled data. However, for the partial labeled data containing both labeled and unlabeled data instances, the studies of uncertain learning methods with shadowed sets are very limited. Aiming at the requirement, in this paper, we propose a novel semi-supervised shadowed set on partial labeled data and thereby construct semi-supervised shadowed neighborhoods to implement the three-way classification of uncertain data. To construct the semi-supervised shadowed set, we reformulate the objective function of shadowed sets, in which the membership loss in fuzzy-rough transformation is weighted by labeled and unlabeled data. We also analyze the influence of labeled data to the shadowed set construction. Experiments validate that the proposed three-way classification method with semi-supervised shadowed sets is effective to utilize partial labeled data to achieve low-risk uncertain data classification. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel multiattribute decision making (MADM) approach based on Fermatean hesitant fuzzy sets (FHFSs) and the modified VIKOR method. Firstly, we propose the definition of distance measures of FHFSs and present its properties. Further, taking the effectiveness of FHFSs for dealing with ambiguous and imprecise data in MADM problems, this paper proposes the remoteness index-based Fermatean hesitant fuzzy-VIKOR (FHF-VIKOR) MADM method. The generalized distance measure for FHFSs is subsequently employed to establish the notions of remoteness indices with the positive ideal and the negative ideal remoteness indices. The objective weighting procedure is developed using the maximum deviation principle and the generalized distance measure to obtain the attributes' weights. Some examples are discussed to reveal the performance of the proposed MADM method. Finally, the advantages of the proposed MADM method in terms of the robustness and the flexibility are shown by a comparative study. The proposed MADM method based on FHFSs and the modified VIKOR method can overcome the drawbacks of the existing MADM methods. [ABSTRACT FROM AUTHOR]

• Maximal and minimal membership degrees are combined to a general adjustable measure. • DT-MFRSs extend, improve VP-MFRSs to have systematicness, robustness, applicability. • DT-MFRSs' measure, region gain granulation nonmonotonicity/uncertainty for reduction. • DT-MFRSs attribute reduction has optimization definitions, system connections on 3WD. Variable precision multigranulation fuzzy rough sets (VP-MFRSs) use two direct integrations: the multigranulation maximum and minimum. Their optimistic and pessimistic models facilitate uncertainty informatization but also imply a potential limitation of extremes. This study improves VP-MFRSs for extension and balance, and thus, decision-theoretic multigranulation fuzzy rough sets (DT-MFRSs) are proposed by systematically fusing the multigranulation maximum and minimum. For DT-MFRSs, their tri-level analysis of measurement, modeling, and reduction is deeply acquired via three-way decisions. First, maximum and minimum membership degrees are linearly combined, and the weight parameter guides a generalized multigranulation membership degree. This adjustable measure motivates DT-MFRSs with positive, negative, and boundary regions, while attitude-preference values of 1, 0, and 0.5 respectively produce optimistic, pessimistic, and compromised models. Then, nonmonotonicity and uncertainty of membership degrees and model regions are determined, and these fundamental characteristics induce new reduction criteria of region preservations. Also, three-way attribute reducts are proposed by preserving positive, negative, and positive–negative regions, and their systematic relationships are obtained. Finally, tri-level results of measures, models, and reducts are validated by table examples and data experiments. In this study, DT-MFRSs extend and improve VP-MFRSs via systematic fusion of membership measurement, and contain optimistic, pessimistic, compromised models, etc., thereby exhibiting extended diversity and applied robustness. Their three-way attribute reduction also perfects uncertainty optimization. [ABSTRACT FROM AUTHOR]

Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on finding solutions and solvability criteria for such systems. The situation is quite different with the so-called weakly linear systems, in which an unknown fuzzy relation is on both sides of the equation or inequality. Precisely, the scholars have only given the characterization of the set of exact solutions to such systems. This paper describes the set of fuzzy relations that solve weakly linear systems to a certain degree and provides ways to compute them. We pay special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. We establish additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. We demonstrate the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks. [ABSTRACT FROM AUTHOR]

Many network-based tasks need powerful feature expression to capture the diversity of networks. It can be provided by network embedding learning of nodes. The related researches led to a significant progress so far. Nevertheless, suffering from paying unique attention on the structure or neighborhoods of nodes, most of methods cannot describe the vector representations of nodes well enough. In addition, random walk plays a key role during the learning procedure in many advanced methods. The successor node is selected according t the proximity of its direct prior node in each walk step. It is easy to lead a similarity drifting in a long walk sequence and influence the sequence quality. To address these issues, a fuzzy hierarchical network embedding approach (FHNE) is put forward to learn the vector expression fusing structure and neighbor information. Firstly, a multi-granular graph is constructed by a fuzzy k -core decomposition to encode the structural and neighborhood information of nodes. Then, inspired by the epidemic method, a biased random walk is designed to solve the similarity drifting. Many numerical experiments demonstrate that our method exhibits superior performance on various tasks in some real datasets. It verifies that FHNE can learn the efficient network representations. [ABSTRACT FROM AUTHOR]