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Many attempts have been made to generalize the concept of intuitionistic fuzzy sets (IFS) like Pythagorean (PFS), q-rung orthopair (q-OFS), and linear Diophantine (LDFS). However, these generalizations have many advantages and disadvantages. Among the disadvantages, the main concern with these sets is that they cannot capture the situation where both or at least one of the memberships and non-membership grades are equal to 1. Secondly, how to reduce the dependency between the membership and non-membership grades. Thus, any data in the form X = {< x1; (1,0.9) > , < x2; (0.3,1) > , < x3; (1,1) >} is not handled by the IFS and other versions of IFS because 1 + 0.9 = 1.9 > 1, 0.3 + 1 = 1.3 > 1, and 1 + 1 = 2 > 1. We propose the new idea of the q-fractional fuzzy set ( q f r s ), which can handle all such situations, using the q-intercept of the straight line and letting both membership and non-membership grades approach 100% without depending on each other. The q = 2 is the smallest value for which all the situations in the first quadrant are tackled, and the sum of membership and non-membership grades is near 1. For all other values of q > 2, the sum of membership and non-membership grades approaches 0, i.e., the larger the value of q, i.e., the intercepts, the sum of memberships and non-membership grades approaches 0. For q = 1, the first intercept is simply the intuitionistic fuzzy set. We provide the basic properties of the q-fractional fuzzy set using the extension principle of fuzzy sets and develop some aggregation operators. We also developed a new q-fractional fuzzy neural network and provided an example. [ABSTRACT FROM AUTHOR]

A novel score function based on the Poincaré metric is proposed and applied to a decision-making problem. Decision-making on Fuzzy Sets (FSs) has been considered due to the flexibility of the data, and it is applied to the decisionmaking. However, decisions with FSs are sometimes nondecisive even for different membership degrees. Hence, Intuitionistic Fuzzy Sets (IFSs) data is applied to design a score function for the decision-making with the Poincaré metric. This function is supported by the profound information of IFSs; IFSs include hesitation degree together with membership and non-membership degree. Hence, IFS membership and non-membership degree are expressed as two-dimensional vectors satisfying the Poincaré metric for simplification. At the same time, the proposed approach addresses the hesitation information in the IFS data. Next, a score function is proposed, constructed and provided. The proposed score function has a strict monotonic property and addresses the preference without resorting to the accuracy function. The strict monotonic property guarantees the preference of all attributes. Additionally, the existing problem of score function design in IFSs is addressed: they return zero scores even with different meanings for the same membership and non-membership degree. The advantages of the proposed score function over existing ones are demonstrated through illustrative examples. From the calculation results, the proposed decision score function discriminates between all candidates. Hence, the proposed research provides a solid foundation for the hesitation analysis on the decision-making problem. [ABSTRACT FROM AUTHOR]

We explore switching techniques between q-fractional fuzzy sets (qFr sets) and various other classes of fuzzy sets to establish connections and provide a comprehensive framework. In particular, we examine the relationships between qFr sets and interval-valued fuzzy sets (IVFS), type 2 fuzzy sets(T2FS), intuitionistic fuzzy sets(IFS), Pythagorean fuzzy sets(PFS), q-rung orthopair fuzzy sets (q-ROFS), and linear diophantine fuzzy sets(LDFS). By examining these interconnections, we aim to understand better qFr sets and their applications in a wide range of fuzzy systems. It is possible to convert qFr sets into other fuzzy set models using the derived switching techniques, facilitating the utilization of existing methods and algorithms. The versatility of qFr sets, combined with the bridging techniques presented, holds promise for addressing complex problems in decision-making, pattern recognition, and other applications where uncertainty and imprecision play significant roles. Through case studies and practical applications, we illustrate the effectiveness and usefulness of the proposed switching techniques, showcasing their potential impact on real-world scenarios. [ABSTRACT FROM AUTHOR]

Recently, many studies have been conducted on developing cubic sets, like cubic intuitionistic fuzzy sets, cubic Pythagorean fuzzy sets, and cubic q-rung orthopair fuzzy sets. However, the cubic set combines interval-valued and fuzzy sets. But both parts of these fuzzy sets cannot attain the maximum value (equal to 1) due to the restriction at the sum of memberships and non-membership grades. For example if one has data of the form Ξ = { < x ; ([ 0.5 , 1 ] , [ 0.1 , 1 ]) , (1 , 0.9) > | x ∈ X } , then clearly these data cannot be handled through cubic q-rung orthopair fuzzy sets. To cover this situation, we introduce the notions of cubic q-fractional fuzzy sets ( C q f r F s ), combining the interval-valued q-fractional fuzzy sets ( I V q f r F s ) and q-fractional fuzzy sets ( q f r F s ) and allowing them to attain the maximum value by introducing a new parameter q ≥ 2. We first introduce the concept of interval-valued q-fractional fuzzy sets ( I V q f r F s ) with elemental properties. Then we propose the novel idea of cubic q-fractional fuzzy sets ( C q f r F s ) and discuss their sensitivity analysis. We also provide the fundamental arithmetic operations of cubic q-fractional fuzzy sets ( C q f r F s ) and properties. In the end, we propose the correlation coefficients to measure the relationship between cubic q-fractional fuzzy sets ( C q f r F s ). Finally, we presented a numerical example of the evaluation of using a digital library in the education department by considering its advantages and disadvantages using the developed correlation coefficients for user X. Thus, by knowing a user's priorities, the digital library database can be updated. [ABSTRACT FROM AUTHOR]

Knowledge discovery through spatial and temporal aspects of data related to occurrences of events has many applications in digital forensics. Specifically, in electronic surveillance, it is helpful to construct a timeline to analyze information. The existing techniques only analyze the occurrence and co-occurrence of events; however, in general, there are three aspects of events: occurrences (and co-occurrences), nonoccurrences, and uncertainty of occurrences/non-occurrences with respect to spatial and temporal aspects of data. These three aspects of events have to be considered to better analyze periodicity and predict future events. This study focuses on the spatial and temporal aspects given in intuitionistic fuzzy (IF) datasets using the granular computing (GrC) paradigm; formal concept analysis (FCA) was used to understand the granularity of data. The originality of the proposed approach is to discover the periodicity of events data given in IF sets through FCA and the GrC paradigm that helps to predict future events. An experimental evaluation was also performed to understand the applicability of the proposed methodology. [ABSTRACT FROM AUTHOR]

Linguistic variables are flexible and intuitive attraction for expressing the wording of decision makers. This paper introduces a new type of linguistic fuzzy sets called linguistic dual hesitant fuzzy sets to express the hesitancy of decision makers' qualitative preferences and non-preferences. Considering the application in decision making, linguistic dual hesitant fuzzy preference relations (LDHFPRs) are introduced that permit the decision makers to apply several linguistic variables to indicate a qualitative preferred judgment and a qualitative non-preferred judgment, respectively. To rank objects from LDHFPRs rationally, a consistency concept is first presented. Then, two optimal models are built to judge the consistency of LDHFPRs. When LDHFPRs are inconsistent, an optimal model-based iteration algorithm for obtaining consistent LDHFPRs is offered. Based on consistent linguistic intuitionistic fuzzy preference relations, a method for calculating the weighted linguistic intuitionistic fuzzy priority vector is introduced. In the setting of group decision making (GDM), a consensus measure based on individually weighted consistent reverse complementary linguistic intuitionistic fuzzy preference relations is defined. When the consensus does not satisfy the requirement, a two-step optimal model-based method for increasing the consensus level is offered. Furthermore, an approach for GDM with LDHFPRs is developed. Finally, an illustrative example concerning the evaluation of basic services internet enterprise websites is provided to show the efficiency of the new method. [ABSTRACT FROM AUTHOR]

Numeric models (including fuzzy models) produce numeric results. There are no ideal models that deliver a complete match with the data. In this study, we advocate that a way of evaluating the quality of models can be realized at the higher level of abstraction by developing a concept of granular prediction. In this way, modeling results are expressed in the form of information granules, in particular as intervals or fuzzy sets. The study formulates a general conceptual and algorithmically supported statement: a meaningful evaluation framework to assess the quality of numeric models is the one engaging information granules. This general observation comprises a special case commonly investigated in regression analysis, where the quality of numeric results is expressed via granular constructs, namely, confidence or prediction intervals. The original design of prediction information granules is formulated as an optimization problem, in which the criteria of coverage of data and specificity of granular results are considered. In the optimization process, we also engage some nonlinear transformation of the level of information granularity depending upon the value of the numeric result. The proposed development is model agnostic and can support a variety of modeling architectures; the experimental part of the study is focused on rule-based models. Further generalizations of prediction information granules are covered by involving granular parameters in the design process. [ABSTRACT FROM AUTHOR]

Every real-world physical problem is inherently based on uncertainty. It is essential to model the uncertainty then solve, analyze and interpret the result one encounters in the world of vagueness. Generally, science and engineering problems are governed by differential equations. But the parameters, variables and initial conditions involved in the system contain uncertainty due to the lack of information in measurement, observations and experiment. However, It is necessary to develop a comprehensive approach for solving differential equations in an uncertain environment. The purpose of this work is to study and investigate the fuzzy solution of linear third-order fuzzy differential equations using the concept of strongly generalized Hukuhara differentiability (SGHD). To make our analysis possible, we apply the first and second differentiability up to the third-order fuzzy derivative of the fuzzy-valued function. Moreover, we develop an important result concerning the relationship between Laplace transform of fuzzy-valued function and third-order derivative. We construct an algorithm to determine a potential solution of linear third-order fuzzy initial-value problem using the Laplace transform technique. All these solutions are represented in terms of the Mittag-Leffler function involving a single series. Furthermore, we discuss the switching points of linear third-order differential equations and their corresponding solutions in fuzzy environments. To enhance the novelty of the proposed technique, some illustrative examples are presented as applications are analyzed to visualize and support theoretical results. [ABSTRACT FROM AUTHOR]

Interpretability of fuzzy rule-based models has always been of significant interest to the research community and the research in this area led to a number of far-reaching results. In this study, we briefly revisit the methodology and concepts of interpretability of Takagi–Sugeno (T-S) rule-based models and develop a conceptual framework involving several levels at which rules are interpreted. The layers at which interpretability is positioned are structured hierarchically by starting with the initial fuzzy set level (originating from the design of the rules), moving to information granules of finite support (where interval calculus is engaged) and finally ending up with symbols built at the higher level. As T-S rule-based models are endowed with local functions forming the conclusion parts of the rules, with the use of the principle of justifiable granularity, we develop a way of forming an interpretable conclusion in the form of information granule. To facilitate interpretability of conditions of the rules, multidimensional fuzzy sets (coming as a result of clustering) are decomposed into a Cartesian product of 1-D fuzzy sets and the quality of the resulting decomposition is evaluated. The quality of granular rules is assessed by analyzing the relationship between specificity of condition and conclusion information granules. The rules emerging at the level of symbols are further interpreted by engaging linguistic approximation, which helps approximate a collection of linguistic terms of subconditions producing a linguistic summarization in the form τ (inputs are A) consisting of a certain linguistic quantifier τ. The performance of summarization is provided in the form of ranking of the relevance of the rules. Experimental studies using publicly available data are completed and analyzed. [ABSTRACT FROM AUTHOR]

The Bandler–Kohout subproduct (BKS) method is one of the two widely acknowledged fuzzy relational inference (FRI) schemes. The previous works related to its stability and robustness mainly concentrated on how the output values were changed with perturbation parameters of input values. However, the works on estimating oscillation bounds of output values with regard to varying limits of input, are lacking. In this study, we investigate the oscillation-bound estimation of perturbations for BKS. First, the BKS output variation scopes are acquired for interval perturbation, where the $R$ -implication, ($S, N$)-implication, QL-implication, and $t$ -norm implication are adopted. Second, in allusion to the more sophisticated problem of the fuzzy reasoning chain with BKS, the oscillation bounds of BKS output resulting from input interval perturbation are offered. Third, we construct the upper and lower bounds of BKS output deviation originated in the simple perturbation of the input fuzzy set, in which the situations of one rule and multiple rules are both dissected. Finally, the stable properties of all these BKS strategies are confirmed. It is emphasized that interval perturbation and simple perturbation are more general ways to give expression describing the robustness issue, and the obtained oscillation bounds also deliver more detailed characterization of the output deviation along with the input perturbation. This study further validates the sound properties of the BKS method. [ABSTRACT FROM AUTHOR]

Rule-based fuzzy models play a dominant role in fuzzy modeling and come with extensive applications in the system modeling area. Due to the presence of system modeling error, it is impossible to construct a model that fits exactly the experimental evidence and, at the same time, exhibits high generalization capabilities. To alleviate these problems, in this study, we elaborate on a realization of granular outputs for rule-based fuzzy models with the aim of effectively quantifying the associated modeling errors. Through analyzing the characteristics of modeling errors, an error model is constructed to characterize deviations among the estimated outputs and the expected ones. The resulting granular model comes into play as an aggregation of the regression model and the error model. Information granularity plays a central role in the construction of granular outputs (intervals). The quality of the produced interval estimates is quantified in terms of the coverage and specificity criteria. The optimal allocation of information granularity is determined through a combined index involving these two criteria pertinent to the evaluation of interval outputs. A series of experimental studies is provided to demonstrate the effectiveness of the proposed approach and show its superiority over the traditional statistical-based method. [ABSTRACT FROM AUTHOR]

This article presents a novel network, contribution-degree-based spiking neural network (CDSNN), which combines ideas of spiking neural network (SNN) and fuzzy set theory. In this framework, two types of information, interval and instantaneous information conveyed by the membrane potential are described by two concepts such as area under membrane potential (AUM) and firing strength. Given that the neuron with large AUM or strong firing strength would enhance the frequency of action potentials of its postsynaptic neurons, the connection between the neuron and its postsynaptic neurons should be strengthened. Combined with an idea of membership function, three contribution degrees ($\boldsymbol{\mu}_E$ , $\boldsymbol{\mu}_S$ , and $\boldsymbol{\mu}_{ES}$) are defined to quantify the ability of a neuron to provide information for postsynaptic neurons. According to these three degrees, the corresponding SpikeProp learning algorithms, referred to as SPE, SPS, and SPES, are developed. Experimental results obtained on ten benchmark datasets, one high-dimensional feature dataset, one big dataset, and one time series dataset with some commonly used algorithms, networks and CDSNN demonstrate that CDSNN can achieve improved performance in terms of accuracy, generalization, precision, recall and F-measure. The article demonstrates that the mechanism by which interval-instantaneous information is simultaneously learned in a SNN is feasible. [ABSTRACT FROM AUTHOR]

Hesitant fuzzy linguistic term sets (HFLTSs) and double hierarchy hesitant fuzzy linguistic term sets (DHHFLTSs) are two frequently used linguistic information forms in uncertain decision-making environments. However, they only include membership grades and cannot yield fuzzy information from a negative aspect. A bipolar fuzzy set can quantify evaluation information from positive and negative sides using positive and negative memberships, respectively. To address this issue, double hierarchy bipolar hesitant fuzzy linguistic term sets (DHBHFLTSs) are proposed, which can highlight the importance of the negative membership degree, and the objects can be evaluated from positive and negative aspects. Furthermore, DHBHFLTSs increase the reasonableness and comprehension of the evaluation information in the process of optimal talent selection. This paper proposed a framework involving the stepwise weight assessment ratio analysis (SWARA) method and the extended weighted aggregated sum product assessment (WASPAS) method. The extended WASPAS method is utilized to aggregate the evaluation information of all the alternatives under the DHBHFLTSs context. So, this proposed method increases the ranking accuracy. The SWARA method is extended to DHBHFLTSs to rank and determine the criteria. This weight determination method is helpful for coordinating and gathering data from experts. Therefore, the proposed method can obtain the weight values efficiently. Subsequently, a case of talent selection is utilized to show the feasibility and applicability of the proposed framework. Finally, the accuracy and comparison analyses with other methods illustrate the superiority of this framework. [ABSTRACT FROM AUTHOR]

In this article, we present a semantic semisupervised learning (Semantic SSL) approach targeted at unifying two machine-learning paradigms in a mutually beneficial way, where the classical support vector machine (SVM) learns to reveal primitive logic facts from data, while axiomatic fuzzy set (AFS) theory is utilized to exploit semantic knowledge and correct the wrongly perceived facts for improving the machine-learning model. This novel semisupervised method can easily produce interpretable semantic descriptions to outline different categories by forming a fuzzy set with semantic explanations realized on the basis of the AFS theory. Besides, it is known that disagreement-based semisupervised learning (SSL) can be viewed as an excellent schema so that a co-training approach with SVM and the AFS theory can be utilized to improve the resulting learning performance. Furthermore, an evaluation index is used to prune descriptions to deliver promising performance. Compared with other semisupervised approaches, the proposed approach can build a structure to reflect data-distributed information with unlabeled data and labeled data, so that the hidden information embedded in both labeled and unlabeled data can be sufficiently utilized and can potentially be applied to achieve good descriptions of each category. Experimental results demonstrate that this approach can offer a concise, comprehensible, and precise SSL frame, which strikes a balance between the interpretability and the accuracy. [ABSTRACT FROM AUTHOR]

Information granulation and degranulation play a fundamental role in granular computing (GrC). Given a collection of information granules (referred to as reference information granules), the essence of the granulation process (encoding) is to represent each data (either numeric or granular) in terms of these reference information granules. The degranulation process (decoding) that realizes the reconstruction of original data is associated with a certain level of reconstruction error. An important issue is how to reduce the reconstruction error such that the data could be reconstructed more accurately. In this study, the granulation process is realized by involving fuzzy clustering. A novel neural network is leveraged in the consecutive degranulation process, which could help significantly reduce the reconstruction error. We show that the proposed degranulation architecture exhibits improved capabilities in reconstructing original data in comparison with other methods. A series of experiments with the use of synthetic data and publicly available datasets coming from the machine-learning repository demonstrates the superiority of the proposed method over some existing alternatives. [ABSTRACT FROM AUTHOR]

Querying and reporting from large volumes of structured, semistructured, and unstructured data often requires some flexibility. This flexibility provided by fuzzy sets allows for categorization of the surrounding world in a flexible, human-mind-like manner. Apache Hive is a data warehousing framework working on top of the Hadoop platform for big data processing. Hive allows executing queries and aggregating and analyzing data stored in Hadoop distributed file system and other repositories. Hive responds to the current needs for efficient big data warehousing, which is impossible with traditional data warehouses due to their rigid nature. This article presents the FuzzyHive library that extends the Hive framework with fuzzy sets based techniques for querying, analyzing, and reporting on big data warehouses. We formalize the fuzzy techniques used while operating on Hive-based data warehouses (including fuzzy filtering on dimensional attributes, projection with fuzzy transformation, fuzzy grouping, and joining). We also show how we embedded these operations in Hive query language, which was not studied so far. Such extensions make big data warehousing more flexible and contribute to the portfolio of tools used by the community of people working with fuzzy sets and data analysis. The FuzzyHive library complements the spectrum of available solutions for fuzzy data processing and querying in large datasets. We investigate Hive fuzzy querying performance, effectiveness, and scalability for various data storage formats (text, Avro, and Parquet). Our experiments demonstrate that the proposed extensions introduce more elasticity and are also efficient for big data warehousing, which is the first such kind of solution for this environment. [ABSTRACT FROM AUTHOR]

The architecture of AND/OR fuzzy neural networks exhibits outstanding learning abilities and significant interpretation capabilities. However, AND/OR networks suffer from structure-related problems namely low efficiency and slow convergence of learning due to several reasons such as high dimensionality and gradient-based learning algorithms which lead to a visible computing overhead. In this paper, we present a two-phase fuzzy logic-oriented network design that is composed of AND/OR neurons. This design takes advantages of Randomized Neural Network (RNN) to achieve higher convergence while exhibiting good nonlinear approximation capabilities. A gradient–based learning algorithm is implemented in the second phase of the design to further reduce values of performance index. The quality of the proposed design and resulting architecture is quantified through the use of numeric data along with fuzzy sets (information granules). Experimental results meet the research's objectives and the proposed design methodology opens up new future directions for proceeding with more improvements. [ABSTRACT FROM AUTHOR]

Transfer learning is aimed at supporting the design of machine learning models in the target domain D t , given that the knowledge (model) has already been constructed in the source domain D s. The domains D t and D s (as well as the corresponding tasks T s and T t) are similar, yet not identical. As a result, the model transferred from D s to D t in this new environment exhibits its relevance (credibility) only to some limited extent. In this study, we develop an original approach, where we advocate that the knowledge transfer (model transfer) gives rise to a granular model where the level of information granularity associated with the produced results quantifies the relevance (quality or credibility) of the transferred model. In other words, we stress that the quality of knowledge transferred to D t becomes captured through a granular generalization of the original numeric model. The overall systematic design process is elaborated on by focusing on the development process of granular neural networks carried out on a basis of the numeric neural networks coming from D s. The key aspect of the design is to elevate the existing numeric neural network to its granular counterpart by admitting that the connections of the developed model come in the form of information granules, in particular intervals and fuzzy sets. The optimization process is guided by adjusting (optimizing) the level of information granularity being regarded as an essential design asset. The optimized performance index builds upon the descriptors of information granules commonly encountered in Granular Computing. In particular, coverage and specificity measures are treated as sound performance indicators of the quality of knowledge transfer (viz. the performance of the granular neural network expressed in the target domain). Several illustrative examples are provided to visualize the performance of the established design environment. [ABSTRACT FROM AUTHOR]

While granular computing has experienced rapid growth in the past decades and some milestones have been reached, a comprehensive study of the representation capabilities delivered by numeric prototypes and granular prototypes produced by different techniques still calls for comprehensive research and a comparative analysis. Well-constructed information granules are reflective of the nature of the numeric evidence and serve as backbones of granular classifiers and granular models. The objective of this study is to review a number of clustering paradigms aimed at the construction of information granules, discuss the development of granular prototypes, and conduct a comprehensive evaluation of quality of numeric prototypes and their corresponding augmentations coming in the form of granular prototypes. We have been witnessing many studies devoted to the construction of information granules, but a comparative analysis of the quality of information granules constructed on a basis of prototypes produced by different clustering algorithms is still lacking. In this regard, the review of the clustering algorithms supporting the formation of information granules and the comprehensive comparative study of their usefulness in classification and modeling tasks offered in this study make sense. This will promote the usage of information granules in various future works, especially classification problem and system modeling. [ABSTRACT FROM AUTHOR]

This article is concerned with a problem of relational factorization which engages fuzzy relational calculus. It forms an interesting alternative to the method of nonnegative matrix factorization that has been commonly discussed and found in numerous applications. The relational factorization takes original n-dimensional data located in the unit hypercube and factorizes it into data of lower dimensionality and some fuzzy relations. Owing to the logic nature of processing delivered by relational calculus, the dimensionality reduction exhibits transparency as the reduction mechanism itself is described in terms of logic expressions. Two types of factorizations mechanisms are investigated by using s–t and t–s composition operators where t and s are triangular norms and conorms, respectively. A two-level process of factorization is designed. A gradient-based learning scheme is developed. The quantification of the performance of the factorization process is realized by bringing a concept of information granularity: The obtained fuzzy relations are formed based on granular constructs and the quality of the produced factorization is assessed in terms of the coverage and specificity of the obtained granular results. A collection of experiments is included to present the performance of factorization and its parametric analysis. In addition, the proposed algorithm comes with sound interpretability in terms of both the structure of the model and an intuitive meaning of the fuzzy relations being the result of factorization. [ABSTRACT FROM AUTHOR]

Financial time series are generally high-dimensional, nonstationary, and exhibit heteroscedasticity. To derive a suitable way to cluster financial time series, these characteristics have to be taken into consideration. With this aim, in this article, the financial time series is firstly modeled using generalized autoregressive conditional heteroscedasticity (GARCH) models, where the parameters of GARCH models can represent the dynamic feature of the volatility in each time series. Therefore, the following clustering is realized based on the GARCH model parameters, which can help reduce the dimensionality of the original time series at the same time. Then, to produce semantically sound clustering results, we granulate the parameters based on the axiomatic fuzzy set (AFS) theory and structure them into a collection of meaningful and semantically sound entities, i.e., AFS information granules. Furthermore, the hierarchical structure of AFS information granules is built to realize time series clustering under the framework of granular computing. In the proposed approach, the characteristics of financial time series is fully considered to proceed dimensionality reduction, and the semantic clustering results obtained for different numbers of clusters are guaranteed to be the most informative. In the experiments, an application for clustering the time series coming from Chinese Yuan exchange rates against international currencies is presented to demonstrate the performance of the proposed clustering method. The results of clustering of the proposed method are the same as those of the fuzzy C-means algorithm and the hierarchical clustering with ward linkage, where the clustering results produced by the AFS hierarchical clustering exhibit well-articulated semantics at each level of the hierarchy. [ABSTRACT FROM AUTHOR]

This article investigates group decision-making (GDM) problems, where the decision makers’ (DMs) preference information is represented by incomplete interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, a multiplicative consistency property and an acceptably multiplicative consistency property for IVIFPRs are offered. Then, an optimization model to estimate the missing values in an incomplete IVIFPR is constructed. Subsequently, two optimization models are, respectively, established to derive a perfectly consistent IVIFPR and an acceptably consistent IVIFPR from a given inconsistent IVIFPR. Furthermore, a model is offered to gain the DMs’ weights. Afterward, the consensus index is defined. When the consensus for IVIFPRs is unacceptable, a model is presented to reach the consensus requirement. Moreover, a novel GDM method for incomplete IVIFPRs is presented. Finally, the presented method is applied to an illustrative example that shows the feasibility of the offered method. [ABSTRACT FROM AUTHOR]

A large number of clustering methods exist including deterministic, probabilistic, and fuzzy clustering. All these methods are devoted to handling different types of uncertainty. No studies have been encountered on clustering taking into account a confluence of probabilistic and fuzzy information. In the existing studies, the reliability of extracted knowledge is one of the important issues to be investigated. The concept of Z-number arises as a formal construct that expresses reliability of information under bimodal distribution. In this article, we propose an approach to construction of Z-number-valued clusters of a dataset for evaluation of reliability of extracted data-driven knowledge. Real-world applications are given that confirm the usefulness of the proposed method. [ABSTRACT FROM AUTHOR]

Orderly Atanassov-type intuitionistic membership grades would be required in decision-making problems, however, sometime they are not completely ordered. To solve this problem, in this article we propose a quantification method for Atanassov-type intuitionistic membership grades, and use it to rank them. According to the majority voting rules, we introduce the measurement function for membership degree. We quantify the uncertainty of information and the preferences of decision-makers conveyed through intuitionistic fuzzy sets. We then use the introduced surrogates to construct the measurement for membership grades. The properties and some logical operations of measurement value are also studied. We recommend using the Takagi–Sugeno model and method to assign values to tuning parameters $K$. Moreover, we present two models for multicriteria decision-making problem, which use the measurement to determine the ranking between sets. Finally, a numerical example of supplier selection is given to show the competitive performance of the proposed method in terms of efficiency and feasibility. [ABSTRACT FROM AUTHOR]

Dual hesitant fuzzy elements (DHFEs) are suitable to express hesitant possible preferred and nonpreferred judgments of decision makers. Preference relation is an important tool in decision making that only needs the decision makers to compare a pair of objects at one time. This study focuses on decision making with dual hesitant fuzzy preference relations (DHFPRs). Considering the consistency, an additive consistency concept is defined. Meanwhile, the property of the new concept is studied. Using this consistency concept, a method for assessing the additive consistency of DHFPRs is offered. To extend the application of DHFPRs, a programming model to determine the missing DHFEs in incomplete DHFPRs is built, which have the highest additive consistency level for the known ones. Two equivalent methods to calculate the priority vector are offered. One method obtains the probabilistic dual hesitant fuzzy priority vector, and the other derives the intuitionistic fuzzy priority vector. Furthermore, a consensus index is defined to measure the consensus of individual opinions in group decision making (GDM), and an interactive method for increasing the consensus level is offered. On the basis of the additive consistency and consensus, an algorithm to GDM with DHFPRs is offered that can address inconsistent and incomplete cases. Finally, a practical example about evaluating color TV is provided to demonstrate the usefulness of the new procedure. [ABSTRACT FROM AUTHOR]

The probabilistic linguistic term set (PLTS), composed by linguistic terms and their probabilities, is effective to represent uncertain evaluations. Considering that interval probability is more powerful than the precise form in describing uncertainty, this study introduces the PLTS with interval probabilities. Based on belief and plausibility measures, in this article, we discuss how to translate complex qualitative evaluations, which may be composed by both interval probabilities and interval linguistic terms, to the PLTS with interval probabilities. Utility-based translation approaches are proposed, which further shows the ability of the PLTS with interval probabilities in representing quantitative information. In addition, a probabilistic linguistic dominance method is developed to compare PLTSs. Integrating optimization models with the Dempster–Shafer theory, we present an aggregation method to estimate the maximum and minimum PLTSs obtained from the combination. Furthermore, a multicriteria decision-making method is introduced considering both the comprehensive evaluations of alternatives and the ability to achieve the tolerance and expectation values of criteria. The applicability of the proposed approach is illustrated by a case study of shelter selection. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with a problem of aggregation of order-2 information granules, and fuzzy sets, in particular. When processing order-1 fuzzy sets, the structural information about the space over which fuzzy sets are defined is not taken into account at all. In contrast, the aggregation of order-2 fuzzy sets requires a careful attention that needs to be paid both to the closeness determined in the space of membership degrees and the collection of information granules over which such fuzzy sets are defined. We formulate an original optimization problem that simultaneously involves considerations of distances in the membership space (space of membership grades) and some measure of resemblance formed in the space of relationships of reference information granules. The gradient-based learning scheme is constructed. Some illustrative examples are included. [ABSTRACT FROM AUTHOR]

In this article, we advocate that a knowledge tidbit residing in the output space could be helpful in improving the performance (accuracy) of the fuzzy rule-based model. It states that if two outputs are far apart from each other, it is advisable to place their corresponding inputs in different clusters when forming subspaces of the input space. Considering this knowledge guidance mechanism, we propose two different methods to partition the input space. In the first method, input data are first partitioned with the use of the standard clustering algorithm, say fuzzy C-means; here, a constructed partition matrix is reflective of the structure present in the input space. Then, the knowledge tidbit is used to adjust the entries of the original partition matrix in such a way that those input data whose corresponding output data are far apart from each other are assigned with low values of proximity. In the second method, we propose two strategies to modify the distance between input data and a prototype (cluster center) identified in the input space. The crux of this method is that if there are many input data (which, in virtue of the knowledge tidbit, are regarded as being far-apart from the input data of interest) around a certain prototype, the distance between the input data of interest and this prototype should be penalized. Thus, the membership of these input data to the prototype is reduced. The comprehensive experimental studies carried out on both synthetic and publicly available data are used to examine the usefulness of the proposed methods. [ABSTRACT FROM AUTHOR]