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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]

This study elaborates on a novel Hidden Markov Model (HMM)-based fuzzy model for time series prediction. Fuzzy rules (rule-based models) are employed to describe and quantify the relationship between the input and output time series, while the HMM is regarded as a vehicle for efficiently capturing the temporal behavior or changes of the multivariate time series which are not capable to capture through commonly encountered fuzzy rule-based models. Essentially, the proposed strategies control the contribution of different fuzzy rules so that the proposed model can well model the dynamic behavior of time series. Fuzzy C-Means clustering technique is an alternative way to construct fuzzy rules. Particle swarm optimization serves as a tool to optimize the parameters of the model (e.g., transition matrix and emission matrix). We construct and investigate the performance of the HMM-based fuzzy model by using a series of synthetic and publicly available multivariate time series. Experimental results demonstrate that the proposed model shows better performance than the fuzzy rule-based models used without the involvement of HMMs. [ABSTRACT FROM AUTHOR]

As an important technology in artificial intelligence, Granular Computing has emerged as a new multi-disciplinary paradigm and received much attention in recent years. Information granules forming an abstract and efficient characterization of large volumes of numeric data have been considered as the fundamental constructs of Granular Computing. By generating centroids (prototypes) and partition matrix, fuzzy clustering is a commonly encountered way of information granulation. As a reverse process of granulation, degranulation involves data reconstruction completed on a basis of the granular representatives (decoding information granules into numeric data). Previous studies have shown that there is a relationship between the reconstruction error and the performance of the granulation process. Typically, the lower the degranulation error is, the better performance of granulation process becomes. However, the existing methods of degranulation usually cannot restore the original numeric data, which is one of the important reasons behind the occurrence of the reconstruction error. To enhance the quality of reconstruction (degranulation), in this study, we develop an augmented scheme through modifying the partition matrix. By proposing the augmented scheme, we elaborate on a novel collection of granulation-degranulation mechanisms. In the constructed approach, the prototypes can be expressed as the product of the dataset matrix and the partition matrix. Then, in the degranulation process, the reconstructed numeric data can be decomposed into the product of the partition matrix and the matrix of prototypes. By modifying the partition matrix, the new partition matrix is constructed through a series of matrix operations. We offer a thorough analysis of the developed scheme. The experimental results are in agreement with the underlying conceptual framework. The results obtained on both synthetic and publicly available datasets are reported to show the enhancement of the data reconstruction performance thanks to the proposed method. It is pointed out that by using the proposed approach in some cases the reconstruction errors can be reduced close to zero by using the proposed approach. [ABSTRACT FROM AUTHOR]

Many oversampling methods applied to imbalanced data generate samples according to local density distribution of minority samples. However, samples generated by these methods can only present a non-deterministic relationship between the local and global distributions. A generative adversarial network (GAN) is a suitable tool to learn an unknown global probability distribution. In this paper, we propose an improved GAN (I-GAN) to oversample according to the global underlying structure of minority samples. The originality of I-GAN stems from the fact it provides additional density distribution information of minority samples for GAN and generated samples. By building on this idea, three detailed strategies are presented: input random vectors of the generator are sampled from a rough estimate of the distribution of minority samples to orientate fake samples more believable; a residual about minority samples is added into the discriminator to strengthen the constraint of loss function; generated samples are redistributed with a reshaper. These three strategies provide innovative methodologies at various stages of GANs for the oversampling task. Compared with 22 classical and popular imbalanced sampling methods under metrics of G m , F 1 , and A U C on 24 benchmark imbalanced datasets, it is shown that I-GAN is effective and robust. The I-GAN implementation line procedure has been uploaded to Github (https://github.com/flowerbloom000/I-GAN). [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]

It is an interesting topic to interpret artificial neural networks (ANNs) by considering some change various approaches. This paper explores the relationship between the input and output units of the simplest ANN, a single layer perceptron for the binary classification problem, from the probability point of view. If the feature variables of datasets follow independent normal distribution and outputs are activated by sigmoid function or smooth Relu function, we advocate that the probability density function (pdf) of the output variable is an exponential family distribution. Furthermore, by introducing an intermediate variable, the pdf of the output variable can be written as a linear combination of three normal distributions with same spread but different centers. Based on these results, the probability of the predicted class label can be written as a standard normal cumulative distribution function (cdf). The originality of this paper comes with interesting theoretical results to provide ANNs with a new description of the relationship between input variables to output variables, which can enable ANNs to be understood from a new perspective. Extensive experiments based on one artificial synthesized dataset and ten real‐world benchmark datasets validate the reasonability of those results. [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]

• Extension of binary options to fuzzy options in GMCR. • Analysis of the power asymmetry conflict completed from the perspective of options. • Proposal of a fuzzy truth value option prioritization realized with the use of fuzzy truth values. • Stability analysis completed to reflect the dynamic interaction between DMs. Asymmetric power conflicts occur frequently. Because of the complexity of the conflict as well as the vagueness of the decision makers' cognition, it becomes urgent and highly motivated to propose an appropriate method to solve power asymmetry conflict. In this study, we consider that decision makers provide option choices quantified by some degrees of membership. The choice of an option is determined by the thresholds of selection degree. At the same time, due to the influence of the power, the follower adjusts its degree of option choice to reach consensus with the leader. The computational rules determining fuzzy truth value are given, and a fuzzy truth value option prioritization method is proposed to calculate the ranking of the states, where the states ordering is related to the fuzzy degree of option selection. Different from the previous studies, this paper is the first one to study the asymmetric power conflict from the perspective of options, considering the psychological threshold of decision maker for option selection, and pointing out that the option choice is described with the fuzzy values rather than being treated as two-valued (Boolean). Furthermore, the introduced stability analysis also reflects the interaction of the options of different decision makers, which makes the proposal being more in rapport with real-world scenarios. Finally, a case study of carbon emission reduction power asymmetry conflict in supply chain is studied to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Three-way approximations of fuzzy sets are an important scheme of granular computing, by abstracting a fuzzy set to its discrete three option-alternatives which adhere to human cognitive behaviors and reduce the computational burden. The key point of such three-way approximations of fuzzy sets is how to choose a suitable design leading to their realization. Undesired three-way approximations might be produced if the selected mechanism is unsuitable to data distribution. In this study, the principles for constructing three-way approximations of fuzzy sets are summarized. The following taxonomy of these principles is provided, namely (i) uncertainty balance-based principle, (ii) prototype-based principle, and (iii) model-based invoking the tradeoff between classification error and the number of data that have to be classified. A number of detailed optimization models are discussed in detail. To evaluate the performance of different construction principles, a general unsupervised learning framework based on three-way approximations of fuzzy sets is exhibited. Some synthetic data sets along with sixteen data sets from UCI repository are involved for experiments. Friedman testing followed by Holm-Bonferroni testing are exploited to test the performance significance of the proposed criteria, which can provide insights and deliver guidance when choosing a principle for constructing three-way approximations of fuzzy sets in the real-world scenarios. The research methods in this paper can also be extended to supervised and semi-supervised learning areas. [ABSTRACT FROM AUTHOR]

Fuzzy clustering is regarded as an unsupervised learning process that constitutes a prerequisite for many other data mining techniques. Deciding how to classify data efficiently and accurately has been one of the topics pursued by many researchers. We anticipate that the classification performance of the clustering is strongly dependent on the boundary data (viz. data located at the boundaries of the clusters). The boundary data hold some levels of uncertainties and as such contain more information than others. Usually the greater the uncertainty, the more information contained in such data. To improve the quality of clustering, this study develops an augmented scheme of fuzzy clustering, in which a novel weighted data-based fuzzy clustering is proposed. In the introduced scheme, a dataset is composed of boundary data and non-boundary data. The partition matrix is used to determine the boundary data and the non-boundary data to be next considered in the clustering process. Then, we assign different weights to each datum to construct the weighted data. During this process, we make the weights for the boundary data and the non-boundary data different, which makes the contributions of the boundary data and the non-boundary data to the prototypes being reduced and enhanced, respectively. Furthermore, we build a weighting function to determine the weights of the data. The weighted data are used to optimize the prototypes. With the optimized prototypes, the partition matrix can be refined, which ultimately makes the boundaries of the clusters optimized. Finally, the classification performance of fuzzy clustering is enhanced. We offer a thorough analysis of the developed scheme. Comprehensive experimental studies involving synthetic and publicly available datasets are reported to demonstrate the performance of the proposed approach. [ABSTRACT FROM AUTHOR]

Drones play a pivotal role in various industries of Industry 4.0. For achieving the application of drones in a dynamic environment, finding a clear path for their autonomous flight requires more research. This paper addresses the problem of finding a navigation path for an autonomous drone based on visual scene information. A deep learning-based object detection approach can localize obstacles detected in a scene. Considering this approach, we propose a solution framework that includes masking with a color-based segmentation method to identify an empty area where the drone can fly. The scene is described using segmented regions and localization points. The proposed approach can be used to remotely guide drones in dynamic environments that have poor coverage from global positioning systems. The simulation results show that the proposed framework with object detection and the proposed masking technique support drone navigation in a dynamic environment based only on the visual input from the front field of view. [ABSTRACT FROM AUTHOR]

In this study, reinforced fuzzy radial basis function neural networks (FRBFNN) classifier driven by feature extracted data completed with the aid of effectively preprocessing techniques and evolutionary optimization, and its comprehensive design methodology are introduced. An Overall structure of the reinforced FRBFNN comprises the preprocessing part, the premise part and the consequence part of fuzzy rules of the network. In the preprocessing part, four types of preprocessing algorithms such as principal component analysis (PCA), linear discriminant analysis (LDA), combination of PCA and LDA (Hybrid PCA) and fuzzy transform are considered. To extract feature data suitable to characterize signal data, the feature extraction of information data is carried out through the dimensionality reduction done by the preprocessing technique, and then the reduced data are used as the input to the FRBFNN classifier. In the premise part of fuzzy rules of the network, the number of fuzzy rules is determined according to the number of clusters by fuzzy c-means (FCM) clustering. The fitness values of individual fuzzy rules are obtained based on data distribution. In the consequence part of fuzzy rules of the network, the parameters of connection weights located between the hidden layer and the output layer of FRBFNN classifier are estimated by means of the least square estimation. Particle swarm optimization (PSO) is exploited for structural as well as parametric optimization in the FRBFNN classifier. The parameters to be optimized by PSO are related to six factors such as the determination of whether to use data preprocessing, the type of data preprocessing technique, the number of input variables reduced by the preprocessing technique, fuzzification coefficient and the number of fuzzy rules used in fuzzy c-means (FCM) clustering, and the type of connection weights. By using diverse benchmark dataset obtained from UCI repository, the classification performance of the reinforced FRBFNN classifier was evaluated. Through a variety of classification algorithms existed in the Weka data mining software (Weka), the classification performance of the reinforced FRBFNN classifier was compared as well. The superiority of the proposed classifier is demonstrated through Friedman test. Furthermore, we assessed the classification performance of the reinforced FRBFNN classifier applied to black plastic wastes spectral data acquired from Raman and Laser induced breakdown spectroscopy equipment for the practical application of the material sorting system of the black plastic wastes. [ABSTRACT FROM AUTHOR]

In literature granular computing and formal concept analysis algorithm use only single-value attributes to knowledge discovery for the data of spatio-temporal aspects. However, most of the datasets like forest fires and tornado storms involve multiscale values for attributes. The limitation of single-value attributes of the existing approaches indicates only the data related to event occurrence which may be missing the elicitation of important knowledge related to severity of event occurrence. Motivated by these limitations, this research article proposes a novel and generalized method which uses ordinal semantic weighted multiscale values for attributes in formal concept analysis with granular computing measures especially when spatio-temporal attributes are not given. The originality of proposed methodology is using ordinal semantic weighted multiscale values for attributes that give complete information of event occurrences. Moreover, the use of ordinal semantic weighted multiscale values improves the results of granular computing measures. The significance of proposed approach is well explained by experimental evaluation performed on publicly available datasets on storm occurring in different States of America. [ABSTRACT FROM AUTHOR]

Probability weighting function (PWF) is the psychological probability of a decision-maker for objective probability, which reflects and predicts the risk preferences of decision-maker in behavioral decisionmaking. The existing approaches to PWF estimation generally include parametric methodologies to PWF construction and nonparametric elicitation of PWF. However, few of them explores the combination of parametric and nonparametric elicitation approaches to approximate PWF. To describe quantitatively risk preferences, the Newton interpolation, as a well-established mathematical approximation approach, is introduced to task-specifi- cally match PWF under the frameworks of prospect theory and cumulative prospect theory with descriptive psychological analyses. The Newton interpolation serves as a nonparametric numerical approach to the estimation of PWF by fitting experimental preference points without imposing any specific parametric form assumptions. The elaborated nonparametric PWF model varies in accordance with the number of the experimental preference points elicitation in terms of its functional form. The introduction of Newton interpolation to PWF estimation into decision-making under risk will benefit to reflect and predict the risk preferences of decision-makers both at the aggregate and individual levels. The Newton interpolation-based nonparametric PWF model exhibits an inverse S-shaped PWF and obeys the fourfold pattern of decision-makers' risk preferences as suggested by previous empirical analyses. [ABSTRACT FROM AUTHOR]

• Time series Covid-19 prediction problem, a case study in Vietnam. • A dual attention mechanism with RNN-based auto-encoding for handling chaotic time series prediction task. • Extensive experiments with realistic Covid-19 data in Vietnam. • Datasets and implementation source code are available for future studies. For preventing the outbreaks of Covid-19 infection in different countries, many organizations and governments have extensively studied and applied different kinds of quarantine isolation policies, medical treatments as well as organized massive/fast vaccination strategy for over-18 citizens. There are several valuable lessons have been achieved in different countries this Covid-19 battle. These studies have presented the usefulness of prompt actions in testing, isolating confirmed infectious cases from community as well as social resource planning/optimization through data-driven anticipation. In recent times, many studies have demonstrated the effectiveness of short/long-term forecasting in number of new Covid-19 cases in forms of time-series data. These predictions have directly supported to effectively optimize the available healthcare resources as well as imposing suitable policies for slowing down the Covid-19 spreads, especially in high-populated cities/regions/nations. There are several progresses of deep neural architectures, such as recurrent neural network (RNN) have demonstrated significant improvements in analyzing and learning the time-series datasets for conducting better predictions. However, most of recent RNN-based techniques are considered as unable to handle chaotic/non-smooth sequential datasets. The consecutive disturbances and lagged observations from chaotic time-series dataset like as routine Covid-19 confirmed cases have led to the low performance in temporal feature learning process through recent RNN-based models. To meet this challenge, in this paper, we proposed a novel dual attention-based sequential auto-encoding architecture, called as: DAttAE. Our proposed model supports to effectively learn and predict the new Covid-19 cases in forms of chaotic and non-smooth time series dataset. Specifically, the integration between dual self-attention mechanism in a given Bi-LSTM based auto-encoder in our proposed model supports to directly focus the model on a specific time-range sequence in order to achieve better prediction. We evaluated the performance of our proposed DAttAE model by comparing with multiple traditional and state-of-the-art deep learning-based techniques for time-series prediction task upon different real-world datasets. Experimental outputs demonstrated the effectiveness of our proposed attention-based deep neural approach in comparing with state-of-the-art RNN-based architectures for time series based Covid-19 outbreak prediction task. [ABSTRACT FROM AUTHOR]

At present, existing methods carry out imbalanced classification at two levels, namely a sampling level and an algorithmic level. However, different sampling methods provide different data distributions for classifiers, which may lead to unreliable decision-making. The algorithmic level methods are often limited to a given classifier. In this paper, we propose a new iterative ensemble classifier for imbalanced data with iterative learning process and ensemble operating process (C-ILEO). There is an ensemble base classifier in each iteration of learning and its elementary classifier learns datasets in a low dimensional space composed of a small number of features which are selected with a new rule based on mutual information. Samples classified into negative class by the ensemble base classifier at each iteration are removed because these samples can no longer provide significant information for the following learning. In addition, the class weight of each elementary classifier is optimized with a new strategy which is similar to the half-interval search algorithm. The ensemble operating process is carried out on what all ensemble base classifiers learned on all the training samples including removed samples during the iteration. Compared with 16 commonly used imbalanced learning methods on 15 real world imbalanced datasets, C-ILEO outperforms most algorithmic, data level and ensemble approaches on metrics of G m e a n , F 1 , A c c , and P r e c i s i o n. [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]

In virtue of omnipresence of temporal data (time series), a quest for efficient and interpretable models of such data becomes of paramount relevance. The accuracy-interpretable dilemma becomes more visible in light of the current trend of producing accurate, yet comprehensible models supporting users in making sound and highly actionable conclusions. This study is positioned in the realm of human-centric granular representation of temporal data (time series) where the logic settings of relational calculus is realized at the level of information granules. We establish modeling pursuits and resulting constructs at a certain level of abstraction delivered by information granules expressed as fuzzy sets. While have been some studies involving a framework of granular computing applied to granular models of time series, in this study we advocate a unified and original environment of logic blueprint of relational dependencies among information granules articulated by means of weighted logic expressions where both the components of the expressions as well as their associations are quantified (calibrated) in the presence of available experimental data. An overall processing scheme is structured as the following sequence of main functional modules: granulation (encoding) – relational computing – logic interpretation – numeric characterization of results (produced through decoding). Relational computing engages a suite of architectures that capture temporal dependencies describing predictions at the level of granular amplitudes and granular trends. The quality of the relational model is quantified at the level of information granules (which emphasizes the interpretability aspect) as well as. A series of experiments is reported to demonstrate the performance of the relational models and their performance impacted by the parameters of the relational architectures. [ABSTRACT FROM AUTHOR]

• New concepts for ATrFPRs are defined that avoid the issues in previous ones. • New methods can cope with incomplete ATrFPRs while all previous research cannot. • New methods based the acceptable consistency have some merits than previous ones. • Optimization models are more reasonable than the iterative method. • Numerical examples and comparative analysis are made. Trapezoidal fuzzy numbers offer a suitable and flexible way to express vague judgments of decision makers. This paper studies decision making with additive trapezoidal fuzzy preference relations. It first analyzes the issues encountered in previous consistency concepts for additive trapezoidal fuzzy preference relations. Then, two new consistency concepts, namely, an additive consistency concept and a multiplicative consistency concept, are introduced. To verify the rationality, several of their properties are discussed. To efficiently assess the consistency of additive trapezoidal fuzzy preference relations, several corresponding optimization models are formed. When the objective function values of the built models are zero, the associated additive trapezoidal fuzzy preference relations are consistent. Considering the case where additive trapezoidal fuzzy preference relations are incomplete, optimization models based on the consistency analysis are built, by which identified unknown judgments have the highest consistency level with the known ones. In general, additive trapezoidal fuzzy preference relations are unacceptably consistent. To rank objects from additive trapezoidal fuzzy preference relations, several optimization models for improving the consistency level are built, which consider the self-confidence of the decision makers, the total adjustment and the number of adjusted elements. According to the above discussion, two frameworks for ranking objects from additive trapezoidal fuzzy preference relations are provided that are based on the additive and multiplicative consistency analysis, respectively. Finally, numerical examples are provided to highlight the concrete application and to deliver the comparative analysis. [ABSTRACT FROM AUTHOR]

In group decision-making problems, experts often form multiple subgroups depending upon their backgrounds or attitudes. To explore the effect of different types of subgroups' formation on the final decision and to make the decision process more conforming to human custom, we propose a comprehensive study on the effect of "multi-subgroup background" on a basis of weights optimization and Granular Computing techniques. Firstly, a comparison of different types of clustering strategies is realized to find the most suitable method for clustering experts. Since Analytical Hierarchy Process is selected as the fundamental model, it becomes a reciprocal matrices' or eigenvectors' clustering task. Secondly, allocation strategies of information granularity to each expert along with weight to each subgroup are carefully designed and optimized. Information granularity is viewed as a design asset to provide experts with some flexibility to adjust their evaluation results, whereas weights reflect the importance degrees of different subgroups and further help to increase the consensus. A granulation-degranulation process is developed to evaluate the performance of a set of weights and granularities under a multi-criteria objective's guidance. Thirdly, a proper evolutionary optimization method like particle swarm optimization is redesigned to iteratively generate the best set of information granularities and weights under an equality constraint condition. A series of numerical studies on synthetic data and real-world data are executed to verify the effectiveness of our method. The results show same trend present across experts. [ABSTRACT FROM AUTHOR]

In this study, we introduce and develop a novel decision-making model that provides an original solution to problems of renewable energy site planning. The main idea is to construct a comprehensive and systematic methodology of ranking alternatives encountered in the environment of multicriteria group decision making. The proposed framework is systematically structured with the aid of information granules, in particular, intervals and fuzzy sets. An overall architecture is developed in a comprehensive manner. The inherent facet of uncertainty, it is formalized and processed with the aid of information granules. The two main design phases involve the determination of preference degrees of alternatives with respect to the set of criteria and the weights of the corresponding criteria. The underlying estimation process is realized with the use of the pairwise comparison method (analytical hierarchy process-AHP) resulting in information granules (fuzzy sets) quantifying degrees of preference and relevance of the weights. In light of the group nature of the decision process and diversity of views and opinions conveyed by the individual decision-makers, the results provided by them are aggregated and the diversity (variability) in the individual assessments is captured through information granules of type-2. Finally, a variety of ranking procedures is analyzed and carefully assessed. A case study of selection of solar site is provided to demonstrate the usefulness of the developed approach. Compared with the existing decision-making scenarios, we show that the new model exhibits a significant level of reliability and is characterized by better interpretability. [ABSTRACT FROM AUTHOR]

In time-series forecasting, it is an important task to make an accurate and interpretable long-term prediction. In this article, we present a novel approach developed from the perspective of granular computing (GrC) to realize the long-term prediction of time series. The proposed method first employs a sliding window strategy to smooth on the raw time series. Subsequently, the smoothed time series is transformed into the corresponding granular time series that is depicted by evolving shape with the aid of the clustering algorithm based on the dynamic time warping (DTW) distance. Finally, a Takagi–Sugeno (TS) architecture-like granular model (GrM) is formed by deriving the relations implying in the granular time series and offers the granular output in the numeric vector format. The GrM adopts the pattern-to-pattern inference mechanism to realize the long-term prediction of time series at the vector level. Experiments on several datasets demonstrate that the proposed method not only has the ability to circumvent the cumulative error but also makes the resulting GrM equip better interpretability. [ABSTRACT FROM AUTHOR]

Considering the conditions that: 1) same linguistic term means different things for different people; 2) flexible semantics cannot be represented by original linguistic term; and 3) some semantics given by decision makers are possible to be changed during the consistency improving process, we bring some flexibility and personality into hesitant fuzzy linguistic preference matrix structures by allowing the linguistic preference matrices to be granular rather than numeric, providing a new characterization of linguistic preference matrices. Inspired by the thought of granular computing, this article proposes a new hesitant fuzzy linguistic method to deal with issues when a lot of decision makers provide hesitant and uncertain preference information in the decision-making process. First, we design a multiplicative consistency index and calculate its thresholds corresponding to different dimensions of preference matrix by the Monte Carlo experiment. Then, we construct a hesitant fuzzy linguistic model with granularity level, so as to recharacterize original assessment information and improve the consistency of preference matrices as far as possible. Considering the features of some large-scale group decision-making situation, where the decision makers have little opportunity to take part in multiple consensus reaching processes, hesitant fuzzy linguistic fuzzy $C$ -means clustering algorithm is developed to integrate the assessment information given by decision makers. Finally, the final decision-making results are derived. An illustrative example of assessing psychological situation of some COVID-19 infected persons clarifies the reasonability of the proposed method. Finally, we complete some comparative studies and simulation experiments to demonstrate the method’s validity and advantages. [ABSTRACT FROM AUTHOR]

This paper investigates decision making with normalized probabilistic linguistic preference relations (NPLPRs). Consistency analysis is indispensable for deriving the reasonable ranking. After recalling previous research, we find that all previous concepts cannot fully define consistent NPLPRs. As a fundamental topic of decision making with preference relations, it is necessary to further study the consistency of NPLPRs. For this purpose, an interactive algorithm for deriving disjunctive probabilistic additive linguistic preference relations (DP-ALPRs) is provided, by which an additive consistency concept for NPLPRs is defined. When NPLPRs are unacceptably consistent, models for obtaining acceptably additively consistent NPLPRs are built. Considering the situation where only incomplete NPLPRs are obtained, a disjunctive probability and additive consistency based interactive algorithm for ascertaining missing judgments is provided. Meanwhile, we discuss group decision making (GDM) with NPLPRs and offer a distance measure based formula to determine the weights of the decision makers. In addition, the method defines a consensus index and builds models for improving the consensus level. Under the additive consistency and consensus discussions, an interactive algorithm for GDM with NPLPRs is proposed. Finally, the new method is applied to select green raw material suppliers to illustrate the application and compared with several previous ones. [ABSTRACT FROM AUTHOR]

Information granules have been considered as the fundamental constructs of Granular Computing. As a useful unsupervised learning technique, Fuzzy C-Means (FCM) is one of the most frequently used methods to construct information granules. The FCM-based granulation–degranulation mechanism plays a pivotal role in Granular Computing. In this paper, to enhance the quality of the degranulation (reconstruction) process, we augment the FCM-based degranulation mechanism by introducing a vector of fuzzification factors (fuzzification factor vector) and setting up an adjustment mechanism to modify the prototypes and the partition matrix. The design is regarded as an optimization problem, which is guided by a reconstruction criterion. In the proposed scheme, the initial partition matrix and prototypes are generated by the FCM. Then a fuzzification factor vector is introduced to form an appropriate fuzzification factor for each cluster to build up an adjustment scheme of modifying the prototypes and the partition matrix. With the supervised learning mode of the granulation–degranulation​ process, we construct a composite objective function of the fuzzification factor vector, the prototypes and the partition matrix. Subsequently, the particle swarm optimization is employed to optimize the fuzzification factor vector to refine the prototypes and develop the optimal partition matrix. Finally, the reconstruction performance of the FCM algorithm is enhanced. Overall, we show that the enhanced version of the degranulation process is beneficial to reduce the deterioration of the reconstruction results and improve the performance of the mechanism of granulation–degranulation, which is also meaningful for transforming data between numeric form and granular format. We offer a thorough analysis of the developed scheme. In particular, we show that the classical FCM algorithm forms a special case of the proposed scheme. Experiments completed for both synthetic and publicly available datasets demonstrate that the proposed approach outperforms the generic data reconstruction approach. [ABSTRACT FROM AUTHOR]

This research is a review and analysis paper that offers a transdisciplinary, methodological, and strategic vision for soft computing development towards a wider favorable impact in data analytics. Strategies are defined, explained, and illustrated by examples. The paper also shows how these strategies are expressed in three dimensions of an ambitious actions plan. They are all integrated into a master strategy called wide knowledge discovery, which offers a way towards the augmented analytics paradigm. Some contributions of this work are defining what kind of mathematical elements should be introduced into soft computing towards a better impact on the area of data analytics, offering orientation towards building new mathematical elements, and defining why and how they can be introduced. [ABSTRACT FROM AUTHOR]

Fuzzy rule-based models are widely employed to tackle regression problems due to their simplicity and comprehensibility. Numeric functions (e.g., linear ones) are generally utilized to represent the conclusion part of a rule, but this simple module cannot well describe the behavior of data located within local area defined by the premise part. In this study, a novel reinforced fuzzy clustering-based rule model (RFCRM) is presented to address this issue. First, information granulation (Fuzzy C-Means) approach is used to reveal the structure in the data and divide the input space into local regions, as well as form the premise parts. Second, the augmented Random Vector Functional Link Networks (RVFLNs) are employed as the conclusion parts to strengthen the description and representation of the behavior of data positioned within the local regions. Weighted Least Squares Error estimation with regularization is exploited to estimate the coefficients of the connection weights of RVFLN. Also, an exponential weight method benefited from the weight function theory encountered in harmonic analysis is introduced to improve the penalty strategy of regularization in coefficient estimation. More specifically, an exponential weighted ℓ 2 regularization (EW ℓ 2) strategy equipped with the exponential functions as the penalty terms is proposed to realize the targeted penalty on coefficients. The merit of EW ℓ 2 over ordinary ℓ 2 is that different types of polynomial terms can be identified and penalized separately in coefficient estimation. This strategy not only prevents the decline of generalization ability, but also effectively enhances the prediction potential of the model. By combining weighted LSE based on EW ℓ 2 with FCM partition and the augmented RVFLN leads to better prediction accuracy as well as lower complexity (viz., number of rules). Finally, detailed experimental studies are conducted to demonstrate the effectiveness of the proposed RFCRM. [ABSTRACT FROM AUTHOR]

• We propose a joint framework that integrates estimation, consistency improvement and consensus reaching into one phase. • We propose a minimum adjustment model with completeness, consistency, and consensus control (MAM-CCCC). • We present a completeness, consistency, and consensus reaching process (CCCRP) to support the GDM problems with incomplete fuzzy preference relations (FPRs). • We verify the effectiveness of MAM-CCCC through simulation and comparison analyses. In group decision making (GDM), there exists a situation in which experts may not able to provide comparative judgements regarding all pairs of alternatives, and therefore leading to the expression of incomplete preference. The existing frameworks to manage the GDM problem with incomplete preference relations have been investigated intensively and followed a common resolution scheme composed of three processes: estimation, consistency improvement, and consensus reaching. However, the consensus reaching process is applied after the estimation process and consistency improvement processes, leading to the destruction of the consistency of estimated preference relations, which will duplicate the process of improving consistency and consensus, resulting in significant adjustments of experts' preferences. To avoid this issue, we first propose a joint framework to integrate the consensus reaching process with estimation and consistency improvement. This framework is designed based on a minimum adjustment optimization model with completeness, consistency, and consensus control. It can produce complete preference relations with acceptable consistency and consensus, and simultaneously guarantee the minimum adjustment of experts' preferences. Finally, simulation and comparative analyses are used to justify the validity of the proposal. [ABSTRACT FROM AUTHOR]

• A novel forecasting model of LSTM-FPNN based on E_RWS is proposed to enhance the generalization abilities for time-series forecasting. • A new neuron driven by deep learning of LSTM is proposed to effectively learn temporal features. • A fuzzy polynomial neuron is used to capture the nonlinear characteristics of the data. • A hybrid selection strategy is developed to effectively increase the diversity of neurons. • The proposed LSTM-FPNN is applied to two practical domain problems to demonstrate its superiority. In this study, we propose a new self-organizing deep network architecture of fuzzy polynomial neural networks (FPNN) based on Fuzzy rule-based Polynomial Neurons (FPNs) and a long short-term memory (LSTM) network to solve the task of time-series forecasting. In the existing regression model based on polynomial neural networks (PNN), it is difficult to achieve high quality performance when predicting time series data, because this model lacks the ability to extract temporal and spatial information. Therefore, we propose a new architecture consisting of one LSTM (temporal) layer and several fuzzy polynomial (spatial) layers to overcome the above-mentioned shortcomings of PNN and enhance its predictive ability to approximate the data. The temporal layer consists of LSTM neurons that have inherently strong modeling capabilities to learn sequential information. The spatial layers are composed of Rule-based Polynomial Neurons (FPNs) that can effectively reflect the complex nonlinear structure found in the input space and granulate it using of the Fuzzy C-Means (FCM) clustering method. An elitism-driven roulette-wheel selection (E_RWS) is used to select appropriate neurons. E_RWS not only ensures that the neuron with the strongest fitting ability is selected but also increases the diversity of candidate neurons. According to the experimental results, the proposed model has a high prediction performance and outperforms many state-of-the-art prediction methods when applied to the real-world time-series. [ABSTRACT FROM AUTHOR]

This paper introduces some recent approaches to the fuzzy fractional Abel integral equations with respect to another function, which is called the fuzzy fractional Abel k −integral equations. The problem proposed here allows for the interpolation of different types of fuzzy classical fractional Abel integral equations, and the solvability of each type of integral equation is also discussed. Additionally, a new approach to the linear fuzzy fractional differential equations in the space of generalized fuzzy functions is introduced. Unlike previous approaches with gH-differentiability, this approach avoids making any prior assumptions about the solution's monotonic behavior during the solving process. It also overcomes the problem of multiple solutions for fuzzy fractional differential equations. To demonstrate the benefits of this approach, two initial value problems of linear fuzzy differential equations are compared using two different concepts of fractional derivatives: the Caputo fractional gH-differentiability and the Caputo fractional derivative in the generalized fuzzy space. Furthermore, some applications are provided in a viscoelastic material model and a population growth with harvesting model. [ABSTRACT FROM AUTHOR]

The product–service system (PSS) is a strategic design approach proposed to address sustainability in socio-economic systems amidst rapid industrialization and transition. Evaluating the concept design of a PSS is a crucial and initial step prior to implementation. This study presents an innovative framework for evaluating concept designs of sustainable PSS based on a well-defined evaluation index system via integrating quality function deployment (QFD) and the technique for order preference by similarity to ideal solution (TOPSIS) while accommodating extended basic uncertain linguistic information (EBULI). Specifically, a QFD-based framework is first developed to identify the requirements of various stakeholders and then to establish the multi-dimensional criteria for evaluating sustainable PSS. Then, a House of Quality-based relationship matrix is introduced to determine the weights of criteria more accurately. Further, an adaptive consensus-reaching process method based on an expert weighting optimization model is proposed to ensure a collective outputs recognized by multiple involved stakeholders. Finally, an improved EBULI-based TOPSIS method is presented to determine the priority ranking of alternative sustainable PSS concepts. A case study on a car-sharing PSS project demonstrates the viability and effectiveness of the proposed QFD–TOPSIS integrated approach under EBULI settings. The alternative PSS concept design, which demonstrates relatively good performance in criteria of high importance, is selected as the most suitable option. Moreover, relevant comparative and sensitivity analyses reveal that the proposed approach exhibits superiorities in appropriate criteria elicitation, accurate weights determination, and high consensus ranking outputs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a cascade AdaBoost neural network (CANN) based on concepts and construct of AdaBoost neurons and cascade structure. Compared with AdaBoost, CANN can represent complex relationships between features. In CANN, representation learning is performed through AdaBoost, and the method of random selection features is utilized to encourage the diversity of AdaBoost neurons. Through the cascade structure, CANN has the context structure for complex feature representation. At the same time, in order to avoid the problem of feature disappearance, shortcut connection is used to add the previous information to the later nodes. Furthermore, particle swarm optimization (PSO) algorithm is utilized to optimize the structure of CANN, it can obtain the number of iterations to achieve better performance. Two types of CANN are proposed based — binary-classification CANN (BCANN) or multi-classification CANN (MCANN). The performance of CANN is evaluated with two kinds of data sets: machine learning data sets and atrial fibrillation data set. A comparative analysis illustrates that the proposed CANN leads to better performance than the models reported in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we relax the definition of Rényi information dimension. The power law of the Entropy-Layer in the Galton board is discovered and we calculate its information fractal dimension. When the Galton board is extended to bias or three-dimensional space, we get the same fractal features. In addition, according to the connection between Pascal's triangle and the Poisson distribution, we find constrained Poisson distribution groups with the same information dimension. This is the first time the information entropy is utilized to explore the fractal features of the Galton board and Pascal's triangle. [ABSTRACT FROM AUTHOR]

Data representation is of significant importance in minimizing multi-label ambiguity. While most researchers intensively investigate label correlation, the research on enhancing model robustness is preliminary. Low-quality data is one of the main reasons that model robustness degrades. Aiming at the cases with noisy features and missing labels, we develop a novel method called robust global and local label correlation (RGLC). In this model, subspace learning reconstructs intrinsic latent features immune from feature noise. The manifold learning ensures that outputs obtained by matrix factorization are similar in the low-rank latent label if the latent features are similar. We examine the co-occurrence of global and local label correlation with the constructed latent features and the latent labels. Extensive experiments demonstrate that the classification performance with integrated information is statistically superior over a collection of state-of-the-art approaches across numerous domains. Additionally, the proposed model shows promising performance on multi-label when noisy features and missing labels occur, demonstrating the robustness of multi-label classification. [ABSTRACT FROM AUTHOR]

In this study, a design methodology based on fuzzy relation-based neural networks for stabilized network structure is introduced to cope with over-fitting as well as multi-collinearity problems which generally appear in conventional fuzzy neural networks. The design method of the proposed self-organizing network structure provides an efficient solution to construct the stabilized Fuzzy Relation-based Neural Networks (FRNN) through a synergy of multi-techniques such as ensemble neurons/layers, L2-norm regularization, compromise technique, and multi-optimization, in order to generate the structure of the multi-layered self-organizing network designed with the aid of the learning as well as novel structural design. The overall network structure is realized with the aid of parallel network structure with newly added layers as well as effective node selection method through the combination technique of both a sum of squared coefficients (SSC) and performance index (PI) as a node selection criterion for each layer in FRNN. Ensemble neurons consist of the current inputs and the original inputs, and they serve to reduce the bias of the proposed FRNN by maintaining the information of the original inputs. Also, ensemble layers stand for the combination of the current layer and the front layer, and they play a role for control the variance of the model by taking into account the output of the previous layers. The least square error estimation (LSE)-based learning method with L2-norm regularization is used for constructing the stabilized network architecture, and their ensuring design methodologies result in alleviating the overfitting phenomenon and also enhancing the generalization ability. For the performance enhancement of FRNN directly affected by some parameters such as the number of input variables, collocation of the specific subset of input variables, the number of membership functions per each variable, and the order of polynomial in the consequent parts of the fuzzy rules, multi-particle swarm optimization (MPSO) is exploited for the effectively structural as well as parametric optimization of the proposed networks. That is, the multi-optimization helps achieve a compromise between the better generation performance and the alleviated over-fitting leading to the stabilization of the proposed multi-layered self-organizing network structure realized with the aid of synergistic multi-techniques and ensemble structures such as a) L2-norm regularization-based LSE learning, b) combination technique for effective node selection through the combination of SSC and PI, and c) a novel parallel network structure including newly added layers and node selection method. The performance of the proposed network structure is quantified by comprehensive experiments and comparative analysis. [ABSTRACT FROM AUTHOR]

In computing with words, it has been stressed that words mean different things for different people, which entails that decision makers (DMs) have personalized individual semantics (PISs) attached to linguistic expressions in linguistic group decision making (GDM). In particular, the PISs of DMs are not fixed, and they will be changing during the consensus building process, which indicates the necessary of continual PIS learning. Therefore, in this article, we propose a continual PIS-learning-based consensus approach in linguistic GDM. Specifically, a continual PIS learning model with the consistency-driven methodology is proposed to update the PISs taking into account all the linguistic preference data given by DMs during the consensus process. Then, the consensus measurement and feedback recommendation based on PIS are developed to detect the consensus process. Finally, numerical examples and simulation analysis are presented to illustrate and justify the use of the continual PIS-learning-based consensus approach. [ABSTRACT FROM AUTHOR]

The study is concerned with the problem of classification of multivariate time series using convolutional neural networks (CNNs). As CNNs regard inputs in the form of images, an original image-like format of temporal data is proposed. Along this line, several design alternatives are studied by forming images with the two corresponding coordinates built by the original temporal data and their differences and second differences. An overall design process is presented with a focus on investigating time series-image transformations. Experimental studies involving publicly available data sets are reported, along with a slew of comparative analyses. [ABSTRACT FROM AUTHOR]

In uncertainty reasoning, conjunctive and disjunctive combination rules are the core tools for information updates. As an extension of evidential reasoning, random permutation set reasoning models uncertain information based on ordered focal sets. Existing combination rules in random permutation set theory, orthogonal sums, do not satisfy the commutativity of one of the original distributions and overly obey its order information. In this paper, the random permutation set theory is interpreted as an refined extension of Dempster–Shafer theory, and a layer-2 belief structure is proposed to describe the permutation event space. Compared with the traditional belief structure, the proposed structure can model both symbolic and numerical uncertainty. Based on the above, the conjunctive and disjunctive combination rules in Dempster–Shafer theory are extended to random permutation set theory. Through properties analysis and simulation demonstration, we demonstrate that the proposed methods can not only resolve the counter-intuitive results of orthogonal sums, but make full use of order information in distributions as well. In addition, we also extend the product space operations and discounting methods based on the proposed methods, and give a general framework of multi-source information fusion under the random permutation set theory. • First interpret the random permutation set from the layer-2 belief structure perspective. • First discuss the requirements of combination rules in random permutation set theory. • First propose the conjunctive and disjunctive rules in random permutation set theory. • First propose the multi-source information fusion approach in random permutation set theory. [ABSTRACT FROM AUTHOR]

In this paper, we present a method of extending t-norms and t-conorms to a given closed subinterval [ a , b ] in [ − ∞ , + ∞ ] , which preserves the basic properties of these fuzzy connectives and their one-to-one correspondence with operations in a I-semigroup. Subsequently, the generalized De Morgan triples, weighted generalized interval t-norms and t-conorms are provided to construct the aggregation function for extended basic uncertain information. Eventually, we apply the proposed aggregation function to establish the basic uncertain linguistic information-based aggregation operator. [ABSTRACT FROM AUTHOR]

Inspired by the collaborative mechanism among biological nervous, endocrine and immune systems, this paper proposes an algorithm of adaptive evolutionary based on biological cooperation (BCAE). This method can solve the dynamic multi-objective optimization problem of Industrial Internet of Things (IIoT) services to reduce the total service cost and service time. The BCAE algorithm consists of two parts: bottom level and top level. In the bottom level, different Pareto frontiers are obtained by coevolution of multiple subpopulations. In the top level, according to the distance between the service request and the service provider and the unit energy consumption of the service provider, the connection weight sequence is designed, and then the affinity matrix is constructed according to the connection weight sequence. Finally, the multi factor genetic algorithm (MFEA-II) is used to mate and imitate the service providers with different affinity, and the total service cost and total service time of the optimal solution are obtained, which are recorded in the top-level optimal antigen solution set. On the basis of single service strategy and collaborative service strategy, the IIoT services with dynamic requests are studied under different distributions. The obtained simulation results show that the performance of BCAE is better than the performance of the four existing algorithms, especially when solving high-dimensional problems. [ABSTRACT FROM AUTHOR]

• We propose fuzzy reinforced polynomial neural networks based on PNN architecture. • We propose fuzzy reinforced polynomial neurons (FRPNs) in the design of FRPNNs. • The proposed FRPNNs is a generalization of hybrid predictors (HPs). In the field of dynamic system identification and prediction, linear models (e.g., autoregressive models), nonlinear models (namely, neural networks models), and hybrid predictors (HPs) that are a hybridization of linear and nonlinear models have been proposed in the past. However, they are not completely free from limitations: they exhibit difficulties to describe high-order nonlinear relations between input and output variables. In this study, we propose fuzzy reinforced polynomial neural networks (FRPNNs), which are polynomial neural network architecture-based on fuzzy reinforced polynomial neurons (FRPNs) to overcome this limitation. The proposed FRPNs that consist of approximation part (AP) and compensation part (CP) arise as novel HPs. Here the CP for modeling nonlinear patterns can be regarded as forming the reinforced part for the AP that aims at capturing linear patterns, while AP is the linear polynomial neuron used in the conventional polynomial neural networks. In some sense, the overall FRPNNs are essentially generalized polynomial neural network architecture with novel HPs. The parameters considered in the design of the proposed fuzzy reinforced polynomial neural networks are optimized with the aid of the particle swarm optimization (PSO). The performance of FRPNNs is discussed involving time series and system identification datasets. Experimental results demonstrate that the proposed FRPNNs achieve at most the accuracy of 43.6% higher in comparison with the accuracy produced by some classical models reported in the literature. [ABSTRACT FROM AUTHOR]

With the increase in the number of optimization objectives, balancing the convergence and diversity in evolutionary multiobjective optimization becomes more intractable. So far, a variety of evolutionary algorithms have been proposed to solve many-objective optimization problems (MaOPs) with more than three objectives. Most of the existing algorithms, however, find difficulties in simultaneously counterpoising convergence and diversity during the whole evolutionary process. To address the issue, this paper proposes to solve MaOPs via multistage evolutionary search. To be specific, a two-stage evolutionary algorithm is developed, where the convergence and diversity are highlighted during different search stages to avoid the interferences between them. The first stage pushes multiple subpopulations with different weight vectors to converge to different areas of the Pareto front. After that, the nondominated solutions coming from each subpopulation are selected for generating a new population for the second stage. Moreover, a new environmental selection strategy is designed for the second stage to balance the convergence and diversity close to the Pareto front. This selection strategy evenly divides each objective dimension into a number of intervals, and then one solution having the best convergence in each interval will be retained. To assess the performance of the proposed algorithm, 48 benchmark functions with 7, 10, and 15 objectives are used to make comparisons with five representative many-objective optimization algorithms. [ABSTRACT FROM AUTHOR]

· Proposing a novel 3WD-GMCR framework under hesitant fuzzy environment. · Improving the option prioritization to explain the ranking strength of states. · Redefining the concepts of conflict stability. · Embedding the DMs' preference differences into the stability analysis. Multi-criteria decision aiding/making (MCDA/M) approach and three-way decisions (3WD) are embedded into the framework of graph model for conflict resolution (GMCR). The objective of this study is to develop more reasonable preference ranking for decision makers (DMs) produced from the perspective of options. This implies further simplification of the underlying computing behind conflict resolution. More specifically, DMs' original option statements are evaluated by hesitant fuzzy sets (HFSs), then a three-way decisions approach is used to screen out the infeasible and rank the feasible option statements, and the feasible option statements' loss values of three-way decisions approach are determined based on the hesitant fuzzy (HF) evaluation values. In addition, based on the value of conditional probability of the feasible option statements, an improved option prioritization technique with objective weights is developed to efficiently rank the conflict states, so that it can efficiently reflect the intensity of the ranking. Then, considering the preference thresholds of DMs, new definitions of threshold stabilities are proposed. As demonstrated by the case study in the problem of a carbon emission conflict in supply chain under "3060 carbon peak and neutrality" goal in China, the proposed novel three-way decisions graph model for conflict resolution (3WD-GMCR) framework can be widely applied to the realistic scenarios of decision-making. Compared with previous studies, the proposed approach can not only obtain the intensity ranking of conflict states, but also resolve complex large-scale conflicts effectively. [ABSTRACT FROM AUTHOR]

Bisimulation techniques have been recently introduced to fuzzy transition systems to compare their behavior. When studying bisimulation, one of the most popular methods is relational lifting. In this paper, we first realize the relational lifting through a function S ˜. The new relational lifting method relaxes the condition of lifting relation from using weight functions to function S ˜ defined by residua (implications) in complete residuated lattices. We first discuss some properties of the lifting operation, and then define an α -bisimulation under S ˜ , where α ∈ [ 0 , 1 ] , for nondeterministic fuzzy transition systems (NFTSs). The new bisimulation can handle the case that two states are equivalent intuitively but are simply treated as distinguishable by some behavioral measures, and so it is much more natural and robust. Then we provide some properties of α -bisimulation under S ˜. The parallel composition operator of NFTSs is addressed and we show that α -bisimilarity under S ˜ is nonexpansive with respect to the operator, which makes compositional verification possible. In addition, we give a fixed-point characterization of α -bisimulation under S ˜ and compute α -bisimilarity under S ˜. Finally, we introduce a real-valued logic to characterize α -bisimilarity under S ˜. [ABSTRACT FROM AUTHOR]