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Picture fuzzy set is an efficient tool for dealing with uncertainty and vagueness, particularly in situations that require assimilation of more dimensions of linguistic assessment such as human voting, feature selection, etc. The correlation coefficient of picture fuzzy sets is a tool to determine the association of two picture fuzzy sets. It has several applications in various disciplines like science, engineering, and management. The prominent applications include decision-making, pattern recognition, clustering analysis, medical diagnosis, etc. In this paper, we introduce a new correlation coefficient for picture fuzzy sets with the justification of its advantages. This correlation coefficient is better than the existing correlation coefficients and other such measures in the picture fuzzy theory because it considers the picture fuzzy set as a whole and also expresses the nature (positive or negative) as well as the extent of association between two PFSs. By performing some comparative analysis based on the computation of correlation degree and linguistic hedges, we establish the effectiveness of the suggested correlation measure over some available correlation measures in a picture fuzzy environment. Further, in the context of pattern recognition, we examine the performance of the proposed correlation measure over some existing picture fuzzy correlation measures. Finally, we apply the suggested picture fuzzy correlation coefficient to a decision-making problem involving the selection of an appropriate COVID-19 mask.

Fermatean fuzzy set theory is emerging as a novel mathematical tool to handle uncertainties in different domains of real world. Fermatean fuzzy sets were presented in order that uncertain information from quite general real-world decision-making situations could be mathematically tractable. To that purpose, these sets are more flexible and reliable than intuitionistic and Pythagorean fuzzy sets. This paper presents a novel hybrid model, namely, the Fermatean fuzzy bipolar soft set (FFBSS, in short) model as a general extension of two powerful existing models, that is, fuzzy bipolar soft set and Pythagorean fuzzy bipolar soft set models. Some fundamental properties of the proposed FFBSS model, namely, subset-hood, equal FFBSSs, relative null and relative absolute FFBSSs, restricted intersection and union, extended intersection and union, AND operation and OR operation are investigated along with numerical examples. In addition, certain basic operations, including Fermatean fuzzy weighted average and score function of FFBSSs are proposed. Furthermore, two applications of FFBSS are explored to deal with different multiattribute decision-making situations, that is, selection of best surgeon robot and analysis of most affected country due to COVID-19 (‘CO’ stands for corona, ‘VI’ for virus, ‘D’ for disease, and ‘19’ stands for its year of emergence, that is, 2019). The proposed methodology is supported by an algorithm. At the end, a comparison analysis of the proposed hybrid model with some existing models, including Pythagorean fuzzy bipolar soft sets is provided.

Picture Fuzzy Sets (PFSs) originated by Cuong and Kreinovich are more capable to capture uncertain, inconsistent and vague information in multi-criteria decision making. In this paper, we propose a new picture fuzzy divergence measure based on Jensen-Tsallis function between PFSs. Further, the concept has been extended from fuzzy sets to novel picture fuzzy divergence measure. Besides the validation of the proposed measure, some of its key properties with specific cases are additionally talked about. The performance of the proposed measure is compared with other existing measures in the literature. Some illustrative examples are provided in the context of novel rapacious COVID-19 and pattern recognition which demonstrate the adequacy and practicality of the proposed approach in solving real-life problems.

Fuzzy soft graphs are effective mathematical tools that are used to model the vagueness of the real world. A fuzzy soft graph is a fusion of the fuzzy soft set and the graph model and is widely used across different fields. In this current research, the concept of picture fuzzy soft graphs is presented by combining the theory of picture soft sets with graphs. The introduction of this new picture fuzzy soft graphs is an emerging concept that can be rather developed into various graph theoretical concepts. Since soft sets are most usable in real-life applications, the newly combined concepts of the picture and fuzzy soft sets will lead to many possible applications in the fuzzy set theoretical area by adding extra fuzziness in analyzing. The notions of picture soft graphs, strong and complete picture soft fuzzy graphs, a few types of product picture fuzzy soft graphs, and regular, totally regular picture fuzzy soft graphs are discussed and validated using real-world scenarios. In addition, an application of decision-making for medical diagnosis in the current COVID scenario using the picture fuzzy soft graph has been illustrated.

Fuzzy mixed graph (FMG) can be used to model some graphical intercommunication network systems if there are many directed and undirected relations between some vertices. In inverse fuzzy graph, the membership value of edges are greater than or equal to the minimum of the membership value of the corresponding vertices. In inverse fuzzy mixed graph (IFMG), directed and undirected relations exist between some vertices and it can be used to analyze many graphical problems of real life such that the membership values of edges are greater than or equal to the minimum of the membership value of the corresponding pair of vertices. In this article, the concept of IFMG is introduced first with some of its properties. Then some isomorphic properties are studied and complement of IFMG is given. Different types of operations like union, intersection, product and join between two IFMGs are defined and investigated some of their related results. An algorithm of the proposed method is executed to identify some vertices. An application is depicted using the concept IFMG to examine the order of vertices in a social network group according to communication gaps.

Spherical fuzzy set (SFS) is also one of the fundamental concepts for address more uncertainties in decision problems than the existing structures of fuzzy sets, and thus its implementation was more substantial. The well-known sine trigonometric function maintains the periodicity and symmetry of the origin in nature and thus satisfies the expectations of the experts over the multi parameters. Taking this feature and the significance of the SFSs into the consideration, the main objective of the article is to describe some reliable sine trigonometric laws for SFSs. Associated with these laws, we develop new average and geometric aggregation operators to aggregate the Spherical fuzzy numbers. Then, we presented a group decision-making strategy to address the multi-attribute group decision-making problem using the developed aggregation operators. To verify the value of the defined operators, a MAGDM strategy is provided along with an application for the selection of an authentic COVID-19 laboratory. Moreover, a comparative study is also performed to present the effectiveness of the developed approach.

Neighborhood systems are used to approximate graphs as finite topological structures. Throughout this article, we construct new types of eight neighborhoods for vertices of an arbitrary graph, say, j-adhesion neighborhoods. Both notions of Allam et al. and Yao will be extended via j-adhesion neighborhoods. We investigate new types of j-lower approximations and j-upper approximations for any subgraph of a given graph. Then, the accuracy of these approximations will be calculated. Moreover, a comparison between accuracy measures and boundary regions for different kinds of approximations will be discussed. To generate j-adhesion neighborhoods and rough sets on graphs, some algorithms will be introduced. Finally, a sample of a chemical example for Walczak will be introduced to illustrate our proposed methods.

The linguistic variables provide a powerful and efficient tool for describing qualitative information in many complex real-life decision situations. The notion of the linguistic interval-valued Pythagorean fuzzy sets (LIVPFSs) has been introduced to more flexibly deal with the decision–information under a qualitative environment. It generalizes the existing theories, such as linguistic Pythagorean fuzzy sets (LPFSs) and linguistic interval-valval intuitionistic fuzzy sets (LIVIFSs). The paper’s main objective is to present a detailed study of operational laws and aggregation operators (AOs) for linguistic interval-valued Pythagorean fuzzy numbers (LIVPFNs). This work uses Archimedean t-norm (ATN) and Archimedean t-conorm (ACN) to define new and generalized operational laws between LIVPFNs. The paper also proves many properties and special cases of them. Next, we define some new AOs for aggregating different LIVPFNs, such as the generalized Archimedean linguistic interval-valued Pythagorean fuzzy weighted average (GALIVPFWA) operator, the generalized Archimedean linguistic interval-valued Pythagorean fuzzy ordered weighted average (GALIVPFOWA) operator, and the generalized Archimedean linguistic interval-valued Pythagorean fuzzy ordered weighted average weighted average (GALIVPFOWAOWA) operator. The properties and special cases of developed AOs are discussed in detail. We use the linguistic scale function (LSF) to accommodate the decision-makers’ different translation requirements during the aggregation process. It is worth noting that the developed AOs have a parameter in their formulations to consider the attitudinal character of the decision-makers. The work designs a new and flexible decision-making approach utilizing the proposed AOs to solve multiple attribute group decision-making (MAGDM) problems with LIVPF information. A real-life problem of robot selection is given to illustrate the effectiveness of the developed approach in complex real-life situations. Finally, we present the sensitivity analysis and the comparison of the characteristic to show the superiority over existing methods.

The growth of online shopping is rapidly changing the buying behaviour of consumers. Today, there are challenges facing buyers in the selection of a preferred item from the numerous choices available in the market. To improve the consumer online shopping experience, recommender systems have been developed to reduce the information overload. In this paper, a cognitive comparison-enhanced hierarchical clustering (CCEHC) system is proposed to provide personalised product recommendations based on user preferences. A novel rating method, cognitive comparison rating (CCR), is applied to weigh the product attributes and measure the categorical scales of attributes according to expert knowledge and user preferences. Hierarchical clustering is used to cluster the products into different preference categories. The CCEHC model can be used to rank and cluster product data with the input of user preferences and produce reliable customised recommendations for the users. To demonstrate the advantages of the proposed model, the CCR method is compared with the rating approach of the analytic hierarchy process. Two recommendation cases are demonstrated in this paper with two datasets, one collected by this research for laptop recommendation and the other an open dataset for workstation recommendation. The simulation results demonstrate that the proposed system is feasible for providing personalised recommendations. The significance of this research is the provision of a recommendation solution that does not depend on historical purchase records; rather, one wherein the users’ rating preferences and expert knowledge, both of which are measured by CCR, is considered. The proposed CCEHC model could be further applied to other types of similar recommendation cases such as music, books, and movies.

The dual hesitant fuzzy set (DHFS) is an effective mathematical approach to deal with the data which are imprecise, uncertain or incomplete information. DHFS is an extension of hesitant fuzzy sets (HFS) which encompass fuzzy sets (FS), intuitionistic fuzzy sets (IFS), HFS, and fuzzy multisets as a special case. DHFS consist of two parts, that is, the membership and non-membership degrees which are represented by two sets of possible values. Therefore, in accordance with the practical demand these sets are more flexible and provide much more information about the situation. The aim of this paper is to develop an effective methodology for solving matrix games with payoffs of triangular dual hesitant fuzzy numbers (TDHFNs). The flaws of the existing approach to solve matrix games with TDHFNs payoffs are pointed out. Moreover, to resolve these flaws, novel, general and corrected approach called Mehar approach is proposed to obtain the optimal strategies for TDHFNs matrix games. In this methodology, the concepts and ranking order relations of TDHFNs are defined. A pair of bi-objective linear programming models for matrix games with payoffs of TDHFNs is derived from two auxiliary dual hesitant fuzzy programming models based on the ranking order relations of TDHFNs defined in this paper. An effective methodology based on the weighted average method is developed to determine optimal strategies for two players. In this approach, it is verified that any matrix game with TDHFNs payoffs always has a TDHFNs equilibrium value. Finally, a numerical experiment is incorporated to illustrate the applicability and feasibility of the proposed Mehar approach in TDHFNs matrix game. The obtained results are compared with the results obtained by the previous approaches for solving TDHFNs matrix game.

In the last few years, the number of applications of the multidimensional fuzzy arithmetic (MFA) and the multidimensional interval arithmetic is expanding. Authors of new papers about applications of MFA are often faced with comments from other researchers, especially the gradual arithmetic (GA) proponents, that the horizontal membership function (HMF) used in MFA is the same as the inverse membership function (InvMF) used in GA, and that MFA itself adds nothing new to the fuzzy arithmetic. This view leads to unfair evaluations of scientific papers about MFA applications submitted to scientific journals and to unnecessary disagreements between MFA and GA proponents. The purpose of this paper is to carefully analyze the two types of functions (HMF and InvMF) and to demonstrate their important differences. The basic and decisive difference is the dimensionality of both functions, which is illustrated by examples. It should also be added that HMF has proven its usefulness in solving difficult problems such as: systems of fuzzy equations or fuzzy differential equations, which is confirmed by numerous publications. The paper enable the reader to have a deeper understanding of the multidimensional fuzzy arithmetic.

Human facial expression recognition has been treated as a multi-class classification problem in the field of artificial intelligence. The main difficulty lies in how to distinguish the different categories of expression features. In this paper, we identify common facial expressions by fusing multiple weak classifiers. It compensates for the disadvantage of single classifier in weak generalization ability and low recognition rate for different datasets and different environments. This paper integrates the prediction results of each classifier through improved weighted mean value method and proposes an expression feature extraction method based on keypoint detection. Classifier fusion methods enable each classifier to perform at its best in order to improve overall expression recognition. Keypoint detection is used to improve the model’s attention on the expression features. Convolution neural network is selected as the model for feature extraction and classification, and the model structure is adjusted. Experiments show that the recognition accuracy of this method used on datasets FER 2013 and CK+ are 70.7% and 95.4% respectively, which are better than that of a single classifier, which shows that the keypoint extraction feature and classifier fusion method used in this paper have a good effect on facial expression recognition.

The objective of this paper is to develop Pythagorean fuzzy (PF) aggregation operators, utilizing the concept of power aggregation operators through Schweizer and Sklar (SS) operations. A series of aggregation operators, viz., PF SS power average operator, PF SS power weighted average operator, PF SS power geometric operator, and PF SS power weighted geometric operator under PF environment is proposed in this paper. The developed operators possess the capacity to make information aggregation technique more flexible than other existing operators due to the presence of SS t-norms and t-conorms in PF environment. Also, for the appearance of power aggregation operator, the developed operators contain the capability to eliminate effects of unreasonable data from biased decision makers by considering interrelationships among the fused arguments. Several properties of the proposed operators are studied and a method for solving multi-attribute decision-making problems under PF context is developed. To illustrate the proposed method and to show its efficiency, an example, studied previously, is solved and compared with existing methods.

The intuitionistic fuzzy sets are an important way to express uncertain information, and they are superior to the fuzzy sets. In this paper, we propose a new measure of intuitionistic fuzzy sets along with various properties. The new intuitionistic Renyi’s–Tsallis discriminant information measure has some desirable merits, in which it meets the distance measurement axiom and can better indicate the discrimination degree of IFSs. Then further, the idea has been generalized from fuzzy sets to well-known novel intuitionistic discriminant information measure. Some of the major properties are discussed to show the effectiveness of our proposed divergence measure. A comparison study is conducted to illustrate the validity and superiority of the proposed discriminant measure. The proposed measure is utilized to calculate the degree of discrimination between intuitionistic fuzzy sets. Finally, some practical examples are employed in the field of medical diagnosis and pattern recognition. In this paper, we also propose an algorithm for pattern recognition using proposed discriminant information measure. This algorithm is demonstrated by some examples and outcome is compared with other existing methods.

This paper proposes an approach for behavioural analysis on continuous data based on sequential pattern mining (SPM) and classification. Discretization and granularity issues associated with SPM and classification with time series data are examined, and a thorough examination of the many factors and parameters that are used to control the granularity of the various dimensions of the problem is conducted. The key contribution of the paper is the seminal examination of machine learning-based classification driven by sequential pattern mining to classify behaviour patterns. Results are reported on the effectiveness of various levels and settings on acceleration data collected from the walking activity of 17 subjects obtained from a publicly available dataset. Using the optimal set of parameter settings identified, the classifier was able to correctly identify 70% percent of walking data segments from a held-out test set as belonging to a particular subject, and 78% of those not belonging to the subject. Moreover, a number of interesting findings were identified when varying the granularity and other settings for the various discretization parameters, solidifying the work as a significant advancement in the area of behavioural analysis with sequential pattern mining and classification on time series data.

This paper presents and evaluates a method to combine time-based granulation and three-way decisions to support decision makers in understanding and reasoning on the learned granular structures conceptualising spatio-temporal events. The method uses an existing approach to discover periodic events in the data, such as periods of intense traffic in a city, and provides an original approach to conceptualize such events to support decision makers in: (i) better comprehending the causes that lead to the repetition of such events and/or (ii) increasing the awareness of their effects and consequences. The formal concept analysis is the central tool of the proposed method. This tool is used as a guide in the phase of time-based granulation, which relies on the principle of justified granularity, and as a support for reasoning and making three-way decisions. The main contribution of the paper is an effective and simple method for time-based granulation of events, their observation, and interpretation to support decision making. The method is described with an illustrative example and evaluated on a real data set on forest fires, showing how to define a spatio-temporal DSS model to support decisions in environmental monitoring problems.

Aggregation operators are mathematical functions and essential tools of unifying the several inputs into single valuable output. The purpose of this paper is to analyze the aggregation operators (AOs) under the q-rung orthopair fuzzy environment with the help of Einstein norms operations. This paper presents AOs, namely, q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG), q-rung orthopair fuzzy Einstein ordered weighted geometric (q-ROFEOWG), generalized q-rung orthopair fuzzy Einstein weighted geometric (Gq-ROFEWG), generalized q-rung orthopair fuzzy Einstein ordered weighted geometric (Gq-ROFEOWG) operators. Some properties of these operators are explained. An algorithmic model to deal with multi-attribute decision making problems in q-rung orthopair fuzzy(q-ROF) environment using generalized q-ROF Einstein weighted geometric operator is established. These operators can remunerate for the possible asymmetric roles of the attributes that represent the problem. At the end, to prove the validity and feasibility of the proposed model, we give applications for the selection of location of thermal power station and selection of best cardiac surgeon. The comparison analysis with other existing operators shows the reliability of our work.

A new similarity measure has been proposed in this paper for generalized trapezoidal fuzzy numbers (GTrFNs). Here, the proposed similarity measure has been devised based on exponent distance, area, perimeter and height of a GTrFN. Most of the researchers described the similarity measure between two GTrFNs whose components belong to [0, 1] only. But the components of GTrFNs should be any real numbers belonged to R. For this reason, in this paper a similarity measure technique has been framed newly on GTrFN whose components are any real number which is the most important consideration here. Depending on the proposed method of similarity measure, some essential properties have been illustrated in this paper. Also, this method has been compared with some existing techniques of similarity measure taking twenty five different sets of GTrFNs. Henceforth, it is obtained that our proposed technique is better than other existing techniques. Finally, the proposed similarity measure has been applied in fuzzy risk analysis in a production system which has been described with numerical illustration.

The main objective of this paper is to develop a sophisticated mathematical expression which can carry much more information than the general intuitionistic fuzzy set (IFS), interval-valued intuitionistic fuzzy set (IVIFS) and cubic intuitionistic fuzzy set (CIFS). CIFS is one of the powerful tools to handle uncertainty in complex situation. It is the simultaneous consideration of both the IVIFS and IFS. As in many real life situation, interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) are more capable than IVIFS and IFS to represent the vagueness or ill-defined information; therefore, it motivates us to enhance the capability of CIFS in complex decision-making problems. This paper presents a novel notion of cubic Pythagorean fuzzy set (CPFS) incorporating IVPFS and PFS simultaneously, to encounter uncertainty in a more specific manner. Furthermore, a family of distance measures for CPFSs is defined and applications of the proposed distance measures are shown in medical decision-making problem.

As a medical system with a long history, traditional Chinese medicine (TCM) owns extremely rich and valuable experience and knowledge. In this paper, the granulation-based attribute partial ordered structure diagram (APOSD) is proposed to achieve knowledge discovery for spleen Yang deficiency syndrome in TCM. APOSD is discussed with the perspective of granular computing. Each layer of APOSD is viewed as a level of granularity and the nodes in APOSD layers are viewed as information granules. Granulation based generation method of APOSD reduces the complexity of the algorithm and can extract more comprehensive information from multi-level and multi-scale. The proposed method is testified through analysis of prescriptions of spleen Yang deficiency syndrome screening from the comprehensive database of TCM. Symptom patterns, import herbs and their relationship are extracted from different layers and clusters in generated APOSD. Compared with commonly used Apriori association rule analysis algorithm, the method proposed in this paper can not only extract herbs with high confidence but also extract herbs with specific effect, and the association relationships can also be shown on the generated APOSD.

Recently, the probabilistic fuzzy set has been applied by the researchers in various domains to model the uncertainties in the system due to both fuzziness and randomness. In this research paper, we propose a novel high-order probabilistic fuzzy set-based forecasting method in the environment of both non-probabilistic and probabilistic uncertainties. We have also proposed a novel probability-based discretization approach to model probabilistic uncertainty during partitioning of time series data. Gaussian probability distribution function is used in this research paper to associate probabilities to membership grades and probabilistic fuzzy elements are aggregated to a fuzzy row vector using an aggregation operator. Major advantages of the proposed method are that it includes both types of uncertainties in a single framework and enhances accuracy in forecast as well. To show its suitability and outperformance over other existing forecasting methods, the proposed method is implemented in University of Alabama enrolments and TAIFEX time series datasets. Various statistical parameters, e.g., coefficient of correlation, coefficient of determination, performance parameter, evaluation parameter and tracking signal are used to verify the validity of proposed PFS-based high-order time series forecasting method.

This year, various researchers paid the attention towards the analysis of m-polar fuzzy attributes for knowledge-processing tasks. In this process, a problem addressed while dealing with acceptation, rejection and uncertain parts of m-polar fuzzy attributes. One of suitable example is classifying the potential researchers of the given field to upscale the university ranking is a major issue for the academic or research team as it is based on multi-valued parameters. This becomes more complex when several random, ghost or fake researchers exist in the university. These types of researchers used to have several papers in most of the research areas rather than a specialized field. Hence, it is difficult for any expert to characterize them based on their truth, false or indeterminant areas based on the given n-number of papers to upscale the university ranking. To solve this problem, two methods are proposed in this paper using the algebra of n-valued neutrosophic set and its Euclidean distance with an illustrative example.

A novel interval-based approach for a gain-scheduled controller synthesis is presented in this paper which aims at stabilizing continuous-time dynamic systems of finite dimension in a guaranteed manner over predefined finitely long time horizons. The fundamental aim of the design is the temporal adaptation of the feedback gain in combination with a reduction of the interval widths which characterize outer enclosures of the states that are reachable in the worst case at a certain time instant while ensuring asymptotic stability of the closed-loop dynamics. For that purpose, feedback gains are computed first for an initial state enclosure. Second, they are tested for validity in such a way that the parameterization of the controller shall be applicable over the whole time discretization step. In case of the verification failing, the gain is modified after determining a bounding box for those states that can be reached over the considered prediction window. That means, an offline calculation of controller gains is possible so that prespecified performance indicators with respect to the closed-loop dynamics are guaranteed to be satisfied. The robust and/or optimal control problem is solved efficiently using linear matrix inequality (LMI) techniques, while methods from interval analysis are furthermore employed during the underlying reachability analysis. Additionally, efficient approaches are presented which reduce overestimation due to the unavoidable wrapping effect. Hence, the proposed design method aims, simultaneously, at the reduction of overestimation and a guaranteed stability verification. To conclude this paper, the resulting control strategy is verified numerically for an inverted pendulum as a prototypical benchmark application.

In this study, we establish a connection between rough soft set (Shabir et al., Knowl Base Syst 40:72–80, 2013) and fuzzy set. Based on the novel granulation structure called modified soft rough approximation space, fuzzy rough soft set is introduced. The important basic properties of fuzzy rough soft set are studied and supported by illustrative examples. Moreover lattice theory is studied on fuzzy rough soft set. The definitions and propositions presented in this paper enrich the soft set theory, rough set theory and fuzzy set theory, and also extend their application scopes. The paper ends with conclusions having future investigations of the study.

The hesitant 2-tuple linguistic set (H2TLS), a special case of hesitant 2-tuple linguistic representation model, not only can express the qualitative preferences of the decision makers (DMs), but can effectively reflect their uncertainty and hesitancy with the help of translation parameter containing more than one element assigned to each linguistic variable of the linguistic term set (LTS). This paper develops an outranking method to address the solution to multi-criteria group decision-making (MCGDM) problems in the context of hesitant 2-tuple linguistic information. The main contribution of this paper is twofold. One is to propose the outranking relations of H2TLEs based on the proposed directional Hausdorff distance. Another is to establish an outranking method, similar to the ELECTRE method, to solve MCGDM problems in which attributes are evaluated by hesitant 2-tuple linguistic arguments under certain criteria. Last but not the least, a numerical example and a comparison analysis of the results obtained both in the proposed method and fuzzy TOPSIS (Technique for order preference by similarity to ideal solution) method for H2TLSs based on the generalized distance measure are conducted to verify the practicality and effectiveness of the work.

Facing the frontier challenges of big data, the alarm analysis of sensors becomes more significant in manufacturing industries. However, many relevant and irrelevant alarms from unpredictable occurrences, unlimited sensors, unknown values, undetermined causes, and uncertain relevance impose their information useless. This paper proposes a big data analytics to solve damage cause with alarms. It constructs granules to reduce data size and explores a baseline to make inference. For illustrating the analytical technique, a case study of chemical vapor deposition within a solar company is presented. Its results provide inferential knowledge for the cause of damage and low performance. The contribution of this paper lies in integrating interdisciplinary techniques and fulfilling analytics with evidential inference.

Multi-label decision data serve as an important extension of decision data, where several labels associate with each object. In classical multi-granulation rough set models, granulation selection is always related to positive region. However, some positive region reducts are not appropriate for multi-label data due to the fact that the uncertainties implied by labels are not totally considered. To overcome this deficiency in this paper, we propose the optimal granulation selection in multi-granulation and multi-label decision table. First, two kinds of fine and coarse decision functions are introduced, which represent all certainly and all possibly associated labels for the objects in the same class from optimistic and pessimistic viewpoints, respectively. Some relevant propositions and theorems are also carefully examined. Based on granular significance, a novel approach of optimal granulation selection is discussed. Finally, a heuristic algorithm is designed and an illustrative example is given to show the effectiveness of this algorithm. The main contribution of this paper is to extend the attribute reduction of multi-label data based on rough set theory, and overcome the limitation of the positive region reducts, in which the uncertainties implied by labels are not totally considered.

This paper firstly reviews the existing divergence measures and presents some counter-intuitive cases. To avoid the drawback of existing measures, a new divergence measure for intuitionistic fuzzy sets (IFSs) is pioneered and afterwards, an entropy measure is originated from the proposed divergence measure. Numerical example illustrates the efficiency of the proposed intuitionistic fuzzy entropy. As in recent times, multi-criteria decision making (MCDM) problems with IFSs have widely been applied by many authors in different fields. Consequently, copious numbers of MCDM techniques have been pioneered in intuitionistic fuzzy environment. So, in this paper, a MCDM technique named as TOmada de Decisao Interativa e Multicrit’erio (TODIM) is presented under intuitionistic fuzzy information. The proposed TODIM approach is developed for correlative MCDM problems, in which the weights of the criteria are calculated in terms of Shapley values and the dominance matrices are evaluated based on Shapley weighted divergence measure with intuitionistic fuzzy information. Furthermore, the efficacy of the technique is demonstrated through a selection problem of senior executive person of a telecommunication company. To validate the result, a comparative analysis with existing methods is presented.

One of the main aims of a theory of three-way decisions is to model a cohort of human ways of dealing with problems and how to go about information processing considering applications in designing and implementing intelligent systems. Three-way decision with decision-theoretic rough sets is a new model characterized by the conditional risk, the loss functions and the minimum-risk decision rules. We provide a three-way decisions approach using loss functions evaluated by multiple experts based on the Bayesian decision theory. In this paper, we propose the methodologies of the aggregation of loss functions of decision-theoretic rough sets under group decision-making environment. Then we further compare them with that of the existing papers, which focus on the aggregation of the thresholds. We also construct basic model of three-way group decisions. Finally, an empirical study of evaluating the loss functions validates the reasonability and effectiveness of the proposed models. The contributions of the study are as follows: group decision-making is regarded as a knowledge based, which makes the decision more convincing as compared to single-person methods. Considering the different situations, we mainly design the corresponding aggregation operators of loss functions. Our new approach provides the methodologies of the aggregation of loss functions which in turn are used to compute the thresholds for the determination or construction of the basic rules of three-way group decision-making.

This paper contains discussion about the paper of Kreinovich (Granular Computing 1(3):171–179, 2016) in which the author suggests the thesis that in conditions of uncertainty “solving different practical problems we get different solutions of the same system of equations with the same granules”. Authors of the present paper are of the opinion that the above thesis is result of one-dimensional approach to interval analysis prevailing at present in scientific community and also used by Kreinovich. According to this approach the direct result of any arithmetic operation $$*\in \{+,-,\times ,/\}$$ on intervals is also an interval. The main scientific contribution of the paper is showing that a granular problem analysis in an incomplete, low-dimensional space can lead to untrue conclusions because picture of the problem in this space is too poor and part of valuable, important information is lost. What is seen in the full-dimensional space of the problem space cannot be seen in an incomplete, low-dimensional space. The paper also shortly describes a multidimensional approach to interval arithmetic that is free of the above-described deficiencies.

2016 is the 60 anniversary of the birth of artificial intelligence (AI). In the past 60 years, many AI theoretical models have been proposed and great achievements have been accomplished. Most intelligent computing models are inspired by various human/ natural/social intelligence mechanisms. Three main schools of artificial intelligence are formed, that is, symbolism, connectionism and behaviorism. Artificial intelligence is intelligence exhibited by machines. It is applied when a machine mimics cognitive functions that humans associate with other human minds. Achievements of cognitive science could give much inspiration to AI. Cognitive computing is one of the core fields of artificial intelligence. It aims to develop a coherent, unified, universal mechanism inspired by human mind’s capabilities. It is one of the most critical tasks for artificial intelligence researchers to develop advanced cognitive computing models. Cognitive computing is the third and most transformational phase in computing’s evolution, after the two distinct eras of computing—the tabulating era and the programming era. Inspired by human’s granularity thinking, problem solving mechanism and the cognition law of “global precedence”, a new powerful cognitive computing model, data-driven granular cognitive computing (DGCC), is proposed in this paper. It takes data as a special kind of knowledge expressed in the lowest granularity level of a multiple granularity space. It integrates two contradictory mechanisms, namely, the human’s cognition mechanism of “global precedence” which is a cognition process of “from coarser to finer” and the information processing mechanism of machine learning systems which is “from finer to coarser”, in a multiple granularity space. It is also based on the idea of data-driven. The research issues of DGCC to be further addressed are discussed. Based on DGCC, deep learning is neither classified into symbolism, nor connectionism. It is taken as a combination of symbolism and connectionism, and named hierarchical structuralism in this paper. The HD $$^3$$ characteristics (hierarchical, distributed, data-driven, and dynamical) of the hierarchical structuralism are analyzed. DGCC provides a granular cognitive computing framework for efficient knowledge discovery from big data.

In the last few decades, there has been an increase in the interest of the scientific community for multivariate statistical techniques of data analysis in which the data are affected by uncertainty, imprecision, or vagueness. In this context, following a fuzzy formalization, several contributions and developments have been offered in various fields of the multivariate analysis. In this paper—to show the advantages of the fuzzy approach in providing a deeper and more comprehensive insight into the management of uncertainty in this branch of Statistics—we present an overview of the developments in the exploratory multivariate analysis of imprecise data. In particular, we give an outline of these contributions within an overall framework of the general fuzzy approach to multivariate statistical analysis and review the principal exploratory multivariate methods for imprecise data proposed in the literature, i.e., cluster analysis, self-organizing maps, regression analysis, principal component analysis, multidimensional scaling, and other exploratory statistical approaches. Finally, we point out certain potentially fruitful lines of research that could enrich the future developments in this interesting and promising research area of Statistical Reasoning. All in all, the main purpose of this paper is to involve the Granular Computing scientific community and to stimulate and focus its interest on these statistical fields.

Machine learning has become a powerful approach in practical applications such as decision making , sentiment analysis and ontology engineering. In order to improve the overall performance in machine learning tasks, ensemble learning has become increasingly popular by combining different learning algorithms or models. Popular approaches of ensemble learning include Bagging and Boosting, which involve voting towards the final classification. The voting in both Bagging and Boosting could result in incorrect classification due to the bias in the way voting takes place. In order to reduce the bias in voting, this paper proposes a probabilistic approach of voting in the context of granular computing towards improvement of overall accuracy of classification. An experimental study is reported to validate the proposed approach of voting by using 15 data sets from the UCI repository. The results show that probabilistic voting is effective in increasing the accuracy through reduction of the bias in voting. This paper contributes to the theoretical and empirical analysis of causes of bias in voting, towards advancing ensemble learning approaches through the use of probabilistic voting.

A rule based system is a special type of expert system, which typically consists of a set of if-then rules. Such rules can be used in the real world for both academic and practical purposes. In general, rule based systems are involved in knowledge discovery tasks for both purposes and predictive modelling tasks for the latter purpose. In the context of granular computing, each of the rules that make up a rule based system can be seen as a granule. This is due to the fact that granulation in general means decomposition of a whole into several parts. Similarly, each rule consists of a number of rule terms. From this point of view, each rule term can also be seen as a granule. As mentioned above, rule based systems can be used for the purpose of knowledge discovery, which means to extract information or knowledge discovered from data. Therefore, rules and rule terms that make up a rule based system are considered as information granules. This paper positions the research of rule based systems in the granular computing context, which explores ways of achieving advances in the former area through the novel use of theories and techniques in the latter area. In particular, this paper gives a certain perspective on how to use set theory for management of information granules for rules/rule terms and different types of computational logic for reduction of learning bias. The effectiveness is critically analysed and discussed. Further directions of this research area are recommended towards achieving advances in rule based systems through the use of granular computing theories and techniques.

In decision-making problems, q-rung orthopair fuzzy sets are considered as a more effective tool than intuitionistic fuzzy sets and Pythagorean fuzzy sets. This article develops some aggregation operators based on Frank t-norm and t-conorm for fusing q-rung orthopair fuzzy (q-ROF) information. Then a multiple attribute decision-making (MADM) approach is introduced based on the proposed operators. The Frank operations of t-norm and t-conorm can have the advantage of good flexibility with the operational parameter. From that point of view, in this paper, we extend the ideas of Frank t-norm and t-conorm to the q-ROF environment and introduce some aggregation operators. Moreover, we illustrate the compatible properties of the proposed operators. Generally, the attribute weights are unknown in the MADM problems. The analytical hierarchy process and entropy methods are efficient tools to handle such MADM problems with unknown attribute weights. So, we present an MADM approach with unknown attribute weights under q-ROF environment using proposed operators. Then to elaborate the flexibility and validity of the proposed model, we discuss and solve a numerical problem concerned with a government project of choosing the best way of industrialization. Next, we show how the involvement of the parameters in our proposed model affects the decision-making results. Finally, to exhibit the superiority of our proposed methodology, the obtained results are compared with the existing ones.

This paper considers pollution sensitive economic order quantity (EOQ) inventory model for items under several styles of managerial leadership via lock fuzzy game theoretic approach. The decision-maker (DM) controls several cost components by playing as Player 1 in one side and the consumers who may accept/reject those items (unwilling to buy those commodities) stand as Player 2 in the other side. At first, we develop a new pollution function through modeling, then we study a real case study in a brick factory. Based on this case study, we formulate a cost minimization EOQ model where the environmental pollution is incorporated. Because of the fuzzy flexibility of the model parameters, we also develop a fuzzy mathematical model by considering the demand and all cost parameters of the inventory system as triangular lock fuzzy numbers. Then, we develop a game problem by applying 5-stage leadership style employing several key vectors into the model itself. Also, we reformulate the proposed model into bi-objective game model with different strategies and solve it by an existing methodology. The problem has been solved for crisp, general fuzzy, bi-objective strategic fuzzy, and models of several leadership styles also. Numerical results show that for cooperative game, inventory cost function reaches its minimum rather than the non-cooperative game with the use of proper keys of the fuzzy locks. A comparative study between proposed method and existing method has been done for showing the superiority of the proposed methodology. Finally, in the sensitivity analysis, graphical illustrations are made to justify the new approach and a conclusive remark has been added followed by scope of future work of the proposed study.

In system modeling, traditional machine learning methods aim to make a model’s output fit the real output as well as possible. However, sometimes they fail to reach the goal, especially when the quantity of samples is small (which often leads to the occurrence of overfitting). Therefore, researchers start to explore new approaches for system modeling with few samples. An effective alternative to solve this problem is to reduce the fitting expectation and make predictions from a certain range of samples. Granular computing techniques simulate human’s thinking rules at a higher level and thus can be applied to few samples learning (FSL). In this paper, we realize granular neural network (GNN) modeling with few samples. A conceptually simple yet powerful method to learn some coarse-grained information from few samples is proposed. The output of the GNN model is fuzzy (represented by information granules). It can predict a new sample’s output with a rough range which makes the model own stronger robustness especially when the quantity of training samples is small. The precondition is that it is not necessary to accurately determine the explicit output of the samples. In the experiment, we compare the coverage of the models on the same test set in which the models are built using different percentage of training samples. The results show that the model built on few samples can also achieve good performance.

In this paper, similarity measures have been discussed which has the potential to discriminate analogous but contrary picture fuzzy sets (PFSs). We have also described their properties along with their implementation in pattern recognition by taking numerical examples. To implement the application of similarity measures in real life problems, we have taken real data from the repository of machine learning. Next, the real data set has been transformed into picture fuzzy (PF)- environment. Thereafter, by using the idea of degree of confidence (DOC), we have compared the proposed measures with existing measures and the potential of proposed measures have been discussed. Furthermore, by extending the idea of maximum spanning tree (MST) and clustering algorithm, a picture fuzzy maximum spanning tree clustering method has been proposed. Although, existing PF-clustering methods can also give reasonable results but proposed method is simple and having less computational cost. Additionally, we have compared the proposed measures with existing PF-similarity measures in terms of linguistic hedges. Proposed measures meet all the conditions as compared to existing measures from linguistic hedges point of view. Thus, comparative results of pattern recognition problems, DOC and linguistic hedges shows the superiority of proposed measures over existing measures.

In this paper, based on the concept of Renyi–Tsallis entropy, we propose an inaccuracy measure for a pair of probability distribution and discuss its relationship with mean codeword length. Furthermore, we propose a new fuzzy entropy measure in the setting of fuzzy set theory and its several properties are examined. Comparison with several existing entropies shows that the proposed fuzzy information measure has a greater ability in discriminating different FSs (fuzzy sets). Furthermore, we introduce a new fuzzy mean codeword length and give their relationship with fuzzy information measure. The upper bounds of these entropies in terms of mean codeword lengths have been provided and some basic properties of the proposed codeword length have been studied. In addition, we introduce a new similarity measure for fuzzy sets and give its applications in pattern recognition and cluster analysis. To implement the application of proposed similarity measure in real life problem, we have taken real data from the repository of machine learning. These practical examples are given to support the findings and also show the availability of similarity measure between fuzzy sets.

Pythagorean fuzzy set (PFS) is an advanced version of generalized fuzzy sets. It has a better applicative expression in decision-making because of its capability in curbing fuzziness embedded in decision science. Correlation coefficient is a reliable measuring operator for the applicability of generalized fuzzy sets in decision-making. Some approaches of estimating correlation of PFSs have been explored, albeit with certain setbacks. This paper introduces some methods of calculating the correlation coefficient of PFSs which resolve the setbacks in the existing methods. Some numerical examples are supplied to confirm the superiority of the novel methods over the existing correlation coefficient measures. In addition, certain decision-making problems such as marital choice-making, classification of building materials, and electioneering process represented in Pythagorean fuzzy values are resolved using the proposed correlation measure. Specifically, the objectives of this work are to (1) introduce some new triparametric methods of computing correlation coefficient of PFSs, (2) characterize their theoretic properties, (3) ascertain their advantages over the existing methods, and (4) explore the application of the proposed methods in certain decision-making problems. From the study, it is observed that the new Pythagorean fuzzy correlation coefficients give reliable outputs compared to the existing ones and, hence, can suitably handle multiple criteria decision-making effectively.

Recently, most of the researchers focused on dealing with uncertainty and unclear data beyond the three-way fuzzy space and its graphical visualization. An issue arises when the bipolar attributes exist beyond the three-way or m-way fuzzy space. One of suitable examples is colors consist of several opposite and non-opposite side of red–green–blue to produce a new color. To deal with these types of bipolarity exists in multi-valued attribute, a method is proposed in this paper using the connection of bipolar fuzzy set with multi-fuzzy set. Same time graphical structure visualization of bipolar multi-fuzzy set is also introduced for precise analysis of knowledge discovery and representation tasks. In addition, an another method is proposed to refine the bipolar multi-fuzzy context at user defined multi-granulation with an illustrative example. The analysis derived from the proposed method is also compared with recently available approaches on bipolar multi-fuzzy set.

The purpose of this paper is to introduce the concepts of Dombi t-norm and Dombi t-conorm to aggregate trapezoidal neutrosophic fuzzy information. First, we have proposed some new operational laws of trapezoidal neutrosophic fuzzy number based on Dombi t-norm and t-conorm. Furthermore, based on these operational laws, we have introduced trapezoidal neutrosophic fuzzy Dombi weighted arithmetic averaging operator, trapezoidal neutrosophic fuzzy Dombi order weighted arithmetic averaging operator, and trapezoidal neutrosophic fuzzy Dombi hybrid arithmetic averaging operator. Moreover, some suitable properties of these operators are also discussed. Then, utilizing these proposed operators, we have presented an algorithm to solve multiattribute decision-making (MADM) problems under trapezoidal neutrosophic fuzzy environment. Finally, we have utilized a numerical example to compare the flexibility of the proposed method with the other existing methods.

Uncertain and vague information plays a crucial role in multicriteria decision making (MCDM) problems. Quantification of such information proves to be very helpful in selection of the most suitable alternative. Picture fuzzy set (PFS) studied by Cuong and Kreinovich has been proved to be a very powerful tool in capturing such vague information. In this paper, based on Renyi’s concept, a new picture fuzzy information measure is introduced. Besides its validation, some major properties of the proposed measure are also discussed. Apart from this, a novel decision making method using picture fuzzy information and weighted correlation coefficients based VIKOR (Vlsekriterijumska Optimizacija i Kompromisno Resenje) approach is put forward. The proposed decision making method is well explained through two numerical examples based on supplier selection.

Many fuzzy time series (FTS) methods have been developed by the researchers without including non-determinacy caused using single function for both membership and non-membership grades. Optimum length of intervals and inclusion of non-determinacy have been two important key issues for the researchers for long time. In this paper, we propose a novel hybrid FTS forecasting method using particle swarm optimization (PSO) and intuitionistic fuzzy set (IFS). PSO determines optimum length and IFS includes non-determinacy during fuzzification of time series data. To show the applicability and suitability of proposed forecasting method, it is implemented on three time series data of the University of Alabama, State bank of India (SBI) share price at Bombay Stock Exchange (BSE), India and car sells in Quebec. Performance of proposed method is measured using mean square error (MSE). Reduced amount of MSE confirms outperformance of proposed FTS forecasting method over various existing fuzzy set, IFS, hesitant and probabilistic hesitant fuzzy set-based FTS forecasting methods in forecasting the University of Alabama enrollments, SBI share price and car sells in Quebec. Validity of the proposed FTS forecasting method is also verified using tracking signal.

The similarity measures have been extensively used as an efficient tool to analyze the degree of similarity between two sets of objects. The linguistic q-rung orthopair fuzzy set provides a flexible model to represent uncertain qualitative information more precisely. This paper defines some new generalized trigonometric similarity measures between two linguistic q-rung orthopair fuzzy sets. We use the linguistic scaling function to accommodate the different semantic levels of the linguistic terms during the similarity measuring process. Several basic and advanced mathematical properties are proved in detail. Next, we present a generalized hybrid trigonometric similarity measure between two linguistic q-rung orthopair fuzzy sets. The weighted version of the developed similarity measures is also defined and extends them to a continuous domain. It is worth mentioning that all the proposed similarity measures are reduced into the similarity measures between linguistic intuitionistic fuzzy sets and the linguistic Pythagorean fuzzy sets when we take $$q=1$$ and $$q=2$$ , respectively. A new decision-making method based on developed generalized hybrid trigonometric similarity measure is formulated to solve multiple attribute decision-making problems under the linguistic q-rung orthopair fuzzy environment. Finally, a numerical example is given to illustrate the flexibility and effectiveness of the developed method in solving real-world decision problems.

This paper develops fuzzy multi-criteria decision making (MCDM) approach to evaluate performance of airports according to the passengers’ perspective. Assessments for qualitative measures and quantitative measures are represented by intuitionistic cubic fuzzy numbers (ICFNs), and crisp numbers represents criteria weights, which are obtained via surveys. To construct a strategy for solving MCDM problems, we have developed extended forms of VIKOR method and TOPSIS method for ICFNs and constructed ICF-VIKOR and ICF-TOPSIS algorithm under ICF information in group decision environment. We have applied these methods for an experimental analysis of 7 significant international airports in the Asia-Pacific region to evaluate their performance with conflicting criteria under ICF environment. To show the supremacy of the proposed techniques, we have presented comparative analysis with the existing techniques.

Hesitant fuzzy (HF) entropy and HF-knowledge measures are two dual concepts that have similar practical applications despite different mathematical structures. In some real-life scenarios, one particular entropy/knowledge measure may not be reasonable due to some undesirable and counter-intuitive situations. In this paper, we introduce a one-parametric generalized knowledge measure in the HF-setting. Such a generalization provides a class of HF-knowledge measures and hence the flexibility in the practical problems. We show the advantages of the generalized measure over the existing HF-entropy/knowledge measures in view of weight computation in the decision-making problems and ambiguity computation of two different hesitant fuzzy elements. At last, we apply the proposed generalized knowledge measure of HFSs in multi-criteria decision-making (MCDM) using the bidirectional projection method in the hesitant fuzzy environment.

In real-world scenario, mostly, the datasets are either imprecise or uncertain in their original form. Due to this reason, the clustering of such datasets is unsatisfactory and we often get compromised results. The information present in the dataset can be made precise and useful with the popular generalization of fuzzy sets known as Atanassov intuitionistic fuzzy sets (AIFSs). Therefore, the AIFS-based c-means clustering algorithm becomes a convenient way for clustering uncertain and vague datasets. In the paper, we propose an intuitionistic fuzzy-based algorithm, namely Generalized Intuitionistic Fuzzy c-Means (G-IFCM) clustering algorithm which uses an adaptive AIFS Euclidean distance measure in its criterion function to cluster the dataset under intuitionistic fuzzy environment. The proposed intuitionistic fuzzification method incorporates a technique to transform the dataset into AIFS and maintains its original structure that tends to change during any fuzzification process. Further, simulation experiments are also conducted on few UCI machine learning repository datasets using G-IFCM and its performance is compared with some known clustering algorithms. The efficacy of the algorithm is tested with some popular benchmark indexes that check the cluster validity and clustering performance of G-IFCM. Finally, the computation suggests efficient performance of G-IFCM where the best choice of fuzzy factor, m probably lies in the interval [1, 2] that indicates reduction in the expense of computation.

The recurrent fuzzy time series function method can be obtained in two ways. First is using a similar model to the autoregressive moving average to obtain fuzzy functions and the second one is using recurrent connections in the combining equation. The recurrent structure provides less number of inputs and more accurate forecasts. The contribution of the paper is proposing a recurrent fuzzy time series function method and its bootstrapped version. The recurrent models are used to obtain fuzzy functions in the proposed method. The classical bootstrap method is applied and bootstrap samples are generated from learning samples. The bootstrap method provides lower forecast error variance for the proposed method. The performance of new methods is compared with some fuzzy function methods and classical approaches. First, Turkey electrical consumption data time series are analyzed. Second, the Australian beer consumption time series are analyzed. As a result of applications, new methods have good forecasting performance if compare to established benchmarks.

A complex intuitionistic fuzzy set (CIFS) has an ability to represent the problems with intuitionistic uncertainty and periodicity, simultaneously. In this paper, we present a new framework for handling complex intuitionistic fuzzy information by combining the CIFSs with competition graphs. We first introduce the concept of complex intuitionistic fuzzy competition graphs along with its two worthwhile extensions, namely, complex intuitionistic fuzzy k-competition and complex intuitionistic fuzzy p-competition graphs. Further, we present complex intuitionistic fuzzy neighborhood graphs and m-step complex intuitionistic fuzzy competition graphs with some of their remarkable results. We also describe an application of complex intuitionistic fuzzy competition graphs along with an algorithm which finds the competition among the applicants competing for the particular jobs possessing the two-dimensional information. Moreover, we provide a comparison analysis of proposed competition graphs with existing graphs to find the validity and superiority of proposed competition graphs.