In this paper, we introduce a new form of describing fuzzy sets (FSs) and a new form of fuzzy rule-based (FRB) systems, namely, empirical fuzzy sets (εFSs) and empirical fuzzy rule-based (εFRB) systems. Traditionally, the membership functions (MFs), which are the key mathematical representation of FSs, are designed subjectively or extracted from the data by clustering projections or defined subjectively. εFSs, on the contrary, are described by the empirically derived membership functions (εMFs). The new proposal made in this paper is based on the recently introduced Empirical Data Analytics (EDA) computational framework and is closely linked with the density of the data. This allows to keep and improve the link between the objective data and the subjective labels, linguistic terms, and classes definition. Furthermore, εFSs can deal with heterogeneous data combining categorical with continuous and/or discrete data in a natural way. εFRB systems can be extracted from data including data streams and can have dynamically evolving structure. However, they can also be used as a tool to represent expert knowledge. The main difference from the traditional FSs and FRB systems is that the expert does not need to define the MF per variable; instead, possibly multimodal, densities will be extracted automatically from the data and used as εMFs in a vector form for all numerical variables. This is done in a seamless way whereby the human involvement is only required to label the classes and linguistic terms. Moreover, even this intervention is optional. Thus, the proposed new approach to define and design the FSs and FRB systems puts the human “in the driving seat.” Instead of asking experts to define features and MFs correspondingly, to parameterize them, to define algorithm parameters, to choose types of MFs, or to label each individual item, it only requires (optionally) to select prototypes from data and (again, optionally) to label them. Numerical examples as well as a naïve empirical fuzzy (εF) classifier are presented with an illustrative purpose. Due to the very fundamental nature of the proposal, it can have a very wide area of applications resulting in a series of new algorithms such as εF classifiers, εF predictors, εF controllers, and so on. This is left for the future research.

On the basis of the hesitant fuzzy membership, this study proposes the extended intuitionistic fuzzy set (EIFS) and the extended intuitionistic fuzzy number (EIFN) to synthesize the characters of the intuitionistic fuzzy set and the hesitant fuzzy set. We further develop two simplified and applied EIFSs, namely the credible EIFS (C-EIFS) and the possible EIFS (P-EIFS), to comprehensively mine the hesitant fuzzy membership information and to avoid the logical difficulty of simultaneously providing the membership and non-membership in each EIFS or EIFN. Then we investigate the foundations of C-EIFS and P-EIFS, including their expressions, operations, functions, differences, and selection rules. The corresponding aggregation operators are also proposed, and the calculation and relationships of these operators are proven. The prominent properties of C-EIFNs and P-EIFNs are focused on the boundary and average values, respectively; that is, the C-EIFN tends to aggregate the extreme information, whereas the P-EIFN prefers aggregating complete information. Therefore, applying them to decision making with risk preference is suitable, and two risk preference investment cases are provided to demonstrate the applications of these concepts and approaches.

In the real multi-attribute group decision making (MAGDM), there will be a mutual relationship between different attributes. As we all know, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between parameters. In addition, in describing uncertain information, the eminent characteristic of *q*-rung orthopair fuzzy sets (*q*-ROFs) is that the sum of the *q*th power of the membership degree and the *q*th power of the degrees of non-membership is equal to or less than 1, so the space of uncertain information they can describe is broader. In this paper, we combine the BM operator with *q*-rung orthopair fuzzy numbers (*q*-ROFNs) to propose the *q*-rung orthopair fuzzy BM (*q*-ROFBM) operator, the *q*-rung orthopair fuzzy weighted BM (*q*-ROFWBM) operator, the *q*-rung orthopair fuzzy geometric BM (*q*-ROFGBM) operator, and the *q*-rung orthopair fuzzy weighted geometric BM (*q*-ROFWGBM) operator, then the MAGDM methods are developed based on these operators. Finally, we use an example to illustrate the MAGDM process of the proposed methods. The proposed methods based on *q*-ROFWBM and *q*-ROFWGBM operators are very useful to deal with MAGDM problems.

The main feature of Pythagorean fuzzy sets is that it is characterized by four parameters, namely membership degree, nonmembership degree, strength of commitment about membership, and direction of commitment. In this paper, we propose a variety of distance measures for Pythagorean fuzzy sets and Pythagorean fuzzy numbers, which take into account the four parameters of Pythagorean fuzzy sets. Finally, a numerical example is provided to illustrate the validity and applicability of the presented distance measures.

In this paper, we introduce the addition and g-difference of continuous *Z*-numbers at first. Then, we define a new partial order to rank the *Z*-numbers with generalized centroids. The g-derivative of *Z*-number function based on g-difference is proposed. It is well known that convexity plays a vital role in optimization problems; consequently, we present the convexity of *Z*-number function. Furthermore, we provide the optimality conditions for optimization problems based on *Z*-numbers. Finally, the validity of the discussion is illustrated by an example.

Data envelopment analysis (DEA) is a widely used technique in decision making. The existing DEA models always assume that the inputs (or outputs) of decision-making units (DMUs) are independent with each other. However, there exist positive or negative interactions between inputs (or outputs) of DMUs. To reflect such interactions, Choquet integral is applied to DEA. Self-efficiency models based on Choquet integral are first established, which can obtain more efficiency values than the existing ones. Then, the idea is extended to the cross-efficiency models, including the game cross-efficiency models. The optimal analysis of DEA is further investigated based on regret theory. To estimate the ranking intervals of DMUs, several models are also established. It is founded that the models considering the interactions between inputs (or outputs) can obtain wider ranking intervals.

]]>The q-rung orthopair fuzzy sets (q-ROFs) are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets and the Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the *q*th power of the membership degree and the *q*th power of the degrees of non-membership is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we propose the q-rung orthopair fuzzy weighted averaging operator and the q-rung orthopair fuzzy weighted geometric operator to deal with the decision information, and their some properties are well proved. Further, based on these operators, we presented two new methods to deal with the multi-attribute decision making problems under the fuzzy environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.

The Pythagorean fuzzy set introduced by R. R. Yager in 2014 is a useful tool to model imprecise and ambiguous information appearing in decision and clustering problems. In this study, we present a general type of distance measure for Pythagorean fuzzy numbers (PFNs) and propose a novel ratio index-based ranking method of PFNs. The novel ranking method of PFNs has more powerful ability to discriminate the magnitude of PFNs than the existing ranking methods for PFNs, which is further extended to compare the magnitude of interval-valued Pythagorean fuzzy numbers (IVPFNs). The IVPFN is a new extension of PFN, which is parallel to interval-valued intuitionistic fuzzy number. We introduce a general type of distance measure for IVPFNs. Afterwards, we study a kind of clustering problems in Pythagorean fuzzy environments in which the evaluation values are expressed by PFNs and/or IVPFNs and develop a novel Pythagorean fuzzy agglomerative hierarchical clustering approach. In the proposed clustering method, we define the concept of the dissimilarity degree between two clusters for each criterion and introduce the clustering procedure in the criteria level. To take all the criteria into account, we also introduce the overall clustering procedure, which is based on the overall dissimilarity degrees for a fixed aggregation operator such as the commonly used weighted arithmetic average operator or the ordered weighted averaging operator. In the overall clustering process, (1) we present a deviation degree-based method to derive the weights of criteria and further obtain the overall clustering results if the weights of criteria are completely unknown; (2) we employ the ratio index-based ranking method of IVPFNs to obtain the overall clustering results if the weights of criteria are given in advance and are expressed by IVPFNs. The salient feature of the proposed clustering method is that it not only can address the clustering problems in which the weights of criteria are not given precisely in advance but also can manage simultaneously the PFNs and IVPFNs data.

]]>In the era of Big Data, users prefer to get knowledge rather than pages from Web. Linked Data, a rather new form of knowledge representation and publishing described by RDF, can provide a more precise and comprehensible semantic structure to satisfy the aforementioned requirement. Besides, as the standard query language for RDF data, SPARQL has become the foundation protocol of Linked Data querying. The core idea of RDF Schema (RDFS) is to extend upon RDF vocabulary and allow attachment of semantics to user defined classes and properties. However, RDFS cannot fully utilize the potential of RDF since it cannot express the implicit semantics between linked entities in Linked Data sources. To fill this gap, in this paper, we design a new semantic annotating and reasoning approach that can extend more implicit semantics from different properties. We firstly establish a well-defined semantically enhanced annotation strategy for Linked Data sources. In particular, we present some new semantic properties for predicates in RDF triples and design a Semantic Matrix for Predicates (SM_{P}). We then propose a novel general Semantically Extended Scheme for Linked Data Sources (SES_{LDS}) to realize the semantic extension over the target Linked Data source through semantically enhanced reasoning. Lastly, based on the experimental analyses, we verify that our proposal has advantages over the initial Linked Data source and can return more valid results.

Analyzing demand in environments with incomplete information is a challenging task. This paper proposes a novel agent-based Pythagorean fuzzy approach for analyzing this kind of demand. First, a Bayesian game is described with a large number of finite players, and this is followed by a Pythagorean fuzzy-based decision mechanism. Unlike the classical methods in the literature, the proposed method in this paper neither assumes nor forecasts the demand in a system. Instead, it tries to analyze the demand when there is limited availability of input data, or processing data are computationally expensive. The study ends with an application of the proposed system to an electricity grid. Electricity prices used as an incentive to construct an agent-based system that efficiently reduces the peak amounts in a smart grid by analyzing the demand. Test results provide evidence that the proposed approach is promising to design demand response systems.

]]>Imprecision and uncertainty appear together in many situations of real life and therefore soft computing techniques must be studied to tackle this problem. Imprecise and uncertain values are usually expressed by means of linguistic terms, especially when they have been provided *by* a human being. This is also the case of temporal information where, in addition to handling time constraints, we may also have both uncertainty and imprecision in the description, like in the sentence “It is very possible that Giotto's Crucifix was painted by 1289.” To manage both uncertainty (very possible) and imprecision (by 1289) in a separate way would lead to a quite complicated computation and a lack of comprehension by the users of the system. Because of these reasons, it is very desirable that both sources of imperfection of time values are combined into a single value that appropriately describes the intended information. In this work, we extend our previous research on this topic and we study how to adapt it to relational systems in order to be useful. The final goal is obtaining normalized fuzzy values that provide an equivalent information about the described temporal fact than the original ones, for making it possible to store and manage them in a fuzzy relational database. On the other hand, there will be some situations where more than one expert opinion about a time period must be taken into account and we need to find a representative value of them all in order to be stored and managed. For the sake of simplicity, comprehensibility, and the efficiency in computation (when using trapezoidal representation), the fuzzy average is used to find such a representative value.

Group recommender systems (GRSs) recommend items that are used by groups of people because certain activities, such as listening to music, watching a movie, dining in a restaurant, etc., are social events performed by groups of people sharing their tastes, and their choices affect all of them. GRSs help groups of people making choices in overloaded search spaces according to all group members preferences. A common GRS scheme aggregates users preferences to generate a group preference profile. However, the aggregation process may imply a loss of information, negatively affecting different properties of the GRS such as *diversity* of group recommendations, which is an important quality factor because of such recommendations are targeted to groups formed by users with individual and possibly conflicting preferences. To avoid and manage the loss of information caused by aggregation, this paper proposes to keep all group members preferences by using hesitant fuzzy sets (HFSs) and interpreting such information like the group hesitation about their preferences that will be used in the group recommendation process. To evaluate the performance and rank quality of the HFS GRS proposal, a case study is carried out.

Social media has become very popular nowadays by spreading plentiful human-centric data to a large number of audience. A rich body of literature has studied the influence maximization problem in social networks under certain propagation models. However, existing models suffer from the assumption of unlimited propagation time and some of them are nondeterministic. Influence maximization algorithms within those frameworks are often trading off between performance guarantee and computational cost. In this paper, we try to formulate the influence maximization problem in another way, where we limit the propagation time to a predefined propagation round and in each round maintain the probability to make the propagation process more tractable. We introduce a new diffusion model called a *probability propagation model* and formulate this optimization problem as a *holistic probability maximization problem*. We show that information diffusion estimation in the proposed framework is not NP-hard. Hence, any algorithm within it will be more efficient. However, the maximization problem is still NP-hard. After proving the submodularity in the proposed framework, we design a *partial-updating greedy algorithm* and its heuristic extension to solve the maximization problem. Extensive experiments on four synthetic data sets and four real-world data sets from Facebook, Wikipedia, arXiv, and Epinions demonstrate the effectiveness of the proposed algorithm.

In decision making, a widely used methodology to manage unbalanced fuzzy linguistic information is the linguistic hierarchy (LH), which relies on a linguistic symbolic computational model based on ordinal 2-tuple linguistic representation. However, the ordinal 2-tuple linguistic approach does not exploit all advantages of Zadeh's fuzzy linguistic approach to model uncertainty because the membership function shapes are ignored. Furthermore, the LH methodology is an indirect approach that relies on the uniform distribution of symmetric linguistic assessments. These drawbacks are overcome by applying a fuzzy methodology based on the implementation of the type-1 ordered weighted average (T1OWA) operator. The T1OWA operator is not a symbolic operator and it allows to directly aggregate membership functions, which in practice means that the T1OWA methodology is suitable for both balanced and unbalanced linguistic contexts and with heterogeneous membership functions. Furthermore, the final output of the T1OWA methodology is always fuzzy and defined in the same domain of the original unbalanced fuzzy linguistic labels, which facilitates its interpretation via a visual joint representation. A case study is presented where the T1OWA operator methodology is used to assess the creditworthiness of European bonds based on real credit risk ratings of individual Eurozone member states modeled as unbalanced fuzzy linguistic labels.

]]>This paper advocates the use of weighted ordered weighted averaging (WOWA) functions in decision-making processes, where the alternatives are not directly comparable. In particular, WOWA allows one to compare the strongest points of each alternative, also weighted by the importance of each criterion. Four different approaches to applying weights in OWA functions are reviewed. Torra's method based on interpolating regular increasing monotone quantifier and the pruned n-ary tree are compared to the WOWA obtained from recently proposed implicit averaging. Computationally efficient algorithms are outlined. The use of WOWA is illustrated in several examples.

]]>In this paper, we propose a generic recommender system that combines opinion mining and fuzzy quantification methods for qualitative data. The proposed system has two novel aspects. First, it employs a novel semantic orientation (SO) computation method to reduce the number of extracted features and opinion expressions. By using this new SO computation method, the proposed recommender system finds out the most related features and opinion expressions. Second, the proposed system generates short summary sentences from qualitative data using fuzzy quantification. The proposed system is evaluated using a restaurant review dataset. The results present that fuzzy quantified sentences offer brief information about the restaurant features from customers’ feedback. In addition, opinion mining extracts positive, negative, and neutral emotions from reviews.

]]>This paper presents the induced heavy ordered weighted moving average (IHOWMA) operator. It is an aggregation operator that uses the main characteristics of three well-known techniques: the moving average, induced operator, and heavy aggregation operator. This operator provides a parameterized family of aggregation operators that include the minimum, the maximum, and total operator as special cases. It can be used in a selection process, considering that not all decision makers have the same knowledge and expectations of the future. The main properties of this operator are studied including a wide range of families of IHOWMA operators, such as the heavy ordered weighted moving average operator and uncertain induced heavy ordered weighted moving average operator. The IHOWMA operator is also extended using generalized and quasi-arithmetic means. An example in an investment selection process is also presented.

]]>The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator used in modern information fusion theory, which is suitable to aggregate numerical values. The prominent characteristic of the MSM operator is that it can capture the interrelationship among the multiinput arguments. In this paper, we extend MSM to Pythagorean fuzzy environment to propose the Pythagorean fuzzy Maclaurin symmetric mean and Pythagorean fuzzy weighted Maclaurin symmetric mean operators. Then, some desirable properties and special cases of these operators are discussed in detail. Finally, a numerical example is provided to illustrate the feasibility of the proposed methods and deliver a comparative analysis.

]]>Formal concept analysis theory (FCA) classically relies on the use of the Galois powerset operator. Formal similarities between possibility theory and formal concept analysis have led to the use of possibilistic operators in FCA, which were ignored before. In this paper, an approach based on the use of asymmetric composition of the two most usual possibilistic operators is proposed. It enables us to complement the stem base, by deriving attribute implications with disjunctions on both sides of the implications. Besides, the approach is also generalized to incomplete contexts involving explicit positive and negative information. We outline the potential application of these results to the completion of TBoxes in description logic.

]]>The hesitant fuzzy sets are a new efficient mathematical approach to study imprecise, uncertain, or incomplete knowledge. This paper focuses to extend this approach on a lattice and shows that two large application concepts of the fuzzy set theory namely resolution identity and representation theorem are true under this extended definition.

]]>The linguistic aggregation operator is an important decision-making model that is proving effective for dealing with the input data that takes the form of uncertain information. In this paper, considering the principal component of the intuitionistic fuzzy linguistic variables, we develop a new intuitionistic fuzzy linguistic hybrid aggregation (NIFLHA) operator to solve group decision-making problems under the situation with intuitionistic fuzzy linguistic information. Then, we study some of its main properties by utilizing some operational laws of intuitionistic fuzzy linguistic variables and the different families of the NIFLHA operator. Moreover, the multiperson NIFLHA (MP-NIFLHA) operator is introduced to evaluate the opinions of experts. Finally, an illustrative example about a multiperson decision-making problem is developed to reveal the applicability and the availability of the raised operator.

]]>To generalize the Rand index (*RI*) from crisp partitions to fuzzy partitions, we first propose a graph method in which color edges in the graph for crisp partitions are used to determine the relation matrix between objects such that the matrix trace can be employed to calculate the *RI*. This approach is then introduced into fuzzy partitions to generalize the *RI* to the fuzzy *RI* (*FRI*). Compared with previous fuzzy generalizations, the most unique aspect of our method has the following important characteristics that for any two partition matrices *M*^{(1)} and *M*^{(2)}, the result with is the necessary and sufficient condition for the result that the *FRI* is equal to 1. This important characteristic renders our fuzzy generalization of the *RI* is not only able to determine the similarities between fuzzy partitions and crisp reference partitions, but also to identify the similarity between fuzzy partitions and fuzzy reference partitions. The method can even be used to explore and compare the similarities between various data sets and the same fuzzy reference partition. Finally, we use synthetic data and real data to give more demonstrations, and further perform comparisons of our method with those existing fuzzy extensions of the *RI*.

In this paper, we propose a novel metric called *MetrIntPair* (*Metric for Pairwise Intelligence Comparison of Agent-Based Systems*) for comparison of two cooperative multiagent systems problem-solving intelligence. *MetrIntPair* is able to make an accurate comparison by taking into consideration the variability in intelligence in problem-solving. The metric could treat the outlier intelligence indicators, intelligence measures that are statistically different from those others. For evaluation of the proposed metric, we realized a case study for two cooperative multiagent systems applied for solving a class of NP-hard problems. The results of the case study proved that the small difference in the measured intelligence of the multiagent systems is the consequence of the variability. There is no statistical difference between the intelligence quotients/level of the multiagent systems. Both multiagent systems should be classified in the same intelligence class.

In this paper, we propose three similarity measure methods for single-valued neutrosophic refined (SVNR) sets and interval neutrosophic refined (INR) sets based on Jaccard, Dice, and Cosine similarity measures of SVN-sets and interval neutrosophic sets. Furthermore, we suggest two multicriteria decision-making (MCDM) methods under SVNR environment and INR environment, and give applications of proposed MCDM methods. Finally, we suggest a consistency analysis method for proposed similarity measures between INR-sets and give an application to demonstrate process of the method.

]]>Credit lenders utilize credit rating approaches to provide a classification system for characterizing credit borrowers. In order to measure the borrowers’ credibility, that is, ability and willingness to repay the debt, there are many financial and non-financial criteria that should be considered. The basic aim of this study is to propose a multiple-criteria credit rating approach that integrates different kinds of information and represents the borrowers’ credibility as a distribution among all the credit ratings. The cumulative belief degree approach is proposed for this purpose. Since all the available information is used in the final representation, a distribution-based credit rating approach is expected to strengthen the lender's inference competency. In order to eliminate subjectivity in the weighting of criteria, an ordered weighted averaging operator is used. Additionally, the credit rating distribution can be transformed into a single credit rating by considering a threshold value. This study proposes a goodness-of-fit test to handle the subjectivity and difficulty of setting the threshold value. The applicability of the proposed approach is demonstrated by analyzing the credibility of selected Turkish firms from the stock exchange market of Turkey.

]]>Due to the limitation of knowledge and the vagueness of human being thinking, decision makers prefer to use hesitant fuzzy linguistic sets (HFLSs) to estimate alternatives. Some methods of HFLSs have been researched based on the more familiar means such as the arithmetic mean and the geometric mean; however, Maclaurin symmetric mean (MSM) that can be used to reflect the interrelationships among input arguments have not been applied to solve hesitant fuzzy linguistic multi-criteria decision-making problems. In this paper, two hesitant fuzzy linguistic harmonic averaging operators are proposed: the hesitant fuzzy linguistic MSM (HFLMSM) operator and the hesitant fuzzy linguistic weighted MSM (HFLWMSM) operator. Furthermore, an approach based on the HFLWMSM operator is proposed. Finally, to verify the validity and feasibility of the proposed approach, an illustrative example and corresponding comparison analysis are presented in the end.

]]>In this paper, we propose an approach to data analysis, which is based entirely on the empirical observations of discrete data samples and the relative proximity of these points in the data space. At the core of the proposed new approach is the *typicality*—an empirically derived quantity that resembles probability. This nonparametric measure is a normalized form of the *square centrality* (*centrality* is a measure of closeness used in graph theory). It is also closely linked to the *cumulative proximity* and *eccentricity* (a measure of the tail of the distributions that is very useful for anomaly detection and analysis of extreme values). In this paper, we introduce and study two types of *typicality*, namely its local and global versions. The *local typicality* resembles the well-known probability density function (*pdf*), probability mass function, and fuzzy set membership but differs from all of them. The *global typicality*, on the other hand, resembles well-known histograms but also differs from them. A distinctive feature of the proposed new approach, empirical data analysis (EDA), is that it is not limited by restrictive impractical *prior* assumptions about the data generation model as the traditional probability theory and statistical learning approaches are. Moreover, it does not require an explicit and binary assumption of either randomness or determinism of the empirically observed data, their independence, or even their number (it can be as low as a couple of data samples). The *typicality* is considered as a fundamental quantity in the pattern analysis, which is derived directly from data and is stated in a discrete form in contrast to the traditional approach where a continuous *pdf* is assumed *a priori* and estimated from data afterward. The *typicality* introduced in this paper is free from the paradoxes of the *pdf*. *Typicality* is objectivist while the fuzzy sets and the belief-based branch of the probability theory are subjectivist. The *local typicality* is expressed in a closed analytical form and can be calculated recursively, thus, computationally very efficiently. The other nonparametric ensemble properties of the data introduced and studied in this paper, namely, the *square centrality*, *cumulative proximity*, and *eccentricity*, can also be updated recursively for various types of distance metrics. Finally, a new type of classifier called naïve typicality-based EDA class is introduced, which is based on the newly introduced *global typicality*. This is only one of the wide range of possible applications of EDA including but not limited for anomaly detection, clustering, classification, control, prediction, control, rare events analysis, etc., which will be the subject of further research.

To enable inference in hybrid Bayesian networks (BNs) containing nonlinear deterministic conditional distributions, Cobb and Shenoy in 2005 propose approximating nonlinear deterministic functions by piecewise linear (PL) ones. In this paper, we describe a method for finding PL approximations of nonlinear functions based on a penalized mean square error (MSE) heuristic, which consists of minimizing a penalized MSE function subject to two principles, domain and symmetry. We illustrate our method for some commonly used one-dimensional and two-dimensional nonlinear deterministic functions such as , , , and . Finally, we solve two small examples of hybrid BNs containing nonlinear deterministic conditionals that arise in practice.

]]>The objective of this work is to present an improved accuracy function for the ranking order of interval-valued Pythagorean fuzzy sets (IVPFSs). Shortcomings of the existing score and accuracy functions in interval-valued Pythagorean environment have been overcome by the proposed accuracy function. In the proposed function, degree of hesitation between the element of IVPFS has been taken into account during the analysis. Based on it, multicriteria decision-making method has been proposed for finding the desirable alternative(s). Finally, an illustrative example for solving the decision-making problem has been presented to demonstrate application of the proposed approach.

]]>The properties of negation of a probability distribution recently defined by Yager are studied. Furthermore, the negation of joint and marginal probability distributions in the bivariate case has been defined and their properties are studied. Finally, we have defined a new entropy function for determination of uncertainty associated with the negation of a probability distribution and the events associated with it.

]]>We study the properties of OWA multiplication monoid. By introducing and-accumulation vectors of OWA operators, we consider the set of all *n*-dimensional OWA operators as a lattice. Then, we analyze some monotonicity properties of OWA operators based on an ordering induced by and-accumulation vectors. We also show that the lattice-theoretical operations are a kind of counterpart of the OWA multiplication monoid (and its dual, additive monoid). An example of using the OWA multiplication monoid and the lattice-theoretical structure in decision making problem is provided.

The aim of this paper is to develop a Pythagorean fuzzy multiattribute group decision making (MAGDM) method based on probabilistic information and the ordered weighted averaging (OWA) approach. The Pythagorean fuzzy probabilistic ordered weighted averaging (PFPOWA) operator is presented. It is a new aggregation operator that considers the probabilities and the OWA in the same formulation. Therefore, it is able to take into account the degree of importance that each concept has in the particular problem considered. Some main properties and different particular cases of the PFPOWA operators are studied. Moreover, a method based on the proposed operator for multiattribute group decision making is put forward. Finally, an example showing analysis of a supplier selection is given to verify the effectiveness and practicability of the proposed method.

Ordered weighted averaging (OWA) operator has been received increasingly widespread interest since its appearance in 1988. Recently, a topic search with the keywords “ordered weighted averaging operator” or “OWA operator” on Web of Science (WOS) found 1231 documents. As the publications about OWA operator increase rapidly, thus a scientometric analysis of this research field and discovery of its knowledge domain becomes very important and necessary. This paper studies the publications about OWA operator between 1988 and 2015, and it is based on 1213 bibliographic records obtained by using topic search from WOS. The disciplinary distribution, most cited papers, influential journals, as well as influential authors are analyzed through citation and cocitation analysis. The emerging trends in OWA operator research are explored by keywords and references burst detection analysis. The research methods and results in this paper are meaningful for researchers associated with OWA operator field to understand the knowledge domain and establish their own future research direction.

Assessing semantic similarity is a fundamental requirement for many AI applications. Crisp ontology (CO) is one of the knowledge representation tools that can be used for this purpose. Thanks to the development of semantic web, CO-based similarity assessment has become a popular approach in recent years. However, in the presence of vague information, CO cannot consider uncertainty of relations between concepts. On the other hand, fuzzy ontology (FO) can effectively process uncertainty of concepts and their relations. This paper aims at proposing an approach for assessing concept similarity based on FO. The proposed approach incorporates fuzzy relation composition in combination with an edge counting approach to assess the similarity. Accordingly, proposed measure relies on taxonomical features of an ontology in combination with statistical features of concepts. Furthermore, an evaluation approach for the FO-based similarity measure named as FOSE is proposed. Considering social network data, proposed similarity measure is evaluated using FOSE. The evaluation results prove the dominance of proposed approach over its respective CO-based measure.

In this paper, we initiate a new axiomatic definition of Pythagorean fuzzy distance measure, which is expressed by Pythagorean fuzzy number that will reduce the information loss and remain more original information. Then, the objective weights of various criteria are determined via grey system theory. Combining objective weights with subjective weights, we present the combined weights, which can reflect both the subjective considerations of the decision maker and the objective information. Meanwhile, a novel score function is proposed. Later, we present two algorithms to solve stochastic multicriteria decision making problem, which takes prospect preference and regret aversion of decision makers into consideration in the decision process. Finally, the effectiveness and feasibility of approach is demonstrated by a numerical example.