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In this article, we propose an axiomatic system for fuzzy “cardinality” measures (referred to as fuzzy c-measures for short) assigning to each finite fuzzy set a generalized cardinal that expresses the number of elements that the fuzzy set contains. The system generalizes an axiomatic system introduced by J. Casasnovas and J. Torrens (2003). We show that each fuzzy c-measure is determined by two appropriate homomorphisms between the reducts of residuated-dually residuated (rdr-)lattices. For linearly ordered rdr-lattices, we prove that each fuzzy c-measure is a product of a non-decreasing and a non-increasing fuzzy c-measure, which indicates that there is a close relation between fuzzy c-measures and FGCount, FLCount and FECount provided by L.A. Zadeh (1983) and generalized by M. Wygralak (2001). Finally, the relationship of fuzzy c-measures to graded equipollence introduced in the first part of this contribution is analyzed.

As an important part of English teaching, oral English evaluation plays an important role in promoting students to learn English. The establishment of a diversified oral college English evaluation system is conducive to changing the traditional summative evaluation model, promoting the smooth progress of college English reform, and facilitating the in-depth development of the overall education reform. Fuzzy measure theory abandons the requirement of additivity in classical measure theory. On the basis of the concept of generalized additivity, the condition of additivity is weakened to make it additive in the new sense. With the development of deep learning, speech recognition technology has undergone tremendous technological changes, in which the acoustic model has gradually developed from the traditional Gaussian mixture model to the neural network model. In this paper, the speech recognition technology and fuzzy measure rules are analyzed, and the evaluation system of spoken English is constructed.

We deal with solutions of classical linear equations a · x = b and y · a = b , applying a particular lattice valued fuzzy technique. Our framework is a structure with a binary operation · (a groupoid), equipped with a fuzzy equality. We call it a fuzzy quasigroup if the above equations have unique solutions with respect to the fuzzy equality. We prove that a fuzzy quasigroup can equivalently be characterized as a structure whose quotients of cut-substructures with respect to cuts of the fuzzy equality are classical quasigroups . Analyzing two approaches to quasigroups in a fuzzy framework, we prove their equivalence. In addition, we prove that a fuzzy loop (quasigroup with a unit element) which is a fuzzy semigroup is a fuzzy group and vice versa. Finally, using properties of these fuzzy quasigroups, we give answers to existence of solutions of the mentioned linear equations with respect to a fuzzy equality, and we describe solving procedures. Quasigroups and other related structures are an algebraic tool successfully applied up to now in coding theory and cryptology. In our work we propose related applications.

In this paper, a new type of integral of a fuzzy-valued function relative to a non-negative set function is proposed as a new useful tool in modeling uncertainties. Various properties of this new type of integral are established concerning the behavior with respect to the fuzzy function or to the set function. Several examples linking our integral with Birkhoff and Lebesgue integrals are provided. Also, problems related to atomicity are addressed.

In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets., 30 pages, final version

International audience; Necessity (resp. possibility) measures are very simple min-decomposable (resp. max-decomposable) representations of epistemic uncertainty due to incomplete knowledge. They can be used in both quantitative and qualitative settings. In the present work, we revisit Choquet and Sugeno integrals as criteria for decision under uncertainty and propose new axioms when uncertainty is representable in possibility theory. First, a characterization of Choquet integral with respect to a possibility or a necessity measure is proposed. We respectively add an optimism or a pessimism axiom to the axioms of the Choquet integral with respect to a general capacity. This new axiom enforces the maxitivity or the minitivity of the capacity without requiring the same property for the functional. It essentially assumes that the decision-maker preferences only reflect the plausibility ordering between states of nature. The obtained pessimistic (resp. optimistic) criterion is an average maximin (resp. maximax) criterion of Wald across cuts of a possibility distribution on the state space. The additional axiom can be also used in the axiomatic approach to Sugeno integral and generalized forms thereof to justify possibility and necessity measures. The axiomatization of these criteria for decision under uncertainty in the setting of preference relations among acts is also discussed. We show that the new axiom justifying possibilistic Choquet integrals can be expressed in this setting. In the case of Sugeno integral, we correct a characterization proof for an existing set of axioms on acts, and study an alternative set of axioms based on the idea of non-compensation.

This contribution first addresses axiomatic definitions of some important information measures of Atanassov's intuitionistic fuzzy sets (AIFSs), such as, distance measure, similarity measure, entropy measure, and recently developed knowledge measure. After that, we study the relationship between similarity measures and knowledge measures, and that between distance measures and knowledge measures for AIFSs. In particular, we investigate the transformations of similarity measures into knowledge measures for AIFSs and vice versa. These transformations can induce more formulae to define the distance measures, the similarity measures, and the knowledge measures for AIFSs. In addition, to support these processes, suitable examples are provided for each of the introduced transformations. The contribution ends by introducing numerical examples with a discussion of results by using several types of knowledge measures and similarity measures.

In this study, we establish the existence of the α-core in fuzzy games. Using the existence results, we identify a class of fuzzy games comprising a dense residual subset of fuzzy games such that every point of the α-core is essential. We also show that every fuzzy game in this class has an essential component of its α-core.

A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. The best known of these allocations is the Shapley value. A communication structure defines the feasible bilateral communication relationships among the agents in a cooperative situation. Some solutions incorporating this information have been defined from the Shapley value: the Myerson value, the position value, etc. Later fuzzy communication structures were introduced. In a fuzzy communication structure the membership of the players and the relations among them are leveled. Several ways of defining the Myerson value for games on fuzzy communication structure were proposed, one of them is the Choquet by graphs (cg) version. Now we study in this work the cg-position value and its calculation. The cg-position value is defined as a solution for games with fuzzy communication structure which considers the bilateral communications as players. So, the Shapley value is applied for a new game (the link game) over the fuzzy sets of links in the fuzzy communication structure and the profit obtained for each link is allocated between both players in the link. As we see in our examples and results the cg-position value is more concerned with the graphical position of the players and their communications than the other cg-values. In this paper we also introduce a procedure to compute exactly the position value, avoiding to calculate the characteristic function of the link game for all coalitions. This procedure is used to determine the cg-position value. Finally we compare the new value with other cg-values in an applied example about the power of the groups in the European Parliament.

This paper presents an output-feedback exponential stabilization condition of sampled-data polynomial fuzzy control systems under variable sampling rates. Compared with previous work, the proposed method is less conservative because of the newly developed time-dependent fuzzy Lyapunov–Krasovskii functional that is based on the conventional fuzzy Lyapunov function. Moreover, the controller is allowed to contain polynomial gain matrices, thereby improving the control performance and design flexibility. This is realized by assuming the difference between the continuous- and discrete-time state vectors as time-varying norm-bounded uncertainties, which are manipulated using a robust control technique. A new sufficient condition is introduced to cast the stability condition containing the integral term as the sum-of-square conditions. Finally, the effectiveness of the proposed method is validated by simulations.

This paper presents a new parametrized family of joint possibility distributions that can be used to control the Pompeiu–Hausdorff norm of the sum resulting from the extension principle. We are able to generate sums with norms covering the entire interval ranging from the smallest possible value to the largest one. Finally, we apply our approach to numerical methods for solving first-order fuzzy initial value problems.

In this paper, an algebraic construction–called rotation–is introduced, which produces a fuzzy implication from a fuzzy implication. This construction method is similar to the rotation construction for triangular norms. An infinite number of new families of such fuzzy implications can be constructed in this way which provides a broad spectrum of choices for e.g. fuzzy connectives in fuzzy set theory. A preservation of the logical properties of the initial implication in the final one is investigated.

We consider multiple criteria decision aiding (MCDA) in the case of interactions between criteria. In dealing with interactions between criteria, fuzzy measures and integrals have demonstrated great advantages. Nevertheless, the determination of fuzzy measures has proven difficult because the capacities of not only single criterion but also all subsets of criteria need to be identified. Due to the value judgment essence of MCDA, the attitudes of the decision maker (DM) are typically modeled to identify fuzzy measures. In this paper, the tolerance attitudes of the DM, which implies a direct requirement instead of partial preference, are modeled with regard to the determination of fuzzy measures for the first time. With two scales developed in this paper, the DM can directly express the tolerance attitudes to certain criteria other than providing partial preference through pairwise comparison. As a result, it requires less prior knowledge and is more efficient to some extent. Further, the inherent interacting mechanism of criteria under different tolerance attitudes is explored. At last, the tolerance attitudes are applied to the process of multiple criteria analysis using a Choquet integral. A classic student evaluation problem is given as an example. The evaluation results are compared with additive models. This paper not only provides a new inspiration to the determination of fuzzy measures but also improves the descriptive capacity of fuzzy measures to the real world.

In assessing fuzzy numbers to model imprecise data associated with random experiments, trapezoidal fuzzy numbers are often considered. Such an assessment is mainly due to easing both interpretation and computation. This becomes especially noticeable when those assessing fuzzy numbers to data have a weak knowledge, low background and little or no expertise in using fuzzy sets (as it happens when questionnaires whose responses involve a free fuzzy rating are conducted), since the required training to explain the meaning and use of trapezoidal fuzzy numbers is definitely lower than that associated with other shapes. Nevertheless, a question that constantly arises in connection with this trapezoidal assessment is whether it can importantly affect the conclusions of the study involving such data. This paper aims to answer the last question from a statistical perspective. More concretely, the analysis of the influence of the choice of trapezoidal fuzzy numbers to model data is to be based on the conclusions from statistical hypothesis testing about the mean values of the involved fuzzy datasets. For this purpose, the p-values of tests have been compared for trapezoidal assessment vs. other frequently used ones, like some LU assessments. The analysis is first performed by developing simulation-based pairwise comparisons, and it is later illustrated and corroborated to some extent with a real-life example. The analysis indicates that the shape of the fuzzy assessment scarcely affects statistical conclusions.

In this paper, we generalize classical results on risk aversion by analyzing the case, when the available data about cash flows are fuzzy numbers, random fuzzy numbers, type-2 fuzzy sets, random type-2 fuzzy sets, etc. We prove that the ArrowPratt measures are also risk aversion measures when an agent measures her risk using the Jensen-type operators. We also provide numerous examples of the Jensen-type operators.

This study presents new fuzzy versions of Minkowski and Beckenbach’s integral inequalities without making use of the Sugeno integral. These new inequalities generalize the interval versions of the Minkowski and Beckenbach inequalities recently published and are obtained by introducing the p − F A − integrability concept for fuzzy-interval-valued functions by means of the Kaleva integral and a fuzzy order relation. This fuzzy order relation is defined level-wise through the Kulisch–Miranker order relation given on the interval space. Numerical examples that illustrate the applicability of the theory developed in this study are presented.

Certain extensions of the classical fuzzy sets have been studied in depth since they have a remarkable importance in many practical situations. We focus on finite interval-valued hesitant fuzzy sets, as they generalize the most usual sets (fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets), so the results obtained can be immediately adapted to these types of sets. In addition, their membership functions are much more manageable than type-2 fuzzy sets. In this work, the cardinality of finite interval-valued hesitant fuzzy sets is studied from an axiomatic point of view, together with several properties that this definition satisfies, which enable to relate it to the classical definitions of cardinality given by other authors for fuzzy sets.

This paper presents an extension to fractional optimal control problems with ambiguity. As the ambiguity is modeled with fuzzy method, we encounter with a fuzzy fractional optimal control problem. The objective in fuzzy fractional optimal control problem is to determine the best possible fuzzy control which satisfies the related fuzzy fractional dynamic systems and minimizes the fuzzy performance index. Here, the fractional derivative is described in the Caputo sense. To find the solution, first we state some definitions and prove some required theorems. Then, we employ the obtained result to determine the necessary conditions. Furthermore, we show that the obtained necessary optimality conditions become sufficient by considering some extra assumptions. Finally, some examples are presented for more illustration of the subject.

Estimation of fuzzy quantities is a topic that often appears in decision making. Some of the well-known approaches are based on the integration done with respect to the probability measure and, later on, on the possibility measure and the necessity measure. The approach considered here is focused on a fuzzy measure in general as a base of integration.

We present two concrete methods for the computation of the Choquet integral of a positive measurable function. Both methods are based upon the truncation and the discretization of the function involved, the second method being associated with an iterative topological procedure. Computer aided procedures (with preassigned precision), illustrating the aforementioned methods, are introduced.

In the paper, we characterize the class of all pairs of fuzzy implications and dual fuzzy implications such that the q-integral and q-cointegral coincide. This gives a solution to the problem posed by Didier Dubois during the conference FSTA 2016, the 13th International Conference on Fuzzy Set Theory and Applications. As a corollary, we obtain the necessary and sufficient conditions under which the seminormed fuzzy integral and the semiconormed fuzzy integral are equal for all measurable functions.

Introduction to Generalized Orthopair Fuzzy Sets.Discussion Knowledge Representation With Generalized Orthopair Fuzzy Sets.Inference Mechanism for Orthopair Approximate Reasoning System.Formulation of the Ideas of Possibility and Certainty Using Orthopair Fuzzy Sets. We introduce the idea of generalized orthopair fuzzy sets, which provide an extension of intuitionistic fuzzy sets. The basic properties of these generalized orthopair fuzzy sets are discussed. We discuss the use of these sets in knowledge representation. We consider the use of these types of orthopair fuzzy sets as a basis for the system of approximate reasoning introduced by Zadeh. This is referred to as OPAR. The basic operations of OPAR are introduced. A reasoning mechanism in OPAR, based on the idea of entailment, is provided. We look at the formulation of the ideas of possibility and certainty using these orthopair fuzzy sets.

This paper proposes a modified particle swarm optimization (PSO) algorithm that can be used to solve a variety of fuzzy nonlinear equations, i.e. fuzzy polynomials and exponential equations. Fuzzy nonlinear equations are reduced to a number of interval nonlinear equations using alpha cuts. These equations are then sequentially solved using the proposed methodology. Finally, the membership functions of the fuzzy solutions are constructed using the interval results at each alpha cut. Unlike existing methods, the proposed algorithm does not impose any restriction on the fuzzy variables in the problem. It is designed to work for equations containing both positive and negative fuzzy sets and even for the cases when the support of the fuzzy sets extends across 0, which is a particularly problematic case.

This study presents fuzzy versions of Jensen inequalities type integral for convex and concave fuzzy-interval-valued functions. To this end, the concepts of fuzzy inclusion order relation, convexity, and concavity for fuzzy-interval-valued functions are used. Some examples showing the applicability of the theory developed in this study are presented. Since the fuzzy results are obtained through level sets of fuzzy-interval elements, the versions of these results in the interval context are presented here for the first time.

The principle of inclusion and exclusion is one of basic tools of classical set theory. Many previous papers have presented its generalizations to fuzzy sets or IF-sets (Atanassov's intuitionistic fuzzy sets) with Godel or product operations. In our previous work we clarified these questions for fuzzy sets with intersections and unions based on continuous triangular norms and conorms. Here we extend these results to interval-valued fuzzy sets and IF-sets.

Trafficking in persons is the most heinous and condemnable organized crime of our time. The eradication of this crime must involve the full cooperation of all. In this paper, we provide a mathematical method to model such cooperation. We use a new concept in fuzzy graph theory, namely, that of incidence. We introduce fuzzy influence graph and characterize influence cutpairs in fuzzy influence graphs since their removal increases the number of connected components of a fuzzy network and thus weakens the potential flow in the network. Some of the properties of influence pairs and influence cutnodes are also studied.

In this paper, a notion of fuzzy copula function is introduced by defining joint distribution function of two fuzzy random variables. Using some lemmas, it is proven that the extended fuzzy copula satisfies many desired properties used for non-fuzzy data. The proposed fuzzy copula is then applied to construct some common non-parametric measures of association between two fuzzy random variables. The proposed methods is then illustrated via some numerical examples.

As an extension of the classical averaging operators, Choquet integral has been shown a powerful tool for decision theory. In this paper, a method based on the generalized interval-valued intuitionistic fuzzy Choquet integrals w.r.t. the generalized interaction indices is proposed for multiattribute group decision making problems, where the importance of the elements is considered, and their interactions are reflected. Based on the given operational laws on interval-valued intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy Choquet integrals with respect to the generalized Shapley and Banzhaf indices are defined. Moreover, some of their properties are studied, such as idempotency, boundary, comonotonic linearity and μ–linearity. Furthermore, a decision procedure based on the proposed operators is developed for solving multi-attribute group decision making under interval-valued intuitionistic fuzzy environment. Finally, a numerical example is provided to illustrate the developed procedure.

In fuzzy neural networks, it is usual to perform rule matching by employing fuzzy similarity measures. A fuzzy similarity measure is reserved for a binary fuzzy relation. In fuzzy relational calculus, the notion of T-transitivity is indispensable because it is strongly related to the concept of a metric. In this study, we consider whether the two types of fuzzy similarity measures given by Li et al. (2014) [24] satisfy T L -transitivity. We also investigate the T L -transitivity of fuzzy equivalencies.

In the paper, we describe the new measure of matching fuzzy sets. The introduced measure of perturbation of one fuzzy set by another fuzzy set is considered instead of commonly used distance between two fuzzy sets. The operations known in the fuzzy set theory are used and the perturbation of one fuzzy set by another fuzzy set is understood as a measure describing changes of the first fuzzy set after adding the second one. Obviously, the opposite case can also be considered wherein the second fuzzy set is perturbed by the first one. In general, the new measure is asymmetric and can provide more information compare to a distance between fuzzy sets. The values of such measures of fuzzy sets perturbation are in range between 0 and 1. In this paper several mathematical properties of the measure of fuzzy sets perturbation are studied, and the measure of sets perturbation is compared to other selected measures.