Fuzzy Number Space Research Articles

Inequalities of Hlawka type for fuzzy real numbers

In this paper, we give and prove inequalities of Hlawka type and related results in the fuzzy real number space. Mathematics subject classification (2010): 26D15.

Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms

In this paper, we introduce polygonal fuzzy numbers to overcome the operational complexity of ordinary fuzzy numbers, and obtain two important inequalities by taking advantage of their fine properties. By presenting an actual example, we demonstrate that the approximation capability of polygonal fuzzy numbers is efficient. Furthermore, the concepts of K-quasi-additive integrals and K-integral norms are introduced. Whenever the polygonal fuzzy numbers space satisfies separability, the density problems for several functions spaces can be studied, by means of fuzzy-valued simple functions and fuzzy-valued Bernstein polynomials. We establish that the class of the integrally-bounded fuzzy-valued functions spans a complete and separable metric space in the K-integral norms. Finally, in the sense of K-integral norms, the universal approximation of fourlayer regular polygonal fuzzy neural networks for fuzzy-valued simple functions is discussed. Furthermore, we show that this type of networks also possesses universal approximation for the class of integrally-bounded fuzzyvalued functions. This result indicates that the approximation capability which regular polygonal fuzzy neural networks for continuous fuzzy systems can be extended as for general integrable systems.

Discontinuity of the trapezoidal fuzzy number-valued operators preserving core

We prove that any trapezoidal fuzzy number-valued operator preserving core is discontinuous with respect to any weighted metric on the space of fuzzy numbers. As an application, we obtain the discontinuity of the weighted trapezoidal approximation operator preserving core.

Fuzzy Optimal Control Problem with Non-linear Functional

In this paper, we introduce a new space of fuzzy numbers equipped with a scalar product defined in this space. The notion of a derivative of a fuzzy function in this space is defined. By employing these notions, an optimal control problem with non-linear functional is formulated and an optimality condition is obtained in the form of maximum principle. Using this result, the numerical algorithm is offered for the solution of such problems.

Fuzzy approximation relations on fuzzy n-cell number space and their applications in classification

In this paper, the problems of fuzzy binary relations on fuzzy n-cell number space and their applications are investigated. Firstly, we have defined some fuzzy approximation relations on fuzzy n-cell number space, and studied their properties. Secondly, as application, we have developed an algorithmic version of classification in an imprecise or uncertain environment by using the fuzzy approximation relations. Practical examples are provided to show the application and rationality of the proposed techniques.

Pareto-optimal security strategies in matrix games with fuzzy payoffs

We present a new methodology for the analysis of fuzzy payoff matrix games. The main difficulty that appears in the study of these games is the comparison between the payoff values associated to the strategies of the players because these payoffs are fuzzy quantities. Our approach does not transform the fuzzy payoffs to crisp numbers via standard defuzzyfication but we use standard fuzzy orders which allows us to find solutions within the same space of fuzzy numbers. Moreover, we provide a method to solve these games finding equivalent fuzzy linear programs whose maximal solutions give the solutions of the games.

A characterization theorem for levelwise statistical convergence

In the present paper, we prove a characterization theorem which gives a necessary and sufficient condition for a sequence of fuzzy numbers to be levelwise statistically convergent in the space of fuzzy numbers. As an application of this theorem we utilize the idea of statistical equi-continuity in order to obtain a condition which guarantees the set of levelwise statistical cluster points of a statistically bounded sequence to be nonempty and a levelwise statistically Cauchy sequence to be levelwise statistically convergent.

On Some Double Lacunary Sequence Spaces of Fuzzy Numbers

In this paper we introduce a new concept for lacunary strong P-convergent with respect to an Orlicz function and examine some properties of the resulting sequence space of fuzzy numbers. We also show that if a sequence is lacunary strong Pconvergence with respect to an Orlicz function then it is S θ r s (F) -convergent.

Optimal synthesis problem for the fuzzy systems

In this paper, first a space of fuzzy numbers is constructed and a scalar product is introduced. The derivative of fuzzy function in this space is defined. Further, a synthesis problem for fuzzy systems is considered. A formula for optimal control is obtained. Copyright © 2010 John Wiley & Sons, Ltd.

On Some Subspaces of Entire Sequence Space of Fuzzy Numbers

On Some Subspaces of Entire Sequence Space of Fuzzy Numbers

On Some Subspaces of Entire Sequence Space of Fuzzy Numbers

In this paper we introduce some subspaces of fuzzy entire sequence space. Some general properties of these sequence spaces are discussed. Also some inclusion relation involving the spaces are obtained. Mathematics Subject Classification: 40A05, 40D25. Keywords—Fuzzy Numbers, Entire sequences, completeness, Fuzzy entire sequences

Hybrid fuzzy support vector classifier machine and modified genetic algorithm for automatic car assembly fault diagnosis

This paper presents a new version of fuzzy support vector machine to diagnose automatic car assembly fault diagnosis, the input and output variables are described as fuzzy numbers and the metric on fuzzy number space is defined. Then by combining the fuzzy theory with v-support vector machine, the fuzzy v-support vector classifier machine (F v-SVCM) is proposed. A fault diagnosis method based on F v-SVCM and its relevant parameter-choosing algorithm is put forward. The results of the application in car assembly diagnosis confirm the feasibility and the validity of the diagnosis method. Compared with the fuzzy neural network (FNN) model, F v-SVCM method requires fewer samples and has better estimating precision.

Fuzzy robust ν-support vector machine with penalizing hybrid noises on symmetric triangular fuzzy number space

In view of the shortage of ε-insensitive loss function for hybrid noises such as singularity points, biggish magnitude noises and Gaussian noises, this paper presents a new version of fuzzy support vector machine (SVM) which can penalize those hybrid noises to forecast fuzzy nonlinear system. Since there exist some problems of hybrid noises and uncertain data in many actual forecasting problem, the input variables are described as fuzzy numbers by fuzzy comprehensive evaluation. Then by the integration of the triangular fuzzy theory, ν-SVM and loss function theory, the fuzzy robust ν-SVM with robust loss function (FRν-SVM) which can penalize those hybrid noises is proposed. To seek the optimal parameters of FRν-SVM, particle swarm optimization is also proposed to optimize the unknown parameters of FRν-SVM. The results of the application in fuzzy sale system forecasts confirm the feasibility and the validity of the FRν-SVM model. Compared with the traditional model and other SVM methods, FRν-SVM method requires fewer samples and has better generalization capability for Gaussian noise.

Fuzzy support vector regression machine with penalizing Gaussian noises on triangular fuzzy number space

In view of the shortage of ε-insensitive loss function for Gaussian noise, this paper presents a new version of fuzzy support vector machine (SVM) which can penalize Gaussian noise to forecast fuzzy nonlinear system. Since there exist some problems of finite samples and uncertain data in many forecasting problem, the input variables are described as crisp numbers by fuzzy comprehensive evaluation. To represent the fuzzy degree of these input variables, the symmetric triangular fuzzy technique is adopted. Then by the integration of the fuzzy theory, ν-SVM and Gaussian loss function theory, the fuzzy ν-SVM with Gaussian loss function (F g-SVM) which can penalize Gaussian noise is proposed. To seek the optimal parameters of F g-SVM, genetic algorithm is also proposed to optimize the unknown parameters of F g-SVM. The results of the application in sale system forecasts confirm the feasibility and the validity of the F g-SVM model. Compared with the traditional model, F g-SVM method requires fewer samples and has better generalization capability for Gaussian noise.

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R 3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies.

Bounded variation double sequence space of fuzzy real numbers

In this paper we have introduced the notion of fuzzy real-valued bounded variation double sequence space b 2 v F . We have studied some of its properties like convergence free, solidness, symmetricity, monotonicity, etc. We have proved some inclusion results too.

A note on the sendograph metric of fuzzy numbers

In this paper, by studying the attainable properties of sendograph metric in fuzzy number space, we generalize Monotone convergence theorem and Nested theorem of intervals from the space of real numbers to the space of fuzzy numbers.

The equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets

In this paper, with the help of the convergence of sequences of fuzzy numbers with respect to the Lebesgue measure, we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the L p metric. We prove that these convergences are equivalent under proper conditions. In addition, by applying our result, we give a new characterization of compact sets in fuzzy number space with the sendograph metric.

Representation of Uncertain Multichannel Digital Signal Spaces and Study of Pattern Recognition Based on Metrics and Difference Values on Fuzzy $n$-Cell Number Spaces

In this paper, we discuss the problem of characterization for uncertain multichannel digital signal spaces, propose using fuzzy n-cell number space to represent uncertain n-channel digital signal space, and put forward a method of constructing such fuzzy n-cell numbers. We introduce two new metrics and concepts of certain types of difference values on fuzzy n -cell number space and study their properties. Further, based on the metrics or difference values appropriately defined, we put forward an algorithmic version of pattern recognition in an imprecise or uncertain environment, and we also give practical examples to show the application and rationality of the proposed techniques.

Comparison and Ordering of Fuzzy Numbers Based on Method of Structured Element

Abstract According to the homeomorphic property between fuzzy number space and the family of standard monotone functions on [−1,1], comparison of fuzzy numbers can be transformed into comparison of standard monotone functions on [−1,1]. Some order relations on the fuzzy number space are defined by using the method of fuzzy structured element, and some properties of the order relations are discussed. At last, an example is designed.