Fuzzy Number Space Research Articles

Inclusion theorems for (C,k) summability method of sequences of fuzzy numbers

We introduce (C,k) summability method of sequences of fuzzy numbers and compare (C,k) summability method with Abel summability method, Holder summability method and with higher order Cesaro method of sequences of fuzzy numbers. Some Tauberian conditions under which (C,k) summability of sequences of fuzzy numbers imply convergence in fuzzy number space are also obtained as corollaries.

On generalized fuzzy numbers

This paper first improves Chen and Hsieh’s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the λ-cutting sets or generalized λ-cutting sets, so that the generalized fuzzy numbers are integrated into a partial ordering set, a semi-ring and a complete non-separable metric space. Again, through two isomorphism theorem, the relationship between generalized fuzzy number space and fuzzy number space is established. Finally, as an application of generalized fuzzy numbers, the concept of continuous generalized fuzzy number-valued functions is introduced. And it is pointed out that many results of trapezoidal generalized fuzzy numbers can also be extended to generalized fuzzy numbers.

Systems of implicit fractional fuzzy differential equations with nonlocal conditions

In this paper, two types of fixed point theorems are employed to study the solvability of nonlocal problem for implicit fuzzy fractional differential systems under Caputo gH-fractional differentiability in the framework of generalized metric spaces. First of all, we extend Krasnoselskii?s fixed point theorem to the vector version in the generalized metric space of fuzzy numbers. Under the Lipschitz conditions, we use Perov?s fixed point theorem to prove the global existence of the unique mild fuzzy solution in both types (i) and (ii). When the nonlinearity terms are not Lipschitz, we combine Perov?s fixed point theorem with vector version of Krasnoselskii?s fixed point theorem to prove the existence of mild fuzzy solutions. Based on the advantage of vector-valued metrics and convergent matrix, we attain some properties of mild fuzzy solutions such as the boundedness, the attractivity and the Ulam - Hyers stability. Finally, a computational example is presented to demonstrate the effectivity of our main results.

Signed-Distance Measures Oriented to Rank Interval-Valued Fuzzy Numbers

This paper proposes a class of signed-distance measures for interval-valued fuzzy numbers based on specific bivariate functions, called kernels, and $\alpha \beta$ -values of interval-valued fuzzy numbers. The main properties, such as robustness, of the proposed signed-distance measure are also studied in the space of interval-valued fuzzy numbers. Then, the proposed signed-distance is applied to rank a set of interval-valued fuzzy numbers using an axiomatic approach. The proposed method is examined and compared to several existing methods, and the feasibility and effectiveness of the proposed methods are cleared via numerical and theoretical comparisons. The results indicate that the proposed method is simple for ranking all kinds of $LR$ -interval-valued fuzzy numbers, and it can overcome drawbacks of the existing methods based on reasonable axioms expected for a ranking criterion in the space of interval-valued fuzzy numbers. Finally, the proposed method is applied to a case study related to a multicriteria group decision making.

Applying the concept of null set to solve the fuzzy optimization problems

The concept of null set in the space of fuzzy numbers is introduced. Based on this concept, we can define two partial orderings according to the subtraction and Hukuhara difference between any two fuzzy numbers. These two partial orderings will be used to define the solution concepts of fuzzy optimization problems. On the other hand, we transform the fuzzy optimization problems into a conventional vector optimization problem. Under these settings, we can apply the technique of scalarization to solve this transformed vector optimization problem. Finally, we show that the optimal solution of the scalarized problem is also the optimal solution of the original fuzzy optimization problem.

A note on Ulam stability of some fuzzy number-valued functional equations

This note investigates the Ulam stability of the following fuzzy number-valued functional equations in Banach spaces by using the metric defined on a fuzzy number space:f(x+y2)+f(x−y2)=f(x)andf(ax+by)=rf(x)+sf(y).

Fuzzy Quantile Linear Regression Model Adopted with a Semi-Parametric Technique based on Fuzzy Predictors and Fuzzy Responses

As an extension to statistical techniques, fuzzy regression analysis estimates the relationship between fuzzy independent variables and non-fuzzy and/or fuzzy dependent variables. However, facing skewed or non-normal data may lead to vague results if the conventional least square models are used. However, there are many situations in applications of intelligence systems where regression components are fuzzy quantities instead of exact values. The present work is aimed to propose a fuzzy semi-parametric quantile regression model with fuzzy predictors, a fuzzy smooth function, exact coefficients and fuzzy responses. The proposed method is based on a new class of novel kernel-based signed-distance measures in space of fuzzy numbers. Main features of the proposed signed-distance including its robustness were studied. It was shown that the proposed signed-distance measure is more reasonable and effective than a common signed-distance which extensively has been used in fuzzy set theory. This is the reason why we decided to apply the proposed signed-distance to extend the classical quantile regression model. Then, a hybrid algorithm was suggested to evaluate optimal bandwidth, non-fuzzy regression coefficients and fuzzy smooth function at a specific quantile level. The proposed method was then compared with some common fuzzy regression models based on two data sets including a simulation study and an applied example. The numerical results clearly indicated that the proposed fuzzy semi-parametric quantile linear regression model performs better than the other available methods in terms of some common goodness-of-fit criteria in cases where the data set includes skewness and/or some outlier data point. The main advantages of the proposed method are as follows: 1-it is not so sensitive to the outlier data points and/or skewed data, 2- it possesses better fitting effect compared to other existing fuzzy regression models. Thus, the proposed quantile regression model could be successfully applied in many practical studies of fuzzy regression model in real-world applications.

Fuzzy absolute error distance measure based on a generalised difference operation

ABSTRACTThe main goal of this paper is to introduce a notion of fuzzy absolute error distance measure between two fuzzy numbers. For this purpose, a notion of generalised difference operation between two fuzzy numbers and absolute value of a fuzzy number was first introduced. The proposed methods were conducted on the basis of α-values of fuzzy numbers. Main properties of the proposed fuzzy distance measure was also verified in the space of fuzzy numbers. The proposed fuzzy distance measure evaluates the fuzzy distance between the two fuzzy numbers as a fuzzy number. Notably, the main advantage of such generalised difference operation is that it always exists. Therefore, it improves the shortcoming of a well-known generalised difference operation called Hakuhara difference. Some of the main properties of the proposed fuzzy absolute error distance measure including robustness were also studied in the space of fuzzy numbers. Several fuzzy distance measures, especially fuzzy absolute error distance, have been proposed so far. However, none of them save all reasonable properties required for an absolute error distance measure in fuzzy environment. Shortcomings relevant to other methods and advantages of the proposed method were also discussed.

Near Fixed Point Theorems in the Space of Fuzzy Numbers

The fuzzy numbers are fuzzy sets owning some elegant mathematical structures. The space consisting of all fuzzy numbers cannot form a vector space because it lacks the concept of the additive inverse element. In other words, the space of fuzzy numbers cannot be a normed space even though the normed structure can be defined on this space. This also says that the fixed point theorems established in the normed space cannot apply directly to the space of fuzzy numbers. The purpose of this paper is to propose the concept of near fixed point in the space of fuzzy numbers and to study its existence. In order to consider the contraction of fuzzy-number-valued function, the concepts of near metric space and near normed space of fuzzy numbers are proposed based on the almost identical concept. The concepts of Cauchy sequences in near metric space and near normed space of fuzzy numbers are also proposed. Under these settings, the existence of near fixed points of fuzzy-number-valued contraction function in complete near metric space and near Banach space of fuzzy numbers are established.

Completeness, metrizability and compactness in spaces of fuzzy-number-valued functions

Fuzzy-number-valued functions, that is, functions defined on a topological space taking values in the space of fuzzy numbers, play a central role in the development of Fuzzy Analysis. In this paper we study completeness, metrizability and compactness of spaces of continuous fuzzy-number-valued functions.

Tauberian theorems for statistical summability methods of sequences of fuzzy numbers

We give Tauberian conditions of slow decrease type under which statistical Cesaro summability and statistical logarithmic summability of sequences of fuzzy numbers imply convergence in fuzzy number space. Besides, we obtain a sufficient condition for statistically convergent sequences of fuzzy numbers to converge in the ordinary sense.

Linear Model With Exact Inputs and Interval-Valued Fuzzy Outputs

This paper considers the classical semiparametric partially linear model with exact inputs and interval-valued fuzzy outputs. For this purpose, the most commonly used classical two-phase procedure is extended to estimate an interval-valued fuzzy smooth function using nonparametric kernel methods at phase 1 and a least absolute deviation method at phase 2 to estimate the interval-valued fuzzy coefficients. A potential application of the proposed method is presented by a simulated data in hydrology study and an applied example. The proposed interval-valued semiparametric partially linear model is also examined to compare with the interval-valued fuzzy linear regression model via some extended goodness-of-fit criteria into the space of interval-valued fuzzy numbers.

Optimality conditions for fuzzy constrained programming problems

This paper solves optimization problems where both the objective and constraints are given by fuzzy functions. In order to get it, we first prove that these problems are equivalent to optimization problems where the constraints functions are non-fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far. This new stationary point concept is based on the gH-differentiability and has many computational advantages that we describe. It is well-known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity. And we are in that case due to the fact that the fuzzy numbers (intervals) space is a nonlinear one. In this direction, the gH-derivative for fuzzy functions is a concept that is more general than Hukuhara and level-wise derivatives that are usually used in fuzzy optimization so far, in the sense that they can be applied to a wider number of fuzzy function classes than above concepts. With this new differentiability concept, we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far. Moreover, due to the fact that we do not have a linear space for fuzzy numbers, the convex concepts and generalized convex fuzzy function notion are very restrictive, also. This implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful.

A foundation on semigroups of operators defined on the set of triangular fuzzy numbers and its application to fuzzy fractional evolution equations

In this article, we study the strongly continuous semigroups of fuzzy-valued operators defined on the space of triangular fuzzy numbers. New notions of fuzzy infinitesimal generators and fuzzy resolvent operators of fuzzy semigroups are established in the sense of the generalized Hukuhara difference. A Hille–Yosida-like theorem in the framework of fuzzy metric space is investigated. We demonstrate the efficiency of theoretical results by studying the existence of mild solutions of fuzzy nonlinear fractional evolution equations.

On statistical convergent sequence spaces of intuitionistic Fuzzy numbers

In the present paper we introduce the classes of sequence stcIFN, stc0IFN and st∞IFN of statistically convergent, statistically null and statistically bounded sequences of intuitionistic fuzzy number based on the newly defined metric on the space of all intuitionistic fuzzy numbers (IFNs). We study some algebraic and topological properties of these spaces and prove some inclusion relations too.

Record value based on intuitionistic fuzzy random variables

ABSTRACTThis paper extends some basic concepts associated to record value based on intuitionistic fuzzy random variables. In this approach, αβ-values of intuituinistic fuzzy numbers are employed to construct intuitionistic fuzzy cumulative distribution function and its common estimator, an extended entropy and its estimator, intuitionistic fuzzy (upper) record value and its common estimator. Main property of the proposed concepts include large sample properties which are investigated in the space of intuitionistic fuzzy numbers. Some numerical examples are also illustrated to clarify the concepts and methods.

A Preference Index for Ranking Closed Intervals and Fuzzy Numbers

This paper proposes a preference index for ranking the most applicable vague data, i.e. closed intervals and LR-fuzzy numbers. In this approach, first a preference degree is suggested to compare two closed intervals. Then, we extended the proposed criterion from the space of closed intervals into the the space of fuzzy numbers. Using some theorems and lemmas, it will be proved that the extended preference degrees will satisfy many common and desired properties based on the proposed axiomatic definitions introduced for similarity measures. Some numerical examples are also used to illustrate the proposed ranking methods. Moreover, the proposed methods are examined to compare other existing methods and the feasibility and effectiveness of the proposed methods will be cleared via some numerical comparisons. Comparative examples show that the proposed method overcomes certain unreasonable and shortcomings exhibited by some existing methods in some common situations.

Some Generalizations of Fixed Point Theorems in Partially Ordered Metric Spaces and Applications to Partial Differential Equations with Uncertainty

Some generalized contractions using altering distances in partially ordered metric spaces are investigated and their applications to fuzzy partial differential equations are considered. Starting from the Banach contraction principle, our theorems presented here generalize, extend, and improve different results existing in the literature on the existence of coincidence points for a pair of mappings. In terms of their applicability, this might constitute the first paper dealing with the solvability of fuzzy partial differential equations from the point of view of considering the structure of the fuzzy number space as a partially ordered space. Under the generalized contractive-like property over comparable items, which is weaker than the Lipschitz condition, we show that the existence of just a lower or an upper solution is enough to prove the existence and uniqueness of two types of fuzzy solutions in the sense of gH-differentiability.

Tauberian conditions for double sequences which are statistically summable (C,1,1) in fuzzy number space

Tauberian conditions for double sequences which are statistically summable (C,1,1) in fuzzy number space

Characterizations of the Borel \(\sigma\)-fields of the fuzzy number space

In this paper, we characterize Borel \(\sigma\)-fields of the set of all fuzzy numbers endowed with different metrics. The main result is that the Borel \(\sigma\)-fields with respect to all known separable metrics are identical. This Borel field is the Borel \(\sigma\)-field making all level cut functions of fuzzy mappings from any measurable spaces to the fuzzy number space measurable with respect to the Hausdorff metric on the cut sets. The relation between the Borel \(\sigma\)-field with respect to the supremum metric \(d_{\infty}\) is also demonstrated. We prove that the Borel field is induced by a separable and complete metric. A global characterization of measurability of fuzzy valued functions is given via the main result. Applications to fuzzy valued integral are given, and an approximation method is presented for integrals of fuzzy valued functions. Finally, an example is given to illustrate the applications of these results in economics. This example shows that the results in this paper are basic to the theory of fuzzy-valued functions, such as the fuzzy version of Lebesgue-like integrals of fuzzy-valued functions, and are useful in applied fields. doi:10.1017/S1446181117000189