Approximation Of Fuzzy Numbers Research Articles

Symmetric trapezoidal approximations of fuzzy numbers under a general condition

Symmetric trapezoidal approximations of fuzzy numbers under a general condition

A generalized and unified approach to the approximation of fuzzy numbers and its arithmetic and characteristics

In this paper we propose a general method for the approximation of an arbitrary fuzzy number. This method, which is constructive, recovers and properly extends some well-known approximations such as those obtained in terms of polygonal fuzzy numbers or simple fuzzy numbers. We prove the convergence of the general method and study the properties of the approximation operator, such as its compatibility with arithmetic operations of fuzzy numbers and with some of their important characteristics. In addition to this, we illustrate the method with some particularly interesting cases by providing algorithms, of great simplicity for practical use and apply them to some numerical examples. Furthermore, the approximations we construct are particularly simple from the point of view of fuzzy arithmetic and preserve some of their most important characteristics.

Teaching assistant selection in Thailand by using an extended VIKOR based on piecewise linear approximation of fuzzy numbers

Because of the COVID-19 situation, selection for a teaching assistant position to get a TA scholarship in a university in Thailand needs to be performed online by the formed committee. Due to the online process and the limited number of scholarships offered by the university, beyond the face-to-face interview, multiple-criteria decision analysis can help to select a proper student. In this study, we use the extended VIKOR method with fuzzy numbers to help committees to select the students from the applicants. The criteria and the weights of the criteria are provided with the help of committees. Both trapezoidal and triangular linguistic variables are used to find the solution and to observe the range of the possible result. The different weights supporting the strategy of maximum group utility are varied to detect the potential alternatives. The ranking results are also compared with the one obtained from the TODIM approach to illustrate the appropriate alternative.

The Best Approximation of Generalized Fuzzy Numbers Based on Scaled Metric

The ongoing study has been vehemently allocated to propound an ameliorated α-weighted generalized approximation of an arbitrary fuzzy number. This method sets out to lessen the distance between the original fuzzy set and its approximation. In an effort to elaborate the study, formulas are designed for computing the ameliorated approximation by using a multitude of examples. The numerical samples will be exemplified to illuminate the improvement of the nearest triangular approximation (Abbasbandy et al., Triangular approximation of fuzzy numbers using α-weighted valuations, Soft Computing, 2019). A variety of features of the ameliorated approximation are then proved.

Approximations of Fuzzy Numbers by Using r-s Piecewise Linear Fuzzy Numbers Based on Weighted Metric

Using simple fuzzy numbers to approximate general fuzzy numbers is an important research aspect of fuzzy number theory and application. The existing results in this field are basically based on the unweighted metric to establish the best approximation method for solving general fuzzy numbers. In order to obtain more objective and reasonable best approximation, in this paper, we use the weighted distance as the evaluation standard to establish a method to solve the best approximation of general fuzzy numbers. Firstly, the conceptions of I-nearest r-s piecewise linear approximation (in short, PLA) and the II-nearest r-s piecewise linear approximation (in short, PLA) are introduced for a general fuzzy number. Then, most importantly, taking weighted metric as a criterion, we obtain a group of formulas to get the I-nearest r-s PLA and the II-nearest r-s PLA. Finally, we also present specific examples to show the effectiveness and usability of the methods proposed in this paper.

Weighted polygonal approximation of fuzzy numbers preserving their main characteristics

In this article we obtain an approximation of a given fuzzy number by a unique polygonal fuzzy number preserving its main characteristics such as core, support, and expected interval. For this, we consider the well-known weighted L2 metric. We show that this weighted polygonal approximation is equivalent to a strict convex quadratic optimization problem finite dimensional with linear inequality and equality constraint. Then, we present efficient and practice solution methods illustrated with several examples. Some properties of invariance of the approximation operator are also obtained. The results presented here extend and improve the methods obtained for weighted trapezoidal approximation and piecewise linear approximation of fuzzy numbers.

Fuzzy Data Envelopment Analysis with Ordinal and Interval Data

In this paper, we reformulate the conventional DEA models as an imprecise DEA problem and propose a novel method for evaluating the DMUs when the inputs and outputs are fuzzy and/or ordinal or vary in intervals. For this purpose, we convert all data into interval data. In order to convert each fuzzy number into interval data, we use the nearest weighted interval approximation of fuzzy numbers by applying the weighting function, and we convert each ordinal data into interval one. In this manner, we could convert all data into interval data. The presented models determine the interval efficiencies for DMUs. To rank DMUs based on their associated interval efficiencies, we first apply the Ω-index that is developed for ranking of interval numbers. Then, by introducing an ideal DMU, we rank efficient DMUs to present a complete ranking. Finally, we use one example to illustrate the process and one real application in health care to show the usefulness of the proposed approach. For this evaluation, we consider interval, ordinal, and fuzzy data alongside the precise data to evaluate 38 hospitals selected by OIG. The results reveal the capabilities of the presented method to deal with the imprecise data.

Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

Numerous research papers and several engineering applications have proved that the fuzzy set theory is an intelligent effective tool to represent complex uncertain information. In fuzzy multi-criteria decision-making (fuzzy MCDM) methods, intelligent information system and fuzzy control-theoretic models, complex qualitative information are extracted from expert’s knowledge as linguistic variables and are modeled by linear/non-linear fuzzy numbers. In numerical computations and experiments, the information/data are fitted by nonlinear functions for better accuracy which may be little hard for further processing to apply in real-life problems. Hence, the study of non-linear fuzzy numbers through triangular and trapezoidal fuzzy numbers is very natural and various researchers have attempted to transform non-linear fuzzy numbers into piecewise linear functions of interval/triangular/trapezoidal in nature by different methods in the past years. But it is noted that the triangular/trapezoidal approximation of nonlinear fuzzy numbers has more loss of information. Therefore, there is a natural need for a better piecewise linear approximation of a given nonlinear fuzzy number without losing much information for better intelligent information modeling. On coincidence, a new notion of Generalized Hexagonal Fuzzy Number has been introduced and its applications on Multi-Criteria Decision-Making problem (MCDM) and Generalized Hexagonal Fully Fuzzy Linear System (GHXFFLS) of equations have been studied by Lakshmana et al. in 2020. Therefore, in this paper, approximation of nonlinear fuzzy numbers into the hexagonal fuzzy numbers which includes trapezoidal, triangular and interval fuzzy numbers as special cases of Hexagonal fuzzy numbers with less loss/gain of information than other existing methods is attempted. Since any fuzzy information is satisfied fully by its modal value/core of that concept, any approximation of that concept is expected to be preserved with same modal value/core. Therefore, in this paper, a stepwise procedure for approximating a non-linear fuzzy number into a new Hexagonal Fuzzy Number that preserves the core of the given fuzzy number is proposed using constrained nonlinear programming model and is illustrated numerically by considering a parabolic fuzzy number. Furthermore, the proposed method is compared for its efficiency on accuracy in terms of loss of information. Finally, some properties of the new hexagonal fuzzy approximation are studied and the applicability of the proposed method is illustrated through the Group MCDM problem using an index matrix (IM).

Trapezoidal approximations of fuzzy numbers using quadratic programs

In this paper we will prove that the nearest trapezoidal approximation of fuzzy numbers with respect to weighted L2-type metrics with or without additional constraints can be obtained via quadratic programs. Actually, the approach is even more general based on so called finite polyhedral subsets of fuzzy numbers which include most of the important special classes of fuzzy numbers available in the literature. In particular, we will recapture the algorithm to compute the nearest weighted trapezoidal approximation of a fuzzy number by a method which we believe that has the potential to be extended to more complex approximation problems. Then, we will improve the Lipschitz constant of the trapezoidal approximation operator preserving the ambiguity. To achieve this improved result we will exploit the fact that we have an analytical expression for this operator. However, note that the same result is obtained if this solution function is described by quadratic programs. Therefore, for similar problems we still can obtain Lipschitz constants for the approximation operator even if an analytical expression of this operator is not available.

Approximation of fuzzy numbers by using multi-knots piecewise linear fuzzy numbers1

In this paper, the problem of approximating general fuzzy number by using multi-knots piecewise linear fuzzy number is studied. First, r - s-knots piecewise linear fuzzy numbers are defined, and the conceptions of the I-nearest r - s-knots piecewise linear approximation and the II-nearest r - s-knots piecewise linear approximation are introduced for a general fuzzy number. Then, most importantly, we set up the methods to get the I-nearest r - s-knots piecewise linear approximation and the II-nearest r - s-knots piecewise linear approximation for a general fuzzy number. And then, we investigate some properties of the new approximation operators. Finally, we also present specific examples to show the effectiveness, usability and advantages of the methods proposed in this paper, and compare the methods with some other approximation algorithms.

Piecewise linear approximation of fuzzy numbers: algorithms, arithmetic operations and stability of characteristics

The problem of the piecewise linear approximation of fuzzy numbers giving outputs nearest to the inputs with respect to the Euclidean metric is discussed. The results given in Coroianu et al. (Fuzzy Sets Syst 233:26–51, 2013) for the 1-knot fuzzy numbers are generalized for arbitrary n-knot (nge 2) piecewise linear fuzzy numbers. Some results on the existence and properties of the approximation operator are proved. Then, the stability of some fuzzy number characteristics under approximation as the number of knots tends to infinity is considered. Finally, a simulation study concerning the computer implementations of arithmetic operations on fuzzy numbers is provided. Suggested concepts are illustrated by examples and algorithms ready for the practical use. This way, we throw a bridge between theory and applications as the latter ones are so desired in real-world problems.

Multiobjective security game with fuzzy payoffs

A multiobjective security game problem with fuzzy payoffs is studied in this paper. The problem is formulated as a bilevel programming problem with fuzzy coefficients. Using the idea of nearest interval approximation of fuzzy numbers, the problem is transformed into a bilevel programming problem with interval coefficients. The Karush-Kuhn-Tucker conditions is applied then to reduce the problem to an interval multiobjective single-level problem. It is shown that the solutions of this problem are obtained by solving a single-objective programming problem. Validity and applicability of the method are illustrated by a practical example.

Weighted pseudometric approximation of 2-dimensional fuzzy numbers by fuzzy 2-cell prismoid numbers preserving the centroid

Up to now, there have been many research results about approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators. However, there are few investigations about approximations of n-dimensional fuzzy numbers. The main reason is that the structure of n-dimensional fuzzy numbers are more complex. This paper devotes to seek an approximation operator that produces a special kind of 2-dimensional fuzzy number which is the nearest one to the given 2-dimensional fuzzy number along the direction k. The concept of four edges approximation for 2-dimensional fuzzy numbers is defined, which is a special kind of 2-dimensional fuzzy numbers. At the same time, the concept of fuzzy 2-cell prismoid numbers is introduced, and the nearest fuzzy 2-cell prismoid numbers of 2-dimensional fuzzy numbers which preserve the centroid of the core for the 2-dimensional fuzzy numbers with respect to the weighted pseudometric are discussed. Finally, with respect to the criteria: scale invariance, identity criterion and nearness criterion, some properties of the approximation operator are presented.

A fast analytical approximation type-reduction method for a class of spiked concave type-2 fuzzy sets

In this paper, an analytical type-reduction method called concave analytical type-reduction method with spikes (CATRS) of two kinds of spiked concave type-2 fuzzy sets is proposed. The concave type-2 fuzzy sets which are considered in the paper include triangular type-2 fuzzy set, trapezoidal type-2 fuzzy set and its spiked versions. The analytical type-reduction method is free of the following steps, such as discourse partition and discourse refinement for algorithms' discrete implementation, thus its calculation complexity is reduced. To obtain the formulae of concave triangular type-2 fuzzy set, the method proposed by Starczewski is extended from the convex case to the concave one, secondary membership function with spikes is considered as a special form of triangular fuzzy set. Centroid set is formed by all the type-1 fuzzy sets that is derived from the convex part of the primary membership with or without spikes at a given discourse partition point. Moreover, the method proposed for concave triangular type-2 fuzzy set is extended to concave trapezoidal type-2 fuzzy set with a trapezoidal fuzzy number approximation operator. The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets, it is more convenient for type-2 fuzzy modeling and inference, and the application of type-reduction method to variable universe of discourse controller design shows that the approximation method is applicable for the adaptive fuzzy controller design.

An Approximation of Fuzzy Numbers Based on Polynomial Form Fuzzy Numbers

In this paper, we approximate an arbitrary fuzzy number by a polynomial fuzzy number through minimizing the distance between them. Throughout this work, we used a distance that is a meter on the set of all fuzzy numbers with continuous left and right spread functions. To support our claims analytically, we have proven some theorems and given supplementary corollaries.

Note on “symmetric triangular approximations of fuzzy numbers under a general condition and properties”

We show by a counterexample that Theorem 2 in Ban, Coroianu [Soft Computing (2016) 20:1249-1261] is not always the symmetric triangular fuzzy number nearest to a given fuzzy number. In addition, a corrected version is provided.

Solving Fully Fuzzy Multi-objective Linear Programming Problem Using Nearest Interval Approximation of Fuzzy Number and Interval Programming

This paper focuses on Fully Fuzzy Multi-Objective Linear Programming (FFMOLP) problem in which all the coefficients and decision variables are LR flat fuzzy numbers, the more generalized version of fuzzy numbers and all the constraints are fuzzy inequalities. A new algorithm is proposed for solving FFMOLP problem which first converts it into the Multi-Objective Interval Linear Programming (MOILP) problem. Further, taking the help of fuzzy slack variable, fuzzy surplus variables, nearest interval approximation of fuzzy numbers and scalarization technique, MOILP is then converted into the Crisp Linear Programming (CLP) problem. It is shown that the optimal solution of CLP problem is the fuzzy Pareto optimal solution of FFMOLP problem. The main advantage of the proposed algorithm is that it transforms FFMOLP problem into Crisp Linear Programming problem. Moreover, to apply algorithm, only the knowledge of arithmetic operations of LR flat fuzzy numbers, centre and width of the closed intervals are required. At the end, to illustrate the proposed method and its effectiveness over the existing method, numerical examples are solved and compared.

Approximations of fuzzy numbers by step type fuzzy numbers

In this paper, the problems of approximations in membership functions space are studied. Firstly, the concept of simple fuzzy numbers is defined, which is a special kind of fuzzy numbers with simple structure, and give the expressions of the membership function of thus fuzzy numbers. Then, we prove that a fuzzy set of real number field R is a simple fuzzy number if and only if it is a normal step type fuzzy set, and obtain a result of that simple fuzzy number space is dense in the fuzzy number space with respect to some metric. And then, we give two theorems about approximations of fuzzy number membership functions by using simple fuzzy number membership functions. At last, we investigate the properties of approximation operators given by us.

Approximation of fuzzy numbers using the convolution method

In this study, we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number. We show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non-differentiable points. In previous studies, this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers, the possible non-differentiable points of which were the two endpoints of the 1-cut. The construction of smoothers is a key step in the process for producing approximations. We also show that if appropriate smoothers are selected, then we can use the convolution method to provide approximations that are differentiable, Lipschitz, and that also preserve the core.

Width invariant approximation of fuzzy numbers

In this paper, we consider the width invariant trapezoidal and triangularapproximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserve the expected value too.