Fuzzy numbers, definitions and properties
Jose Luis Verdegay1994
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Two dierent denitions of a Fuzzy number may b e found in the literature. Both fulll Goguen's Fuzzication Principle but are dierent in nature because of their dierent starting points. The rst one was introduced b y Z adeh and has well suited arithmetic and algebraic properties. The second one, introduced by Gantner, Steinlage and Warren, i s a good and formal representation of the concept from a topological point of view. The objective of this paper is to analyze these denitions and discuss their main features.
Related papers
On some structures of fuzzy numbers
Iranian Journal of Fuzzy Systems, 2009
The operations in the set of fuzzy numbers are usually obtained by the Zadeh extension principle. But these definitions can have some disadvantages for the applications both by an algebraic point of view and by practical aspects. In fact the Zadeh multiplication is not distributive with respect to the addition, the shape of fuzzy numbers is not preserved by multiplication, the indeterminateness of the sum is too increasing. Then, for the applications in the Natural and Social Sciences it is important to individuate some suitable variants of the classical addition and multiplication of fuzzy numbers that have not the previous disadvantage. Here, some possible alternatives to the Zadeh operations are studied.
Definition of the Complement of Fuzzy Number: An Exposition
In this article, our main intention is to revisit the existing definition of complementation of fuzzy sets and thereafter various theories associated with it are also commented on. The main contribution of this paper is to suggest a new definition of complementation of fuzzy sets on the basis of reference function. Some other results have also been introduced whenever possible by using this new definition of complementation.
A new definition for defuzzification of generalized fuzzy numbers and its application
Applied Soft Computing, 2015
In this paper, a new approach for defuzzification of generalized fuzzy numbers is established. This method uses the incentre point of a triangle where the three bisector lines of its angles meet. Coordinates of incentre point can also be easily calculated by the "Mathematica" package to solve problems of defuzzification and ranking fuzzy numbers. Some numerical examples are illustrated to show the utility of proposed method.
Oriented Fuzzy Numbers vs. Fuzzy Numbers
Mathematics, 2021
A formal model of an imprecise number can be given as, inter alia, a fuzzy number or oriented fuzzy numbers. Are they formally equivalent models? Our main goal is to seek formal differences between fuzzy numbers and oriented fuzzy numbers. For this purpose, we examine algebraic structures composed of numerical spaces equipped with addition, dot multiplication, and subtraction determined in a usual way. We show that these structures are not isomorphic. It proves that oriented fuzzy numbers and fuzzy numbers are not equivalent models of an imprecise number. This is the first original study of a problem of a dissimilarity between oriented fuzzy numbers and fuzzy numbers. Therefore, any theorems on fuzzy numbers cannot automatically be extended to the case of oriented fuzzy numbers. In the second part of the article, we study the purposefulness of a replacement of fuzzy numbers by oriented fuzzy numbers. We show that for a portfolio analysis, oriented fuzzy numbers are more useful than ...
Fuzzy Numbers
Springer eBooks, 2019
In this chapter, preliminaries related to fuzzy numbers have been discussed. Fuzzy numbers and fuzzy arithmetic may be considered as an extension of classical real numbers and its arithmetic. As such, we may understand fuzzy arithmetic as basics for handling fuzzy eigenvalue problems, nonlinear equations, system of nonlinear equations (Abbasbandy and Asady 2004), differential equations (Chakraverty et al. 2016), etc. There exist different types of fuzzy numbers as discussed in Hanss (2005), but for the sake of completeness of the chapter, triangular, trapezoidal, and Gaussian fuzzy numbers based on the membership functions have only been included here. Further, the conversions of these fuzzy numbers to fuzzy intervals with respect to the concept of intervals (Chap. 1) are incorporated. In this regard, the interval arithmetic mentioned in Chap. 1 has been further extended to fuzzy intervals in Sect. 3.4. 3.1 Preliminaries of Fuzzy Numbers A convex fuzzy setà is a fuzzy set having membership function μÃ(x), satisfying μÃ(λx 1 + (1 − λ)x 2) ≥ min(μÃ(x 1), μÃ(x 2)), (3.1) where x 1 , x 2 ∈ X and λ ∈ [0, 1]. Figure 3.1 depicts convex and non-convex fuzzy sets. Convex fuzzy sets defined with respect to universal set (set of all real numbers) may be interpreted as fuzzy numbers. In this respect, the classical definition of fuzzy number is given below. Fuzzy number: A fuzzy setà is referred as a fuzzy numberã if the following properties are satisfied:
On the Issue of Comparison of Fuzzy Numbers
Mathematical Optimization Theory and Operations Research, 2019
In the class of decision-making problems with fuzzy information concerning criterion values, the problem of comparing fuzzy numbers is relevant. There are various approaches to solving it. They are determined by the specific character of the problem under consideration. This paper proposes one approach to comparing fuzzy numbers. The proposed approach is as follows. At first, a rule is constructed for comparing a real number with a level set of a fuzzy number. Then, with the help of a procedure for constructing the exact lower approximation for the collection of sets, a fuzzy set is constructed. This fuzzy set determine the rule for comparing a real number with a fuzzy number. Using this rule and the approach based on separating two fuzzy numbers with a real number, the procedure is chosen for comparing two fuzzy numbers. As an example, fuzzy numbers with trapezoidal membership functions are considered, and the geometric interpretation of the results being given.
A Brief Analysis and Interpretation on Arithmetic Operations of Fuzzy Numbers
Results in Control and Optimization, 2023
Fuzzy set theory is a generalized form of crisp set theory where elements are binary inclusion forms. In fuzzy set, it differs with degree of membership for every element in the set. There are several strategies for arithmetic operations on fuzzy numbers. Previous studies show that there are many approaches, such as the α-cut technique, extension principle, vertex method, etc., to execute arithmetic operations on fuzzy numbers. In this study we perform details analysis and interpretation on arithmetic operations based on the α-cut method in a new way.
THEORY OF TRIANGULAR FUZZY NUMBER
The fuzzy numbers are defined in uncertainty situation and applied in real world problems of science and engineering. In earlier days, there was no mathematical concept to define vagueness. The laws of logic, the Law of Identity, the Law of Non-Contradiction, and the Law of Excluded Middle were introduced, and can be applied in any kind of situation. This logic is the origin of Fuzzy. In this paper, the number theoretical aspect of Fuzzy Number and Triangular Fuzzy number have been established.
Fuzzy Numbers: Positive and Nonnegative
2008
In this paper we emphasize that the definition of positive fuzzy number in D.Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980 (page 49), and also recently quoted for solving fully fuzzy linear systems in M. Dehghan et al., Applied Mathematics and Computation 179 (2006) 328-343, is in fact a definition for the nonnegative fuzzy number. Moreover, we give two different definitions for nonnegative and positive triangular fuzzy numbers to eliminate this shortcoming.
The opposite of quasi-triangular fuzzy number
This paper presents the geometrical interpretation of the quasitriangular fuzzy number, being also shown that the opposite of quasi-triangular fuzzy number is a quasi-triangular fuzzy number with centre symmetrical to the origin and spread rotated with 90 0. 2 Preliminaries This section reviews the definitions and basic propositions applied in this paper. Definition 2.1 (Fuzzy set). Let be X a set. A mapping μ : X → [0, 1] is called membership function, and the set A = { (x, μ (x)) / x ∈ X} is called fuzzy set on X. The membership function of A is denoted by μ A. The collection of all fuzzy set on X is denoted by F(X). Triangular norms were introduced by K. Menger [10] and studied first by B. Schweizer and A. Sklar [11], [12], [13] to model distances in probabilistic metric spaces. In fuzzy sets theory triangular norms are extensively used to model the logical connection and.
Related topics
Related papers
On a canonical representation of fuzzy numbers
Fuzzy Sets and Systems, 1998
Fuzzy numbers, and more generally linguistic values, are approximate assessments, given by experts and accepted by decision-makers when obtaining more accurate values is impossible or unnecessary. To simplify the task of representing and handling fuzzy numbers, several authors have introduced real indices in order to capture the information contained in a fuzzy number. In this paper we propose two parameters, value and ambiguity, for this purpose. We use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision-making problems. Several examples illustrate these ideas.
A new similarity measure for generalized fuzzy numbers
Neural Computing and Applications, 2012
In this paper, first of all the distance measure entitled generalized Hausdorff distance is defined between two trapezoidal generalized fuzzy numbers that has been introduced by Chen [9]. Then using a other distance and combining with generalized Hausdorff distance, the similarity measure is defined. The basic properties of the above-mentioned similarity measure are proved in detail. Finally, the concept is illustrated by solving several testing examples.
The Complement of Normal Fuzzy Numbers: An Exposition
International Journal of Intelligent Systems and Applications, 2013
In this article, our main intention is to revisit the existing definition of complementation of fuzzy sets and thereafter various theories associated with it are also commented on. The main contribution of this paper is to suggest a new definition of complementation of fuzzy sets on the basis of reference function. Some other results have also been introduced whenever possible by using this new definition of complementation.
Fuzzy Number -A New Hypothesis and Solution of Fuzzy Equations
Mathematics and Statistics, 2022
In this paper, a new hypothesis of fuzzy number has been proposed which is more precise and direct. This new proposed approach is considered as an equivalence class on set of real numbers 𝑅 with its algebraic structure and its properties along with theoretical study and computational results. Newly defined hypothesis provides a well-structured summary that offers both a deeper knowledge about the theory of fuzzy numbers and an extensive view on its algebra. We defined field of newly defined fuzzy numbers which opens new era in future for fuzzy mathematics. It is shown that, by using newly defined fuzzy number and its membership function, we are able to solve fuzzy equations in an uncertain environment. We have illustrated solution of fuzzy linear and quadratic equations using the defined new fuzzy number. This can be extended to higher order polynomial equations in future. The linear fuzzy equations have numerous applications in science and engineering. We may develop some iterative methods for system of fuzzy linear equations in a very simple and ordinary way by using this new methodology. This is an innovative and purposefulness study of fuzzy numbers along with replacement of this newly defined fuzzy number with ordinary fuzzy number.
Towards the properties of fuzzy multiplication for fuzzy numbers
Kybernetika
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
On discriminating fuzzy numbers
Ranking fuzzy numbers plays a very important role in linguistic decision making and some other fuzzy application systems such as data analysis, artificial intelligence and socio economic systems. Various approaches have been proposed in the literature for the ranking of fuzzy numbers and most of the methods seem to suffer from drawbacks. In this paper a new method is proposed to rank fuzzy numbers. This method is based on the centroid of centroids of generalized trapezoidal fuzzy numbers and allows the participation of decision maker by using an index of optimism to reflect the decision maker’s optimistic attitude and also an index of modality that represents the importance of considering the areas of spreads by the decision maker. This method is relatively simple and easier in computation and ranks various types of fuzzy numbers along with crisp fuzzy numbers as special case of fuzzy numbers.
Merkezi̇ Olmayan Sonsuz Sevi̇yeli̇ Aralik Sayilari İle Fuzzy Sayilari / Fuzzy Numbers with Non-Central Infinite-Level Interval Numbers
2001
G.Bojadziev ve M. Bojadziev farkli bir sekilde bir merkezli ve bir maksimumlu sonsuz seviyeli aralik sayilarina limit uygulayarak fuzzy number ve aritmetigini tanimlamistir. Biz de bu yontemi merkezi olmayan ve cok maksimumlu sonsuz dereceli aralik sayilarina genellestirdik. G.Bojadziev and M. Bojadziev defined differently a fuzzy number and its arithmetic from central infinite-level interval numbers with one maximum by using the limiting process. We generalised this process to the non-central and infinite-level interval numbers with more than one maximum.
Introduction to fuzzy arithmetic: Theory and applications
International Journal of Approximate Reasoning, 1987
This book provides an introduction to fuzzy numbers and the operations using them. The basic definitions and operations are clearly presented with many examples. However, despite the title, applications are not covered. A fuzzy number is defined as a fuzzy subset of the reals that is both normal and convex; fuzzy numbers may also be defined over other sets of numbers, including the integers. Fuzzy arithmetic may be regarded as a fuzzy generalization of interval arithmetic, which has been extensively studied. However, the connections between fuzzy arithmetic and interval arithmetic are not acknowledged here. Because a number of the results for fuzzy arithmetic duplicate those previously obtained for interval arithmetic, this is inappropriate. Intervals of confidence are used in Chapter 1 to introduce fuzzy numbers. The extension of basic arithmetic operations to fuzzy numbers is presented. Several restricted sets of fuzzy numbers are defined; these include L-R fuzzy numbers, triangular fuzzy numbers, and trapezoidal fuzzy numbers. A fuzzy number may be combined with a random variable to form a hybrid number. Operations using such hybrid numbers are covered in Chapter 2. Also covered in this chapter are sheaves, or samples, of fuzzy numbers and a measure of dissimilarity between fuzzy numbers referred to as a dissemblance index. Additional classes of fuzzy numbers are described: multidimensional fuzzy numbers and fuzzy numbers whose defining membership functions are either fuzzy or random. Fuzzy versions of modular arithmetic and complex numbers are presented in Chapter 3. Sequences and series of fuzzy numbers are discussed, and fuzzy factorials are defined. Properties of functions of fuzzy numbers are presented, with emphasis on exponential, trigonometric, and hyperbolic functions; derivatives are also mentioned. Several ways to describe and compare fuzzy numbers are covered in Chapter 4. These include deviations, divergences, mean intervals of confidence, agreement indices, and upper and lower bounds. However, the general problems
Algebraic characteristics of extended fuzzy numbers
Information Sciences, 1991
Extended fuzzy numbers, previously called fuzzy intervals, are discussed by using the resolution identity and the extension principle. The regularity and the spread are defined for describing the algebraic properties of extended fuzzy numbers. Arithmetic operations on a-level set intervals are suggested instead of general set operations in order to reduce the amount of computation. A sufficient and necessary condition for solving A + X= C is derived. Tbe exact solution for A + X = C is obtained. Finally, A -A = 0 (a fuzzification of the crisp O), which is a natural extension from the nonfuzzy field, is proved.
A fuzzy distance measure for generalized fuzzy numbers
Data Envelopment Analysis and Decision Science, 2014
In this paper, first of all the distance measure entitled generalized Hausdorff distance is defined between two generalized fuzzy numbers that has been introduced by Chen [3]. Then using another distance and combining it with the generalized Hausdorff distance, a fuzzy distance measure is introduced with the help of fuzzy distance measure proposed in [1] to generalize fuzzy numbers. The concept is illustrated by solving several testing examples.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.