New Operations on Fuzzy Numbers and Intervals (original) (raw)
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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.
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 interval fuzzy S-implications
Information Sciences, 2010
This paper presents an analysis of interval-valued S-implications and interval-valued automorphisms, showing a way to obtain an interval-valued S-implication from two S-implications, such that the resulting interval-valued S-implication is said to be obtainable. Some consequences of that are: (1) the resulting interval-valued S-implication satisfies the correctness property, and (2) some important properties of usual S-implications are preserved by such interval representations. A relation between S-implications and interval-valued Simplications is outlined, showing that the action of an interval-valued automorphism on an interval-valued S-implication produces another interval-valued S-implication.
Fuzzy numbers, definitions and properties
1994
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.
Lecture Notes in Computer Science, 2005
Algebra of ordered fuzzy numbers (OFN) is defined to handle with fuzzy inputs in a quantitative way, exactly in the same way as with real numbers. Additional two structures: algebraic and normed (topological) are introduced to define a general form of defuzzyfication operators. A useful implementation of a Fuzzy Calculator allows counting with the general type membership relations.
An alternative method of finding the membership of a fuzzy number
In this article, it has been shown that the Dubois-Prade left and right reference functions of a fuzzy number viewing as a distribution function and a complementary distribution function respectively, leads to a very simple alternative method of finding the membership of any function of a fuzzy number. This alternative method has been demonstrated with the help of different examples.
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.
Evaluation and interval approximation of fuzzy quantities
Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology, 2013
In this paper we present a general framework to face the problem of evaluate fuzzy quantities. A fuzzy quantity is a fuzzy set that may be non normal and/or non convex. This new formulation contains as particular cases the ones proposed by Fortemps and Roubens [7], Yager and Filev [12, 13] and follows a completely different approach. It starts with idea of "interval approximation of a fuzzy number" proposed, e.g., in [4, 8, 9].
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