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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.
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.
The new concept of generalized fuzzy numbersis proposed and treated in this paper by removing the “normality” on the definition of fuzzy numbers. We introduce the notion of isosceles triangle fuzzy number. We also discuss about the algebra of this fuzzy numbers. Ranking of fuzzy numbers play an important role in decision making. Though it is difficult to rank between fuzzy numbers like crisp. In this paper we got some inequality relations to obviate this problem. A new concept ranking index, is defined and employed to describe this method. Some relevant numerical examples are also included.
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.
2012
The fuzzy set theory has been applied in many fields such as operation research, control theory and management sciences etc. The fuzzy numbers and fuzzy values are widely used in engineering applications because of their suitability for representing uncertain information. In standard fuzzy arithmetic operations we have some problem in subtraction and division operations. In this paper, a new operation on Triangular Fuzzy Numbers is defined, where the method of subtraction and division has been modified. These modified operators yield the exact inverse of the addition and multiplication operators.
International Journal of Intelligent Computing and Cybernetics, 2013
PurposeThe purpose of this paper is to study a nascent theory and an emerging concept of solving a fully fuzzy linear system (FFLS) with no non negative restrictions on the triangular fuzzy numbers chosen as parameters. Two new simplified computational methods are proposed to solve a FFLS without any sign restrictions. The first method eliminates the non‐negativity constraint from the coefficient matrix while the second method eliminates the constraint of non‐negativity on the solution vector. The methods are introduced with an objective to broaden the domain of fuzzy linear systems to encompass a wide range of problems occurring in reality.Design/methodology/approachThe design of numerical methods is motivated by decomposing the fuzzy based linear system into its equivalent crisp linear form which can be further solved by variety of classical methods to solve a crisp linear system. Further the paper investigates Schur complement technique to solve the crisp equivalent of the FFLS.F...
2014
The global solution of a fuzzy linear system contains the crisp vector solution of a real linear system. So discussion about the global solution of a n\times n fuzzy linear system A\tilde{x}=\tilde{b} with a fuzzy number vector b in the right hand side and crisp a coefficient matrix A is considered. The advantage of the paper is developing a new algorithm to find the solution of such system by considering a global solution based upon the concept of a convex fuzzy numbers. At first the existence and uniqueness of the solution are introduced and then the related theorems and properties about the solution are proved in details. Finally the method is illustrated by solving some numerical examples.
Computational Mathematics and Modeling, 2015
This paper proposes a new computational method to obtain a positive solution for arbitrary fully fuzzy linear system (FFLS). The new method transforms the coefficients in FFLS to a one-block matrix. As a result, none of the fuzzy operations are needed. This method can provide a solution regardless of the size of a system. Some necessary theorems are proved and new numerical examples are presented to illustrate the proposed method.
International Journal of Fuzzy Logic and Intelligent System, 2022
This study proposes a new straightforward approach for solving a fully fuzzy algebraic system of linear equations. The proposed technique was based on a convex combination approach. Here, elements of the fuzzy coefficients matrix, fuzzy unknown vector, and right-hand side fuzzy vector are considered non-negative. Using the fuzzy arithmetic and convex combination concepts, the fuzzy system is converted into an equivalent crisp system. After solving the corresponding system for any two distinct parametric values of the convex combination, the final solution is obtained. Various example problems were solved and compared with existing results for validation.
Information Sciences, 1991
This paper deals with algebraic equations involving generalized fuzzy numbers with continuous membership functions. The generalized fuzzy number defined in this paper is a general name for fuzzy numbers, fuzzy intervals, crisp numbers, and interval numbers. Tbree important properties of the generalized fuzzy numbers with continuous membership functions are introduced as the basis for deriving the solvability criterion. The sufficient and necessary condition for solving A & X = C is derived. The sufficient and necessary condition for solving A X X = C and A + X = C when A and C are nonzero generalized fuzzy numbers is also derived in this paper. Instead of the inverse operation method, which fails for fuzzy sets, inverse operations on the boundary points of cy-level set intervals for continuous membership functions are used when the derived criteria of solvability are satisfied. A unique solution rather than a solution set is obtained.
Sadhana, 2015
This paper proposes two new methods to solve fully fuzzy system of linear equations. The fuzzy system has been converted to a crisp system of linear equations by using single and double parametric form of fuzzy numbers to obtain the non-negative solution. Double parametric form of fuzzy numbers is defined and applied for the first time in this paper for the present analysis. Using single parametric form, the n×n fully fuzzy system of linear equations have been converted to a 2n×2n crisp system of linear equations. On the other hand, double parametric form of fuzzy numbers converts the n × n fully fuzzy system of linear equations to a crisp system of same order. Triangular and trapezoidal convex normalized fuzzy sets are used for the present analysis. Known example problems are solved to illustrate the efficacy and reliability of the proposed methods.
Journal of Intelligent & Fuzzy Systems, 2017
In this paper, Fully Fuzzy Linear Equation System (FFLS) that all parameters and variables are represented by triangular fuzzy numbers is discussed. FFLS has many important applications to branches of science, engineering and other disciplines. The objective of this paper is to find the feasible (strong) and approximate solution with a proposed method based on a mixed integer modeling of the nonsquare FFLS by removing all restrictions on the parameters and variables. The method is illustrated with numerical examples. Results of the numerical examples show that this method has the ability to generate a feasible (strong) and an approximate fuzzy solution and also to indicate no solution case of a nonsquare or square FFLS.
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:
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.
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.
Anais do VI Workshop-Escola de Informática Teórica (WEIT 2021), 2021
This work deals with the study of a new total order Triangular Fuzzy Numbers and arithmetic properties that are maintained in relation to the operations of addition and subtraction. Additionally, we present as example of application the shortest path solution for the Travelling Salesman Problem with fuzzy distances. Resumo. Este trabalho trata do estudo de uma nova ordem total para números fuzzy triangulares e propriedades aritméticas que são mantidas em relaçãoàs operações de adição e subtração. Além disso, apresentamos como exemplo de aplicação a solução do caminho mais curto para o Problema do Caixeiro Viajante com distâncias fuzzy.
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.
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.
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
The use of new concept of fuzzy number as an equivalence class on for solving fuzzy equation is proposed. This approach of solving fuzzy equations by new concept of fuzzy number has a considerable advantage. Through theoretical analysis, an illustrative example and computational results the paper shows that the proposed approach is more general and straightforward for solving fuzzy algebraic equations.
Fuzzy Sets and Systems, 2011
The aim of this paper is to analyse the solution of a fuzzy system when the classical solution based on standard fuzzy mathematics fails to exist. In particular we analyse the solution of the system Ax=b with A squared matrix with positive fuzzy coefficients and y crisp vector of positive elements. This system is particularly important for financial applications. We propose two different solution methods that are based respectively on the work of and Friedman, . An application to an important financial problem, the derivation of the artificial probabilities in a lattice framework, is provided.
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