Equality and Identity of Fuzzy Numbers and Fuzzy Arithmetic with Equality Constraints

Abstract In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets.We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.

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Fuzzy Reliability of an Imprecise Failure to Start of an Automobile Using Pseudo-Parabolic Fuzzy Numbers

New Mathematics and Natural Computation ◽

10.1142/s1793005718500205 ◽

2018 ◽

Vol 14 (03) ◽

pp. 323-341 ◽

Cited By ~ 1

Author(s):

F. Abbasi

Keyword(s):

Fuzzy Systems ◽

Fuzzy Numbers ◽

Extension Principle ◽

Fuzzy Arithmetic ◽

Fuzzy Reliability ◽

Fuzzy Environment ◽

Distributed Components ◽

Proposed Model ◽

New Type ◽

The Membership Function

In this paper, we propose the notion of pseudo-parabolic fuzzy numbers and the component failure probabilities are considered as a new type of fuzzy number as pseudo-parabolic to incorporate the uncertainties in the parameter, due to a more realistic estimate of them. Then, we analyze the reliability of fuzzy system (particularly, series and parallel system) with independent and non-identically distributed components, and using the new operations of TA [F. Abbasi et al., Journal of Intelligent and Fuzzy Systems 29 (2015) 851–861], due to the smaller results support, easier calculations and special properties than fuzzy arithmetic operations based on the extension principle (in the domain of the membership function) and the interval arithmetic (in the domain of the [Formula: see text]-cuts). We provide a more realistic fuzzy reliability analysis. Finally, an imprecise failure to start of an automobile is considered in fuzzy environment. The reliability of components of the proposed model is considered as pseudo-parabolic fuzzy numbers.

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EXPERT ESTIMATES AVERAGING BY CONSTRUCTING INTUITIONISTIC FUZZY TRIANGLES

Mathematical Modelling and Analysis ◽

10.3846/13926292.2015.1049970 ◽

2015 ◽

Vol 20 (3) ◽

pp. 409-421 ◽

Cited By ~ 1

Author(s):

Aleksandras Krylovas ◽

Natalja Kosareva

Keyword(s):

Decision Making ◽

Monte Carlo ◽

Monte Carlo Simulations ◽

Fuzzy Numbers ◽

Fuzzy Arithmetic ◽

Intuitionistic Fuzzy Numbers ◽

Intuitionistic Fuzzy ◽

Probability Matrices

The problem of ranking (sorting) of m alternatives is considered when experts evaluate each alternative according to k criteria. Functions of arithmetic and geometric averages are constructed for decision making. We present a generalization of this scheme when there are evaluation matrices of several experts and this information is aggregated in the form of triangular intuitionistic fuzzy numbers. Fuzzy triangles were constructed with different uncertainty levels, experts decision matrices and the number of experts varied from 2 to 5. Moreover, method for construction of experts decision probability matrices is proposed in the paper. Ranking results obtained by performing Monte Carlo simulations. Probabilities of errors are compared for arithmetic, geometric, fuzzy arithmetic and fuzzy geometric averages.

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THREE-PARAMETER FUZZY ARITHMETIC APPROXIMATION OF L-R FUZZY NUMBERS FOR FUZZY NEURAL NETWORKS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488506003959 ◽

2006 ◽

Vol 14 (02) ◽

pp. 211-233 ◽

Cited By ~ 1

Author(s):

HSIAO-FAN WANG ◽

CHING-YI KUO

Keyword(s):

Approximation Method ◽

Fuzzy Numbers ◽

Learning Algorithm ◽

Error Function ◽

Back Propagation ◽

Fuzzy Neural Networks ◽

Fuzzy Arithmetic ◽

Learning Procedure ◽

New Learning ◽

Fuzzy Neural

In this study, we proposed an alternative operation of fuzzy arithmetic on L-R fuzzy numbers by three parameters of mode, left spread and right spread. Then, based on this approximation method, a new learning algorithm of a fully fuzzified neural network was developed in which the L-R fuzzy numbers were considered as the fuzzy signals. While the forward operations of fuzzy signals were based on the proposed three-parameter fuzzy arithmetic approximation method, the backward learning adopted a back-propagation learning procedure with a measurable error function. The learning algorithm was illustrated by an example of the recognition of fuzzy IF-THEN rules. The simulation result showed that the proposed approximation method used in such learning model was efficient and accurate.

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COMBINATION OF FUZZY ARITHMETIC AND A FAST BOUNDARY ELEMENT METHOD FOR ACOUSTIC SIMULATION WITH UNCERTAINTIES

Journal of Computational Acoustics ◽

10.1142/s0218396x09003811 ◽

2009 ◽

Vol 17 (01) ◽

pp. 45-69 ◽

Cited By ~ 2

Author(s):

M. JUNGE ◽

D. BRUNNER ◽

J. BECKER ◽

M. MAESS ◽

J. ROSEIRA ◽

...

Keyword(s):

Boundary Element Method ◽

Boundary Element ◽

Sound Velocity ◽

Fuzzy Numbers ◽

Structural Parameters ◽

Acoustic Properties ◽

Fuzzy Arithmetic ◽

Driving Frequency ◽

Radiated Sound ◽

Element Method

A so-called FuzzBEM methodology for analyzing the influence of uncertain acoustic and structural parameters on the radiated sound field of vibrating structures combining fuzzy arithmetic and fast multipole boundary element method is introduced. Uncertainties in acoustic properties may result from uncertain parameters of the vibrating mechanical structures, e.g. material density or geometry, as well as from uncertainties in the acoustic domain, e.g. sound velocity. The use of the transformation method in the proposed approach allows to employ simulation tools based on the crisp number arithmetic by appropriate preprocessing of the fuzzy numbers modeling the uncertain input parameters and postprocessing of the simulation results to determine the fuzzy numbers for the considered output quantities. In this contribution, the proposed FuzzBEM procedure is applied to a sound radiating, vibrating stiffened cylindrical shell where the investigated uncertainties include the shell wall thickness and the driving frequency of a monofrequency point load and the air density and sound velocity. As exemplary output quantities of acoustic performance, the acoustic pressure at multiple field points and the radiated sound power are evaluated. The proposed coupling of fuzzy arithmetic and acoustic boundary elements yields run times two orders of magnitudes or more longer than a single BEM calculation. Nevertheless, the systematic parameterization obtained by the proposed fuzzy analysis has the potential to reveal input–output relationships difficult to identify with individual conventional BEM simulation runs.

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Solving Discounting Problem Using Piece-Wise Quadratic Fuzzy Numbers

International Journal of Fuzzy System Applications ◽

10.4018/ijfsa.2021100101 ◽

2021 ◽

Vol 10 (4) ◽

pp. 1-13

Author(s):

Hemiden Abd El-Wahed Khalifa ◽

Pavan Kumar

Keyword(s):

Portfolio Selection ◽

Fuzzy Numbers ◽

Interval Number ◽

Fuzzy Arithmetic ◽

Investment Portfolio ◽

Numerical Example ◽

Research Article ◽

Close Interval ◽

Interval Approximation

The discounting problem is one of the important aspects in investment, portfolio selection, purchasing with credit, and many other financial operations. In this paper, a discounting problem using piecewise quadratic fuzzy numbers is proposed. The implementation of piecewise quadratic fuzzy numbers is described based on such operations. Fuzzy arithmetic and interval number arithmetic are used for computation. The close interval approximation of piecewise quadratic fuzzy numbers is used for solving the proposed discounting problem. This research article addresses the discounted investment for Year 1, Year 2, and Year 3. Additionally, the authors determine the cumulative discounted investment for different values of the parameter α ranging from 0 to 1. A discounting problem using piecewise quadratic fuzzy numbers is solved as a numerical example to illustrate the proposed procedure.

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