An Application of Interval Arithmetic for Solving Fully Fuzzy Linear Systems with Trapezoidal Fuzzy Numbers

Conjugate gradient is an iterative method that solves a linear system Ax=b, where A is a positive definite matrix. We present this new iterative method for solving linear interval systems Ãx̃=b̃, where Ã is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of Ãx̃ and b̃ at every step while the norm is sufficiently small. In addition, we present another iterative method that solves Ãx̃=b̃, where Ã is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution x̃ for linear interval systems, where Ãx̃=b̃; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.

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“…In the following two theorems, we show that operations "⊖" and "⊘" are inverse of operations "+" and "×," respectively [34].…”

Section: Methodsmentioning

confidence: 91%

Two Iterative Methods for Solving Linear Interval Systems

Siahlooei

Fazeli

2018

Applied Computational Intelligence and Soft Computing

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“…The multiplication of two fuzzy sets could be achieved using interval arithmetic, as follow [54][55][56]:…”

Section: Proposed Fuzzy-based Approachmentioning

confidence: 99%

Improving Human Reliability Analysis for Railway Systems Using Fuzzy Logic

Ciani¹,

Guidi

Patrizi

et al. 2021

IEEE Access

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The International Union of Railway provides an annually safety report highlighting that human factor is one of the main causes of railway accidents every year. Consequently, the study of human reliability is fundamental, and it must be included within a complete reliability assessment for every railway-related system. However, currently RARA (Railway Action Reliability Assessment) is the only approach available in literature that considers human task specifically customized for railway applications. The main disadvantages of RARA are the impact of expert's subjectivity and the difficulty of a numerical assessment for the model parameters in absence of an exhaustive error and accident database. This manuscript introduces an innovative fuzzy method for the assessment of human factor in safety-critical systems for railway applications to address the problems highlighted above. Fuzzy logic allows to simplify the assessment of the model parameters by means of linguistic variables more resemblant to human cognitive process. Moreover, it deals with uncertain and incomplete data much better than classical deterministic approach and it minimizes the subjectivity of the analyst evaluation. The output of the proposed algorithm is the result of a fuzzy interval arithmetic, α-cut theory and centroid defuzzification procedure. The proposed method has been applied to the human operations carried out on a railway signaling system. Four human tasks and two scenarios have been simulated to analyze the performance of the proposed algorithm. Finally, the results of the method are compared with the classical RARA procedure underline compliant results obtain with a simpler, less complex and more intuitive approach.

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“…There are many authors have written about developing fuzzy analysis, significantly solving fully fuzzy linear and nonlinear systems. Different methods have been used to solve these two types of fully fuzzy systems, including a method for computing the positive solution of a fully fuzzy linear system (Ezzati et al, 2012); a method used to solve a fully fuzzy linear system via decomposing the symmetric coefficient matrix into two equations systems with the Cholesky method (Senthilkumar and Rajendran, 2011); the Jacobi iteration method for solving a fully fuzzy linear system with fuzzy arithmetic (Marzuki, 2015) and triangular fuzzy number (Megarani et al, 2022); a method for finding a positive solution for an arbitrary fully fuzzy linear system with a one-block matrix (Malkawi et al, 2015a); the singular value decomposition method for solving a fully fuzzy linear system (Marzuki et al, 2018); the Gauss-Seidel method for solving a fully fuzzy linear system via alternative multi-playing triangular fuzzy numbers (Deswita and Mashadi, 2019); the Jacobi, Gauss-Seidel, and SOR iterative methods for solving linear fuzzy systems (Inearat and Qatanani, 2018);a linear programming approach utilizing equality constraints to find non-negative fuzzy numbers (Otadi and Mosleh, 2012); combining interval arithmetic with trapezoidal fuzzy numbers to solve a fully fuzzy linear system (Siahlooei and Fazeli, 2018); using an ST decomposition with trapezoidal fuzzy numbers to solve dual fully fuzzy linear systems via alternative fuzzy algebra (Safitri and Mashadi, 2019), using LU factorizations of coefficient matrices for trapezoidal fuzzy numbers to solve dual fully fuzzy linear systems (Marni et al, 2018); combining QR decomposition with trapezoidal fuzzy numbers (Gemawati et al, 2018); and using an ST decomposition with trapezoidal fuzzy numbers to solve a dual fully fuzzy linear system (Jafarian, 2016). In the case of fully fuzzy linear matrix equations, several studies have identified methods for solving them, including a method that utilizes fully fuzzy Sylvester matrix equations (Daud et al, 2018;Elsayed et al, 2022;He et al, 2018;Malkawi et al, 2015b), and a method that finds fuzzy approximate solutions (Guo and Shang, 2013).…”

Section: Introductionmentioning

confidence: 99%

Computational Mathematics: Solving Dual Fully Fuzzy Nonlinear Matrix Equations Numerically using Broyden’s Method

Zakaria

Megarani

Faisol

et al. 2023

Int. j. math. eng. manag. sci.

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Fuzzy numbers have many applications in various mathematical models in both linear and nonlinear forms. In the form of a nonlinear system, fuzzy nonlinear equations can be constructed in the form of matrix equations. Unfortunately, the matrix equations used to solve the problem of dual fully fuzzy nonlinear systems are still relatively few found in the publication of research results. This article attempts to solve a dual fully fuzzy nonlinear equation system involving triangular fuzzy numbers using Broyden’s method. This article provides the pseudocode algorithm and the implementation of the algorithm into the MATLAB program for the iteration process to be carried out quickly and easily. The performance of the given algorithm is the fastest in finding system solutions and provides a minimum error value.

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An Application of Interval Arithmetic for Solving Fully Fuzzy Linear Systems with Trapezoidal Fuzzy Numbers

Cited by 10 publications

References 23 publications

Two Iterative Methods for Solving Linear Interval Systems

Two Iterative Methods for Solving Linear Interval Systems

Improving Human Reliability Analysis for Railway Systems Using Fuzzy Logic

Computational Mathematics: Solving Dual Fully Fuzzy Nonlinear Matrix Equations Numerically using Broyden’s Method

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