Numerical methods for fuzzy system of linear equations

doi:10.1016/S0096-3003(03)00793-8

Applied Mathematics and Computation

Volume 155, Issue 2, 6 August 2004, Pages 493-502

https://doi.org/10.1016/S0096-3003(03)00793-8 Get rights and content

Abstract

In this paper numerical algorithms for solving `fuzzy system of linear equations' (FSLE) are considered. Schemes based on the iterative Jacobi and Gauss Sidel methods in detail are discussed and there are followed by convergence theorems. Algorithms are illustrated by solving some numerical examples.

Introduction

Systems of simulations linear equations play major role in various areas such as mathematics, physics, statistics, engineering and social sciences. Since in many applications at least some of the system's parameters and measurements are represented by fuzzy rather than crisp numbers, it is important to develop mathematical models and numerical procedures that would appropriately treat general fuzzy linear systems and solve them. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [6], [2] and … One of the major applications using fuzzy number arithmetic is treating linear systems their parameters are all or partially represented by fuzzy numbers [1], [4] and … A general model for solving an n×n FLSE which coefficient matrix is crisp and the right hand side is arbitrary fuzzy number vector were first proposed by Friedman et al. [3]. They used the embedding method and replaced the original n×n fuzzy linear system AX=Y by a 2n×2n crisp linear system with a matrix S which may be singular even if A be nonsingular. The structure of this paper is organized as follows.

In Section 2, we bring some basic definitions and results on fuzzy numbers and fuzzy system of linear equation and (FSLE). In Section 3, we apply the iterative Jacobi and Gauss Sidel methods for solving FSLE with convergence theorems. The proposed algorithms are illustrated by solving some examples in Section 4 and conclusions are drawn in Section 5.

Section snippets

Preliminaries

We represent an arbitrary fuzzy number by an ordered pair of functions $(u (r), u (r)),0⩽r⩽1$ , which satisfy the following requirements:

1.
$u (r)$ is a bounded left continuous nondecreasing function over [0,1].
2.
$u (r)$ is a bounded left continuous nonincreasing function over [0,1].
3.
$u (r)⩽ u (r)$ , 0⩽r⩽1.

A crisp number α is simply represented by $u (r)= u (r)=α,0⩽r⩽1$ . By appropriate definitions the fuzzy numbers space ${u (r), u (r)}$ becomes a convex cone E¹ which is then embedded isomorphically and isometrically in to a

The Jacobi and Gauss Sidel iterative techniques

First we are going to proof the following theorems.

Theorem 3.1

Let S be nonsingular. Then the unique solution X of Eq. (2.5) is always a fuzzy vector for arbitrary vector Y, if S⁻¹ is nonnegative.

Proof

It is sufficient to show that Definition 2.2 is hold for X. It is clear that S⁻¹ have the same structure like S, i.e. $S^{−1} = T_{1} T_{2} T_{2} T_{1},$ from X=S⁻¹Y, we have $T_{1} Y −T_{2} Y = X, −T_{2} Y +T_{1} Y = X .$ Thus $(X − X)=(T_{1} +T_{2})(Y − Y)⩾0,$ because (S⁻¹)_ij=t_ij⩾0 and $(Y − Y)⩾0$ . Since $Y$ is monotonically decreasing and $Y$ is monotonically increasing, Eq. (3.1) is

Examples

Example 4.1

Consider the 2 × 2 fuzzy system $x_{1} −x_{2} =(r,2−r), x_{1} +3x_{2} =(4+r,7−2r).$

The extended 4 × 4 matrix is $S= 100 −1 13000 −1 100013,$ $X= x_{1} (r) x_{2} (r) x_{1} (r) x_{2} (r) =S^{−1} Y= +1.1250 −0.1250 −0.3750 +0.3750 −0.3750 −0.3750 −0.3750 −0.1250 −0.3750 +0.3750 +0.3750 −0.1250 +0.1250 −0.1250 −0.3750 +0.3750 r 4+r 2−r 7−2r .$

The exact solutions are $x_{1} =(x_{1} (r), x_{1} (r))=(1.375+0.625r,2.875−0.875r),$ $x_{2} =(x_{2} (r), x_{2} (r))=(0.875+0.125r,1.375−0.375r).$

The exact and approximated solutions are plotted and compared in Fig. 1.

Example 4.2

Consider the 3 × 3 fuzzy system $4x_{1} +x_{2} −x_{3} =(r,2−r), −x_{1} +3x_{2}$

Conclusion

In this work we apply Jacobi and Gauss Sidel iterative methods for approximate of the unique solution of FSLE, since solving FSLE as analytically on the hole is difficult. We have assumed that the proposed matrix S by Friedman et al. [3] be nonsingular and a_ii>0, then we have proved that the matrix A is strictly diagonally dominant iff S be strictly diagonally dominant. If the unique solution of SX=Y is a strong or weak fuzzy number, then the approximate solution of iterative method also would

References (6)

J.J Buckley
Fuzzy eigenvalues and input–output analysis
FSS
(1990)
L.A Zadeh
The concept of a linguistic variable and its application to approximate reasoning
Inform. Sci.
(1975)
M Friedman et al.
Fuzzy linear systems
FSS
(1998)

There are more references available in the full text version of this article.

Cited by (167)

General strong fuzzy solutions of fuzzy Sylvester matrix equations involving the BT inverse
2024, Fuzzy Sets and Systems
This paper aims to solve the fuzzy Sylvester matrix equation (FSME) $A \tilde{X} + \tilde{X} B = \tilde{C}$ , in which A, B are $n \times n$ and $m \times m$ crisp matrices, and $\tilde{C}$ is a $n \times m$ fuzzy matrix. Using the Kronecker product, we transform the FSME into an $m n \times m n$ fuzzy linear system. Firstly, we obtain the necessary and sufficient conditions for the existence of strong fuzzy solutions to the FSME by using the BT inverse of the coefficient matrix. Secondly, we investigate the existence of a nonnegative BT inverse by using its block structure. Next, we derive general strong fuzzy solutions to a class of FSME and establish an algorithm to obtain general strong fuzzy solutions to the FSME by the BT inverse. Finally, we give three numerical examples and an applied example in the economy to illustrate the proposed methods.
An algorithm for solving a system of linear equations with Z-numbers based on the neural network approach
2024, Journal of Intelligent and Fuzzy Systems
Triangular intuitionistic fuzzy linear system of equations with applications: an analytical approach
2024, Applied Mathematics in Science and Engineering
Generalized fuzzy eigenvectors of real symmetric matrices
2023, Journal of Intelligent and Fuzzy Systems
A New Method Based on Jacobi Iteration for Fuzzy Linear Systems
2023, Thai Journal of Mathematics
Fuzzy Introductory Concepts
2023, Studies in Computational Intelligence

View all citing articles on Scopus

View full text