International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 713617, 8 pages

doi:10.1155/2012/713617

Research Article

Solution of Fuzzy Matrix Equation System

Mahmood Otadi and Maryam Mosleh

Department of Mathematics, Islamic Azad University, Firoozkooh Branch, Firoozkooh, Iran

Received 22 March 2012; Revised 30 August 2012; Accepted 30 August 2012

Academic Editor: Soheil Salahshour

Copyright © 2012 Mahmood Otadi and Maryam Mosleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main is to develop a method to solve an arbitrary fuzzy matrix equation system by using the embedding approach. Considering the existing solution to fuzzy matrix equation system is done. To illustrate the proposed model a numerical example is given, and obtained results are discussed.

1. Introduction

The concept of fuzzy numbers and fuzzy arithmetic operations was first introduced by Zadeh [1], Dubois, and Prade [2]. We refer the reader to [3] for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces [4], fuzzy differential equations [5], fuzzy linear systems [6–8], and particle physics [9, 10].

One of the major applications of fuzzy number arithmetic is treating fuzzy linear systems [11–20], several problems in various areas such as economics, engineering, and physics boil down to the solution of a linear system of equations. Friedman et al. [21] introduced a general model for solving a fuzzy linear system whose coefficient matrix is crisp, and the right-hand side column is an arbitrary fuzzy number vector. They used the parametric form of fuzzy numbers and replaced the original fuzzy linear system by a crisp linear system and studied duality in fuzzy linear systems where and are real matrix, the unknown vector is vector consisting of fuzzy numbers, and the constant is vector consisting of fuzzy numbers, in [22]. In [6–8, 23, 24] the authors presented conjugate gradient, LU decomposition method for solving general fuzzy linear systems, or symmetric fuzzy linear systems. Also, Abbasbandy et al. [25] investigated the existence of a minimal solution of general dual fuzzy linear equation system of the form , where and are real matrices, the unknown vector is vector consisting of fuzzy numbers, and the constants and are vectors consisting of fuzzy numbers.

In this paper, we give a new method for solving a fuzzy matrix equation system whose coefficients matrix is crisp, and the right-hand side matrix is an arbitrary fuzzy number matrix by using the embedding method given in Cong-Xin and Min [26] and replace the original fuzzy linear system by two crisp linear systems. It is clear that, in large systems, solving linear system is better than solving linear system. Since perturbation analysis is very important in numerical methods. Recently, Ezzati [27] presented the perturbation analysis for fuzzy linear systems. Now, according to the presented method in this paper, we can investigate perturbation analysis in two crisp matrix equation systems instead of linear system as the authors of Ezzati [27] and Wang et al. [28].

2. Preliminaries

Parametric form of an arbitrary fuzzy number is given in [29] as follows. A fuzzy number in parametric form is a pair of functions , which satisfy the following requirements:

(1) is a bounded left continuous nondecreasing function over , (2) is a bounded left continuous nonincreasing function over , and (3).

The set of all these fuzzy numbers is denoted by which is a complete metric space with Hausdorff distance. A crisp number is simply represented by .

For arbitrary fuzzy numbers , and real number , we may define the addition and the scalar multiplication of fuzzy numbers by using the extension principle as [29]

(a) if and only if and ,(b), and(c)

Definition 2.1. The linear system is as follows:

where the given matrix of coefficients , , is a real matrix, the given , , with the unknowns , is called a fuzzy linear system (FLS). The operations in (2.1) is described in next section.

Here, a numerical method for finding solution [21] of a fuzzy linear system is given.

Definition 2.2 (see [21]). A fuzzy number vector given by

is called a solution of the fuzzy linear system (2.1) if

If, for a particular , for all , we simply get

Finally, we conclude this section by a reviewing on the proposed method for solving fuzzy linear system [21].

The authors [21] wrote the linear system of (2.1) as follows:

where are determined as follows:

and any which is not determined by (2.1) is zero and

The structure of implies that , , and that

where contains the positive entries of , and contains the absolute values of the negative entries of , that is, .

Theorem 2.3 (see [21]). The inverse of nonnegative matrix

is

where

Corollary 2.4 (see [30]). The solution of (2.5) is obtained by

3. Fuzzy Matrix Equation System

A matrix system such as

where , , , are real numbers, the elements in the right-hand matrix are fuzzy numbers, and the unknown elements are ones, is called a fuzzy matrix equation system (FMES).

Using matrix notation, we have

A fuzzy number matrix

is called a solution of the fuzzy matrix system (2.1) if

In this section, we propose a new method for solving FMES.

Theorem 3.1. Suppose that the inverse of matrix exists and is a solution of this equation. Then is the solution of the following systems:

where .

Proof. It is the same as the proof of Theorem 3 in [27].

For solving (3.2), we first solve the following system:

Using matrix notation, we have

Suppose that the solution of (3.7) is as

Let matrices and have defined as Section 2. Now using matrix notation for (3.7), we get in parametric form . We can write this system as follows:

By substituting and in the first and second equation of above system, respectively, we have

therefore, we have

Therefore, we can solve fuzzy matrix equation system (3.2) by solving (3.7)–(3.10).

Theorem 3.2. Let in (3.3) , also and are the number of multiplication operations that are required to calculate

(the proposed method in Friedman et al. [21]) and

from (3.7)–(3.10), respectively. Then and .

Proof. According to Section 2, we have

where

Therefore, for determining , we need to compute and . Now, assume that is matrix and denote by the number of multiplication operations that are required to calculate . It is clear that

and hence

For computing from (3.7) and from (3.10) the number of multiplication operations is and , respectively. Clearly , so

and hence . This proves theorem.

Remark 3.3. In (3.3) if , then this paper is similar to [27].

Example 3.4. Consider the fuzzy matrix equation system as follows:

By using (3.7) and (3.10), we have

and hence

Obviously, and , are fuzzy numbers.

4. Conclusions

In this paper, we propose a general model for solving fuzzy matrix equation system. The original system with matrix coefficient is replaced by two crisp matrix equation systems.

References

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning. I,” Information Sciences, vol. 8, pp. 199–249, 1975.

D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978.

A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold, New York, NY, USA, 1985.

J. H. Park, “Intuitionistic fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1039–1046, 2004.

S. Abbasbandy, J. J. Nieto, and M. Alavi, “Tuning of reachable set in one dimensional fuzzy differential inclusions,” Chaos, Solitons and Fractals, vol. 26, no. 5, pp. 1337–1341, 2005.

S. Abbasbandy, A. Jafarian, and R. Ezzati, “Conjugate gradient method for fuzzy symmetric positive definite system of linear equations,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 1184–1191, 2005.

S. Abbasbandy, R. Ezzati, and A. Jafarian, “LU decomposition method for solving fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 633–643, 2006.

B. Asady, S. Abbasbandy, and M. Alavi, “Fuzzy general linear systems,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 34–40, 2005.

M. S. Elnaschie, “A review of E-infinity theory and the mass spectrum of high energy particle physics,” Chaos, Solitons & Fractals, vol. 19, pp. 209–236, 2004.

M. S. Elnaschie, “The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics,” Chaos, Solitons & Fractals, vol. 22, pp. 495–511, 2004.

T. Allahviranloo, “Numerical methods for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 493–502, 2004.

T. Allahviranloo, “Successive over relaxation iterative method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 162, no. 1, pp. 189–196, 2005.

T. Allahviranloo, “The Adomian decomposition method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 553–563, 2005.

T. Allahviranloo, S. Salahshour, and M. Khezerloo, “Maximal- and minimal symmetric solutions of fully fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4652–4662, 2011.

T. Allahviranloo and S. Salahshour, “Fuzzy symmetric solutions of fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4545–4553, 2011.

T. Allahviranloo, “comment on fuzzy linear systems,” Fuzzy Sets and Systems, vol. 140, no. 3, p. 559, 2003.

M. Otadi and M. Mosleh, “Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network,” Applied Mathematical Modelling, vol. 35, no. 10, pp. 5026–5039, 2011.

T. Allahviranloo and M. Ghanbari, “On the algebraic solution of fuzzy linear systems based on interval theory,” Applied Mathematical Modelling, vol. 36, pp. 5360–5379, 2012.

T. Allahviranloo and S. Salahshour, “Bounded and symmetric solutions of fully fuzzy linear systems in dual form,” Procedia Computer Science, vol. 3, pp. 1494–1498, 2011.

R. Ghanbari and N. Mahdavi-Amiri, “New solutions of LR fuzzy linear systems using ranking functions and ABS algorithms,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3363–3375, 2010.

M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets and Systems, vol. 96, no. 2, pp. 201–209, 1998.

M. Friedman, M. Ming, and A. Kandel, “Duality in fuzzy linear systems,” Fuzzy Sets and Systems, vol. 109, no. 1, pp. 55–58, 2000.

S. Abbasbandy and M. Alavi, “A method for solving fuzzy linear systems,” Iranian Journal of Fuzzy Systems, vol. 2, no. 2, pp. 37–43, 2005.

S. Abbasbandy and M. Alavi, “A new method for solving symmetric fuzzy linear systems,” Mathematics Scientific Journal, Islamic Azad University of Arak, vol. 1, pp. 55–62, 2005.

S. Abbasbandy, M. Otadi, and M. Mosleh, “Minimal solution of general dual fuzzy linear systems,” Chaos, Solitons & Fractals, vol. 37, no. 4, pp. 1113–1124, 2008.

W. Cong-Xin and M. Ming, “Embedding problem of fuzzy number space. I,” Fuzzy Sets and Systems, vol. 44, no. 1, pp. 33–38, 1991.

R. Ezzati, “Solving fuzzy linear systems,” Soft Computing, vol. 15, pp. 193–197, 2011.

K. Wang, G. Chen, and Y. Wei, “Perturbation analysis for a class of fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 54–65, 2009.

M. Ming, M. Friedman, and A. Kandel, “A new fuzzy arithmetic,” Fuzzy Sets and Systems, vol. 108, no. 1, pp. 83–90, 1999.

D. Kincaid and W. Cheney, Numerical Analysis, Brooks/Cole, Pacific Grove, Calif, USA, 1996.