Minimal Solution of Non-square Fuzzy Linear

Available online at www.ispacs.com/jfsva Volume 2012, Year 2012 Article ID jfsva-00105, 10 pages doi:10.5899/2012/jfsva-00105 Research Article

Minimal Solution of Non-square Fuzzy Linear Systems Mahmood Otadi

1∗

, Maryam Mosleh

1

(1) Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

c Mahmood Otadi and Maryam Mosleh. This is an open access article distributed under Copyright 2011 ⃝ the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, the main is to develop a method to solve an arbitrary inconsistent fuzzy linear system by using the embedding approach. Considering the existing of minimal solution to m × n linear fuzzy system is done. To illustrate the proposed model numerical examples are given and obtained results are discussed. Keywords : Fuzzy number; Fuzzy linear systems; Pseudo inverse; Minimal solution.

1

Introduction

The concept of fuzzy numbers and fuzzy arithmetic operations were first introduced by Zadeh [29], Dubois and Prade [16]. We refer the reader to [23] for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces [27], fuzzy differential equations [5], fuzzy linear systems [3, 4, 13] and particle physics [17, 18]. One of the major applications of fuzzy number arithmetic is treating fuzzy linear systems [7, 8, 9, 10, 11, 12, 26], 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. [20] introduced a general model for solving a fuzzy n × n 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 n × n linear system by a crisp 2n × 2n linear system and studied duality in fuzzy linear systems AX = BX + Y where A , B are real n × n matrix, the unknown vector X is vector consisting of n fuzzy numbers and the constant Y is vector consisting of n fuzzy numbers, ∗

Corresponding author. Email address: [email protected], Tel:0989126964202

1

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in [22]. In [1, 2, 3, 4, 13] the authors presented conjugate gradient, LU decomposition method for solving general fuzzy linear systems or symmetric fuzzy linear systems. Also, Abbasbandy et al. [6] investigated the existence of a minimal solution of general dual fuzzy linear equation system of the form A˜ x + f˜ = B x ˜ + c˜, where A, B are real m × n matrices, the unknown vector x ˜ is vector consisting of n fuzzy numbers and the constant f˜, c˜ are vectors consisting of m fuzzy numbers. In this paper, we give a new method for solving a m × n fuzzy linear system whose coefficients matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector by using the embedding method given in Cong-Xin and Min [15] and replace the original m × n fuzzy linear system by two m × n crisp linear systems. It is clear that in large systems, solving m × n linear system is better than solving 2m × 2n linear system. Since perturbation analysis is very important in numerical methods. Recently, Ezzati [19] presented the perturbation analysis for n × n fuzzy linear systems. Now, according to the presented method in this paper, we can investigate perturbation analysis in two m × n crisp linear systems instead of 2m × 2n linear system as the authors of Ezzati [19] and Wang et al. [28].

2

Preliminaries

The minimal solution of an arbitrary linear system is formally defined such that: 1. If the system is consistent and has a unique solution, then this solution is also the minimal solution. 2. If the system is consistent and has a set solution, then the minimal solution is a member of this set that has the least Euclidean norm. 3. If the system is inconsistent and has a unique least squares solution, then this solution is also the minimal solution. 4. If the system is inconsistent and has a least squares set solution, then the minimal solution is a member of this set that has the least Euclidean norm. Parametric form of an arbitrary fuzzy number is given in [25] as follows. A fuzzy number u in parametric form is a pair (u, u) of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfy the following requirements: 1. u(r) is a bounded left continuous non-decreasing function over [0, 1], 2. u(r) is a bounded left continuous non-increasing function over [0, 1], 3. u(r) ≤ u(r), 0 ≤ r ≤ 1. The set of all these fuzzy numbers is denoted by E which is a complete metric space with Hausdorff distance. A crisp number α is simply represented by u(r) = u(r) = α, 0 ≤ r ≤ 1. For arbitrary fuzzy numbers x = (x(r), x(r)), y = (y(r), y(r)) and real number k, we may define the addition and the scalar multiplication of fuzzy numbers by using the extension principle as [25] (a) x = y if and only if x(r) = y(r) and x(r) = y(r),

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(b) x + y = (x(r) + y(r), x(r) + y(r)), { (c) kx =

(kx, kx), (kx, kx),

k ≥ 0, k < 0.

Definition 2.1. The m × n    a11 x1   a21 x1 ..  .    am1 x1

linear system + ··· + ···

+ a12 x2 + a22 x2

+ am2 x2 + · · ·

+ a1n xn + a2n xn .. .

= = .. .

y1 , y2 , .. .

(2.1)

+ amn xn = ym ,

where the given matrix of coefficients A = (aij ), 1 ≤ i ≤ m and 1 ≤ j ≤ n is a real m × n matrix, the given yi ∈ E, 1 ≤ i ≤ m, with the unknowns xj ∈ E, 1 ≤ j ≤ n is called a fuzzy linear system (FLS). The operations in (2.1) is described in next section. In this paper, we assume the matrix A is full rank, i.e., rank(A) = m (for m ≤ n) or rank(A) = n (for n < m). Here, a numerical method for finding minimal solution [6] of a fuzzy m×n linear system based on pseudo-inverse calculation by singular value decomposition, is given when matrix of coefficients has row full rank or column full rank. Definition 2.2. [20], A fuzzy number vector (x1 , x2 , ..., xn )t given by xj = (xj (r), xj (r));

1 ≤ j ≤ n, 0 ≤ r ≤ 1,

is called a solution of the fuzzy linear system (2.1) if  ∑n ∑n  j=1 aij xj = j=1 aij xj = y i ,    ∑n

j=1 aij xj

=

∑n

j=1 aij xj

= yi.

If, for a particular i, aij > 0, for all j, we simply get n ∑

aij xj = y i ,

j=1

n ∑

aij xj = y i .

j=1

Finally, we conclude this section by a reviewing on the proposed method for solving fuzzy linear system in Abbasbandy et al. [6]. The authors of Abbasbandy et al. [6] wrote the linear system of Eq.(2.1) as follows: SX = Y,

(2.2)

where sij are determined as follows: aij ≥ 0 =⇒ sij = aij , si+m,j+n = aij , aij < 0 =⇒ si,j+n = −aij , si+m,j = −aij ,

(2.3)

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and any sij which is not determined by (2.3) is zero and 

  y1 x1  ..   ..  .   .     xn   y    m X= , Y =   −y 1  −x1    ..   . .  .   . −xn −y m

     .    

The structure of S implies that sij ≥ 0, 1 ≤ i ≤ 2m, 1 ≤ j ≤ 2n and that ( ) B C S= , C B where B contains the positive entries of A, and C contains the absolute values of the negative entries of A, i.e., A = B − C. Corollary 2.1. Let T be p × q real column full rank or row full rank. There exists a p × p orthogonal matrix U , a q × q orthogonal matrix V , and a p × q diagonal matrix Σ with ⟨Σ⟩ij = 0 for i ̸= j and ⟨Σ⟩ii = σi > 0 with σ1 ≥ σ2 ≥ · · · ≥ σs > 0, where s = min{p, q}, such that the singular value decomposition T = U ΣV t , is valid. And if Σ+ is that q × p matrix whose only nonzero entries are ⟨Σ+ ⟩ii = 1/σi for 1 ≤ i ≤ s, then T + = V Σ+ U t is the unique pseudo-inverse of T . We refer the reader to [14] for more information on finding pseudo-inverse of an arbitrary matrix, and when we work with full rank matrices, there are not any problem and all calculations are stable and well-posed. Corollary 2.2. The matrix S is row full rank (for m ≤ n) or column full rank (for n < m) if and only if the matrices A = B − C and B + C are both row full rank (for m ≤ n) or column full rank (for n < m). Theorem 2.1. [6], The pseudo-inverse of non-negative full rank matrix ( ) B C S= C B (

is S+ = where

1 D = [(B + C)+ + (B − C)+ ], 2

D E

E D

) ,

(2.4)

1 E = [(B + C)+ − (B − C)+ ]. 2

Corollary 2.3. [24], The minimal solution of (2.2) is obtained by X = S + Y.

(2.5)

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3

The model

In this section, we propose a new method for solving inconsistent fuzzy linear system. Theorem 3.1. Suppose the matrix A in system (2.1) is full rank and x = (x1 , x2 , . . . , xn )T is a minimal solution of this equation. Then x + x = (x1 + x1 , x2 + x2 , . . . , xn + xn )T is the minimal solution of the following system A(x + x) = y + y where y + y = (y 1 + y 1 , y 2 + y 2 , . . . , y n + y n )T . Proof. The same as the proof of Theorem 3 in [19]. For solving Eq. (2.1), we first solve the following system:  a11 (x1 + x1 ) + · · · + a1n (xn + xn ) = (y 1 + y 1 ),        a21 (x1 + x1 ) + · · · + a2n (xn + xn ) = (y + y 2 ),  2 ..    .      am1 (x1 + x1 ) + · · ·

.. .

+ amn (xn + xn ) = (y m + y m ),

and suppose the minimal solution of this system is as:    d1 x1 + x1  d2   x + x2   2  d= . =x+x= .. .  .   . dn

(3.6)

.. .

   . 

xn + xn

Let matrices B and C have defined as Section 2. Now using matrix notation for system (2.1), we get Ax = y and in parametric form (B − C)(x(r) + x(r)) = (y(r) + y(r)). We can write this system as follows: { Bx(r) − Cx(r) = y(r), Bx(r) − Cx(r) = y(r). By substituting of x(r) = d − x(r) and x(r) = d − x(r) in the first and second equation of above system, respectively, we have (B + C)x(r) = y(r) + Cd,

(3.7)

(B + C)x(r) = y(r) + Cd,

(3.8)

x(r) = (B + C)+ (y(r) + Cd)

(3.9)

and therefore, we have

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Journal of Fuzzy Set Valued Analysis

and x(r) = (B + C)+ (y(r) + Cd).

(3.10)

Therefore, we can solve inconsistent fuzzy linear system Eq.(2.1) by solving Eqs. (3.6)(3.7). Theorem 3.2. Assume that g and G are the number of multiplication operations that are required to calculate X = (x1 , x2 , . . . , xn , −x1 , −x2 , . . . , −xn )T = S + Y (the proposed method in Abbasbandy et al. [6]) and X = (x1 , x2 , . . . , xn , x1 , x2 , . . . , xn )T from Eqs. (3.6)-(3.7), respectively. Then G ≤ g and g − G = mn. Proof. According to Section 2, we have ( +

S =

D E E D

) ,

where 1 D = [(B + C)+ + (B − C)+ ], 2

1 E = [(B + C)+ − (B − C)+ ]. 2

Therefore, for determining S + , we need to compute (B + C)+ and (B − C)+ . Now, assume that M is m × n matrix and denote by h(M ) the number of multiplication operations that are required to calculate M + . It is clear that h(S) = h(B + C) + h(B − C) = 2h(A) and hence g = 2h(A) + 4mn. For computing x + x = (x1 + x1 , x2 + x2 , . . . , xn + xn )T from Eq.(3.6) and x = (x1 , x2 , . . . , xn )T from Eq.(3.7) the number of multiplication operations are h(A)+mn and h(B +C)+2mn, respectively. Clearly h(B + C) = h(A), so G = 2h(A) + 3mn and hence g − G = mn. This proves Theorem.

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4

Numerical examples

Example 4.1. Consider the 2 × 3 fuzzy linear system { x1 + x3 = (r, 2 − r), 2x1 − x2 = (2r, 3 − r). By using Eqs. (3.6) and (3.7), we have:    1.3333 + 0.3333r x1 (r) + x1 (r)        x2 (r) + x2 (r)  =  −0.3333 − 0.3333r    x3 (r) + x3 (r) 0.6667 − 0.3333r and

and hence



x1 (r)





0.1111 + 0.7222r

     =  −0.1111 − 0.2222r   −0.1111 + 0.2778r

  , 







   x2 (r)  x3 (r)



1.2222 − 0.3889r

     =  −0.2222 − 0.1111r   0.7778 − 0.6111r

   



   x2 (r)  x3 (r) x1 (r)



  . 

Obviously, x2 is not fuzzy number, therefore the example is not a minimal fuzzy solution. Example 4.2. Consider the 3 × 2 fuzzy linear system   x + x2 = (r, 2 − r),   1  x1 − x2 = (1 + r, 3 − r),     2x + x = (2r, 3 − r). 1 2 By using Eqs. (3.6) and (3.7), we have: ] [ x1 (r) + x1 (r) x2 (r) + x2 (r) and

[

x1 (r)

]

[ =

[

x1 (r)

]

[ =

x2 (r)

=

2.4286 + 0.2857r

]

−1.2857 + 0.1428r

0.7143 + 0.6429r

] ,

−1.1428 + 0.5714r

x2 (r) and hence

[

1.7143 − 0.3571r −0.1429 − 0.4286r

] .

According to this fact that xi ≤ xi , i = 1, 2, are monotonic decreasing functions then the minimal fuzzy solution is (xi , xi ), i = 1, 2.

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Conclusion

In this paper, we propose a general model for solving a system of m fuzzy linear equations with n variables. The original system with matrix coefficient A is replaced by two m × n crisp linear systems.

Acknowledgment We would like to present our sincere thanks to the Editor in Chief, Prof. T. Allahviranloo for their valuable suggestions.

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