In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.
Systems of simultaneous matrix equations are essential mathematical tools in science and technology. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. So, it is very important to develop a numerical procedure that would appropriately handle and solve fuzzy matrix systems. The concept of fuzzy numbers and arithmetic operations were first introduced and investigated by Zadeh [1] and Dubois[2].
Since M. Friedman et al. [3] proposed a general model for solving a n × n fuzzy linear systems whose coefficients matrix is crisp and the right-hand side is a fuzzy number vector in 1998, many works have been done about how to deal with some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), fully fuzzy linear systems (FFLS), dual fully fuzzy linear systems (DFFLS) and general dual fuzzy linear systems (GDFLS), see [4-9]. However, for a fuzzy linear matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past decades. In 2010, Gong Zt [10,11]investigated a class of fuzzy matrix equations by means of the undetermined coefficients method, and studied least squares solutions of the inconsistent fuzzy matrix equation by using generalized inverses. In 2011, Guo X. B. [12] studied the minimal fuzzy solution of fuzzy Sylvester matrix equations. Recently, they [13] considered the fuzzy symmetric solutions of fuzzy matrix equations.
The LR fuzzy number and its operations were firstly introduced by Dubois [2]. In 2006, Dehgham et al. [6] discussed the computational methods for fully fuzzy linear systems whose coefficient matrix and the right-hand side vector are denoted by LR fuzzy numbers. In this paper, we propose a practical method for solving a class of fuzzy matrix system in which A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix. In contrast, the contribution of this paper is to generalize Dubois’ definition and arithmetic operation of LR fuzzy numbers and then use this result to solve fuzzy matrix systems numerically. The importance of converting fuzzy linear system into two systems of linear equations is that any numerical approach suitable for system of linear equations may be implemented. In addition, since our model does not contain parameter r, , its numerical computation is relatively easy.
2. Preliminaries
Definition 2.1. [2] A fuzzy number is said to be a LR fuzzy number if
where m is the mean value of, and and are left and right spreads, respectively. The function, which is called left shape function satisfying: 1); 2) and; 3) is a non increasing on.
The definition of a right shape function is usually similar to that of. A LR fuzzy number is symbolically shown as.
Noticing that in Definition 2.1, which limits its applications, we extend the definition of LR fuzzy numbers as follows.
Definition 2.2. (Generalized LR fuzzy numbers) Let, we define
1) if and, then
, and
2) if and, then
, and
3) if and, then, and
For arbitrary LR fuzzy number and, we have
1)
2)
Definition 2.3. The matrix system
where are crisp numbers and are LR fuzzy numbers, is called a LR fuzzy matrix equation (LRFME).
Using matrix notation, we have
A LR fuzzy numbers matrix
, ,
,
is called a solution of the LR fuzzy matrix systems if satisfies (2).
3. Method for Solving LRFME
In this section we investigate the LR fuzzy matrix system (2). Firstly, we propose a model for solving the LR fuzzy matrix system, i.e., convert it into two crisp systems of matrix equations. Then we define the LR fuzzy solution and give its solution representation to the original fuzzy matrix system. At last, the existence condition of the strong LR fuzzy solution to the original fuzzy matrix system is also discussed.
3.1. Extended Crisp Matrix Equations
By using arithmetic operations of LR fuzzy numbers, we extend the LR fuzzy matrix Equation (2) into two crisp matrix equations.
Theorem 3.1. The LR dual fuzzy linear Equation (2) can be extended into two crisp systems of linear equations as follows:
i.e.,
and
i.e.,
where , are determined as follows:
If, then,; if, then, , and any which is not determined by the above items is zero, ,.
Proof. Let
and
.
Then the fuzzy matrix Equation (1) can be rewritten in the block forms
Thus the original system (1) is equivalent to the following fuzzy linear equations
Now we consider the Equations (4). Let be the ith row of matrix A, , we can represent in the form,.
Denoting and
, we have
.
i.e.,
Consider the given LR fuzzy vector
we can write the system (2) as
Suppose the system, has a solution. Then, the corresponding mean value
of the solution must lie in the following linear system
Meanwhile, the left spread and the right spread of the solution can be derived from solving the following crisp linear system
Finally, we restore the Equation (5) and obtain above matrix Equations (3) and (4).
The proof is completed.
3.2. Computing Model Matrix Equations
In order to solve the original fuzzy linear Equation (2), we need to consider crisp matrix Equations (3) and (4). Since Equations (3) and (4) are crisp, their computation is relatively easy.
In general [14], the minimal solutions of matrix systems (3) and (4) can be expressed uniformly by
and
respectively, no matter the Equations (3) and (4) are consistent or not.
It seems that we have obtained the solution of the original fuzzy linear system (2) as follows:
But the solution vector may still not be an appropriate LR fuzzy numbers vector except for. So we give the definition of the minimal LR fuzzy solution to the Equation (2) as follows:
Definition 3.1. Let
. If is the minimal solution of Equation (3), and are minimal solution of Equation (4) such that, , then we call is a strong LR fuzzy solution of Equation (2). Otherwise, it is a weak LR fuzzy solution of Equation (2) given by
3.3. A Sufficient Condition of Strong Fuzzy Solution
The key points to make the solution vector being a LR fuzzy solution are and. Since
, we know that the non negativities of and are equivalent to the condition now that is known.
By the above analysis, we have the following result.
Theorem 3.2. Let A belong to. If is nonnegative, the solution of the LR fuzzy matrix system (2) is expressed by
and it admits a strong minimal LR fuzzy solution.
The following Theorem gives a result for such to be nonnegative.
Theorem 3.3. [15] Let S be an 2p × 2p nonnegative matrix with rank r. Then the following assertions are equivalent:
1);
2) There exists a permutation matrix P, such that PS has the form
where each has rank 1 and the rows of are orthogonal to the rows of, whenever, the zero matrix may be absent.
3)
for some positive diagonal matrix. In this case,
.
4. Numerical Examples
In this section, we work out two numerical examples to illustrate the proposed method.
Example 4.1. Consider the fuzzy matrix systems:
.
The coefficient matrix A is nonsingular and the extended matrix S is singular. By the Theorem 3.1., the mean value x, the left spread and the right spread of solution are obtained from
and
.
Thus, we have
and
.
Since is negative, according to Definition 3.1., the LR fuzzy approximate solution of the original fuzzy matrix system is
and it admits a strong LR fuzzy solution.
Example 4.2. Consider the following matrix systems:
.
By the Theorem 3.1., the mean value x of solution lies in the following crisp matrix system
.
Meanwhile, the left spread and the right spread of solution are obtained by solving the following crisp matrix system
.
Thus, we have
and
By Definition 3.1., the original fuzzy system has a strong LR fuzzy approximate solution given by
.
5. Conclusion
In this work we proposed a general model for solving the fuzzy matrix equation where A is an m × n crisp matrix and is a n × p arbitrary LR fuzzy numbers matrix. We converted the fuzzy matrix system into two crisp matrix equations and obtained the LR fuzzy solution to the original fuzzy system by solving crisp matrix equations. Moreover, the existence condition of strong LR fuzzy solution was studied. Numerical examples showed that our method is effective to solve LR fuzzy matrix equations.
6. Acknowledgements
The authors are very thankful for the reviewer’s helpful suggestions to improve the paper. This research was financially supported by the National Natural Science Foundation of China (Grant No. 71061013) and the Youth Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN-11-20).
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doi:10.1016/S0019-9958(65)90241-XD. Dubois, and H. Prade, “Operations on Fuzzy Numbers,” International Journal of Systems Science, Vol. 9, No. 3, 1978, pp. 613-626. doi:10.1080/00207727808941724M. Friedman, M. Ma and A. Kandel, “Fuzzy Linear Systems,” Fuzzy Sets and Systems, Vol. 96, No. 2, 1998, pp. 201-209. doi:10.1016/S0165-0114(96)00270-9M. Friedman, M. Ma and A. Kandel, “Duality in Fuzzy Linear Systems,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 55-58. doi:10.1016/S0165-0114(98)00102-XT. Allahviranloo, “Numerical Methods for Fuzzy System of Linear Equations,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 493-502.
doi:10.1016/S0096-3003(03)00793-8B. Asady, S. Abbasbandy and M. Alavi, “Fuzzy General Linear Systems,” Applied Mathematics and Computation, Vol. 169, No. 1, 2005, pp. 34-40.
doi:10.1016/j.amc.2004.10.042K. Wang and B. Zheng, “Inconsistent Fuzzy Linear Systems,” Applied Mathematics and Computation, Vol. 181, No. 2, 2006, pp. 973-981. doi:10.1016/j.amc.2006.02.019M. Dehghan, B. Hashemi and M. Ghatee, “Solution of the Full Fuzzy Linear Systems Using Iterative Techniques,” Chaos, Solitons & Fractals, Vol. 34, No. 2, 2007, pp. 316-336. doi:10.1016/j.chaos.2006.03.085S. Abbasbandy, M. Otadi and M. Mosleh, “Minimal Solution of General Dual Fuzzy Linear Systems,” Chaos, Solitons & Fractals, Vol. 37, No. 4, 2008, pp. 638-652.
doi:10.1016/j.chaos.2006.10.045X. B. Guo and Z. T. Gong, “Undetermined Coefficient Method for Solving Semi-Fuzzy Matrix Equation,” Proceedings of 9th International Conference on Machine Learning and Cybernetics, Qingdao, 11-14 July 2010, pp. 596-600.Z. T. Gong and X. B. Guo, “Inconsistent Fuzzy Matrix Equations and Its Fuzzy Least Squares Solutions,” Applied Mathematical Modelling, Vol. 35, No. 1, 2011, pp. 1456-1469. doi:10.1016/j.apm.2010.09.022X. B. Guo, “Approximate Solution of Fuzzy Sylvester Matrix Equations,” The 7th International Conference on Computational Intelligence and Security, Sanya, 3-4 December 2011, pp. 52-57.X. B. Guo and D. Q. Shang, “Fuzzy Symmetric Solutions of Fuzzy Matrix Equations,” Advances in Fuzzy Systems, 2012, Article ID: 318069. doi:10.1155/2012/318069X. D. Zhang, “Matrix Analysis and Its Applications,” Tsinghua University and Springer Press, Beijing, 2004.A. Berman and R. J. Plemmons, “Nonnegative Matrices in the Mathematical Sciences,” Academic Press, New York, 1979.
by means of the undetermined coefficients method, and studied least squares solutions of the inconsistent fuzzy matrix equation by using generalized inverses. In 2011, Guo X. B. [
. Recently, they [
.
in which A is an m × n crisp matrix and 
is an m × p arbitrary LR fuzzy numbers matrix. In contrast, the contribution of this paper is to generalize Dubois’ definition and arithmetic operation of LR fuzzy numbers and then use this result to solve fuzzy matrix systems numerically. The importance of converting fuzzy linear system into two systems of linear equations is that any numerical approach suitable for system of linear equations may be implemented. In addition, since our model does not contain parameter r, 
, its numerical computation is relatively easy.
is said to be a LR fuzzy number if
, and 
and
are left and right spreads, respectively. The function
, which is called left shape function satisfying: 1)
; 2) 
and
; 3) 
is a non increasing on
.
is usually similar to that of
. A LR fuzzy number
is symbolically shown as
.
in Definition 2.1, which limits its applications, we extend the definition of LR fuzzy numbers as follows.
, we define
and
, then
, and
and
, then
, and
and
, then
, and
and
, we have
are crisp numbers and 
are LR fuzzy numbers, is called a LR fuzzy matrix equation (LRFME).
, 
,
, 
satisfies (2).
, 
are determined as follows:
, then
,
; if
, then
, 
, and any 
which is not determined by the above items is zero, 
,
. 
.
Thus the original system (1) is equivalent to the following fuzzy linear equations 
be the ith row of matrix A, 
, we can represent 
in the form
,
.
and
, we have
.
we can write the system (2) as
, 
has a solution. Then, the corresponding mean value
of the solution must lie in the following linear system
and the right spread 
of the solution can be derived from solving the following crisp linear system
. So we give the definition of the minimal LR fuzzy solution to the Equation (2) as follows:
. If 
is the minimal solution of Equation (3), 
and 
are minimal solution of Equation (4) such that
, 
, then we call 
is a strong LR fuzzy solution of Equation (2). Otherwise, it is a weak LR fuzzy solution of Equation (2) given by
and
. Since
, 
we know that the non negativities of 
and 
are equivalent to the condition 
now that 
is known.
. If 
is nonnegative, the solution of the LR fuzzy matrix system (2) is expressed by
to be nonnegative.
;
where each 
has rank 1 and the rows of 
are orthogonal to the rows of
, whenever
, the zero matrix may be absent.
. In this case,
.
.
and the right spread 
of solution are obtained from
.
.
is negative, according to Definition 3.1., the LR fuzzy approximate solution of the original fuzzy matrix system is
.
.
and the right spread 
of solution are obtained by solving the following crisp matrix system
.
.
where A is an m × n crisp matrix and 
is a n × p arbitrary LR fuzzy numbers matrix. We converted the fuzzy matrix system into two crisp matrix equations and obtained the LR fuzzy solution to the original fuzzy system by solving crisp matrix equations. Moreover, the existence condition of strong LR fuzzy solution was studied. Numerical examples showed that our method is effective to solve LR fuzzy matrix equations.