Advances in Matrices, Finite and Infinite, with ApplicationsView this Special Issue
Fuzzy Approximate Solution of Positive Fully Fuzzy Linear Matrix Equations
The fuzzy matrix equations in which , , and are , , and nonnegative LR fuzzy numbers matrices, respectively, are investigated. The fuzzy matrix systems is extended into three crisp systems of linear matrix equations according to arithmetic operations of LR fuzzy numbers. Based on pseudoinverse of matrix, the fuzzy approximate solution of original fuzzy systems is obtained by solving the crisp linear matrix systems. In addition, the existence condition of nonnegative fuzzy solution is discussed. Two examples are calculated to illustrate the proposed method.
Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modeling, which can formulate uncertainty in actual environment. In many matrix equations, some or all of the system parameters are vague or imprecise, and fuzzy mathematics is a better tool than crisp mathematics for modeling these problems, and hence solving a fuzzy matrix equation is becoming more important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [1, 2], Dubois and Prade , and Nahmias . A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu , Goetschel, Jr. and Voxman , and Wu and Ma [7, 8].
In the past decades, many researchers have studied the fuzzy linear equations such as fuzzy linear systems (FLS), 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). These works were performed mainly by Friedman et al. [9, 10], Allahviranloo et al. [11–17], Abbasbandy et al. [18–21], Wang and Zheng [22, 23], and Dehghan et al. [24, 25]. The general method they applied is the fuzzy linear equations were converted to a crisp function system of linear equations with high order according to the embedding principles and algebraic operations of fuzzy numbers. Then the fuzzy solution of the original fuzzy linear systems was derived from solving the crisp function linear systems. However, for a fuzzy matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past. In 2009, Allahviranloo et al.  discussed the fuzzy linear matrix equations (FLME) of the form in which the matrices and are known and real matrices, respectively; is a given fuzzy matrix. By using the parametric form of fuzzy number, they derived necessary and sufficient conditions for the existence condition of fuzzy solutions and designed a numerical procedure for calculating the solutions of the fuzzy matrix equations. In 2011, Guo et al. [27–29] investigated a class of fuzzy matrix equations by means of the block Gaussian elimination method and studied the least squares solutions of the inconsistent fuzzy matrix equation by using generalized inverses of the matrix, and discussed fuzzy symmetric solutions of fuzzy matrix equations . What is more, there are two shortcomings in the above fuzzy systems. The first is that in these fuzzy linear systems and fuzzy matrix systems the fuzzy elements were denoted by triangular fuzzy numbers, so the extended model equations always contain parameter , which makes their computation especially numerical implementation inconvenient in some sense. The other one is that the weak fuzzy solution of fuzzy linear systems does not exist sometimes see; .
To make the multiplication of fuzzy numbers easy and handle the fully fuzzy systems, Dubois and Prade  introduced the LR fuzzy number in 1978. We know that triangular fuzzy numbers and trapezoidal fuzzy numbers  are just specious cases of LR fuzzy numbers. In 2006, Dehghan et al.  discussed firstly computational methods for fully fuzzy linear systems whose coefficient matrix and the right-hand side vector are LR fuzzy numbers. In 2012, Allahviranloo et al. studied LR fuzzy linear systems  by the linear programming with equality constraints. Otadi and Mosleh considered the nonnegative fuzzy solution  of fully fuzzy matrix equations by employing linear programming with equality constraints at the same year. Recently, Guo and Shang  investigated the fuzzy approximate solution of LR fuzzy Sylvester matrix equations . In this paper we propose a general model for solving the fuzzy linear matrix equation , where , , and are , , and nonnegative LR fuzzy numbers matrices, respectively. The model is proposed in this way; that is, we extend the fuzzy linear matrix system into a system of linear matrix equations according to arithmetic operations of LR fuzzy numbers. The LR fuzzy solution of the original matrix equation is derived from solving crisp systems of linear matrix equations. The structure of this paper is organized as follows.
In Section 2, we recall the LR fuzzy numbers and present the concept of fully fuzzy linear matrix equation. The computing model to the positive fully fuzzy linear matrix equation is proposed in detail and the fuzzy approximate solution of the fuzzy linear matrix equation is obtained by using pseudo-inverse in Section 3. Some examples are given to illustrate our method in Section 4 and the conclusion is drawn in Section 5.
2.1. The LR Fuzzy Number
Definition 1 (see ). A fuzzy number is a fuzzy set like which satisfies the following.(1)is upper semicontinuous,(2) is fuzzy convex, that is, for all , ,(3) is normal, that is, there exists such that ,(4)supp is the support of the , and its closure cl(supp ) is compact.
Let be the set of all fuzzy numbers on .
Definition 2 (see ). A fuzzy number is said to be an LR fuzzy number if
where , , and are called the mean value, left, and right spreads of , respectively. The function , which is called left shape function satisfies(1),
(2) and ,(3) is nonincreasing on .
The definition of a right shape function is similar to that of .
Clearly, is positive (negative) if and only if ().
Also, two LR fuzzy numbers and are said to be equal, if and only if , , and .
Definition 3 (see ). For arbitrary LR fuzzy numbers and , we have the following.(1)Addition: (2)Multiplication:(i)If and , then (ii)if and , then (iii)if and , then (3) Scalar multiplication:
2.2. The LR Fuzzy Matrix
Definition 4. A matrix is called an LR fuzzy matrix, if each element of is an LR fuzzy number.
will be positive (negative) and denoted by if each element of is positive (negative), where . Up to the rest of this paper, we use positive LR fuzzy numbers and formulas given in Definition 3. For example, we represent LR fuzzy matrix , that with new notation , where , and are three crisp matrices. In particular, an dimensions LR fuzzy numbers vector can be denoted by , where , and are three dimensions crisp vectors.
Definition 5. Let and be two and fuzzy matrices; we define which is an fuzzy matrix, where
2.3. The Fully Fuzzy Linear Matrix Equation
Definition 6. The matrix system:
where , , , , , , and , , are LR fuzzy numbers, is called an LR fully fuzzy linear matrix equation (FFLME).
Using matrix notation, we have
A fuzzy numbers matrix is called an LR fuzzy approximate solution of fully fuzzy linear matrix equation (8) if satisfies (9).
Up to the rest of this paper, we will discuss the nonnegative solution of FFLME , where , , and .
3. Method for Solving FFLME
First, we extend the fuzzy linear matrix system (9) into three systems of linear matrix equations according to the LR fuzzy number and its arithmetic operations.
Theorem 7. The fuzzy linear matrix system (9) can be extended into the following model:
Theorem 9. Let , , and be three nonnegative fuzzy matrices, respectively, and let and be the product of a permutation matrix by a diagonal matrix with positive diagonal entries. Moreover, let , , and . Then the systems has nonnegative fuzzy solutions.
Proof. Our hypotheses on and imply that and exist and they are nonnegative matrices. Thus .
On the other hand, because and , so with and , we have and . Thus is a fuzzy matrix which satisfies . Since , the positivity property of can be obtained from the condition .
When or is a singular crisp matrix, the following result is obvious.
Theorem 10 (see ). For linear matrix equations , where , , and . Then
is its minimal norm least squares solution.
By the pseudoinverse of matrices, we solve model (11) and obtain its minimal norm least squares solution as follows: that is,
Definition 11. Let be an LR fuzzy matrix. If is a minimal norm least squares solution of (11) such that , , , and , we call a nonnegative LR fuzzy minimal norm least squares solution of (9).
At last, we give a sufficient condition for nonnegative fuzzy minimal norm least squares solution of FFLME (9) in the same way.
Theorem 12. Let and be nonnegative matrices. Moreover, let , , and . Then the systems has nonnegative fuzzy minimal norm least squares solutions.
Proof. Since and are nonnegative matrices, we have .
Now that and , therefore, with and , we have and . Thus is a fuzzy matrix which satisfies . Since , the nonnegativity property of can be obtained from the condition .
The following Theorems give some results for such and to be nonnegative. As usual, denotes the transpose of a matrix .
Theorem 13 (see ). The inverse of a nonnegative matrix is nonnegative if and only if is a generalized permutation matrix.
Theorem 14 (see ). Let be an nonnegative matrix with rank . Then the following assertions are equivalent:(a). (b)There exists a permutation matrix , such that has the form where each has rank and the rows of are orthogonal to the rows of , whenever , the zero matrix may be absent.(c) for some positive diagonal matrix . In this case,
4. Numerical Examples
Example 15. Consider the fully fuzzy linear matrix system
By Theorem 7, the model of the above fuzzy linear matrix system is made of the following three crisp systems of linear matrix equations
Now that the matrix is singular, according to formula (18), the solutions of the above three systems of linear matrix equations are as follows:
By Definition 11, we know that the original fuzzy linear matrix equations have a nonnegative LR fuzzy solution
since , , , and .
Example 16. Consider the following fuzzy matrix system:
By Theorem 7, the model of the above fuzzy linear matrix system is made of following three crisp systems of linear matrix equations:
By the same way, we obtain that the solutions of the above three systems of linear matrix equations are as follows:
Since , , , and , we know that the original fuzzy linear matrix equations have a nonnegative LR fuzzy solution given by
In this work we presented a model for solving fuzzy linear matrix equations in which and are and fuzzy matrices, respectively, and is an arbitrary LR fuzzy numbers matrix. The model was made of three crisp systems of linear equations which determined the mean value and the left and right spreads of the solution. The LR fuzzy approximate solution of the fuzzy linear matrix equation was derived from solving the crisp systems of linear matrix equations. In addition, the existence condition of strong LR fuzzy solution was studied. Numerical examples showed that our method is feasible to solve this type of fuzzy matrix equations. Based on LR fuzzy numbers and their operations, we can investigate all kinds of fully fuzzy matrix equations in future.
The work is supported by the Natural Scientific Funds of China (no. 71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).
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