Minimal Solution of Singular LR Fuzzy Linear Systems

In this paper, the singular LR fuzzy linear system is introduced. Such systems are divided into two parts: singular consistent LR fuzzy linear systems and singular inconsistent LR fuzzy linear systems. The capability of the generalized inverses such as Drazin inverse, pseudoinverse, and {1}-inverse in finding minimal solution of singular consistent LR fuzzy linear systems is investigated.


Introduction
The concept of fuzzy numbers and fuzzy arithmetic operations was first introduced and investigated by Zadeh [1,2], Dubois and Prade [3], and Nahmias [4]. Some different approaches to fuzzy numbers and the structure of fuzzy number spaces were given by Puri and Ralescu [5], Goetschel and Voxman [6,7], and Cong-Xin and Ming [8,9]. Fuzzy systems are used to a variety of problems ranging from control chaotic systems [10] to fuzzy metric spaces [11], fuzzy linear systems, fuzzy differential equations [12], particle physics [13], and so on. System of simulations linear equations plays major role in various areas such as mathematics, physics, statistics, engineering, and social sciences [14]. Linear systems of equations plays major role in various areas of sciences and engineering such as fuzzy mathematics, and so forth [15]. The linear system of equations̃=̃, (1) where is a × crisp matrix and the right-hand side is a vector of fuzzy numbers in parametric form, is called a fuzzy linear system (FLS). Friedman et al. [14] use an embedding method for solving a special case of (1) when is a nonsingular matrix. They introduce new notions for solution of a FLS (1) and transform the FLS to a crisp linear system. In their method, the ordinary inverse matrix is used to solve the FLS. The fuzzy linear system (1) where is a × singular crisp matrix and the right-hand side is a vector of fuzzy numbers in parametric form is called a singular fuzzy linear system (SFLS) [16]. The capability of pseudoinverse [17] and Drazin inverse [16] in solving fuzzy linear systems and {1}-inverse [18] and {1, 3}-inverse [19] in finding fuzzy leastsquare solutions of inconsistent fuzzy linear systems has been studied.
Ordinary inverse for any × nonsingular matrix exists and is unique. Unlike the case of the nonsingular matrix, which has a single unique inverse for all purposes, there are different generalized inverses for different purposes. For some purposes, as in the examples of solutions of linear systems, there is not a unique inverse, but any matrix of a certain class will do. Generalized inverses have various applications in the theory of finite Markov chains, the study of singular differentia and difference equations, the investigation of Cesaro-Neumann iterations, cryptograph, iterative methods in numerical analysis, multibody system dynamics, and others [20][21][22][23]. The fuzzy linear system (1) with LR fuzzy variables and LR fuzzy right-hand side is called a fuzzy LR linear system (FLRLS). Ghanbari et al. [24,25] give the exact and approximated solutions of fuzzy LR linear systems. They give new algorithms using a least-squares model and the ABS approach. LR-type fuzzy systems of linear algebraic equations are discussed by Abramovich et al. in [26], and based on their work in [27] Allahviranloo et al. study LR fuzzy linear systems. They prove existence of fuzzy number vector solution of LR fuzzy linear system by solving optimization problem and obtain the approximated and exact solutions of mentioned systems. This approximated solution in general 2 The Scientific World Journal is different from the weak fuzzy solution considered by Friedman et al. [14,25].
Here, the singular LR fuzzy linear systems are introduced. Such systems are divided into two parts: singular consistent LR fuzzy linear systems and singular inconsistent LR fuzzy linear systems. In this paper, the capability of the generalized inverses in solving singular consistent LR fuzzy linear systems is investigated. In Section 2, we recall some preliminaries. The purpose of this paper is to propose a method for solving singular LR fuzzy linear systems on the LR fuzzy linear systems discussed in Section 3. The proposed method is given in Section 4. Numerical examples are given in Section 5 followed by suggestions and concluding remarks in Section 6.

Preliminaries
This section gives a brief summary of index of matrix, Drazin inverse, and LR fuzzy numbers. We refer the reader to [28][29][30][31] for more information. (2) Definition 3. Let ∈ × . The matrix of order × is the pseudoinverse of , denoted by + , if satisfies the following conditions: Definition 4. Let ∈ × . The matrix of order × is the {1}-inverse of , denoted by (1) , if satisfies the following condition: Definition 5. The fuzzy number̃is of LR-type if there exist shape functions (for left), (for right), and scalers > 0, > 0 with̃( The mean value of̃, , is a real number and , are called the left and right spreads, respectively.̃is denoted by ( , , ) LR . A crisp number is specified to be ( , 0, 0) LR .

Singular LR Fuzzy Linear Systems
In this section, singular LR fuzzy linear systems are introduced and the existing methods for solving LR fuzzy linear systems are investigated.

Shortcomings of the Existing Methods.
In this subsection, the shortcomings of the existing methods for solving LR fuzzy linear systems are pointed out.
(i) If the crisp system (7) lacks solution, then the FLRLS (6) does not have any solution.
(ii) If is a solution of the system = and , are the solutions of the constrained least-square problem (8), then the corresponding̃= ( , , ) LR is said to be an exact solution of (6).
The Scientific World Journal 3 (iii) If̃> 0 for all̃∈ ( ) LR , then the system (10) lacks a solution, and hence FLRLS (6) lacks an exact solution. In this case, if̃= ( , , ) LR is a solution of the system = and , are the solutions of the constrained least-square problem (8).
(2) Allahviranloo et al. [27] transform the LR fuzzy linear system into the 1-level crisp linear system (7) and the following minimization problem: The spreads of fuzzy number vector solution are obtained from the model (13). Finally, by allocating these spreads to vector solution of 1-level system, the fuzzy number vector solution results and if the computed solution satisfies the FLS, it can be as exact or approximated solution.
(i) If the crisp system (7) lacks solution, then the FLRLS (12) does not have any solution. (ii) If model (13) has more than two optimal solutions then it has a set of solutions.
Example 9. Consider the LR fuzzy linear system Using Allahviranloo's method, the vector solution of the 1-level system is (2, 1) . This fuzzy linear system has infinite number of fuzzy number vector solutions as follows: The existing methods are incapable of finding the minimal solution for the following LR fuzzy linear system:

Singular LR Fuzzy Linear Systems.
In this subsection, the singular consistent or inconsistent LR fuzzy linear systems is introduced.
Definition 10. The × linear system where the × crisp singular matrix = ( ) and LR fuzzy vector̃are given and̃= ( ) LR is an unknown LR fuzzy vector to be found, is called a singular LR fuzzy system of equations (SFLRLS).

Minimal Solution of Singular LR Fuzzy Linear Systems
Let̃=̃ (   20) be a singular consistent LR fuzzy linear system and let (18) be the extended crisp linear system of it. The coefficient matrix system (18) is a singular matrix [16,32]. In this section, the capability of generalized inverses in finding minimal solution of the singular consistent LR fuzzy linear system (20) is investigated. Equivalent systems of linear equations have exactly the same solutions [33]. Therefore, the fundamental theorem for the singular consistent LR fuzzy linear system is given.

4
The Scientific World Journal Theorem 12 (fundamental theorem). The system (20) is equivalent to the singular consistent crisp linear system and the singular consistent constrained linear system

Notes
Note 1. If model (22) has a solution, then (20) has an exact LR fuzzy solution and (22) gives exact solutions; otherwise we get a set of approximated solutions. For computing approximated solution instead of the singular constrained linear system (22), we solve the following singular constrained linear system: where = (1, . . . , 1) ∈ and > 0 is a user-defined parameter.
Note 2. If model (22) has a set of solutions and (21) is a consistent system, then (20) has a set of LR fuzzy exact solutions. (21) is a singular inconsistent linear system, the singular LR fuzzy linear system (20) has no solution.

Note 3. If the crisp linear system
Note 4. We observe that ( , , ) is the exact minimal LR fuzzy solution of the singular consistent LR fuzzy linear system (20) if and only if is a minimal LR fuzzy solution of the singular consistent crisp linear system (21) and the following constrained least-square problem: has a solution with * = 0; otherwise, the singular consistent LR fuzzy linear system has a set of approximated minimal LR fuzzy solutions.

Solution
Strategy. According to Theorem 12, for solving (20) we must solve the system (21) and singular constrained linear system (22). Finding positive solutions for linear systems of equations has been investigated [34]. Nonnegativity of matrices and their group or Drazin [35] inverses and pseudoinverse [36] were investigated.

Solving Singular Constrained Linear System
using generalized inverses are given in the next section. It is possible that the generalized inverses do not obtain the positive solution for the system (27). In this case, we use the lsqlin function of MATLAB software. The capability of the lsqlin function of MATLAB software in finding exact or approximated solutions is illustrated in next the section.

Numerical Results
In this section, solving singular LR fuzzy linear systems is illustrated.
Example 16. Consider the following singular consistent LR fuzzy linear system: According to Tables 1 and 2 Drazin inverse, pseudoinverse, a {1}-inverse and the lsqlin function gives a set of exact LR fuzzy solution for (28).
Example 17. Consider the following singular consistent LR fuzzy linear system: The solutions obtained by generalized inverses for the system (29) are listed in Tables 3 and 4.
The Scientific World Journal 5   The generalized inverses are incapable of finding an exact LR fuzzy solution for (29). Therefore, we use the lsqlin function of MATLAB software for finding a set of approximated solutions (see Table 5).
Example 18. Now consider a system where the coefficient matrix is the same as in the previous system, but with the LR fuzzy vector on the right-hand side leads to an exact minimal LR fuzzy solution According to Tables 6 and 7, the system (31) has a set of the exact minimal LR fuzzy solutions. Hence for ∈ [0, 1], the singular LR fuzzy linear system (31) has a set of exact minimal LR fuzzy solution as follows: Example 19 (singular dual LR fuzzy linear systems). Usually, there is no inverse element for an arbitrary LR fuzzy number [38]. Also, if̃= ( 1 ) LR is an arbitrary LR fuzzy number, then we havẽ⊕ Therefore, the following two fuzzy linear systems are not equivalent In [38] the necessary and sufficient conditions for 1̃= 2̃+̃2 were given, where 1 and 2 are square matrices and in [25] on the DFLRLS (17) was discussed. Now, we consider an economic application of minimal solution of general dual fuzzy linear systems. The market price of goods and the quantity produced are determined by the equality between supply and demand. Suppose that demand and supply are linear functions of the price + = * + , where is the quantity supplied, which is required to be equal to , the quantity requested, is the price, , , , , , and are coefficients to be estimated, where the coefficients , , , and are represented by LR fuzzy numbers, and and are crisp numbers. By imposing the equality between quantity supplied and requested, the following general dual fuzzy linear system should be solved:

Conclusions and Suggestions
In this paper, the × singular LR fuzzy linear system̃=ĩ s transformed into the 3 × 3 singular crisp linear system = . Such systems are divided into two parts: singular consistent LR fuzzy linear systems and singular inconsistent LR fuzzy linear systems. The singular consistent LR fuzzy linear system̃=̃has an exact minimal LR fuzzy solution or a set of the approximated minimal LR fuzzy solutions. It is possible that the generalized inverses be incapable of finding the exact or approximated minimal LR fuzzy solution of the system̃=̃. In this case, we use the lsqlin function of MATLAB software. We hope that this paper will be a new path to improve solving singular consistent LR fuzzy linear systems using preconditioning techniques and iterative methods.