Solving Complex Fuzzy Linear Matrix Equations

Abstract

In this paper, a kind of complex fuzzy linear matrix equation , in which is a complex fuzzy matrix and and are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simplified procedures for solving the original fuzzy equation are proposed and the correspondingly sufficient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to illustrate our proposed method.

1. Introduction

The uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented and computed by the fuzzy numbers. The theory and computation of linear systems related with fuzzy numbers always play an important role in the fuzzy mathematics. In the past decades, there has a great enormous investigation in the study of fuzzy mathematics and its applications. The definition of fuzzy numbers and their arithmetic operations were first introduced by Zadeh [1], Dubois and Prade [2], and Nahmias [3]. A different approach to fuzzy numbers and the fuzzy number spaces was studied by Puri and Ralescu [4], Goetschell and Voxman [5], and Wu and Ma [6, 7].

In 1998, Behra and Chakraverthy [8] investigated fuzzy linear systems by an embedding approach of fuzzy set decomposition theorem. Later, Abbasbandy et al., Allahviranloo et al., and Zheng et al. studied some more complicated fuzzy linear systems [9–17]. In recent years, new approaches and theories for linear systems in which part or all parameters may be uncertain and can be represented and computed by the fuzzy numbers has been emerging one after another [8, 18–20].

It is well known that some matrix systems such as Lyapunov, Sylvester, and Stein matrix equations always have wide use in science and technology field. So, the investigation on fuzzy matrix systems has been paid attention by some scholars in past decades. In 2009, Allahviranloo et al. [21] investigated the fuzzy matrix equation, . In 2018, AmirfakhrianIn et al. [22] presented a new algorithm for calculating the fuzzy linear matrix equation with the form by another way. In 2011, Gong and Guo [23] discussed inconsistent fuzzy linear systems and studied its least squares fuzzy solution. In 2014, Gong et al. [24] studied the general dual fuzzy matrix systems based on the LR fuzzy numbers. In 2017, Guo et al. [25, 26] studied the fuzzy matrix system of the form by a matrix method and made a further investigation to dual fuzzy matrix equation . In 2018, Guo et al. introduced the complex fuzzy matrix equation and proposed a general model to deal with it. At the same year, they considered the approximate fuzzy inverse and its simple application in fully fuzzy linear systems [27]. In 2019, Guo and Shang [28] put up a new method for solving linear fuzzy matrix equations, .

For complex fuzzy linear systems, few researchers have developed methods to investigate them in the past decades. The fuzzy complex numbers were introduced firstly by Buckley [29] in 1989. In 2000, Qiu et al. [30] restudied the sequence and series of fuzzy complex numbers and their convergence by considering the fuzzy complex linear systems. In 2009, Rahgooy et al. [31] applied the fuzzy complex linear system of linear equations to described circuit analysis problem. In 2014, Behera and Chakraverty discussed the fuzzy complex system of linear equations by the embedding method and modified arithmetic operations of the complex fuzzy numbers later [8, 18].

In this paper, we propose a matrix method to deal with complex fuzzy linear matrix equation, . At first, we introduce the complex fuzzy matrix and its operation with the crisp number. Then, we convert the complex fuzzy matrix equation to a model which is a crisp linear system of function matrix equations. The fuzzy solution of the original fuzzy equation is gained by solving the model, that is, a crisp system of matrix equations. Then, conditions of the strong fuzzy solution are also discussed. Finally, two illustrating examples are given. Our results enrich fuzzy linear system theory.

2. Preliminaries

Definition 1 (see [1]). fuzzy number is a fuzzy set such as which satisfies(1) is upper semicontinuous(2) is fuzzy convex, i.e., , for all (3) is normal, i.e., there exists such that (4)supp is the support of the , and its closure cl(supp) is compactLet be the set of all fuzzy numbers on .

Definition 2 (see [22]). A fuzzy number in the parametric form is a pair of functions and , , which satisfies the requirements:(1) is a bounded monotonic increasing left continuous function(2) is a bounded monotonic decreasing left continuous function(3),

Definition 3 (see [8]). An arbitrary complex fuzzy number should be represented as , where and , for all . In this case, can be written as

Definition 4 (see [8]). For any two arbitrary complex fuzzy numbers and , where are fuzzy numbers, their arithmetic operations are as follows:(1)(2)(3)(4)

Definition 5. The matrix system:where are crisp numbers and are complex fuzzy numbers, which is called a complex fuzzy linear matrix equations (CFLMEs).
Using matrix notation, we haveA complex fuzzy matrix,is called a solution of the fuzzy matrix equation (2) if and only if it satisfies .

3. Solving Complex Fuzzy Linear Matrix Systems

Definition 6 (see [32]). A matrix is called a complex fuzzy matrix if each element of is a complex fuzzy number. Let , and the complex fuzzy matrix can be represented by .

Definition 7. For any two arbitrary complex fuzzy matrix and , where are fuzzy numbers matrices; their arithmetic is as follows:(1)(2)(3)(4)

Theorem 1. The fuzzy matrix equation (3) can be extended to a crisp function linear matrix system as follows:whereandin which the elements of matrix and of matrix are determined by the following way: if else ; if else .

Proof. Let , and the unknown complex fuzzy matrix . We also let in which the elements of matrix and of matrix are determined by the following way: if else ; if else and also let in the same way.
For complex fuzzy matrix equation, , i.e.,Supposing and , we haveSinceandSo, equation (9) can be rewritten asIn comparison with the coefficients of , we obtainandi.e.,Denoting in the matrix form, the above matrix equations can be written asandThus, we obtain equations (5)–(7) as follows:whereandBy the matrix operation, the above linear matrix equations are equivalent withIt is completed the proof.
In a similar way, we could obtain another model for solving equation (3).

Theorem 2. The fuzzy linear matrix equation (3) can be extended to a crisp function linear matrix system as follows:whereandin which the elements of matrix and of matrix are determined by the following way: if else ; if else .
By the matrix operation, the above linear matrix equations are equivalent with

Proof. The proof is similar to Theorem 1.

Theorem 3 (see [28]). Let belong to , belong to , and belong to . Then, the minimal solution of the matrix equation is expressed byIn order to solve the fuzzy matrix equation (3), we need to consider the systems of linear equations (21) or (26). It seems that we have obtained the minimal solution of the function linear system (21) asMeanwhile, we obtain the minimal solution of the function linear system (26) aswhere is the Moore–Penrose generalized inverse of matrix .
However, the solution matrixfrom equations (28) or (29), may still not be an appropriate complex fuzzy matrix except that both and are appropriate fuzzy number matrices. Restricting the discussion to the complex triangular fuzzy numbers, i.e., , and consequently are all linear functions of , and having calculated which solves (21) or (26), we define the complex fuzzy minimal solution to the fuzzy linear matrix equation (3) as follows.

Definition 8. Let be the minimal solution of model (21) or (26). The complex fuzzy number matrix defined byandIf , and , are all fuzzy numbers, then , and is called a strong complex fuzzy minimal solution of fuzzy linear matrix systems (3). Otherwise, is called a weak complex fuzzy minimal solution of fuzzy linear matrix systems (3).
To illustrate expression (28) or (29) to be a fuzzy solution matrix, we now discuss the generalized inverses of the nonnegative symmetric matrix,in a special structure [33].

Theorem 4. Ifwherethenwhere are Moore–Penrose inverses of matrices and , respectively.

Proof. LetWe obtainBy adding and then subtracting the two parts of the above equations, we obtainand consequently,The proof is completed.
The key points to make the solution matrix being a strong fuzzy solution is thatorare appropriate fuzzy numbers matrices, i.e., each element in which is an appropriate triangular fuzzy number. By the following analysis, a sufficient condition is that or . It is completed the proof.
For the model equation (26), we have the following result.

Theorem 5. IfandThe complex fuzzy matrix equation (3) has a strong complex fuzzy minimal solution as follows:where

Proof. LetWe know the condition that is equivalent to conditions and by Theorem 4.
Since , is a bounded monotonic increasing left continuous function matrix and is a bounded monotonic decreasing left continuous function matrix with , by Definition 6. For fuzzy matrix , the above property is the same.
According to equation (29), we haveIt admits a bounded monotonic increasing continuous function matrix.
On the contrary, we haveandi.e.,Now that , , andWe know thatThus, the above complex fuzzy matrix equation has a strong complex fuzzy minimal solution asIt is completed the proof.
For the model equation (21), we have the following result by the similar analysis.

Theorem 6. Ifand

The fuzzy matrix equation (3) has a strong fuzzy minimal solution as follows:where

Proof. The proof is similar to Theorem 5.

4. Numerical Examples

Example 1. Consider the following complex fuzzy linear matrix system:By Theorem 2, the original fuzzy matrix equation is extended into the following system of linear matrix equation (17):whereandFrom (31), the solution of the computing model isIt meansandSince both,are appropriate fuzzy numbers matrices, we can assert that theis strong complex fuzzy solution of the original fuzzy matrix system by Definition 8.

Example 2. Consider another complex fuzzy linear matrix system:SupposeandBy Theorem 2, the original fuzzy matrix equation is extended into the following system of linear matrix equation (26):From (32), the solution of the computing model isIt meansandSince bothandare not appropriate fuzzy numbers matrices, we can assert that the fuzzy solution of the original fuzzy system iswhich admits a weak fuzzy minimal solution by Definition 5.

5. Conclusion

In this paper, we put up a scheme for calculating complex fuzzy matrix equations , where and are and crisp matrices, respectively, and is an complex fuzzy matrix. The fuzzy approximate solution of the original fuzzy matrix equation was derived from solving the model which is a function linear matrix system. In addition, the conditions of strong fuzzy approximate solution were analysed. Numerical examples showed the effectiveness of the proposed method. We should point out that a fact, i.e., it is rather difficult to obtain the strong complex fuzzy solution of the system in general. Our method can be applied to solve all kinds of semicomplex fuzzy matrix equations. We will study the fully complex fuzzy matrix equation and its applications on traffic control and decision making on this basis in next step.

Data Availability

No data were used related with practical problems.

Conflicts of Interest

The authors declare that they have no conflicts of interest, financial, or otherwise.

Acknowledgments

The work was supported by the Natural Scientific Funds of PR China (nos. 61967014 and 11861059), Scientific Research Project of Gansu Province Colleges and Universities (no. 2019A-004), and Natural Scientific Funds of Gansu Province (no. 20JR10RA101).