Abstract
This paper investigates the complex fuzzy linear equation in which is a crisp complex matrix and is an arbitrary LR complex fuzzy vector. The complex fuzzy linear system is converted to equivalent high order fuzzy linear system . A new numerical procedure for calculating the complex fuzzy solution is designed and a sufficient condition for the existence of strong complex fuzzy solution is derived in detail. Some examples are given to illustrate the proposed method.
1. Introduction
In the mathematical modeling of physics, engineering computation, and statistical analysis, it is the linear systems that have mature theory and easy computational property. However, the uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented by fuzzy numbers. So the investigation of theory and computing method for fuzzy linear systems plays an important role in the fuzzy mathematics and its applications. The concept of fuzzy numbers and arithmetic operations with these numbers were firstly introduced and investigated by Zadeh [1], Dubois and Prade [2], and Nahmias [3]. A different approach to fuzzy numbers and the structure of fuzzy number spaces were given by Puri and Ralescu [4], Goetschell and Voxman [5], and Wu and Ma [6, 7].
Since Friedman et al. [8] proposed a general model for solving fuzzy linear systems by an embedding approach, in the past decades, a lot of researches about some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), full fuzzy linear systems (FFLS), dual full fuzzy linear systems (DFFLS), and general dual fuzzy linear systems (GDFLS) have emerged. See [9–20]. In general, the uncertain elements of fuzzy linear systems were denoted by the parametric form of fuzzy numbers and the systems were extended into crisp function linear systems. Thus, it may lead to two defects. The one is that the extended linear equations always contain parameter , which makes their computation inconvenient in some sense. The other is that the weak fuzzy solution of fuzzy linear systems does not exist [21] sometimes. To make the multiplication of fuzzy numbers easy, Dubois and Prade [2] introduced the LR fuzzy number. We know that triangular fuzzy numbers are just specious cases of LR fuzzy numbers. In 2006, Dehghan et al. [18] discussed computational methods for fully fuzzy linear systems whose coefficient matrix and the right-hand side vector are LR fuzzy numbers. In the past decade, some researchers paid more attention to LR fuzzy linear systems. In 2013, Guo and Shang [22] proposed a computing method for the fuzzy Sylvester matrix equations with LR fuzzy numbers. Later, Gong et al. [23] studied the general dual fuzzy linear matrix systems based on LR fuzzy numbers.
For complex fuzzy system of linear equations, few researchers have developed methods to solve them. The fuzzy complex numbers were introduced firstly by Buckley [24] in 1989. In 2010, Jahantigh et al. [25] studied firstly the fuzzy complex linear systems. Solution of fuzzy complex linear system of linear equations was described and was applied to circuit analysis problem by Rahgooy et al. [26]. In 2014, Behera and Chakraverty [27] discussed the fuzzy complex system of linear equations by the embedding method and redefined the complex fuzzy number [28]. In this paper the LR complex fuzzy linear system is investigated. A numerical procedure for calculating the fuzzy solution is designed and a sufficient condition for the existence of strong fuzzy solution is derived. Finally, some examples are given to illustrate our method.
2. Preliminaries
There are some basic definitions and results for fuzzy numbers.
2.1. The Fuzzy Number
Definition 1 (see [1]). 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) is the support of , and its closure cl(supp ) is compact.Let be the set of all fuzzy numbers on .
Definition 2 (see [2]). fuzzy number is said to be a LR fuzzy number if where , , and are called the mean value and left and right spreads of , respectively. The function , which is called left shape function, satisfies the following:(1)(2) and .(3) is nonincreasing on The definition of a right shape function is similar to that of
Clearly, two LR fuzzy numbers and are said to be equal, if and only if and Also, is positive (negative) if and only if ().
Definition 3 (see [2]). For arbitrary LR fuzzy numbers and , we have the following:(1) Addition: (2) Subtraction: (3) Scalar multiplication:
Definition 4. An arbitrary LR complex fuzzy number should be represented as , where and . Likely, can be written as .
Definition 5. For any two arbitrary complex fuzzy numbers and where are fuzzy numbers, their arithmetic is as follows:(1),(2),(3).
2.2. Complex Fuzzy Linear Systems
Definition 6. The linear system equation where are LR complex numbers and are complex fuzzy numbers, is called a LR complex fuzzy linear system (CFLS).
Using matrix notation, we have A complex fuzzy numbers vector is called a fuzzy solution of the complex fuzzy linear system (5) if satisfies (6).
3. Solving Complex Fuzzy Linear System
Definition 7. For two LR fuzzy numbers vectors and , we have the following:(1) Addition: (2) Subtraction: (3) Scalar multiplication:
Definition 8. An arbitrary LR complex fuzzy vector should be represented as , where and are two LR fuzzy numbers vectors. Likely, the LR complex fuzzy vector can be written as
3.1. Equivalent Fuzzy Linear System
Theorem 9. The complex fuzzy linear system (5) is equivalent to order fuzzy linear system: where
Proof. We denote and , where and are fuzzy number vectors. We also suppose the unknown vector , where and are two unknown fuzzy number vectors.
Since , we have That is, Comparing with the coefficient of , we have That is,which admits order fuzzy linear system.
We express it in matrix form as follows:
3.2. Solving CFLS
In order to solve the complex fuzzy linear system (5), we need to solve the real fuzzy system of linear equations (12).
Firstly, we set up a computing model for solving LR CFLS. Then we define the complex fuzzy solution of CFLS and obtain its solution representation by the generalized inverses of matrices.
According to operations of LR fuzzy numbers, we have the following results.
Theorem 10. The fuzzy linear system (12) can be extended into the following system of linear matrix equations: where where the elements and are determined as follows: if then ; if then .
Proof. Denoting , where , are the center values and are the left and right spread values of fuzzy number vectors , respectively, then the fuzzy linear system is Let where the elements and are determined as follows: if then ; if then .
Since we have The equation is equivalent to Thus, we have Denoting (27) in a matrix form, we have
By means of calculations, we obtain the minimal solution of the model equation (19) as follows:
It seems that we obtained the fuzzy vector as the above (29). However, the solution vector may not be an appropriate LR fuzzy number one except for . So we give the definition of LR complex fuzzy solution to complex fuzzy linear system (6) as follows.
Definition 11. Let If is the minimal solution of (19) such that , we call a strong LR fuzzy minimal solution of (17). Meanwhile, the complex fuzzy vector is called a strong LR complex fuzzy minimal solution of (5). Otherwise, is said to be a weak LR complex fuzzy minimal solution of (5) given by
3.3. A Sufficient Condition for Strong Fuzzy Solution
Now we give a sufficient condition for strong fuzzy approximate solution to the complex fuzzy linear system by the following analysis.
To illustrate expression (29) to be a LR fuzzy solution vector, we now discuss the generalized inverses of nonnegative matrix in a special structure.
Lemma 12 (see [29]). Let Then the matrix is the Moore-Penrose inverse of the matrix , where are Moore-Penrose inverses of matrices and , respectively.
The key point to make the solution vector be a strong LR fuzzy solution is that is LR fuzzy vector, in which each element is a LR fuzzy number. By the following analysis, we know that it is equivalent to the condition .
Theorem 13. If the fuzzy linear equation (5) has a strong LR complex fuzzy minimal solution as follows: where Further, if , the fuzzy linear equation (5) has a nonnegative strong LR complex fuzzy minimal solution.
Proof. Since and are the left and right spreads fuzzy matrix and . It means is a nonnegative matrix.
Let We know the condition where is equivalent to the fact that and .
Now that and , the product of two nonnegative matrices is nonnegative in nature: that is, and
Since , it means that . Thus, we have that the fuzzy linear equation (5) has a nonnegative strong LR complex fuzzy minimal solution by Definition 8.
The following theorems give some results for such and to be nonnegative. As usual, denotes the transpose of a matrix .
Theorem 14 (see [30]). The inverse of a nonnegative matrix is nonnegative if and only if is a generalized permutation matrix.
Theorem 15 (see [31]). 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
In this section, we will demonstrate the efficiency of the proposed method by two numerical examples.
Example 1. Consider the following CFSLE: Let . Applying Theorem 9., the complex fuzzy linear is equivalent to the following fuzzy linear system : From Theorem 10., we need to solve the following function linear system: By (34), we have Thus, we obtain the solution of the complex fuzzy linear system as
Example 2. Consider a simple RLC circuit with fuzzy current and fuzzy source; see Figure 1. The CFSLE for the circuit is as follows:Let .
According to Theorem 9., the LR complex fuzzy linear is equivalent to the following fuzzy linear system : Applying Theorem 10, we need to solve the following linear systems: Form (29), the solution of the model is obtained as where where Since are nonnegative vectors, we obtain the fuzzy solution of the CFSLE as follows: and it admits a strong complex LR fuzzy solution.
Example 3. We consider an economic application of minimal solution of complex fuzzy linear systems. The market price of a good 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 , and are coefficients to be estimated, where the coefficients , and are represented by LR complex fuzzy numbers, and and are crisp numbers. By imposing the equality between quantity supplied and requested, the following complex dual fuzzy linear system should be solved: In fact, the above system is equivalent to a complex fuzzy linear system as follows: Let .
By using our proposed method, we need to solve the two following fuzzy systems: We can obtain easily via computation thatIt means Since and are appropriate LR fuzzy numbers vectors, we obtained the solution of the complex fuzzy linear system which is and it admits a strong complex fuzzy solution by Definition 11.
5. Conclusion
In this work we presented a matrix method for solving LR complex fuzzy linear equation , where is a crisp complex matrix and is an arbitrary complex fuzzy vector. When a complex fuzzy numbers vector is expressed by this way, its operations for both computation and analysis will become more simple and easy. Numerical examples showed that our method is effective to solve the complex fuzzy linear system.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work is supported by the Natural Scientific Funds of PR China (no. 61262022) and the Youth Scientific Research Promotion Project of Northwest Normal University (NWNULKQN-1120).