Minimal Solution of Complex Fuzzy Linear Systems

Copyright © 2016 X. Guo and K. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the complex fuzzy linear equation C?̃? = ?̃? in which C 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 G?̃? = b. 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.


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][10][11][12][13][14][15][16][17][18][19][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 , 0 ≤  ≤ 1, 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 Ãx = b 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  X + X = C with LR fuzzy numbers.Later, Gong et al. [23] studied the general dual fuzzy linear matrix systems  X+ B =  X + D 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 2 Advances in Fuzzy Systems system of linear equations by the embedding method and redefined the complex fuzzy number [28].In this paper the LR complex fuzzy linear system z = w 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.

Preliminaries
There are some basic definitions and results for fuzzy numbers.
Let  1 be the set of all fuzzy numbers on .
Definition 2 (see [2]). fuzzy number M is said to be a LR fuzzy number if where , , and  are called the mean value and left and right spreads of M, respectively.The function (⋅), which is called left shape function, satisfies the following: (1) () = (−).
Definition 5.For any two arbitrary complex fuzzy numbers x = p + q and ỹ = ũ + Ṽ where p, q, ũ, Ṽ are fuzzy numbers, their arithmetic is as follows: (1)

Complex Fuzzy Linear Systems
Definition 6.The linear system equation where   , 1 ≤ ,  ≤  are LR complex numbers and w , 1 ≤  ≤  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 z satisfies (6).

Equivalent Fuzzy Linear System
Theorem 9.The  ×  complex fuzzy linear system ( 5) is equivalent to 2 × 2 order fuzzy linear system: where Proof.We denote  =  + , ,  ∈  × and w = ũ + Ṽ, where ũ and Ṽ are fuzzy number vectors.We also suppose the unknown vector z = p + q, where p and q are two unknown fuzzy number vectors.Since z = w, we have That is, Comparing with the coefficient of , we have That is, which admits 2 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.
Proof.Denoting z = (,   ,   ) + (,   ,   ), w = (,   ,   ) + (V, V  , V  ), where ,  are the center values and   ,   ,   ,   are the left and right spread values of fuzzy number vectors p, q, respectively, then the fuzzy linear system x = b is  (,   ,   ) = (,   ,   ) . Let where the elements  +  and  −  are determined as follows: if we have Advances in Fuzzy Systems 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 x as the above (29).However, the solution vector x = (,   ,   ) 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.

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.
The key point to make the solution vector be a strong LR fuzzy solution is that x = (,   ,   ) 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  † ≥ .
Proof.Since   and   are the left and right spreads fuzzy matrix b,   ≥  and   ≥ .It means (  ,   ) is a nonnegative matrix. Let We know the condition where  † ≥ 0 is equivalent to the fact that  ≥ 0 and  ≥ 0. Now that  ≥  and  ≥ , the product of two nonnegative matrices is nonnegative in nature: that is, ( −  ) ≥ .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  −1 and  † to be nonnegative.As usual, (⋅) ⊤ denotes the transpose of a matrix (⋅).Theorem 14 (see [30]).The inverse  −1 of a nonnegative matrix  is nonnegative if and only if  is a generalized permutation matrix.
where each   has rank 1 and the rows of   are orthogonal to the rows of   ; whenever  ̸ = , the zero matrix may be absent.

Conclusion
In this work we presented a matrix method for solving LR complex fuzzy linear equation z = w, where  is a crisp complex matrix and w 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.