General Solutions of Fully Fuzzy Linear Systems

and Applied Analysis 3 Definition 6 (see [34, 39]). The united solution set (USS), the tolerable solution set (TSS), and controllable solution set (CSS) for the system (10) are, respectively, as follows: X ∃∃ = {x 󸀠 ∈ R n : (∃A 󸀠 ∈ A) (∃b 󸀠 ∈ b) s.t. A󸀠x󸀠 = b󸀠} = {x 󸀠 ∈ R n : Ax 󸀠 ∩ b 󸀠 ̸ = 0} , X ∀∃ = {x 󸀠 ∈ R n : (∀A 󸀠 ∈ A) (∃b 󸀠 ∈ b) s.t. A󸀠x󸀠 = b󸀠} = {x 󸀠 ∈ R n : Ax 󸀠 ⊆ b} , X ∃∀ = {x 󸀠 ∈ R n : (∀b 󸀠 ∈ b) (∃A 󸀠 ∈ A) s.t. A󸀠x󸀠 = b󸀠} = {x 󸀠 ∈ R n : Ax 󸀠 ⊇ b} . (7) Definition 7. A fuzzy vector X = (x 1 , . . . , x n ) , given by x i (r) = [x i (r), x i (r)], 0 ≤ r ≤ 1, is called the minimal symmetric solution of (5) which is placed in CSS, if for any arbitrary symmetric solution ?̃? = (y 1 , . . . , y n ) t which is placed in CSS and ?̃?(1) = X(1), we have (?̃? ⊇ X) , that is, (y i ⊇ x i ) , that is, (σ yi ≥ σ xi ) , ∀i = 1, . . . , n, (8) where σ yi and σ xi are symmetric spreads of y i and x i , respectively. See [34, 39]. Definition 8. A fuzzy vector X = (x 1 , . . . , x n ) , given by x i (r) = [x i (r), x i (r)], 0 ≤ r ≤ 1, is called the maximal symmetric solution of (5) which is placed in TSS, if for any arbitrary symmetric solution Z = (?̃? 1 , . . . , ?̃? n ) t which is placed in TSS and Z(1) = X(1), we have (X ⊇ Z) , that is, (x i ⊇ ?̃? i ) , that is, (σ xi ≥ σ ?̃?i ) , ∀i = 1, . . . , n, (9) where σ xi and σ ?̃?i are symmetric spreads of x i and ?̃? i , respectively. See [34, 39]. 3. General Solutions In this section, we suggest a novel and practical method to obtain general solutions of FFLS. To this end, we solve the 1-cut system (5), which is a crisp system. So, we solve the following crisp system:


Introduction
Systems of simulations linear equations play major role in various areas such as mathematics, statistics, and social sciences.Since in many applications, at least some of thesystem's parameters and measurements are represented by fuzzy rather than crisp numbers, therefore, it is important to develop mathematical models and numerical procedures that would appropriately treat general fuzzy linear systems and solve them.
The linear system Ã X = b, where the elements, ã , of the matrix Ã and the elements, b , of the vector b are fuzzy numbers, is called a fully fuzzy linear system (FFLS).
Buckley and Qu in their consecutive works [23][24][25] proposed different solutions for FFLSs.Also, they found relation between these solutions.Based on their works, Muzzioli and Reynaerts in [26] studied FFLS of the form  1  +  1 =  2  +  2 , while for implementing their method 2 (+1) crisp systems should be solved.
Consequently, Dehghan et al. have studied some methods for solving FFLS.They have represented Cramer's rule, Gaussian elimination, fuzzy LU decomposition (Doolittle algorithm), and its simplification; they also have showed the applicability of linear programming approach for overdetermined FFLS in [27][28][29].Also, in [30], Allahviranloo and Mikaeilvand proposed an analytical method to obtain solution of FFLS by an embedding method.Their method is constructed based on obtaining a nonzero solution of the FFLS.
Vroman et al. in their continuous works [31][32][33] suggested two methods for solving system.In [33], they have proposed Cramer's rule to solve FFLS approximately, then they proved that their solution is better than Buckley and Qu's approximate solution vector.Furthermore, they have proposed an algorithm to improve their method to solve FFLS by parametric functions [32].
Recently, Allahviranloo et al. [34] have proposed a new practical method to solve an FFLS based on the 1-cut expansion.In their method, some spreads and then some new solutions have been derived that belong to TSS or CSS.Note that they have obtained some spreads which are symmetric.
We show that, using the proposed method in the present paper, we can obtain better solutions.On the other hand, the created errors in some certain cases with respect to the proposed distance are less than the errors that are obtained via Allahviranloo et al. 's method [34].
The structure of this paper is organized as follows.
In Section 2, we discuss concisely some important basic concepts and definitions which will be used later.In Section 3, we present our new method and concentrate on how we could derive the linear general spreads of fuzzy vector solution corresponding to TSS or CSS.The proposed method is illustrated by solving some examples in Section 4, and conclusion is drawn in Section 5.

Preliminaries
Let   (R) denote the family of all nonempty compact convex subset of R.
A nonempty bounded subset  of R is called convex if and only if The basic definition of fuzzy numbers is given in [35][36][37][38].
The set of all fuzzy real numbers is denoted by .
An alternative definition of fuzzy number is as follows.
Also, we define the distance between two fuzzy vectors (each vector with fuzzy components)  and b as follows: where   is the th row of FFLS and b is the th compnent of fuzzy vector b.
where the elements, ã , of the coefficient matrix Ã, 1 ≤ ,  ≤  and the elements, b , of the vector b are fuzzy numbers, are called fully fuzzy linear systems (FFLSs).

General Solutions
In this section, we suggest a novel and practical method to obtain general solutions of FFLS.To this end, we solve the 1-cut system (5), which is a crisp system.So, we solve the following crisp system: where b (1), ã (1) ∈ R and   ,  = 1, . . .,  are unknown crisp variables which determine by solving system (10).Therefore, we fuzzify, the obtained solution from the crisp system (10), by allocating some unknown general spreads (asymmetric spreads) to each row of the system (10).
Then, crisp system (10) is converted to the following system 2 linear equations: In the above system,   ,  = 1, . . .,  are of the obtained of crisp system (11) and   () > 0,   () > 0,  = 1, . . .,  are unknown spreads.However, for obtaining general solutions of the FFLS, first we have to solve the above equation system, which requires finding the general spreads solution.Now, to solve system (11), we suppose the following one type for the components of fuzzy matrix: Remark 9. Without loss of generality, we explain our method with the assumption that, in interval [  −   (),   +   ()], function   −   () is positive.We just remove some types which elements of fuzzy matrixes are negative and positivenegative, and also zero does exists in the support of elements of fuzzy matrixes and fuzzy solution.Moreover, the ordering > means that   > 0 if and only if   (0) > 0 and   < 0 if and only if   (0) < 0.
So, we consider some situations on linear asymmetric spreads of solutions as follows: Hence, the fuzzy vector solution of FFLS, by using general spreads, will be obtained like the following: However, the obtained spreads of the FFLS in crisp manner should be zero, which we will talk about in the following part.
Have in mind this feature is applicable for the mentioned type.
Proof.Based on the proposed method, we have assumed that 1-cut position is crisp (since the fuzzy values are triangular).So, all spreads are zero, that is, and then we deduce that the solutions in this case coincide with the 1-cut solution, that is, which completes the proof.
Theorem 11.The solution of fully fuzzy linear system (10) is a fuzzy vector.
Proof.We state the proof for the solution, X−,,  , and the proof for the other solution, X+,,  , is similar.So, we omit it.It is easy to verify that X−,,  satisfies for each 0 ≤  ≤ 1.Also, let us consider 0 <  1 ≤  2 ≤ 1, then we have So, we deduce that X−,,  is a fuzzy vector which completes the proof.Theorem 12.For given spreads from (26) and the solutions of FFLS given by (27), one has the following properties: which shows that X−,,  ∈ TSS.Similarly, using the proposed method and definition of X+,,  , one has Proof.Based on the definition of maximal and minimal general solutions, the proof is straightforward.Now, we provide some useful result to show the difference between proposed method and the symmetric solutions [34,39].Theorem 14.Assuming that  −,,  ≤  −,,  , then one has: Proof.By comparing obtained results for the symmetric solution  −,, sym , proposed in [34,39], the proof is straightforward.
The following results show that, under certain conditions, the approach suggested in the present paper has less errors than the Allahviranloo et al. 's method [34].
Proof.Based on definitions of  −,, sym and  −,,  , we have Clearly, in the mentioned fuzzy vector solutions, the lower functions of each component,   −  −,,  ,  = 1, . . ., , are equivalent.So, we can state that  −,, sym =  −,,  .Now, we discuss the upper functions of the fuzzy vector Then, after simple calculations we obtain: ( which completes the proof.
We depict all the solutions via Figures 5 and 6.

Conclusion
In this paper, we presented a practicl method for determining the general solutions of a fully fuzzy linear system.To do so, we firstly solved the system in 1-cut form, then we fuzzify 1-cut solution of the FFLS by devoting general spreads.Therefore, the crisp system was changed into a new system that we should have obtained its spreads.Moreover, we have discussed the obtained result which was placed in the TSS and CSS.Furthermore, we have established that, under certain conditions, proposed method has less errors than the previously reported symmetric solutions.This method is a new approach to find the general solutions of the fully fuzzy linear systems.Also, the presented method always gives a fuzzy vector solution.