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We propose a method to approximate the solutions of fully fuzzy linear system (FFLS), the so-called

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 system of linear equations

The linear system

Buckley and Qu in their consecutive works [

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 [

Vroman et al. in their continuous works [

Recently, Allahviranloo et al. [

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 [

The structure of this paper is organized as follows.

In Section

Let

A nonempty bounded subset

A fuzzy number is a function such as

An alternative definition of fuzzy number is as follows.

A fuzzy number

The Hausdorff distance between fuzzy numbers given by

Also, we define the distance between two fuzzy vectors (each vector with fuzzy components)

The

A fuzzy vector

The united solution set (USS), the tolerable solution set (TSS), and controllable solution set (CSS) for the system (

A fuzzy vector

A fuzzy vector

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 (

Then, crisp system (

Without loss of generality, we explain our method with the assumption that, in interval

Consider the linear asymmetric spreads (

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,

The solution of fully fuzzy linear system (

We state the proof for the solution,

For given spreads from (

Let us consider

Consider

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 [

Assuming that

By comparing obtained results for the symmetric solution

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 [

Assuming that

By comparing obtained results in [

If

Based on definitions of

Now, we discuss the upper functions of the fuzzy vector solutions

In this section, we take an example which has been solved in [

Consider the following FFLS:

Compare

Compare

Furthermore, the solutions of the suggested method are plotted to compare with Dehghan's method (

Compare the proposed solution

Compare the proposed solution

Note that Dehghan's solution [

Consider the following fully fuzzy linear system:

The 1-cut solution of system is

So, we have

Compare

Compare

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.

The reviewers' comments, which have improved the quality of this paper, are greatly appreciated.