Fuzzy linear system of the form $\tilde{A}_{1} X\Theta_{gH}\tilde{A}_{2} X=\tilde{b}$

(1)

Fuzzy linear system of the form

A X

₁

Θ A X b

_gH ₂



Kh. Sabzi1_{, M. Afshar}2_{, M. Keshavarz}1*

(1) Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. (2) Department of Mathematics, Islamic Azad University, Tehran, Iran.

Copyright 2016 © Kh. Sabzi, M. Afshar and M. Keshavarz. 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.

Abstract

In this paper, we shall propose a new method to obtain solutions of a fully fuzzy linear system (FFLS) based on concept of generalized Hakuhara difference. FFLS of the form

A X

₁



A X b

₂



is converted to

1 gH 2

A XΘ A X b that is solved by two kind of solutions. The first kind of FFLS A X₁ Θ A X b_gH ₂  is the same as the FFLS



A

₁



A

₂



X



b

will be the same. But in the latter the spread of solutions are higher in number.

Keywords: Fully Fuzzy Linear System (FFLS), Maximal Symmetric Solution, Minimal Symmetric Solution, Generalized Hukuhara Differentiability.

1 Introduction

Linear system have important application in many branches of science and engineering. the system of linear equations

AX



b

where the elements, aij, of the matrix A are crisp value and the elements,

b

_i are the vector b are fuzzy numbers, is called a fuzzy linear system (FSLE).The system of linear equation

AX



b

where the elements

b

_i of the vector b are fuzzy number, is called a fully fuzzy linear system (FFLS).FFLSs have been studied by many authors. Buckley and Qu in their continuous work [5-7] proposed different solution for FFLSs. Fried man et al. [11] use the embedding method given in [15]. Allahviranloo [2-4] use the iterative Jacobi and Gauss Siedel method, and the Adomian method respectively. Ma et al. [14] starting from the work by Friedman et al. [11], analyse the solution of fuzzy sysems of the form A1x=A2x+b. they remark that the system A1x=A2x+b is not equivalent to the system (A1-A2) x=b since for an arbitrary fuzzy number u there exite no element v such that u+v=0. They give the conditions under which the new system has a solution.

Volume 2016, Special Issue 1, Year 2016 Article ID jfsva-00279, 13 Pages

doi:10.5899/2016/jfsva-00279

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Also, Wang et al. [16] presented an iterative algoritm for solving dual linear system of the form x=Ax+u, where A is a real n*n matrix, the unknown vector x and the constant u are all vectors consisting of fuzzy numbers. Recently, Muzziloi and Reynaerts [13] considered fuzzy linear systems of the form A1x+b1=A2x+b2 with A1, A2 square matrices of fuzzy coefficients and b1, b2 fuzzy number vectors. In srction 2, we present some basic definitions. In section 3, we propose our new method to solve a FFLS. Finally, numerical example are given in section 4 and conclusions are drawn in section 5. Consequently, Dehghan et al. [8,9], proposed Cramer rule, Gaussian elimination, LU decomposition (Doolittle algoritm) and its simplificationthis paper is to decide how to deal with decision making units that have negative and interval inputs and outputs.

2 Preliminaries

The basic definitions of a fuzzy number are given in [10, 12, 17] as follows:

Definition 2.1.A fuzzy number u in parametric form is a pair



u , u



of functions

u r , u r , 0

   

 

r 1

, which satisfy the following requirements:

1.

u r

 

is a bounded non-decreasing left continuous function in



0,1



, and right continuous at 0, 2.

u r

 

is a bounded non-increasing left continuous function in



0,1



, and right continuous at 0, 3.

u r

   



u r , 0

 

r 1

.

The trapezoidal fuzzy number

u





x , y , ,

₀ ₀

 



, with two defuzzifiers

x , y

₀ ₀ and left fuzziness

 

0

and right fuzziness

 

0

is a fuzzy set where the membership function is as

 









0 0 0

0 0

0 0 0

1

x x x x x ,

1 x x y ,

u x 1

y x y x y ,

0 otherwise.

 _ _{ } _{   }

 

  

 

 _{  } _{ } _{ }

  

If

x

₀



y

₀, then u is called a triangular fuzzy number and we write

u





x , ,

₀

 



. The support of fuzzy number u is defined as follows: sup p u

 





x | u x

 

0 ,



Where



x | u x

 

0



is closure of set



x | u x

 



0 

.

For arbitrary fuzzy numbers u r

 

 _u r , u r

   

_ and 

 

r  _

   

r , r _, we shall define addition, subtraction and multiplication as follows for

0  

r 1

:

1- Addition:



u 

 

r _u r

       

  r , u r , r _

2- Subtraction:



u 

 

r _u r

       

  r , u r   r _

3- Multiplication:

(3)

Definition 2.2. The

n n



linear system of equations

11 1 12 2 1n n 1

21 1 22 2 2n n 2

n1 1 n2 2 nn n n

a x

... a x

b

a x

a

x

... a

x

b

a x

a

x

... a

x

b





 







 









_

_{ }

_



where the elements,

a

_ij of the coefficient matrix A,1 i , j n, and the elements

b

_i, of the vector b are fuzzy numbers are called fully fuzzy linear systems (FFLS).

Definition 2.3. A fuzzy vector X



x ,..., x₁ _n



t given by x_i 



x_i

   

r , x_i r



, 1 i n , 0 r 1

is called the solution of (3.2) if

n n n n

ij j ij j i ij j ij j i

j 1 j 1 j 1 j 1

a x

b

,

a x

b .

   





Definition 2.4.Let A B,  _f. If there existsC f such that A B C, then C is called the Hukuhara

difference of A and B, it is denoted by B.

Definition 2.5. The generalized Hukuhara difference of two fuzzy number A B,  _f is defined as follows:

 

A=B+C, B=C

B=A+ .

gH

i A

or ii C

  



 1

(i) = + = -

= + = -

A B C C A B

A B C

A B C C A B 

    

 

(ii)

 

= - = -

B A C C A B B A

B A C C A B

 

     

 

1

3 Main Section

In this section we consider two FFLS:

AXb (3.1)

1 2

A xA x b  (3.2)

11 1n 11 1n 11 1n

1 2

n1 nn n1 nn n1 nn

a a a a a a

A A A

a a a a a a

   

     

     

_ _ _ _ _ _

   _ _   _ _ 

     

For finding the solutions system (3.2) according to definition gH difference have:

 

1 2

1 gH 2

2 1

i A X

A X

b

A X

Θ A X b

or

ii A X

A X

1 b









  



_

_{ }



(4)

Step1: solve the 1-cut of the FFLS to obtain a crisp solution of the FFLS.

 





 

n

ij ij j i

j 1

a 1 a 1 x b 1 , i 1,..., n



    



(3.3)

Where, b 1 , a_i

 

_ij

 

1 R and

x

_j is an unknown crisp variable which will be determined by solving system (3.3). Regarding to the gH difference consider the following system:

 

_n

11 1 1n n gH 11 1 1n n 1

21 1 2n n gH 21 1 2n n 2

n1 1 nn n gH n1 1 nn n

a

r x

a

r x

Θ

a r x

b

a

r x

a

r x

Θ

a r x

a

r x

b

a

r x

a

r x

Θ

a r x

b

 



_

 





_



_



 





_













_

 



_



_

 



_







 



_

 





_



_



 







_





(3.4)

Hence, we set:

(3.5)

(3.6)

Let

(3.7)

Step 2. Allocate some unknown symmetric spreads to each row of a 1-cut of a FFLS. Using Eq. (3.5), (3.6), (3.7).have:

 







 





 





 





 



 







   



 







 





 





 





 



 







 



a₁₁ r a₁₁ r , a₁₁ r a₁₁ r x₁ ₁ r , x₁ ₁ r ... a_1n r a_1n r , a_1n r a_1n r

x_n ₁ r , x_n ₁ r b₁ r , b₁ r

a₂₁ r a₂₁ r , a₂₁ r a₂₁ r x₁ ₂ r , x₁ ₂ r ... a_2n r a_2n r , a_2n r a_2n r

x_n ₂ r , x_n ₂ r b₂ r , b₂ r

a n

                 

    



                

    



   

   

 

  _ _

 







 





 





 





 



 







 



r a_n1 r , a_n1 r a_n1 r x₁ _n r , x₁ _n r ... a r a_nn r , a_nn r a_nn r

1 nn

x_n _n r , x_n _n r b_n r , b_n r

      

         

    

         

  __ __

 

 

(3.8)

Where

 

_i

0 , i 1, 2,..., n



are unknown spreads which will be determined by the solution solving 2n above equations and

x , j 1,..., n

_j



are elements of the obtained vector solution of system (3.3).

We consider three separate case for the elements of matrix A , A as follow: ₁ ₂

 





1 n n ij ij

1. I



i, j



N



N | a , a

 



0 ,

 





2 n n ij ij

2. I



i, j



N



N | a , a

 



0 ,

3 1 2

3. I



I

I .

Where,

N

_n





1,..., n .



 





 





n

ij ij j i

j 1 n

ij ij j i

j 1

a

r

a

r x

b i=1,...,n

a

r

a

r x

b i=1,...n





























11 11 nn nn

a

a ,..., a

a

,

a

a ,..., a

a













(5)

Remark 3.1. Notice that we consider that

 

i i1

x

 

r

and

x

_i

 

_i2

 

r

are positive, and describe our theory by such an assumption. However, we just omitted some cases where zero does not exists in the support of elements of fuzzy matrices and fuzzysolutions.

Case (1):

I

₁





 

i, j



N

_n



N |a , a

_n _ij

 

_ij



0 , I



₁



n

2

Here, the fuzzy matrix

A , A ,

₁ ₂ is assumed positive. So, the i-th row of system (3.8) is presented as following:

 



 



 





   

i1 1 i 1 i in

i1 in

n i n i

i1 i1 in in

i i

a r , a r x r , x r ... a r , a r

x r , x r r , b r

a r a r a r a r

b

        

    

  _ _ _ _  __ _ _ _ _ __

 

 

 

The

compact form of above equation is given below:

 







 



 



 





 



 



n

ij ij j i i

j 1

i 1 1 n i1 ij in ij i

a

r

a

r

x

r

b r

r

f

x ,..., x , a

r

a

r ,..., a

r

a

r , b r









 



















(3.9)

(3.10)

We replace

α r

_i1

 

with

α r

_i

 

in Eq. (3.10) and replace

α r

_i2

 

with

α r

_i

 

in Eq. (3.11).

 



 





 



 



 



 





 



 



i1 1 1 n i1 i1 in in i

i2 2 1 n i1 in in in i

r f x ,..., x , a r a r ,..., a r a r , b r ,

r f x ,..., x , a r a r ,..., a r a r , b r

   

   

   

    (3.11)

Step 3. Finding the symmetric spread

We suggest two total procedures to determine the symmetric spread of solutions of the FFLS.

 



 



s i1 i2

0 r 1

α r



min α r , α r , i=1,...,n

 



(3.12)

 



 



s i1 i2

0 r 1

α r



max α r , α r , i=1,...,n

 



(3.13)

So,

X r

 





x r ,..., x

₁

 

_n

 

r



t is obtained as a follow:

 

i i s i s

x r _x α r ,x α r    _ (3.14) and

(3.15)

Sometimes, obtaining the minimum of several functions on the interval

 

0,1

is not easy work, also, even if we do it, the final spreads



_s

 

r ,_s

 

r



are not linear functions. So, in order to achieve linear spreads, we carry out some changes in the structure of the obtained spreads Eqs. (3.12)-(3.13). By using Eq. (3.11),

 







 



 



 





 



 



n

ij ij j i i

j 1

i 2 1 n i1 ij in ij i

a r a r x r b r

r f x ,..., x , a r a r ,..., a r a r , b r .



     

   

    



 

i i s i s

(6)

the linear form of the spreads of fuzzy symmetric solutions are determined as follows:

 





 





 





 





n

ij j i i i

ij j 1

i1 _n

ij ij

j 1

n

ij ij j i i i

j 1

i2 _n

ij ij

j 1

a

r

a

r

x

b 1

r

,

a

r

a

r

a

r

a

r

x

b 1

r

.

a

r

a

r











   



















  













(3.16)

let a_ij

 

r _a_ij

   

0 , a_ij 1 _, a_ij

 

r  a_ij

   

0 , a 1_ij _ where supp

 

   

ij ij ij ij ij ij

a





_



a



0 , a



0 , a



_



 



a



0 , a



0 

_

, then :

   







   





   



n n n

ij ij ij

0 r 1 _{j 1} _{j 1}0 r 1 _{j 1}

min

a

r , a

r

min

a

r , a

r

a

0 , a

0 , i 1,..., n

  _ _   _







_

_



_

_

_

_

_

_

_















   







   





   



n n n

ij ij ij

0 r 1

0 r 1 j 1 j 1 j 1

max

a

r , a

r

min

a

r , a

r

a

1 , a

1 , i 1,..., n ,

 

  _ _ _







_

_



_

_

_

_

_

_

_















   







   





   



n n n

ij ij ij ij ij ij

0 r 1 _{j 1} _{j 1}0 r 1 _{j 1}

min

a

r , a

r

min

a

r , a

r

a

1 , a

1 , i 1,..., n ,

  _ _   _







_

_



_

_

_

_

_

_

_















 







 





 



n n n

ij ij ij ij ij ij

0 r 1 _{j 1} _{j 1}0 r 1 _{j 1}

max

a

r

a

r

max

a

r

a

r

a

0 a

0 , i 1,..., n ,

  _ _   _







_

_

_



_

_

_

_

_

_

_

_

_















Hence we set:

 





 





n

ij j i

ij j 1 l

i1 _n

ij ij

j 1

a

r

a

r

x

b 1

r

, i 1,..., n ,

a

1 a

r





_

_

_













(3.17)

 





 





n

ij j i

ij j 1 u

i1 _n

ij ij

j 1

a

r

a

r

x

b 1

r

, i

1,..., n ,

a

0 a

0

















_

_



(7)

 





 





n

ij ij j i

j 1 l

i2 _n

ij ij

j 1

a

r

a

r

x

b 1

r

, i 1,..., n ,

a

0 a

0























(3.19)

 





 





n

ij ij j i

j 1 u

i2 _n

ij ij

j 1

a

r

a

r

x

b 1

r

, i 1,..., n ,

a

1 a

1























(3.20)

Linear symmetric spreads of solutions of the FFLS are as follow:

 



 



,l l l

s i1 i2

0 r 1

r

min

r ,

r

,



 







(3.21)

 



 



,u u u

s i1 i2

0 r 1

r

min

r ,

r

,



 







(3.22)

 



 



,l l l

s i1 i2

0 r 1

r

max

r ,

r

,



 







(3.23)

 



 



,u u u

s i1 i2

0 r 1

r

max

r ,

r

,



 







(3.24) Therefore, by applying various symmetric spreads, the corresponding solutions are derived as follows:

 



 



t

 

,l ,l ,l ,l ,l ,l

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

(3.25)

 



 



t

 

,u ,u ,u ,u ,u ,u

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

(3.26)

 



 



t

 

,l ,l ,l ,l ,l ,l

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

(3.27)

 



 



t

 

,u ,u ,u ,u ,u ,u

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

(3.28)

(8)

 







 



 



 



 





n

ij ij j i1 i

j 1

n

i ij ij j

j 1

i1 _n

ij ij j 1

a r a r x r b r

b r a r a r x

r , i 1,..., n ,

a r a r



     

 

 

   

  



(3.29)

 







 



 





 





n

ij ij j i2 i

j 1

n

ij ij j i

j 1

i2 _n

ij ij j 1

a r a r x r b r

a r a r x b r

r , i 1,..., n

a r a r



     

   

   

  



(3.30)

Hence, various spreads of the solution of the FFLS for each row are derived as follows:

 



 



 





n

i ij ij j

j 1 l

i1 _n

ij ij

j 1

b r

a

r

a

r

x

r

, i

1,..., n ,

a

1 a

1























(3.31)

 



 



 





n

i ij ij j

j 1 u

i1 _n

ij ij

j 1

b r

a

r

a

r

x

r

, i

1,..., n ,

a

0 a

0























(3.32)

 





 





n

ij ij j i

j 1 l

i2 _n

ij ij

j 1

a

r

a

r

x

b r

r

, i 1,..., n ,

a

0 a

0























(3.33)

 





 





n

ij ij j i

j 1 u

i2 _n

ij ij

j 1

a

r

a

r

x

b r

r

, i 1,..., n ,

a

1 a

1























(3.34)

Case (3): Without loss of generality, consider the i-th row a_ij

   

r , a_ij r 0 , j N_s and

   

ij ij n s

(9)

 







 





 





 



 

s n

ij j i ij ij j i i

ij

j 1 j s 1

a

r

a

r

x

r

a

r

a

r

x

r

b r , i 1,..., n ,

  







 









 





(3.35)

 







 





 





 



 

s n

ij ij j i ij ij j i i

j 1 j s 1

a

r

a

r

x

r

a

r

a

r

x

r

b r , i 1,..., n .

  







 









 





(3.36)

we get the following,

 





 







 



n

ij ij j i

j 1

i1 _s _n

ij ij ij ij

j 1 j s 1

a

r

a

r

x

b r

r

,

i

1,..., n ,

a

r

a

r

a

r

a

r

   





























(3.37)

 





 







 



n

ij ij j i

j 1

i2 _n _s

ij ij ij ij

j s 1 j 1

a

r

a

r

x

b r

r

,

i

1,..., n

a

r

a

r

a

r

a

r

   





























(3.38)

Thus, various types of linear spreads of solutions are modified as follows:

 





 







 



n

ij j i

ij j 1 l

i1 _s _n

ij ij

j 1 j s 1

a

r

a

r x

b r

r

, i 1,..., n ,

a

1 a

0 a

0

   



_

_

_





















(3.39)

 





 







 



n

ij j i

ij j 1 u

i1 _s _n

ij ij ij ij

j 1 j s 1

a

r

a

r x

b r

r

, i

1,..., n ,

a

0 a

1 a

1

   





























(3.40)

 





 







 



n

ij ij j i

j 1 l

i2 _n _s

ij ij ij ij

j s 1 j 1

a

r

a

r x

b r

r

, i 1,..., n ,

a

0 a

1 a

1

   





























(3.41)

 





 







 



n

ij ij j i

j 1 u

i2 _n _s

ij ij ij ij

j s 1 j 1

a r a r x b r

r , i 1,..., n ,

a 1 a 1 a 0 a 0

                 



(3.42)

(10)

 



 



t

 

,l ,l ,l ,l ,l ,l

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

 



 



t

 

,u ,u ,u ,u ,u ,u

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

 



 



t

 

,l ,l ,l ,l ,l ,l

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

 



 



t

 

,u ,u ,u ,u ,u ,u

n i s i s

1 i

X



r



x



r ,..., x



r

, s.t.x



r





_

x

 



r , x

 



r



_

,

4 Numerical Example

Example 4.1.Consider the following FFLS:



 















4 r , 6 r

5 r ,8 2r

40 10r , 67 17r

A

, b

.

6 r , 7

4,5 r

43 5r , 55 7r























_

_



_

_

















So, crisp solution (1-cut) is obtained as

X

c





x , x

₁ ₂



t, where



x , x

₁ ₂

  

t



4,5

t. Then, crisp system (3.3) is fuzzified as follows:





 





 









 





 





1 1 1 1

2 2 2 2

4 r,6 r 4 r , 4 r 5 r,8 2r 5 r ,5 r 40 10r,67 17r

6 r,7 4 r , 4 r 4,5 r 5 r ,5 r 43 5r ,55 7r .

                    

    



                 

 _ _ _ _



So, in order to determine the spreads of the FFLS, we should construct 4 linear equations as follows:



4 r 4







 

1

 

r



 



5 r 5





 

1

 

r





40 10r





6 r







4  

1

 

r



 



8 2r 5





 

1

 

r





67 17 r





6 r







4  

2

 

r





4 5



 

2

 

r





43 5r



 



2









2

 



7 4

 

r

 

5 r 5

 

r



55 7r.



Then, by solving Eqs. above, we get:

 

11

1 r

r

,

9 2r









 

12

3 3r

r

,

14 3r









 

21

1 r

r

,

10 r









 

22

2 2r

r

.

12 r









In addition, by applying Eqs. (3.21)-(3.24), we obtain various types of linear spreads of solutions of the FFLS as follows:

 

,l ,u ,l ,u

s s s s

1 r

3 3r

r

,

r

,

r

,

r

.

11

10

14

11



























(11)

 

t

,l

1 r

X

r

4 , 4

, 5

,

11



_





_



_



 

_



_

 







_

 

_{ }



_







(4.43)

 

t

,u

1 r

X

r

4 , 4

, 5

,

10



_





_



_



 

_



_

 





_

_

_{ }

_{ }

_

_







(4.44)

 

t

,l

3 3r

X

r

4 , 4

, 5

,

14



_





_



_



 

_



_









_

_

_{ }

_{ }

_

_







(4.45)

 

t

,u

3 3r

X

r

4 , 4

, 5

,

11



_





_



_



 

_



_











_

 

_{ }



_







(4.46) Here matrix A divided two part:



 











4 r , 6 r

5 r ,8 2r

A

6 r , 7

4,5 r













 

_

_









 





 



1

r 3, 4

3, 4 r

A

r 2,3

5, 6 r









 

_









 



2

r 2, r r 4 , r 2

A 4 1          _   









40 10 r , 67 17r

b

43 5 r , 55 7r











_



_









Type (







 





 





 









 



 











 









 









 



 







 





 





 



 











 





 





 



1 1 1 1

2 2 2 2

3 r

4 r

3 5

r

4 r

2 r 5

r

40 10 r

4 4

r

4 r 5

r

2 r

4 r

4 r 5

r

67 17 r

i )

3 4

r

5 5

r

4 4

r

5 r

43 5 r

3 4

r

6 r 5

r

4 4

r

5 r

55 7r

 







 

 









 

 

 

 

 

 

 









 



















 

 

 



 

  







 

11 12 21 22

1 r

3 3r

1 r

2 2r

r

,

r

,

r

,

r

.

9 2r

14 3r

10 r

12 r



























Type (

 

t ,l t ,u t ,l ,u

1 r 1 r 1 r 1 r

X r 4 , 4 , 5 , 5 ,

11 11 11 11

1 r 1 r 1 r 1 r

X r 4 , 4 , 5 , 5

10 10 10 10

i )

3 3r 3 3r 3 3r 3 3r

X r 4 , 4 , 5 , 5

14 14 14 14

3 3r 3 3r

X r 4 , 4

11 11             __   _{ }   __               __   _{ }   __               __   _{ }   __              t

3 3r 3 3r

, 5 , 5

(12)

Type







 





 





 









 



 











 









 









 



 







 





 





 



 











 





 





 



1 1 1 1

2 2 2 2

3 r

4 r

3 5

r

4 r

2 r 5

r

67 17 r

4 4

r

4 r 5

r

2 r

4 r

4 r 5

r

40 10 r

(i i )

3 4

r

5 5

r

4 4

r

5 r

55 7r

3 4

r

6 r 5

r

4 4

r

5 r

43 5 r

 







 

 









 

 

 

 

 

 

 





























 

 

 



 

  







 

11 12 21 22

26 r 26

24 r 24

11r 11

10 r 10

r

,

r

,

r

,

r

.

9 2r

14 3r

10 r

12 r



























 

t

, l

t

, u

, l

26 26 r 26 26 r 26 26 r 26 26 r

X r 4 , 4 , 5 , 5 (4.47)

11 11 11 11

26 26 r 26 26 r 26 26 r 26 26 r

X r 4 , 4 , 5 , 5 (4.48)

9 9 9 9

Type (i i )

24 24 r

X r 4

14





       

    

       

    

   

   

__ _{ }_{ } __

 

    

__ _{ }_{ } __ 

 

 

t

,u

24 24 r 24 24 r 24 24 r

, 4 , 5 , 5 (4.49)

14 14 14

24 24 r 24 24 r 24 24 r 24 24 r

X r 4 , 4 , 5 , 5 (4.50)

11 11 11 11



     

  

       

    

      

 

    

 

 __  _{ } _ _



 __ _{ } _ _

 ___ _{ }_{ } __ _



5 Conclusion

In this paper, we proposed a method to solve fully fuzzy linear system. We saw that with displacement the gH difference, solutions of FFLS

A X

₁



A X b , A

₂





₁



A

₂



X



b

, are the same. The only difference is that different spread of solutions FFLS

A X

₁



A X b

₂



are higher in number.

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