Fuzzy Minimal Solution of Dual Fully Fuzzy Matrix ... - Atlantis Press

International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2016)

Fuzzy Minimal Solution of Dual Fully Fuzzy Matrix Equations Dequan Shang1 and Xiaobin Guo2,* 1

Science Courses Teaching Department, Gansu Traditional Chinese Medicine University, Lanzhou 730000, China 2 College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China *Corresponding author [16] investigated the dual fuzzy matrix equations form AX  B  C X  D based on LR fuzzy numbers.

Abstract—The paper deals with the dual fully fuzzy matrix equation A  X  B  C  X  D . The dual fuzzy matrix equation is converted to a crisp system of matrix equations according to arithmetic operations of fuzzy numbers. The fuzzy approximate solution of fuzzy matrix equation is obtained by solving the model which is made of three linear matrix equations. The existence condition of the nonnegative fuzzy solution is also discussed. A example is given to illustrate the proposed method.

II.PRELIMINARIES A. LR Fuzzy Numbers

Let E1 be the set of all fuzzy numbers on R .

Keywords-fuzzy numbers; matrix analysis; dual fuzzy matrix equations; fuzzy approximate solutions

Definition 2.1. A fuzzy number M

is said to be a

LR fuzzy number if I. INTRODUCTION System of simultaneous linear matrix equations is an essential mathematical tool in science and technology. In practice some or all parameters may be represented by fuzzy numbers rather than crisp ones. Therefore it is necessary to develop mathematical theory and numerical schemes to handle fuzzy matrix systems. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [1], Dubois et al.[2] and Nahmias [3].

uM

(1) L  x   L   x  , (2) L  0   1 and L 1  0 ,



(3) L  x  is nonincreasing on 0,   .

However, for a fuzzy linear matrix equation which always has a wide use in control theory and control engineering, few work has been done in the past decades. In 2009 , Allahviranloo et al. [12] discussed the fuzzy linear matrix

The definition of a right shape function R    is usually similar to that of L    .A LR

equations(FLME) of the form AX B  C where

fuzzy number

symbolically shown as M   m ,  , 

A and B are m  m and n  n real matrices respectively, C is a given m  n fuzzy numbers matrix. In 2011 , Gong and Guo [13]   B and investigated a class of fuzzy matrix equations Ax

Clearly, M   m ,  , 

L R

Definition

arbitrary

M is

L R .

is positive(negative) if and

only if m    0  m    0  .

studied its fuzzy least squares solutions by using generalized inverses of the matrix. Later, Guo and Shang [14-15] proposed a computing method of fuzzy symmetric solutions to fuzzy matrix

2.2.

For

M   m,  ,   LR and

 B and discussed the fuzzy Sylvester matrix LR fuzzy numbers. Recently, Guo and Gong

© 2016. The authors - Published by Atlantis Press

  , x  m ,   0,     , x  m ,   0, 

where m is the mean value of M , and  and  are left and right spreads, respectively. The function L   ; which is called left shape function satisfying:

Since Friedman et al. [4] proposed a general model for solving an fuzzy linear systems Ax=b by an embedding approach in 1998, lots of works have been done about some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), fully fuzzy linear systems (FFLS)and general dual fuzzy linear systems (GDFLS) see [5-8]. And some new theories and methods for fuzzy linear systems still appeared recently [9-11].

equations AX equations with

 mx L    x    R  x  m   

(1) Addition

373

LR fuzzy numbers

N   n,  ,   LR , we have

M  N   m ,  ,   LR   n ,  ,   LR   m  n ,    ,     LR 

(2) Multiplication If M  0 and N  0 ; then

M  N   m,,   LR  n,  ,  LR   mn, m  n, m  n  LR  B. The Dual Fully Fuzzy Matrix Equations

 

Definition 2.3. A matrix A  a ij ,1  i  m,1  j  n is called a LR fuzzy matrix, if each element of fuzzy number.

A is a LR

 c11 c12  c 22 c   21  ... ... c  m1 cm 2

... c1n   x11  ... c2 n   x21  ... ...   ... ... cmn   xn1

 d11  d   21  ... d  m1

... d1 p   ... d 2 p  ... ...  ... d mp 

d12 d 22 ... dm2



LR

with new notation A   A , M , N

where A   aij  , M   ij  and N 

   are three m  n

(2.1)

 2.1

and

fuzzy numbers matrix LR X  xij   xij , yij , zij  ,1 i  n,1 j  p is said to the LR LR A

ij

... a1n   x11 x12 ...  a22 ... a2 n   x21 x22 ...  ... ... ...   ... ... ... am 2 ... amn   xn1 xn 2 ...  b11 b12 ... b1 p     b21 b22 ... b2 p     ... ... ... ...   b  b ... b mp   m1 m 2

,

A  X  B  C  X  D 

,

 

solution of dual fuzzy matrix equation  2.1 if

Definition 2.4. The linear system

a12

xn 2

Using matrix notation, we have

crisp matrices.

 a11   a21  ...  a  m1

...

bij , dij ,1  i  m,1  j  p are LR fuzzy numbers, is called a LR dual fully fuzzy matrix equations(DFFMEs).



aij ,1  i  m,1  j  n of A be positive (negative). We can represent m  n fuzzy matrix A   a ij  , that a ij   a ij ,  ij ,  ij 

x22

... x1 p   ... x2 p  ... ...  ... xnp 

a ij , cij ,1  i  m,1  j  n

where

A LR fuzzy matrix A is said to be positive(negative) and if each element denoted by A  0 A 0

x12

X satisfies

Eq.  2.2  , i.e.,

x1 p   x2 p   ...  xnp 

A  X j  B j  C  X j  Dj , j  1,..., p 

where

  d

X j  x1 j , x 2 j ,..., x nj Dj

1j

 , B  b , b  , j  1,..., p

, d 2 j ,..., d mj

are jth column of respectively.

LR

T

j

1j

2j



T

,..., bmj

,

T

fuzzy matrices X , B and D

Up to rest of this paper we want to find the positive solution X   X , Y , Z   0 of nonnegative  2.2  , where

A   A, M , N   0, B   B , E , F   0, C   C , G , H   0 and D   D , P , Q   0 .

III. SOLVING DUAL FULLY FUZZY MATRIX EQUATION

Theorem 3.1. The dual fully fuzzy matrix system

 2 .2  can be extended into the following model system which is made of three linear matrix equations

374

AX  B  CX  D ,    AY  MX  E  CY  GX  P ,   AZ  NX  F  CZ  HX  Q.  Proof.

We

denote

A   A, M, N  , B   B, E, F  , C   C, G, H  , D   D, P, Q

 † X   A  C  D  B ,  † †  Y   A  C  P  E   G  M  A  C  B  D ,  3.3   Z   A  C†  Q  F    H  N  A  C†  B  D . 

 

 

Definition 3.2. Let X   X , Y , Z  . If X   X , Y , Z 

and assume X   X , Y , Z   0 , then

is the minimal solution of Eqs.(3.1) such that X  0, Y  0, Z  0, then we say X   X , Y , Z  is a LR fuzzy minimal solution of Eqs.(2.2). If X   X , Y , Z  also satisfies X  Y  0 , we say it is a

A X  B  C  X  D  i.e.,

, , N  X,Y, Z  B, E, F  C,G, H  X,Y, Z  D, P,Q .  AM

nonnegative LR fuzzy minimal solution of Eqs.(2.2). Theorem 3.3. Let

According to multiplication of nonnegative LR fuzzy numbers of Definition 2.3 , we have

A   A, M , N  , B   B , E , F



and C   C , G , H

 be three nonnegative LR fuzzy † , ,   CX,CYGX,CZHX  DPQ , ,  . matrices, respectively, and  A  C  be nonnegative  AX, AYMX, AZNX  BEF 1 matrix. Let D  B  0, P  E   G  M  A  C   D  B and Thus we obtain a model for solving DFFMEs (2.2) as 1 follows: Q  F   H  N  A  C   D  B  . Then the dual fuzzy systems A  X  B  C  X  D has a LR fuzzy minimal solution. If the condition  D  B  I   G  M  A  C 1   P  E  holds,

AX  B  CX  D,    AY  MX  E  CY  GX  P,    AZ  NX  F  CZ  HX  Q. 



then the dual fuzzy systems (2.1) has a nonnegative LR fuzzy minimal solution.

In order to solve the dual fuzzy system (2.2), we need to consider the crisp system of linear matrix equation (3.1). For example, when A  C is a nonsingular crisp matrix, we obtain its solution as follows:

Proof. Since

 AC

†

and D  B are nonnegative

matrices. Thus the fact X   A  C  apparent. On the P  E  G  M

 1 X   A C  D  B ,  1 1  Y   A C  P  E   G  M  A C  B  D ,(3.2)   Z   A C1  Q  F    H  N  A C1  B  D . 

 



 

†

 D  B  0

other hand, †  A  C   D  B 

is

because

and Q  F 

H

 N

 A  C   D †

 B  , 



so with Y   A  C †  P  E    G  M  A  C †  B  D 

Theorem 3.2([17]). Let A belong to R mn and B belong to R m p . Then the minimal solution X  of the matrix equation AX  B is expressed by



and Z   AC

X   A† B. 

†

Q  F    H  N  A  C   D  B  ,  †

we have Y  0 and Z  0. Thus X   X , Y , Z  is a LR fuzzy solution of the dual fuzzy systems A  X  B  C  X  D. Since

When A  C is a singular matrix, by the Theorem 3.2., we solve model  3.1 and obtain its minimal solution as follows:

375





X Y  AC  DB  PE  GM AC  DB , †

†

and

 z11 z12   x11 x12   2 3 1  1 2 0    1 2     z21 z22      x21 x22      1 1 1 1 1  3 2 2       x x   z31 z32   31 32 

the nonnegative property of the solution X to dual fuzzy systems (2.1) can be obtained from the condition

 I  G  M  A  C   D  B    P  E  . †

IV. NUMERICAL EXAMPLE

 z11  2 1 1    z 21  1 2 3  z  31  4 2  . 3 0

Example 4.1. Consider the dual fully fuzzy matrix system

  2,1,1 LR    3,1,1 LR   2, 2,1 LR     2,1,1 LR

 x11

 3,1, 2  LR 1, 0, 0  LR   x  2, 0,1 LR  2,1,1 LR   21  x31 

x12   x22   x32  

z12   x11  1 0 1  z 22     x21 1 1 1     z32   x31

x12   x22  x32  

According to formula (3.3), the minimal solutions of above three systems of linear matrix equations are as follows:

1,1, 2  LR    3,1,1 LR 

X   A  C   D  B   †

 x11 1,1,1    LR  x  3, 0,1LR   21  x31 

  2,1,1 LR    1, 0,1 LR

1, 0, 0 LR  2,1,1 LR

  4, 2, 2  LR     5, 4,3 LR

 3, 2, 2  LR  .  3, 2, 0  LR 

x12   x22   x32  

†

  2 3 1   1 0 1    4 3   2 1            3 2 2  1 1 1    5 3   2 3     2.0571 1, 2286      1.3143 0.8429   1.1327 2.2857   

By Theorem 3.1., the model to above fuzzy linear matrix system is made of following three crisp systems of linear matrix equations

 x11 x12   x11 x12  2 3 1  2 1 2 1 1  4   x21 x22     x21 x22  3 2 2  2 3 1 2 3 x x  5  x31 x32   31 32 

Y   AC

†

 P  E   G  M  X  

2 3 1 1 0 1 2 2 2 1 1 0 1 1 1 0         X  3 2 2 1 1 1 4 2 1 1 0 1 0 1 0 1  †

3 ,  3

 1.0333 0.1236      01258 0.9896   1.1250 1.3333   

 y11 y12   x11 x12  2 3 1  2 1 2 1 1  2 1   y21 y22     x21 x22    3 2 2  2 3 1 2 3 x x  1 1  y31 y32   31 32 

Z   AC

†

Q  F    H  N  X  

2 3 1 1 0 1 4 2 1 2 1 0 1 1 2 0         X 3 2 2 1 1 1 3 0 1 1 1 1 1 1 1 1   1.0333 0.1236      1.1258 0.0333  .   0.1333 1.3333    †

 y11 y12   x11 x12   2 1 1   1 0 1    2 2    y21 y22     x21 x22      1 2 3   0 1 0   x x   4 2 y y  31 32   31 32 

376

[15] X.B. Guo, D.Q. Shang, “Approximate solutions of LR fuzzy Sylvester matrix equations”, Journal of Applied Mathematics, Volume 2013, Article ID 752760, 10 pages. [16] Z.T. Gong, X.B. Guo, K. Liu, “Approximate solution of dual fuzzy matrix equations”, Information Sciences 266 (2014) 112-133. [17] A. Berman, R.J. Plemmons, “Nonnegative matrices in the Mathematical Sciences”, Academic press, New York, 1979.

By Definition 3.2, we know the original fuzzy linear matrix equations has a nonnegative LR fuzzy solution

x11 x12   2.0571,1,0333,1.0333 1.2286,0.1236,0.1236  LR LR     Xx21 x221.3143,0.1258,0.1258LR  0.8429,0.9896,0.0333LR,   x31 x32  1,1327,1.1250,0.1333LR  2.2857,1.3333,1.3333LR    Since X  0, Y  0, Z  0 and X  Y  0. V. CONCLUSION In this work, we proposed a simple model for solving the general dual fully fuzzy matrix equation. We converted the dual fuzzy matrix equation to a crisp system of linear matrix equations according to the arithmetic operations of LR fuzzy numbers. The existence condition of nonnegative LR fuzzy solution was also studied. Numerical examples showed that our method is effective to solve the dual fully fuzzy matrix equation. ACKNOWLEDGEMENTS This research was supported by the National Natural Science Foundation of China (no.61262022). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

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