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Advances in Pure Mathematics, 2012, 2, 373-378
Published Online November 2012 (http://www.SciRP.org/journal/apm)
http://dx.doi.org/10.4236/apm.2012.26056
Approximate Solution of Fuzzy Matrix Equations
with LR Fuzzy Numbers
Xiaobin Guo1, Dequan Shang2
1College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
2Department of Public Courses, Gansu College of Chinese Medicine, Lanzhou, China
Email: guoxb@nwnu.edu.cn, gxbglz@163.com
Received August 10, 2012; revised September 12, 2012; accepted September 20, 2012
ABSTRACT
In the paper, a class of fuzzy matrix equations AXB where A is an m × n crisp matrix and B is an m × p arbitrary
LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then
the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The ex-
istence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are
given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.
Keywords: LR Fuzzy Numbers; Matrix Analysis; Fuzzy Matrix Equations; Fuzzy Approximate Solution
1. Introduction AXB .
Systems of simultaneous matrix equations are essential The LR fuzzy number and its operations were firstly
mathematical tools in science and technology. In many introduced by Dubois [2]. In 2006, Dehgham et al. [6]
applications, at least some of the parameters of the sys- discussed the computational methods for fully fuzzy lin-
tem are represented by fuzzy rather than crisp numbers. ear systems whose coefficient matrix and the right-hand
So, it is very important to develop a numerical procedure side vector are denoted by LR fuzzy numbers. In this
that would appropriately handle and solve fuzzy matrix paper, we propose a practical method for solving a class
systems. The concept of fuzzy numbers and arithmetic of fuzzy matrix system AXB in which A is an m × n
operations were first introduced and investigated by Za- crisp matrix and B is an m × p arbitrary LR fuzzy
numbers matrix. In contrast, the contribution of this pa-
deh [1] and Dubois[2]. per is to generalize Dubois’ definition and arithmetic op-
Since M. Friedman et al. [3] proposed a general model eration of LR fuzzy numbers and then use this result to
for solving a n × n fuzzy linear systems whose coeffi- solve fuzzy matrix systems numerically. The importance
cients matrix is crisp and the right-hand side is a fuzzy of converting fuzzy linear system into two systems of
number vector in 1998, many works have been done linear equations is that any numerical approach suitable
about how to deal with some advanced fuzzy linear sys- for system of linear equations may be implemented. In
tems such as dual fuzzy linear systems (DFLS), general addition, since our model does not contain parameter r,
fuzzy linear systems (GFLS), fully fuzzy linear systems 01r
(FFLS), dual fully fuzzy linear systems (DFFLS) and , its numerical computation is relatively easy.
general dual fuzzy linear systems (GDFLS), see [4-9]. 2. Preliminaries
However, for a fuzzy linear matrix equation which al-
ways has a wide use in control theory and control engi-
neering, few works have been done in the past decades. Definition 2.1. [2] A fuzzy number M is said to be a LR
In 2010, Gong Zt [10,11] investigated a class of fuzzy fuzzy number if
mx
matrix equations AXB by means of the undeter- Lx,,m 0,
mined coefficients method, and studied least squares so- x
lutions of the inconsistent fuzzy matrix equation by using M
xm
Rx,,m 0,
generalized inverses. In 2011, Guo X. B. [12] studied the
minimal fuzzy solution of fuzzy Sylvester matrix equa-
tions AXXBC. Recently, they [13] considered the where m is the mean value of M , and and are
fuzzy symmetric solutions of fuzzy matrix equations left and right spreads, respectively. The function L.,
opyright © 2012 SciRes. APM
C
374 X. B. GUO, D. Q. SHANG
which is called left shape function satisfying: 1) Lx where a are crisp numbers and b are LR fuzzy num-
ij ij
Lx L 01 L 10; 3) Lx is a non bers, is called a LR fuzzy matrix equation (LRFME).
; 2) and
increasing on 0,. Using matrix notation, we have
The definition of a right shape function L . is usually
AXB . (2)
similar to that of L.. A LR fuzzy numberM is sym-
bolically shown as Mm ,, .
A LR fuzzy numbers matrix
LR
Noticing that 0, 0 in Definition 2.1, which
limits its applications, we extend the definition of LR T lr
Xx , x xxx,, ,
ij ij ij ij ij
fuzzy numbers as follows. np LR
Definition 2.2. (Generalized LR fuzzy numbers) Let , 1jp
1 in
Mm ,, , we define
LR solution of the LR fuzzy matrix systems if
1) 0 and 0, then is called a X
if satisfies (2).
Mm,0,Max , , and
LR
0, x m, 3. Method for Solving LRFME
In this section we investigate the LR fuzzy matrix system
x
xm
M ,.
Rxm
(2). Firstly, we propose a model for solving the LR fuzzy
max ,
matrix system, i.e., convert it into two crisp systems of
2) if 0 and 0, then matrix equations. Then we define the LR fuzzy solution
and give its solution representation to the original fuzzy
Mm,Max , ,0 , and
LR matrix system. At last, the existence condition of the
strong LR fuzzy solution to the original fuzzy matrix
mx
system is also discussed.
m
Lx,,
x max ,
M
0, . 3.1. Extended Crisp Matrix Equations
x m
By using arithmetic operations of LR fuzzy numbers, we
3) if 0 and 0, then Mm,, , and
LR extend the LR fuzzy matrix Equation (2) into two crisp
matrix equations.
mx
Lx,,m
Theorem 3.1. The LR dual fuzzy linear Equation (2)
x can be extended into two crisp systems of linear equa-
M
xm
tions as follows:
Rx,.m
AXB
, (3)
For arbitrary LR fuzzy number Mm ,, and
i.e.,
LR
Nm
,, , we have
LR aa ax xx
11 12 1n 11 12 1p
1) MNmn,, .
LR aa ax xx
21 22 2p
21 22 2n
n,, , 0,
LR
2) N
n,,,0.
aaax xx
RL nn np
mm mn12
12
bb b
Definition 2.3. The matrix system
11 12 1p
bbb
21 22 2p
x xx
aa a
11 12 1n 11 12 1p
x xx
aa a
21 22 2n 21 22 2p
bbb
mm mp
12
x xx
aaa
nn
mm mn12 np
12
(1) and
bb b
11 12 1p ll
XB
bb b
, (4)
21 22 2p SF
rr
XB
bbb
mm mp
12 i.e.,
Copyright © 2012 SciRes. APM
X. B. GUO, D. Q. SHANG 375
ll l ll l
xxx bb b
11 12 1 11 12 1
pp
ll l ll l
xxx bb b
21 22 2 21 22 2
pp
ss s
11 12 1,2n
lll
rr r
ss s
xx x bb b
21 22 2,2n
11 mp
12 mm
nn np
,
rr r rr r
xx x bb b
11 12 1p 11 12 1p
rrr
rr r
ss s
xx x bb b
2,1 2,2 2,2
mm mn 21 22 2p
21 22 2p
rr r rr r
xx x bb b
11
11 mm mp
nn np
where s , 12im, 12jn are determined as fol- Consider the given LR fuzzy vector
lows: ij
lr lrT,
Bb ,,bb,,b,b,b
If a 0 , then s a , s a ; if a 0 ,
j j j j nj nj nj
ij ij ij mi,nj ij ij 111
then s a , s a , and any s which is not LR LR
in, j ij mi ,n ij kl we can write the system (2) as
determined by the above items is zero, 12km ,
1l2n.
lr
axax,,ax ax
Proof. Let XX ,,X X, ik kj ik kj ik kj ik kj
12 p
kQkQ kQkQ
jjjj
T
lr lr
Xx ,,xx,,x,x,x rllr
j 11j j 1j nj nj nj
LR LR ax ax B,,BB
ik kj ik kj j j j LR
kQ kQ
jj
LR
BB ,,B B,
and
12 p
Suppose the system AXB , 1jp has a solu-
j j
lr lrT tion. Then, the corresponding mean value
Bb ,,bb,,b,b,b . T
j 11j j 1j nj nj nj
LR LR Xx ,,x,x of the solution must lie in the
12
jjjnj
Then the fuzzy matrix Equation (1) can be rewritten in following linear system
the block forms aa ax b
11 12 1n 11j j
aa ax b
,
A XX,,X B,B ,B 22j j
21 22 2n
12 p 12 p
. (7)
Thus the original system (1) is equivalent to the fol-
aaax b
mm12 mnnj nj
lowing fuzzy linear equations
T
llll
Meanwhile, the left spread
AXB,1 jp. (5) Xx ,,x,x
jj jj12jnj
T
and the right spread rrrr
of the so-
Now we consider the Equations (4). Let a be the ith Xx ,,x,x
12
i jjjnj
lution can be derived from solving the following crisp
row of matrix A, 1im, we can represent AXji
linear system
1,
in the form AXa X, im 2,, . ll
j ij
i x b
11j j
Denoting and
Qa:a0
iikik ss s
11 12 1,2n
Qa:a0, we have ll
iikik
ss sx b
21 22 2,2n nj nj
. (8)
rr
x b
11
j j
AXaxax,1i,2,,m.
j ikkj ikkj
i
kQkQ ss s
jj 2,mm1 2,2 2,m2n
rr
x b
nj nj
i.e.,
Finally, we restore the Equation (5) and obtain above
l matrix Equations (3) and (4).
AX ax ax
, ax
j ikkj ikkj ikkj
i The proof is completed.
kQ kQ kQ
jjj
(6)
rrl 3.2. Computing Model Matrix Equations
ax
, ax ax
ik kj ik kj ik kj
kQ kQ kQ
jjj
LR In order to solve the original fuzzy linear Equation (2),
Copyright © 2012 SciRes. APM
376 X. B. GUO, D. Q. SHANG
we need to consider crisp matrix Equations (3) and (4). By the above a
nalysis, we have the following result.
mn
Since Equations (3) and (4) are crisp, their computation Theorem 3.2. Let A belong to R . If S is non-
is relatively easy. negative, the solution of the LR fuzzy matrix system (2)
In general [14], the minimal solutions of matrix sys- is expressed by
tems (3) and (4) can be expressed uniformly by lr
XX ,,XX
LR
(13)
X AB (9)
Ad,,IOSFOI SF
nn
and LR
ll and it admits a strong minimal LR fuzzy solution.
XB
The following Theorem gives a result for such S to
(10)
SSF
r
r
XB be nonnegative.
respectively, no matter the Equations (3) and (4) are con- Theorem 3.3. [15] Let S be an 2p × 2p nonnegative
r. Then the following assertions are
sistent or not. matrix with rank
equivalent:
It seems that we have obtained the solution of the 1) S 0;
original fuzzy linear system (2) as follows: 2) T P, such that PS
here exists a permutation matrix
lr has the form
XX ,,XX
LR (11)
Q
Ad,,IOSFOI SF 1
nn
LR
PS
,
But the solution vector may still not be an appropriate
Q
r
LR fuzzy numbers vector except for SF0. So we
O
the definition of the minimal LR fuzzy solution to
give
the Equation (2) as follows: re each Q has rank 1 and the rows of Q are or-
whe i i
lr Q ij
X xxx,, , thogonal to the rows of , whenever , the zero
Definition 3.1. Let
i
ij ij ij LR ay be absent.
1,in1jp. If Xx is the minimal solu- matrix m
ij np TT
HEHF
l rr S
3)
tion of Equation (3), Xx and Xx are
TT
ij np ij np
HFHE
minimal solution of Equation (4) such that Xl 0, for som H . In this case,
e positive diagonal matrix
r lr
X 0, then we call XX ,,XX is a strong
TT
LR EFHEF, EFHEF.
LR fuzzy solution of Equation (2). Otherwise, it is a
weak LR fuzzy solution of Equation (2) given by 4. Numerical Examples
lr l r
xx,,x , x0,x 0,
ij ij ij LR ij ij In this section, we work out two numerical examples to
lr l r illustrate the proposed method.
xx,0,max ,x, x0,x0,
ij ij ij LR ij ij Example 4.1. Consider the fuzzy matrix systems:
x
ij lr lr (12)
xx,max , x,0 , x0,x0,
ij ij ij LR ij ij
1 0 1 xx 2,1,1 3,2,1
11 12 LR LR
rl l r
xx,,x, x0,x0.
110xx 2,1,2 2,1,2.
ij ij ij ij ij
21 22
LR LR LR
211xx 6,3,2 5,2,3
1,in1jp. 31 32
LR LR
3.3. A Sufficient Condition of Strong Fuzzy The coefficient matrix A is nonsingular and the ex-
S is singular. By the Theorem 3.1., the
Solution tended matrix
x, the left spread xl and the right spread
mean value
The key points to make the solution vector being a LR xr of solution are obtained from
fuzzy solution are Xl 0 and Xr 0 . Since
101 23
xx
l r
X IOSF, X OI SF, we know that the 11 12
n l n r
110 22
xx
non negativities of X and X are equivalent to the 21 22
l
211xx 65
31 32
condition S 0 now that X 0 is known.
r
X and
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