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International Journal of Mathematics and Mathematical Sciences
Volume2012,ArticleID713617,8pages
doi:10.1155/2012/713617
Research Article
Solution of Fuzzy Matrix Equation System
MahmoodOtadiandMaryamMosleh
Department of Mathematics, Islamic Azad University, Firoozkooh Branch, Firoozkooh, Iran
CorrespondenceshouldbeaddressedtoMahmoodOtadi,otadi@iaufb.ac.ir
Received22March2012;Revised30August2012;Accepted30August2012
AcademicEditor:SoheilSalahshour
Copyright q 2012 M. Otadi and M. Mosleh. 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.
Themainistodevelopamethodtosolveanarbitraryfuzzymatrixequationsystembyusingthe
embedding approach. Considering the existing solution to n × n fuzzy matrix equation system is
done. To illustrate the proposed model a numerical example is given, and obtained results are
discussed.
1. Introduction
TheconceptoffuzzynumbersandfuzzyarithmeticoperationswasfirstintroducedbyZadeh
1, Dubois, and Prade 2. We refer the reader to 3 for more information on fuzzy numbers
andfuzzyarithmetic.Fuzzysystemsareusedtostudyavarietyofproblemsincludingfuzzy
metric spaces 4, fuzzy differential equations 5, fuzzy linear systems 6–8, and particle
physics 9, 10.
Oneofthemajorapplications of fuzzy number arithmetic is treating fuzzy linear sys-
tems11–20, several problems in various areas such as economics, engineering, and physics
boil down to the solution of a linear system of equations. Friedman et al. 21 introduced a
generalmodelforsolvingafuzzyn×nlinearsystemwhosecoefficientmatrixiscrisp,andthe
right-hand side column is an arbitrary fuzzy number vector. They used the parametric form
of fuzzy numbersandreplacedtheoriginalfuzzyn×nlinearsystembyacrisp2n×2nlinear
systemandstudieddualityinfuzzylinearsystemsAx BxywhereAandBarerealn×n
matrix, the unknown vector x is vector consisting of n fuzzy numbers, and the constant y is
vectorconsistingofnfuzzynumbers,in22.In6–8,23,24theauthorspresentedconjugate
gradient, LU decomposition method for solving general fuzzy linear systems, or symmetric
fuzzy linear systems. Also, Abbasbandy et al. 25 investigated the existence of a minimal
solution of general dual fuzzy linear equation system of the form Ax f Bx c, where A
andBarerealm×nmatrices,theunknownvectorxisvectorconsistingofnfuzzynumbers,
andtheconstantsf andcarevectorsconsistingofmfuzzynumbers.
2 International Journal of Mathematics and Mathematical Sciences
In this paper, we give a new method for solving a n × n fuzzy matrix equation system
whosecoefficientsmatrixiscrisp,andtheright-handsidematrixisanarbitraryfuzzynumber
matrix by using the embedding method given in Cong-Xin and Min 26 and replace the
original n × n fuzzy linear system by two n × n crisp linear systems. It is clear that, in large
systems, solving n × n linear system is better than solving 2n × 2n linear system. Since per-
turbation analysis is very important in numerical methods. Recently, Ezzati 27 presented
the perturbation analysis for n × n fuzzy linear systems. Now, according to the presented
method in this paper, we can investigate perturbation analysis in two crisp matrix equation
systems instead of 2n×2n linear system as the authors of Ezzati 27 and Wang et al. 28.
2. Preliminaries
Parametricformofanarbitraryfuzzynumberisgivenin29asfollows.Afuzzynumberuin
parametric form is a pair u
requirements: , u of functions ur ,ur , 0 ≤ r ≤ 1, which satisfy the following
1 u
r is a bounded left continuous nondecreasing function over 0,1,
2 ur is a bounded left continuous nonincreasing function over 0,1,and
3 ur ≤ ur , 0 ≤ r ≤ 1.
The set of all these fuzzy numbers is denoted by E which is a complete metric space
withHausdorffdistance.Acrispnumberαissimplyrepresentedbyur ur α, 0 ≤ r ≤
1.
For arbitrary fuzzy numbers x xr ,xr ,yyr ,yr , and real number k,we
maydefinetheaddition and the scalar multiplication of fuzzy numbers by using the exten-
sion principle as 29
a x y if and only if xr yr and xr yr ,
b x y xr yr ,xr yr ,and
c kx kx,kx ,k≥ 0,
kx, kx ,k<0.
Definition 2.1. The n × n linear system is as follows:
a x a x ···a x y ,
11 1 12 2 1n n 1
a x a x ···a x y ,
21 1 22 2 2n n 2
. 2.1
.
.
a x a x ···a x y ,
n1 1 n2 2 nn n n
where the given matrix of coefficients A a ,1≤ i, j ≤ n is a real n × n matrix, the given
ij
y ∈ E,1≤ i ≤ n, with the unknowns x ∈ E,1≤ j ≤ n is called a fuzzy linear system FLS .
i j
Theoperationsin2.1 isdescribed in next section.
given. Here, a numerical method for finding solution 21 of a fuzzy n × n linear system is
International Journal of Mathematics and Mathematical Sciences 3
Definition 2.2 see 21 . A fuzzy number vector x ,x ,...,x t given by
1 2 n
x x r ,x r ;1≤j ≤n, 0 ≤ r ≤ 1 2.2
j j j
is called a solution of the fuzzy linear system 2.1 if
n n
a x a x y,
ij j ij j i
j1 j1
2.3
n n
a x a x y.
ij j ij j i
j1 j1
If, for a particular i, a >0,forall j, we simply get
ij
n n
a x y, a x y.
ij j i ij j i 2.4
j1 j1
Finally, we conclude this section by a reviewing on the proposed method for solving
fuzzy linear system 21.
Theauthors21wrotethelinearsystemof2.1 asfollows:
SXY, 2.5
wheresij are determined as follows:
a ≥0⇒s a ,s a ,
ij ij ij in,jn ij 2.6
a <0⇒s a ,s a ,
ij i,jn ij in,j ij
andanysij whichisnotdeterminedby2.1 iszeroand
⎡x1 ⎤ ⎡y1 ⎤
⎢ . ⎥ ⎢ . ⎥
⎢ . ⎥ ⎢ . ⎥
⎢ . ⎥ ⎢ . ⎥
⎢ ⎥ ⎢ ⎥
⎢xn ⎥ ⎢y ⎥
⎢ n ⎥
X⎢ ⎥,Y . 2.7
⎢x1⎥ ⎢y ⎥
⎢ ⎥ ⎢ 1⎥
⎢ . ⎥ ⎢ . ⎥
⎣ . ⎦ ⎢ . ⎥
. ⎣ . ⎦
xn y
n
Thestructure of S implies that sij ≥ 0, 1 ≤ i, j ≤ 2n and that
S BC, 2.8
CB
4 International Journal of Mathematics and Mathematical Sciences
whereBcontainsthepositiveentriesofA,andCcontainstheabsolutevaluesofthenegative
entries of A,thatis,A B C.
Theorem2.3see21 . Theinverse of nonnegative matrix
S BC 2.9
CB
is
S1 DE, 2.10
ED
where
D1 1 B C 1 1 1 B C 1 2.11
BC ,E BC .
2 2
Corollary 2.4 see 30 . The solution of 2.5 is obtained by
XS1Y. 2.12
3. Fuzzy Matrix Equation System
Amatrixsystemsuchas
⎛ ⎛
⎛a a ··· a ⎞ x x ··· x ⎞ y y ··· y ⎞
11 12 1n 11 12 1n 11 12 1n
⎜ ⎜
⎜a a ··· a ⎟ x x ··· x ⎟ y y ··· y ⎟
21 22 2n ⎜ 21 22 2n ⎜ 21 22 2n
⎜ ⎟ ⎟ ⎟
⎜ ⎜ , 3.1
⎜ . . . . ⎟ . . . . ⎟ . . . . ⎟
. . . . ⎝ . . . . ⎝ . . . .
⎝ . . . . ⎠ . . . . ⎠ . . . . ⎠
a a ··· a x x ··· x y y ··· y
n1 n2 nn n1 n2 nn n1 n2 nn
where a ,1≤ i, j ≤ n, are real numbers, the elements y in the right-hand matrix are fuzzy
ij ij
numbers, and the unknown elements x are ones, is called a fuzzy matrix equation system
ij
FMES .
Usingmatrixnotation, wehave
AXY. 3.2
Afuzzynumbermatrix
Xx ,...,x,...,x 3.3
1 j n
is called a solution of the fuzzy matrix system 2.1 if
Ax y, 1≤j≤n. 3.4
j j
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