The pseudo inverse matrices to solve general fully fuzzy linear systems

Joural of Soft Computig ad Applicatios 2013 (2013) 1-11 Available olie at www.ispacs.com/jsca Volume 2013, Year 2013 Article ID jsca-00012, 11 Pages doi:10.5899/2013/jsca-00012 Research Article The pseudo iverse matrices to solve geeral fully fuzzy liear systems S. Moloudzadeh 1, P. Darabi 1, H. Khadai 2 (1) Departmet of Mathematics, Sciece ad Research Brach, Islamic Azad uiversity, Tehra, Ira. (2) Departmet of Mathematics, Mahabad Brach, Islamic Azad Uiversity, Mahabad, Ira. Copyright 2013 c S. Moloudzadeh, P. Darabi ad H. Khadai. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper, we preset a solutio of a arbitrary geeral fully fuzzy liear systems (FFLS) i the form à x = b. Where coefficiet matrix à is a m fuzzy matrix ad all of this system are elemets of LR type fuzzy umbers. Our method discuss a geeral FFLS (square or rectagle fully fuzzy liear systems with trapezoidal or triagular LR fuzzy umbers). To do this, we trasform fully fuzzy liear system i to two crisp liear systems, the obtai the solutio of this two systems by usig the pseudo iverse matrix method. Numerical examples are give to illustrate our method. Keywords: Fully fuzzy liear system (FFLS); Overdetermied liear system; Pseudo iverse matrix; Uderdetermied liear system. 1 Itroductio Systems of liear equatios are used to solve may problems i various areas such as structural mechaic applicatios i civil ad mechaical structures, heat trasport, fluid flow, electromagetic ad etc. I may applicatios, at least oe of the system s parameters ad measuremets are vague or imprecise ad we ca preset them with fuzzy umbers rather tha crisp umbers. Hece, it is importat to develop mathematical models ad umerical procedure that would appropriately treat geeral fuzzy systems ad solve them. Fuzzy liear system A x = b, where A is a crisp matrix ad b is a fuzzy umber vector has bee solved by Friedma ad his colleagues. Usig the embeddig approach Friedma et. al. proposed a geeral model to solvig such a fuzzy liear systems (see [9]). Asady et al. [4], who merely discussed the full row rak system, used the same method to solve the m fuzzy liear system for m. Zheg ad wag [12, 13] discussed the solutio of the geeral m cosistet ad icosistet fuzzy liear system. The system of liear equatios à x = b where ã i j ad b i are the elemets of the matrix A ad vector b respectively, ad are fuzzy umbers, is called a fully fuzzy liear systems (FFLS). FFLS has bee studied by several authors. Dehgha et. al. [5, 6, 7] has studied some methods for solvig FFLS. They have represeted fuzzy umbers i LR form ad applied approximate operators betwee fuzzy umbers to fid positive solutios of FFLS, so fidig the solutios of FFLS is trasformed to fidig the solutios of three crisp systems. Allahviraloo et. al. [2, 3] proposed a ew method to obtai symmetric solutios (bouded ad symmetric solutios) of a fully fuzzy liear systems (FFLS) based o a 1-cut expasio. The purpose of this paper is to preset a solutio of a arbitrary geeral (square or rectagle) FFLS. To do this, the origial m fully fuzzy liear system Correspodig author. Email address: saeidmoloudzadeh@gmail.com, Tel:+989143891299

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 2 of 11 à x = b is trasformed i to two crisp liear systems, the the solutio of this two systems are discussed by usig the pseudo iverse (or Moore-Perose iverse) matrix method. This paper is orgaized as follows: I sectio 2, the basic results of the fuzzy umbers ad fuzzy calculus are discussed ad a summary of the fuzzy matrix will be illustrated. I sectio 3, a arbitrary geeral FFLS are discussed ad our method for solvig FFLS (Ax = b) is itroduced. The preseted method are illustrated by some examples i sectio 4. coclusio are draw i sectio 5, i Appedix a brief discussio of the o square liear system of equatios is give, I sectio 6. 2 Prelimiaries ad otatios I this sectio, we give some ecessary defiitios ad otatios which will be used throughout the paper. Defiitio 2.1. Let X be a oempty set. A fuzzy set u i X is characterized by its membership fuctio u : X [0,1]. Thus u(x) is iterpreted as the degree of membership of a elemet x i the fuzzy set u for each x X. Let deote by E the class of fuzzy subsets of the real axis (i.e. u : R [0,1]) satisfyig the followig properties: u is ormal, that is, there exists s 0 R such that u(s 0 ) = 1, u is covex fuzzy set (i.e. u(ts + (1 t)r) mi{u(s),u(r)}, t [0,1], s,r R), u is upper semi-cotiuous o R, cl{s R u(s) > 0} is compact, where cl deotes the closure of a subset. Amog the ifiite umber of possible fuzzy sets i u X that qualify as fuzzy umbers, some types of membership fuctios u(x) are of particular importace, especially with respect to the use of fuzzy umbers i applied fuzzy arithmetic. I this paper we will use LR fuzzy umber. Defiitio 2.2. A fuzzy umber à is called a LR fuzzy umber or LR fuzzy itervals if its membership fuctio µã : R [0,1] has the followig form: L( a x α ),(a α) x < a,α > 0, 1,a x b µã(x) = R( x b β ),b < x (b + β),β > 0 0, otherwise, where [a,b] is the peak or core of Ã, L : [0,1] [0,1] ad R : [0,1] [0,1] are cotiuous ad o-icreasig shape fuctios with L(0) = R(0) = 1 ad L(1) = R(1) = 0 The LR fuzzy umber à as described above will be represeted as à = (a,b,α,β) LR. Here L ad R are called as the left ad right referece fuctios, a ad b are respectively called startig ad ed poits of the flat iterval, α is called the left spread ad β is called the right spread. Clearly à = (a,b,α,β) LR is positive if ad oly if a α > 0 (ote that L(1)=0). Let L(x) = R(x) = 1 x the istead à = (a,b,α,β) LR we simply write à = (a,b,α,β) ad called a trapezoidal fuzzy umber ad also if a = b we write à = (a,α,β) ad say that à is a triagular fuzzy umber. Note that we use a fixed fuctio L(.) ad a fixed fuctio R(.) for all fuzzy umbers i each problem. Defiitio 2.3. Two LR fuzzy umber à = (a,b,α,β) ad B = (c,d,γ,δ) are said to be equal, if ad oly if a = c, b = d, α = γ ad β = δ. Remark 2.1. We cosider 0 = (0,0,0,0) as a zero LR fuzzy umber. For two fuzzy umber, we defied the followig operatios [8]. Defiitio 2.4. For two LR fuzzy umbers A = (a,b,α,β) ad B = (c,d,γ,δ) the formula for the exteded additio becomes: (a,b,α,β) (c,d,γ,δ) = (a + c,b + d,α + γ,β + δ). (2.2) (2.1) Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 3 of 11 The formula for the exteded opposite becomes: A = (a,b,α,β) LR = ( b, a,β,α) RL. (2.3) The approximate formula for the exteded multiplicatio of two fuzzy umbers ca be summarized as follows: If A > 0 ad B > 0, the (a,b,α,β) (c,d,γ,δ) (ac,bd,aγ + cα,bδ + dβ). If A > 0 ad B < 0, the (a,b,α,β) (c,d,γ,δ) (bc,ad,bγ cβ,aδ dα). If A < 0 ad B > 0, the (a,b,α,β) (c,d,γ,δ) (ad,bc, aδ + dα, bγ + cβ). If A < 0 ad B < 0, the (a,b,α,β) (c,d,γ,δ) (bd,ac, bδ dβ, aγ cα). For scalar multiplicatio: (2.4) { (λa,λb,λα,λβ), λ 0, λ A = λ (a,b,α,β) = (λb,λa, λβ, λα), λ < 0. (2.5) Defiitio 2.5. A matrix à = (ã i j ) is called a fuzzy matrix if each elemet of à is a fuzzy umber [8]. à will be positive (egative) ad deoted by à > 0 (à < 0) if each elemet of à be positive (egative). Similarly oegative ad o-positive fuzzy matrices will be defied. We may represet m fuzzy matrix à = (ã i j ) m, such that ã i j = (a i j,b i j,α i j,β i j ), with the ew otatio à = (A,B,M,N), where A = (a i j ),B = (b i j ),M = (α i j ) ad N = (β i j ) are four m crisp matrices. Clearly A,B are called the mea value matrix, M ad N are called the left ad right spread matrices, respectively. 3 Geeral fully fuzzy liear systems ad its solutios Defiitio 3.1. Cosider the m liear system of equatios: (ã 11 x 1 ) (ã 12 x 2 ) (ã 1 x ) = c 1, (ã 21 x 1 ) (ã 22 x 2 ) (ã 2 x ) = c 2,..... (ã m1 x 1 ) (ã m2 x 2 ) (ã m x ) = c m. (3.6) Where ã i j, x j ad b i,1 i m,1 j are all fuzzy umbers. This system is called a m fully fuzzy liear system (FFLS). If m = this system is called a fully fuzzy square liear system (FFSLS) ad also if m is a fully fuzzy osquare liear system or fully fuzzy rectagle liear system (FFRLS). The matrix form of this fully fuzzy liear system is à x = c, (3.7) or simply à x = c where the coefficiet matrix à = (ã i j ) = (A,B,M,N), 1 i m,1 j is a m fuzzy matrix, x = ( x j ) = (x,y,w,z), 1 j is a 1 fuzzy matrix ad c = ( c i ) = (c,d,g,h), 1 i m is a m 1 fuzzy matrix. So if all fuzzy umbers of FFLS are trapezoidal (or triagular) fuzzy umbers the (3.7) is of the form (A,B,M,N) (x,y,w,z) = (c,d,g,h) (or (A,M,N) (x,w,z) = (c,g,h)). Now we are goig to solve FFLS (FFSLS or FFRLS) by usig the approximate multiplicatio rule for LR trapezoidal (or triagular) fuzzy umbers. For all 1 i m ad 1 j, ã i j = (a i j,b i j,m i j, i j ) of fuzzy matrix à we defie ã + i j = (a+ i j,b+ i j,m+ i j,+ i j ) ad ã i j = (a i j,b i j,m i j, i j ) as follows: ã + i j = (a i j,b i j,m i j, i j ) ad ã i j = (0,0,0,0) i f a i j 0, ã + i j = (0,0,0,0) ad ã i j = (a i j,b i j,m i j, i j ) i f b i j 0, (3.8) ã + i j = (0,b i j,0, i j ) ad ã i j = (a i j,0,m i j,0) i f a i j < 0 ad b i j > 0. Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 4 of 11 This implies ã i j = ã + i j ã i j that is ã+ i j ad ã i j are oegative iterval mea value ad egative iterval mea value respectively. Therefore fuzzy matrix à = (ã i j ) = (A,B,M,N) ca be writte as à = à + à where à + = (ã + i j ) = (A+,B +,M +,N + ) ad à = (ã i j ) = (A,B,M,N ). Clearly crisp matrices A = (a i j ) ad B = (b i j ) are o-positive ad residue crisp matrices A +,M +,N +,M ad N are o-egative. Also we have A = A + + A,B = B + + B,M = M + + M ad N = N + + N. Let fuzzy umber vector x = (x,y,w,z) = ( x 1, x 2,..., x ) T be solutio of FFLS (3.6) where x j = (x j,y j,w j,z j ),1 j, the usig the same method we defie x j = x + j x j,1 j such that x+ j = (x + j,y+ j,w+ j,z+ j ) ad x j = (x j,y j,w j,z j ). Therefore we ca be writte as x = x+ + x where x j = x + j + x j ad also y = y + + y,w = w + + w,z = z + + z. I the followig theorems, we give a arbitrary solutio of FFRLS ad the fuzzy umbers are trapezoidal. Theorem 3.1. A fuzzy umber vector x = ( x 1, x 2,..., x ) T drive by x j = (x j,y j,w j,z j ),1 j is called a solutio of the FFRLS (3.6) if A + x + + B + x + A y + + B y = c, B x + + A x + B + y + + A + y = d, A + w + + B + w A z + B z = g M + x + + N + x M y + + N y, B w + A w + B + z + + A + z = h N x + + M x N + y + + M + y. Where A = A + + A, B = B + + B, M = M + + M, N = N + + N, x = x + + x, y = y + + y, w = w + + w ad z = z + + z. Proof. By applyig the approximate multiplicatio rule for i th row system (3.6), we get: (ã i1 x 1 ) (ã i x ) = c i, [(ã + i1 ã i1 ) ( x+ 1 x 1 )] [(ã+ i ã i ) ( x+ x )] = [(ã + i1 x+ 1 ) (ã+ i1 x 1 ) (ã i1 x+ 1 ) (ã i1 x 1 )] [(ã+ i x+ ) (ã + i x ) (ã i x+ ) (ã i x )] = c i, [(a + i1 x+ 1,b+ i1 y+ 1,a+ i1 w+ 1 + m+ i1 x+ 1,b+ i1 z+ 1 + + i1 y+ 1 ) (b+ i1 x 1,a+ i1 y 1,b+ i1 w 1 + i1 x 1,a+ i1 z 1 m+ i1 y 1 ) (a i1 y+ 1,b i1 x+ 1, a i1 z+ 1 + m i1 y+ 1, b i1 w+ 1 + i1 x+ 1 ) (b i1 y 1,a i1 x 1, b i1 z 1 i1 y 1, a i1 w 1 m i1 x 1 )] [(a+ i x+,b + i y+,a + i w+ + m + i x+,b + i z+ + + i y+ ) (b + i x,a + i y,b + i w + i x,a + i z m + i y ) (a i y+,b i x+, a i z+ + m i y+, b i w+ + i x+ ) (b i y,a i x, b i z i y, a i w m i x )] = (c i,d i,g i,h i ), (3.9) where: a + i1 x+ 1 + b+ i1 x 1 + a i1 y+ 1 + b i1 y 1 + + a+ i x+ + b + i x + a i y+ + b i y = c i, b + i1 y+ 1 + a+ i1 y 1 + b i1 x+ 1 + a i1 x 1 + + b+ i y+ + a + i y + b i x+ + a i x = d i, (a + i1 w+ 1 + m+ i1 x+ 1 + b+ i1 w 1 + i1 x 1 a i1 z+ 1 + m i1 y+ 1 b i1 z 1 i1 y 1 ) + + (a + i w+ + m + i x+ + b + i w + i x a i z+ + m i y+ b i z i y ) = g i, (b + i1 z+ 1 + + i1 y+ 1 + a+ i1 z 1 m+ i1 y 1 b i1 w+ 1 + i1 x+ 1 a i1 w 1 m i1 x 1 ) + + (b + i z+ + + i y+ + a + i z m + i y b i w+ + i x+ a i w m i x ) = h i. Usig summatio otatio, we have: a + i j x+ j + b + i j x j + a i j y+ j + b + i j y+ j + a + i j w+ j + b + i j z+ j + a + i j y j + m + i j x+ j + + i j y+ j + b i j x+ j + b + i j w j a + i j z j b i j y j = c i, a i j x j = d i, + i j x j m + i j y j a i j z+ j + b i j w+ j + m i j y+ j i j x+ j b i j z j a i j w j i j y j = g i, m i j x j = h i. (3.10) Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 5 of 11 By usig matrix otatio we have: A + x + + B + x + A y + + B y = c, B x + + A x + B + y + + A + y = d, A + w + + B + w A z + B z = g M + x + + N + x M y + + N y, B w + A w + B + z + + A + z = h N x + + M x N + y + + M + y. Therefore, we get the solutio of FFRLS (3.6) ad the proof is completed. Corollary 3.1. Usig matrix otatio by theorem 3.3, if the system (3.6) be a osquare fully fuzzy liear system (FFRLS) with trapezoidal fuzzy umbers, the (3.9) systems is trasformed i to two followig crisp liear systems: { RX = U, (3.11) SZ = T. Where matrices R,S ad vectors X,Z,U,T are give as the followig: [ ] [ ] [ ] S1 S R = 2 S1 S, S = 2 x, X =, [ S 3 ] S 4 [ ] S 3 [ S 4 ] [ y ][ w c g M1 M Z =, U =, T = + 2 x z d h M 3 M 4 y ]. (3.12) Where S i,m i (1 i 4) are m 2 crisp matrices ad x,y,w,z are 2 1 crisp vectors ad c,d,g,h are m 1 crisp vectors are defied as follow: S 1 = [ A + B + ], S 2 = [ A B ], S 3 = [ B A ], S 4 = [ B + A + ], M 1 = [ M + N + ], M 2 = [ M N ], M 3 = [ N M ], M 4 = [ N + M + ], [ ] [ ] [ ] [ ] x + y + w + z + x =, y =,w =, z =. x y Brifly where the system (3.6) is FFSLS or FFRLS ad the related fuzzy umbers are trapezoidal or triagular we have four diffret cases that oe of them discussed above (case 1). Now we discuss three other cases as follows. Corollary 3.2. (case 2: the system (3.6) is FFRLS ad the fuzzy umbers are triagular) I this case the matrices R,S ad vectors X,Z,U,T i (3.11) are defied i the followig forms: [ A + A R = A, X = x, U = c, S = ] A A +, [ ] [ ] [ ] [ ] w g M N Z =, T = x z h N + + x M. w z (3.13) Where A,A +, A,M,N are m crisp matrices ad x,w,z,x +,x are 1 crisp vectors ad c,g,h are m 1 crisp vectors. So crisp liear systems (3.11), is trasformed i to the followig crisp liear systems: { Ax = c, (3.14) SZ = T. Corollary 3.3. (case 3: the system (3.6) is FFSLS ad the fuzzy umbers are trapezoidal) I this case the matrices R,S ad vectors X,Z,U,T i (3.11) are defied as (3.12). Where S 1,S 2,S 3,S 4,M 1,M 2,M 3,M 4 are 2 crisp matrices ad x,y,w,z are 2 1 crisp vectors ad also c,d,g,h are 1 crisp vectors. Corollary 3.4. (case 4: the system (3.6) is FFSLS ad the fuzzy umbers are triagular) I this case the matrices R,S ad vectors X,Z,U,T i (3.11) are defied as (3.13) ad (3.14). Where A,A +, A,M,N are crisp matrices ad x,w,z,x +,x,c,g,h are 1 crisp vectors. Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 6 of 11 To clearly metio the order of related matrices, the above cases is briefed as follows: { R2m 4 X case 1: 4 1 = U 2m 1, { S 2m 4 Z 4 1 = T 2m 1, Am x case 2: 1 = c m 1, { S 2m 2 Z 2 1 = T 2m 1, R2 4 X case 3: 4 1 = U 2 1, { S 2 4 Z 4 1 = T 2 1, A x case 4: 1 = c 1, S 2 2 Z 2 1 = T 2 1, (3.15) Now to fid the solutio of case1 i (3.15), we costitute the pseudo iverse of the matrices R,S ad have the followig four cases. { X = R 1 U, Z = S 1, if m = 2 = rak(r) = rak(s); { T, X = R T (RR T ) 1 U = R + U, Z = S T (SS T ) 1 T = S +, if 2 > m = rak(r) = rak(s); { T, X = (R T R) 1 R T U = R + U, (3.16) Z = (S T S) 1 S T T = S +, if m > 2 = rak(r) = rak(s); T, X = k u T i b i=1 σ i v i U = R + U, Z = k u T i b, if rak(r) = rak(s) = k mi(m, 2). i=1 σ i v i T = S + T, where the type of the matrices R ad S ca be square systems with full rak matrices, uderdetermied systems with full rak matrices, overdetermied systems with full rak matrices ad geeral case. We treat the other three cases Similarly. The pseudo iverse of the matrix S i secod ad fourth cases is defied as follow: Theorem 3.2. If S + exists it must have the same structure as S (see [1, 9]), i.e : [ ] S + D E =, (3.17) E D where { D = 1 2 [(A + A ) + + (A + + A ) + ], E = 1 2 [(A+ A ) + (A + + A ) + ]. (3.18) (Replace (.) + by (.) 1 whe S is square matrix.) We obtai x,y,w,z via equatios systems (3.15). If the left ad right spreads w,z are egative, the the followig defiitio is give. Defiitio 3.2. Let X = (x j,y j,w j,z j ),1 j deote the uique solutio of systems (3.15). The fuzzy umber vector Ũ = (p j,q j,u j,v j ),1 j defied by p j = x j,1 j q j = y j,1 j u j = w j,1 j v j = z j,1 j, (3.19) is called the fuzzy solutio of systems (3.15). O the other had if (x j,y j,w j,z j ),1 j are all fuzzy umbers (i.e. w j,z j 0) the u j = w j,v j = z j,1 j ; otherwise u j = w j,v j = z j,1 j ad U is called a fuzzy solutio. Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 7 of 11 4 Numerical examples I this sectio, we take some umerical example. Example 1. Cosider the followig FFLS { (1,1.5,0.1,0.2) x = (2,4.5,1.2,2.1), ( 2, 1,0.7,0.3) x = ( 6, 2,2.3,1.6). It is clear that this system is a FFRLS with trapezoidal fuzzy umbers (case 1). To fid the solutio of this system, we solve two crisp liear systems (3.11) (i.e. RX = U, SZ = T ), where: 1 1 0 0 1 1 0 0 2 0.4 R = 0 0 2 1 0 0 1.5 1,S = 0 0 2 1 0 0 1.5 1,U = 6 4.5,T = 0.4 3.0. 1 2 0 0 1 2 0 0 2 0.4 Sice rak(r)=rak(s)=4 the this systems are square with full rak matrix, The by usig the iverse of matrices R ad S, we have: x + 2 w + 0.4 X = x y + = R 1 U = 0 3, Z = w z + = S 1 T = 0 5.2. y 0 z 10.8 Sice x = x + +x = 2+0 = 2,y = y + +y = 3+0 = 3,w = w + +w = 0.4+0 = 0.4,z = z + +z = 5.2+10.8 = 5.6, therefore the solutio of this FFRLS is x = (2,3,0.4,5.6). Example 2. Let à = (A,B,M,N) ad c = (c,d,g,h) be a fully fuzzy matrix ad a fully fuzzy vector, respectively with [ ] [ ] [ ] [ ] 1 1.5 1 2 2 1.5 0.1 0.3 0.2 0.4 0.1 0.2 A =, B =, M =, N =, 1 2 0 0 3 1 0.3 0.4 0.1 0.2 0.3 0.1 [ 1 c = 7.5 ] [ 5.25, d = 0.5 ] [ 1.6, g = 2.65 ] [ 3.15, h = 1.75 The we defie the solutio of systems (3.7) (i.e. à x = c). It is clear that this system is a FFRLS with trapezoidal fuzzy umbers (case 1). To fid the solutio of this system, we solve two crisp liear systems (3.11) (i.e. RX = U, SZ = T ), where R = S = 1 1.5 1 2 2 1.5 0 0 0 0 0 0 0 2 0 0 3 1 1 0 0 0 0 0 0 0 0 0 0 0 2 2 1.5 1 1.5 1 0 0 0 1 0 0 0 3 1 0 2 0 1 1.5 1 2 2 1.5 0 0 0 0 0 0 0 2 0 0 3 1 1 0 0 0 0 0 0 0 0 0 0 0 2 2 1.5 1 1.5 1 0 0 0 1 0 0 0 3 1 0 2 0, U =, T = ], 1 7.5 5.25 0.5 2.0055 2.1330 2.6151 2.0037 Sice rak(r)=rak(s)=4 the this systems are uderdetermied with full rak matrices ad also by usig the pseudo iverse of the matrices R ad S, we have:,. Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 8 of 11 ad also x = x + + x = X = w = w + + w = x+ 1 x 2 + x 3 + x + x y + y w+ 1 w + 2 w + 3 = R+ U = + x 1 x2 x3 + = w 1 w 2 w 3 0.4753 0.8529 0.4753 1.1151 1.3982 0.0700 1.5294 0.2528 0.3953 0.3732 0.2307 0.3732 1.59.4 2.2511 0.4053 =, Z = 0.3191 0.5917 0.3135 w + w z + z,y = y + + y =,z = z + + z = = S+ T = y+ 1 y + 2 y + 3 z+ 1 z + 2 z + 3 Therefore the solutio of this FFRLS is x 1 x = x 2 x 3 = (1.5904,1.9026,0.3191,0.5792) ( 2.2511, 0.4835, 0.5917, 0.5956). (0.4053, 0.7685, 0.3135, 0.4412) Example 3. Cosider the followig FFLS ( test 5.5. i the [5] ) (6,1,4) (5,2,2) (3,2,1) x 1 (12,8,20) (14,12,15) (8,8,10) x 2 = (58,30,60) (142, 139, 257) (24,10,34) (32,30,30) (20,19,24) x 3 (316, 297, 514). + + 0.1084 0.2475 0.1084 0.2107 0.3442 0.2051 0.4003 0.3395 0.2623 0.1789 0.2561 0.1789 y 1 y 2 y 3 z 1 z 2 z 3 = =, 1.9026 0.4835 0.7685 0.5792 0.5956 0.4412 Sice this system is square ad all of elemets is triagular, thus accordig to the fourth case of (3.15) we solve two crisp liear system Ax = c, SZ = T, where 6 5 3 0 0 0 A = 6 5 3 12 14 8,c = 58 12 14 8 0 0 0 142,S = 24 32 20 0 0 0 24 32 20 316 0 0 0 6 5 3. 0 0 0 12 14 8 0 0 0 24 32 20 O the first system Ax = c, we have x = [4 5 3] T ad so T = [10 23 50 31 72 156] T ad by SZ = T, Z is obtaied: Z T = [w 1 w 2 w 3 z 1 z 2 z 3 ] = [1 0.5 0.5 3 2 1] Thus the solutio is: x = (4,1,3) (5,0.5,2). (3,0.5,1) Sice x is zero vector, it is ot ecessary to write it i the calculatio. Now we cosider that our solutio is the same as the solutio i [5]. +,. Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 9 of 11 5 Coclusio I this paper a solutio of geeral fully fuzzy liear system (FFLS) was foud. Ideed a method was itroduced, where the solutio is a LR fuzzy umber vector. By our method every FFLS ca be coverted to two crisp liear systems. For this purpose, we first have to look at the FFLS ad i accordig to the type of this system (square or rectagle fully fuzzy liear systems with trapezoidal or triagular LR fuzzy umbers) four cases (FFRLS with trapezoidal fuzzy umbers, FFRLS with triagular fuzzy umbers, FFSLS with trapezoidal fuzzy umbers, FFSLS with triagular fuzzy umbers) are obtaied. To fid the solutio of this crisp liear systems, with regard the related type of liear systems (square systems with full rak matrices, uderdetermied systems with full rak matrices, overdetermied systems with full rak matrices ad geeral case) we costitute the pseudo iverse matrices for the related systems ad give the solutios. Appedix There are several methods for solvig the liear system Ax = b where A was assumed to be square ad osigular. However, i several practical situatios, we eed to solve a system where the matrix A is osquare ad/or sigular. I such cases, solutios may ot exist at all; i cases where there are solutios, there may be ifiitely may. For example, whe A is m ad m >, we have a overdetermied system ad a overdetermied system typically has o solutio. I cotrast, a uderdetermied system (m < ) typically has a ifiite umber of solutios. I this system, with regard to matrix A, we shall cosider the followig cases: 1. m = ad rak(a) = (square system with full rak matrix); 2. m < ad rak(a) = m (uderdetermied system with full rak matrix); 3. m > ad rak(a) = (overdetermied system with full rak matrix); 4. rak(a) = k mi(m, ) (geeral case). Defiitio 5.1. (see [11], Geeral Least Squares Problem) For A R m ad b R m, let ε = ε(x) = Ax b. The geeral least squares problem is to fid a vector x that miimizes the quatity m εi 2 = ε T ε = (Ax b) T (Ax b). i=1 Ay vector that provides a miimum value for this expressio is called a least squares solutio. The set of all least squares solutios is precisely the set of solutios to the system of ormal equatios A T Ax = A T b. There is a uique least squares solutio if ad oly if rak (A) =, i which case it is give by x = (A T A) 1 A T b If Ax = b is cosistet, the the solutio set for Ax = b is the same as the set of least squares solutios. If the least squares problem has more tha oe solutio, the oe havig the miimum Euclidea orm is called miimum legth solutio or the miimum orm solutio. Rectagular matrices do ot have iverses or determiats. A partial replacemet for the iverse is provided by the pseudo iverse or Moore-Perose iverse which is calculated by the piv fuctio i MAT LAB. Defiitio 5.2. (see [10, 11]) The pseudo iverse (or Moore-Perose iverse) of ay m matrix A (whether it has full rak or ot) is a m matrix A + satisfyig the followig coditios: AA + A = A, A + AA + = A +, (AA + ) T = AA +, (A + A) T = A + A The matrix A + is uique ad does always exist. (Replace (.) T by (.) whe A is complex matrix.) Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 10 of 11 Defiitio 5.3. (see [10]) The matrix A + = (A T A) 1 A T, whe A is m (m ) ad rak(a) =, or the matrix A + = A T (AA T ) 1, whe A is m (m < ) ad rak(a) = m, is called the pseudo iverse of A. A full rak uderdetermied system has a ifiite umber of solutios. I this case, the complete set is give as follows: Corollary 5.1. (see [10]) Let Ax = b be a full rak uderdetermied system. The the solutio set is give by x = A T (AA T ) 1 b + (I A T (AA T ) 1 A)y = A + b + (I A + A)h, where h is a arbitrary vector i R 1. Theorem 5.1. (see [10]) Let A be m (m < ) ad have full rak.the the miimum orm solutio x to the uderdetermied system Ax = b is give by x = A T (AA T ) 1 b = A + b. Whe A is ot full rak, the the upo Corollaries ca ot be used. More geerally, the pseudo iverse is best computed usig the sigular value decompositio (SVD) reviewed below. Theorem 5.2. (see [10, 11]) Let A be a m real matrix ad rak(a) = k,0 < k mi(m,). There exists a m m orthogoal matrix U, a orthogoal matrix V, ad a m quasi-diagoal matrix Σ = diag(σ 1,σ 2,,σ k,0,,0), where σ 1 σ 2 σ k > 0, such that the sigular value decompositio A = UΣV T is valid. The uique pseudo iverse of A is A + = V Σ + U T where Σ + = diag( 1 σ 1, 1 σ 2,, 1 σ k,0,,0), is m quasi-diagoal matrix. A differet way is this: There do always exist two matrices C m k ad D k of rak k, such that A = CD. Usig these matrices it holds that A + = D T (DD T ) 1 (C T C) 1 C T. Corollary 5.2. For the cosistet (icosistet) liear system A m, x = b m, x = A + b is the solutio of miimal Euclidea orm (the least squares solutio of miimal Euclidea orm) (see [11]). Ad ay pseudo iverse A + of the coefficiet matrix A for the four cases may be summarized as follows: A 1, square system with full rak matrix; A + A T (AA T ) 1, uderdetermied system with full rak matrix; = (A T A) 1 A T, overdetermied system with full rak matrix; k u T i b i=1 σ i v i, geeral case. Refereces [1] S. Abbasbady, M. Otadi, M. Mosleh, mimal solutio of geeral dual fuzzy liear systems, Chaos Solitos ad Fractals, 37 (2008) 1113-1124. http://dx.doi.org/10.1016/j.chaos.2006.10.045 [2] T. Allahviraloo, S. Salahshour, M. Khezerloo, Maximal- ad miimal symmetric solutios of fully fuzzy liear systems, Joural of Computatioal ad Applied Mathematics, 235 (2011) 4652-4662. http://dx.doi.org/10.1016/j.cam.2010.05.009 [3] T. Allahviraloo, S. Salahshour, Bouded ad symmetric solutios of fully fuzzy liear systems i dual form, Procedia Computer Sciece, 3 (2011) 14941498. http://dx.doi.org/10.1016/j.procs.2011.01.038 [4] B. Asady, S. Abbasbady, M. Alavi, Fuzzy geeral liear systems, Applied Mathematics ad Computatio, 169 (2005) 34-40. http://dx.doi.org/10.1016/j.amc.2004.10.042 [5] M. Dehgha, B. Hashemi, M. Ghatee, Computatioal methods for solvig fully fuzzy liear systems, Applied Mathematics ad Computatio, 179 (2006) 328-343. http://dx.doi.org/10.1016/j.amc.2005.11.124 Iteratioal Scietific Publicatios ad Cosultig Services

Joural of Soft Computig ad Applicatios http://www.ispacs.com/jourals/jsca/2013/jsca-00012/ Page 11 of 11 [6] M. Dehgha, B. Hashemi, Solutio of the fully fuzzy liear systems usig the decompositio procedure, Applied Mathematics ad Computatio, 182 (2006) 1568-1580. http://dx.doi.org/10.1016/j.amc.2006.05.043 [7] M. Dehgha, B. Hashemi, M. Ghatee, Solutio of the fully fuzzy liear systems usig iterative techiques, Chaos Solitos ad Fractals, 34 (2007) 316-336. http://dx.doi.org/10.1016/j.chaos.2006.03.085 [8] D. Dubois, H. Prade, Fuzzy Sets ad Systems: Theory ad Applicatios, Academic Press, New York, (1980). [9] M. Friedma, M. Mig, A. Kadel, Fuzzy liear systems, Fuzzy Sets ad Systems, 96 (1998) 201-209. http://dx.doi.org/10.1016/s0165-0114(96)00270-9 [10] B. N. Data, Numerical liear Algebra ad Applicatios, cole publisher compay-first editio, (1995). [11] C. D. Meyer, Matrix Aalysis ad Applied liear Algebra, siam, (2000). http://dx.doi.org/10.1137/1.9780898719512 [12] K. Wag, B. Zheg, I cosistet fuzzy liear systems, Applied Mathematics ad Computatio, 181 (2006) 937-981. http://dx.doi.org/10.1016/j.amc.2006.02.019 [13] B. Zheg, K. Wag, Geeral fuzzy liear systems, Applied Mathematics ad Computatio, 181 (2006) 1276-1286. http://dx.doi.org/10.1016/j.amc.2006.02.027 Iteratioal Scietific Publicatios ad Cosultig Services