Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 9-9 Vol., Issue (Jue 8), pp. 77 88 (Peviously, Vol., No. ) Applicatios ad Applied Mathematics: A Iteatioal Joual (AAM) Siged Decompositio of Fully Fuzzy Liea Systems Tofigh Allahvialoo Depatmet of Mathematics, Islamic Azad Uivesity Sciece ad Reseach Bach, Teha, Ia Tofigh@allahvialoo.com Nasse Mikaeilvad Depatmet of Mathematics, Islamic Azad Uivesity Sciece ad Reseach Bach, Teha, Ia Mikaeilvad@AOL.com Nasis Aftab Kiai Depatmet of Mathematics, Islamic Azad Uivesity Sciece ad Reseach Bach, Teha, Ia Nasiskiai@Yahoo.com Rasol Mastai Shabestai Depatmet of Mathematics,Islamic Azad Uivesity Sofiya Bach, Sofiya, Ia a_mastai@yahoo.com Received May 9, 7; accepted Febuay, 8 Abstact System of liea equatios is applied fo solvig may poblems i vaious aeas of applied scieces. Fuzzy methods costitute a impotat mathematical ad computatioal tool fo modelig eal-wold systems with ucetaities of paametes. I this pape, we discuss about fully fuzzy liea systems i the fom AX b (FFLS). A ovel method fo fidig the o-zeo fuzzy solutios of these systems is poposed. We suppose that all elemets of coefficiet matix A ae positive ad we employ paametic fom liea system. Fially, Numeical examples ae peseted to illustate this appoach ad its esults ae compaed with othe methods. Keywods: Fuzzy umbes, Fully Fuzzy Liea Systems, Systems of Fuzzy Liea Equatios, No-zeo Solutios, Decompositio Method AMS Subect lassificatio Numbes: F, 8A7, D, A 77
78 Allahvialoo et al.. Itoductio Systems of liea equatios ae used fo solvig may poblems i vaious aea such as stuctual mechaics applicatios, heat taspot, fluid flow, electomagetism...i may applicatios, at least oe of the system's paametes ad measuemets ae vague o impecise ad we ca peset them with fuzzy umbes athe tha cisp umbes. Hece, it is impotat to develop mathematical models ad umeical pocedue that would appopiately teat geeal fuzzy system ad solve them. The system of liea equatios AX b whee the elemets, a i, of the matix A ae cisp umbes ad the elemets, b i, of b ae fuzzy umbes, is called fuzzy system of liea equatio (FSLE). The (FSLE) has bee studied by may authos (Abbasbay et al. (, ), Allahvialoo et al. (,, a, b,, ), Asady et al. (), Dehgha et al. (), Fiedma et al.(998, ), Ma et al. (), Wag et al. (), Xizhao et al. (), Zheg et al. ()). Fiedma et al. (998) poposed a geeal model fo solvig such fuzzy liea systems by usig the embeddig appoach. Followig Fiedma et al. (998), Allahvialoo et al. i (,, a, b,, ) ad othe authos i Abbasbay et al. (, ), Asady et al. (), Dehgha et al. (), Wag et al. (), Zheg et al. () ae desig some umeical methods fo calculatig the solutios of FSLE. The system of liea equatios AX B whee the elemets, a i, of the matix A ad the elemets, b i, of the vecto b ae fuzzy umbes, is called Fully Fuzzy Liea System FFLS. The FFLShas bee studied by may authos (Buckley ad Qu (99, 99a, 99b), Dehgha et al. (, ), Muzzioli ad Reyaets(, 7), Voma et al. (, 7a, 7b)). Buckley ad Qu i thei sequetial woks (99, 99a, 99b) suggested diffeet solutios fo FFLS. Also, they foud elatio betwee these solutios. Based o thei woks, Muzzioli ad Reyaets i (, 7) studied FFLS of the fom A X b A X b. The lik betwee iteval liea systems ad fuzzy liea systems is claified by them. Thei appoach cotais ( ) solvig of cisp systems fo all [,]. Dehgha et al. (, ) have studied some methods fo solvig FFLS. They have epeseted fuzzy umbes i L. R. fom ad applied appoximately opeatos betwee fuzzy umbes the foud positive solutios of FFLS (Dubois ad Pade (98), Zimmema (98)), so calculatig the solutios of FFLS is tasfomed to calculate the solutios of thee cisp systems. I thei appoach, esult of multiplyig two tiagula fuzzy umbes is a tiagula fuzzy umbe which is ot good appoximatio. Voma et al. i thei cotiuous wok (, 7a, 7b) suggested two method fo solvig FFLS. I (7a) they have poposed a method to solve FFLS appoximately the they pove that thei solutio is bette tha Buckley ad Qu s appoximate solutio vecto X. Futhemoe B
AAM: Ite, J., Vol., Issue (Jue 8) [Peviously, Vol., No. ] 79 i (, 7b) they have poposed a algoithm to impove thei method to solve FFLS by paametic fuctios. I may applicatios, which ca be modeled by a system of liea equatios, system's paametes ae positive ad we iteest to fid its o-zeo solutios, so it is impotat to popose a method to fid o-zeo solutios of FFLS, whee system's paametes ae positive. I this pape, we ae goig to fid o-zeo solutios of FFLS. We eplace the oigial FFLSby a paametic liea system the a umeical method fo calculatig the solutios is poposed. The est of pape is ogaized as follows: I Sectio, we discuss some basic defiitios, esults o fuzzy umbes ad FFLS. I Sectio, the umeical pocedue fo fidig o-zeo solutios of FFLS is peseted. The poposed algoithm is illustated by solvig some umeical examples i sectio. oclusios ae daw i sectio.. Pelimiaies The basic defiitio of fuzzy umbes is give i (Goetschel et al.(98)) as follow: Defiitio : A fuzzy umbe is a fuzzy set u : [,] which satisfies [.] u is uppe semi cotiuous; [.] u ( x) outside some iteval [ c, d] ; [.] thee ae eal umbes a, b; c a b d fo which (i). u ( x ) is mootoic iceasig o [ c, d] ; (ii). u ( x ) is mootoic deceasig o [ b, d] ; (iii). u ( x), a x b. The set of all fuzzy umbes is deoted by E. A alteative defiitio of fuzzy umbe is: Defiitio : (Goetschel(98),Kaleva(987)) A fuzzy umbe u is a pai ( u( u( ) of fuctios u( u( which satisfy the followig equiemets: ( i ) u( is a bouded mootoic iceasig left cotiuous fuctio; ( ii ) u( is a bouded mootoic deceasig left cotiuous fuctio; ( iii ) u( u(. A cisp umbe k is simply epeseted by k ( k( k ad is called sigleto. The fuzzy umbe space { u( u( } becomes a covex coe E which is the embedded isomophic ally ad isometic ally ito a Baach space (og-xig (99, 99)). A fuzzy umbe a ca be epeseted by its -cuts ad sup a l({ x x ad a ( x) }) [ a(), a()].
8 Allahvialoo et al. Note that the -cuts of a fuzzy umbe ae closed ad bouded itevals. Also the fuzzy aithmetic based o the Zadeh extesio piciple ca be calculated by applyig iteval aithmetic o the -cuts. Thee ae diffeet defiitios of opeatios betwee two itevals. Fo fuzzy umbeu ( u( u( ), we wite () u, if u(), () u, if u(), () u, if u(), ad () u, if u ( ). If u o u this fuzzy umbe is called o-zeo fuzzy umbe. Fo abitay u ( u( u( ), v ( v( v( )) ad k we defie additiou v, subtactio u v, multiplicatio u. v ad scala poduct o by k as Additio: uv () u () v (), uv () u () v (). () Subtactio: uv () u () v (), uv () u () v (). () uv() mi{()(), u v u()(), v u()(), v u()()}, v Multiplicatio: uv() max{()(), u v u()(), v u()(), v u()()}. v () Multiplicatio of two fuzzy umbes fo two impotat cases i moe detail is as follows: ase : u ad v uv( u( v( uv( u( v( ). ase : u ad v uv( u( v( uv( u( v( ). Scala poduct: ( ku( ku( ), k, ku () ( ku( ku( ), k. Note that, distibutive law fo fuzzy umbe's multiplicatio is ivalid. Also cacellatio law does ot hold i fuzzy umbes aithmetic i.e. if u, v ad w ae fuzzy umbes whee u v w, if v is o- sigleto the u w v. Defiitio : The liea system of equatios ax ax a x b, ax ax a x b, ax ax ax b, () whee the elemets, a, i,, of the coefficiet matix A ad the elemets, b, of the vecto i ae fuzzy umbe is called a fully fuzzy liea system of equatios FFLS. i
AAM: Ite, J., Vol., Issue (Jue 8) [Peviously, Vol., No. ] 8 If A is a cisp matix, fuzzy liea system FSLE. AX b is called Fuzzy System of Liea Equatios Defiitio : Fo ay ( FFLS) AX b ad fo all [,], a x a x a x b a x a x a x b a x a x a x b,,, () whee a i, x ad ] b i, i,, [, ae -cut sets of fuzzy umbes a i, x ad i,, espectively, is called -cut system of liea system ad epeseted by b i A X b,. A iteval [ a, b] is o zeo if [ a, b] [, ) o [ a, b] (,]. Some authos ivestigate (FFLS) usig this fact that -cut of fuzzy umbes is iteval. Fo moe ifomatio, see (Muzzioli ad Reyaets (, 7)). Whe iteval aithmetic is used to solve FFLS, the poblem of fidig the solutio of A X b covet to a multi obective oliea optimal poblem which fo solvability eeds to satisfy cetai coditios. Hece this appoach is ot suitable. t Defiitio : A fuzzy umbe vecto ( x x,,, x ) give by x ( x ( x ( ) i, i i is called a solutio of (FFLS), if i ax() ax() b(), ax() ax() b(). (7) i i i i i i i i i i We defie o-zeo solutio of (FFLS) as follows: Defiitio : A solutio vecto x, i,,,. t ( x, x,, x ) of ) (FFLS is ozeo if x o i Necessay ad sufficiet coditio fo the existece of a No-zeo fuzzy solutio of (FFLS ) is: Theoem: If (FFLS ) AX b has a solutio which it is a fuzzy umbe, the AX b will have a ozeo fuzzy umbe solutio if ad oly if -cut system of liea system epeseted by A X b has ozeo solutio.
8 Allahvialoo et al. t Poof: Let ( x x,,, x ) is fuzzy umbe solutio of AX b. AX b Has ozeo solutio, if ad oly if, x o x, i,,,, if ad oly if i sup p a l({ x x ad a ( x) }) [ a(), a()] ae o- zeo if ad oly if A X b has o- zeo solutio. I fact, may simultaeous system of liea equatio have o-zeo solutio.. No-Zeo Solutio of (FFLS ) I this sectio, we ae goig to fid the solutio of (FFLS ). To do that, if (FFLS ) have fuzzy o-zeo solutio, we eplace oigial (FFLS ) by paametic system the by poposed algoithm it is fouded. Let AX b be (FFLS ). oside i th equatio of this system: Now, let i ax b,,,,. (8) i i AX b has o-zeo solutio, we defie J { ad x }. (9) Hece Eq. (8) ca be tasfomed to ax ax ax b, i,,,. () i i i i i J J We defie two -vectos Y ( ) t y, y,, y ad Z ) t ( z, z,, z as follows: x, J, x, J, z y, J,, J. () This is obvious that z y x,. () By eplacig () ad () i ()
AAM: Ite, J., Vol., Issue (Jue 8) [Peviously, Vol., No. ] 8 a x ax ax az ay i i i i i i J J J J a z a y b, i,,,. i i i i i () By applyig ()-() ad (7) ad eplacig i () we have: ad Sice b() a z () a y (), i,,,, i i i () b( a z ( a y ( i,,,. i i i () z, y, a, i, by () we have i az() a() z(), ay() a() y(), i i i i a z () a () z (), a y () a () y (). i i i i () By eplacig () i () ad (), ad b() a () z () a () y (), i,,,, i i i (7) b( a ( z ( a ( y ( i,,,. i i i (8) If ad ae paametic matices by elemets ( ) a () ( ) a () (9) i i i i ad if Z, Z, Y, Y, B ad B ae paametic -vectos by elemets ( Z ) z ( ( Z ) z ( ( Y) y ( ( Y ) y ( ( B ) b ( ( B ) b ( () the, Matix epesetatio of ( FFLS) AX b covets to
8 Allahvialoo et al. Z Z B, Y B Y () whee this coefficiet matix epeset i matix fom. But i fact, by defiitio of Y ad Z, elemet of vaiable matix is zeo ad hece colum of coefficiet matix ae omitted. Hece, we eplace (FFLS) by system of liea paametic equatios. If we solve this system, its solutio is o-zeo solutio of (FFLS). Poposed algoithm to fid FFLS's o-zeo solutio is as follow: No-Zeo Solutio of (FFLS) s Algoithm: Suppose AX b is a (FFLS) whee have a fuzzy umbe solutio.. Solve A X b system. If this system has o-zeo solutio the go to else go to.. Tasfom AX b to () system.. Omit colums of coefficiet matix.. Solve paametic system ().. This solutio is a o-zeo solutio of ( FFLS) AX b go to 7.. This system has a zeo solutio ad this algoithm caot solve it. 7. Ed.. Examples Example. Dehgha() oside the fully fuzzy liea system AX b, whee (, (,8 (,7 7 A ad b. (,7) (, (, 7 Dehgha et al. i () have solved this system ad its solutio vecto has peseted as x (,) ad x (, ), which ae tiagula fuzzy umbes. We solve this system by ou algoithm as follows: This system has a positive solutio hece z x, z x, y, y. Sice 8 ad ad 7
AAM: Ite, J., Vol., Issue (Jue 8) [Peviously, Vol., No. ] 8 7 8 7 8. Sice y y, coefficiet matix is tasfomed to: 7 8. Ou solutios vecto is: ) 89, 7 8 ( x Ad ) 9 7, 7 8 7 ( x. Example. oside the fully fuzzy liea system b AX, whee ), ( ), ( ), ( ), ( A ad. ) 9, ( ), ( b Dehgha's appoach, caot fid this systems solutio, because:. oefficiets of system have vaious L.R. fom.. Solutio of systems is ot positive. If we solve -cut of fully fuzzy liea system, we have.,,, x y y z x z Ad ad. With (), ou coefficiets matix is.
8 Allahvialoo et al. Sice z y, coefficiet matix is tasfomed to:. Ou solutio vecto is x (, x (, ).. oclusio I this pape, we popose a ovel method fo solvig the fully fuzzy liea system AX b whee all elemets of coefficiet matix ae positive.its o-zeo solutio is foud by eplacig oigial system by paametic fom system. We fist, detemie positive ad egative solutios of this system by solvig -cut system ad the eplace fuzzy coefficiet matix by paametic coefficiet matix. Fially, we omit colums of this matix ad solve this system. We compae ou appoach by Dehgha's (, ) appoach. I case which FFLS has egative solutio, Dehgha's appoach caot obtai the solutio but the poposed appoach fid it easily. Futhemoe, the esult of poposed method is moe pecise ad stable tha othe methods. REFERENES Abbasbady, S., R. Ezzati, A. Jafaia (). LU decompositio method fo solvig fuzzy system of liea equatios, Applied Mathematics ad omputatio 7, -. Abbasbady, S., A. Jafaia (). Steepest descet method fo system of fuzzy liea equatios, Applied Mathematics ad omputatio 7, 8-8. Allahvialoo, T. (). Discussio: A commet o fuzzy liea systems, Fuzzy Sets ad Systems, 9. Allahvialoo, T. (). Numeical methods fo fuzzy system of liea equatios, Applied Mathematics ad omputatio, 9-. Allahvialoo, T. (a). Successive ove elaxatio iteative method fo fuzzy system of liea equatios, Applied Mathematics ad omputatio, 89-9. Allahvialoo, T. (b). The Adomia decompositio method fo fuzzy system of liea equatios, Applied Mathematics ad omputatio, -. Allahvialoo, T., M. Afsha Kemai (). Solutio of a fuzzy system of liea equatio, Applied Mathematics ad omputatio 7,9-. Allahvialoo, T., E. Ahmady, N. Ahmady, Kh. Shams Alketaby (). Block Jacobi two stage method with Gauss Sidel ie iteatios fo fuzzy systems of liea equatios, Applied Mathematics ad omputatio 7,7-8.
AAM: Ite, J., Vol., Issue (Jue 8) [Peviously, Vol., No. ] 87 Asady, B., S. Abasbady, M. Alavi (). Fuzzy geeal liea systems, Applied Mathematics ad omputatio 9, -. Buckley, J. J., Y. Qu (99). Solvig liea ad quadatic fuzzy equatios, Fuzzy Sets ad Systems 8, -9. Buckley, J. J., Y. Qu (99a). Solvig fuzzy equatios: a ew solutio cocept, Fuzzy Sets ad Systems 9, 9 -. Buckley, J. J., Y. Qu (99b). Solvig systems of liea fuzzy equatios, Fuzzy Sets ad Systems,-. og-xi, Wu, Ma Mig (99). Embeddig poblem of fuzzy umbe space: Pat I, Fuzzy Sets ad Systems, -8. og-xi, Wu, Ma Mig (99). Embeddig poblem of fuzzy umbe space: Pat III, Fuzzy Sets ad Systems, 8-8. Dehgha, M., B. Hashemi, M. Ghatee (). omputatioal methods fo solvig fully fuzzy liea systems, Applied Mathematics ad omputatio, 79, 8 -. Dehgha, M., B. Hashemi (). Iteative solutio of fuzzy liea systems, Applied Mathematics ad omputatio 7, -7. Dehgha, M., B. Hashemi (). Solutio of the fully fuzzy liea systems usig the decompositio pocedue, Applied Mathematics ad omputatio, 8, 8-8. Dubois, D., H. Pade (98). Fuzzy sets ad systems: theoy ad applicatios, Academic pess. Fiedma, M., M. Mig, A. Kadel (998). Fuzzy liea systems, Fuzzy Sets ad Systems 9, -9. Fiedma, M., M. Mig, A. Kadel (). Discussio: Autho`s eply, Fuzzy Sets ad Systems,. Goetschel, R., W. Voxma (98). Elemetay calculus, Fuzzy Sets ad Systems 8, -. Kaleva, O., (987). Fuzzy diffeetial equatios, Fuzzy Sets ad Systems, -7. Ma, M., M. Fiedma, A. Kadel (). Duality i fuzzy liea systems, Fuzzy Sets ad Systems 9, -8. Mooe, R., Iteval Aithmetic (99). Petice-Hall, Eglewood liffs, NJ, USA. Muzzioli, S., H.Reyaets (). Fuzzy liea system of the fom A X b A X b, Fuzzy Sets ad Systems7, 99-9. Muzzioli, S., H.Reyaets (7). Disciplies the solutio of fuzzy liea systems by o-liea pogammig: a fiacial applicatio, Euopea Joual of Opeatioal Reseach, 77, 8 -. Pedycz, W. (987). O solutio of fuzzy fuctioal equatios, J. Math. Aal. Appl., 89-. Sachez, E. (98). Solutios of fuzzy equatios with exteded opeatios, Fuzzy Sets ad Systems,7-8. Voma, A., G. Deschive, E. E. Kee (). A solutio fo systems of liea fuzzy equatios i spite of the o-existece of a field of fuzzy umbes" Iteatioal Joual of Ucetaity, Fuzziess ad Kowledge-Based Systems, () -. Voma, A., G. Deschive, E. E. Kee (7a). Solvig systems of liea fuzzy equatios by paametic fuctios, IEEE Tasactios o Fuzzy Systems,, 7-8. Voma, A., G. Deschive, E. E. Kee (7b). Solvig systems of liea fuzzy equatios by paametic fuctios- a impoved algoithm, Fuzzy Sets ad Systems, 8, -. Wag, Ke, Big Zheg (). Icosistet fuzzy liea systems, Applied Mathematics ad omputatios, 8, 97-98.
88 Allahvialoo et al. Waga, Xizhao, Zimia Zhog, Mighu Ha (). Iteatio algoithms fo solvig a system of fuzzy liea equatios, Fuzzy Sets ad Systems 9,-8. Zheg, Big, Ke Wag (). Geeal fuzzy liea systems, Applied Mathematics ad omputatio, 8, 7-8. Zimmema, H. J. (98). Fuzzy set theoy ad applicatios, Kluwe, Doecht.