Fuzzy Minimal Solution of Dual Fully Fuzzy Matrix Equations

Transcription

1 Iteratioal Coferece o Applied Mathematics, Simulatio ad Modellig (AMSM 2016) Fuzzy Miimal Solutio of Dual Fully Fuzzy Matrix Equatios Dequa Shag1 ad Xiaobi Guo2,* 1 Sciece Courses eachig Departmet, Gasu raditioal Chiese Medicie Uiversity, Lazhou , Chia 2 College of Mathematics ad Statistics, Northwest Normal Uiversity, Lazhou , Chia *Correspodig author [16] ivestigated the dual fuzzy matrix equatios form AX B C X D based o fuzzy umbers Abstract he paper deals with the dual fully fuzzy matrix equatio A X B C X D he dual fuzzy matrix equatio is coverted to a crisp system of matrix equatios accordig to arithmetic operatios of fuzzy umbers he fuzzy approximate solutio of fuzzy matrix equatio is obtaied by solvig the model which is made of three liear matrix equatios he existece coditio of the oegative fuzzy solutio is also discussed A example is give to illustrate the proposed method IIPRELIMINARIES A Fuzzy Numbers Let E1 be the set of all fuzzy umbers o R Keywords-fuzzy umbers; matrix aalysis; dual fuzzy matrix equatios; fuzzy approximate solutios Defiitio 21 A fuzzy umber M is said to be a fuzzy umber if I INRODUCION System of simultaeous liear matrix equatios is a essetial mathematical tool i sciece ad techology I practice some or all parameters may be represeted by fuzzy umbers rather tha crisp oes herefore it is ecessary to develop mathematical theory ad umerical schemes to hadle fuzzy matrix systems he cocept of fuzzy umbers ad arithmetic operatios with these umbers were first itroduced ad ivestigated by Zadeh [1], Dubois et al[2] ad Nahmias [3] um (1) L x L x, (2) L 0 1 ad L 1 0, (3) L x is oicreasig o 0, However, for a fuzzy liear matrix equatio which always has a wide use i cotrol theory ad cotrol egieerig, few work has bee doe i the past decades I 2009, Allahviraloo et al [12] discussed the fuzzy liear matrix he defiitio of a right shape fuctio R is usually similar to that of L A equatios(flme) of the form AX B C where fuzzy umber symbolically show as M m,, A ad B are m m ad real matrices respectively, C is a give m fuzzy umbers matrix I 2011, Gog ad Guo [13] B ad ivestigated a class of fuzzy matrix equatios Ax Clearly, M m,, L R Defiitio arbitrary M is L R is positive(egative) if ad oly if m 0 m 0 studied its fuzzy least squares solutios by usig geeralized iverses of the matrix Later, Guo ad Shag [14-15] proposed a computig method of fuzzy symmetric solutios to fuzzy matrix 22 For M m,, ad B ad discussed the fuzzy Sylvester matrix fuzzy umbers Recetly, Guo ad Gog 2016 he authors - Published by Atlatis Press, x m, 0,, x m, 0, where m is the mea value of M, ad ad are left ad right spreads, respectively he fuctio L ; which is called left shape fuctio satisfyig: Sice Friedma et al [4] proposed a geeral model for solvig a fuzzy liear systems Ax=b by a embeddig approach i 1998, lots of works have bee doe about some advaced fuzzy liear systems such as dual fuzzy liear systems (DFLS), geeral fuzzy liear systems (GFLS), fully fuzzy liear systems (FFLS)ad geeral dual fuzzy liear systems (GDFLS) see [5-8] Ad some ew theories ad methods for fuzzy liear systems still appeared recetly [9-11] equatios AX equatios with m x L x R x m (1) Additio 373 fuzzy umbers N,,, we have

2 ,,,,,, M N m m (2) Multiplicatio If M 0 ad N 0 ; the,,,,,, MN m m m m B he Dual Fully Fuzzy Matrix Equatios Defiitio 23 A matrix A a,1 i m,1 is called a fuzzy matrix, if each elemet of A is a fuzzy umber A fuzzy matrix A is said to be positive(egative) ad deoted by A 0A 0 if each elemet a,1 im,1 of A be positive (egative) We ca represet m fuzzy matrix A a, that a a,, with ew otatio A A, M, N, where A a, M ad N are three m crisp matrices Defiitio 24 he liear system a11 a12 a x 1 11 x12 x 1 p a21 a22 a x 2 21 x22 x 2 p a 1 2 x1 x2 x m am a m p b11 b12 b b21 b22 b2 p bm1 bm2 b mp c11 c12 c x 1 11 x12 x c21 c22 c x 2 21 x22 x 2 p c m cm c x x m x p, (21) d11 d12 d d21 d22 d2 p 21 dm 1 dm2 d mp where a, c,1 im,1 ad b, d,1 im,1 p are fuzzy umbers, is called a dual fully fuzzy matrix equatios(dffmes) Usig matrix otatio, we have A X BCX D A fuzzy umbers matrix X x x, y, z,1 i,1 p is said to the solutio of dual fuzzy matrix equatio 21 if X satisfies Eq22, ie, AX B CX D, 1,, p X x1 x2 x B b1 b2 bm where D d1, d2,, dm, 1,, p,,,,,,,, are th colum of fuzzy matrices X, B ad D respectively Up to rest of this paper we wat to fid the positive solutio X, Y, Z 0 22, where X of oegative A A M N B B E F C C G H ad D D, P, Q 0,, 0,,, 0,,, 0 III SOLVING DUAL FULLY FUZZY MARIX EQUAION heorem 31 he dual fully fuzzy matrix system 22 ca be exteded ito the followig model system which is made of three liear matrix equatios 374

3 AX B CX D, AY MX E CY GX P, AZ NX F CZ HX Q Proof We deote A AM,, N, B BEF,,, C CGH,,, D DPQ,, ad assume X X, Y, Z 0, the is the miimal solutio of Eqs(31) such that X 0, Y 0, Z 0, the we say X X, Y, Z is a AX BCX D fuzzy miimal solutio of Eqs(22) If X X, Y, Z also satisfies X Y 0, we say it is a ie, oegative fuzzy miimal solutio of Eqs(22) AMN,, XYZ,, BEF,, CGH,, XYZ,, DPQ,, Accordig to multiplicatio of oegative fuzzy heorem 33 Let A A, M, N, B B, E, F umbers of Defiitio 23, we have ad C C, G, H be three oegative fuzzy AXAY, MX, AZ NX B, E, F CX, CYGX, CZHX D, P, Q matrices, respectively, ad A C be oegative hus we obtai a model for solvig DFFMEs (22) as matrix Let D B0, PEGMAC 1 D B ad 1 Q F H N A C D B he the dual AX B CX D, AY MX E CY GX P, AZ NX F CZ HX Q I order to solve the dual fuzzy system (22), we eed to cosider the crisp system of liear matrix equatio (31) For example, whe A C is a osigular crisp matrix, we obtai its solutio as Y A C P E G M A C B D Z AC QF HN AC BD 1 XAC DB, ,(32) m heorem 32([17]) Let A belog to R ad B m p belog to R he the miimal solutio X of the matrix equatio AX B is expressed by X AB Whe A C is a sigular matrix, by the heorem 32, we solve model 31 ad obtai its miimal solutio as Y A C P E G M A C B D Z AC QF HN AC BD X AC DB,,33 Defiitio 32 Let X X, Y, Z If X X, Y, Z fuzzy systems A XBCX D has a fuzzy miimal solutio If the 1 coditio D BI GMAC P E holds, the the dual fuzzy systems (21) has a oegative fuzzy miimal solutio Proof Sice A C ad D B are oegative matrices hus the fact X A C D B 0 is apparet O the other had, because P E G M A C D B ad Q F H N A C D B, so with Y AC PEGMAC B D ad Z AC QF H N AC DB, we have Y 0 ad 0 X XYZ,, is a fuzzy solutio of the dual fuzzy systems AX BCX D Sice Z hus 375

4 XY AC DB PE GM AC D B the oegative property of the solutio X to dual fuzzy systems (21) ca be obtaied from the coditio I GMAC DB P E IV NUMERICAL EXAMPLE Example 41 Cosider the dual fully fuzzy matrix system x11 x 12 x x 2,1,1 3,1, 2 1, 0, 0 3,1,1 2, 0,1 2,1, x31 x32 2, 2,1 1,1, 2 2,1,1 3,1,1 2,1,1 1, 0, 0 1,1,1 1, 0,1 2,1,1 3, 0,1 4, 2, 2 3, 2, 2 5, 4,3 3, 2, 0, x11 x 12 x x x31 x32 By heorem 31, the model to above fuzzy liear matrix system is made of followig three crisp systems of liear matrix equatios x x x x x21 x22 x21 x22, x31 x 32 x31 x 32 y y x x y y x x y21 y22 x21 x y11 y12 x11 x y21 y22 x21 x y y x x ad z11 z12 x11 x z21 z22 x21 x z z x x z11 z12 x11 x z z x x z31 z 32 x31 x 32 Accordig to formula (33), the miimal solutios of above three systems of liear matrix equatios are as X AC DB , Y AC PE GM X X Z AC QF H N X X

5 By Defiitio 32, we kow the origial fuzzy liear matrix equatios has a oegative fuzzy solutio x11 x 12 X x x 13143,01258, ,09896,00333, 20571,1,0333, ,01236, ,1327,11250, ,13333, x31 x 32 Sice X 0, Y 0, Z 0 ad X Y 0 [15] XB Guo, DQ Shag, Approximate solutios of fuzzy Sylvester matrix equatios, Joural of Applied Mathematics, Volume 2013, Article ID , 10 pages [16] Z Gog, XB Guo, K Liu, Approximate solutio of dual fuzzy matrix equatios, Iformatio Scieces 266 (2014) [17] A Berma, RJ Plemmos, Noegative matrices i the Mathematical Scieces, Academic press, New York, 1979 V CONCLUSION I this work, we proposed a simple model for solvig the geeral dual fully fuzzy matrix equatio We coverted the dual fuzzy matrix equatio to a crisp system of liear matrix equatios accordig to the arithmetic operatios of fuzzy umbers he existece coditio of oegative fuzzy solutio was also studied Numerical examples showed that our method is effective to solve the dual fully fuzzy matrix equatio ACKNOWLEDGEMENS his research was supported by the Natioal Natural Sciece Foudatio of Chia (o ) REFERENCES [1] LA Zadeh, he cocept of a liguistic variable ad its applicatio to approximate reasoig, Iformatio Sciece 8 (1975) [2] D Dubois, H Prade, Operatios o fuzzy umbers, Joural of Systems Sciece 9 (1978) [3] S Nahmias, Fuzzy variables, Fuzzy Sets ad Systems, 1(2)(1978) [4] M Friedma, M Ma, A Kadel, Fuzzy liear systems, Fuzzy Sets ad Systems 96 (1998) [5] M Ma, M Friedma, A Kadel, Duality i Fuzzy liear systems, Fuzzy Sets ad Systems 109 (2000) [6] S Abbasbady, M Otadi, M Mosleh, Miimal solutio of geeral dual fuzzy liear systems, Chaos, Solitios ad Fractals 37 (2008) [7] B Zheg, K Wag, Geeral fuzzy liear systems, Applied Mathematics ad Computatio, 181 (2006) [8] M Dehgha, B Hashemi, M Ghatee, Solutio of the full fuzzy liear systems usig iterative techiques, Chaos, Solitos ad Fractals, 34 (2007) [9] Allahviraloo, M Ghabari, A ew approach to obtai algebraic solutio of iterval liear systems, Soft Computig 16 (2013) [10] R Ghabari, N Mahdavi-Amiri, New solutios of fuzzy liear systems usig rakig fuctios ad ABS algorithms, Applied Mathematical Modellig, 34 (2012) [11] R Ghabari, Solutios of fuzzy algebraic liear systems usig liear programs, Applied Mathematical Modellig, 39 (2015) [12] Allahviraloo, N Mikaeilvad, M Barkhordary, Fuzzy liear matrix equatios, Fuzzy Optimizatio ad Decisio Makig 8 (2009) [13] Z Gog, XB Guo, Icosistet fuzzy matrix equatios ad its fuzzy least squares solutios, Applied Mathematical Modellig 35 (2011) [14] XB Guo, DQ Shag, Fuzzy symmetric solutios of fuzzy matrix equatios, Advaces i Fuzzy Systems, Volume 2012, Article ID , 9 pages 377