Fully Fuzzy Linear Systems Solving Using MOLP

Transcription

1 World Appled Sceces Joural 12 (12): , 2011 ISSN IDOSI Publcaos, 2011 Fully Fuzzy Lear Sysems Solvg Usg MOLP Tofgh Allahvraloo ad Nasser Mkaelvad Deparme of Mahemacs, Islamc Azad Uversy, Scece ad Research Brach, Tehra, Ira Absrac: As ca be see from he defo of exeded operaos o fuzzy umbers, subraco ad dvso of fuzzy umbers are o he verse operaos o addo ad mulplcao, respecvely. Hece for solvg equaos or sysem of equaos, we mus use mehods whou usg verse operaors. I hs paper, we propose a ovel mehod o fd he ozero soluos of fully fuzzy lear sysems (show as FFLS). We suppose ha all elemes of marx of coeffces are posve. We employ embeddg mehod o rasform (FFLS) o 2 2 paramerc form lear sysem. We decompose soluos o o egaves ad o posves usg soluos of oe MOLP he al sep of hs mehod. Fally, umercal examples are used o llusrae hs approach. Key words: Fuzzy umbers fully fuzzy lear sysems sysems of fuzzy lear equaos o-zero soluos decomposo mehod INTRODUCTION Sysem of equaos s he smples ad he mos useful mahemacal model for a lo of problems cosdered by appled mahemacs. I pracce, uforuaely, he exac values of coeffces of hese sysems are o a rule kow. Ths uceray may have eher probablsc or o probablsc aure. Accordgly, dffere approaches o he problem ad dffere mahemacal ools are eeded. I hs arcle, sysem of lear equaos whose coeffces ad rgh had sdes are fuzzy umbers s cosdered. The sysem of lear equaos AX = b where he elemes, elemes, a, of he marx A ad he b, of b are fuzzy umbers, s called Fully Fuzzy Lear Sysem (FFLS). Abramovch ad hs colleagues [1], Buckley ad Qu [5-7], Muzzol ad Reyaers [14, 15], Dehgha ad hs colleagues [8-10], Vroma ad her colleagues [16-18], Allahvraloo ad hs colleagues [2-4] suggesed dffere approaches o solvg (FFLS). I may applcaos, ha ca be solved by solvg sysem of lear equaos, sysem's parameers are posve ad hece s mpora o propose mehod o fd o-zero soluos of (FFLS), where sysem's parameers are posve. Allahvraloo e al. [3] suggess a ew mehod for solvg (FFLS) usg decomposo mehod. I he al sep of her mehod, hey solve 0-cu erval lear sysem for decompose varables o o posves ad o egaves. Base o her work, hs paper, we are gog o fd o-zero soluos of hs (FFLS). For hs reaso, we decompose varables wo groups: o posves ad o egaves usg soluos of oe MOLP he al sep ad rasform mulplcaos of fuzzy umbers o mulplcaos of fucos. We use embeddg approach o replace he orgal (FFLS) by a 2 2 paramerc lear sysem ad desg umercal mehod for calculag he soluos. The srucure of hs paper s orgazed as follows: I seco 2, we dscuss some basc defos, resuls o fuzzy umbers ad (FFLS). I seco 3, we dscuss our umercal procedure for fdg o-zero soluos of (FFLS) ad he proposed algorhm are llusraed by solvg some umercal examples. Cocluso are draw seco 4. PRELIMINARIES The se of all fuzzy umbers s deoed by E ad defed as follows: Defo 1: [12, 13] A fuzzy umber u s a par (u(r),u(r)) of fucos u(r),u(r);0 r 1 whch sasfy he followg requremes: u(r) s a bouded moooc creasg lef couous fuco; u(r) s a bouded moooc decreasg lef couous fuco; u(r) u(r),0 r 1. Correspodg Auhor: Dr. Nasser Mkaelvad, Deparme of Mahemacs, Islamc Azad Uversy, Scece ad Research Brach, Tehra, Ira 2268

2 World Appl. Sc. J., 12 (12): , 2011 A crsp umber k s smply represeed by k(r)=k(r)=k; 0 r 1 ad called sgleo. The fuzzy umber space {u(r),u(r)} becomes a covex coe E whch s he embedded somorphcally ad somercally o a Baach space. A fuzzy umber a ca be represeed by s λ-cus (0<λ 1): ad 0 λ a ={x x R, a(x) λ} supp a=a =Cl({x x R, a(x)>0})=[a(0),a(0)] Case 2: u 0 ad ν 0 uv(r)=u(r)v(r), uv(r)=u(r)v(r) Defo 2: The lear sysem of equaos a x + a x + + a x =b a x + a x + + a x =b a x + a x + + a x =b (4) Noe ha he λ-cus of a fuzzy umber are closed ad bouded ervals. The fuzzy arhmec based o he Zadeh exeso prcple ca also be calculaed by erval arhmec appled o he λ-cus. For fuzzy umber u=(u(r),u(r)), 0 r 1 we wll wre (1)u>0 f u(0)>0,(2)u 0 f u(0) 0,(3)u<0 f u(0)<0,(4)u 0 f u(0) 0. If u 0 or u 0 hs fuzzy umber s called o-zero fuzzy umber. For arbrary u=(u(r),u(r)), v=(v(r),v(r)) ad k>0 we defe addo (u + v), subraco (u v) ad mulplcao (u.v) as: Addo: (u+ v)(r)=u(r) + v(r), (u + v)(r)=u(r) + v(r) (1) Subraco: (u v)(r)=u(r) v(r), (u v)(r)=u(r) v(r) (2) Mulplcao: (uv)(r)=m{u(r)v(r),u(r)v(r),u(r)v(r),u(r)v(r)}, (uv)(r)=max{u(r)v(r),u(r)v(r),u(r)v(r),u(r)v(r)} The mpora cases where we wa o calculae mulplcao of wo fuzzy umbers are as follows: (3) where he elemes, a, of he coeffce marx A, 1, ad he elemes, b, of he vecor b are fuzzy umbers s called a fully fuzzy lear sysem of equaos (FFLS). Defo 3: A fuzzy umber vecor ( x, x,...,x ) 1 2 x = (x (r),x (r)) 1, 0 r 1 gve by s called a soluo of (FFLS), f =1 =1 a x (r)= a x (r)= =1 =1 a x (r)=b(r), a x (r)=b(r), =1,, (5) Now, we defe o-zero fuzzy umber soluo of (FFLS) as follows: Defo 4: A fuzzy umber soluo vecor ( x 1, x 2,...,x ) of (FFLS) s called o-zero fuzzy umber soluo f for all, (=1,2,...,) x are o-zero fuzzy umbers. Necessary ad suffce codo for he exsece of a No-zero fuzzy umber soluo of (FFLS) s: Theorem: [3] If (FFLS) AX = b has a fuzzy umber soluo, he (FFLS) AX = b has o-zero fuzzy umber soluo f ad oly f 0-cu sysem of lear sysem represeed by A 0 X 0 = b 0 has o-zero soluo. THE MODEL Case 1: u 0 ad ν 0 uv(r)=u(r)v(r), uv(r)=u(r)v(r) 2269 As ca be see from he defo of exeded operao o fuzzy umbers, subraco ad dvso of fuzzy umbers are o he verse operaos o

3 World Appl. Sc. J., 12 (12): , 2011 addo ad mulplcao, respecvely. Hece for solvg equaos or sysem of equaos, we mus use mehods whou usg verse operaors. I hs seco, we are gog o replace a orgal (FFLS) wh a 2λ2 paramerc sysem ad he dscuss our algorhm o fd hs soluo. Le AX = b s (FFLS). Cosder -h equao of hs sysem: =1 a x = b,=1,..., (6) Now, le AX = b has a o-zero fuzzy umber soluo, we defe If we use defos 3 ad 4 ad f x=(x,,x ) 1 s fuzzy umber soluo of AX = b, eq (11) ca be rewre as: ad (12) b(r)= a w (r) + a v (r) =1,..., =1 =1 (13) b(r)= a w (r) + a v (r) =1,..., =1 =1 Sce w 0, v 0, (1 ) ad by applyg (3) we ca wre: J = { 1, x 0} (7) Hece we ca decompose eq. (6) ad rewre as: a x = a x + ax=b,=1,..., (8) =1 J J We defe wo -vecor V=(v,v,...,v 1 2 ) where Ths s obvous ha If we replace rewre as follows: x f J w = 0 f J x f J v = 0 f J 1 2 ad W = ( w, w,...,w ) (9) w + v =x, 1 (10) x (8) wh w + v ax= ax + ax =1 J J = aw + av J J = aw + av =1 =1 =b,=1,...,, ca be (11) 2270 a w ( r ) = a (r).w (r) a w ( r ) = a (r).w (r) a v ( r ) = a (r).v (r) a v ( r ) = a (r).v(r) (14) Now, f we replace above expressos (12) ad (13), hey ca be rewre as: a x ( r ) = a (r).w (r) + a (r)v (r)=b(r), (15) =1 =1 =1 ad =1,...,, a x (r)= a (r)w(r) + a (r).v (r)=b(r), (16) =1 =1 =1 =1,...,. Noe ha before solvg (FFLS) AX = b sce we decompose marx of coeffces, we mus have formao abou s soluo such as: 1-Does hs sysem have a ozero soluo? ad 2-Is x posve fuzzy umber? I [3] Allahvraloo e al. fd her soluos by solvg 0-cu sysem of AX = b o fd (FFLS) AX = b soluo suppors. Afer solvg hs erval sysem hey fd our quesos'aswers. Now, we propose aoher mehod for fd her soluos whou solvg 0-cu sysem of AX = b o fd (FFLS) AX = b soluo suppors. We defe a crsp umber y, 1 as follows: Ad defe z 0 Hece y = m{x R x 0, x + v 0} (17) z = v + y 0, 1 (18) v = z y, 1 (19)

4 Ths s obvous ha f y = 0 he x 0 ad hece v=0 ad w =x. bu f y > 0 we ca o fer x egave fuzzy umber ad we oly ca fer x s o posve. By applyg (1)-(4) ad (19) ad replacg (15) ad (16) we ca rewre hey as follows: ad a x (r)= a (r).w(r ) + a (r).(z (r) y) (20) =1 =1 =1 =b(r) World Appl. Sc. J., 12 (12): , 2011 =1,..., a x (r)= a (r).w(r ) + a (r).(z (r) y) (21) =1 =1 =1 Ad hece, ad =b(r), =1,..., a (r).w (r) + a (r).z (r) =1 =1 =b (r ) + a (r)y, =1,..., =1 a (r).w (r) + a (r).z (r) =1 =1 =b (r ) + a (r)y, =1,..., =1 s (22) (23) Now we llusrae hese equaos marx forms. If A 1 ad A 2 are paramerc marces by elemes ( A 1) = a (r), (A 2) = a (r), 1, (24) ad f W 1, W 2, Z 1, Z 2, B 1 ad B 2 are paramerc - vecors by elemes ( W ) = w (r), ( W ) = w (r), 1 2 (Z 1) = Z (r), (Z 2) = Z (r), 1, ( B ) = b ( r ), (B ) = b (r), 1 2 (25) Ad Y = (y 1,, y ), marx represeao of (FFLS) AX = b rasform o 2271 W1 A1 0 A2 0 W2 B1+ A2Y 0 A2 0 A 1 Z 1 = B2 + AY 1 Z 2 (26) where hs coeffces marx represes 2 4. Bu fac, by defo of Z 1, Z 2, W 1 ad W 2, 2 elemes of varable marx are zero ad hece 2 colums of coeffce marx are omed ad hece we replace (FFLS) by a 2 2 sysem of lear paramerc equaos. We do o kow ay formao abou x ad hece we ca o defe w, z ad y, ( = 1,,). By defo of posve fuzzy umber ad defo of y, for fdg y ad hece fdg w ad z ( = 1,,) we solve 0-cu of fully fuzzy lear sysem where rasform o (26) ad sasfy (19). Hece we oly requre o solve he followg (MOLP): =1 =1 M y M y M y s.. (a (0).w (0) + a (0).(z (0) y) =1 =1 1 2 =b(0), =1,...,, a (0).w (0) + a (0).(z (0) y ) = b (0), =1,...,, z (0) z (0),=1,...,, w (0) w (0),=1,...,, w (0) 0,,=1,...,, z (0) 0,,=1,...,, y(0) 0,,=1,...,, (27) If hs MOLP does o have ay feasble soluo, 0-cu of fully fuzzy lear sysem would o have ay soluo ad hece hs (FFLS) s usolvable. If above MOLP has a feasble soluo, we ca fd z ( = 1,,) by solvg (26) 2 4 paramerc lear sysem. Maybe, hs MOLP have alerave soluos, bu we oly requre a soluo whch sasfy he followg codo: f y (0) = 00 he v = 0 ( = 1,,) (because hs MOLP's soluo sasfy (15),(16) ad (19).) Afer solve above MOLP ad fd requred soluo ad replace (26) ad solve hs sysem, f

5 World Appl. Sc. J., 12 (12): , 2011 s soluos defe fuzzy umber, (FFLS) AX = b has fuzzy umber soluo where x =w + v (=1,...,) Example [8] Cosder he sysem of equaos (4 + r,6 r)x 1 + (5 + r,8 2r)x 2 =( r,67 17r) (6 + r,7)x 1+ (4,5 r)x 2 = (43 + 5r,55 7r) (28) Based o her work, hs paper, we sugges he al sep of algorhm, replace solvg 0-cu sysem by solvg MOLP (27). Ths sugges has useful ad harmful because, f y = 0 he x 0 ad hece v ~ = 0 ad w ~ = ~ x. bu f y > 0 we ca o fer x s egave fuzzy umber ad we oly ca fer x s o posve. Maybe 2λ2 paramerc sysem has soluo, bu s soluos ca o defe fuzzy umbers ad hece (FFLS) does o have ay fuzzy umber soluo. Dehgha e al. [8] solved hs sysem. Ther soluo s ad 43 1 x=( 1 + r,4) x 2 = ( + r, r) Where hese are approxmaed soluos. We solve hs sysem by our algorhm as follow's : Frs, we solve he MOLP. The soluo of hs sysem s y 1 = y 2 = 0. Hece ν 1 = ν 2 = 0 ad we ca replace 2 2 coeffce marx by 4 4 paramerc coeffce marx; The soluo vecor-exac soluo of hs (FFLS)-s: ad r 28r 55 3r + 14r 105 x=(, ) r 7r 14 r + 3r r 37r 68 7r + 22r 139 x = (, ) r 7r 14 r + 3r 26 CONCLUSION Oe of he mpora applcaos of mahemacs s sysem of lear equaos ha whe all sysem's parameers are fuzzy s called fully fuzzy lear sysem. Allahvraloo e al. [3], desg a ovel mehod for solvg (FFLS) ad for fdg s o zero soluos usg embeddg mehod. I her mehod, hey decompose varables o wo groups: o egaves ad o posves ad f sysem's soluos s wo groups hey solve hs sysem ad fd s soluos. For hs mea, he al sep, hey solve 0-cu sysem of (FFLS) ad fd s soluos. If s soluos are o zero, hey use embeddg mehod ad rasform fuzzy sysem o 2 2 paramerc sysem ad solve. Sce 0-cu sysem s erval lear sysem, for solvg, 2, (+1) crsp sysem mus be solved REFERENCES 1. Abramovch, F., M. Wagekech ad Y.I. Khurg, Soluo of LR-ype fuzzy sysems of lear algebrac equaos. Busefal, 35: Allahvraloo, T., N.A. Ka ad M. Mosleh, Homomorphc Soluo of Fully Fuzzy Lear Sysem. Joural of Compuaoal Mahemacs ad Modelg, Sprger, 19 (3): Allahvraloo, T. ad N. Mkaelvad, Sged Decomposo of Fully Fuzzy Lear Sysems. A Ieraoal Joural of Applcaos ad Appled Mahemacs, (I Press). 4. Allahvraloo, T. ad N. Mkaelvad, No-zero soluo of fully fuzzy lear sysem. Submed for Appear. 5. Buckley, J.J. ad Y. Qu, Solvg lear ad quadrac fuzzy equaos. Fuzzy Ses ad Sysems, 38: Buckley, J.J. ad Y. Qu, Solvg fuzzy equaos: A ew soluo cocep. Fuzzy Ses ad Sysems 39: Buckley, J.J. ad Y. Qu, Solvg sysems of lear fuzzy equaos. Fuzzy Ses ad Sysems, 43: Dehgha, M., B. Hashem ad M. Ghaee, Compuaoal mehods for solvg fully fuzzy lear sysems. Appled Mahmacs ad Compuao, 179: Dehgha, M. ad B. Hashem, Soluo of he fully fuzzy lear sysems usg he decomposo procedure. Appled Mahemacs ad Compuao, 182: Dehgha, M., B. Hashem ad M. Ghaee, Soluo of he fully fuzzy lear sysems usg erave echques Compuaoal. Chaos Solos ad Fracals, 34: Dubos, D. ad H. Prade, Fuzzy ses ad sysems:heory ad applcaos. Academc press. 12. Goeschel, R. ad W. Voxma, Elemeary calculus, Fuzzy Ses as Sysems, 18:

6 World Appl. Sc. J., 12 (12): , Kaleva, O., Fuzzy dffereal equaos. Fuzzy Ses ad Sysems, 24: Muzzol, S. ad H. Reyaers, Fuzzy lear sysem of he form A 1 x+b 1 = A 2 x+b 2. Fuzzy Ses ad Sysems, 157: Muzzol, S. ad H. Reyaers, The soluo of fuzzy lear sysems by o-lear programmg: A facal applcao. Europea Joural of Operaoal Research, 177: Vroma, A., G. Deschrver ad E.E. Kerre, A soluo for sysems of lear fuzzy equos spe of he o-exsece of a feld of fuzzy umbers. Ieraoal Joural of Uceray, Fuzzess ad Kowledge-Based Sysems, 13 (3): Vroma, A., G. Deschrver ad E.E. Kerre, Solvg sysems of lear fuzzy equaos by paramerc fucos. IEEE Trasacos o Fuzzy Sysems, 15: Vroma, A., G. Deschrver ad E.E. Kerre, Solvg sysems of lear fuzzy equaos by paramerc fucos-a mproved algorhm. Fuzzy Ses ad Sysems, 158: Zmmerma, H.J., Fuzzy se heory ad applcaos, Kluwer, Dorrech. 20. Allahvraloo, T., N. Mkaelvad, N.A. Ka ad R. Masa, Sged Decomposo of Fully Fuzzy Lear Sysems. I Joural of Applcaos ad Appled Mahemacs: A Ieraoal Joural (AAM), 3 (1): Allahvraloo, T. ad N. Mkaelvad, Posve Soluos of Fully Fuzzy Lear Sysems. I Joural of Appled Mahemacs. IAU of Laha, 3 (11). 22. Allahvraloo, T. ad N. Mkaelvad, No zero soluos of he fully fuzzy lear Sysems. I Ieraoal Joural of Appled ad Compuaoal Mahemacs. 23. Allahvraloo, T. ad S. Salahshour, Maxmal-ad mmal symmerc soluos of fully fuzzy lear sysems. I Joural of Compuaoal ad Appled Mahemacs, (I Press). 24. Allahvraloo, T. ad S. Salahshour, Bouded ad symmerc soluos of fully fuzzy lear sysems dual form. I Joural of Proceda-Compuer Scece Joural. 2273