Solve Multi Linear Programming Problem By Taylor Polynomial solution

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1 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0 Solv Mult Lr Progrmmg Problm By ylor Polyoml soluto Wld Khld Jbr Collg of Comutr S.& Mthmt Comutr Drtmt h-qr Uvrsty انخالصت تض انبحج حم يسائم انبشيجت انخط ت ان تعذدة باستخذاو يتسهسهت تا ه س انحص ل عهى انحم األيخم. ح ج تى يماس ت ان تائج انت تى انحص ل عه ا باستع ال ز انطش م يع انحم انذل ك ن ز ان سائم بانطشق االعت اد ت ان ستع هت نحم يسائم انبشيجت انخط ت اظ شث ان تائج ا ز انطش م يتماسبت كف ءة. Abstrt: W hv roosd soluto to Mult Lr Progrmmg Problm (MLPP) by dg th ordr st ylor olyoml srs. h ylor srs s srs so tht rrstto of futo ths obtv futos t otml ots of h lr obtv futos fsbl rgo. hus th roblm s rdud to sgl obtv. Numrl ml s rovdd to dmostrt th ffy d fsblty of th roosd roh. 6

2 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0 Itrdto I th modlg of th rl word roblms l fl d orort lg roduto lg mrtg d md slto uvrsty lg d studt dmssos hlth r d hostl lg r for mt uts b brhs t. frqutly my b fd u wth dso to otmz dt/quty rto roft/ost vtory/sls tul ost/stdrd ost outut/mloy studt/ost urs/tt rto t. rst to som ostrts (L d Hwg [996]).[6] I th ltrtur dffrt rohs r to solv dffrt modls of Lr Progrmmg Problm (LPP). Bus rogrmmg solvs mor fftly th bov roblms wth rst to Lr Progrmmg Problm (LPP). I ths rs LPP r dsussd dtls. It s showd tht LPP b otmsd sly. But th grt sl dso roblms thr s mor th o obtv whh must b stsfd t th sm tm s ossbl. Howvr most of ths r lr obtvs. It s dffult to tl bout th otml solutos of ths roblms. h solutos srhd for ths roblms r w fft or strog fft. If rqurd o omroms soluto b rhd by th ffto of th modls wth th dso mrs (DMs). hr st svrl mthodologs to solv mult obtv lr rogrmmg roblm (MoLPP) th ltrtur. Most of ths mthodologs r omuttolly burdsom (Chrborty d Gut [00]). Korbluth d Stur [98] Y.J.L d C.L.Hwg [996] hv dvlod lgorthm for solvg th MoLPP for ll w-fft vrts of th fsbl rgo. Nyows.olws [978] d Dutt t ll [99] hv roosd omroms rodur for MoLPP. Choo d Ats [98] hv gv lyss of th brtr LPP.[3].. Lr Progrmmg [][][3][4][8] A grl modl of rs lr rogrmmg s formultd (Stdrd formulto): M Subt to A b 0...() whr d r dmsol vtors b s m dmsol vtor d A s (m ) mtr. 6

3 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0.. Dfto lt hm m rolm : m m () 0 () /... / A b; 0 0 ( ) t R d m t... t 0 t... ; A b; 0 (..) (..)... horm Proof: If ( o ) solv (..) th solv (..) d o= () If s fsbl soluto to (..) th ( () ) s fsbl soluto to (..) (S A = b ; 0 d () ( (t)) 0 ; t = ). o () o () (..) o ( (t)) ; t = o () () From () d () w hv o = () d () o = () for y fsbl solutos to (..) ().. Modl Dvlomt I ths r w osdr th Mult Obtv Lr Progrmmg Problm (MoLPP) M () = M { () () () } S.t. X = { R A b 0 }.. wth b R A Rm d () = whr R Lt th mmum vlu th obtv futo M () = o th fsbl rgo t ours wh = ( ) for =. 6

4 6 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0 Suos tht () d ll of t's rtl drvtv of ordr lss th or qul ( + ) r otuous o th fsbl rgo X X. By dg th st ordr ylor olyoml srs for obtv futo () bout obtv futo () s obtd from () =P () + P() Whr futo P () s lld th frst ylor olyoml vrbls bout d P () s th rmdr trm ssotd wth P () w hv : for O O h h... (..) whr O(h) s ordr of th mmum rror. hs olyoml gvs urt romto to () wh s los to. By rlg () = I MOLPP ll of th obtv futos () = bom th st ordr lr futos s M followg so MOLPP rdus th R... :... X = { R A b 0 } (..3) wth b R A Rm If w ssum tht th wghts of obtv futos roblm (..3) r qul th roblm (..3) s wrtt s follows: (..4)... M

5 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0 X = { R A b 0 } wth b R A Rm I roblm (..4) st X s o-mty ov st hvg fsbl ots. h otml soluto of roblm (..4) gvs th fft soluto of MoLPP (..3). Bus wghts of obtv futos tht r dd ylor srs r qul d (..3) s osdrd th wghtd obtv futo. Eml: M { = + = 3 - } S.t Now to solv obto futo () h Stdrd form of obto futo () s : - - = S = S = 0 W hv th otml soluto of r: =.7 =.5749 th M = 4.8 Now to solv obto futo () M = 3 - S.t h Stdrd form of obto futo () s: = S = S = 0 w hv th otml soluto of r: = 3 = 0 03

6 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0 M = 9 It's obsrvd tht & 0 for h M (.7.57) = 4.8 & M ( 3 0 ) = 9 By dg th st ordr olyoml srs for obtv futo () d () bout ots (.7.57 ) d ( 3 0 ) s : ().57 () [ 33] [ 0 3 ( )] So ll obtvs r trsformd to th L.P. h obtd MoLP s quvlt followg: th M { () + () } = 4 S.t h otml soluto for ths MoLP s: M { () + () } (30) = = 3 d = 0 w hv M = 3 d M = 9 03

7 Jourl of h-qr Uvrsty umbr Vol.6 Mrh/0 Coluso I ths r w hv roosd soluto to Mult Obtv Lr Progrmmg Problm (MoLPP) usg ylor olyoml srs. Wth th hl of th ordr st ylor olyoml srs t otml ots of h lr obtv futo fsbl rgo. h obtd MoLPP s solvd ssumg tht wghts of ths lr obtv r qul d osdrg th sum of th lr obtv futos. h roosd soluto to MoLPP lwys ylds fft soluto v strog-fft soluto. hrfor th omlty solvg MoLPP hs rdud sy omuttol. Rfrs: أ.د. نطف ن ز س ف بح ث انع ه اث ان ج انك التخار انمشاساث داس انجايعاث ان صش ت [ 7 ]. ][. د. الل ادي صانح د. خانذ جشج س عب - ح اء سش ذ صادق بح ث انع ه اث تطب مات ا لسمى عهمى انحاسمباث - ][. انجايعت انتك ن ج ت [77]. [3]. Chrbty M. GUPA S Fuzzy mthmtl rogrmmg for mult obtv lr frtol rogrmmg roblm Fuzzy Sts d systms 5: [00]. [4]. Glmor AC Gomory RE A lr rogrmmg roh to th uttg sto roblm II Ortol Rsrh: [963]. [5]. Hmdy A.h Orto Rsrh hrd dto [98]. [6]. L Y Hwg CL Fuzzy Multl Obtv Dso Mg Srgr[996]. [7]. Mutu E Rdo I Clulul Srlr lor m oom l utorl dtot fot stud s rtr mtmt lu fsol XI 49-58[960]. [8]. Pd Vst Otmzto Produt M Problm Usg Fuzzy Lr Progrmmg Drtmt of Mthmts Amr Dgr Progrm Nl Itrtol Collg Mlys 06