An Alternative Method to Find the Solution of Zero One Integer Linear Fractional Programming Problem with the Help of -Matrix

Transcription

1 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 ISSN A Altertve Method to Fd the Soluto of Zero Oe Iteger Ler Frctol Progrg Prole wth the Help of -Mtr VSeeregsy *, DrKJeyr ** * Professor of Mthetcs, PSNA College of Egeerg d Techology, Ddgul-64 6, Tl Ndu, Id ** De of Scece d Hutes, PSNA College of Egeerg d Techology, Ddgul-646, Tl Ndu, Id Astrct- To ze the coputtol effort eeded to solve zero oe Iteger Ler Frctol progrg prole ew pproch hs ee proposed Here we use tr for fdg the soluto of the teger ler frctol progrg proles Ide Ters- oe Iteger Ler Frctol Progrg Proles, tr d Prosg vrles Z I INTRODUCTION ero-oe Iteger Ler Frctol Progrg Prole s specl cse of Iteger Ler Frctol Progrg Prole There re vrous ethods used to solve such proles To reduce the coputtol effort ew lgorth hs ee proposed to solve Zero-Oe Iteger Ler Frctol Progrg Prole I ths lgorth the decso vrles re rrged sed o the u cotruto to the oectve fucto The rrged vrles re the llowed to eter to the ss Ths leds to reduce the coputtol te y es of reducg the terto The descrptos of ew lgorth hve ee dscussed ths chpter II STRUCTURE OF ZERO-ONE INTEGER LINEAR FRACTIONAL PROGRAMMING PROBLEM I Iteger Ler Frctol Progrg Prole, f ll the vrles re restrcted to tke the vlues ether or oly, the the gve prole s kow s Zero-Oe Iteger Ler Frctol Progrg Prole The geerl Zero-Oe Iteger Ler Frctol Progrg Proles s gve y C D X c Etreze Z AX P T T Suect to d X or X d Where A X 3 P 3 Let the colus correspodg to the tr A e deoted y P, P, P 3 P where wwwsrporg

2 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 ISSN wwwsrporg P 3 P 3 P P 3 C T ( c, c, c 3 c ), D T ( d, d,d 3 d ) d c, d re sclrs III ZERO-ONE INTEGER LINEAR FRACTIONAL PROGRAMMING ALGORITHM Ths lgorth cossts of three phses I frst phse prosg vrles re rrged, secod phse rrged vrles re llowed to eter to the ss d flly deterto of ew soluto vector to the teger ler frctol progrg prole The step y step procedure s s gve elow Step : Let terto Step : Perfor phse I Step 3 : Perfor phse II Step 4 : If the set J s epty the, Perfor phse III Step 5 : stop Phse I - Orderg of Prosg vrles Step Usg the tercepts of the decso vrles log the respectve es wth respect to the chose ss tr s clled tr s to e costructed A typcl tercept for the th vrle, due to the th the resource, s > The epded for of tr s S S S S Ech row of the tr cossts of uer of tercepts of the decso vrle log ther respectve es d ech colu cossts of tercepts fored y the uer of prosg decso vrles ech of the costrts Step The u tercept d ts posto ech row of tr s fd out If there re ore oe u tercept the oe of the s selected rtrrly Multply the u tercept of the vrle correspodg to row wth the correspodg cotruto coeffcet the oectve fucto

3 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 3 ISSN oth the uertor d deotor d the oectve c c fucto vlue d d s clculted Step 3 Repet step tll the u for ech row s well s ts cotruto to the oectve fucto re clculted Step 4 Let J s set cosstg of the suscrpt of the prosg vrles c c Step 5 Select the vrle whose d d vlue s c c the lrgest If the se lrgest d d occurs, for ore th oe vrle the the vrle tht hs u cotruto cludg the frctol vlue s tke s the prosg vrleif tht s lso se the select y oe rtrrly Step 6 Let t e vrle R The R s selected s the prosg Step 7 Icreet y The suscrpt of the vrle th s stored s the l eleet set J Step 8 The row correspodg to the vrle R s well s the other rows whose u occurs the colu t whch the u for R occurs re deleted Step 9 Step 5 to 8 re repeted tll ether ll the rows or ll the colus re deleted Step The set of vrles collected Steps 5 to 8 re the ordered prosg vrles Let J { Suscrpts of the prosg vrles rrged the descedg c c order d d vlue } Let e the totl uer of eleets the set J Phse II Arrged vrles re llowed to eter to the ss The rrged prosg vrles re llowed to eter to the ss oe y oe sed o the eterg crterthe step y step procedure s gve elow Step Let k, pos d X B s the soluto vector d flg() s the flg vector R flg ; P old P Step If Iterto the P old P Step 3 Iterto s creeted y th Step 4 The k eleet the set J s selected d let t e The the eterg vrle s Step 5 Assg to the th eleet of the vector X B Step 6 P vector s odfed usg the relto P ew ( P old ) - ( P ),,3 Step 7 Replce P old y P ew Step 8 If pos the t pos k, else t pos If pos the Perfor Phse I Step 9 t t + Step Let the t th eleet the set J e r r { t { Pos t Step If cr r c d r r d ( P old ( P ) r the goto Step ) } ; ( P ) > } Else f t < the goto Step 9 Step Let the suscrpts of the vrle correspodg to pos e r d goto Step 5 Phse III Deterto of ew ( proved ) soluto vector to the Zero-Oe Iteger Ler Frctol Progrg Proles Ecept for the ost prosg vrle the soluto set oted phse II the vlues of reg vrles re set to zero Tkg ths s strtg soluto, phse I d II re perfored utl proved soluto s oted If there s o proveet the et prosg vrle vlue log wth the ost prosg vrle lso s reted d the reg sc vrles de to zero Phse III s repeted utl the sc vrles lst ehusted wwwsrporg

4 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 4 ISSN Algorth Stge I The sc vrles re rrged ccordg to the descedg order of ther cotruto to the oectve fucto Step, k, s s the uer of sc vrles hvg o zero vlues the soluto Step X s the soluto vector oted phse II Step 3 If Multply th eleet X, e X >, the c d c d, let t e stored s k th row th colu eleet of rry W d s stored s k th row th colu eleet of rry W k s creeted y oe Step 4 s creeted y oe Step 5 If < the goto step 3 Step 6 The rry W s sorted the descedg order sed o the th colu vlues of W Stge II Fdg the soluto y ssgg ll the vrle vlues ecept oe the ss to zero level Step 7 k Step 8 Step 9 Step If > k the J W X Step s creeted y oe Step If < the goto step Step 3 Now P c s the curret resource vector or ( RHS ) d correspodg oectve fucto vlue Z s clculted Stge III Fd the ew soluto Step 4 Use phse I d phse II d fd the ew soluto X whch s stored s Y( ) d the correspodg oectve fucto vlue Z s stored s V ( ) Step 5 s creeted y oe Step 6 If < the goto step 9 Step 7 Fd the lrgest of V ( ) d ts posto pos, where ( < ), Let t e stored Z 3 Step 8 If Z 3 > Z, the Replce X y Y ( pos) goto step else f k < the creet k y goto step 8 IV NUMERICAL EXAMPLE Solve the followg Iteger ler Progrg Prole Mze Z Suect to the costrts Where,, 3, 4, 5 d ll re tegers Soluto A P, P, P, P 3 X, P 4, P 5 C T (4,7,4,3,9 ), D T (,3,4,6,3),C 3, D 5 Phse - I To fd Mtr d c wwwsrporg

5 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 5 ISSN Arrgeet of prosg vrles J {3,, 5} Phse II s greter th So, X eter to the ss wth vlue X P X 3 eter to the ss wth vlue Z 77 P - ew P -old P _ Z 5 P - ew P -old P _ P -old P - ew 3 rd eleet the set J s r (e) r X 5 s greter th So, X 5 eter to the ss wth vlue P -old P - ew d eleet the set J s r (e) r X,Z 5,P - ew P -old P u u (9,3,7) 3,P -old P - ew 9 Phse - I To fd Mtr d c Arrgeet of prosg vrles J {} wwwsrporg

6 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 6 ISSN Phse II X B P -old P X eter to the ss wth vlue Arrgeet of prosg vrles J { } The curret soluto s Z 8 P - ew P -old P Mu Z 8 Phse-III s reted d reg vrles re set to zero (e) s Phse - I To fd Mtr Now P d c Phse - I To fd Mtr d c Arrgeet of prosg vrles J { 3,,} Phse II X B P -old P Z 4 X 3 eter to the ss wth vlue Z 8,P - ew P -old P d eleet the set J s r (e) r X u u (8,, 4) s greter th 66 wwwsrporg

7 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 7 ISSN So, X eter to the ss wth vlue Mu Z 8 Z 5 P - ew P -old P V CONCLUSION I ths pper ew pproch to solve Zero Oe Iteger Ler Frctol Progrg prole hs ee dscussed The ove lgorth redered est optl soluto I Future ths ethod c e ppled dfferet types of Ler Frctol progrg prole to get etter optl soluto 3 rd eleet the set J s r (e) r X u u (3,, ) 3 s greter th or equl to So, X eter to the ss wth vlue Z 8,P - ew P -old P,P -old P - ew Phse - I To fd Mtr d c Arrgeet of prosg vrles J { } Here the Phse III wll ot prove the soluto So, the Optl Soluto s ) Reserch Elortos ) Results or Fdg 3) Coclusos REFERENCES [] CAudet, PHse, BJurd d GSvrd, Jourl of Optzto theory d Applcto Vol93, No, (997) 73-3 [] AIBrros, JBG Frek, SSchle d S Zhg, A ew lgorth for geerlzed frctol progrs Mthetcl Progrg 7 (996),, [3] AChrles, WW Cooper A eplct geerl soluto ler frctol progrg, Vol Septeer 973 [4] Fegquyou & Igco Gross Solvg Med-Iteger Ler Frctol Progrg Proles wth Dkelch s Algorth d MINLP ethods [5] HIsh, T Irk d HMe, Frctol kpsck proles, Mthetcl Progrg 3 (976), 3, 55-7 [6] Kt Swrup, Gupt PKMoh, Opertos Reserch, Sult Chd d Sos, [7] GKrthKey, Desg of ew coputer oreted lgorth to solve ler progrg proles, PhD, thess, Algpp Uversty, Id, My [8] Pt JC Operto Reserch, Itroducto to optzto, 7th edto 8 [9] Er Pre Kur Gupt d Dr DS Hr, Proles Operto Reserch (Prcples d solutos) S Chd & copy, R Ngr, New Delh [] SeeregsyV, DrKJeyr A effecet ethod for esy coputto y -tr y cosderg the teger vlues for solg teger ler frctol progrg proes,ijsrp Volue 3, Issue 5, My 3 edto [] Stcu Ms, IM Frctol progrg theory, ethods d pplctos seres, Mthetcs d ts Applcto Vol 49 (997)43p [] Suresh Chdr, M Chdr Moh, A ote o teger ler frctol progrg, Volue 7 (98)7-74 [3] LVcete, GSvrd d SJudcs, Jourl of Optzto Theory d Applctos, 89, No3 (996) [4] Wukfred Cdler d Roert Towsley, Coputers d Opertos Reserch, 9(98) AUTHORS Frst Author VSeeregsy, Professor of Mthetcs, PSNA College of Egeerg d Techology, Ddgul-64 6, Tl Ndu, Id Secod Author DrKJeyr, De of Scece d Hutes, PSNA College of Egeerg d Techology, Ddgul-646, Tl Ndu, Id wwwsrporg

8 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 8 ISSN wwwsrporg