Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2)

Transcription

1 A ew Algorth for er Progrg Dhy P. ehedle Deprtet of Eletro See, Sr Prhurhu College, lk Rod, Pue-00, d dhy.p.ehedle@gl.o Atrt- th pper we propoe ew lgorth for ler progrg. h ew lgorth ed o tretg the oetve futo preter. We trfor the tr of oeffet repreetg th yte of equto the redued row ehelo for otg oly oe vrle, ely, the oetve futo telf, preter whoe optl vlue to e detered. We lyze th tr d develop ler ethod to fd the optl vlue for the oetve futo treted preter. We ee tht the etre optzto proe evolve through the proper ly of the d tr the redued row ehelo for. t wll e ee tht the optl vlue e oted ) y olvg ert uyte of th yte of equto through proper utfto for th t, or ) By kg pproprte d legl row trforto o th tr the redued row ehelo for o tht ll the etre the utr of th tr, oted y olletg row whh the oeffet of o lled ukow preter d whoe optl vlue to e detered, eoe oegtve d th ew tr ut e equvlet to orgl tr the ee tht the oluto et of the tr equto wth orgl tr d tr equto wth trfored tr re e. We the proeed to how tht th de turlly eted to del wth oler d teger progrg prole. For oler d teger progrg prole we ue the tehque of Groer e e Groer equvlet of redued row ehelo for for yte of oler equto, d the ethod of olvg ler Dophte equto repetvely. Key Word: Oetve futo preter, oetve equto, oetve ple, otrt ple, ehelo for, Groer e, Dophte equto. RODUCO here re two type of ler progr (ler progrg prole):. ze: C Suet to: A 0 Or. ze: C Suet to: A 0 where olu vetor of ze of ukow. Where C olu vetor of ze of proft (for zto prole or ot (for zto prole oeffet, d C row vetor of ze oted y tr trpoto of C. Where A tr of otrt oeffet of ze. Where olu vetor of ott of ze repreetg the oudre of otrt. By trodug the pproprte lk vrle (for zto prole d urplu vrle (for zto prole, the ove etoed ler progr get overted to tdrd for : ze: Suet to: A + = (.) 0, 0 Where lk vrle vetor of ze. h zto prole. Or C ze: C Suet to: A = (.) 0, 0 Where urplu vrle vetor of ze. h zto prole. geoetrl lguge, the otrt defed y the equlte for rego ouded y ove polyhedro, rego ouded y the otrt ple A =, lled fele rego d t trghtforwrd to hek tht there et t let oe verte of th polyhedro t whh the optl oluto for the prole tuted whe the prole t hd ot uouded or fele. here y e uque optl oluto d oete there y e ftely y optl oluto, e.g. whe oe of the otrt ple prllel to the oetve ple we y hve ulttude of optl oluto. A etre ple or etre edge ottute the optl oluto et. We eg wth oe oo oto d defto tht re prevlet the lterture. A vrle lled vrle gve equto f t pper wth ut oeffet tht equto d wth zero oeffet ll other equto. A vrle whh ot lled o vrle. A equee of eleetry row operto tht hge gve yte of ler equto to equvlet yte (hvg the e oluto et) d whh gve o vrle e de vrle lled pvot operto. A equvlet yte otg d o vrle oted y pplto of utle eleetry row operto lled ol yte. At te, the troduto of lk vrle for otg tdrd for utotlly produe ol yte, otg t let oe vrle eh equto. Soete equee of pvot operto eeded to e perfored to get ol yte. he oluto oted fro ol yte y ettg the o vrle to zero d olvg for the vrle lled oluto d ddto whe ll the vrle hve oegtve vlue the oluto tfyg ll the poed otrt lled fele oluto. Beue of the fr gret prtl porte of the ler progr d other lr prole the operto reerh t ot dered thg to hve lgorth whh work gle tep, f ot, few tep pole. o

2 ethod h ee foud whh wll yeld optl oluto to ler progr gle tep ([], Pge 9). We to propoe lgorth for ler progrg whh t fulfllg th requreet et pole d ovel wy.. A EW AGORH FOR EAR PROGRAG We trt wth the followg equto: C = d (.) d ll t oetve equto. he (pretr) ple defed y th equto wll e lled oetve ple. hu, we hve tke the oetve futo ew ukow preter lled d d the prole of ler progrg to fd the optl vlue of th ukow preter. We du frt the zto prole. A lr pproh for zto prole wll e dued et. Gve zto prole, we frt otrut the oed yte of equto otg the oetve equto (.) d the equto defed y the otrt poed y the prole uder oderto, oed to gle tr equto follow: ( 0( et E= ( ( 0( = d (.) ( d, d let F = R=rref([E,F]) (.) ote tht the ugeted tr [E, F] well t redued row ehelo for R ot oly oe vrle, ely, d d ll other etre re ott. Fro R we detere the oluto et S for every fed d, S = { /( fed) d rel}. he uet of th oluto et of vetor whh lo tfe the oegtvty otrt the et of ll fele oluto for tht d. t ler tht th uet e epty for prtulr hoe of d tht de. he zto prole of ler progrg to detere the uque d whh provde fele oluto d h u vlue for d,.e., to detere the uque d whh provde optl oluto. the e of uouded ler progr there o upper (lower, the e of zto prole lt for the vlue of d, whle the e of fele ler progr the et of fele oluto epty. he tep tht wll e eeuted to detere the optl oluto hould lo tell y plto whe uh optl oluto doe ot et the e of uouded or fele prole. he geerl for of the tr R repreetg the redued row ehelo for et [E, F] deote the ugeted tr oted y ppedg the olu vetor F to tr E lt olu. We the fd R, the redued row ehelo for ([], pge 7-75) of the ove ugeted tr [E, F]. hu, R = ( d + e d + e d + e d + e d + e + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) Aog frt ( + olu of R ert frt olu orrepod to vrle (olu tht re ut vetor) d the reg oe to o vrle (olu tht re ot ut vetor). For olvg ler progr we eed to detere the vlue of o vrle uh tht the vlue of d optl, fro whh we detere the vlue of ll the vrle y uttuto d the ler progr thu olved opletely. ote tht the row of R tully repreet equto wth vrle, =,, d vrle, =,, o left de d epreo of type kd + ek, k =,, ( + ) otg the vrle d o the rght de. he row wth potve oeffet for the preter d repreet thoe equto whh the preter d e reed rtrrly wthout voltg the oegtvty otrt o vrle,. So, thee equto wth potve oeffet for the preter d re ot plyg y upper oud o the u pole vlue of preter d however; thee row re ueful ert tuto. he row wth egtve oeffet for the preter d repreet thoe equto whh the preter d ot e reed rtrrly wthout voltg the oegtvty otrt o vrle,. So, thee equto wth egtve oeffet for the preter d re plyg upper oud o the u pole vlue of preter d d o portt oe th repet. So, we ow proeed to fd out the utr of R, ely, R, de up of ll olu of R d otg thoe row of R for whh the oeffet of the preter d re egtve. et,,, k re ll d re oly egtve rel uer the row of R olleted R d the oeffet of d ll other row of R re greter th or equl to zero. Our ethod for olvg ler progrg prole eetlly ot of olvg ert uyte of equto repreeted

3 y ert tow of R or R, or ltertvely pultg etre the olu of R through ert utle row trforto o tht the trfored tr tll repreet the e yte of equto eee d ow the etre the olu orrepodg to ll o vrle R re oegtve. Algorth. (zto: Step : Epre the gve prole tdrd for: ze: C Suet to: A + = 0, 0 Step : Cotrut the ugeted tr [E F], where E = C ( ) 0 d (, d F = ( d ot the redued row ehelo for: R = rref ([E, F]). ote: () We ll tht vrle vrle for whh the orrepodg olu (olu vetor) of R repreetg oeffet for tht vrle ut vetor. () We ll tht vrle o vrle for whh the orrepodg olu (olu vetor) of R repreetg oeffet for tht vrle ot ut vetor. Step : f there row (or row) of zeroe t the otto of R the frt olu d otg ozero ott the lt olu the delre tht the prole otet d top. Step : Ele f the oeffet of d the lt olu re ll potve or f there et olu of R orrepodg to oe o vrle wth ll etre egtve the delre tht the prole t hd uouded d top. Step 5: Ele f for y hoe vlue of d oe oerve tht oegtvty otrt for oe vrle get volted y t let oe of the vrle the delre tht the prole t hd fele d top. ote: () felty of ler progrg prole lo e deded y uouded ture of t dul prole Step 6: Ele fd the utr of R, y R de up of thoe row of R for whh the oeffet of d the lt olu egtve. Step 7: Solve r d + e r = 0 for eh uh ter the lt olu of R d fd the vlue of d = for d r r =,,, k d fd d = { } d d r d = { d }. r ote: (d) he optl vlue of d optl, le etwee d = { } d d = { } d r d r Step 8: Chek the olu of R orrepodg to o vrle. Fd out the olu wth ll etre oegtve d et thee o vrle to zero. Step 9: f ll thee olu orrepodg to o vrle ot oly oegtve etre the ( per Step 7) et ll o vrle to zero. Suttute d = d the lt olu of R. Detere the fele oluto (whh wll e the optl oluto for the prole d top. Step 0: Flly, whe there et olu R orrepodg to oe o vrle otg potve etre oe row d oe egtve etre oe other row d the tl d vlue re uh tht they pproh eh other whe the vlue of thee o vrle re reed fro zero the we tke followg two pprohe: (A) We for d olve utle uyte of equto ly otg thee o vrle d y oete otg oe vrle tht led to l vlue for d d thu olve the prole opletely. uh e we y requre to hek optlty y eprtely ppedg eh vrle. (B) We rry out ert utle row trforto o the tr R o tht the ewly trfored tr equvlet to R d ow ll the etre the olu of ewly trfored R ( effet of thee trforto rred out o R ) orrepodg to o vrle re oegtve. We lo del wth zto prole o lr le d develop lr lgorth for ther oluto. Eple.: We ow oder prole for whh the ple terto re epoetl futo of the ze of the prole. A prole elogg to the l dered y Klee d ty otg vrle requre ple tep. We ee tht the ew ethod doe t requre y pel effort ze: Suet to: ,, 0 Soluto: he followg the tr R :

4 0 0 0 /0 / (/00)d / (/ 5) d R = d /0 /00 9 (/00) d So lerly, / (/ 5) d R = /0 /00 9 (/00) d he frt four olu of R, R orrepod to vrle, whle the et two olu orrepod to o vrle. he lt olu orrepod to etre of type d + e r r. he olu for vrle, of R ot ll potve etre o we et = 0. he olu for vrle ot etre wth egtve g, o for thee row the vlue of preter d uouded, e y gg y lrge (potve) vlue to th o vrle we ree the vlue of preter d thee row to y hgh vlue wthout voltg the oegtvty otrt. So, etoed ove, we eed to pped row fro R wth potve g for d d otrut the yte Pz = Q. So, y ppedg frt row of R d ut vetor for we hve 0 /0 /00 P = 0 / 5, z =, d /0 /00 d 90 Q = he oluto for th yte yeld =, = 00, d = he uttuto of thee vlue d the lredy fed vlue of, ely, = 0, we get the oplete oluto follow: = 0, = 0, = 0000, =, = 00, = 0, d the u vlue of d = R k repreet k-th row of R. Altertve Soluto: et Wth the of hevg the oegtvty of etre the olu orrepodg to o vrle R we perfor followg utle row trforto o R, ely, R 0 R + R, R R + R. Wth thee trforto o R we get the ew (trfored) R follow: R /0 (000 (/ 0) d) = ow, ug oegtvty of d d 6 th olu fro frt row we hve = 0, = 0, d o, d = Further ug thee vlue ewly trfored R we fd the oplete oluto d hek tht t tur out e we oted ove y olvg uyte of equto.. A EW AGORH FOR OEAR PROGRAG We ow how tht we del wth oler progrg prole ug the lr tehque. Here the ethod developed y Bruo Buherger whh trfored the trt oto of Groer to fudetl tool oputtol lger wll e utlzed. he tehque of Groer e eetlly vero of redued row ehelo for (ued ove to hdle the ler progr de up of ler equto) for hgher degree equto []. A typl oler progr e tted follow: ze: f () Suet to: h ( ) = 0, =,,, g ( ) 0, = +, +,, p k 0, k =,,, Gve oler optzto prole we frt otrut the followg oler yte of equto: f ( ) d = 0 (.) h ( ) = 0, =,,, (.) g ( ) + = 0, = +, +,, p (.) where d the ukow preter whoe optl vlue to e detered uet to oegtvty odto o prole vrle d lk vrle. For th to heve we frt trfor the yte of equto to equvlet yte of equto erg the e oluto et uh tht the yte eer to olve. We hve ee o fr tht the effetve wy to del wth ler progr to ot the redued row ehelo for for the oed yte of equto orportg oetve equto d otrt equto. We wll ee tht for the oler e the effetve wy to del wth to ot the equvlet of redued row ehelo for, ely, the Groer repreetto for th yte of equto (.)-(.). We the et up the equto oted y equtg the prtl dervtve of d wth repet to prole vrle d lk vrle to zero d utlze the tdrd theory d ethod ued lulu. We deotrte the eee of th ethod y olvg eple: Eple.: ze:

5 Suet to: , 0 Soluto: We uld the followg yte of equto: d = = = 0 + = 0 We ow trfor the ove yte of equto d ot t Groer repreetto follow: 50 9d = 0 (..) = 0 (..) = 0 (..) = 0 (..) fro frt equto (..), order to ze d, we detere the vlue of, follow: d f we et = 0 we get the vlue of tht ze d, d ely, =. Slrly, f we et = 0 we get the 7 vlue of tht ze d, ely, =. Puttg thee vlue of, the frt d eod equto we get repetvely the u vlue of d = 67 d the vlue of =. Ug further thee vlue the thrd d fourth equto we get =.5,. 75. = ote: t trutve to ote tht the optl oluto tht we hve oted ot o the oudry of fele rego ( oe lwy get the e of ler progrg prole ut the teror of the fele rego. V. A EW AGORH FOR EGER PROGRAG We ow proeed to del wth teger progrg prole ug the lr tehque. he eetl dfferee th e tht we ot teger oluto y tretg the yte of equto et of Dophte equto We eg ( doe prevouly for ler progrg prole) wth the followg equto: C = d (.) where d ukow preter, d ll t oetve equto. he (pretr) ple defed y th equto wll e lled oetve ple. et C e row vetor of ze d de up of teger opoet,,,, ot ll zero. t ler tht the oetve equto wll hve teger oluto f d oly f gd (gretet oo dvor) of,,, dvde d. We ow proeed to for the followg Dophte yte of equto gle tr equto: We du the zto prole. A lr pproh for zto prole e developed. Gve zto prole, we frt otrut the oed yte of equto otg the oetve equto d the equto defed y the otrt poed y the prole uder oderto, oed to gle tr equto, vz., ( 0( ( d = (.) d ot P-relto oluto. h P-relto oluto provde the upper oud tht ut e tfed y the optl teger oluto. he we proeed to olve the yte yte Dophte yte of equto follow: order to olve th yte Dophte yte of equto we ue the tdrd tehque gve ([], pge -). Frt y ppedg ew vrle u, u,, u( + d rryg out pproprte row d olu trforto dued ([], pge 7, ) we ot the pretr oluto for the yte. hu, we trt wth the followg tle: ( 0( (( + ( + ( d d trfor the yte of equto to equvlet yte tht dgol. hu, we hve the followg pretr oluto: u k = d (for oe k ) (.) u = h r r (where h r re ott for r = to, r k ), d = α + δ r= r= u r r = β + η u r r (where (where α, δ re ott.) r r β, η re ott.) We etup proedure to lyze th pretr oluto d detere the optl tegrl oluto. Eple.: ze: + 0 5

6 Suet to: , 0, d teger. Soluto: We frt fd the P-relto optl vlue, whh for th prole. Ad the oplete Prelto optl oluto,, ) = (8.66, 6.77, 0, 0) (, hu, the upper lt for optl vlue for teger progr, d, e 58. Strtg wth the tle (.5) etoed y opt. ove d rryg out the pproprte row-olu trforto we get the followg pretr oluto: u = d (..) u = 5 (..) u = (..) = d + 0u (..) = u (..5) = d + 5u + 5 (..6) = d u + (..7) Ug the upper lt o the optl vlue, ely, d = d opt. = 58 the lt equto ove we ee tht the u vlue tht u tke (to t oegtvty of ) 6. he forth d th equto gve ove for d repetvely ot d wth egtve oeffet ( = ). o uttutg = 0 d u = 6 equto (..6) we get the dered optl oluto d = 55, = 0, =, = 5, = u = 6 Akowledgeet thkful to Dr.. R. odk d Dr. P. S. Jog for ueful duo. Referee [] Hdley G., er Progrg, ro Pulhg Houe, ew Delh 0 07, Eghth Reprt, 997. [] Golderg Jk., tr heory wth Applto, Grw-Hll, tertol Edto, 99. [] Wll W. Ad, Phlppe outuu, A troduto to Groer Be, Grdute Stude thet, Volue, Aer thetl Soety. [] ve., Zuker H., otgoery H., A troduto to the heory of uer, Joh Wley & So, Pvt. td., 99. 6