International Journal of Scientific and Research Publications, Volume 3, Issue 5, May ISSN

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1 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 1 A Effcet Method for Esy Coputto y Usg - Mtr y Cosderg the Iteger Vlues for Solvg Iteger Ler Frctol Progrg Proles 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 solvg 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- Iteger Ler Frctol Progrg Proles, tr d Prosg vrles T I INTRODUCTION o solve Iteger Ler Frctol Progrg Proles wth reduced coputtol effort, ew ethod of C D X c Etreze Z Suect to T T X d pproch hs ee proposed I ths ethod, og the decso vrles, the vrles whch c eter to the ss re detfed d ordered sed o the u cotruto to the oectve fucto The ordered decso vrles oe y oe re llowed to eter to the ss y checkg whether t s stll gvg proved soluto II GENERAL INTEGER LINEAR FRACTIONAL PROGRAMMING PROBLEMS IN MATRIX FORM The geerl Iteger Ler Frctol Progrg Proles s gve y AX ( ) P X d re teger Where A X P 1 3 Let the colus correspodg to the tr A e deoted y P 1, P, P 3 P where wwwsrporg

2 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 P P 1 3 P P 1 3 C T ( c 1, c, c 3 c ), D T ( d 1, d,d 3 d ) d c, d re sclrs III APPROACH I ths ew pproch to solve Iteger Ler Frctol Progrg Proles, three phses re cluded d those phses re gve elow Phse I: Prosg decso vrles re detfed to eter the ss d those prosg vrles re ordered sed o the cotruto to the oecto fucto The ove three phses re repeted tll the optu soluto reched I ths ethod, the prosg vrles re detfed d rrged sed o the u cotruto to the oectve fucto y cosderg the teger vlues of the tr etres The step y step procedure s s gve elow Step 1: 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 II: The rrged prosg vrles re llowed to eter to the ss the rrged order fter checkg whether the ewly eterg vrles wll prove the oectve fucto of the prole, keepg the feslty Phse III: Fdg the proved soluto vector Phse I - Orderg of Prosg vrles Step 1 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 1 S 1 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 decso uer of prosg vrles ech of the costrts Step The u teger tercepts d ts posto ech row of tr s foud out If there re ore oe u teger tercept the oe of the s selected rtrrly Multply the u tercept of the vrle wwwsrporg

3 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 3 correspodg to row wth the correspodg cotruto coeffcet the oectve fucto oth the uertor d c c deotor d the oectve fucto vlue d d s clculted Step 3 Let J s set cosstg of the suscrpt of the prosg vrles c c Step 4 Select the vrle whose d d vlue s c c the lrgest teger If the se lrgest teger d d vlue occurs, for ore th oe vrle the the vrle tht hs u cotruto cludg the frctol vlue s tke s the prosg vrle If tht s lso se the select y oe rtrrly Step 5 Let t e vrle R The R s selected s the prosg Step 6 Icreet y 1The suscrpt of the vrle th s stored s the l eleet set J Step 7 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 8 Step 4 to 7 s repeted tll ether ll the rows or ll the colus re deleted Step 9 The set of vrles collected Steps 4 to 7 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 crter The step y step procedure s gve elow Step 1 Let k 1, X B s the soluto vector d flg () s the flg vector R flg 1 ; P old P Step Iterto s creeted y 1 th Step 3 The k eleet the set J s selected d let t e The the eterg vrle s Step 4 Coputto of vlue The vlue wth whch c eter to the ss s coputed y usg the followg forul, k { t { ( P old ( P ) ) } ; ( P ) > } 1,,3 Step 5 If k 1 d k1 the the vlue of 1 else the vlue of s chose etwee to 1 (Let 5 ) Step 6 Copute S t ( k ) k t ( 1- ) k 1 s dded to flg S s dded to the th eleet of the vector X B d Step 7 P vector s odfed usg the relto (P ew ) (P old ) - (P ) S 1,, Step 8 (P old ) s replced y ( P ew If k1 or S 1 go to step 16 ) Step 9 Check whether k th eleet s stll prosg og the reg lst of ( -k ) prosg vrles set J usg the followg steps Step 1 Let r 1 Step 11 Select the ( k + r ) th eleet ths set J Let t e q The the vrle correspodg posto q Step 1 Fd q usg the forul ( P old ( P ) q { t { ) } ; ( P ) > } wwwsrporg

4 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 4 q cq q c d d q q q Step 13 If k q < goto step 15 Step 14 Icreet r y oe If r ( -k ) the goto step 11 Else goto step 16 Step 15 If k < goto step 1 k s replced y k + r d k s replced y Step 16 If flg k 1 goto step 3 Else goto Perfor Phse I Phse III Deterto of ew (proved) soluto vector to the 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 IV ALGORITHM Stge I The sc vrles re rrged ccordg to the descedg order of ther cotruto to the oectve fucto Step 1, k, 1 s s the uer of sc vrles hvg ozero 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 1 th colu eleet of rry W k s creeted y oe Step 4 s creeted y oe Step 5 If < 1 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 1 If k the J W 1 X Step 11 s creeted y oe Step 1 If < 1 the goto step 1 Step 13 Now P c s the curret resource vector or ( RHS ) d correspodg oectve fucto vlue Z 1 s clculted Stge III Fd the ew soluto Step 14 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 15 s creeted y oe Step 16 If < 1 the goto step 9 Step 17 Fd the lrgest of V ( ) d ts posto pos, where ( < 1 ), Let t e stored Z 3 Step 18 If Z 3 > Z, the Replce X y Y ( pos) goto step 7 else f k < 1 the creet k y 1 goto step 8 V NUMERICAL EXAMPLES Solve the followg Iteger ler frctol Progrg Prole Mze Z Suect to the costrts Where 1,, 3, 4, 5, 6 d ll re tegers Soluto A P, P 1, P, P 3 wwwsrporg

5 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 5 X, P 4, P 5, P 6 C T (4,17,4,3,19,13 ), D T (,3,4,6,3,5,5 ),C, D 5 Phse - I To fd Mtr d c Arrgeet of prosg vrles J {, 3, 5} Phse II X prosg vrle u u (16,,141,141) 141 S 7 Z 4585 (P ew) (P ew) (P ew) (P ew) X prosg vr le u u (9, 13, 71, 71) 71 S 35 Z 4896 (P ew) 1 115,(P ew) 19,(P ew) 3 11,(P ew) 4 11 Repetg ths procedures Phse I d Phse II we get the soluto s The curret soluto s 13, 3 13 Mu Z 513 wwwsrporg

6 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 6 Phse - III s reted 13 d reg vrles re set to zero,tht s 3 Now P Followg slrly we get the fl soluto The OptPhse - I To fd Mtr d c Arrgeet of prosg vrles J { 3, } Phse II X 3 prosg vrle u u (13, 35, 35, 35) 13 S 6 Z 58 (P ew) 1 35,(P ew) 116,(P ew) 3 9,(P ew) 4 9 Followg slrly we get X 3 prosg vrle u u (1, 3, 3, 3) 1 S 1 Z 513 (P ew) 1,(P ew) 88,(P ew) 3,(P ew) 4 Phse - I To fd Mtr d c wwwsrporg

7 Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 7 Arrgeet of prosg vrles J { } Here the Phse III wll ot prove the soluto so, the optl soluto s 13, 3 13 Mze Z 513 VI CONCLUSION I ths ew pproch to solve Iteger Ler Frctol Progrg prole hs ee dscussed The ove lgorth redered est optl soluto I Future ths ethod c e ppled Med Iteger Ler Frctol progrg prole to get etter optl soluto REFERENCES [1] CAudet, PHse, BJurd d GSvrd, Jourl of Optzto theory d Applcto Vol93, No, (1997) 73-3 [] AIBrros, JBG Frek, SSchle d S Zhg, A ew lgorth for geerlzed frctol progrs Mthetcl Progrg 7 (1996),, [3] AChrles, WW Cooper A eplct geerl soluto ler frctol progrg, Vol Septeer 1973 [4] Erk BBlov Ler Frctol progrg theory, ethods, Applctos d Sftwre [5] Fegquyou & Igco Gross Solvg Med-Iteger Ler Frctol Progrg Proles wth Dkelch s Algorth d MINLP ethods [6] Hdy ATh, Opertos Reserch- A Itroducto, Seveth Edto, Pretce-Hll of Id Prvte Lted, 4 [7] HIsh, T Irk d HMe, Frctol kpsck proles, Mthetcl Progrg 13 (1976), 3, [8] Kt Swrup, Gupt PKMoh, Opertos Reserch, Sult Chd d Sos,1 [9] GKrthKey, Desg of ew coputer oreted lgorth to solve ler progrg proles, PhD, thess, Algpp Uversty, Id, My 11 [1] Pt JC Operto Reserch, Itroducto to optzto, 7 th edto 8 [11] Er Pre Kur Gupt d Dr DS Hr, Proles Operto Reserch (Prcples d solutos) S Chd & copy, R Ngr, New Delh [1] Stcu Ms, IM Frctol progrg theory, ethods d pplctos seres, Mthetcs d ts Applcto Vol 49 (1997)43p [13] Suresh Chdr, M Chdr Moh, A ote o teger ler frctol progrg, Volue 7 (198) [14] LVcete, GSvrd d SJudcs, Jourl of Optzto Theory d Applctos, 89, No3 (1996) [15] Wukfred Cdler d Roert Towsley, Coputers d Opertos Reserch, 9(198) 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