Dual-Matrix Approach for Solving the Transportation Problem

Transcription

1 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 ul-mtr Aroh for Solvg the Trsortto Prole Vy Shr r Chr Bhus Shr ertet of Mthets, BBM College r, Jeh, (MU), INIA E-Prl, SS College Jeh, (MU), INIA Astrt A Algorth hs ee eveloe for solvg sel tyes of trsortto role hvg totl e ore th or equl to the totl suly Ths lgorth s useful for solvg oth le ule trsortto roles The ul-tr roh s use the reset lgorth Keywors: Trsortto role, ul-tr roh, steg-stoe etho, sle etho, ler rogrg oels Itrouto The u ost lg lys ortt role for solvg the trsortto roles fro orgs to fferet esttos, suh s fro ftores to wrehouses, or fro wrehouses to suerrets, et The shg ost fro oe loto to other s usully ler futo of the uer of uts she Sle etho whh utlzes ler rogrg oels e effetvely use for solvg trsortto roles Pg CHU(00) showe tht ul-tr roh s ore effet orso to sle etho the otet of trsortto ul- tr roh ws le suessfully erler for solvg the trsortto roles[] (g CHU, 00) hvg e s less th or equl to the suly However the reset stuy tes tht eve the Pg CHU s roh lso suffers fro ert ostrts hvg e s ore th or equl to the suly The ew roh slr to ul-tr roh reresets the lgorth for solvg trsortto role hvg e s greter th or equl to the suly The ew roh osers the ul of the trsortto role ste of rl ots the otl soluto of the ul y the use of tr oertos The ew lgorth s etle the er, flly uerl ele s gve to llustrte the roh The Moel ts ul The trsortto role s usully resete s tr s show Fgure The ut trsortto ost geerlly te o the orthest orer eh ell Ths role e eresse s ler rogrg oel s follows: ze Suet to (,,, ) Here ll (,,, ) LP() s 0 (,,,,,,, ) s re ssue to e ostve, lle sules es resetvely, s show Fgure, The ost re ll oegtve, s oto for le trsortto role If ISSN: -57 htt://wwwttourlorg Pge 45

2 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 ths oto s ot et, uy org or estto s geerlly troue to e the role le The ew ul-tr roh resete ow oes ot requre trsortto role to e le The roh e le to oth le ule roles, o uy org or estto s troue, so te se re sve The ul tr roh osers the ul oel of the trsortto role, the oertos re lulte o tr Aother vtge s tht egeery oes ot est the ul-tr roh If oel LP s osere s the rl, the ts ul e forulte s follows: ze u v Suet to u v (,,,,,,, ) u, v 0 (,,,,,,, ) LP() Here ll s s re ssue to e ostve, lle sules es resetvely, s show fgure The ost re ll oegtve, s oto for le trsortto role If ths oto s ot et, uy org or estto s geerlly troue to e the role le The ew ul-tr roh resete ow oes ot requre trsortto role to e le The roh e le to oth le ule roles, o uy org or estto s troue, so te se re sve The rl role hs + ostrts vrles The ul hs + vrles ostrts Beuse of fewer vrles ul role, the ul role wll e solve y the ul-tr roh quly However, ll ostrts re ot eessry to e resete eltly s LP All these ostrts re et orgl trsortto tr, s te Fgure The ul-mtr Aroh The e of the ul-tr roh s to ot frst fesle soluto to the ul role ts orresog tr The the ulty theory s use to he the otlty oto to get the levg ell All o-s ells re evlute orer to get the eterg ell Flly, the eterg ell reles the levg ell the tr s ute The ul-tr roh s resete s follows: Ste 0 Itlzto Ste 0 Set A,,,,,,, Ste 0 Set v 0;,,, ISSN: -57 htt://wwwttourlorg Pge 46 u,,,, ;,,, let t Tes e roe rtrrly The orresog ell to resetvely re,,,,, Ste 0 Let the s ell set,,,,,,, 0,, 0,,, 0, The ells 0,, 0,,, 0, re lle vrtul ells euse they o ot est the orgl trsortto role tr Ste 04 Let the tr,,,,, ; where,,,, ;,,, ;,,,,,,, 0 otherwse

3 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 Ste Ste Ste A oute the oetve u v eterto of levg ell: Ste Coute Y A Ste F the sllest vlue y the eleets of Y, tht s the vlue of the th eleet Y s the sllest Tes e roe rtrrly Ste If y 0, the soluto s otl(oth the ul rl), sto therwse, levg ell s th ell tht s, : eterto of eterg ell: Ste Let, q,, q, P Q, q,, q, Ste For ll o s ells, f q 0, the the ul role s ot oue, the orgl rl role hs o fesle soluto, sto therwse, oute u v f q 0 Ste F the sllest vlue st ll, the ell s, t s the eterg ell Tes e roe rtrrly Utg Ste Ute the tr Ste For the eleets of olu ˆ l,,, l l Ste For the eleets of the other olus ˆ,,,,,, ;, ˆ r lr lr t r sr l l,,, th Ste Ute the s ell set : rele the ell, wth the eterg ell s, t Ste Ute the oetve vlue: Coute the oetve u v go to ste uˆ u,,, st vˆ v q,,, st ISSN: -57 htt://wwwttourlorg Pge 47

4 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 estto rg e Fgure : trsortto role Mtr Suly The tlzto roeure s to ot tl fesle soluto By settg v 0 u eg the sllest ost the row, ovously they eet the ostrt set() the ul role The tr s tr, whh e ve to four su-tres s follows: The uer left su tr s etty tr The uer rght su-tr s tr: If the ell, s s ell (orresog to the the eleet, ths su-tr s - All other eleets ths su-tr re 0 The lower left su-tr s zero tr 4 The lower rght su-tr s egtve etty tr urg the roeure of the ul-tr roh, ste s to get the levg ell, slr to gettg levg vrle the sle etho As tter of ft, the tl fesle soluto the ul-tr roh s very goo strtg ot Fro the oetve futo the ul LP, t s ovous tht v shoul e sller, the etter The sllest s 0 for ll ue to the ostrts set () v the other h, u oly e the u vlue of ) u shoul e lrger, the etter However the row Ste s to ot the eterg ell y evlutg ll o-s ells, whh s slr to steg-stoe etho (Chres & Cooer, 954) The equto ste s the se s the oe steg-stoe etho (MI, resely) eet the sg of u Flly the tr other relevt t re ute orgly To el ths roh thetlly, ell, e reresete s equto, tht s u v The thetl grou of the ul-tr roh s to f equtos, e, s ells fro ostrts set() the ul If these equtos ot ze the oetve of the ul, tht s soluto s ot otl, f other ell (equto), e, the eterg ell, to rele oe equto set utl otl soluto s fou So, the lgorth of ths roh s very sle s show revously ISSN: -57 htt://wwwttourlorg Pge 48

5 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 Nuerl Ele The followg ule trsortto role s osere Ste 0 rg Itlzto estto e Suly ,, 4,,,,, 0,, 0,, 0,, 0, 4 A 00, 500, 600, 00, 00, 500, 500 v v v4 0 u 4, u 0, u 8 Ste u v 4900 eterto of levg ell: Y A 00, 500, 600, 00, 00, 00, 00 so =5 the levg ell s (0,) Ste Ste eterto of eterg ell,5 0 P,5 0,5 q 4,5 0 q,5 Q q 6,5 0 q 0 4 7,5,,,,,, 4 hve o ostve q,,,,, 4 hve ostve q Aog ll the o-s ells whle the ells st 4 Now the eterg ell s, t, s, s so,,, 0 0,8 0 0, ,8,4 5 Utg ISSN: -57 htt://wwwttourlorg Pge 49

6 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 ˆ 5 ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ ˆ 55 ˆ ˆ 0 0 ˆ ˆ 6 65 ˆ 0 0 ˆ ˆ 7 75 Slrly other eleets e ute ˆ ˆ ˆ , 4,,,,, 0,,,, 0,, 0,4 uˆ u uˆ u 0 5 5, st uˆ u vˆ v q vˆ v q 0 5 st 5 vˆ v q vˆ v q u v 5,000 Y A 00, 00, 600,00, 00, 00, 00 Se oe eleet of Y s egtve so the role s ot otl, So we go to Ste reet the roeure After oe terto, fl soluto e wrtte s: , 4,,,,, 0,,,, 0,, 0,4 ISSN: -57 htt://wwwttourlorg Pge 50

7 Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 uˆ u 0 uˆ u 4 5, st uˆ u u v 5,500 Y A vˆ v q 0 vˆ v q 0 vˆ v st q vˆ v4 q4 vˆ v q , 00, 500,00, 00,00,00 So the otl soluto s ot wth the oetve 5, 500, wth 00, 00, 500, 00, 00, 00, Colusos Ths etho s ore effet for the roles hvg totl e ore th the totl suly The ultr roh e le to oth le ule trsortto roles A ule trsortto role s ot requre to e overte to le role Aother vtge s tht t oes ot hve the egeery role o th trg Referees: [] Chres A W W, Cooer (954), The steg stoe etho for elg ler Progrg lultos trsortto roles Mgeet See, [] Pg JI KF CHU (00), A ul-mtr Aroh to the trsortto role As Pf Jourl of ertol Reserh 5-45 [] S Shr (00), ertos Reserh, Ker Nth R Nth & Coy, I [4] For, L R R Fulerso (956), Solvg Trsortto Prole, Mgeet See, 4- [5] Goyl S K (984), Irovg VAM for ule trsortto roles Jourl of ertol Reserh Soety 5, -4 [6] Shft, A S K Goyl, (988), Resoluto of egeery trsortto Prole, Jourl of ertol Reserh Soety 9, 4-4 ISSN: -57 htt://wwwttourlorg Pge 5