An Algorithm for Solving Bi-Criteria Large Scale Transshipment Problems

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1 Glol Jourl of Reserhes Egeerg: B Automotve Egeerg Volume 4 Issue 4 Verso 0 Yer 04 Type: Doule Bl Peer Revewe Itertol Reserh Jourl Pulsher: Glol Jourls I USA Ole ISS: & Prt ISS: A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems By Khl Alhulf Jsem AlRh Elsye E M Ellmoy Mohse AlArh & Hll A Aelwl College of Tehologl Stues Kuwt Astrt- Ths pper esres lgorthm for solvg ert lss of -rter multstge trsportto prolems wth trsshpmet BMTSP A severl -rter multstge trsportto prolem wth trsshpmet re formulte The presete lgorthm s mly se o pplto of the methos of solvg -rter sgle stge trsportto prolems utlzg vlle eomposto tehques for solvg lrge-sle ler progrmmg prolems the methos of tretg the trsshpmet prolems The mthemtl formulto of the presete lss oes ot ffet the spel struture of the trsshpmet prolem for eh of the vul stges A llustrtve emple s troue to vlte tht the mplemetto of the lgorthm Keywors: lrge sle trsportto prolem trsshpmet prolem mult-oetve eso mg eomposto tehque of ler progrmmg GJRE-B Clssfto : FOR Coe: AAlgorthmforSolvgBCrterLrgeSleTrsshpmetProlems Strtly s per the omple regultos of : 04 Khl Alhulf Jsem AlRh Elsye E M Ellmoy Mohse AlArh & Hll A Aelwl Ths s reserh/revew pper strute uer the terms of the Cretve Commos Attruto-oommerl 30 Uporte Lese org/leses/y-/30/ permttg ll o ommerl use struto reprouto y meum prove the orgl wor s properly te

2 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Khl Alhulf α Jsem AlRh σ Elsye E M Ellmoy ρ Mohse AlArh Ѡ & Hll A Aelwl Astrt- Ths pper esres lgorthm for solvg ert lss of -rter multstge trsportto prolems wth trsshpmet BMTSP A severl -rter multstge trsportto prolem wth trsshpmet re formulte The presete lgorthm s mly se o pplto of the methos of solvg -rter sgle stge trsportto prolems utlzg vlle eomposto tehques for solvg lrge-sle ler progrmmg prolems the methos of tretg the trsshpmet prolems The mthemtl formulto of the presete lss oes ot ffet the spel struture of the trsshpmet prolem for eh of the vul stges A llustrtve emple s troue to vlte tht the mplemetto of the lgorthm Keywors: lrge sle trsportto prolem trsshpmet prolem mult-oetve eso mg eomposto tehque of ler progrmmg I Itrouto W he shpmets go retly from supply pot to em pot e shpmets o ot te ple etwee orgs or etwee esttos or from esttos to orgs t s lle lssl trsportto prolem I my rel lfe stutos shpmets re llowe etwee supply pots or etwee em pots There re my pots lle trsshpmet pots through whh goos e trsshppe o ther ourey from supply pot to em pot Shppg prolems wth y or ll of these hrtersts re osere s trsshpmet prolems It ws frst troue y Ore 965 [] whh he troue eteso of the orgl trsportto prolem to lue the posslty of trsshpmet The prolem of etermg smulteously the flow of prmry prouts through proessors to the mret of fl prouts hs ee formulte ltertvely s trsshpmet moel y mult-regol mult-prout mult-plt prolem formulte the form of geerl ler progrmmg moel hs ee propose y Juge et l 965 [4] Author α : Deprtmet of Power Ar Cotog College of Tehologl Stues PAAET Kuwt e-mls: lhulf@peteuw ellmoy@hotmlom Author σ ρ Ѡ: Automotve Mre Deprtmet College of Tehologl Stues PAAET Kuwt e-mls: mlrh@peteuw helwl@hotmlom lrh@peteuw Lter vrous ltertve formultos of the trsshpmet prolem wth the frmewor of the trsportto moel tht permts soluto of prolems of the type susse y Kg Log wthout the ee for sutrto of rtfl vrles were susse y Hurt Trmel 965 [5] O the other h Grg Prsh 985 [6] stue tme mmzg trsshpmet prolem The ym trsshpmet prolem ws stue y Herer Tzur 00 [7] Ozemr 006 stue Mult loto trsshpmet prolem wth ptte prouto lost sles fterwrs [8] Furthermore Osm et l 984 [9] troue lgorthm for solvg -rter multstge trsportto prolems Reetly Khur et l 0 [0] stue trsshpmet prolem wth me ostrts He troue lgorthm for solvg tme mmzg ptte trsshpmet prolem [] Ao-elg et l 0 [] troue trust rego glolzto strtegy to solve mult-oetve trsportto ssgmet trsshpmet prolems Khur 03 [3] troue Mult-e fe hrge -rtero trsshpmet prolem Rer et l [4] 0 presete A ew metho mely splttg metho to solve fully tervl trsshpmet prolems et l [5] 0 use the geet lgorthm for solvg trsportto ssgmet trsshpmet prolems Oh et l [6] 0 formulte sgle mult-oetve trsportto moels wth fuzzy reltos uer the fuzzy log Sr et l [7] 00 solve the mult oetve trsportto prolem MOTP uer fuzzess usg tervl umers A El-Whe [8] 00 presete mult-oetve trsportto prolem uer fuzzess Ds et l [9] 999 troue mult-oetve trsportto prolem wth tervl ost soure estto prmeters I ths pper formulto of fferet strutures of -rter lrge-sle trsshpmet prolems lgorthm for solvg lss of them whh e solve usg the eomposto tehque of ler progrmmg y utlzg the spel ture of trsshpmet prolems s presete The ew lgorthm etermes the pots of the o-omte set the oetve spe The metho ossts of solvg the sme multstge trsshpmet prolem repetely ut wth fferet oetves eh Yer 04 Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I 04 Glol Jourls I US

3 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I Yer 04 terto gves ether ew o-omte etreme pot or hges the reto of serh the oetve spe A llustrtve emple s presete ths pper II Formulto of B-rter Multstge Trsshpmet Prolems The formulto of fferet -rter multstge trsportto prolems wth trsshpmet presete ths pper overs severl rel stutos s show the followg ses B-Crter Multstge Trsportto Prolem wth trsshpmet of the Frst K BMTSP Ths se represets multstge trsshpmet prolems wthout y restrtos o termete stges I orer to evelop mthemtl formulto of the prolems t s ssume tht the vlltes re "" where = 3 "" s the umer of soures esttos Where s the requremets re " " = 3 The mmum trsportto osts eterortos from to re " "" " where = 3 X eotes the qutty shppe from to " " s the et mout trsshppe through pot where 0 The the prolem tes the form: M Where = 0 for the qutty shppe from the soure "S" to tself from estto "D" to tself: 0 for ll B-rter Multstge Trsportto Prolem wthtrsshpmet of the Seo K BMTSP : Ths se represets -rter multstge trsshpmet prolems whh the trsportto t y stge s epeet of the trsportto of the other stges I orer to ot the mthemtl formulto of the prolem whh represets ths se t s ssume tht for th stge =3 The vlltes re: 3 s the umer of soures esttos t the th stge the requremets re: 3 the 04 Glol Jourls I US trsportto osts eterortos re where = 3 = 3 eotes the qutty shppe from to s the et mout trsshppe through pot 0 The the prolem tes the form: M Where for the qutty shppe from 0 the soure S to tself from the estto D to tself s follows: the mmum trsportto ost s gve y: B-Crter Multstge Trsportto Prolem Wth Trsshpmet of the Thr K BMTSP 3: Ths se represets -rter multstge trsshpmet prolems wth some tol trsportto restrtos o the termete stges whh oes ot ffet the trsshpmet prolem formulto t eh stge The mthemtl formulto of the prolem represetg ths se s gve s: M 0 for ll M M

4 Where = 0 for the qutty shppe from the soure to tself from the estto to tself: = For ll where: re ler futos represetg the tol trsportto restrtos r s the umer of ths ler futos t the th stge B-Crter Multstge Trsportto Prolem wth trsshpmet of the Fourth K BMTSP 4 Ths se represets -rter multstge trsshpmet prolems whh the fferee etwee the put output trsportto ommoty s ow t the soures esttos of eh termete stge The ssume trsportto restrtos ths se ffet the trsshpmet formulto of eh vul stge The mthemtl formulto of the prolem represetg ths se s gve s: Where for the qutty shppe from the soure to tself from the estto to tself: = S D 0 r F K K F r M 0 S D 0 for ll = BMTSP s solve s -rter sgle stge trsshpmet prolem BMTSP e solve s sgle stge rter trsshpmet prolems the mmum vlue of the totl trsport osts eterortos re ote s the sum of the mmum trsportto osts eterortos for eh vul stge BMTSP 3 e solve usg the eomposto tehque utlzg the spel ture of trsshpmet prolems The et seto wll e evote to the soluto of ths type of prolems BMTSP 4 s solve usg y metho for solvg -rter ler progrmmg prolems A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems 04 Glol Jourls I US Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I 3 Yer 04

5 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I Yer 04 4 e A Algorthm for Solvg BMTSP 3 The eomposto tehque of ler progrmmg e use to solve the -rter multstge trsshpmet prolems espelly for the BMTSP 3 type Ths type of -rter multstge trsshpmet prolems eompose to [ 3 5 8]: Su prolems orrespog to every stge A mster progrm whh tes together the su prolems Let: D e the mtr osstg of the oeffets of th suprolem ostrts A the mtr osstg of the oeffets of th stge te- ostrts the vetor of ostt oeffets the te- ostrts the vetor osstg of the vlltes requremets of th su-prolem R o the mtr osstg of the frst m o olums of B - m o eotes the umer of elemets of B e the urret ss mtr the vetor of frst oetve oeffets of th suprolem the vetor of seo oetve oeffets of th suprolem B the orrespog vetor of s vrles oeffets the umer of su- prolems The followg seto presets lgorthm for etermg ll o omte etreme pots for the BMTSP 3 moel from whh the soluto for BMTSP BMTSP moels e eue from t s spel ses Assumg tht epeet ostrts re: D = s the tehologl mtr of the th stge tvty D s m + * m + mtr s the umer of stges m s the umer of soures t th stge s the umer of esttos s the olum vetor osstg of the vlltes requremets of the th su-prolem s m + * olum vetor It follows tht eh set of epeet ostrts e wrtte s: D = = represet the vetor of the orrespog vrles s m + * olum vetor Assumg tht ommo ostrts re: A th whh represets the tehogl mtr of stge tvty A s m 0 * m * mtr m 0 s the umer of ommo ostrts 0 s the orrespog ommo resoures vetor whh e wrtte s m 0 * Ths gves: A + A + + A + +A = 0 Assumg tht the oetve futos re: K whh represet the vetor of the frst rtero oeffets for the th stge tvty s *m * row vetor represet the vetor of the seo rtero oeffets for the th stge tvty s *m * row vetor Let: For the mster progrm: B e the s mtr ssote wth the urret s soluto B s mo* * m o + mtr C B the row vetor of the orrespog oeffets the oetve futo C B s *m o + row vetor R o the mtr of sze m o + *m o osstg of the frst mo olums of B - v the m o + th olum of the sme mtr B - The lgorthm presete here s ve to two phses Phse : To eterme the o-omte etreme pots the oetve spe Ths lgorthm s vlte y the followg theorem [] Theorem Pot z q q q = o-omte etreme pot s z z the oetve spe f oly f zq s reore y the lgorthm Phse II: The eomposto lgorthm e fou [7] Se the spel struture of the BMTSP 3 moel my llow the etermto of the optml soluto y eomposg the prolem to smll su-prolems the y solvg those su-prolems lmost epeetly the eomposto lgorthm for solvg lrge sle ler progrmmg prolems utlzg the spel ture of trsshpmet prolem e use to solve t Phse I: Step : From phse II we f: z M z / M z M z / z z M z z re ote q s set to Smlrly we f: z M z / M z M z / z z M If z z z stop z 04 Glol Jourls I US

6 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Otherwse reor z z set q = q+ Defes sets L = {} E = go to step Step : Choose elemet rs L set Go to phse II to ot the optml soluto = to the multstge trsshpmet prolem Mmze Suet to If there re ltertve optm hoose optml soluto = for whh r r If z s equl to or z z z z Set E = E {rs} go to step 3 q q Otherwse reor z z suh tht q q z z z z set q q L = L {rq} qs} go to step 3 Step 3 : Set L = L - {r-s} If L = stop Otherwse go to step Phse II: Step : Reue the orgl prolem to the mofe form terms of the ew vrles Step : F tl s fesle soluto to the mofe prolem Step 3 : Solve the su-prolems Suet to: r s r s z z s s z z r r r s r s e M o Let z = z = OR R m A w B o D o s z s ote: s use wth the frst rter s use wth the seo rter I orer to ot ˆ l w * y usg the trsportto tehque go to step 4 Step 4 : For the urret terto f: The eterme * w If o the urret soluto s optml the proess e termte the optml soluto to multstge trsportto prolem s: Otherwse go to step 5 B v Step 5 : Itroue the vrle orrespog to to the s soluto Determe the levg vrle usg the feslty oto ompute the et B - usg the revse smple metho tehque go to step 3 Illustrtve Emple The suggeste lgorthm for solvg prolem of the type BMTSP 3 s llustrte the followg emple: Coser the followg -rter two-stge trsshpmet prolem For eh stge the vlltes requremets osts eterortos for eh stge re gve y: = 6 = 6 = 4 L L M 3 = = 3 = 4 K L K l = L = 9 = = 3 Tle - : Trsportto ost t stges D D S S S 3 S S S D D Tle - : Trsportto ost t stges D D D 3 S S S S D D D Yer 04 5 Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I 04 Glol Jourls I US

7 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Tle - : Deterorto ost t stges D D S S S 3 S S S D D Suet to: = Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I Yer 04 6 Tle - : Deterorto ost t stges D D D 3 S S S S D D D Oe requremet s e to the ove prolem: It s requre tht the qutty shppe from the frst soure to the frst estto the frst stge s equl to the qutty shppe from the frst soure to the frst estto the seo stge The mthemtl moel s gve s follows: Mmze = Tle 3 : Set of o omte etreme pots = = = = = = = = = = = = = = = = = = = = = Iterto L E Reore Pot {} {} {3 3} {3 4 43} { } { } {4 43} {4} {5} {5 35} { } { } = 356 =740 3 = 4 4 = = = = = = Glol Jourls I US

8 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Tle 4 : o zero vlue of X for eh o omte pot III Iterto o omte o zero vlue of X = 356 X =6 X =4 X 3=8 X 3=4 X 4= X 3= X 35= X 4= X 5= X 5= X =6 X 3=4 X 4= X = X 4= X 5= X 3= X 4= X 53= =740 X =6 X = X 3=0 X =3 X = X 4= X 33= X 35= X 4= X 5= X =6 X 3=3 X 4= X = X 3= X 5= X 3= X 4= X 53= 3 3 = 4 X =6 X =3 X 3=9 X =3 X 3= X 4= X 33= X 35= X 4= X 5= X =6 X 3=4 X 4= X = X 4= X 5= X 3= X 4= X 53= 4 4 = 549 X =6 X =3 X 3=9 X = X 3=3 X 4= X 3= X 35= X 4= X 5= X =6 X 3=4 X 4= X = X 4= X 5= X 3= X 4= X 53= 5 5 = 440 X =6 X =3 X 3=9 X =3 X 3= X 4= X 33= X 35= X 4= X 5= X =6 X 3=3 X 4= X = X 3= X 5= X 3= X 4= X 53= 6 6 = 440 X =6 X =3 X 3=9 X =3 X 3= X 4= X 33= X 35= X 4= X 5= X =6 X 3=3 X 4= X = X 3= X 5= X 3= X 4= X 53= 7 7 = 440 X =6 X =3 X 3=9 X =3 X 3= X 4= X 33= X 35= X 4= X 5= X =6 X 3=3 X 4= X = X 3= X 5= X 3= X 4= X 53= 8 8 = 549 X =6 X =3 X 3=9 X = X 3=3 X 4= X 3= X 35= X 4= X 5= X =6 X 3=4 X 4= X = X 4= X 5= X 3= X 4= X 53= 9 9 = 356 X =6 X =4 X 3=8 X 3=4 X 4= X 3= X 35= X 4= X 5= X 5= X =6 X 3=4 X 4= X = X 4= X 5= X 3= X 4= X 53= Coluso A lgorthm for solvg ert lss of rter multstge trsportto prolems wth trsshpmet BMTSP s presete The presete lgorthm eles solvg suh prolems more relstlly It e use for etermg ll effet etreme pots The m vtge of ths pproh s tht the -rter two stge trsshpmet prolem e solve usg the str form of trsshpmet prolem t eh terto Goos trsportto my ot operte lwys retly mog supplers ustomers I suh prolems t s possle to optmze the trsshpmet prolem to two stges From the pplto eso mer wll hve ll effet etreme pots ther relte strutos Therefore y pot e hose whh wll prove ther poly Referees Référees Referes Ore A 956 Trsshpmet prolem Mgemet See 3: Kg G Log S 964 Optmum loto umer sze of proessg plts wth rw prout fl prout shpmets Jourl of Frm Eooms 46: Rhoy D 963 Iterregol ompettve posto of the hog-por ustry the southest Ute Sttes PhD thess Iow Stte Uversty 4 Juge G Hsvle J Rze R 965 A terregol moel: Its formulto pplto to the lve-sto ustry Agrulture Eoomy Revso 7 :-9 5 Hurt V Trmel T 965 Altertve formulto of the trsshpmet prolem Jourl of Frm Eooms 47 3: Grg R Prsh S 985 Tme mmzg trsshpmet prolem I Jourl of Pure Apple Mthemts 6 5: Here Y Tzur M 00 The ym trsshpmet prolem vl Reserh Logsts Qurterly 48: Ozemr D Yues E Here Y 006 Mult loto trsshpmet prolem wth ptte prouto lost sles Proeeg of the 006 Wter Smulto Coferee Pges Osm MSA Ellmoy EEM 984 O rter multstge trsportto prolems Frst Coferee o Opertos Reserh ts Mltry Appltos Pge Khur A Aror S 0 Solvg trsshpmet prolems wth me ostrts Itertol Jourl of Mgemet See Egeerg Mgemet 6 4: Pge 9-97 Khur A Trpt V Aror S 0 A lgorthm for solvg tme mmzg ptte trsshpmet prolem Itertol Jourl of Mgemet See Egeerg Mgemet 7 3: Pge 9-99 Yousr A Both E H 0 Trust rego lgorthm for mult-oetve trsportto Yer 04 7 Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I 04 Glol Jourls I US

9 A Algorthm for Solvg B-Crter Lrge Sle Trsshpmet Prolems Glol Jourl of Reserhes Egeerg B Volume XIV Issue IV Verso I Yer 04 8 ssgmet trsshpmet prolems Lfe See Jourl 9 3: Pge Arh Khur Jue 03 Mult-e fe hrge -rtero trsshpmet prolem OPSEARCH Volume 50 Issue pp P Rer P P 0 Solvg Fully Itervl Trsshpmet Prolems Itertol Mthemtl Forum Vol 7 o Sye A A Allh A Mous Hmy M Geee Ael Y Elmewy 0 Effet Multoetve Geet Algorthm for Solvg Trsportto Assgmet Trsshpmet Prolems Apple Mthemts Vol 3 o Artle ID: pges 6 Aupm Oh Shyml Kr Mol Mor Mt 0 Trsportto poles for sgle mult-oetve trsportto prolem usg fuzzy log Mthemtl Computer Moellg 0/0 53: Msour Sr Feryl Mshoorzeh Septemer 00 Solvg Mult Oetve Trsportto Prolem MOTP Uer Fuzzess o Usg Itervl umers AIP Cof Pro 8 37 Rhoes Greee 8 Wel F A El-Whe Jury 00 A multoetve trsportto prolem uer fuzzess Fuzzy Sets Systems Volume 7 Issue Pges SK Ds A Goswm SS Alm August 999 Multoetve trsportto prolem wth tervl ost soure estto prmeters Europe Jourl of Opertol Reserh Volume 7 Issue Pges Glol Jourls I US