A Decomposition Algorithm for Solving A Class of Bi-Criteria Multistage Transportation Problem With Case Study

Transcription

1 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 A Decoposto Algorth for Solvg A Class of B-Crtera Multstage Trasportato Proble Wth Case Study Dr. Jase M.S. Al-Rah Dr. Hlal A. Abdelwal Dr. Elsayed E.M. Ellaoy Dr. Swle A.M. Swle* Assstat Professor Departet of Autootve ad Mare College of Techologcal Studes PAAET Kuwat. Abstract I trasportato probles wth oe or ultstages sgle crtero of g the total cost s usually cosdered. But certa practcal stuatos two or ore obectves are relevat. For eaple the obectves ay be atos of the total cost cosupto of certa scarce resources such as eergy total deterorato of goods durg trasportato etc. Clearly ths proble ca be solved usg ay of the ultobectves lear prograg techques but the coputatoal efforts eeded would be prohbtve ay cases. The coputatoal coplety these techques arses fro the fact that each of the ethods fds the set of effcet etree pots the decso space where such etree pots are geerally ay. Therefore ths paper develops a ethod of fdg the odoated etree pots the crtera space for a certa odel of ultstage trasportato probles. Ths paper presets the atheatcal forulato of dfferet b-crtera ultstage trasportato proble ad a algorth for solvg a class of the. Ths class ca be solved usg the decoposto techque of lear prograg utlg the specal ature of the trasportato proble ad the ethod of fdg the o-doated etree pots the crtera space. A case study s cluded ths paper to fd the effcet dstrbuto of wheat ad flour Mddle Egypt ll-stoes copay. Keywords Trasportato Proble Multstage Trasportato Proble. B-Crtera Trasportato Proble Multobectve Decso Mag Decoposto Techque. I. ITRODUCTIO The stadard for of the trasportato type of lear progra wth a sgle obectve of g the total cost was frst forulated ad solved by Fra L. Htchoc ad T.C. Koopas the 940s. Multplcty of crtera s relevat a varety of lear prograg probles. I certa trasportato probles two or ore obectves are relevat. For eaple the two lear obectves ay be ato of the total cost ad ato of the total deterorato. Deterorato s relevat the case of certa pershable or decayg tes. The degree of deterorato ay be depedet o the route ode ad te of trasportato. Practcally the trasportato processes are doe ay stages ad the optal soluto for these probles dffers whe usg the optal soluto for each sgle stage. Iva Brea ad Adraa Istvaova [] preseted a way of solvg two-stage trasport proble. Osa M.S.A. ad Ellaoey E.E.M. [] preseted a algorth for solvg a class of ultstage trasportato probles. Dfferet ultstage bcrtera trasportato probles ca be arse practcally depedg o the type of trasportato whether classcal or ot uber of stages the coecto betwee trasportatos at each subsequet stage ad so o. Copyrght to IJIRSET 493

2 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 Itesve vestgato o ultobectve lear trasportato probles (MOLTP) have bee ade by several researchers. Aea ad ar [3] preseted a bcrtera trasportato proble odel. Lee ad Moor [4] studed the optato of trasportato probles wth ultobectves. Da [56] ad Isera [7] proposed the procedures to geerate all o-doated solutos to the MOLTP. Curret et al [8] prepared a revew of ultobectve desg of trasportato etwors. Rguest J.L. ad Rs D.B. [9] proposed the teractve solutos for lear ultobectve trasportato proble. Recetly Yousra Abo-elaga et al (0) [] troduced a trust rego globalato strategy to solve ult-obectve trasportato assget ad trasshpet probles. Khuraa et al (0) [3] studed a trasshpet proble wth ed costrats. Also. I (0) Khuraa et al [4] they troduced a algorth for solvg te g capactated trasshpet proble. I ths paper forulatos of dfferet bcrtera ultstage trasportato probles ad a algorth for solvg a class of the are preseted ths class of bcrtera ultstage trasportato probles ca be solved usg the decoposto techque of lear prograg [0] utlg the specal ature of the trasportato probles. The preseted algorth deteres the pots of the odoated set the obectve space stead of the decso space. The ethod cossts of solvg the sae ult stage trasportato proble repeatedly but wth dfferet obectves ad each terato gves ether a ew odoated etree pot or chages the drecto of search the obectve space. Also the ethod cossts of decopose the bcrtera ultstage trasportato probles to: - Subprobles correspodg to every stage - A aster progra whch tes together the subprobles. Every subproble costtutes a bcrtera trasportato proble ad the algorth terates whe o ore ew odoated etree pot s avalable. II. FORMULATIO OF BICRITERIA MULTI-STAGE TRASPORTATIO PROBLEMS The forulatos of dfferet bcrtera ultstage trasportato probles whch are preseted ths paper alost cover several real stuatos. The proposed forulatos eable the decso aer dscussg such trasportato probles to fd hs goal. A. BICRITERIA MULTISTAGE TRASPORTATIO PROBLEMS OF THE FIRST KID (BMTP ): Ths case represets bcrtera ultstage trasportato probles wthout ay trasportato restrctos o the teredate stages. I order to obta the atheatcal forulato of the probles represetg ths case let us assue that: c : u trasportato costs betwee the frst stage sources ad the last stage destatos. d : u deterorato of a ut whle trasportg fro frst stage sources to last stage destatos. a : Avalabltes of the frst stage sources. b : Requreets of the last stages destatos. : Aouts trasported fro to. The the proble taes the for: Copyrght to IJIRSET

3 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 Me Subect to: a b o for d c all. B. BICRITERIA MULTISTAGE TRASPORTATIO PROBLEMS OF THE SECOD KID (BMTP ): Ths case represets bcrtera ultstage trasportato probles whch the trasportato at ay stage s depedet of trasportato at the other stages. I order to obta the atheatcal forulato of the probles represetg ths case let us assue that for K th stage a K= the avalabltes of the sources are ( ) = the requreets at the destatos are b ( c ) =... the cost of trasportg oe ut fro source ( ) to destato ( ) s ( ) ad the d deterorato of oe ut whle trasportg fro ( ) to ( ) s ( ). The atheatcal forulato for ths case ca be wrtte as () sgle stage bcrtera trasportato probles the for:. Subect to Copyrght to IJIRSET a c c...

4 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 Copyrght to IJIRSET b... all for o The u trasportato cost s gve as: * Ad the u deterorato s gve as: * C. BICRITERIA MULTISTAGE TRASPORTATIO PROBLEMS OF THE THIRD KID (BMTP 3): Ths case represets bcrtera ultstage trasportato probles wth soe addtoal trasportato restrctos o the teredate stages whch does ot effect the bcrtera trasportato proble forulato at each stage. The atheatcal forulato of the proble represetg ths case s gve us: K c K d Subect to... a... b o K r b F ) (... all for o Where: r F = are lear fuctos represetg the addtoal trasportato restrctos at the () stages ad r s the uber of ths lear fuctos at the th stage. D. BICTERIA MULTISTAGE TRASPORTATIO PROBLEMS OF THE FOURTH KID (BMTP4): Ths case represets bcrtera ultstage trasportato probles whch the subtracto betwee the put ad the output trasportato coodty s ow at the sources (destatos) of each teredate stage. The assued trasportato restrctos ths case affect the trasportato forulato of each dvdual stage.

5 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 Copyrght to IJIRSET The atheatcal forulato of the proble represetg ths case s gve as: K c K d Subect to:... a b b... all for o (BMTP) Is solved as a sgle stage bcrtera trasportato proble. (BMTP) Ca be solved as sgle stage bcrtera trasportato probles ad the odoated solutos ca be obtaed fro the su of the solutos for each dvdual stage. (BMTP3) Ca be solved usg the decoposto techque utlg the specal ature of the trasportato probles for deterg the pots of the odoated set the obectve space. (BMTP4) Ca be solved usg ay algorth for solvg ultobectve lear prograg probles. A effcet soluto for ay odel s defed as that pot the feasble doa of that odel for whch there s o other feasble pot whch doates t for all the obectves. Its age the obectve space s called a odoated soluto. I the followg we wll preset a algorth for deterg all odoated etree pots of the (BMTP3) odel fro whch the soluto of the (BMTP) ad the (BMTP) odels ca be deduced as specal cases fro t. Let: for depedet costrats: D = be the techologcal atr of the th stage actvty D s ( + ) * ( + ) atr s the uber of stages s the uber of sources at th stage s the uber of destatos. b be the colu vector cosstg of the avalabltes ad requreets of the th subproble b s a ( + ) * colu vector.

6 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 It follows that each set of depedet costrats ca be wrtte as: D = b = represet the vector of the correspodg varables s ( + ) * colu vector. Let: For the coo costrats: A be the techogcal atr of th stage actvty A s 0 * ( * ) atr 0 be the uber of coo costrats. b 0 be ts correspodg coo resources vector b 0 s ( 0 * ) colu vector. Ths gves: A + A +.+ A +. +A = b 0 Let: for the obectve fuctos: c K represet the vector of the frst crtero coeffcets for the th stage actvty c s *( * ) row vector. d represet the vector of the secod crtero coeffcets for the th stage actvty d s *( * ) row vector. Let: For the aster progra: B be the basc atr assocated wth the curret basc soluto B s ( 0 *) * ( 0 +) atr. C B be the row vector of the correspodg coeffcets the obectve fucto C B s *( 0 +) row vector. R 0 s the atr of se ( 0 + )* 0 cosstg of the frst 0 colus of B - ad v s the ( 0 + ) th colu of the sae atr B - The algorth preseted here s dvded to two phases. Phase : III. A ALGORITHM FOR SOLVIG (BMTP3) Detere the odoated etree pots the obectve space. Ad the algorth s valdated by the followg theore [3]. Theore: A pot (q) = the algorth. q q s a odoated etree pot s the obectve space f ad oly f (q) s recorded by Phase II: Phase II Is the decoposto algorth whch ca be foud []. Sce the specal structure of the (BMTP3) odel ay allow the deterato of the optal soluto by frst decoposg the proble to sall subprobles ad the solvg those subprobles alost depedetly the the decoposto algorth for solvg large scale lear prograg probles utlg the specal ature of trasportato proble ca be used to solve t. Copyrght to IJIRSET

7 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 A. Phase I: Step : Go to phase II fd () M. / M Ad fd () () Record ( ) ad set q =. Slarly go to phase II fd Ad fd () () () () M. / ad M. () Otherwse record ( ) ad set q = q+ Defes sets L = {()} ad E = ad go to step. M. / M () () M. / ad M. If () () () () ( ) ( ) stop. Step : Choose a eleet (rs) L ad set ( s) ( s) a a r ( r) ( r s) ( s) ( r) ad ad Go to phase II to obta the optal soluto ( =.. ) to the ultstage trasportato proble. ( r s) ( r s) M ( e c a d ) Subect to: If there are alteratve opta choose a optal soluto M o.. Let: Z ( c =... for whch.) Copyrght to IJIRSET c

8 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 If ( Step 3: Set L = L - {(r-s)}. If L = stop. Otherwse go to step. B. Phase II: Ad Z ( r) ( d ( r) ( s) ( s) ) s equal to ) or ( ) Set E = E {(rs)} ad go to step 3. ( q) ( q) Otherwse record ( ) such that ( q) ( q) ad set q q L = L {(rq)} (qs)} ad go to step 3. Step : Reduce the orgal proble to the odfed for ters of the ew varables Step : Fd a tal basc feasble soluto to the odfed proble. Step 3: Solve the subprobles w ( c cb Ro A ) Obtag ˆl Subect to D b o.... ad optal obectve values w * by usg the trasportato techque Go to step 4. Step 4: For the curret terato fd ad the detere o * w c B v... Copyrght to IJIRSET M ( ) If the curret soluto s optal ad the process s terated the optal soluto to ultstage trasportato proble s: L L K L AK L...

9 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 Otherwse go to step 5. Step 5: Itroduce the varable L correspodg to to the basc soluto. Detere the leavg varable usg the feasblty codto ad copute the et B - usg the revsed sple ethod techque go to step3. IV. A CASE STUDY: EFFICIET DISTRIBUTIO OF WHEAT AD FLOUR I MIDDLE EGYPT MILLSTOES CO. A. DATA COLLECTIO The preseted algorth s appled to Mddle Egypt Mllstoes Copay (El-Ma Sector) to obta the odoated etree pots the obectve space for trasportg wheat ad flour. The proble s a b-crtera two-stage trasportato proble to trasport wheat ad dfferet types of flour betwee a stores (supplers) to llstoes (warehouses) ad fally to destatos (custoers). The collected data for trasportato cost (LE/to) trasportato deterorato (losses) (LE/to) avalabltes of a sources requreets of warehouses ad destatos (to) are preseted Tables () (3) for the frst ad the secod stages respectvely. Table () Trasportato cost deterorato avalabltes ad requreets for stage (). M. Stoes Stores Maghagha Be Maar Saalout El-Ma AbuKoras Mallawy Avalabltes Maghagha (0 0) (9 ) ( 3) (5 4) (8 5) ( 6) 4350 Saalout ( 3) (9 ) (0 0) (9 ) ( 3) (5 4) 5340 El-Ma (5 4) (3 3) (9 ) (0 0) (9 3) ( 4) 530 Deerawas ( 5) (0 5) (7 3) (3 ) ( ) (9 ) 407 Requreets ( ) Table (3) Trasportato cost deterorato avalabltes ad requreets for stage (). Destatos Stores El- Edwa Maghagha Be Maar Mata Saalout El Ma Abu- Koras Mallawy Deerawas Avalabltes Maghagha (9 ) (0 0) (9 ) ( ) ( 3) (5 4) (8 5) ( 6) ( 5) 900 Be Maar (9 ) (9) (0 0) (9 ) (9 ) (3 3) (5 ) (9 3) (0 5) 64 Saalout (3 4) ( 3) (9 ) (9 ) (0 0) (9 ) ( 3) (5 4) (7 3) 3560 El Ma (63) (5 4) (3 3) ( 4) (9 ) (0 0) (9 3) ( 4) (3 ) 43 Abu Koras (95) (8 5) (5 ) (4 3) ( 3) (9 3) (0 0) (9 ) ( ) 379 Mallawy (5) ( 6) (9 3) (7 4) (5 4) ( 4) (9 ) (0 0) (9 ) 40 Requreets ( ) Copyrght to IJIRSET

10 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 B. APPLICATIO RESULTS The proble s solved by the preseted algorth ad Tables (4) (5) llustrate the o-doated etree pots as well as the effcet dstrbuto accopayg each terato. Choosg a o-doated pot ad ts related dstrbuto fro Tables (4) ad (5) should be doe accordg to the decso aer pot of vew. Table (4): Soluto of the real Proble the obectve space. Iterato L E Recorded Pots {()} Φ () Z = (094330) () Z = ( ) {(3)(3)} Φ (3) Z = ( ) 3 {(3)} {(3)} (3) Z = ( ) 4 Φ {(3) (3)} (3) Z = ( ) Iterato Table (5): The effcet dstrbuto for both stages () ad () for all teratos. Effcet Dstrbuto for Stage () Effcet Dstrbuto for Stage () = 900 = 450 = 74 3 = = = = 8 46 = 693 = 900 = 450 = 74 3 = = = = = 693 = 47 = 853 = 30 3 = = = = = = = = = = 557 = 47 = 853 = 30 3 = = = = = = = = = = 557 Copyrght to IJIRSET 494

11 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber = 900 = 450 = 74 3 = = = = 8 46 = 693 = 900 = 450 = 74 3 = = = = = 693 = 47 = 853 = 30 3 = = = = = = = = = = 557 = 47 = 853 = = = = = = = = = = = 557 V. COCLUSIO I certa stuatos two obectves are relevat trasportato probles also the goods trasportato ay ot operate always drectly aog supplers ad custoers. I such probles t s possble to opte the trasportato o two stages. The preseted algorth ths paper eables solvg such probles ore realstcally. It ca be used for deterg all effcet etree pots. The a advatage of ths approach s that the b-crtera two stage trasportato proble ca be solved usg the stadard for of a trasportato proble at each terato. Fro the applcato decso aer wll have all effcet etree pots ad ther related dstrbutos. Therefore he ca choose ay pot whch provdes hs polcy. REFERECES [] Bera Istraova The way of solvg two-stage trasportato probles Matheatcal Methods Ecoocs [] Osa M.S.A. Ellaoy E.E.M. O b-crtera ult-stage trasportato probles Proceedgs of the frst ORMA coferece pp ov [3] Aea Y.P. ar K.P.K. B-crtera trasportato Probles Maageet Scece vol. 5 pp 979. [4] Lee S.M. Moore L.J. Optg trasportato probles wth ultple obectves AIIE Trasactoal vol. 5 pp [5] Da J.A. Solvg ult-obectve trasportato proble Eooco- Mateatcy Obor vol. 4 pp [6] Da J.A. Fdg a coplete descrpto of all effcet solutos to a ult-obectve trasportato proble Eooco- Mateatcy Obor vol. 5 pp [7] Isera H. The euerato of all effcet solutos aval Research Logstc Quarterly vol. 6 pp [8] Curret J. M H. Mult-obectve desg of trasportato etwors: Taooy ad aotato Europea Joral of Operatos Research vol [9] Rguest J.L. Rs D.B. Iteractve solutos for the lear ultobectve trasportato proble Eur. J. Oper. Res [0] Taha H.A. Operatos Research A Itroducto Seveth Edto Pretce Hall ISB [] Hller F.S. Lebera G.J. Itroducto to Operatos Research Seveth Edto Mc Graw-Hll ISB [] Yousra A. Botha E. ad Haad Z. Trust rego algorth for ult-obectve trasportato assget ad trasshpet probles Lfe Scece Joural 9 (3): Page [3] Khuraa A. ad Arora S. Solvg trasshpet probles wth ed costrats Iteratoal Joural of Maageet Scece ad Egeerg Maageet 6 (4): Page [4] Khuraa A. Trpt V. ad Arora S. A algorth for solvg te g capactated trasshpet proble Iteratoal Joural of Maageet Scece ad Egeerg Maageet 7 (3): Page Copyrght to IJIRSET 494

12 ISS: Iteratoal Joural of Iovatve Research Scece Egeerg ad Techology (A ISO 397: 007 Certfed Orgaato) Vol. Issue 9 Septeber 03 BIOGRAPHY Dr. Jase M. Alrah s Assstat Professor ad Head of Departet of Autootve ad Mare Egeerg Techology at the College of Techologcal Studes (CTS) sce 00. He receved hs B.Sc. Mechacal Egeerg fro Dayto Uversty USA 994 ad M.Sc. wth hoor degrees Autootve Egeerg (Brae ad Safety Vehcle Syste) fro Uversty of Leeds UK 999. The he receved hs Ph.D. Mechacal Egeerg Vehcle Accdet Recostructo ad Safety Syste to Reduce Rear Ipact c Iury fro Moash Uversty Australa 006. Dr. Hlal A. Abdelwal s Assstat Professor Departet of Autootve ad Mare Egeerg Techology at the College of Techologcal Studes (CTS) PAAET Kuwat. He receved hs B.Sc. Autootve Egeerg fro Ma Uversty Egypt 988 ad M.Sc. Autootve Egeerg fro Ma Uversty Egypt 99. The he receved hs Ph.D. Autootve Egeerg fro Ma Uversty Egypt Chael Schee wth the Uversty of ebrasa Lcol (UL) USA 997. Dr. Hlal s a SAE eber sce 00 to ow. Dr. Elsayed E.M. Ellaoy s Assstat Professor Departet of Autootve ad Mare Egeerg Techology at the College of Techologcal Studes (CTS) PAAET Kuwat. He receved hs B.Sc. Autootve Egeerg fro Helwa Uversty Egypt 975 ad M.Sc. Autootve Egeerg fro Helwa Uversty Egypt 98. The he receved hs Ph.D. Autootve Egeerg fro Helwa Uversty Egypt 985. Dr. Swle A.M. Swle s Assstat Professor Departet of Autootve ad Mare Egeerg Techology at the College of Techologcal Studes (CTS) PAAET Kuwat. He receved hs B.Sc. aval Archtecture ad Mare Egeerg fro Aleadra Uversty Egypt 98 ad M.Sc. Structural Desg fro Aleadra Uversty Egypt 986. The he receved hs Ph.D. Structural Desg fro Hrosha Uversty Japa 99. Research assstat at Hrosha Uversty Copyrght to IJIRSET