A Two Objective Model for Location-Allocation in a Supply Chain

Transcription

1 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) The Joural of Mathematcs ad Computer Scece Avalable ole at The Joural of Mathematcs ad Computer Scece Vol. 4 No.3 (22) A Two Objectve Model for LocatoAllocato a Supply Cha AmrHosse Nobl, Abolfazl Kazem 2, Alreza Alejad 3,2,3 Qazv Brach, Islamc Azad Uversty, Qazv, Ira, amrhosse.obl@yahoo.com Receved: February 22, Revsed: May 22 Ole Publcato: July 22 Abstract I today's compettve world, locatoallocato (LA) decsos are oe of the most mportat aspects of supply cha (SC) optmzato. Ths LA decsos are cludg selecto of kow stes for costructo of facltes ad allocato of the dstrbuto etwork betwee the levels of SC. I ths paper, a olear programmg model to locato facltes ad allocate the supply cha dstrbuto etwork order to mmze both the cost ad tme of threeechelo are preseted. The proposed model due to computatoal complexty hgh dmesos caot be solved wth covetoal ad accurate methods, Therefore to acheve a soluto of a method metaheurstc called geetc algorthm s used. Fally, to exame ad the effectveess of the proposed algorthm, computatoal results obtaed are compared wth output of lgo 2 software. Keywords: Locatoallocato; Supply cha; olear programmg; Geetc Algorthms Itroducto I recet decades, may of whch reduce costs due to locato models have bee cosdered by researchers []. Locato theory by Fota 826 was rased for the frst tme agrcultural actvtes [2]. But as formulated frst by Alfred Weber [3] 99. Hs Locato for a stock wth the am of mmzg the total dstace traveled betwee warehouses ad customers offered. Faclty Locato problems o ste from amog a set of kow ad costat. Ths problems for the costructo of facltes such as hosptals, fre statos ad emergecy servces are very useful. I 963, Cooper [4] developed the locato problems. He sad the frst problems the faclty Locatoallocato. LA problem of effectve decso makg problem assocated wth the faclty desg. Ths problem, the optmal umber ad locato of facltes accordg to dstrbuto etwork ad the objectve fucto (maxmum/ mmum) to be determed. 392

2 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) I 982, Murtagh ad Nwattsyawog [5] proposed the capactated faclty locatoallocato (FLA) problem, Ulke most prevous doe s that research s ot wthout the capacty of the faclty. Ther model s cosdered to be oe of the most mportat FLA researches focuses o capacty of faclty. I 994, Lu ad et al. [6] mmzg the total weghted dstace from supply ceters to customers wth regard to the rectlear dstaces, whch s cosdered as oe of the most mportat researches ths LA problems. I 23, Lu ad Zhou [7] proposed the stochastc model for LA problems wth the capacty. I 29, Bschoff ad Dächert [8] compared to tradtoal search methods for LA problems wth the ew search methods ad studed the effectveess of these methods to each else. I recet years, combed FLA problems wth supply cha approach have bee cosdered by researchers. Because of creasg customer expectatos ad dustry competto, prevous producto maagemet methods that focus o teral processes of the orgazato have lost ther effectveess ad tegrated supply cha approach s of terest to dustral maagers. Supply cha s the commucato betwee compaes that create products ad servces drectly ad drectly relate to each other [9] ad these compaes ted to work together to acheve ther strategc objectves more effcet. I 28, Ho ad et al. [] optmze the FLA problem a customerdrve supply cha. Ths study of the aalytc herarchy process (AHP) ad goal programmg (GP) order to maxmze profts s used, cosders both quattatve ad qualtatve factors. I 2, Wag ad et al. [] preseted LA decsos the twoechelo supply cha wth both proft ad cost objectves. Ther locato decsos, whch are related to wth each other wth the supply cha cossts of three strategc decsos below stated. * Locate facltes * Dstrbuto assgmet * Dstrbuto quatty Based o lterature revew studes, regard to the atteto of researchers has multple objectves most models. Also, those studes that was used of the multple objectve fucto s usually set up cost, shppg ad proft a twoechelo supply cha were the real world s usually more tha two echelo the supply cha s avalable. I ths paper, a teger programmg model for twoobjectve LA problem of mmzg the cost to set up factores, trasportato, maufacturg, overtme ad mmze producto tme the supply cha s proposed. Ths model s preseted three echelos for supply cha supplers, maufacturers, dstrbutors. I ths paper, the proposed model s solved usg a metaheurstc method s called geetc algorthm. The proposed soluto algorthm ca ear optmal soluto for largesze as compared to a optmzato tool called lgo. Ths paper s orgazed as follows. Secto 2, Modelg LA decsos wll be descrbed order to mmze total costs ad tme. Secto 3 s expressed by the geetc algorthm approach to solve the proposed model. Secto 4 shows computatoal results ad performace of the proposed algorthm. Fally, Secto 5 cocludes the study. 2 The proposed model I ths secto, a mathematcal model of two objectves LA to mmze total costs ad tme threeechelo supply cha are preseted. I ths paper, LA problem s a teger olear programmg model. The model s the mmum set up cost, trasportato, producto, maufacturg, overtme, ad also the producto of goods the supply cha. Ths supply cha cossts of three levels of suppler, factory ad dstrbutor are lked together. Ths proposed model solves the followg mportat decsos: 393

3 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) How may stes should be selected for settg up the factory? Whch dstrbuto ceters should be coectos wth each factory? Whch factores should be coectos wth each suppler? The otatos of the locatoallocato model the supply cha are defed as followg: D : Demad of dstrbuto j Q : Quatty of product produced the factory f : Fxed cost of settg up a factory at ste S : Capacty of factory located at ste G : Quatty of product excess of producto capacty the factory U : Quatty of raw materal produced suppler V : Producto cost per ut product factory a : Producto cost per ut product excess of factory capacty R : Producto cost per ut raw materal suppler k C : Trasportato cost per ut dstrbuted from factory to dstrbuto j B : Trasportato cost per ut delvered from suppler k to factory T : Producto tme per ut of product produced factory L : Producto tme per ut of raw materal produced suppler k The decso varable s deoted as follows: x : Cofgurato relatoshp betwee suppler ad factory, x ε{,} where x = f suppler k delvers raw materals to fulfll the demad to factory, x = otherwse. y : Decso to set up a factory, y ε{,} where y = f a factory s set up at ste, y = otherwse. z : Cofgurato relatoshp betwee factory ad dstrbuto, z ε{,} where z = f factory dstrbutes products to fulfll the demad to dstrbuto j, z = otherwse. I ths model, the followg assumptos are used: () there are a sgle perod; (2) there s a sgle product ad raw materal; (3) there s threeechelo supply cha, suppler, factory ad dstrbuto; (4) suppler s capacty s ulmted; (5) factory s capacty s lmted but ca be compesated wth overtme (6) a factory ca serve several dstrbuto whle each dstrbuto ca be served oly by a factory; (7) a suppler ca serve several factory whle each factory ca be served oly by a suppler; (8) there s eed to satsfy all demads of a dstrbuto ad a factory; (9) Products are produced a factory as a lear, also raw materals are produced a suppler as a lear. Fg. shows a llustrato of threeechelo supply cha. 394

4 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) Suppler Factory Dstrbuto Fg.. A llustrato of threeechelo supply cha The objectve fuctos ad costrats are as follows: m F f y S V R U ag C z D B x Q 2 K m K k k j j j k k k j k K m F ( Q ( Q ) / 2) T ( U ( U ) / 2) L K k x z Q k j x z k j k k k k y y m j j j z D U x Q k k G max{, Q S } y x k {,} y {,} z j {,}, 2,..., j, 2,..., m k, 2,..., K,, 2,...,, 2,...,, j, 2,..., m, 2,..., k,2,..., K, 2,...,, 2,...,, k, 2,..., K, 2,...,, 2,...,, j, 2,..., m () (2) (3) (4) (5) (6) (7) (8) (9) () () (2) (3) The objectve fucto () mmzes total cost of SC. Note that t s the sum of the factory setup, the producto cost, the trasportato cost ad overtme cost. The objectve fucto (2) s mmum total producto tme factores ad supplers. Costrat (3) specfes that a factory ca be served by oly oe suppler. Costrat (4) specfes that a dstrbuto ca be served by oly oe factory. Costrat 395

5 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) (5) states that f a factory was set up, suppler do ot delver raw materal from the ste. Costrat (6) states that f a factory was set up, dstrbutors do ot sed product to the ste. Costrat (7) specfes the amout of produced per factory. Also, costrat (8) specfes the amout of produced per suppler. Costrat (9) wll determe the amout of product that should be excess of factory capacty to produce. Costrats () specfes that at least oe factory should be set up to meet demad dstrbutors. Fally, Costrats (), (2) ad (3) are specfed type of decso varables. 3 Soluto method I ths paper, a LA programmg model s proposed the supply cha of three echelo. The LA problems large szes are cosdered as the complcated ad dffcult oes, exact soluto methods for solvg them are usually tmecosumg [4, 2]. Hece, ths artcle come from a metaheurstc method called geetc algorthm s used. Geetc algorthm s a stochastc optmzato method for solvg dffcult optmzato problems s very useful [3, 4]. I 975, geetc algorthm were frst expressed by Hollad [5]. I 23, Zhou ad Lu [6] usg geetc algorthms to solve the problem of capacty locatos. I 22, Zarrpoor ad et al. [] to solve the maxmal coverg locatoallocato problem wth geetc algorthm the compettve ad userchoce evromet. 3.. Procedure of geetc algorthm I Ths paper, the proposed geetc algorthm for olear programmg s preseted accordg to the objectves ad costrats. The proposed GA cosst of makg three decsos: how may ad whch factores should be set up, whch factores should be served by whch suppler ad whch dstrbutos should be served by whch factory. Frst, the algorthm radomly geerates the tal geerato of chromosomes. Ths proposed chromosome cossts of three parts. P, paret populato s produced each geerato. P2 ad P3, respectvely, the populato of chldre that come from crossg ad mutato. The, the algorthm calculates the ftess fucto ad operators rus for each populato. As far as the stop codto s acheved. If stoppg codtos are ot acheved, the algorthm deals wth the mechasm chose to produce a ew geerato. The flowchart of the proposed algorthms s preseted Fg Chromosome structure The frst step ths proposed algorthm, Set of parameters s ecoded as a gee [2]. The coected them together to create a seres of chromosomes. Each chromosome s made by three set of gees, the bary code of x, y ad z that each of three gees the tal geerato radomly arse. These gees must to satsfy costrats 3 utl 6. Fg. 3 shows a llustrato of a chromosome wth K =, = 2 ad m =. 396

6 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) Fg. 2. The flowchart of the proposed algorthms x() x(2) y() y(2) z() Fg. 3. llustrato of a chromosome z(2) 3.3. Ital geerato At ths stage, the umber of chromosomes (POP) are radomly geerated ad stored P Ftess fucto Mathematcal model preseted ths paper has two objectve fuctos that must be mmzed. Therefore, the LPmetrc method has bee used ths model. I ths method, each objectve fucto to be optmzed separately ad the the dstace betwee objectve ad model s mmzg [7]. Ftess fucto s based o LPmetrc method as follows: F ( E(( f y SV R U ag k k k C z D B x Q F )/ F )) j j j k k j k (( E) (( ( Q ( Q )/2) T k K m K K ( U ( U )/2) L F )/ F )) k k k * * * * 2 (4) where E s a value that s gve to decso maker o prortes. 397

7 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) Crossover operator Oepot crossover operato s used to the proposed GA for every three parts of chromosome. Frst, oe radom umber betwee ad ( s umber of ste ) s selected ad employed as crossover cuttg pot for every three parts of chromosome, Because all three parts of the chromosome s the umber of factory stes (). The, two chromosome are selected from P ad ther correspodg parts of chromosome are exchaged. The chromosomes after crossover are repared to commt the costrats 3 utl 6. The umber of chromosomes resultg from crossover P2 are stored. The procedure of crossover could be llustrated Fg. 4. P P2 Fg. 4. The procedure of crossover 3.6. Mutato operator Radom mutato operato s used to the proposed GA. Frst, oe radom umber betwee ad ( s umber of ste ) s selected ad employed as mutat gee for secod part (y ) of chromosome. The, oe chromosome s selected from P ad ts correspodg part (y ) of chromosome s exchaged. Two other parts of chromosome after chage are repared to commt the costrats 3 utl 6. The umber of chromosomes resultg from mutato P3 are stored. The procedure of mutato could be llustrated Fg. 5. P 3.7. Selecto mechasm P3 Fg. 5. The procedure of mutato Selecto mechasm s based o selecto the best chromosomes each geerato. Chromosomes wth the P, P2 ad P3 are merged ad the ftess fucto are arraged from lowest to hghest. The, the populato of chromosomes wth the lowest ftess fucto for the ext tal geerato are selected. The scheme of selecto mechasm s show Fg Stop codto Fg. 6. The scheme of selecto mechasm I ths algorthm, s acheved stopped f the umber of geeratos s or better s ot doe. 398

8 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) Parameter settg Geetc algorthm parameters (POP: umber of populato, Pcr: crossover rate ad Pmu: mutato rate) are specfed by RSM method. For settg the parameters of the RSM method has bee used Mtab4 program that sze of problem uder vestgato s of 6 supplers, 6 factores ad dstrbutos. Solutos optmzed parameters of the algorthm s show Table. All the rest of parameters are show Appedx. Table. The optmzed parameter Optmal POP Pcr Pmu Hgh..8 Curret Low Computatoal results I ths paper, computatoal results of the geetc algorthm s compared wth the output Lgo 2 software. The proposed geetc algorthm has bee mplemeted Matlab software for the LA problem threeechelo supply cha. These comparsos Table 2 s show. As show Table 2, whe the problem scale grows to 2 supplers, 2 factores ad 2 dstrbutos, Lgo software s ot able to fd the optmum soluto. But the geetc algorthm proposed approprate tme obtas a acceptable aswer. Table 2. The computatoal results Problem sze Lgo 2 GA K M Ftess fucto CPU tme (s) Ftess Fucto CPU tme (s) Coclusos I ths paper, a LA model for mmzg the cost ad tme was used the threeechelo supply cha. The proposed model cludes three levels of supplers, factores ad dstrbutors the supply cha. I ths paper has bee effort that locate a umber of factores amog a fte set of stes, ad the effort to allocate task assgmet betwee supplers, factores ad dstrbutos. Frst, the model used to solve the lgo 2, but cosderg hgh complexty ad olear problems, ths software s oly able to resolvg smallsze model. The large sze, was used to acheve a soluto of a method metaheurstc called geetc algorthm. Ad compare the results of geetc algorthm wth output lgo 2 software showed that the proposed geetc algorthm s capable of good performace. 399

9 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) Ths research ca be exteded as follows: Frst, perodcal demad could ad Several product could be cosdered the model. Secod, other objectves, such as proft ad vetory could be cosdered the model. Fally, Costs such as purchase, orders ad mateace could be cosdered. Appedx. Parameters settg Parameters Value the fxed cost of settg up a factory at f ~ (,2) ste ( thousads) U capacty of factory located at ste ( S ~ U (5,7) pece) demad of dstrbuto j ( pece) Dj ~ U (6,8) Producto cost per ut product produced factory ( thousads/pece) Producto cost per ut raw materal suppler k ( thousads/pece) Producto cost per ut product excess of factory capacty ( thousads/pece) producto tme per ut of product produced factory I ( hours /pece) producto tme per ut of raw materal produced suppler k ( hours/pece) trasportato cost per ut dstrbuted from factory to dstrbuto j ( thousads/pece) trasportato cost per ut delvered from suppler k to factory ( thousads/pece) V ~ U (5,2) R ~ U (5,8) k a ~ U (5,8) T Lk Cj Bk U U (.75,.25) (.5,.75) ~ U(,2) ~ U(5,) Refereces [] Zarrpour, N., Shavad, H., & Bagherejad, J., Exteso of the Maxmal Coverg Locato Allocato Model for Cogested System the Compettve ad Userchoce Evromet, Iteratoal Joural of Idustral Egeerg & Producto Maagemet, Vol. 22, No. 4, pp , 22. [2] Jabalamel, M. S, Shahaagh, K., Hosav, R., & Nasr, M. R., A Combed Model for Locatg Crtcal Ceters (HAPIT), Iteratoal Joural of Idustral Egeerg & Producto Maagemet, Vol. 2, No. 4, pp. 6576, 2. [3] Weber, A., Über de Stadort der Idustre. Tübge, 99. Alferd Weber s theory of the Locato of Idustres, Uversty of Chcago Press (Eglsh traslato by C.J. Fredrch, 929). [4] Cooper, L., Locatoallocato problems, Operatoal Research, Vol., No. 3, pp , 963. [5] Murtagh, B.A., & Nwattsyawog, S.R., Effcet method for the multdepot locatoallocato problem, Joural of the Operatoal Research Socety, Vol. 33, N. 7, pp , 982. [6] Lu, CM., Kao, RL., & Wag, AH., Solvg locatoallocato problems wth rectlear dstaces by smulated aealg, Joural of Operatoal Research, Vol. 45, No., pp , 994. [7] Zhou, J., & Lu, B., New stochastc models for capactated locatoallocato problem, Computers & Idustral Egeerg, Vol. 45, No., pp. 25, 23. 4

10 AmrHosse Nobl, Abolfazl Kazem,Alreza Alejad/ TJMCS Vol. 4 No. 3 (22) [8] Bschoff, M., & Dächert, K., Allocato search methods for geeralzed class of locatoallocato problems, Europea Joural of Operatoal Research, Vol. 92, No. 3, pp , 29. [9] Stadtler, H., & Klger, C., "Supply cha maagemet ad advaced plag", Sprger, 2. [] Ho, W., Lee, C. K. M., & Ho, G. T. S., Optmzato of the faclty locatoallocato problem a customer drve supply cha, Operatos Maagemet Research, Vol., pp. 6979, 28. [] Wag, K.J., Makod, B., & Lu, S.Y., Locato ad allocato decsos a twoechelo supply cha wth stochastc demad A geetcalgorthm based soluto, Expert Systems wth Applcatos, Vol. 38, pp , 2. [2] Taghavfard, M. T., & Shahsavar, A., MultObjectve LocatoAllocato Problems Usg Smulated Aealg, Iteratoal Joural of Idustral Egeerg & Producto Maagemet, Vol. 9, No. 4, pp. 935, 29. [3] Beasley, D., Bull, D. R., & Mart, R. R., A overvew of geetc algorthms: part, fudametals, Uversty Comput, Vol. 5, No. 2, pp. 5869, 993. [4] Reer, G., & Ekart, A., Geetc algorthms computer aded desg, ComputerAded Desg, Vol. 35, No. 8, pp , 23. [5] Hollad, J. H., Adaptato atural ad artfcal systems, A Abor, MI: Uversty of Mchga Press, 975. [6] Zhou, J., & Lu, B., New stochastc models for capactated locatoallocato problem, Computers & Idustral Egeerg, Vol. 45, No., pp. 25, 23. [7] Taylor, D., Global Cases Logstcs ad Supply Cha Maagemet, Idustral Thompso Busess, Bosto,