Multiple Batch Sizing through Batch Size Smoothing

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1 Jounal of Indual Engneeng (9)-7 Mulple Bach Szng hough Bach Sze Smoohng M Bahadoghol Ayanezhad a, Mehd Kam-Naab a,*, Sudabeh Bakhh a a Depamen of Indual Engneeng, Ian Unvey of Scence and Technology, Tehan, Ian Receved 3 Nov., 8; eved Jan., 9; acceped 9 Jan. 9 Abac Bach zng n dffeen plannng peod caegozed a a clacal poblem n poducon plannng, ha o many eac & heuc mehod have been popoed o olve h poblem, each of whch condeng vaou apec of he ognal poblem. The oluon obaned fom majoy e.g. MRP n h foma ha hee may be ome peod of dlene o each peod hould poduce a needed n dffeen adjacen peod If hee ae moe he one fnal ndependen poduc o be poduced n a facoy, h make he poducon plannng epeence ong vaaon n bach ze fo dffeen peod, whch poducon manage ae oppoed o hee popoed poducon plan In h pape, ome of he model ae popoed o olve h hocomng of he poducon plan o mooh he vaaon of bach ze and conequenly o mee he manage deal. Fnally all of he popoed model ae ued n a eal cae poblem and he be model noduced n ha cae. Keywod:Ideal Bach Sze, Maeal Requemen Plannng (MRP), Bach Szng, Poducon Smoohng;. Inoducon Poducon manage uually dee o poduce n a monoone ae fo dffeen peod [,7,9]. One of he pfall of he common poducon model ha do no pay aenon o h wllng a well [,7,4]. Conequenly he opmal oluon would be wh lo of vaaon n bach ze of dffeen peod Mo of he me, hee vaaon make mange no o accep he poducon plan. In ecen yea, vaou model ae popoed o olve h poblem and o afy manage Mo of hee model ae eekng a way o deemne an deal poducon level ha vaaon of bach ze would be a lle a poble n a naow band aound h deal level [5,,3]. Th band uually labeled a deal poducon band. Some of he model n he leaue aemp o foce a /ome dummy objecve() o he clac bach ze model, o becaue of he conflc beween dffeen goal of he model, hoe model ae o be olved by goal pogammng /game heoy, ec [6,8,,]. Ohe ed o oban he naowe deal band a poble [,3,4]. Few of hem amed o foce he model o oban an deal poducon band lmed o mamum poducon capacy. Fgue how an eample of wha mean a deal level and deal poducon band. The dahed lne n he deal band he deal poducon level. In fgue (), he foecaed demand (D ), bach ze * ( ) and moohed bach ze ) of =,,..., peod ae ploed. I noewohy ha a fundamenal aumpon ha value of demand n each peod gven o foecaed by a good mehod uch a eponenal moohng a well. The uppe do lne ndcae he mamum poducon capacy. Anohe fundamenal aumpon ha he mamum poducon capacy of all peod conan... Poducon Smoohng and JIT One of he opc n leaue he elaon beween JIT and bach ze moohng appoache Fo eample queoned whehe o no o ue bach ze moohng model va poducon n he famewok of JIT? If o, how h hould be done no o eceed fom he phloophy of JIT? In ho anwe, hould be condeed ha JIT need o have daa of demand of dffeen peod o be foecaed n a eaonable neval uch a week/ day/ hou, o o be aken fom cuome hemelve Now f demand daa have ong vaaon fom one peod o anohe *E-mal:mehdkamnaab@yahoo.com, Tel:

2 M Bahadoghol Ayanezhad e al. /Mulple Bach Szng hough Bach Sze Smoohng peod, o a lea one daa eceed he mamum poducon capacy, how hould he poducon manage plan he bach ze? Fo eample, conde he daa e of, 3, 8, 5,, 35, 38, 645, 7, 46, 58 and a he demand of poducon peod by he mamum poducon capacy of 55. A we know, one of he equemen o poduce n JIT famewok no o have ong vaaon n make demand Now fo olvng h dlemma and neang o he phloophy of JIT, ome of he ck could be aken. Fo eample, cuome demand could be me n mamum one peod delay. In he ohe wod, f he demand of a peod le han he mamum poducon capacy, ha peod hould poduce a JIT, bu n ohe cae, he dffeence beween demand and mamum poducon capacy could be poduced n he adjacen peod. And he am hould be no o ue he lae mehod a poble. All of hee ae becaue of hgh lackne co and low holdng co condon The hgh lackne co would be fo he mpoance/ aenon pad o cuome So hee cenao would depend on holdng/ lackne/ delay/ co, whch wll be condeed. Fuhe moe, facoe no poducng n he famewok of JIT, hould no be woed abou ung he popoed model Thee poblemace make u o ue mahemacal model nead of decon makng baed on mple condon o conde dffeen cenao of poducon plannng.. The Popoed Model Becaue of ndependency of he fnal poduc, a could be magned a plan wokng wh paallel poducon lne, popoed o mooh bach ze of each poduc ndependenly. Uually ha been obeved ha h pocedue make he fnal cumulave poducon plan o be a mooh a poble. I hould be noed ha befoe ung each model fo poducon moohng, effcency hould be condeed by vaou cea and hen a model whch he mo favoe by condeng all of cea hould be ued. So f ome of he model have weakne & engh n dffeen cea, a good mulple cea decon makng mehod hould be ued o deemne he fe model o demand daa. Model () a he ba of he ohe ha could be ued even a a complemen fo clac bach zng model Paamee & vaable of Model () ae a follow: vaable of bach ze n peod Δ vaable of deal band lengh plannng peod ( =,,.., T) Fg.. Ideal Poducon Band C D Mn. mamum poducon capacy demand of peod Δ C n Model () One fundamenal aumpon n all of model of h pape ha ummaon of all demand hould be equal o he ummaon of all poducon In he ohe wod, he conan D ) = Indcae h dea a well. Mo of he me f h conan no condeed n he model, he eul would be of no fndng any deal poducon band, becaue hee may be lle lo ale co ha no poducng wll be he

3 be acon. Model () could be conveed o Model () n ode o mamzng he numbe of moohed bach ze n he opmaly. Ma ( λ ). C n Mλ, λ {,} Jounal of Indual Engneeng (9)-7 Model () In Model (), M a bg pove numbe and λ a bnay vaable ha f he bach ze of peod eceed fom he deal poducon band, wll be, and f ele wll be zeo. In Model () aumed ha Δ a a paamee. An neeng wok o olve Model () n ode o oban Δ* and ung ha value a a paamee n Model (). Auho have epemenally obeved ha h pocedue make naowe deal band when dealng wh devaonal D. Model (3) yng o mnmze he mamum devaon of:. The dffeence of demand and poducon of each peod, and. The dffeence of poducon of neghbo peod One of he mo ccal apec n ung Model (3) ha no common ofwae package able o olve becaue of f conan, bu could be ealy conveed o Model (3-) by noducng new vaable and conan: Mn Δ. Ma n Mn Δ. y {, D } C C y = ( λ ) D D y, D ; =,, L, T M ( λ ) Mλ λ n, λ {,} Model (3) Model (3-) 3 Anohe model ha could be conuced accodng o Model (3) a Model (4) ha no decly eekng o oban he wdh of he deal band bu levelng devaon of he poduc level n all of peod fom deal level. Mn. { Ma )} C n Model (4) Th model could be conveed o he followng Model (4-): Mn. Δ C n Model (4-) Model (5) he genealzed fom of Model (3), (4) whch need le numbe of conan/ vaable: Mn. { Ma, D )} C n Model (5) Th model could be conveed ealy o Model (5-): Mn Δ. D C n Model (5-)

4 M Bahadoghol Ayanezhad e al. /Mulple Bach Szng hough Bach Sze Smoohng Model (6) a mue of pevou one and yng o mamze he followng mulaneouly:. The numbe of me ha he dffeence of poducon n wo adjacen peod le han he wdh of he deal band, and. The numbe of me ha he dffeence of poducon and demand of each peod le han he wdh of he deal band. Ma. C γ M ( λ λ ) D γ, Δ n ( λ )( λ ) γ Mλ γ Mλ ; λ, λ {,} Model (6) Model (6) could be olved n anohe way: (a) agnng value o γ, Δ befoe unnng he model and he olvng he model, (b) agnng an abay value o γ and agnng he be obaned value of Δ by olvng Model () o (5). Afe olvng each/all of Model () o (6) ung daa of demand of peod, hee would be ome of poducon plan by dffeen wdh of deal band If b he un co of hoage and h condeed a he un holdng co, he oal co (TC) of each poducon plan could be elaboaed a follow: D b f D C = D h ele TC = C In above-menoned fomula C he coepondng co of peod f hee wa no e up co Now f elaboang he andad devaon of each plan, hee would be a mulple-cea decon makng poblem n ode o fnd he be poducon plan, whee choce ae he plan obaned fom olvng he model and cea ae: Sandad devaon of λ n each plan, whch he le andad devaon, he le devaonal plan and he bee plan, Wdh of poducon band, whch he naowe band he mo waned, Toal co of each poducon plan, whch he mnmal he be So a good decon makng mehod hould be ued o olve h poblem. Auho hemelve popoe TOPSIS becaue of mplcy fo h pupoe. Afe dong o, he deal poducon band could be lluaed a follow: Sep. Sep. Sep 3. By condeng he obaned eul, elaboae he coepondng value of Δ, By aveagng he bach ze n dffeen peod whch ae obaned fom he be model -ha deemned pevouly- and noduce h value a deal. Plo a hozonal lne of deal and deal.5δ a he uppe bound and deal -.5Δ a he lowe bound of he deal poducon band epecvely. 3. Numecal Eample In Table (), demand daa of an auomoble facoy n an epecal ype fom Januay o Decembe 9 ae foecaed. In h able, a he ndcao of plannng peod (=,,,T), D a he demand of peod and Ma Level a he mamum poducon capacy. Then f aumed ha b=3$ and h=$ we would have a decon ma a povded n able (), whee SD he ndcao of andad devaon and Δ he ndcao of he wdh of poducon band coepondng o each poducon plan. Model and Model (6) ae olved by dffeen value of Δ ha ae hown a - o - and 6- o 6-6 n Table () and (3) epecvely. Table Daa of plannng peod Peod () A B D C D E A Ma B Level C D E

5 Jounal of Indual Engneeng (9)-7 Table Decon ma obaned fom olvng 6 moohng model Model A B TC C D E A B SD C D E A B Δ C D E Now fo olvng h mulple cea decon makng poblem by TOPSIS, fly he pove deal oluon (PIS) and negave deal oluon (NIS) hould be deemned fom he decon ma of able () and he coodnae could be elaboaed ealy a follow: Each coodnae of PIS he be value of each ceon among poducon plan accodng o he ognaly of ha abue. Each coodnae of NIS he wo value of each ceon among poducon plan accodng o he ognaly of ha ceon. Then by condeng value of TC, SD and a he e of cea, clea ha all of TC, SD and ae of co ype. Theefoe PIS would have he malle coodnae, and NIS would have he lage coodnae among dffeen poducon plan Then clea ha: PIS(A) ={TC PIS = 96, SD PIS =.668, Δ PIS =.8} NIS(A)={TC PIS =48, SD NIS =5.434, Δ NIS =8} PIS(B)={TC PIS = 8, SD PIS =.797, Δ PIS =.8} NIS(B)={TC PIS =38, SD NIS =5.877, Δ NIS =5} PIS(C)={TC PIS = 8, SD PIS =.668, Δ PIS =.8} NIS(C)={TC PIS =6,SD NIS =4.939, Δ NIS =} PIS(D)={TC PIS = 8, SD PIS =.54, Δ PIS =.8} NIS(D)={TC PIS =348,SD NIS =9.3, Δ NIS =3} PIS(E)={TC PIS = 76, SD PIS =.389, Δ PIS =.8} NIS(E)={TC PIS =48, SD NIS =7.57, Δ NIS =} Now he poducon plan obaned fom olvng each model a able () could ge coe by CL, CL a beng neghbo o PIS and NIS epecvely, and fnally could ge an aggegae coe a CL whch hown n able (3): CL = ( TC PIS TC ) ( SDPIS SD ) ( Δ PIS Δ CL = ( TC NIS TC ) ( SDNIS SD ) ( Δ NIS Δ CL CL = CL CL I clea ha CL would aggegaecl, ) ) CL uch ha he moe CL, he moe favoe ne of he poducon plan. So could be obeved fom able (3) ha he poducon plan obaned fom model 6-, 5, 6-, 5 and 6- ae he be one fo poduc A, B, C, D and E epecvely a hey have been eleced by TOPSIS a he mo favoe poducon plan, whch he bach ze ae hown n able (4). Fgue o 7 how he gaphcal eul of ung he popoed appoach fo deemnng he mo favoe bach ze moohng model. Table 3 Value of CL of each poduc planned by each of 6 model Model A B C D E Poduc 5

6 M Bahadoghol Ayanezhad e al. /Mulple Bach Szng hough Bach Sze Smoohng Table 4 Bach ze of each poduc hough plannng peod Peod () A Poduc B C D E oal poduc oal demand Peod Fg.. Plo of oal poduc and oal demand n plannng peod Poducon of A Demand of A Peod Fg. 3. Plo of poducon plan of A veu demand of A n plannng peod Poducon of B Demand of B Peod Fg. 4. Plo of poducon plan of B veu demand of B n plannng peod Poducon of C Demand of C Peod Fg. 5. Plo of poducon plan of C veu demand of C n plannng peod Poducon of D Demand of D Peod Fg. 6. Plo of poducon plan of D veu demand of D n plannng peod Poducon of E Demand of E Peod Fg. 7. Plo of poducon plan of E veu demand of E n plannng peod 4. Concluon A ha been hown n h pape, hee ae o many poenal poducon plan obaned fom olvng numeou mahemacal model, whch ae eekng dffeen goal The man goal of h pape wa o model he wllngne of poducon mange o poduce a monoone a poble. Fuhemoe a became clea, hee model could be ued fo dealng wh eaonal demand daa o hey could be eended ung fuzzy deal band becaue of vaguene embedded n he meanng of deal. Refeence [] [] M.A. Azoza, M.C. Bonney, Daggegang aggegae poducon plan o mooh wokload vaaon; Engneeng Co and Poducon Economc, vol. 9, 5-3, 99. J.W.M. Beand, H.P.G. Van-Oojen, Inegang maeal coodnaon and capacy load moohng n mul-poduc mulphae poducon yem; Inenaonal Jounal of Poducon Economc, vol. 47, -, 996.

7 Jounal of Indual Engneeng (9)-7 [3] [4] [5] [6] [7] [8] [9] A.K. Chakavay, A. Shub, Balancng med model lne wh npoce nvenoe; Managemen Scence, vol. 3, 6 74, 985. E.M. Dael, R.F. Cohe, Aembly lne equencng fo model m; Inenaonal Jounal of Poducon Reeach, vol. 3, , 975. E.M. Dael, M. Rabnovch, Opmal plannng and chedulng of aembly lne; Inenaonal Jounal of Poducon Reeach, vol. 6, , 988. F.Y. Dng, L. Cheng, A mple equencng algohm fo medmodel aembly lne n ju-n-me poducon yem; Opeaon Reeach Lee, vol. 3, 7 36, 993. K.A. Dowland, Nue chedulng wh abu each and aegc ocllaon; Euopean Jounal of Opeaon Reeach, vol. 6, , 998. R. Ehhad, Fnhed good managemen fo JIT poducon: New model fo analy; Inenaonal Jounal of Compue Inegaed Manufacung, vol., 7-5, 998. G.J. Salegna, P.S. Pak, Wokload moohng n a boleneck job hop; Inenaonal Jounal of Opeaon and Poducon Managemen, vol. 6, 9-, 996. [] [] [] [3] [4] M. Yavuz, S. Tufekc, Analy and oluon o he ngle-level bach poducon moohng poblem; Inenaonal Jounal of Poducon Reeach, vol. 45, , 7. M. Yavuz, E. Akcal, Poducon moohng n ju-n-me manufacung yem: A evew of he model and oluon appoache; Inenaonal Jounal of Poducon Reeach, vol. 45, , 7. M. Yavuz, S. Tufekc, Dynamc pogammng oluon o he bachng poblem n ju-n-me flow-hop; Compue and Indual Engneeng, vol. 5, 46-43, 6. M. Yavuz, S. Tufekc, A bounded dynamc pogammng oluon o he bachng poblem n med-model ju-n-me manufacung yem; Inenaonal Jounal of Poducon Economc, vol. 3, 84-86, 6. M. Yavuz, E. Akcal, S. Tufekc, Opmzng poducon moohng decon va bach elecon fo med-model ju-n-me manufacung yem wh abay eup and poceng me; Inenaonal Jounal of Poducon Reeach, vol. 44, 36-38, 6. 7

8 M Bahadoghol Ayanezhad e al. /Mulple Bach Szng hough Bach Sze Smoohng 8