AN MIP APPROACH TO THE U-LINE BALANCING PROBLEM WITH PROPORTIONAL WORKER THROUGHPUT. Reyhan Erin Magna PowerTrain Troy, Michigan

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1 AN IP APPROACH TO THE U-LINE BALANCING PROBLE WITH PROPORTIONAL WORKER THROUGHPUT Ryhan Erin agna PowrTrain Troy, ichigan USA Andrs L. Carrano Dpartmnt of Industrial and Systms Enginring, Rochstr Institut of Tchnology, Rochstr, NY USA. oiss Sudit Cntr for ultisourc Information Fusion Dpartmnt of Industrial and Systms Enginring. Univrsity at Buffalo. Buffalo, NY USA. Abstract On of th maor challngs facd by manufacturing companis is to rmain comptitiv in dynamic nvironmnts, whr fluctuations in customr dmand and production rats rquir systms capabl of adapting in a practical and conomical way. A U-shapd production cll is considrd on of th most flxibl dsigns for adapting th workforc lvl to varying conditions. Howvr, r-balancing fforts ar tim consuming and oftn rquir a nw work allocation and lin dsign. In this papr, a two-stag IP modl to dtrmin th bst cll dsign undr varying workforc lvls is proposd. Th modl sks to maintain proportionality btwn throughput and th numbr of workrs. Computational xprimnts considring various lin configurations (up to 9 stations) and workloads (up to 79 tasks) ar prformd. Th rsults show th proposd algorithm provids xcllnt rsults for all small and mdium siz problms addrssd in this study, as wll as for crtain configurations of larg problms. This approach can b usd to gnrat lookup tabls of lin dsigns to hlp with quick rallocation of workr assignmnts on th shop floor and with minimal disruption.

2 . Introduction U-shapd production lins, whr workrs handl on or mor machins, ar widly usd in cllular manufacturing and lan production systms. Th main advantags driv from a flxibl lin with cross-traind workrs capabl of adapting to changs in dmand and production pac. In lan systms, production lins typically striv to mt a prdtrmind, dmand-drivn cycl tim dnotd as takt tim. This cycl tim is achivd through a lin balancing xrcis that involvs grouping tasks into stations and assigning workrs to tnd on or mor stations. Whn th dmand rat or workforc lvls chang, th lin balancing is rvisitd and modifid with nw task groupings and workr assignmnts. This procss of adapting th production rsourcs to th changing dmand pattrns is known as Shoinka []. On th othr hand, th implications of not appropriatly addrssing lin balancing span across a wid rang of issus, from production control (.g. ovrproduction, idl workrs, tc.) to accounting (.g. variabl dirct labour cost pr unit, tc.). Currnt r-balancing practics in industry oftn involv trial-and-rror approachs. Such itrativ approachs, howvr, can b tdious and tim consuming. Additionally, most lin balancing algorithms for U-shapd lins proposd in litratur tnd to addrss stabl nvironmnts and assum littl or no variation with rspct to workforc lvls. Som fforts hav addrssd workforc issus but hav typically sought to minimiz th numbr of workrs for a givn workload condition [2]. Ths algorithms may not b practical in a dynamic nvironmnt whr vry frqunt, daily somtims, U-lin rdsigns ar ndd. Th rasons for such frqunt rdsigns ar typically found whnvr a volatil dmand nvironmnt is coupld with an unstabl workforc in turn a consqunc of absntism and high lvls of tmporary workrs. This papr was motivatd by th nds of a tir-on automotiv supplir with a similar nvironmnt. Th obctiv of this papr is to dvlop a modl for dsigning and balancing U- shapd production lins that maximizs th linarity of workr throughput. Such modl aims to produc th dsign that is most conduciv to lin balancing whil facing a varying numbr of availabl lin workrs. 2. Litratur Rviw Extnsiv rsarch has bn don on lin balancing for traditional production lins ovr th past four dcads. Howvr, rsarch fforts on U-shapd lins startd with th widsprad adoption of JIT and lan manufacturing concpts and philosophy in th arly 990s. Prvious rsarch mainly considrs two typs of lin balancing problms: (i) minimization of th rquird numbr of stations for a fixd cycl tim or, (ii) minimization of th maximum cycl tim for a fixd numbr of stations.

3 iltnbrg and Wingaard [3] introducd th U-lin assmbly lin balancing problm (ULB) whr assignmnts can b don on both sids of th lin. In this work, a dynamic programming modl was dvlopd for small siz (up to tasks) problms to minimiz th rquird numbr of stations for a givn cycl tim. For mdium siz problms, a maximum rankd positional wight huristic procdur was proposd. This work, howvr, did not considr walking tims or crossovr issus on walk-paths. Svral fforts hav sinc addrssd th ULB with diffrnt approachs and various dgrs of succss. Urban [4] usd an intgr programming modl with a branch and bound prliminary sarch to solv problms with mor than 2 tasks. Anblit and Wainwright [5] usd a gntic algorithm approach with six diffrnt task assignmnt mthods on largr problms (up to tasks). Scholl and Klin [6] dvlopd a branch and bound procdur for simpl assmbly lin balancing. In this work, modls with up to 297 tasks wr solvd with optimal or bst solutions. Erl t al. [7] proposd solving th ULB problm with a simulatd annaling algorithm. artinz and Duff [8] proposd a gntic algorithm approach to improv rsults obtaind from 0 diffrnt huristics. Gokcn and Agpak [9] proposd a goal programming approach for simpl U-lin balancing problms. In this work, up to 30 tasks with conflicting goals, including minimization of th numbr of work-stations, cycl tim and th numbr of tasks, wr considrd. Thr is archival litratur on production lins that hav considrd walking tims, waiting tims and walk-paths. Nakad and Ohno [0] dvlopd a procdur to calculat cycl tims undr optimal workr allocation scnario whil considring walking and waiting tims. Nakad and Ohno [] proposd a modl for dtrministic walking and procss tim scnarios. In this papr, th minimum numbr of workrs undr a givn cycl tim is first dtrmind. Thn, an optimal workr allocation with a minimum numbr of oprators is obtaind. Stockton t al. [2] modlld flxibl walk cycls that includd walking tims using gntic algorithms. Th obctivs wr to minimiz th numbr of oprators and to rduc th smallst oprator cycl tim. Shwchuk [2] addrssd th workr allocation problm for U-shapd lins with obctivs of minimizing th numbr of workrs whil maximizing full work. Th proposd modl incorporats circular walking paths whr crossovrs ar not prmittd. Th maority of th xisting rsarch on lin balancing has bn dvlopd for stabl production nvironmnts whr th rbalancing frquncy is vry low. Dynamic nvironmnts, du to dmand volatility and workforc instability, rquir ngaging on mor frqunt lin balancing of U-lins, thus th chosn approach should b asy and conomical. This work addrsss this gap.

4 3. Production Systm Dscription Figur illustrats a simpl u-shapd production lin. ultipl stations ar closly locatd in a u-shap to form a production cll. Tasks ar allocatd to stations according to prcdnc rlationships. Typically, onc tasks ar allocatd to stations, thn assignmnts of workrs ar stablishd. ultipl stations can b assignd to th sam workr as long as crossovr and cycl tim constraints ar satisfid. Th trm walk-path is usd to dfin th ara of rsponsibility of ach workr. Figur rprsnts a U-shapd systm with 7 stations and 4 workrs (A, B, C and D). With rprsnting th numbr of stations and N rprsnting th numbr of workrs, th systm in Figur can b dscribd as (7, N4). In this Figur, workr A is rsponsibl for stations and 7, workr B is rsponsibl for stations 2 and 3, workr C is rsponsibl for stations 4 and 5, whil workr D only tnds to station 6. Figur : U-shapd production lin (7, N4) In a U-shap cll, parts flow squntially through th stations in th systm. Aftr procssing at station is compltd, workr A dlivrs th part to workr B for procssing at station 2. Aftr that is compltd, and sinc workr B is rsponsibl for station 3 as wll, h thn walks th assmbly down to station 3 to prform furthr procssing. Aftr dlivring th part to workr C at station 4, workr B thn rturns to station 2. A crossovr occurs whn a workr, by dsign, systmatically intruds into th work zon (as dfind by th walk-path) of anothr workr. This would occur if, for xampl, in Figur, workr A wr in charg of stations and 3, whil workr B was in

5 charg of stations 2 and 6. Crossovrs ar undsirabl for svral rasons, including safty and productivity considrations. 4. athmatical modl As mntiond bfor, th obctiv of this modl is to dtrmin th cll configuration that, for a variabl numbr of workrs, allows for maximum linarity of workr throughput. This is accomplishd with a two-stag IP modl. Th first stag sks to allocat th tasks with th obctiv of balancing th workload across th stations. In th scond stag, workrs ar assignd to stations with th goal of balancing walk-path tims (i.. task tim + walking tim). Th scond stag is run for diffrnt lvls of workforc, from 2 workrs to - (a practical maximum) and itratd until th proposd linarity mtric is accptabl. During this itrativ procss, th obctiv valu is rlaxd (stag ) and th oprators ar rassignd (stag 2), until th targt linarity is achivd. Although not part of th IP modl, a third stag is usd to calculat th linarity of th rsults. Figur 2 shows th logic of th modl. Figur 2: Flowchart of th two-stag modl

6 In this modl, th following assumptions ar mad: ) Task procssing tims and workr walking tim btwn stations ar considrd dtrministic. 2) Workrs ar cross-traind on all tasks. 3) Th production protocol is that of on-pic flow for a singl modl. 4) Oprators ar not allowd to chas ach othr or to shar stations. 5) achin brakdowns ar not considrd. 6) Thr is no machin waiting tim. 7) Thr is an infinit supply of parts in front of th first station. Also, it is assumd that th stations ar locatd in a grid arrangmnt and that it taks t units of tim to travl to adacnt stations (in th sam row) or dirctly across. This assums qual and constant travl spd with no significant acclration/dclration. It was also assumd that it would tak twic that amount of tim (2t) to rach a station locatd diagonally across. This assumd a worst cas scnario (sinc 2 t > 2 t ). Ths travl assumptions ar mappd to othr stations locatd farthr away. Figur 3 dpicts th travl tims for a u-cll configurations with vn numbr of stations. Figur 3: Travl tims to stations (vn numbr of stations, 0) 4. Stag- mathmatical modl Th modl in this stag sks to distribut th tasks to stations without initially considring workr assignmnts. Th rquird inputs for this stag ar: (i) numbr of

7 stations, (ii) numbr of tasks, (iii) prcdnc rlationships, and (iv) task procssing tims. Stag- nomnclatur T Total numbr of tasks Total numbr of stations i Indx for tasks ( i, 2,, T) Indx for stations (, 2, 3, ) D i Standard tim to procss th task i (dtrministic) P (i, h) A st of prcdnc of tasks, P{(i, h) / task i must b compltd bfor task h} Stim Sum of all th task procssing tims assignd to station axf aximum of th tim diffrnc btwn station tims. (Stim ) Stag- dcision variabls To stablish th allocation of th tasks to th stations, dfin Stag- obctiv function: X i, if task i is assignd to station 0, othrwis Th obctiv function in this stag sks to balanc th workload among stations by minimizing th maximum diffrnc btwn station tims (Stim ) pairs. inimiz Z axf Stag- constraints: [C] Each task can b assignd to only on station.

8 Xi i... T [C2] All stations must hav an assignmnt. Dummy stations ar not allowd. T i Xi... [C3] This constraint nsurs that prcdnc rlations ar maintaind. P {(h,i) / task h must b compltd bfor task i}. If task h is assignd to station, task i can only b assignd to station k, k { k / k, +, + 2,... }. Xh Xik k ( h, i)... P [C4] Th following constraints ar usd to compar th tims (Stim ) btwn stations. Th variabl axf gts assignd to th largst valu of th gaps. Th two lft trms in th constraints ach giv th total procssing tim at station and g. axf is consquntly minimizd through th obctiv function. T T Di * Xi i i Di * Xig axf...( g ( +)... ) ( a) T T Di * Xig i i Di * Xi axf...( ) g ( +)... ( b) Stag- output: Stim T Di * Xi... i

9 4.2 Stag-2 athmatical odl This stag is usd to crat walk-paths by allocating workrs to th stations and with th aim of balancing walk-path tims. A walk-path tim is th sum of th task procssing plus walking tims assignd to th sam workr. Th modl for this phas considrs two maor constraints whn dvloping th walk-paths: (i) path crossovr and, (ii) balancd workloads. Workrs cannot b assignd to th stations that rquir crossing th walk-path of anothr workr. In ordr to gnrat a masur of throughput linarity, Stag-2 is rpatd for diffrnt numbrs of workrs :, N, N { /,2,3,... }. Stag-2 nomnclatur N Total numbr of workrs Total numbr of stations Indx for workrs (i, 2,, N) Indx for stations (, 2, 3.., ) Stim Total procssing tim at station L i Walking tim btwn station i and. LS max An input usd to prvnt crossovrs. LS max / 2 / 2,, if if is vn numbr is odd numbr HS max An input usd to prvnt crossovrs. HS max ( / 2 ) / 2 +, +, if if is vn is odd numbr numbr Stag-2 dcision variabls: ) This variabl is usd to rprsnt station assignmnts to workrs. Y, if station is assignd to workr 0, othrwis

10 2) This variabl is usd to dtrmin th walk-path for ach workr. Wh, if station and station h ar assignd to sam workr 0, othrwis 3) Variabl I h is rfrrd as immdiat station. It is usd to dtrmin th squnc of stations through which workr movs in his walk-path. I h, if station and h ar assignd to sam workr, and if workr movsto station h immdiatly aftr station 0, othrwis 4) Variabl FS dnots th first station visitd in th walk-path. FS, if station is th first station in th walkpath of workr 0, othrwis 5) Variabl LS h dnots th last station visitd in th walk-path. LS h, if station h is th last station in th walkpath of workr 0, othrwis 6) Variabl B h is cratd to dtrmin th nd of th cycl. If B h, thn is th last station and h is th first station in th squnc. B h, if station and h ar assignd to sam workr, and if workr gos back from th last station() to th first station (h) 0, othrwis 7) Wtim Sum of all th task procssing tims and walking tims btwn stations assignd to workr, also rfrrd as walk-path tim of workr. 8) maxt aximum of walk-path tims Stag-2 obctiv function: Th obctiv function of th scond stag sks to balanc walk-path tims by minimizing th maximum walk-path tim.

11 inimiz Z 2 maxt Stag-2 constraints: [C6] Each station can only b assignd to on workr. N Y... [C7] On workr can b assignd to many stations if th numbr of workrs is lss than th numbr of stations. Th following constraints dtrmin th stations in th walk-path of ach workr. + W g Y g + Y g...( ( + ) N ) (a) W g Y g g...( ( + ) N ) (b) W g Y g...( ( + ) N ) (c) [C8] Th following constraints prvnt crossovrs. Nithr diagonal nor horizontal crossovrs ar allowd in this modl. Constraint (C8-a) prvnts diagonal crossovrs (across th main aisl in th cll) whil constraints C8 (b, c) prvnt horizontal crossovrs (along th row of stations on th sam sid of th cll). 3Y Y g Y hf + Y rf h... ( + )...( g )... N g r f ( + )... ( g + ) N, f (a)

12 ( g )*( Y + Y g ) ( g) k ( + ) Y k g...( LS... N max ( + 2 )... LS 2) max ( b) ( g ) *( Y + Y g ) ( g ) Yk k ( + ) g HS... N max...( ( + 2 )... 2) (c) [C9] Th following constraints ar usd to dtrmin th squnc of stations in ach walk-path. Constraints C9 (a, b, c) dtrmin th nxt immdiat station in th squnc for workr (if multipl stations ar assignd to him). Constraints C9 (d,, f) ar usd to dcid th last station in th squnc to complt on cycl. Constraints C9 (g, h, i) ar cratd to dtrmin th first station in th squnc. Rturn tim from last station to first station is calculatd through constraints C9 (, k, l). I g W g ( g ) h ( + ) W h g...( ) ( + ) N (a) W g I g... g ( + ) N (b) ( g ) h ( + ) W h *( I g )...( ) g ( + ) N (c)

13 ... (d)... ) * ( ) * ( N Y LS g g g ()... N LS (f)......n Y LS )* ( ) * ( N Y FS g g g (g)... N FS (h) N Y FS (i)... )...( 2... N g LS FS B g g + + () )...( g N FS B g g (k)

14 B g LS... N 2... g...( ) (l) [C0] This constraint is dvlopd to balanc walk-path tims by driving workr tim to a minimum. Th trms at th lft sid of th constraint rprsnt th task procssing tims, th walking tims from station to station, and th rturn tim aftr compltion of on cycl. Th sum of all thr givs th ovrall walk-path tim for ach workr. Stim ( ) Y + Lg * Ig + Lg * Bg maxt 2 g ( + ) g *... N Stag-2 output: Wtim ( ) () Stim * Y + Lg * Ig + Lg * Bg... g ( + ) 2 g N 5. Computational xprimnts Th data usd to tst this modl was obtaind from two distinct sourcs. Th first-tir automotiv supplir, which motivatd this work, providd on st of data. Th othr sts wr obtaind from archival litratur availabl in th Hompag for Assmbly Lin Optimization Rsarch [3], which hosts a collction of data sts usd in simpl-lin and U-lin balancing problms. In this papr, a total of 8 diffrnt data sts wr usd. Each data st consistd of a collction of task procssing tims and prcdnc diagrams. Tabl prsnts a classification of th data sts. All ight data sts wr tstd with small and mdium siz configurations ( 6), whil four data sts wr usd to tst larg configurations (>6). For practical rasons, th xtrm points in th numbr of workrs (i.. N and N) wr xcludd from th analysis. Total of 08 diffrnt configurations wr tstd for all lvls of workforc btwn [2, -], for a total of 008 xprimnts. All xprimnts wr conductd on a 3.20GHz procssor with GB of RA. Th proposd modl was solvd with OPL Studio using a CPLEX program. A branch and cut modl was usd in sarching for an optimal solution.

15 Tabl : Summary of data sts (numbr of tasks in parnthsis) Small / dium Siz Larg Siz (5-6 stations) (7-9 stations) itchll (2) Sawyr (30) Roszig (25) Warnck (58) Sawyr (30) Tong (70) Kilbrid (45) Automotiv Supplir (79) Warnck (58) Tong (70) W-ag (75) Automotiv Supplir (79) 5.. Linarity masurmnt With th output from th modl, a scattr plot btwn throughput and th numbr of workrs is dvlopd. A linar rgrssion with a bst fit lin that intrcpts at th origin is obtaind and its corrsponding cofficint of dtrmination R 2 thn calculatd. In this work, R 2 is usd to masur th linarity of throughput pr workr. 6. Rsults and discussion Tabl 2 prsnts th computational rsults. A cursory xamination of th summary data suggsts that th proposd procdur prforms wll within th boundaris of th xprimnt.

16 Tabl 2: Computational rsults (R 2 ) Numbr of Numbr of Numbr of Numbr of Problm Computational rsults (R 2 ) stations () data sts tasks (T) workrs (N) siz Avrag Rang S/ S/ S/ S/ S/ S/ S/ S/ S/ S/ S/ S/ L L L All th scnarios in th small/mdium siz rang wr solvd with high avrag linarity (R 2 є [ ]) whil th larg siz scnarios prsntd a satisfactory cofficint of dtrmination (R 2 є [ ]). Howvr, it can b obsrvd that ovrall prformanc of th algorithm dgrads with th siz of th problm (i.. T and ). Basd on ths rsults, it appars th algorithm works vry wll for all small and mdium siz problms and for som configurations of larg problms. Tabl 3 shows th workload assignmnts obtaind from th modl and with th providd data sts. Th prcntags shown in th tabl dpict of th amount of work to b allocatd to ach station such that, aftr accounting for walking tims, th linarity (R 2 ) falls in th rangs shown in Tabl 2. This allocation can b usd as a gnral guidlin for a prmannt work distribution among stations whn th linarity of th throughput pr workr is paramount.

17 Tabl 3: Bst solutions for work allocation (prcntags by station) Numbr of Stations Finally, onc all tasks ar allocatd, th various workr assignmnts can b dpictd for all scnarios of workforc lvls btwn 2 and Conclusions A 2-stag IP modl is dvlopd for U-shapd production lins for nvironmnts with unstabl workforc. Th modl sks to optimiz th linarity of throughput pr workr whil maintaining th lin balancing within accptabl limits. Th algorithm is abl to solv problms up to 79 tasks and 9 stations. Th mthod tabulats th bst work allocation by station that is conduciv to both high throughput-workr linarity and lin balancing. Tabls of lin dsign for varying numbr of stations and workforc lvls can b drivd. Ths look-up tabls can b vry usful on th shop floor, whr a quick rdsign rspons may b ndd to adapt to daily changs in workforc lvls.

18 Rfrncs [] Y. ondn. Toyota Production Systm. 2 nd Ed. Industrial Enginring and anagmnt Prss, Institut of Industrial Enginrs. Norcross, GA [2] J.P. Shwchuk. Workr allocation in lan U-shapd production lins. Intrnational. Journal of Production Rsarch (2008). 46(3), [3] J. iltnbrg, J. Wingaard. Th U-lin balancing problm. anagmnt Scinc (994). 40(0), [4] T. Urban. Optimal balancing of U-shapd assmbly lins, anagmnt Scinc (998). 44(5), [5] D.A. Anblit, R.L. Wainwright. Applying gntic algorithms to th U-shap assmbly lin balancing problm. Procdings of th 998 IEEE Confrnc on Evolutionary Computation, ICEC. (998). Anchorag, Alaska, ay [6] Scholl, R. Klin. ULINO: Optimally balancing U-shapd JIT assmbly lins, Intrnational Journal of Production Rsarch (999). 37(4), [7] E. Erl, I. Sabuncuoglu, B.A. Aksu. Balancing of U-typ assmbly systms using simulatd annaling. Intrnational Journal of Production Rsarch (200). 39(3), [8] U. artinz, W. Duff. Huristic approachs to solv th U-shapd lin balancing problm augmntd by gntic algorithms. IEEE Systms and Information Enginring Dsign Symposium (2004).Charlottsvill, Virginia, April [9] H. Gokcn, K. Agpak. A goal programming approach to simpl U-lin balancing problm. Europan Journal of Oprational Rsarch (2006), 7(2), [0] K. Nakad, K. Ohno. Stochastic analysis of a U-shapd production lin with multipl workrs. Computrs & Industrial Enginring (997), 33 (3-4), [] Nakad, K. and Ohno, K. An optimal workr allocation problm for a U-shapd production lin. Intrnational Journal of Production Economics (999), 60-6,

19 [2] D.J. Stockton, J. Ardon-Finch, R. Khalil. Walk cycl dsign for flxibl manpowr lins using gntic algorithms. Intrnational Journal of Computr Intgratd anufacturing (2005), 8(), [3] S. Armin. Hompag for assmbly lin optimization rsarch [onlin]. URL: [accssd arch 2007].

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