One way to organize workers that lies between traditional assembly lines, where workers are specialists,

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1 MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 7, No. 2, Sprg 2005, pp ss ess forms do /msom INFORMS Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les Joh J. Barthold III School of Idustral ad Systems Egeerg, Georga Isttute of Techology, Atlata, Georga , Doald D. Eseste Graduate School of Busess, Uversty of Chcago, Chcago, Illos 60637, Oe way to orgaze workers that les betwee tradtoal assembly les, where workers are specalsts, ad craft assembly, where workers are geeralsts, are bucket brgades. We descrbe how oe frm used bucket brgades as a termedate strategy to mgrate from craft assembly to assembly les. The adopto of bucket brgades led to a arrowg of tasks for each worker ad thus accelerated learg. The creased producto more tha compesated for the tme lost whe workers walk back to get more work, whch was sgfcat ths mplemetato. To uderstad the trade-offs mgratg from craft to assembly les, we exted the stadard model of bucket brgades to capture had-off ad walk-back tmes. Key words: bucket brgades; assembly le; work sharg; dyamcal system; self-orgazg system Hstory: Receved: August 27, 2002; accepted: November 1, Ths paper was wth the authors 16 moths for 2 revsos. TUG s a compay that assembles about 10 models of dustral tractor of the type commoly used at arports to pull luggage tras. Itally, the compay was small ad prvately held but, whe the fouder retred, t was acqured by a professoal maagemet group whose goal was to grow the compay aggressvely. Ideed, the ew owers had secured some large orders ad, to meet sales commtmets, had to crease producto sgfcatly ad quckly. Ufortuately, the way whch maufacturg was orgazed at TUG made t dffcult to crease producto. Each tractor was bult by a sgle perso (or, for the larger models, a two-perso team) from start to fsh. Assembly bega after a fork-lft truck delvered a ege ad a locker of parts to the work area ad the placed a ew tractor frame o four jack stads. Wth a day or two, depedg o the model, the tractor was assembled ad drve off to the pat booth whle work was begu o aother. Uder ths style of assembly, each worker had to (1) kow how to buld a tractor from start to fsh, ad (2) have a complete set of expesve tools. Requremet (1) meat that t took moths for a ew employee to become profcet. TUG tred to hre aggressvely, but requremet (2) arrowed the pool of avalable labor because each worker was expected to provde hs ow tools. Furthermore, uder pressure to crease producto, workers were pressed to so much overtme that the compay suffered a 60% turover rate. As a result, TUG faced a crss: How to crease producto quckly whe t was so hard to tra ad reta workers? TUG was tryg to covert to a tradtoal assembly le the hopes that specalzato of labor would make t easer to tra ew workers as well as relax the tool requremet. TUG had worked for almost a year to buld specalzed jgs, hosts, coveyors, ad other materal-hadlg equpmet for evetual coverso to a assembly le, but ths process took some of ther best labor away from assembly. To further complcate matters, there was stll o formalzato of task prmtves, o work cotet models, ad o work stadards o whch to base ay sort of tradtoal assembly le. I addto, t was estmated that t would take moths to documet assembly of eve the smplest tractor. TUG was dauted: They 121

2 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les 122 Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS had to crease producto but the oly way forward seemed to be to shut dow producto for moths whle they documeted work ad reorgazed assembly. Wth a few hours, we coverted the maufacturg of ther most popular tractor to a bucket brgade assembly le, whch has prevously bee show to be self-balacg (Barthold et al. 1999, 2001; Barthold ad Eseste 1996, 1999). The self-balacg propertes of bucket brgades had may useful cosequeces for TUG. Frst, because the le s self-balacg, formal task descrptos or work stadards are uecessary ad so the bucket brgade le was easy to set up. Moreover, self-balacg arrowed the rage of tasks requred of each worker; ths specalzato resulted a mmedate crease producto. Furthermore, the self-balacg property of bucket brgades spotaeously reallocated work as ew hres mproved ther sklls. Fally, all ths creased the retamet of workers, ad so TUG was able to ejoy the beefts of trag ts workforce. We had to customze bucket brgades to work the specal crcumstaces at TUG, ad ths provded the occaso to exted the stadard model of bucket brgades, o whch all prevous mplemetatos had bee based. I the followg we descrbe the techcal detals of those chages, aalyze ther mplcatos, ad report performace at TUG; but the ew dea here s a maageral oe: that bucket brgades ca serve as a meas of smoothly mgratg a orgazato from craft maufacturg to assembly le. 1. Bucket Brgades at TUG Bucket brgades are a way of coordatg workers o a assembly le. As orgally coceved, each worker moves from work stato to work stato progressvely assemblg a product. Whe the last worker fshes, he walks back ad takes over the product of the worker mmedately upstream, who smlarly walks back to get more work, utl the frst worker returs to the startg work stato ad begs a ew product. Workers rema sequece because o passg s allowed. Elsewhere we have aalyzed varous models of bucket brgades ad show that uder very geeral codtos, f workers are sequeced from slowest to fastest alog the drecto of materal flow, the assembly le wll balace tself: That s, wthout maagemet terveto, the workers wll gravtate to a optmal dvso of the work ad the producto rate wll be maxmzed (Barthold ad Eseste 1996, 1999; Barthold et al. 2001, 2002). To work the TUG evromet we had to adapt bucket brgades some ew ad terestg ways. Most mmedately, t was ot possble to move tractors--process from work-stato to work-stato because ths would have requred expesve specal purpose materal-hadlg equpmet. Cosequetly, we requred that the workers move from tractor to tractor. At TUG, four work statos are clustered aroud a shared crae. The movemet of the four-worker bucket brgade s llustrated Fgure 1, where we have umbered the workers so that worker s faster tha worker 1. Each worker depedetly assembles a tractor. If, as s the ormal state of affars, worker s tractor s earer completo tha worker 1 s, the we would evetually observe the followg sequece of evets: Completo Worker 4 fshes hs tractor ad t s drve off the assembly area; Walk-back Worker 4 collects hs tools ad rolls hs tool box to the tractor of worker 3; Had-off Workers 3 ad 4 revew ad agree o what work remas to complete assembly of the tractor. Whe worker 4 uderstads exactly what Fgure 1 Successve Sapshots of the Four-Worker Bucket Brgade at TUG (a) (b) (c) (d) Notes. (a) Each of the four workers s assemblg a tractor. (The work statos are clustered so that they share a crae.) (b) Whe 4, the fastest, fshes hs tractor, t s drve to the pat booth ad he pushes hs tools to the stato of hs predecessor, 3, ad takes over 3 s tractor. (c) Worker 3 the moves back to take over 2 s tractor. (d) Worker 2 tur moves back to take over worker 1 s tractor. (e) Worker 1, the slowest, starts a ew tractor. (e) - 4

3 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS 123 remas to be doe, he assumes resposblty for the tractor ad takes over ts assembly; Remag walk-backs/had-offs Worker 3 collects hs tools, walks back to worker 2, ad takes over assembly of hs tractor, after whch worker 2 goes back to take over assembly of the frst tractor; Worker 1 moves hs tools to the empty area vacated by the tractor ewly completed by worker 4 ad begs assemblg a ew tractor. Ths process the repeats deftely, wth each completo followed shortly by the start of a ew tractor. There are three sgfcat dffereces ths behavor from that of the prevous models of bucket brgades (Barthold et al. 1999, 2001, 2002; Barthold ad Eseste 1996). Frst, a slower worker may pass a faster worker. Because each worker has hs ow work stato, the tractor of a slower worker may overtake that s, advace closer to completo tha the tractor of a faster worker, f the latter s delayed for some reaso. Secod, walk-back tme s sgfcat. At TUG, each worker must gather hs tools ad push hs tool box to the assembly area of hs predecessor, whch takes three to fve mutes. Thrd, had-off tme s sgfcat. The tme to had off work from oe worker to aother s sgfcat. Both workers must wat utl the dowstream worker assumes resposblty for the tractor, whch takes 10 to 20 mutes because the worker assumg resposblty must uderstad exactly what remas to be completed. 2. A Model of Bucket Brgades at TUG Because walk-back ad had-off tmes are sgfcat, ad passg s allowed, we must exted the ormatve model of bucket brgades to uderstad the effects of these ew ssues. Key questos clude: Wll the bucket brgade reta the property of selfbalace? How wll the tme lost to walk-back ad had-off chage the balace? How wll the lost tme reduce the producto rate? 2.1. Revsed Bucket Brgade Rules for TUG We beg by formalzg the revsed bucket brgade rules that each TUG worker must follow. Let the workers be umbered from slowest to fastest 1. Worker s the predecessor of worker + 1, ad worker +1 s the successor of worker. Uder bucket brgades each worker follows these rules. Work forward: Cotue to assemble your tem as quckly as possble. If you complete your tem the wat; f you are preempted by your successor, had off your work ad walk back to get more work. Wat: If you are the last worker (worker ), remove your fshed tem from the assembly area, the walk back to get more work as soo as o other worker s walkg back. If you are ot the last worker, the wat utl your successor takes over your tem ad the walk back to get more work. Walk back: If you are the frst worker (worker 1), walk back to the assembly area vacated by the last worker (worker ) ad beg assemblg a ew tem. If you are ot the frst worker, walk back to the work stato of your predecessor ad take over assembly of hs tem. I ether case beg to work forward. Note that oly the frst, slowest worker s allowed to beg work o a ew tractor, ad oly the last, fastest worker s allowed to tate a walk-back. Furthermore, the last worker ca beg a walk-back oly whe o other walk-back or had-off s progress. Ths dffers from prevous specfcatos of bucket brgade behavor, all of whch walk-backs ad had-offs were assumed to be stataeous. The watg rule s also ew. It restrcts behavor so that o more tha oe worker at a tme ca walk back to get more work. Ths cofers aalytcal tractablty, because t prevets degeerate stuatos such as whe worker + 1 walks back to take over the tractor of worker but fds worker the process of takg over the tractor of worker 1. The watg rule prevets ths by requrg that ay worker (except the last) who completes a tractor must wat utl the fastest worker completes hs tractor, ad, f a walkback or had-off s progress, the last worker must wat utl the frst worker begs a ew tractor. Ths has the potetal of wastg productve effort because t requres some workers to stad dle. However, uder ormal operato of a properly cofgured bucket brgade le, the watg rule wll ever be voked, so the aalytc tractablty t cofers costs us othg practce. Ideed, as we shall show later, vocato of the watg rule meas that there are too may workers the bucket brgade, ad so too may had-offs.

4 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les 124 Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS Fally, we have stated the bucket brgade rules wth more geeralty tha oe mght thk ecessary, so that they cotue to work eve whe thgs go wrog o the shop floor. For example, t s possble that oe tractor may pass aother; that s, the slower worker may advace assembly of hs tractor beyod that of the faster worker. Ths could happe f the faster worker expereced some dffculty durg assembly, such as advertetly strppg a thread. Passg was forbdde prevous models of bucket brgade, but ths was because of the cotext whch they operated, whch cluded assembly alog a sequece of work statos (Barthold ad Eseste 1996) or order-pckg a warehouse (Barthold et al. 2001). Our revsed rules esure that, after passg, workers are retured to ther proper sequece of slowest to fastest. We shall show that, eve though our model of work cotet does ot capture the possblty of radom dsruptos, the resultat behavor of the bucket brgade does deed compesate for such evets Adaptg the Normatve Model To uderstad the behavor of bucket brgade assembly les at TUG we here exted the ormatve model (Barthold et al. 1999, 2001). Ths model s a partcularly smple oe, cosstg of three dealzed codtos that guaratee the performace of bucket brgades. The smplcty of ths model makes t a useful bechmark to gude mplemetatos. A reasoable frst approach s to try to make the work evromet match the ormatve model as much as possble. Two of the three fudametal assumptos of the ormatve model are reasoably satsfed by TUG: Assumpto 1. Smoothess ad Predctablty of Work. The omal work cotet of the product s a costat (whch we ormalze to 1), ad cotuous. Ths s reasoable wheever assembly cossts of may tasks, oe of whch takes too much tme compared wth the total work cotet. Also mplct here s that each successve tem s assembled the same sequece, whch was ot strctly true at TUG but teded to be eforced by precedece costrats o assembly (for example, oe must mout the axles before the wheels). Uder ths assumpto, we ca represet the state of a partally completed tractor as a umber betwee 0 ad 1 gvg the fracto of stadard work-cotet that has bee completed. (Note that bucket brgades do ot requre kowledge of ether the stadard work cotet or the fracto completed for ay tractor.) Assumpto 2. Total Orderg of Workers by Velocty. Each worker = 1 s characterzed by a dstct, costat work velocty v. Ths remas, our experece, commo assembly systems, cludg TUG, because of the costat ecoomc pressure to smplfy work ad reduce varace. (Furthermore, t s show Barthold et al that the essetal behavor of bucket brgades remas uchaged eve uder models of stochastc work cotet.) The remag assumpto of the ormatve model s the oe we must relax order to uderstad the performace of bucket brgades at stes such as TUG. I all prevous applcatos we have assumed the followg: Assumpto 3. Isgfcat Walkg Tme. The total tme to assemble a product s sgfcatly greater tha the tme to walk the legth of the flow le. Therefore all had-offs occur, for all practcal purposes, smultaeously, sychrozed by tem completos of the last worker. Ths was a questoable assumpto at TUG, where t requred as log as fve mutes to pack up ad move tools ad 10 to 20 mutes to egotate the had-off of a partally completed tractor. Wth four workers egaged the bucket brgade, there would be three had-offs per tractor, each volvg two workers, whch represets oe to two perso-hours of lost producto per tractor. The smplest model of tractor requred about eght perso-hours to assemble, so the producto capacty lost due to walk-backs ad had-offs represeted about 12% 25% of the total producto capacty. It was ot clear that bucket brgades could overcome such a dsadvatage. It s the relaxato of Assumpto 3 that s the prmary cocer of ths paper. To model the bucket brgade we mplemeted at TUG, we replace Assumpto 3 wth the followg two assumptos, whch descrbe the detals of how work s trasferred from oe worker to aother. Assumpto 3.1. Costat Walk-Back Tmes. The tme requred for worker to walk back to hs predecessor s a costat b 0.

5 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS 125 Assumpto 3.1 dffers from the ormatve model that the tme b to walk back ca be ozero; the tme depeds oly o the detty of the worker. Ths made sese the cotext of TUG because the work statos were at the corers of a square ad so the walkg dstaces were all detcal, but some workers get ther tools together ad move to the prevous work stato faster tha others. Assumpto 3.2. Costat Had-off Tmes. After worker has walked back to take over the work of hs predecessor, the tme requred to egotate the had-off s a costat h 0, after whch worker assumes work o the tem for whch he has just take resposblty ad worker 1 begs walkg back to take over the tem of hs predecessor. Assumpto 3.2 does ot specfy how h s determed. Here are some possbltes that le wth our model Cosder a had-off to cosst of two subtasks: The upstream worker must relqush the tractor, ad the dowstream worker must accept resposblty for the tractor. Let r be the tme requred by worker to relqush hs tractor to the dowstream worker; ad let s be the tme requred by worker to accept work from the upstream worker. Four plausble ways of determg had-off tme are h = max r 1 s (the had-off s complete oly whe both workers have agreed to the had-off); h = s (the had-off s complete oly whe worker, the dowstream worker, has assumed resposblty for the tem); h = r 1 (the had-off s complete oly whe worker 1, the upstream worker, has relqushed the tem); ad h = m r 1 s (the had-off s complete as soo as ether worker s satsfed). Fally, t s worth remarkg that we caot smply sum the walk-back tme b ad had-off tme h to a sgle costat term because the state of the system evolves dfferetly durg walk-back tmes ad hadoff tmes. For example, worker 1 cotues to make progress whle worker s walkg back for durato b, but they both susped work for durato h. For otatoal coveece let B = =1 b be the total tme requred to perform all walk-backs (excludg had-offs), ad let H = =1 h be the total tme requred to perform all had-offs betwee the tme the last worker completes a product ad the frst worker begs a ew oe. (Note that we take h 1 = 0 because ay setup tme may be assumed to be part of the work cotet of the product.) We shall fd that, eve wth the waste of ozero walk-back ad had-off tmes, a bucket brgade assembly le remas self-balacg. Furthermore, eve though t loses some producto capacty theory (the amout of whch we shall make precse what follows), we were surprsed to fd that practce bucket brgades were evertheless a mprovemet. 3. Dyamcs of the TUG Assembly Le I our model, all tractors share a commo work cotet, so the state of the system at ay tme may be summarzed by gvg, for each worker, the fracto of work cotet that has bee completed. As Barthold ad Eseste (1996), we refer to the fracto completed by a worker as hs posto ad we restrct our atteto to the sequece of worker postos x 0 x 1 x 2 x k at those stats whe a ew tractor s begu by the frst worker. The vector x k s the kth terate of the bucket brgade system. Because our rules madate that o worker s walkg back or preemptg wheever a ew tractor s begu, the all workers have a tem at the start of a terato, ad, therefore, the worker postos x k 1 x k 2 x k are well-defed over the work cotet 0 1 for all k. Let f be the fucto that maps the vector of worker postos at the start of oe terato to that at the subsequet terato, so that x k+1 = f x k. We study the behavor of a bucket brgade le by studyg ts trajectory x k = f k x 0 k=0. The fucto f expresses the followg dyamcs. A terato begs whe the frst worker begs a ew tractor, ad thus x k+1 1 = 0. Durg a terato, worker 1 progresses at rate v 1 for tme 1 x k /v, utl worker completes hs tractor. The, ulke behavor prevous models of bucket brgades, worker 1 cotues at rate v 1 utl he s preempted; ths wll requre tme j= b j + j=+1. After worker 1 s preempted by worker, worker

6 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les 126 Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS works at rate v whle worker 1 walks back ad preempts worker 2, ad so forth, utl worker 1 begs a ew tractor; ths requres tme 1 b j + 1. However, the forward progress of a worker may be restrcted by the ed of the le, whch case he wats at posto 1 utl hs tem s take over by hs sucessor. These dyamcs are summarzed by: x k+1 1 = 0 x k+1 { = m 1 x k 1 +( 1 x k ( + b j + j= )v 1 v )v 1 + j=+1 ( 1 1 b j + ) v } for =2 (3.1) 3.1. Balace A assembly le s balaced f each worker repeats the same terval of work cotet o successve tems. For a bucket brgade, ths meas that there exsts a fxed pot x = f x, such that, f the workers beg at these postos, the, after completo of each tem, they wll walk back to exactly these same postos to beg work o the subsequet tem. From Equatos (3.1), we ca express the fxed pot posto x of each worker terms of x to get, x 1 = 0 { x = m 1 1 x 1 1( ) 1 } v v j + B j + H j vj +v b j + for =2 (3.2) where we defe B = B b, ad H = H h +1 h, ad for otatoal coveece we defe h +1 = 0. Solvg for x yelds x = m {1 1 =1 v + v =1( B + H ) v =1 v } (3.3) We ow have the followg property for ay fxed pot. Lemma 3.1. For ay fxed pot x, the workers are postoal sequece, so that 0 = x 1 x 2 x 1. Proof. Follows from Equatos (3.2). The followg lemma establshes that the fxed pot for our system s uque: Lemma 3.2. There s a uque fxed pot x to the system of Equatos (3.1). Proof. Equato (3.3) returs a uque value for x, whch s the substtuted to Equatos (3.2). The tme B + H that s lost walk-back ad hadoff durg assembly of each tractor creases wth the umber of workers a bucket brgade. I fact, f there are too may workers, the the wasted persohours could exceed the tme requred to assemble a tractor. I such case, the watg rule would keep some workers permaetly dle, watg at 1 for the curret had-off to rpple upstream. I the extreme case of may workers, the fxed pot could degeerate to x = so that oly the frst ad possbly secod workers cotrbute to producto. We say that the bucket brgade s overstaffed f x = 1, for the the efforts of at least oe worker are completely forfet because the accumulated waste exceeds the work cotet of the product. The mplcato for practce s clear: Oe must ot overstaff the bucket brgade. Equato (3.3) gves the precse codto ecessary for x < 1. We ote that x < 1 oly f the term =1 B + H v < 1. Ths term accumulates the total amout of work accomplshed over all workers durg had-offs ad walk-backs. Thus, f a ut ca be completed durg the tme requred to complete the seres of walk-backs ad had-offs, the the le s overstaffed, or, roughly speakg, the bucket brgade s overstaffed f the uproductve tme exceeds the productve tme. The followg lemma provdes a smple, suffcet codto for a le ot to be overstaffed. Ths ca be helpful as a maageral frst check. Lemma 3.3. The bucket brgade s ot overstaffed (that s, x < 1) fb + H<1/ 1 v max. Proof. From Equato (3.3), x < 1 as log as =1 B + H v < 1, so the lemma holds because =1 B + H v < B+ H 1 v max. We determed that overstaffg would ot be a problem at TUG, where four workers assembled ts most popular tractor. The TUG assembly le would ot be overstaffed utl there were about 17 workers, whch would mea 16 walk-backs ad had-offs per tractor, each volvg 2 workers for about 15 mutes, whe the total waste would match the 8 persohours of work cotet.

7 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS Self-Balace If the workers are sequeced from slowest to fastest, the all trajectores coverge to the uque fxed pot. Theorem 3.4. If the workers are sequeced from slowest to fastest, v 1 v 2 v 3 v 1 <v, the ay trajectory of worker postos x k = f k x 0 k=0 coverges to the uque fxed pot. Proof. See the appedx Producto Rate The followg lemma gves the producto rate of the bucket brgade at the fxed pot. Lemma 3.5. If the le s overstaffed (that s, f x = 1), the the producto rate at the fxed pot s 1 B + H If the le s ot overstaffed (that s, f x < 1), the producto rate at the fxed pot s =1 v 1 + =1 v b + h + h +1 (3.4) Proof. We let t deote the cycle tme, or tme betwee successve tem completos, whe the system s at ts fxed pot. Thus t = 1 x /v + B + H. If x < 1, the from Equato (3.3) we get t = 1 + =1 v b + h + h +1 / =1 v. Lemma 3.5 shows that, whe the le s overstaffed (equvaletly, whe the tme to walk back ad had off the product s very large), the producto rate s determed etrely by the materal hadlg tme (walk-back ad had-off). Whe the le s ot overstaffed, the producto rate s gve by Equato (3.4). Thus, f walk-back ad had-off tmes were eglgble, the, as would be expected, the producto rate s smply the sum of the worker veloctes. Otherwse, we terpret b + h + h +1 as the uproductve tme worker must cur for each ut produced, ad thus v b +h +h +1 s hs lost work. Therefore, Equato (3.4) ca be terpreted as the total productve capacty dvded by the total work cotet requred (oe ut of productve work plus =1 v b + h + h +1 uts of uproductve work). We also ote that the frst ad last workers are somewhat uderrepreseted the deomator of Expresso (3.4). Ths s because h 1 = h +1 = 0; the frst worker ad last worker are each volved a sgle had-off, whereas all other workers partcpate two. Ths s more easly see by cosderg the smple case where all walk-back tmes are equal, b = b, ad all had-off tmes are equal, h = h. I ths case the producto rate smplfes to 1 1 h v 1 + v / =1 v + 2h + b (3.5) The terms v 1 ad v are represeted a extra term. Ths reflects the fact that the frst (slowest) ad last (fastest) workers cotrbute more of ther tme to producto tha do the other workers. Ths suggests addtoal ways that oe mght try to crease producto rate of a bucket brgade. For example, creases the work veloctes of the frst ad last worker are more valuable tha creases the veloctes of ay other workers. Smlarly, Expresso (3.4) shows that the greatest opportuty for mprovg the producto rate les reducg the tme to walk back ad had off work for the fastest workers. 4. Bucket Brgades Eable Rapd Learg Uder bucket brgades, the workers at TUG lost tme walk-backs ad had-offs ad stll creased the producto rate by about 10% the frst week. We beleve that ths was due to creased rate of learg amog workers, whch was made possble by the specalzato that emerged as the le balaced tself. Image a sgle ew worker jog three expereced workers, each producg a tractor wth eght stadard perso-hours of work-cotet. If the ew worker performed at half the speed of a seasoed worker, uder craft-maufacturg he would complete oe tractor every two days. But f the four workers were orgazed as a bucket brgade, the ew worker would repeat the same tal porto of workcotet at least three tmes a day. Ths frequet repetto s the bass for the faster learg. I add-

8 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les 128 Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS to, because of the self-balacg ature of bucket brgades, the ew worker wll spotaeously assume a larger porto of work cotet as hs sklls crease. Gve eough tme, the workers at TUG could have become eve more productve uder craft maufacturg tha uder bucket brgades because of the tme lost uder bucket brgades to walk-backs ad had-offs but the urget eed to respod to customer orders dd ot allow TUG the luxury of watg. The producto rate had to be creased as soo as possble. The hgh productvty of bucket brgades uder learg s cosstet wth recet work of Vllalobos ad Muoz (1999) ad Muoz ad Vllalobos (2002), who cocluded from smulato studes that bucket brgades outperform alteratve ways of orgazg workers whe there s hgh labor turover. (I related work, Armbruster ad Gel 2002 used smulato to study the detaled dyamcs of bucket brgades uder some models of learg.) 5. Coclusos Bucket brgades ca be vewed as a way to orgaze workers; t les betwee tradtoal assembly, where workers are specalsts fxed subtervals of work cotet, ad craft assembly, where workers are geeralsts. At TUG, bucket brgades provded a way for the orgazato to evolve gracefully from craft assembly toward assembly les to ga worker specalzato ad cosequet learg. Importatly, TUG could take ths step wthout sacrfcg producto rates. Ths gave TUG tme to buld a workcotet model ad pla a assembly le. (Ideed, the bucket brgade may be maged to do some of the plag tself: Because the bucket brgade s selfbalacg, the emerget allocato of work s strogly suggestve of how work should be dstrbuted o the evetual assembly le.) The success of ths mplemetato depeded o oe sght ad oe pece of seredpty. At frst t dd ot seem to us that bucket brgades could be used at TUG because the tractors had to rema statoary whle uder costructo. It was surprsgly dffcult to see what s ow obvous, that the workers could rotate amog the work statos. However eve the we feared that bucket brgades would be less productve tha craft assembly due to the cosderable tme lost to walk-backs ad had-offs. Fortuately ad, to us, surprsgly ths was more tha made up for by creased learg. TUG evetually dd move to a assembly-le but ths was for a mx of reasos, oe havg to do wth the techcal aspects of bucket brgades. Most mmedately, there was a complete chage maagemet after TUG was acqured by a larger compay, Stewart & Steveso Servces, Ic., whch has partcular stregths paced assembly. That compay vested the egeerg ad captal requred to covert etrely to tradtoal assembly les. It remas possble for Stewart & Steveso to revert to bucket brgades, especally ow that the work cotet has bee carefully documeted, makg had-offs more effcet. Ackowledgmets The authors thak the maagemet of TUG for mplemetg bucket brgade assembly. They also apprecate the support of the Offce of Naval Research through Grat N (Barthold), the Natoal Scece Foudato through Grat DMI (Barthold), The Logstcs Isttute at Georga Tech ad ts sster orgazato The Logstcs Isttute Asa Pacfc Sgapore (Barthold), ad the Uversty of Chcago Graduate School of Busess (Eseste). Appedx. Proof of Theorem 3.4: Covergece to the Fxed Pot We troduce a ew coordate system that measures the tme allocato betwee worker ad + 1 at the start of terato k: t k = x k +1 x k ( b j + ) v+1 v + b j + for = 1 1 t k = 1 x k v + B + P Combg ths coordate system wth Equatos (3.1) yelds x k+1 = x k 1 + ( ( t k B P) v 1 + b j + )v 1 j= j=+1 ( b j + )v = x k 1 + ( 1 t k b j ( 1 )v b j + )v

9 Barthold ad Eseste: Usg Bucket Brgades to Mgrate from Craft Maufacturg to Assembly Les Maufacturg & Servce Operatos Maagemet 7(2), pp , 2005 INFORMS 129 Now we ca rewrte the tme allocato as t k+1 = 1 [ ( x k + t k v +1 b j )v ( + b j + 1 [ ( x k 1 v + t k 1 + b j ( 1 ( b j + ) v+1 + v ) v +1 ] )v 1 1 b j + = x k x k 1 ( 1 b j + 1 ) v + v ( 1 b j ) v 1 v = v 1 /v t k v 1/v t k ) v ] b j v 1 /v t k Ad, thus, rewrtg these equatos as a lear system t k+1 = T t k we observe that whe v 1 v 2 v 1 <v, the sequece of terates coverges, because T s the trasto matrx of a fte state Markov cha that s rreducble ad aperodc. Now, f at some teratos workers are held at the ed of the le (ad thus ther x s are dampeed to 1), the the trasto matrx wll be substochastc. Because the cycle tmes are bouded below, however, the system coverges to the fxed pot. Refereces Armbruster, D., E. S. Gel, J. Murakam Bucket brgades wth learg. Workg paper. Barthold, III, J. J., D. D. Eseste A producto le that balaces tself. Oper. Res. 44(1) Barthold, III, J. J., D. D. Eseste The bucket brgade home page. jjb/bucket-brgades.html. Barthold, III, J. J., L. A. Bumovch, D. D. Eseste Dyamcs of 2- ad 3-worker bucket brgade producto les. Oper. Res. 47(3) Barthold, III, J. J., D. D. Eseste, R. A. Foley Behavor of bucket brgades whe work s stochastc. Oper. Res. 49(5) Barthold, III, J. J., D. D. Eseste, Y. F. Lm Bucket brgades o -tree assembly etworks. A. Dolgu, guest ed. Eur. J. Oper. Res. (Specal Issue o Assembly Le Balacg). Forthcomg. Muoz, L. F., J. R. Vllalobos Worker allocato strateges for seral assembly les uder hgh labor turover. Iterat. J. Producto Res. 40(8) Vllalobos, J. R., L. F. Muoz Assembly le desgs that reduce the mpact of persoel turover. Proc. IIE Solutos Coferece, Phoex, AZ. Yelle, L. E The learg curve: Hstorcal revew ad comprehesve survey. Decso Sc