BALANCING OF U-SHAPED ASSEMBLY LINE

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1 BALANCING OF U-SHAPED ASSEMBLY LINE Nuchsara Krengkorakot, Naln Panthong and Rapeepan Ptakaso Industral Engneerng Department, Faculty of Engneerng, Ubon Rajathanee Unversty, Thaland Emal: ABSTRACT In recent year, many manufactures have adopted a Just-ntme (JIT) approach to manufacturng. One of the mportant changes resultng from JIT mplementaton s the replacement of the tradtonal straght lnes wth U- shaped producton lnes. The mportant characterstc of these new confguratons s that multsklled workers perform varous tasks of dfferent statons along the producton lne. Ths feature allows assgnng tasks both from the begnnng and the end of the precedence dagram to the same statons. On the other hand, the problem of assgnng tasks on a U-lne confguraton s more complex than the straght lne due to the much larger search space. Ths paper addresses the U-shaped assembly lne balancng problem type I (UALBP-; the number of workstatons s mnmzed for a gven cycle tme) usng heurstc methods. To measure performance of the heurstcs, they are appled to solve a large number of benchmark problems. The results are compared wth the optmal solutons obtaned from the prevous publshed research. The computatonal results ndcate that the maxmum task tme rule s able to dentfy better solutons and performs qute effectvely. Fnally, concluson and future research are also presented n ths paper. KEY WORDS Just-In-Tme (JIT), U-Shaped Assembly Lne Balancng Problem (UALBP), Optmzaton Problem, Heurstc Methods. Introducton A producton lne s often used to take advantage of mass producton. As a consequence of the mplement of justn-tme (JIT) prncples nto manufacturng, many companes are organzng ther producton processes nto U-shaped producton lnes (Fgure ) rather than tradtonal straght producton lnes []. When compared to straght lnes they typcally have, better balancng, mproved vsblty and communcatons, fewer work statons, more flexblty for adjustment, mnmzaton of operaton travel, and easer materal handlng []. The tradtonal lne or straght lne assembly lne balancng problem consders a producton lne n whch statons are arranged consecutvely n a lne. A balance s determned by groupng tasks nto statons whle movng forward through a precedence dagram. However, the U- lne assembly lne balancng problem s more complex than the straght lne because tasks can be assgned by movng forward, backward, or smultaneously n both drectons through the precedence dagram. Fgure : U-shaped producton lne, [0] In terms of the soluton of the lne balancng problem, ths mples that the soluton of the U-shaped lne confguraton domnates the soluton of the tradtonal straght lne confguraton due to the number of statons on a U-lne s less than or equal to the number of statons requred on a straght lne [].. Lterature Revew Sngle model and mxed model straght lne assembly lne balancng have been thoroughly researched snce the frst publshed work n 9. However, the frst publshed work on U-shaped lnes was not untl 99. In comparson to the well studed straght assembly lne balancng problem, there are many areas n U-lne assembly lne balancng whch requre further research []. The frst UALBP study n the lterature was by Mltenburg and Wjngaard [9], who developed a DP formulaton for the sngle-model U-lne to mnmze the number of statons. The authors presented a Ranked Postonal Weght Technque (RPWT)-based heurstc for larger sze problems (-tasks problems). Later, Mltenburge and Sparlng [8] developed three exact algorthms to solve the UALBP. The frst was based on a reachng DP formulaton, whereas the other two were breadth- and depth-frst branch-and bound (B&B) algorthms.

2 Later, Urban [] developed an nteger lnear programmng formulaton to solve small- to medumszed of UALBP wth up to tasks. Scholl and Klen [0] developed a branch-and-bound procedure to solve, ether optmally or suboptmally, problem wth up to 9 tasks. Mxed-model U-lnes were studed by Sparlng and Mltenburg []. They developed a heurstc procedure for the U-lne by whch dfferent products were assembled smultaneously. Ther approxmate soluton algorthm that merges each model s precedence dagram nto a sngle precedence dagram solved problems wth up to tasks. Mltenburg [] proposed a DP formulaton for a U-lne faclty that conssted of numerous U-lnes connected by multlne statons. Sparlng [] developed heurstc soluton procedures for a U-lne faclty consstng of ndvdual U-lnes operatng at the same cycle tme and connected wth multlne statons. Ajenblt and Wanwrght [] developed a genetc algorthm. Erel et al. [] proposed smulated annealng as soluton methodologes for larger U-lne and Had Gokcen et al. [] presented a shortest route formulaton of smple U- type assembly lne balancng problem and llustrated on a numercal example. Ths paper addresses the U-shaped assembly lne balancng problem type I (UALBP-; the number of workstatons s mnmzed for a gven cycle tme) usng heurstc methods. To measure performance of the heurstc rules, they are appled to solve a large number of benchmark problems and compared wth the optmal solutons obtaned from the prevous publshed research.. A Defnton of UALBP The U-lne assembly lne balancng problem (UALBP) s an extenson of smple assembly lne balancng problem (SALBP) whch s based on a U-shaped assembly lne nstead of a seral lne. As n the case wth SALBP, t can defne three problem versons of UALBP (cf. Mltenburg and Wjngaard [9]) as well as Scholl and Klen [0]. UALBP- : Gven the cycle tme (c), mnmze the number of staton (m) UALBP- : Gven the number of statons (m), mnmze the cycle tme (c). UALBP-E :Maxmze the lne effcency (E) for c and m beng varable. Snce models for UALBP dffer from those for SALBP only wth respect to the precedence constrants. In SALBP all (drect and ndrect) predecessors of a task j performed at a staton k must be assgned to one of the statons,,k. In UALBP, each task n prncple can share a staton wth any of ts predecessors or successors. However, all predecessors or (and) all successors of a task j performed at a staton k must be assgned to one of the staton,,k. In many cases, a hgher effcency s possble wth UALBP. Note that ncreasng the lne effcency has the further postve effect of smoothng the levels of staton utlzaton,.e., the statons get more equally loaded. The smple U-lne assembly lne balancng problem defned by Mltenburg and Wjngaard [9] s gven as follows : Mltenburg and Wjngaard s defnton follows from that gven by Gutjahr and Nemhauser [] for the tradtonal lne balancng problem. Gven set of tasks F = } =,,, n}, a set of precedence constrants P = {(x,y) task x must be completed before task y}, a set of task tmes T = {t =,,,n}, cycle tme c and a number of workstaton m, fnd a collecton of subsets of F, (S, S,,Sn) where S k = } task s done at a workstaton k}, that satsfy the followng condtons: Ι S k S k j Υ m k = j S k = F () = t c, S k k =,,,n For each task y, f (x,y) P, x S k, y S j, then k j, for all x; or f (y,z) P, y S j, z S, then j, for all z. m mc t s mnmzed. k= Sk Condton ensures that all tasks are assgned to a workstaton. As a result of condton, each task s assgned only once. Condton ensures that the work content of any workstaton does not exceed the cycle tme. Condton ensures that the precedence constrants are not volated on the U-lne. As a result of the objectve functon, the number of workstatons wll be mnmzed [9].. Heurstcs. Heurstcs rules Two heurstc rules (maxmum task tme and mnmum task tme) are used n ths research to fnd solutons to the UALBP-. These heurstc rules were prevously used to solved the SALBP. However, to allow them to work for UALBP some modfcatons were made because each task n prncple can share a staton wth any of ts predecessors or successors. The proposed heurstcs are descrbed below. () () () ()

3 Let U p be the set of tasks whch must precede task, Let U s be the set of tasks whch must succeed task, Then at any nstant the set of assgnable tasks, V = } all x U p or all y U s have already been assgned}..) Maxmum task tme The prorty functon p maxt (), called the U-lne maxmum task tme p maxt () = t() ().) Mnmum task tme The prorty functon p mnt (), called the U-lne mnmum task tme p mnt () = t() (). Numercal llustraton An example problem wth tasks and a cycle tme of 0 unts s consdered. The precedence dagram and straght lne confguraton results are shown n Fgure (a), (b) respectvely. It can be seen that all tasks are performed at statons n tradtonal lne balancng. The llustraton example for assgnng tasks to work statons n U-lne by usng maxmum task tme rule s shown n Table. 8 total tasks tme =, assume c = 0 (a) precedence dagram problem st. st. st. st. st. (b) straght lne confguraton results Fgure : An Example Problem and Straght Lne Results Table : Assgnng Tasks to Statons Usng Max. Task Tme n U-Lne staton Canddate lst assgned Task tme Idle tme,,, 0,,,, 8,,,,,, 0 Table shows the llustraton for assgnng tasks to the statons n U-lne by usng maxmum task tme rule. The assgnable tasks for the frst staton are V = {,}. Consder max. task tme, max(t(), t()) = max(,) =, we assgn task, then V = {,,} as task has the hghest prorty and suffcent cycle tme remanng, t s also assgned to workstaton, the remanng assgnment process s descrbed on Table. From the results, the U- lne confguraton s depcted n Fgure. st. st. st. Fgure : U-Lne Confguraton Results of Example Problem Fgure llustrates the soluton of UALBP- wth four statons : S = {,}, S = {}, S = {,}, S = {,} and wth staton tmes of 0, 8, 9 and 0, respectvely. In U- lne, statons can nclude tasks located on dfferent parts of the producton lne. For example the frst staton conssts of tasks and, where task s located at the begnnng of the lne whle task s located at the end of the lne. In ths case, the number of statons requred on a U-lne s less than one statons on a straght lne.. Computatonal Results Several well-known test problems were consdered by Mltenburg and Wjngaard [9]. We consder only the larger problems ( or more tasks) n U-lne. Table presents the results of the proposed heurstcs and compares the results wth [9] by usng a maxmum ranked postonal weght heurstc and the optmal solutons of the U-shaped lne balancng problem []. As seen by the results, problems of up to tasks can be solved wth two prorty rules of heurstcs (maxmum task tme and mnmum task tme) for assgnng tasks to statons. The problems were solved on a personal computer usng FoxPro.0 wth a Pentum,.0 GHz, MB RAM and the operatng system on wndows XP. The computatonal results ndcated that the maxmum task tme rule was able to dentfy better solutons than [9] on out of problems and take less than ten seconds n each nstance problems (ths take about. sec. on average). The mnmum task tme rule gave very poor results when compared to the others. It only obtaned st.

4 optmal solutons out of nstances. Concernng the average relatve devatons from the best known soluton, the rankng s a max. task tme (rank ), max. RPW (rank ) and a mnmum task tme (rank) respectvely.. Concluson Recently, U-shaped lne has been utlzed n many producton lnes n place of the tradtonal straght lne confguraton due to the use of just-n-tme prncple. The shape of U-lnes mproves vsblty and allows the constructon of statons contanng tasks on both sdes of the lne. Ths arrangement, combned wth cross-traned operators, provdes greater flexblty n staton constructon than s avalable on a comparable straght producton lne. Ths paper studes the U-shaped assembly lne balancng problem and two prorty rules of heurstcs for assgnng tasks to statons are presented. The performance of the heurstcs are appled to solve a large number of benchmark problems. Its results are compared wth the optmal solutons and a maxmum ranked postonal weght heurstc whch obtaned from the prevous publshed research. The computatonal results ndcated that one of the heurstcs rules (a maxmum task tme heurstc) can produce good solutons and the computatonal requrements are not hgh. Ths study has taken a step n the drecton of fndng good heurstc rules to solve the UALBP. It s possble that dfferent heurstc rules wth dfferent problems may produce dfferent results. Because there are a large varety of UALBP. For further research, t would be nterestng to use other heurstcs, metaheurstcs (e.g. tabu search, genetc algorthms, ant system etc.) and fnd more flexble soluton approaches n the larger U-shaped assembly lne balancng problem. Table : Solutons of the Test Problems for U- Shaped Lne Balancng Problems Cycle tme IP soluton Max. RPW soluton Max. task tme soluton Mn. task tme soluton m m % m % Cal. Tme (sec.) m % Cal. Tme (sec.) Mtchell Heskaoff Sawyer Klbrdge & Wester Total optmal solutons (or lower bound) found from nstances problem 8. /nst. 8. /nst. Integer Programmng soluton, results from Urban[] ( n table, p.0) Maxmum Ranked Postonal Weght heurstc researched by Mltenburg and Wjngaard [9]. m s the number of statons. % s the average relatve devaton from the best known soluton. /nst. 8. /nst.. /nst.

5 Acknowledgements The authors would lke to thank the faculty of Engneerng, Ubon Rajathanee Unversty for the fnancal support. References [] Ajenblt, D.A. and Wanwrght, R.L., Applyng genetc algorthms to the U-shaped assembly lne problem. In proceedngs of the 998 IEEE Internatonal Conference on Evolutonary Computaton. Anchorage, AK, 998, 9-0. [] Chen, S., Just-In-Tme U-Shaped Assembly Lne Balancng. PhD. Dssertaton. Lehgh Unversty, 00, 9. [] Cheng, C.H., Mltenburg, J. and Motwan, J., The effect of straght- and U-shaped lnes on qualty, IEEE Transactons on Engneerng Management, (), 000, -. [] Erel E., I. Sabuncuoglu and B.A. Aksu, Balancng of U-type Assembly Systems Usng Smulated Annealng. Internatonal Journal of Producton Research, 9(), 00, []Gutjahr, A. L., Nemhauser, G.L., An algorthm for the lne balancng problem. Management Scence, (), 9, 08-. [] Had Gökçen, Kürşat Ağpak, Cevrye Gencer and Emel Kzlkaya., A shortest route formulaton of smple U-type assembly lne balancng problem. Appled Mathematcal Modellng, 9, 00, -80. [] Mltenburg, J., Balancng U-lnes n a multple U-lne faclty. European Journal of Operatonal Research, 09, 998, -. [8] Mltenburg, J. and Sparlng, D., Optmal soluton algorthms for the U-lne balancng problem. Workng Paper. McMaster Unversty, Hamlton, 99. [9] Mltenburg, J. and J. Wjngaard, The U-lne Lne Balancng Problem. Management Scences, 0(0), 99, [0] Scholl, A., and Klen, R., ULINO-Optmally balancng U-shaped JIT assembly lne. Internatonal Journal of Producton Research, (), 999, -. [] Sparlng, D., Balancng just-n-tme producton unts: the N U-lne balancng problem. Informaton Systems and Operatonal Research,, 998, -. [] Sparlng, D. and Mltenburg, J., The mxed-model U- lne balancng problem. Internatonal Journal of Producton Research, (), 998, 8-0. [] Urban, T.L., Note. Optmal Balancng of U-Shaped Assembly Lnes. Management Scences, (), 998, 8-.