SOLUTION APPROACHES FOR THE CLUSTER TOOL SCHEDULING PROBLEM IN SEMICONDUCTOR MANUFACTURING

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1 SOLUTION APPROACHES FOR THE CLUSTER TOOL SCHEDULING PROBLEM IN SEMICONDUCTOR MANUFACTURING Heko Nedermayer Olver Rose Computer Networks and Internet Wlhelm-Schckard-Insttute for Computer Scence Dstrbuted Systems Department of Computer Scence Unversty of Tübngen Unversty of Würzburg D Tübngen, Germany D Würzburg, Germany E-mal: E-mal: KEYWORDS smulaton, manufacturng, semconductor, cluster tools, schedulng ABSTRACT Wth the ncrease n computer performance schedulng complex manufacturng plants wth global heurstcs s becomng a realstc opton. The complete factory schedulng problem conssts of the global problem and ts subproblems for each work center and machne. In semconductor manufacturng a lot of processng s done usng cluster tools. Cluster tools are a specal knd of machne that can be descrbed as a small factory. We dscuss solutons for the optmzaton of schedules for cluster tools whch s a subproblem of the factory schedulng problem. Our man dea s to use rules based on slow-down factors or search approaches based on the use of slow-down factors for predctng the cycle tmes. We use smulaton to evaluate the schedules. INTRODUCTION Semconductor manufacturng plants are often consdered to be the most complex factores that currently exst. Even gnorng the sze of these manufacturng plants the schedulng problem s extremely complex. It s a job shop schedulng problem that ncludes batch machnes, sequence-dependent setups, recrculatng flows, and cluster tools. Whle batch machnes and setups are wdely studed n the lterature (Pnedo 2001) the problem wth cluster tools s not. Cluster tools are machnes that consst of loadlocks, handlers and other machnes to process the wafers of the lots n the loadlocks. The advantage of cluster tools s that the processng of the wafers s ppelned. Thus compared wth the cycle tme of a lot usng consecutve machnes the cycle tme of a lot n a cluster tool s reduced. As cluster tools are usually pumped to vacuum the yeld may also be mproved and less clean-room space may be requred. cluster tools wth respect to the cycle tme s complex. In ths paper we usually consder cluster tools to have two loadlocks whch s a vald assumpton for most cluster tools currently n use. RELATED WORK Typcal work center problems that are addressed n the lterature are sngle machne work centers, parallel machnes, batch machnes, and machnes wth sequencedependent setups. Solutons and approaches for many of these problems can be found n the book by Pnedo (Pnedo 2001). An mportant objectve of the optmzaton problem s the total weghted tardness (TWT). The FORCe schedulng project ams at globally schedulng a complete factory usng the Shftng Bottleneck heurstc (Fowler et al. 2002a, Fowler et al. 2002b). For the soluton of the work center problems they use the BATCS rule whch s an adaptaton of the ATC rule (Apparent Tardness Cost) for batch machnes and setups (Fowler et al. 2002b, Pabst et al. 2002). For rather smple cluster tools Perknson et al (Perknson et al. 1994, Perknson et al. 1996) analyzed ther throughput and cycle tme behavor analytcally. Others used smulaton and optmzed schedules wth genetc algorthms (Dümmler 1999). Dümmler already ntroduces the noton of a slow-down factor, but dd not use t for schedulng. We analyzed and smulated cluster tools to study slow-down factors ourselves (Nedermayer and Rose 2003, Nedermayer and Rose 2004). FACTORY SCHEDULING Schedulng deals wth allocatng resources for tasks over tme. The goal s to fnd a soluton that s optmal or near-optmal wth respect to gven objectves. Intally makespan was the man objectve. The focus of manufacturng today s often more on costumer orders and on-tme delvery, thus t s most mportant to satsfy all or at least the most mportant due dates. Fndng an optmal schedule s not always easy. In fact, most schedulng problems are NP-hard. Snce cluster tools are small factores themselves that may process more than one lot at a tme, the behavor of

2 Defnton: Lateness Let c be the completon tme of lot and date, then the lateness L of lot s L d ts due = c d. Defnton: Tardness Let c be the completon tme of lot and d ts due date, then the tardness T of lot s T max{ 0, c d ) = max{0, L ) =. Defnton: Total Weghted Tardness (TWT) In addton to the due date each lot s assgned a weght w that specfes ts mportance. Then the total weghted tardness of all lots s defned as = TWT w T. In manufacturng t s common to use dspatchng rules at the work centers or machnes. Dspatchng rules are myopc methods. Hence, they are usually not optmal for the soluton of the global problem. It s lkely that they are also not optmal regardng the local objectve of nterest. PROBLEM GRAPH AND SCHEDULE GRAPH Schedulng problems can be transformed nto graph problems. Addtonally, graphs gve a vsual representaton of the problem. The problem graph s a drected graph. It has a source (o) and snk (*) node, and there s a node for each operaton. A detaled descrpton of problem and schedule graphs can be found n the book by Ovack and Uszoy (1997). Fgure 1 shows the problem graph. It ncludes all potental arcs for determnng a schedule. Operatons at one machne or work center buld a clque, except for operatons of one job. Fgure 1: Problem graph The schedule graph (Fgure 2) s a drected acyclc graph. The arcs ndcate the order of the operatons. The longest path to an operaton s the tme at whch the operaton can start. The longest path from source to snk s the makespan. As one can see the schedulng problem qute naturally decomposes nto subproblems at the work centers or machnes. The graph specfes the release dates for the operatons of the subproblem. When a one machne or work center s scheduled, these release dates for other work centers change. Hence, one soluton for the schedulng problem can be to teratvely solve work center subproblems, adapt the graph and solve the next subproblem, and so on. In ths text we focus on the soluton of such subproblems. Fgure 2: Schedule Graph WORK CENTER SUB PROBLEMS The jobs for each work center or machne have to be scheduled. A decomposton of the global schedulng problem usually conssts of such problems. The same subproblem may be optmzed several tmes wth dfferent release dates as arcs n the schedule graph change. The result s a sequence of operatons n the nput queue of the work center or machne. Ths may also nclude the assgnment of an operaton to a specfc resource, e.g. to a partcular machne n parallel machne problems. Typcal problems are sngle machne problems, parallel machne problems, batch machne problems, problems wth sequence-dependent setups, and, as n our case, cluster tool problems. CLUSTER TOOLS Snce the 1990s cluster tools are becomng a more and more ntegral element of wafer processng n semconductor manufacturng. A cluster tool conssts of a manframe wth several machnes (chambers). A machne usually processes one wafer. A cluster tool has handlers to move the wafers from one chamber to another. The lots are stored n the loadlocks. A handler takes a wafer from a lot n the loadlock and moves t to the processng chamber as ndcated by ts recpe. A wafer may vst several machnes untl t s completed and brought back to ts lot. A lot s completed and can leave the cluster tool when all ts wafers are completed. Snce there s vacuum nsde the tool the loadlock has to be pumped after a lot enters and vented before the lot leaves the tool. Fgure 3 shows the model of an Endura cluster tool wth 2 hand-over chambers. A typcal confguraton could be that the chambers n the secton close to the loadlocks are used for pre-processng steps (algnment, heatng) and post-processng steps (coolng) and the chambers of the other secton are the machnes for the long processng steps (Sedel 2001).

3 4 shows the basc algorthm n pseudo code. The functon TmeToNextEvent and the estmaton of the amount of work done durng an overlap need the slowdown factors and the sngle mode cycle tme predctons for the lots. Fgure 3: Model of an Endura cluster tool For our schedulng problem cluster tools wth one loadlock behave exactly lke sngle machnes (wth lotdependent processng tmes). Thus, we focus on parallel mode cluster tools wth two loadlocks. Here, lots can overlap and snce they use common resources they slow each other down durng ther overlap. Dependng on the compatblty of the recpes the slow-downs can range from small (roughly 1) to large (more than 2). The result s a rather complex behavor wth respect to the lot cycle tme. Defnton: Slow-down factor Let CT ( A, f ) be the tme needed for the fracton f of the work for lot A n the cluster tool when lot A s alone. Let f be the fracton of the work done for lot A durng ts overlap wth lot B and CT ( A, A + B, f ) be the length of ths overlap. Then the slow-down factor s CT ( A, A + B, f ) SDF AB = CT ( A, f ) In prevous papers we studed ths slow-down factor and studed how to predct t (Nedermayer and Rose 2003, Nedermayer and Rose 2004). The slow-down factor may be dfferent for dfferent knd of overlaps, e.g. our smulator tends to prefer the lot that entered the tool frst. The partcular tool cycle tmes for lots n a schedule can only be predcted for the complete schedule from the begnnng to the tme of nterest. Ths can be done usng smulaton or approxmaton. An approxmaton can be computed as follows. We assume that we know the cycle tmes of all lots whle they use the cluster tool exclusvely (n sngle mode). Ths can be estmated once for each recpe by smulaton or analytcally. Addtonally, we assume that we know the slow-down factors for the recpe combnatons of nterest. If a lot s processed completely durng an overlap wth another lot ts cycle tme can be approxmated by CT[ A, A + B,100%] = SDF CT[ A,100%] For lots that overlap wth more than one lot or only partally overlap wth one lot, the amount of work that was done durng each overlap has to be determned and the cycle tme can then be computed accordngly. Fgure AB ALGORITHM INPUT: Queue wth lots n the order gven by the schedule, Tme WHILE Queue.notEmpty() and Clustertool.notEmpty() DO FOR all empty loadlocks DO IF Queue.nextLotReady(Tme) THEN Add Q.next() to loadlock and set ts start tme. ENDIF ENDFOR LengthOfOverlap = TmeToNextEvent(all lots n clustertool); Tme = Tme + LengthOfOverlap FOR all lots n cluster tool DO Determne the amount of work done for lot durng the overlap. IF lot s completed THEN Remove lot from loadlock and set ts completon tme. ENDFOR ENDWHILE Fgure 4: Basc algorthm for computng lot cycle tmes of schedule for cluster tools Compared to smulaton ths algorthm s very effcent because t only needs a few floatng-pont operatons per lot. From ths descrpton of cluster tools n parallel mode t becomes obvous that parallel mode cluster tool schedulng s a problem that s dfferent from standard schedulng problems wth batch machnes or sequencedependent setups, etc. CLUSTER TOOL SIMULATION For our studes we used the cluster tool smulator CluSm that was developed by Dümmler et al. (Dümmler 1999, Schmd 1999) at the Unversty of Würzburg and s also used at Infneon Technologes for cluster tool optmzaton. We used smulaton for two purposes. The frst one was to determne slow-down factors for all combnatons of lots. Snce the slow-down factor may vary dependng on how lots overlap and whch lot s processed frst, we smulated dfferent overlaps. We dscussed ths n more detal n a prevous paper (Nedermayer and Rose 2003). Ths s mportant because n the next secton we wll use these slow-down factors for schedulng. The second purpose of smulaton s to evaluate the schedules computed by the dfferent optmzaton approaches. For the evaluaton scenaros were created, for each test set 20 scenaros wth 20 lots, each lot wth

4 weght, release date and due date. The lots were released n a way that the cluster tool dd not run out of work. The results of the smulaton runs are used to determne cycle tmes, makespan and tardness. The cluster tool nput queue n CluSm s a FIFO queue (Frst In Frst Out). To evaluate a partcular schedule the release dates n the nput fle of the smulator have to be adapted to ensure the correct order. The release date of a lot followng another lot has to be hgher than the release date of ts predecessor even though t s actually released before ts predecessor. Smulaton s also mportant for the overall schedulng problem. Once the operatons for a cluster tool are scheduled, the schedule can be smulated and the results can be used to specfy the processng tmes n the schedule graph. CLUSTER TOOL SCHEDULING In ths secton we descrbe two approaches to schedule cluster tools. We assume that we have one nput queue and that we can modfy the order of the lots n the queue. Our frst approach s a dspatchng rule based on the slow-down factors ntroduced n the last two sectons. The dspatchng rule SDFavg works as follows. Let lot A be n the cluster tool. From all lots that are currently released and avalable n the nput queue take the lot B that mnmzes SDF + AB SDF BA 2 To mplement ths rule we need to predct the cycle tmes of the lots already scheduled to determne the current tme. Ths s necessary to dentfy the lots that are released, but not yet scheduled. Our second approach s a Random Search approach. It s actually a varaton of a genetc algorthm. It uses a populaton of ndvduals, eltsm, and the mutaton s based on permutaton of lots. Schedules are evaluated wth cycle tme predctons usng slow-down factors for the lot combnaton and sngle mode cycle tme predctons for the lots. We dd not search for the fastest search algorthm. We smply wanted to study whch soluton such a search algorthm can fnd gven a consderable amount of tme. Snce the predctons dffer from the actual cycle tmes an optmum of the Random Search s not necessarly an optmum of the real schedulng problem.. Table 1: Objectve Makespan Optmzer Test set Normalzed Makespan FIFO Dresden SDFavg Dresden Random Search Dresden FIFO Dresden SDFavg Dresden Random Search Dresden FIFO Vllach SDFavg Vllach Random Search Vllach FIFO Smple SDFavg Smple Random Search Smple For the test sets Smple and Vllach we also computed lower bounds for the makespan for each scenaro usng a work dstrbuton algorthm. Snce precedence constrants are gnored these bounds are true lower bounds. Any schedule has to be worse. Table 2 gves the detals. On average SDFavg s rather close to ths lower bound. Table 2: Comparng the schedules wth a true lower bound Test set Lower Bound SDFavg FIFO (unrealstc) Vllach 47396s 54749s 61822s Smple 67169s 78687s s As second objectve we used the total weghted tardness (TWT). Table 3 shows that the cluster tool throughput s sequence-dependent and that n the schedules produced by EDD the throughput s reduced. As a consequence, a lot of lots are not completed on tme. Our rule SDFavg on the other hand does not know anythng about due dates, but snce ts schedules result n hgh throughput most lots are completed n tme. The Random Search wth objectve TWT was usually slghtly better. We can conclude that dspatchng rules that gnore cluster tool behavor produce poor throughput and may therefore fal to acheve ther prmary objectve. Thus, combnng them wth SDFavg or smlar approaches s necessary. COMPARISON In ths secton we compare SDFavg, the Random Search based on predctons wth slow-down factors, FIFO and EDD (Earlest Due Date). Frst, we examned the qualty of the schedulers for the objectve makespan. Table 1 shows that SDFavg outperforms FIFO and that SDFavg s roughly as good as the Random Search.

5 Table 3: Objectve Total Weghted Tardness Optmzer Test set Normalzed TWT EDD Dresden SDFavg Dresden Random Search Dresden EDD Dresden SDFavg Dresden Random Search Dresden EDD Vllach 1.35 SDFavg Vllach 1.32 Random Search Vllach 1.06 EDD Smple 1.46 SDFavg Smple 1.15 Random Search Smple 1.10 LOCAL IMPROVEMENTS To avod long ndvdual cycle tmes or to avod poor overall throughput t s useful to have the opton to let one lot wat and thus to avod certan combnatons of lots. To do ths t s necessary to have combnaton characterstcs that ndcate for the scheduler whether a combnaton should be avoded. Agan, slow-down factors are a soluton. To avod long ndvdual cycle tmes one could avod lot combnatons for whch the slow-down factor of one lot s larger than a gven threshold, say 2.5. To avod poor overall performance one could avod a lot combnaton when the average slow-down factors are larger than a partcular threshold, say 2.0. CONCLUSIONS We demonstrated that dspatchng rules based on slowdown factors are an nterestng and promsng approach for schedulng cluster tools. Ths approach takes the specal characterstcs of cluster tools nto account and avods poor throughput and long cycle tmes. It s also effcent and schedules can be computed quckly. Larger studes are to be made. In partcular, the mpact of predcton errors has to be analyzed more deeply. Future work may also nclude the adaptaton of the ATC rule for the use wth cluster tools and to nclude slow-down factors. One mght also thnk of combnng EDD or Crtcal Rato wth a rule based on slow-down factors for local optmzaton. Another opton s to analyze and use other, maybe smpler, lot compatblty measures than slow-down factors. Another feld of research could be to fnd other problems where such an approach mght be useful. REFERENCES Dümmler, M Usng smulaton and genetc algorthms to mprove cluster tool performance. In Proceedngs of the 1999 Wnter Smulaton Conference Fowler, J., Carlyle, M., Runger, G., Gel, E., Mason, S., Rose, O. A New Approach for Schedulng Semconductor Wafer Fabs. Semconductor Fabtech, 15th Edton, pp , Fowler, J., Brown, S., Carlyle, M., Gel, E., Mason, S., Mönch, L., Rose, O., Runger, G., Sturm, R. A Modfed Shftng Bottleneck Heurstc for Schedulng Wafer Fabrcaton Facltes. In Proceedngs of the FAIM 2002, July 15-17, Dresden, Germany, pp , Nedermayer, H. and Rose, O. A Smulaton-based Analyss of the Cycle Tme of Cluster Tools n Semconductor Manufacturng In Proceedngs of the 15th European Smulaton Symposum, Delft, Netherlands, Nedermayer, H. and Rose, O. Approxmaton of Cycle Tme of Cluster Tools n Semconductor Manufacturng In Proceedngs of the Annual IIE Industral Engneerng Research Conference, Houston, Texas, Ovack, I.M. and Uzsoy, R Decomposton Methods for Complex Factory Schedulng Problems Kluwer Academc Publshers. Pabst, D., Fowler, J., Pfund, M., Mason, S., Rose, O., Mönch, L., Sturm, R. Determnstc Schedulng of Wafer Fab Operatons. In Proceedngs of the Brooks Worldwde Automaton Symposum 2003, Oct Perknson, T.L., McLarty P.K., Gyurcsk, R.S., Cavn III., R.K "Sngle-Wafer Cluster Tool Performance: An Analyss of Throughput." In IEEE Transactons on Semconductor Manufacturng Vol. 7, August Perknson, T.L., McLarty P.K., Gyurcsk, R.S., Cavn III., R.K ""Sngle-Wafer Cluster Tool Performance: An Analyss of the Effects of Redundant Chambers and Revstaton Sequences on Throughput." In IEEE Transactons on Semconductor Manufacturng Vol. 9, August Pnedo, M Schedulng. Theory, Algorthms, and Systems. 2 nd edton. Prentce-Hall. Schmd, M Modellerung und Smulaton von Cluster Tools n der Halbleterfertgung. Master s thess. Department of Computer Scence. Unversty of Würzburg, Germany. Sedel, G Smulaton und Optmerung von Cluster Tools n der Halbleterfertgung. Master s thess, Insttute of Mathematcs. Techncal Unversty of Graz, Austra. AUTHOR BIOGRAPHIES HEIKO NIEDERMAYER s Ph.D. student at the Department for Computer Networks and Internet at the Unversty of Tübngen. He receved an M.S. degree n Computer Scence from the Unversty of Würzburg. Hs e-mal address s: nedermayer@nformatk.un-tuebngen.de. OLIVER ROSE s assstant professor n the Department of Computer Scence at the Unversty of Würzburg, Germany. He receved an M.S. degree n appled mathematcs and a Ph.D. degree n computer scence from the same unversty. He has a strong background n the modelng and performance evaluaton of hgh-speed communcaton networks. Currently, hs research focuses on the analyss of semconductor and car manufacturng facltes. He s a member of IEEE, ASIM, and SCS. Hs web address s: www3.nformatk.un-wuerzburg.de/~rose.