K vary over their feasible values. This allows

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1 Proceedngs of the 2007 INFORMS Smulaton Socety Research Workshop. MULTI-PRODUCT CYCLE TIME AND THROUGHPUT EVALUATION VIA SIMULATION ON DEMAND John W. Fowler Gerald T. Mackulak Department of Industral Engneerng Arzona State Unversty Tempe, AZ, , U.S.A. Barry L. Nelson Bruce Ankenman Dept. of Industral Engneerng and Management Scences Northwestern Unversty Evanston, IL, , U.S.A. ABSTRACT In ths paper, we wll dscuss our efforts to create the next generaton of semconductor factory smulaton tools, whch we call complete response-surface mappng (crsm). More specfcally, we wll descrbe the basc research and software development necessary to produce the capablty to provde smulaton results on demand for cycle-tme measures as a functon of throughput and product mx. 1 INTRODUCTION Many man-hours are nvested n developng and exercsng smulaton models of wafer fabs, models that nclude crtcal detals that are dffcult or mpossble to ncorporate nto smple load calculatons or queueng approxmatons. Unfortunately, smulaton models can be clumsy tools for plannng or decson makng because even a few mnutes per smulaton run (whch s optmstc) s too slow to allow what-f analyss n real tme. Even optmzaton va smulaton (where some comb naton of smulaton outputs s maxmzed or mnmzed) has drawbacks snce an obectve functon must be specfed and ths hnders the decson maker s ablty to consder trade offs that are not easly quantfed. We are nvestgatng the creaton of the next generaton of smulaton tools for decson support n semconductor manufacturng, whch we call complete response-surface mappng (crsm). crsm explots the avalablty of large quanttes of dle computer resources, whle recognzng the scarcty of decson-maker tme. crsm combnes computng horsepower, adaptve statstcal methods and queueng theory to allow a smulaton to be used for plannng and decson makng n a much dfferent way than before. crsm represents a brdge between the flexblty of smulaton and the nsght provded by an analytcal queueng model by delverng smulaton results on demand. More specfcally, we are performng the basc research and software development to produce a crsm tool that provdes smulaton results on demand for cycletme measures as a functon of throughput and product mx n semconductor manufacturng. Gven a smulaton model of a wafer fab and mnmal nformaton on the controllable parameters, crsm runs an automated sequence of experments to generate a model structure (MS) that represents the frst four moments (equvalently, mean, varance, skewness and kurtoss) of product cycle tme as a functon of product mx and throughput. These experments could use dle computer resources, explot multple processors f they are avalable, and execute wthout human nterventon. The MS s the nput to a smulaton-on-demand query engne (QE) that allows the decson maker to nvestgate optons and trade offs on demand wthout runnng addtonal smulatons. Any questons that can be answered through combnatons of the mean, standard devaton and percentles of the cycle tme as a functon of throughput and product mx are supported, wth results delvered as numercal and graphcal dsplays. To be more precse, let λ1, λ2, K, λk be the throughputs (release rates) of k products nto the factory smulaton. We denote the steady-state cycle tme of the th product by C = C ( λ, K, λ ) = C ( λα, K, α, ) 1 k 1 k where λ s the factory throughput, and α s the fracton of the throughput that wll be product type ( k = 1 α = 1 ). crsm wll produce a MS that approxmates the moments of λα, 1,, αk C as the decson varables K vary over ther feasble values. Ths allows the decson maker to answer questons such as: 1. What s the weghted cycle tme of the factory at a partcular throughput and product mx? 2. What s the 80 th percentle of cycle tme for products at a partcular throughput and product mx? 3. What are the feasble values of throughput and product mx λα, 1, K, αk such that average cycle-tme constrants are met? E[C] c, = 1,2, K, k

2 4. What s the mpact on the cycle tmes of products 1,2,, k-1 of ncreasng the throughput of product k to meet ncreased demand? 5. What product mx maxmzes revenue whle keepng cycle tmes below requred lmts? The crsm that we are developng bulds an MS for a factory smulaton n whch only the throughput and product mx can be altered; however, multple nstances of a crsm can be used for capacty plannng obectves. By lettng crsm buld an MS for factory smulatons wth dfferent levels of capacty, crsm facltates capacty plannng and expanson analyss that takes cycle tme nto account. The emphass of ths approach s dfferent from much smulaton research: Our focus s on the effcency of obtanng useful smulaton results, rather than on the effcency of the smulaton run tself. crsm assumes that the user s wllng to run a substantal number of smulatons to buld the MS, although substantal stll means orders of magntude less tme than was requred to buld the smulaton model. We wll desgn crsm to make these runs effcently, but the real savngs from crsm are most apparent after the MS s avalable, when a decson maker can use the QE to quckly and easly answer a varety of questons on demand, wthout rerunnng the smulaton or even knowng that a smulaton exsts. Our goal s to get more value out of the smulaton, va deeper nsght, more complete exploraton and tmely responses, than s currently possble wth ether smulaton or analytcal models. 2 PRIOR WORK In our prevous NSF-sponsored research we developed effcent tools for accurate and precse estmaton of the mean, standard devaton and percentles of cycle tme as a functon of the overall factory throughput for a fxed product mx. Central to ths work was developng flexble famles of models to represent the frst four moments of steady-state cycle tme as a functon of throughput; percentles of cycle tme then come from a four-moment approxmaton. To desgn smulaton experments to ft these models, we also needed models for the varablty of the moment estmators themselves, because the moments, and the varance of ther estmators, explode as the throughput approaches factory capacty. The form of our models was motvated by heavy-traffc queueng analyss (e.g., Whtt 1989), but our research showed that generalzatons of these smple models were essental when the smulaton ncluded tool falures, dfferent product flows and prorty schemes found n semconductor manufacturng (Allen 2003, Johnson 2003). To defne these models, let x represent the fracton of system capacty n use when the factory throughput capablty s λ (ths allows the maxmum throughput to be standardzed as 1). We developed technques to use wafer fab smulaton outputs to ft the followng model for the mth moment of cycle tme: m E[ C ] = t = 0 ax ( 1 x ) p (1.1) We also needed a model for the varance of the estmator of the mth moment, specfcally Var[C ]= 1 u m = 0 bx ( x) 2q (1.2) For smple networks of frst-n-frst-out queues, heavy traffc analyss suggests that p=m, q=2m+2; however, we showed that ths s not always the case for the queueng networks that are typcal of semconductor manufacturng facltes. Wth both p and q unknown, and up to eght models to be ft smultaneously ((1.1) and (1.2) for each of the frst four moments), we developed technques to effcently and effectvely desgn the smulaton experment and ft the models, and showed that these models gve remarkably accurate predctons of the mean, standard devaton and percentles of cycle tme (McNell et al. 2003ab, Mackulak et al. 2004, Park, et al. 2002, Yang et al. 2004). Our experence buldng CT -TH models for a fxed product mx demonstrates that smulaton output data can be used to ft accurate, and easly manpulated, models of the form (1.1) for the factory as a whole. Once the models are avalable they can be used to quckly and nteractvely evaluate cycle tme -throughput scenaros on demand n the same way we use a queueng model. Our approach s to take these deas to the next level n two mportant ways: (1) To allow product mx, as well as throughput, to be vared; and (2) to develop a smulaton-on-demand QE that uses these models for decson support. 3 MODELS FOR PRODUCT MIX We beleve that the best approach for ncorporatng product mx nto the CT-TH analyss s to leverage, as much as possble, our expertse n developng CT-TH models wth a fxed product mx. For a gven product mx, we have developed procedures to ft mean cycle tme curves as a functon of the throughput, and then to derve cycle-tme percentle curves from these moment models. The procedures make effcent use of the smulaton runs, dagnose and correct for lack of ft, and can be completely automated. Unfortunately, our nvestgaton of analytcally tractable queueng network models convnces us that extendng the moment model (1.1) to nclude product mx as an ndependent varable (as n Lamb and Cheng 2002) s unlkely to be successful because the correct form of the model depends on specfcs of the network topology of the factory, somethng we do not thnk the user of crsm should have

3 to fgure out. Instead, we wll use smulaton to ft CT-TH models for a carefully selected range of product mxes and then nterpolate among these models to derve cycle-tme measures at product mxes that we dd not smulate. In a rough sense, we are lookng for a set of bass functons that span the cycle tme for the product mx space of nterest. Snce we already have the capablty to ft CT-TH slces of ths surface (.e., curves for a fxed product mx), the focus of the research s desgnng the smulaton experment, nterpolatng the curves, and verfyng the accuracy of the results. We next dscuss each of these ssues n turn 3.1 Desgn of the crsm Experment Our prevous research has provded effcent and effectve experment desgn strateges when the product mx s fxed. In ths context a desgn corresponds to settngs of the standardzed throughput at whch to make smulaton runs, and an allocaton of smulaton effort to each desgn pont (Park et al. 2002, Yang et al. 2004). For crsm, the desgn also ncludes the product mx settngs at whch we ft the models. There are a number of research challenges to address: 1. The desgn space s no longer smple as the stablty requrement (throughput must be less than capacty) for dfferent product mxes further complcates the desgn problem. The work center, machne group or staton that frst reaches capacty depends on the product mx ( α1, K, α k ). Fgure 1a shows the CT-TH curves for Product 1 n a two product, mult-staton system when the mx of Product 1 changes. 2. The experment desgn must fll the product mx space n a way that facltates nterpolaton at product mxes not smulated. Thus, a good desgn mght need to nclude settngs of product mx that cause each work center, machne group or staton that could defne the fab capacty to actually defne t. 3.2 Interpolaton A second research challenge s nterpolatng among the ftted CT-TH curves when we encounter a new product mx. For nstance, n the example shown n Fgure 1 the curve for Product 1 at 50% of the mx would be some nterpolaton of the four ftted curves. What knd of nterpolaton wll work best? It s plausble that CT -TH curves derved at a base collecton of product-mx settngs can be used to nfer the entre CT-TH surface, but we cannot expect the nterpolaton to be so easy n a practcal stuaton. The followng s a a = ( α, K, α k ) denote a more realstc approach: Let 1 vector of mx parameters, and let A denote the collecton of product mxes at whch we run smulatons and ft mo d- els. Then for a new mx a that we dd not smulate, the nterpolaton mght take the form Cˆ (, ) D(, ) ˆ λ a = aa C( λ, a ) (1.3) a? where D s a measure of the dstance between the mx of a ftted curve and the desred curve, and C ˆ ( λ, a ) s the ftted curve at mx a. Fgure 1 llustrates how ths mght be done to determne Product 1 s cycle tme when t s 50% of the mx, and the fab throughput s 4. In ths case the values of the four ftted curves at 20%, 40%, 60% and 80% Product 1 provde four values for a quadratc nterpolaton at 50% Product 1 (rght fgure). The nterpolated value s days, whle the true mean cycle tme s days. Although the quadratc dstance measure D works well n ths example, the key research queston wll be the choce of dstance measure for more complex problems. 3.3 Accuracy Snce we requre the experment desgn, smulaton and fttng process (n other words, the constructon of the MS) to be completely automated, we need a way to determne when the MS s complete. We wll agan leverage our ablty to construct accurate and precse CT-TH curves for a fxed product mx, mplyng that we have confdence that each ndvdual curve n our bass s vald. To determne when the bass s complete, we propose usng a crossvaldaton approach. When we have ft g curves, we can test these curves for adequacy by droppng one curve at a tme and measurng the ablty of the remanng g-1 curves to approxmate t va (1.5). When all curves can be accurately approxmated by nterpolatng the other g-1 curves then the MS s complete. Based on our experence fttng CT-TH curves, maxmum relatve error along the curve s a good choce for measurng approxmaton accuracy. 4 SOFTWARE TOOLS To support the development and use of crsm, two software tools must be bult. The frst, the crsm Generator, s for use n creatng the Model Structure (MS) and wll control whch combnatons of throughput rates and product mxes are smulated. The second s the smulaton-ondemand Query Engne (QE), whch s for use after the MS has been generated. The QE answers questons, such as those outlned above, gven a collecton of nonlnear response surface models for each of the frst four moments at varous product mxes. Results wll be provded on demand, wthout requrng any addtonal smulaton effort.

4 4.1 crsm Generator A crtcal step n creatng the crsm that must occur before the MS s generated s performng smulaton runs at a varety of product mx/throughput combnatons. Snce t s mpossble to perform smulaton runs at all possble combnatons, the crsm Generator determnes what desgn ponts (throughputs and product mxes) to run, how much smulaton effort to allocate at each desgn pont and automatcally executes the runs. It requres as nput the product routngs, the processng rates of each product on each machne group, and the number of machnes n each machne group. Our pror work focused on a sngle product mx, whch essentally reduces the problem to a sngle slce through the response surface. Addng the addtonal varable of product mx makes the problem sgnfcantly more dffcult, as t dramatcally ncreases the number of canddate desgn ponts at whch to smulate. Further, t s also mportant to determne the relatve mportance of each desgn pont for developng the MS. Both tasks are accomplshed by the crsm Generator. Fnally, snce the results of all smulaton runs wll be stored n a database, the crsm generator wll have the capablty to reuse exstng smulaton runs. For example, f answerng queston posed to the Query Engne requres addtonal precson beyond that attaned by the ntal crsm runs, then the ntal results wll be retrevable so that duplcate effort s not requred to obtan the new, hgher-precson results 4.2 Query Engne Once the MS has been generated, nonlnear response surface models exst for each of the frst four moments of cycle tme. These models can be used to answer a varety of nterestng questons about the system, and the smulatonon-demand QE wll provde the decson support mechansm by whch these questons are asked and answered. Specfcally, the QE software tool wll consst of a frontend user nterface, whch accepts nputs from the user and dsplays the outputs. Addtonally, to answer several of the questons we antcpate decson makers askng, the QE wll need to contan a response-surface search algorthm that effcently and effectvely searches the CT-TH surfaces to fnd settngs that yeld specfed cycle-tme performance, and locate optmal or near-optmal solutons. There are two types of questons that wll typcally be handled through the QE: those about the mean cycle tme of the system and those about cycle-tme percentles. Specfc soluton approaches for some sample questons are gven below. Queston 1: What are the mean (or P-percentle) product cycle tmes for a gven fab throughput and product mx or vector of start rates? In ths case, the user s nterested n obtanng an estmate of the mean (or P-percentle) cycle tme for each product when the start rates of all products are already known. The nput can be gven as an overall factory throughput and product mx or as a vector of start rates for each product type. To answer ths queston usng the QE, the approprate coordnates on the mean response surface model (the ndependent axes represent the start rates for each of the products) smply need to be dentfed and the assocated response nterpolated for each product. When the user s nterested n obtanng a vector of estmates of a partcular cycle tme percentle, (.e., the vector of the 95th cycle-tme percentles) for a gven set of start rates, the answer reles on our prevous work n percentle estmaton usng the Cornsh-Fsher expanson, whch was found to effectvely estmate percentles from any sample dstrbuton, gven a percentle from the standard normal dstrbuton and estmates of the sample dstrbuton s frst four moments. The only requred user nputs are the desred percentle and the product mx. Further detals on the Cornsh-Fsher expanson as t apples to cycle-tme percentle estmaton can be found n the McNell, et al. (2003ab). Queston 2: What product mx s most proftable that acheves gven mean (percentle) cycle tme targets? The user s now nterested n determnng the product mx (or, equvalently, vector of start rates) that wll maxmze proft for the system, whle stll obtanng no more than a maxmum mean (or percentle) cycle -tme value for each product. To answer ths queston, the user must supply the followng nputs: the vector of profts for each unt of each product produced, the cycle tme requrements for each product; the mnmum start rate for each product (based on producton requrements) the maxmum start rate for each product (based on product demand). Obtanng a soluton to ths queston wll be sgnfcantly more dffcult than the prevous queston, as there may be an nfnte number of product mxes that wll meet the cycle-tme requrements. Therefore, we must fnd an effcent way to navgate the soluton space towards the optmum, or at least toward a soluton wth a hgh total proft. We expect to use nonlnear programmng technques, such as gradent search that s specalzed to explot the fact that these nonlnear functons have a known general form (1.1) and certan propertes (e.g., monotoncally ncreasng as throughput ncreases). If the cycle tme targets are on percentles, an addtonal step, evaluatng the Cornsh-Fsher (C-F) expanson, wll be requred. Once a possble soluton s dentfed, each of the frst four sample moments must be estmated for each product usng the CT-TH surface models. These values wll then be plugged nto the C-F expanson to determne f the percentle estmate for each product meets the constrant. If the constrant vector s met, the proft for ths soluton s stored, and the soluton space search contnues. If the constrant s not met, the soluton s nfeasble, and the obectve functon value need not be calculated. Alternatvely, we wll nvestgate whether t s more

5 effcent to buld an entre response surface of the Cornsh- Fsher expanson for a gven percentle, whch could be searched drectly, rather than evaluatng the expanson ndependently for each throughput and product mx. Clearly, Queston 2 s more dffcult than Queston 1 because t nvolves a complex feasble regon. However, answerng ths type of queston provdes great beneft to the decson maker and searchng the soluton space wll take sgnfcantly less tme than runnng addtonal smulaton models or tryng to do optmzaton va smulaton. Fgure 2 shows the relatonshp between the user nputs, the MS nputs, and the outputs of the QE for the questons dscussed. Nether the lst of questons nor the fgure s ntended to be exhaustve. Rather, they are ntended to provde an example of the types of questons that the QE wll be able to answer. REFERENCES Allen, C The Impact of Network Topology on Ratonal-Functon Models of the Cycle Tme- Throughput Curve, Honors Thess, Department of Industral Engneerng & Management Scences, Northwestern Unversty. Johnson, R Non-Lnear Regresson Fts for Cycle Tme vs. Throughput Curves usng Two Data Sets from Actual Semconductor Manufacturng Facltes, Research Report, Dept of Industral Engneerng & Management Scences, Northwestern Unversty. Lamb, J. and R. Cheng Optmal allocaton of runs n a smulaton metamodel wth several ndependent varables. Operatons Research Letters 30, Mackulak, G., Fowler, J., Park, S., McNell, J.E "A Three Phase Smulaton Methodology for Generatng Accurate and Precse Cycle Tme - Throughput Curves", accepted by Internatonal Journal of Smulaton and Process Modelng, Vol. 1, Nos. 1/2, pp McNell, J.E., Mackulak, G.T., and Fowler, J.W., (2003a) Indrect Estmaton of Cycle Tme Quantles From Dscrete Event Smulaton Models Usng the Cornsh- Fsher Expanson, Proceedngs of the 2003 Wnter Smulaton Conference, S. Chck, P. J. Sánchez, D. Ferrn, and D. J. Morrce, eds., pp McNell, J.E., Mackulak, G.T., Fowler, J.W., and Nelson, B.L. (2003b) Indrect Cycle Tme Quantle Estmaton Usng the Cornsh-Fsher Expanson, ASU Workng Paper Seres. Park, S., Fowler, J., Mackulak, G., Keats, J. and Carlyle, W. (2002) D-optmal sequental experments for generatng a smulaton-based cycle tme -throughput curve, Operatons Research, 50(6), pp Yang, F., Ankenman, B. and Ne lson, B.L Generaton of Cycle Tme-Throughput Curves through Smulaton, Workng Paper, Dept of Industral Engneerng & Management Scences, Northwestern Unversty. Whtt, W Plannng Queueng Smulatons, Management Scence 35, AUTHOR BIOGRAPHIES JOHN W. FOWLER s a Professor of Industral Engneerng at Arzona State Unversty (ASU) and s the Center Drector for the Factory Operatons Research Center that s ontly funded by Internatonal SEMATECH and the Semconductor Research Corporaton. Hs research nterests nclude modelng, analyss, and control of semconductor manufacturng systems. Dr. Fowler s a member of ASEE, IIE, INFORMS, POMS, and SCS. He s an Area Edtor for SIMULATION: Transactons of the Socety for Modelng and Smulaton Internatonal and an Assocate Edtor of IEEE Transactons on Electroncs Packagng Manufacturng. He s an IIE Fellow and s on the Wnter Smulaton Conference Board of Drectors. Hs emal address s <ohn.fowler@asu.edu>. BARRY L. NELSON s the Krebs Professor of Industral Engneerng and Management Scences at Northwestern Unversty, and s Drector of the Master of Engneerng Management Program there. Hs research centers on the desgn and analyss of computer smulaton experments on models of stochastc systems. He has publshed numerous papers and two books. Nelson has served the professon as the Smulaton Area Edtor of Operatons Research and Presdent of the INFORMS (then TIMS) College on Smulaton. He has held many postons for the Wnter Smulaton Conference, ncludng Program Char n 1997 and currently Char of ts Board of Drectors. Hs e-mal address s <nelsonb@northwestern.edu> GERALD T. MACKULAK s an Assocate Professor of Engneerng n the Department of Industral Engneerng at Arzona State Unversty. He s a graduate of Purdue Unversty recevng hs B.Sc., M.Sc., and Ph.D. degrees n the area of Industral Engneerng. Hs prmary area of research s smulaton applcatons wthn manufacturng wth a specal focus on automated materal handlng wthn semconductor manufacturng. Hs emal address s <mackulak@asu.edu>. BRUCE E. ANKENMAN s an Assocate Professor n the Department of Industral Engneerng and Management Scences at the McCormck School of Engneerng at Northwestern Unversty. Hs current research nterests nclude response surface methodology, desgn of experments, robust desgn, experments nvolvng varance components and dsperson effects, and desgn for smulaton experments. He s a past char of the Qualty Statstcs and Relablty Secton of INFORMS, s an Assocate Edtor for Naval Research Logstcs and s a Department Edtor for IIE Transactons: Qualty and Relablty Engneerng. Hs e-mal address s ankenman@northwestern.edu.

6 Fgure 1: (a) CT-TH curves for one product n a two-product system as the mx changes from (left to rght) 20%, 40%, 60% to 80% of Product 1. (b) Interpolaton of CT-TH curves wth 20%, 40%, 60% and 80% Product 1 to determne cycle tme at 50% Product 1. User Inputs d l l max l mn Query Engne QE Example Outputs d a p RS varance RS kurtoss RS mean RS skewness Inputs from MS Fgure 2: Representaton of the QE n terms of potental nputs and outputs