USING INPUT PROCESS INDICATORS FOR DYNAMIC DECISION MAKING

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1 Proceedgs of he 999 Wer Smulao Coferece P. A. Farrgo, H. B. Nembhard, D. T. Surrock, ad G. W. Evas, eds. USING INPUT PROCESS INDICATORS FOR DYNAMIC DECISION MAKING Mchael Fremer School of Operaos Research ad Idusral Egeerg Corell Uversy Ihaca, NY 4853, U.S.A. Lee Schrube Deparme of Idusral Egeerg ad Operaos Research Uversy of Calfora a Berkeley 435 Echeverry Hall Berkeley, CA , U.S.A. ABSTRACT I a coually chagg evrome, a smulao sudy ha egraes he acves of daa colleco, model aalyss, ad decso makg has some dsc advaages. I hs paper we look a smulao projec dyamcs from a hgh-level ad exame some ways for egrag hese acves. From hs perspecve, he processes used o drve a smulao model are forecass of evromeal chages, ad he parameers for models of hese processes are vewed as leadg dcaors. A smple decso-makg scearo havg some of he characerscs of semcoducor maufacurg s used o llusrae he deas. INTRODUCTION From a hgh-level perspecve, a smulao sudy cosss of daa colleco, model aalyss, ad decso makg. Each of hese s geerally vewed as a separae acvy, he eracos of whch are usually o cosdered. The goal of hs paper s o provde a framework for egrag hese acves, wh he am of producg beer decsos. The parcular example we develop s movaed by he problem of ool se seleco semcoducor maufacurg, bu he deas geeralze o ay suao where smulaos are used o make decsos a rapdly chagg evrome. The followg smple decso-makg scearo s used for llusrao. Cosder he suao where we mus make a choce bewee wo alerave sysem cofguraos someme he fuure. The sysem s currely cofgured as A, ad by some fuure dae, T d, (whch, for smplcy, we wll assume s fxed) we mus decde wheher o swch he cofgurao o B or o maa cofgurao A. We would lke o choose he cofgurao ha wll maxmze he eced performace of he sysem (e.g. profs) over a fe me horzo [T d, T h ], codoed o he hsory of he sysem up o me T d. Fgure shows a mele of hs decso process. T represes he curre me. Cofgurao A Swch o B? Performace Fgure : le of he decso process. We wll make he decso bewee cofguraos A ad B based o he resuls of a smulao sudy. The sudy (daa colleco, smulao execuo, ad oupu aalyss) ypcally requres large amous of me, o he order of weeks or mohs. The curre sae of pracce eds o separae daa colleco from he oher asks. I oher words, durg he frs sage of he sudy, say from T o T sm, daa s colleced or a daa base s updaed ad quered o parameerze he model. (See Fgure.) Smulao execuo ad oupu aalyss are performed from T sm o T d, a whch po he decso s made. Ths meas ha he decso s based oly o observaos of he real world up o me T sm, whch may be some me before T d. Uder he presumpo ha he qualy of he decso creases wh he amou of real-world daa, here s a edecy o delay smulao execuo ad oupu aalyss (ad hece he decso) as log as possble. We beleve here s value havg some formao abou he decso earler ha T d. For example, f early codos look favorable, we may wsh o ede he decso. If codos are less favorable, we may wsh o delay he decso ad search for oher aleraves. I parcular, we would lke o ake o accou codos rgh up o me T d. 35

2 Fremer ad Schrube collec daa smulae T ow T sm T d T h mohs ad cosss of a ramp-up phase, a maure phase, ad a ramp-dow phase. (See Fgure 3.) Produco sraeges may be buld-o-order sce veory rapdly becomes obsolee. Fgure : Revsed mele of he decso process. A sequeal approach o aalyss s cumbersome f we desre up-o-dae formao abou he decso as codos chage over he erval T o T d. We are requred o repeaedly parameerze ad execue he smulao model. A alerave s o make mulple rus of he smulao model a he begg of he sudy (.e. a me T ) usg a rage of parameer values. We calculae varous leadg dcaors (e.g. esmaed demad rae) over he course of each sample pah, up o smulaed me T d. The a ay me T bewee T ad T d, we ca cosruc a regresso model o predc performace over he erval [T d, T h ] based o he values of he leadg dcaors over he erval [T, T ]. By calculag he acual values of he leadg dcaors from he real-world daa, we ca predc he eced performace, codoal o he evrome up o me T. The dea s smlar o respose surface mehodologes for predcg smulao resuls, bu for pu processes raher ha decso varables. A dffculy s decdg how o selec he parameer values o use for he smulao rus a me T. Oe mgh sample from a approprae poseror dsrbuo fuco, gve ay avalable formao. Aoher dffculy s choosg approprae leadg dcaors. Whe here s o depedecy he pu process, he dcaors ca be fucos of he curre sysem sae ad he parameer values used o geerae he sample pah. (The real-world dcaors would use esmaes of he parameers.) Whe here s depedecy, he dcaors ca also be fucos of he hsory of he sysem up o he curre me. We hope o make hese deas more cocree wh he example descrbed he ex seco. We descrbe how he sadard approach ad he leadg dcaors mehodology ca be appled o hs example ad compare he qualy of decsos made wh each mehodology a varous me pos bewee T ad T d. EXAMPLE: TOOL SELECTION IN A SEMICONDUCTOR FAB We cosder a smple example ha has some of he characerscs ypcal of he semcoducor dusry. Ths dusry s characerzed by hghly ucera demad for producs, exremely esve equpme, ad log lead mes for orderg equpme. The ecoomc lfe cycle of a ew produc may oly be a maer of several Produc Demad ramp-up maure ramp-dow Fgure 3: Ecoomc lfe cycle of a ew produc. I hs example he maufacurer has sared o erece demad for a ew produc, a he begg of s ramp-up phase. The facory has some exsg capacy for maufacurg he ew produc, however by me T d he maufacurer mus decde wheher o order aoher ool se for he facory or o make do wh he exsg equpme. If a ew ool se s ordered, goes ole mmedaely. The curre cofgurao (A) for our model facory cosss of fve parallel boleeck ools, each wh a oeal processg rae µ. Cofgurao B s obaed by addg a sxh ool o he sysem. The maufacurer s cocered wh he facory s ably o sasfy demad durg he ramp-up phase of he ew produc, whe he marke allows hem o ask a hgh prce for he ew produc. I hs aalyss T h s he ed of he produc s ramp-up phase, he me a whch demad eers he maure phase. (Here we cosder T h o be fxed, alhough would more realscally be a ucera quay.) We would lke o mmze a cos fuco over he perod [T d, T h ], whch cludes he cos of he ool se (f purchased). Sce he maufacurer would lke o fll orders quckly, he cos fuco also pealzes log lead mes (he delay bewee a order s arrval ad fulfllme). The cos fuco s: cos c c [ ool se] purchase + ( average lead me of orders receved) For hs example we ake c.5 ad c. Orders for he ew produc arrve accordg o a osaoary Posso process wh arrval rae λ(). We model λ() wh a logsc fuco: e () λ γ. e 36

3 Usg Ipu Process Idcaors for Dyamc Decso Makg I hs example he acual order arrval process has values -6,., ad 5 for α, β, ad γ. (The arrval rae λ() ad he egraed rae fuco Λ() are show Fgures 4 ad 5.) We suppose ha he maufacurer kows γ (he sze of he marke) bu mus esmae α ad β for he smulao sudy. We descrbe a esmao procedure for hese parameers he ex seco. The values for T d ad T h are 8 ad respecvely. 3 FORECASTING THE ORDER ARRIVAL PROCESS A varous mes T durg he smulao sudy we mus esmae he parameers α ad β from he acual order arrval process. Suppose ha here were order arrvals durg he erval [T, T ]. Le T, ad le,, be he mes of he order arrvals bewee T ad T. The arrval rae has he form λ() λ () γ { {, where α ad β are values ha we wsh o esmae. The egraed rae fuco s Fgure 4: The arrval rae fuco, λ(). Λ() Fgure 5: The egraed rae fuco, Λ(). To udersad how he decso s affeced by he values of α ad β, we smulaed he sysem a a varey of parameer segs. Each replcao had 5 servers up o me T d. Usg CRN we he spl he replcao, eher keepg 5 servers (he o-purchase decso) or addg oe (he purchase decso). Table shows some of he resuls. A Y meas he purchase decso had he lower mea cos, ad a N meas he o-purchase decso had he lower mea cos. The acual parameer values (-6,.) correspod o a purchase decso. Table : Purchase decsos a dffere values of α ad β, wh γ 5. α,β Y -9 N Y -8 Y -7 N Y -6 Y -5 Y Y { { γ + Λ() λ( a) da l β + α Defe radom varables X as: X Λ γ β ( ) Λ( ) + { α + β,..., l { + α + β. The X s are oeally dsrbued wh mea. We oba mome esmaors αˆ ad βˆ by solvg he followg equaos for α ad β : ad { α + β { α + β γ + X l, β + X γ + l β { α + β { α + β. The soluo may be approxmaed usg, for example, he Gauss-Newo mehod. Oher approaches for esmag he parameers of Posso processes are descrbed by Cox ad Lews (966). For a rece paper o he subjec, refer o Kuhl, Damerdj, ad Wlso (977). 4 APPLYING THE LEADING INDICATORS METHODOLOGY To vesgae he performace of he leadg dcaors mehodology, we smulaed s use o he example descrbed he prevous secos. For each replcao we geeraed a acual arrval sample pah usg he values (-6,.) for (α,. A mepos T 4, T 6, ad T 3 37

4 Fremer ad Schrube 8 boh he leadg dcaors mehod ad he sadard approach were used o make a decso. We frs descrbe he mplemeao of he wo mehods, he her relave performace a he hree mepos. The frs sep of he leadg dcaors mehodology s o geerae sample pahs for he regresso model. For each sudy we made smulao rus, drawg values for α ad β from U(-,) ad U(,) dsrbuos, respecvely. Regresso models were he cosruced for each mepo T,,,3 o predc he dfferece performace bewee he wo sysem cofguraos over he erval [T d, T h ]. The depede varables (leadgdcaors) were α, β, ad he sysem sae (umber of free servers ad umber queue) a me T. Fally, a mes T,,,3 he regresso models were used o make he purchase decsos. Esmaors αˆ ad βˆ (based o he acual daa up o me T ) were used for α ad β. For he sadard approach, esmaors αˆ ad βˆ were calculaed a mepos T,,,3 based o he acual daa avalable up o me T. We he used hese values o geerae smulaos of he sysem over he erval [T, T h ] ad chose he cofgurao wh he lowes average cos. I order o deerme he correc decso for each replcao, codoal o he acual sample pah up o me T, we also smulaed each sysem cofgurao over he erval [T, T h ] usg he acual parameer values α - 6 ad β.. Table shows he frequecy of correc decso for each mehod a mepos T, T, ad T 3. The able also shows he average squared dfferece bewee he predced performace over he erval [T, T h ] ad he eced performace gve he correc parameer values α -6 ad β.. As eced, he qualy of decsos made wh he sadard approach mproves wh he amou of daa. The leadg dcaors approach works reasoably well early he me horzo, meag ha may be a useful ool for provdg early formao abou he decso. Table : Relave performace of mehodologes. Sadard Approach (Leadg Idcaors) Early T 4 T 6 T 3 8 Decso % correc 33 (8) 8 (76) 9 (8) MSE.33(.7).5(.4).3(.4) 5 COMMENTS AND TOPICS FOR FURTHER RESEARCH The leadg dcaors mehod descrbe here seems promsg as a ool for provdg early formao abou decsos. Furher vesgao s requred cocerg he choce of leadg dcaors ad he mehod s performace o problems wh depede arrval processes. Aoher area of eres s he mehod s relaoshp o he Bayesa approach o pu dsrbuo seleco developed by Chck (999). Raher ha po esmaes for α ad β, oe would use a approprae poseror dsrbuo based o daa colleced up o he me of he decso. For example, suppose we sar wh pror π ( α, β ). Sce he X s are d oeal ad we have { α + β { α + β γ + X l, β (,..., α, f γ + { α + β { α + β + The poseror dsrbuo s he: h ( α, β,..., ) f γ β. { α + β { α + β f (,..., α, π( α, (,..., α, π( α, d α dβ The leadg dcaors would be approprae fucos of hs dsrbuo. ACKNOWLEDGEMENTS The research repored here was parally suppored by a jo src (fj-49) ad sf (dm ) research projec semcoducor operaos modelg. REFERENCES Chck, S. E Ipu Dsrbuo Seleco for Smulao Expermes: Accoug for Ipu Uceray. Submed for publcao. Cox, D.R. ad P.A.W.Lews The Sascal Aalyss of Seres of Eves. Lodo: Mehue & Co. Kuhl, M.E., H. Damerdj, ad J.R. Wlso Esmag ad Smulag Posso Processes wh Treds or Asymmerc Cyclc Effecs. I Proceedgs of he 997 Wer Smulao Coferece, ed., Sgru Adrador, Kev Healy, Davd Whers, ad Barry Nelso, Isue of Elecrcal ad Elecrocs Egeers, Pscaaway, New Jersey. 38

5 Usg Ipu Process Idcaors for Dyamc Decso Makg AUTHOR BIOGRAPHIES MICHAEL FREIMER s a Ph.D. caddae a Corell Uversy s School of Operaos Research ad Idusral Egeerg. He receved hs udergraduae degree mahemacs from Harvard. Pror o aedg graduae school he worked for a operaos research cosulg frm, Appled Decso Aalyss, Ic. Melo Park, CA. Hs research eress are smulao modelg ad reveue maageme. LEE SCHRUBEN s a Professor he Deparme of Idusral Egeerg ad Operaos a he Uversy of Calfora a Berkeley ad s currely o a leave of absece from Corell Uversy where he s he Adrew S. Schulz Jr. Professor of Idusral Egeerg. Hs research eress are sascal desg ad aalyss of smulao ermes ad graphcal smulao modelg mehods. Hs smulao applcao ereces ad eress clude semcoducor maufacurg, dary ad food scece, healh care, bakg, ad he hospaly dusry. 39

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