Elsevier Editorial System(tm) for Expert Systems With Applications Manuscript Draft

Transcription

1 Elsever Edtoral System(tm) for Expert Systems Wth Applcatons Manuscrpt Draft Manuscrpt Number: ESWA-D tle: An Ordnal Optmzaton heory Based Algorthm for a Class of Smulaton Optmzaton Problems and Applcaton Artcle ype: Full Length Artcle Keywords: ordnal optmzaton, stochastc smulaton optmzaton, artfcal neural network, genetc algorthm, wafer probe testng Correspondng Author: Assstant Professor Shh-Cheng Horng, Ph.D. Correspondng Author's Insttuton: Chaoyang Unversty of echnology Frst Author: Shh-Cheng Horng, Ph.D. Order of Authors: Shh-Cheng Horng, Ph.D.; Sheh-Shng Ln, Ph.D.

2 Cover Letter Dear Edtor, We would lke to submt the enclosed manuscrpt enttled An Ordnal Optmzaton heory Based Algorthm for a Class of Smulaton Optmzaton Problems and Applcaton, whch we wsh to be consdered for publcaton n Expert Systems wth Applcatons. Correspondence and phone calls about the paper should be drected to Shh-Cheng Horng at the followng address, phone and fax number, and e-mal address: Insttute: Department of Computer Scence & Informaton Engneerng, Chaoyang Unversty of echnology Adress: 168 Jfong E. Rd., Wufong ownshp achung County, 41349, awan, R.O.C. Phone: ext 7633 Fax: e-mal : schong.ece90g@nctu.edu.tw hank you very much your consderng our manuscrpt for potental publcaton. I'm lookng forward to hearng from you soon. Sncerely yours, Shh-Cheng Horng

3 * Manuscrpt Clck here to vew lnked References An Ordnal Optmzaton heory Based Algorthm for a Class of Smulaton Optmzaton Problems and Applcaton Shh-Cheng Horng and Sheh-Shng Ln schong@cyut.edu.tw ssln@mal.sju.edu.tw Submtted to the Expert Systems wth Applcatons as a REGULAR PAPER Correspondent : Assstant Professor Shh-Cheng Horng Insttute: Department of Computer Scence & Informaton Engneerng Chaoyang Unversty of echnology Address: 168 Jfong E. Rd., Wufong ownshp, achung County, 41349, awan, R.O.C. Phone: ext 7801 Fax: e-mal : schong@cyut.edu.tw Shh-Cheng Horng s currently an assstant professor of the Department of Computer Scence and Informaton Engneerng at Chaoyang Unversty of echnology, awan, R.O.C. Sheh-Shng Ln s now a professor of the Department of Electrcal Engneerng at St. John's Unversty, awan, R.O.C. hs work was partally supported by Natonal Scence Councl n awan, R.O.C. under Grant NSC E CC3.

4 Abstract In ths paper, we have proposed an ordnal optmzaton theory based two-stage algorthm to solve for a good enough soluton of the stochastc smulaton optmzaton problem wth huge nput-varable space. In the frst stage, we construct a crude but effectve model for the consdered problem based on an artfcal neural network. hs crude model wll then be used as a ftness functon evaluaton tool n a genetc algorthm to select N excellent settngs from. In the second stage, startng from the selected N excellent settngs we proceed wth the exstng goal softenng searchng procedures to search for a good enough soluton of the consdered problem. We appled the proposed algorthm to the reducton of overklls and retests n a wafer probe testng process, whch s formulated as a stochastc smulaton optmzaton problem that conssts of a huge nput-varable space formed by the vector of threshold values n the testng process. he vector of good enough threshold values obtaned by the proposed algorthm s promsng n the aspects of soluton qualty and computatonal effcency. We have also justfed the performance of the proposed algorthm n a wafer probe testng process based on the ordnal optmzaton theory. Key Words: ordnal optmzaton, stochastc smulaton optmzaton, artfcal neural network, genetc algorthm, wafer probe testng. 1

5 1. Introducton Smulaton optmzaton problems could be vewed as optmzaton problems of a system whose outputs can only be evaluated by smulatons (Fu et al., 2005). hus, the objectve of smulaton optmzaton s to fnd the optmal settngs of the nput varables to the smulated system that makes the output varables at ther best or optmal condtons. Varous methods had been developed for ths purpose such as the Gradent Search based methods (Nocedal & Wrght, 2006; Km, 2006), the Stochastc Approxmaton methods (heler & Alper, 2006; Spall, 2003), the Sample Path methods (Hunt, 2005), the Response Surface methods (Myers et al., 2004), and Heurstc search methods. hese methods had been thoroughly dscussed n (Aprl et al., 2003). Among them, the Heurstc search methods ncludng the Genetc Algorthm (GA) (Haupt & Haupt, 2004), the Smulated Annealng (SA) method (Suman & Kumar, 2006), and the abu Search (S) method (Hedar & Fukushma, 2006) are frequently used n smulaton optmzaton (Blum & Rol, 2003; ekn & Sabuncuoglu, 2004). Accordng to an emprcal comparson of these algorthms (Lacksonen, 2001), GA showed the capacty to robustly solve large problems and performed well over the others n solvng a wde varety of smulaton problems. Despte the success of several applcatons of the above heurstc methods (Ahmed, 2007; Fattah et al., 2007), many techncal hurdles and barrers to broader applcaton reman as ndcated n (Dréo et al., 2006). Chef among these s speed, because usng the smulaton to evaluate the output varables for a gven settng of the nput varables s already computatonally expensve not even menton the search of the best settng provded that the nput-varable space s huge. Furthermore, smulaton often faces stuatons where varablty s an ntegral part of the problem. hus, stochastc nose further complcates the smulaton optmzaton problem. he purpose of ths paper s to resolve ths challengng stochastc smulaton optmzaton problem effectvely. he consdered stochastc smulaton optmzaton problem s stated n the followng 2

6 mn J ( ) (1) where s an nput-varable space, and J () s the objectve functon, whch may be an expected output or a functon of expected outputs of the smulated system. o cope wth the computatonal complexty of ths problem, we wll employ the Ordnal Optmzaton (OO) theory based goal softenng strategy (Lau & Ho, 1997; Ho, 1999), whch seeks a good enough soluton wth hgh probablty nstead of searchng the best for sure based on the expectaton that the performance order of the nput-varable settngs s lkely to be preserved even evaluated by a crude model. A crude model s defned as a model that s tolerant of a large modelng nose. From here on, we wll use the word settng to represent the settng of nput varables. he basc dea of the OO theory based goal softenng strategy s to reduce the searchng space gradually, and ts exstng searchng procedures can be summarzed n the followng (Lau & Ho, 1997): () Unformly select N, say 1000, settngs from. () Evaluate and order the N settngs usng a crude model of the consdered problem, then pck the top s, say 50, settngs to form the Selected Subset (SS), whch s the estmated Good Enough Subset (GS). A Good Enough Subset s defned as the subset consstng of the top n % solutons n the nput-varable space. () Evaluate and order all the s settngs n SS usng the exact model, then pck the top k ( 1) settngs. In OO theory (Lau & Ho, 1997), the model nose s used to descrbe the degree of roughness of the crude model. he OO theory had shown that for N =1000 n () and a crude model wth sgnfcant nose n (), the top settng (.e., k=1) selected from () wth s 50 must belong to the GS wth probablty 0.95, where GS represents a collecton of the top 5% actually good enough settngs among N. hs means the actual top settng n SS selected from () s among the actual top 5% of the N settngs wth probablty However, the good enough soluton of problem (1) that we are searchng for should be a good enough settng n nstead of the N settngs unless s as small as 3

7 N (Chen et al., 1999; Ho et al., 2007). As ndcated n a recent paper by Ln and Ho (Ln & Ho, 2002), under a moderate modelng nose, the top 3.5% of the unformly selected N settngs wll be among the top 5% settngs of a huge wth a very hgh probablty ( 0.99), and the best case can be among the top 3.5% settngs of provded that there s no modelng error. However, for wth sze of 30 10, a top 3.5% settng s a settng among the top ones. hs certanly not seems to be a good enough soluton n the sense of practcal optmzaton; however, t s acceptable only when conssts of lots of good settngs so that even f the performance order of the selected settng s not practcally good enough, the correspondng objectve value s. As a matter of fact, most of the practcal stochastc smulaton optmzaton problems do not have lots of good settngs; otherwse, fndng a good enough soluton won t be dffcult. herefore to apply the exstng goal softenng searchng procedures, we need to develop a new scheme to select N excellent settngs from to replace () so as to ensure the fnal selected-settng s a good enough soluton of (1) from the practcal vewpont. Heurstc methods for obtanng N excellent settngs may depend on how well one s knowledge about the consdered system. For nstance n the optmal power flow problems wth dscrete control varables, Ln et al. proposed an algorthm based on the OO theory and engneerng ntuton to select N excellent dscrete control vectors (Ln et al., 2004). However, the engneerng ntuton may work only for specfc systems. hus, n ths paper, we wll propose an OO theory based systematc approach to select N excellent settngs from and combne wth the exstng goal softenng searchng procedures to fnd a good enough soluton of (1). he presentaton of ths OO theory based two-stage algorthm to solve (1) for a good enough soluton s a novel approach n the area of smulaton optmzaton and s one of the contrbutons of ths paper. Reducng overklls and retests s an mportant ssue n semconductor wafer probe testng 4

8 process. akng the chp demand nto account, we have formulated ths problem as a stochastc smulaton optmzaton problem, whch possesses a huge nput-varable space and s most sutable for demonstratng the valdty of the proposed OO theory based two-stage algorthm. hs novel formulaton as well as the novel soluton methodology for ths mportant and practcal stochastc optmzaton problem s another contrbuton of ths paper. We organze our paper n the followng manner. In Secton 2, we wll descrbe the OO theory based two-stage approach and present the proposed two-stage algorthm. In Secton 3, we wll ntroduce the stochastc optmzaton problem of reducng overklls and retests n semconductor wafer probe testng process and present the applcaton of the proposed algorthm. In Secton 4, we wll show the test results of applyng the proposed algorthm on a real case and demonstrate the soluton qualty and the computatonal effcency by comparng wth a vast number of randomly generated solutons and competng methods, respectvely. We have also justfed the performance of the proposed algorthm n a wafer probe testng process based on the ordnal optmzaton theory. Fnally, we wll make a concluson n Secton he OO heory Based wo-stage Approach Apparently the optmzaton problem (1) s a stochastc smulaton optmzaton problem wth huge dscrete nput-varable space. However, to evaluate the true objectve value of a settng, we need to perform a stochastc smulaton of nfnte test samples for the. Although nfnte test samples wll make the objectve value of (1) stable, n fact, ths s practcally mpossble. hus, suffcently large test samples are utlzed n place of nfnte test samples to make the objectve value of (1), J ( ), suffcently stable. he proposed OO theory based approach conssts of two stages to solve (1) for a good enough settng. he frst stage s an exploraton stage. In ths stage, we wll employ a Genetc Algorthm (GA) to search through usng an off-lne traned Artfcal Neural Network 5

9 (ANN) as a crude model for ftness evaluaton and select N (=1024) excellent settngs. he heurstc generaton of N (=1024) s based on the OO theory (Lau & Ho, 1997). he second stage s an explotaton stage to fnd a good enough settng from the N settngs obtaned n frst stage wth more refned crude models. A more refned crude model s defned as a model that s tolerant of a small modelng nose. Suppose we use the exact model to evaluate all the N settngs, we can obtan the best settng n the N, however at the cost of too much computaton tme, whch s aganst our objectve. herefore, we wll dvde the second stage nto multple subphases. he more refned crude models for estmatng J ( ) of a settng employed n these subphases are stochastc smulatons of varous lengths rangng from very short (crude model) to very long (exact model). he canddate soluton set n each subphase (or the estmated good enough subset resulted from prevous subphase) wll be reduced gradually. In the last subphase, we wll use the exact model to evaluate all the settngs n the most updated canddate soluton set, and the one wth smallest J ( ) s the good enough settng that we seek. herefore, the computatonal complexty can be drastcally decreased, because the sze of the canddate soluton set had been largely reduced when the crude model s more refned. In the followng, we wll present the detals of the OO theory based two-stage approach. 2.1 he Frst Stage Approach Snce the order of settngs are relatvely mmune to effects of estmaton nose, performance order of the settngs s lkely to be preserved even evaluated usng a crude model. hus, to select N excellent settngs from wthout consumng much computaton tme, we need to construct a crude but effectve model to evaluate the objectve value J ( ) for a gven settng, and use a selecton scheme to select N excellent settngs. Our crude model s constructed based on ANN (Graupe, 2007), and our selecton scheme s GA (Haupt & Haupt, 2004). 6

10 2.1.1 he Artfcal Neural Network (ANN) Based Model ANN s consdered to be a unversal functon approxmator due to ts genetc and convenent property to model complcated nonlnear nput-output relatonshps. Consderng the nputs and outputs as the settngs and the correspondng objectve values J ( ), respectvely, we can use an ANN to mplement the mappng from the nputs to the outputs (Graupe, 2007). o construct such an ANN, frst of all, we wll select a tranng data set by randomly samplng M settngs wthout replacement from. he formula to calculate the number of random sample (RS) of a gven nput-varable space s as follows (Moore & McCabe, 1999): 2 2 p(1 p) z CI RS (2) 2 2 1[ p(1 p) z CI ] where z s 1.96 and 2.57 for 95% and 99% confdence level, respectvely; p s the percentage pckng a choce, whch s 0.5 used for calculatng sample sze; and CI s confdence nterval whch expresses as decmal. he confdence level s the estmated probablty that a populaton estmate les wthn a gven margn or error. he confdence nterval measures the precson wth whch an estmate from a sngle sample approxmates the populaton value. Consder an nput-varable space wth 30 10, the number of random sample determnng by (2) s for confdence level 99% and confdence nterval 1%. hen we wll evaluate the objectve values of these M =16641 settngs usng an exact model, whch can be a stochastc smulaton wth suffcent large number of test samples as ndcated n (Chen et al., 1999). hese collected M nput-output pars of (, J ( ) ) wll be used to tran the ANN to adjust ts arc weghts. Once ths ANN s traned, we can nput any settng to obtan an estmaton of the correspondng J ( ) from the output of the ANN; n ths manner, we can avod an accurate but lengthy stochastc smulaton to evaluate J ( ) for a gven. hs forms our crude model to roughly estmate the objectve value of (1) for a 7

11 gven settng. Effectveness of ths crude model s justfed by the OO theory as mentoned above, because what we care here are the relatve order of s, not the value of J ( ) s he Genetc Algorthm (GA) GA s a stochastc search algorthm based on the mechansm of natural selecton and natural genetcs. By the ad of the above effectve objectve value (or the so-called ftness value n GA termnology) evaluaton model, we can select N excellent settngs from usng GA, whch s brefly descrbed as follows. Assumng an ntal random populaton produced and evaluated, genetc evoluton takes place by means of three basc genetc operators: (a) parent selecton; (b) crossover; (c) mutaton. he chromosome n GA termnology represents a settng n our problem, and each chromosome s encoded by a strng of 0s and 1s. Parent selecton s a smple procedure whereby two chromosomes are selected from the parent chromosome based on ther ftness values. Solutons wth hgh ftness values have a hgh probablty of contrbutng new offsprng to the next generaton. he selecton rule we used n our approach s a smple roulette-wheel selecton. Crossover s an extremely mportant operator for the GA. It s responsble for the structure recombnaton (nformaton exchange between matng chromosomes) and the convergence speed of the GA and s usually appled wth relatvely hgh probablty, say 0.7. he chromosomes of the two parents selected are combned to form new chromosomes that nhert segments of nformaton stored n parent chromosomes. here are many crossover scheme, we employ the sngle-pont crossover n our approach. Whle crossover s the man genetc operator explorng the nformaton ncluded n the current generaton, t does not produce new nformaton. Mutaton s the operator responsble for the njecton of new nformaton. Wth a small probablty, random bts of the offsprng chromosomes flp from 0 to 1 and vce versa and gve new characterstcs that do not exst n the parent chromosome. In our approach, the mutaton operator s appled wth a relatvely small probablty 0.02 to every bt of the chromosome. 8

12 here are two crtera for the convergence of GA. One s when the ftness value of the best chromosome does not mprove from the prevous generaton, and the other s when evolvng enough generatons. he ntal populatons of the GA employed n our frst stage approach are I, say 5000, randomly selected settngs from. After the appled GA converges, we rank the fnal generaton of these I chromosomes based on ther ftness values and pck the top N chromosomes, whch form the N excellent settngs that we look for. 2.2 he Second Stage Approach Startng from the selected N excellent settngs, n the second stage, we wll proceed drectly wth step () of the exstng goal softenng searchng procedures descrbed n Secton 1. In ths stage, we wll evaluate the objectve value of each settng usng a more refned model than the crude one employed n the frst stage. hs more refned model uses stochastc smulaton wth varous lengths (.e. number of test samples) L. We let L s = represent the suffcently large L. In the sequel, we defne the exact model of (1) as when the smulaton length L. For the sake of smplcty n expresson, we let J ( ) denote the Ls s objectve value of a settng computed by exact model,.e. L Ls. Frst, we defne a basc smulaton length L 0 =500. We set the smulaton length of subphase, denoted by L, to be kl 1 L (or L k L0 ), 1,2,..., where the postve nteger k ( 2) denotes the parameter for controllng the smulaton length L. We let set the sze of the selected estmated good enough subset n subphase to be N1 N and N N 1 / k (or N1 k ), 2,3,... N /. We denote n k as the total number of subphases, and n k s nk 1 nk determned by n arg{ mn( L0k Ls L0k, 1 N n 10)}, where L s = he k nk k above formula determnes n k to be the mnmum of the followng: () the n k such that smulaton length n k L 0 k exceeds the length of exact model, s L, and () the sze of the 9

13 selected estmated good enough subset resulted n subphase n k s small enough,.e. 1 N n 10. Once n k s determned, we set k L k L, whch mply that n the last subphase n s (.e. subphase n k ), the crude model s n fact the exact model of (1), and the settng wth smallest J ( ) s the good enough settng that we seek. Suppose k s very large such that L 1 kl0 L s, then there wll be only one subphase, and each of the N settng wll be evaluated by the exact model, whch wll consume too much computaton tme even though the resulted settng s exactly the best among the N. However, t s not easy to quantfy the tradeoff between the computaton tme and the goodness of the obtaned good enough settng nto an analytcal formula. In fact, what s the best k s really problem dependent, because some problems may care more on computaton tme and some others on the goodness of the obtaned soluton. herefore, we wll show the computaton tme and the goodness of the obtaned good enough soluton of our problem for varous k n Secton he wo-stage Algorthm Now, our OO theory based two-stage algorthm can be stated as follows. Step 1: Randomly select M s from. Compute the correspondng J ( ) for each s usng smulaton length L s. ran an ANN by adjustng ts vector of arc weghts usng the obtaned M nput-output pars,.e. the M pars of (, J ( ) ) s. Let f (, ) denote the functonal output of the traned ANN. Step 2: Randomly select I settngs from as the ntal populatons. Apply a GA wth the followng setup: smple roulette-wheel selecton scheme, sngle-pont crossover scheme wth probablty p c, and mutaton probablty pm to these chromosomes by the ad of the ftness-value evaluaton model, 1 f (, ). After the algorthm converges, we rank all the fnal I chromosomes based on ther ftness values and select the best N chromosomes (.e. s). 10

14 Steps 1 and 2 consttute the frst stage approach. Step 3: Use the stochastc smulaton wth smulaton length L k L0 to estmate the J ( ) of the canddate 1 N / k s, 1,, nk 1 ; rank the canddate N / k 1 s based on ther estmated J ( ) and select the best N / k s as the canddate soluton set for subphase 1. Step 4: Use the stochastc smulaton wth smulaton length L to compute the J ( ) of s s the canddate n k N / k s. he wth the smallest s ( ) J s the good enough that we look for. Steps 3 and 4 represent the procedures of the second stage approach. 3. Applcaton to Reducton of Overklls n Wafer Probe estng Process 3.1 Wafer Probe estng Process he wafer fabrcaton process s a sequence of hundreds of dfferent process steps, whch results n an unavodable varablty accumulated from the small varatons of each process step. Chps are tested multple tmes throughout the desgn and manufacturng process to ensure the ntegrty of the chp desgn and the qualty of the manufacturng process. hus, to avod ncurrng the sgnfcant expense of assemblng and packagng chps that do not meet specfcatons, the wafer probng n the manufacturng process becomes an essental step to dentfy flaws early. he prmary components of a wafer probe testng system nclude probes, probe card, probe staton, and test equpment. Wafer probng establshes a temporary electrcal contact between test equpment and each ndvdual de (or chp) on a wafer to determne the goodness of a de. In general, an 8-nches wafer may consst of 500 to des and each de s a chp of ntegrated crcuts. Although there exst technques such as the statstcal methods and machne learnng methods (Chen et al., 2003; Barnett et al., 2005) for montorng the operatons of the wafer probes, the probng errors may stll occur n many 11

15 aspects and cause some good des beng over klled; consequently, the proft s dmnshed. Fgure 1 shows the Cause-and-Effect dagram of overklls. hus, reducng the number of overklls s always one of the man objectves n wafer probe testng process. he key tool to dentfy or save overklls s retest, whch s an addtonal wafer probng. However, retest s a major factor for decreasng the throughput. hus, the overkll and the retest possess nherent conflctng factors, because reducng the former can gan more proft, however, at the expense of ncreasng the latter, whch wll degrade the throughput and ncrease the cost. What mples s that drawng a fne lne for decdng whether to go for a retest to save possble overklls s an mportant research ssue n ths optmzaton problem of the wafer probe testng process. Consderng the economc stuaton regardng throughput requrement, t would be most benefcal for us to use the trade-off method (Collette & Sarry, 2003) to solve the current problem. hat s to mnmze the overklls subject to a tolerable level of retests provded by the decson maker. Method Probe Staton est Program Customer request Others Eng. mstake Setup ester Probes Devce Materal Operator Probe card Overklls Fgure 1: Cause-and-Effect dagram of overklls. here may be dfferent testng procedures n dfferent chp manufacturers. After the wafer probng, a bn number s used to label each bad de of the wafer. A bn number denotes a classfcaton of crcutry-defect falure n a de. he bn number goes from 1 to a certan number as defned by engneers. But, no matter what testng procedures are used, the decson for carryng out the retest should be based on whether the number of good des and the 12

16 number of bns n a wafer exceed the correspondng threshold values. hus, determnng these threshold values so as to mnmze the overklls under a tolerable level of retests s the man theme of the optmzaton problem consdered here. Furthermore, snce the goodness of a de and the probng errors are of stochastc nature, the consdered problem becomes a stochastc smulaton optmzaton problem. hus, ths computatonally ntractable problem s most sutable for the applcaton of our OO theory based two-stage algorthm to seek for good enough threshold values. 3.2 Problem Statements and Mathematcal Formulaton In ths secton, we employ typcal testng procedures used n a renowned wafer foundry n awan, whch s brefly descrbed n the followng. For every wafer, the wafer probng s performed twce. he second probng apples only to those des faled n the frst one. A de s consdered to be good f t s good n ether probng. We let w ( w ) denote the number of good (bad) des n wafer, and let B j denote the number of bn j n wafer. Assume there are J types of bns n a wafer, then J w B j j1 and w D w, where D denotes the total number of des n wafer. Followng the two tmes of wafer probng, a two-stage checkng on the number of good des s performed to determne the necessty of carryng out a retest,.e. an addtonal wafer probng. We let W mn denote the threshold value of the number of good des n a wafer to determne whether to pass or hold the wafer; we let b, j 1,..., J, denote the threshold j max value of the number of des of bn j n the hold wafer to determne whether to perform a retest. he mechansm of the two-stage checkng can be summarzed below. If w Wmn, we pass wafer ; otherwse, we wll hold ths wafer and check ts bns. For those hold wafers, f Bj b j max, we wll perform retests for all des of bn j to check whether there are probng 13

17 errors that cause overklls. hs partcular class of polces for decdng retest based on the threshold values s commonly practced n wafer fabrcaton processes. hus, the relatonshp between the nputs and the outputs of the consdered problem can be descrbed n Fgure 2, n whch W, b j max, j 1,..., J, are the nput varables, V and R are the output varables, mn and the tested wafers are part of the testng procedures. V 1 L L V 1 and R 1 L L R 1 represent the average overklls and retests per wafer, respectvely, n whch V and R denote the number of overklls and retests n wafer, respectvely, and L denotes the total number of the tested wafers as shown n Fgure 2. W mn b j, j 1,, J max Input varables L tested wafers Wafer probe testng procedures V,R Output varables Fgure 2: Relatonshp between the nputs and the outputs of wafer probe testng procedures. Detals of the testng procedures for a wafer are shown n the flow chart of Fgure 3, n whch the calculatons of the number of overklls and retests are also ncluded. For the purpose of smulatons, we randomly generate Bj based on a Posson probablty dstrbuton wth mean j to represent the results of two tmes of wafer probng, whch are not performed n current computer smulaton and thus shown n the dashed-lne square n Fgure 3. Once B j s generated, we can randomly generate the number of overklls n B j, denoted by o v j, based on a Posson probablty dstrbuton wth mean jbj, where j s the proportonal coeffcent for bn j. he number of overklls n a bn s, n general, proportonal to the number of des of that bn; that means the former wll be less provded that the latter s less. he values of j and j can be found from the real manufacturng data. 14

18 Current wafer Next wafer =+1 wo tmes of wafer probng Randomly generate types of bns; calculate B and the correspondng j J w B j j1 and v for all j J w D w Is w W mn? Yes B b j Are all, j 1,..., J? j max No Yes Pass wafer J V v j R j 1 0 No j=1 Next bn j=j+1 Next bn j=j+1 Is B? j b j max No Pass bn j vj v j r j 0 Yes Perform retests on all des of bn j Is j J? No v j 0 rj B j Yes No Is j J? Yes J V v j j 1 J R r j j 1 Fgure 3: Flow chart of the wafer probe testng procedures. 15

19 In contrast to o v j, we let v j denote the number of overklls for bn j of wafer after completng the testng procedures and let r j denote the correspondng number of retests. In these testng procedures, although we may pass the wafer when the threshold-value test s a success, there may be overklls. As ndcated n Fgure 3, for the passed wafer, the number of overklls J o V v j j1 and the number of retests R =0. he same logc apples to the passed bn j of the hold wafer that v j = v o j and r j =0. However, for any retested bn, the probablty of any undentfed overkll s extremely small, because the des had been probed three tmes, whch nclude two tmes of wafer probng before any retest. hus, for any retested bn j, we have v j =0 and r j = B as ndcated n Fgure 3. he resultng values of j V and R of wafer shown n Fgure 3 wll be used to calculate V and R. From Fgure 3, we see that f we ncrease W whle decreasng b j max, there wll be more mn retests and less overklls. hus, to reduce overklls under a tolerable level of retests, we wll set mnmzng the average number of overklls per wafer, V, as our objectve functon whle keepng the average number of retests per wafer, R, under a satsfactory level. hus, usng the trade-off method (Collette & Sarry, 2003), ths optmzaton problem can be formulated as the followng constraned stochastc smulaton optmzaton problem: mn xx V 1 L L 1 V subject to R 1 L L 1 R r, (3) where x W, b j, j 1,..., J ] denotes the vector of threshold values, that s the vector of [ mn max nput varables; X denotes the nput-varable space; and r denotes the tolerable average-number of retests per wafer. 16

20 Remark 1: he value of r s determned by the decson maker based on the economc stuaton. When the chp demand s weak, the throughput, n general, s not a crtcal problem n the manufacturng process; therefore, we can allow a larger r so as to save more overklls to gan more proft. On the other hand, f the chp demand s strong, then the throughput s more mportant, thus we should set the value of r smaller. akng the chp demand nto account s a dstngushed feature of our formulaton. hs constraned stochastc smulaton optmzaton problem (3) s to fnd an optmal vector of threshold values, x, to mnmze V subject to the employed testng procedures and the constrant on R. herefore, we can use a penalty functon to transform (3) nto the followng unconstraned stochastc smulaton optmzaton problem: mn xx F( x) V P( R r ) ( R r ) (4) where P( R r ) denotes a contnuous penalty functon for the constrant R r, such that P ( R r ) 0 for R r and P ( R r ) 0 for R r. 3.3 Applcaton of the wo-stage Algorthm he stochastc smulaton optmzaton problem (4) has the same form as (1) by treatng x as, X as, and F x) V P( R r ) ( R r ) as J ( ) (. he sze of the nput-varable space X s huge; for example, for an 8-nches wafer, whch conssts of a typcal number of 588 des, the possble ranges of the nteger values W and b j max are [1, mn 588] and [1, 588], respectvely. Consequently for a typcal number of bn types K 10, the sze of X wll be more than hus, ths stochastc smulaton optmzaton problem (4) s most sutable for the applcaton of our two-stage algorthm Applyng Step 1 o apply Step 1 of the two-stage algorthm to problem (4), we need to construct the crude 17

21 model based on the ANN frst, whch conssts of two parts: (A) collectng the tranng data set, and (B) tran the ANNs. We employ two three-layer feed-forward back propagaton ANNs (Graupe, 2007). Assume there are J types of bns n a wafer, J 1, 2 ( J 1) and 1 neurons are used n the nput, hdden and output layers, respectvely. he actvaton functons of the neurons n the hdden and output layers are the hyperbolc tangent sgmod and lnear functons, respectvely. he nputs for both ANNs are x X ; whle for the outputs, one s the correspondng V and the other s R. We obtan the set of tranng data for the two ANNs by the followng two steps. (a) Narrow down the nput-varable space X by excludng the rratonal threshold values and denote the reduced nput-varable space by Xˆ. In general, the yeld rate and statstcal dstrbuton of the number of any bn for typcal products can be collected from a wafer foundry. hus the threshold values, W and b j max, should le n a mn reasonable range determned based on ther correspondng mean values of w and B j, respectvely. (b) Randomly select M =16641 vectors from Xˆ and compute the correspondng outputs V and R usng a stochastc smulaton of large number of test wafers (Chen et al., 1999), that s to perform the smulatons of the testng procedures shown n Fgure 3 for L s = wafers. hs consttutes part (A) of constructng the crude model. We denote the M randomly selected nput vectors by x, 1,..., M, the M correspondng smulated outputs V s by V, 1,..., M, and R s by R, 1,..., M. he tranng problems for adjustng the arc weghts of the above two ANNs are: mn c 1 M 1 [ V f ( x 1 c )] 1 2 (5) and mnc 2 M 1 [ R f 2 ( x 2 c )], (6) 2 where c1 and c2 denote the vectors of the arc weghts of the ANN for V and the ANN 18

22 for R, respectvely; f x ) and f x ) denote the actual outputs of the 1( c1 2( c2 correspondng ANNs when the nput vector s x. hus, the tranng problems are tryng to adjust the vector of arc weghts c1 and c 2 to make the actual outputs f x ) and 1( c1 f x ) as close to the desred outputs V and 2( c2 R as possble. o speed up the convergence of the back propagaton tranng, we employed the BFGS quas-newton method (Gll et al., 1981; Stanevsk & svetkov, 2004) and the one step secant method (Battt, 1992; Fore et al., 2004) to solve (5) and (6), respectvely. Stoppng crtera of the above two tranng algorthms are when any of the followng two condtons occurs: () the sum of the mean squared errors,.e. the objectve value of the tranng problem, s smaller than 10-3, and () the number of epochs exceeds 300. hs consttutes part (B) for constructng the crude model. Once these two ANNs are traned, we can nput any vector x to the two ANNs to estmate the correspondng V and R, whch wll be used to estmate F (x). hs forms our crude but effectve model to estmate F (x) for a gven nput vector x Applyng Step 2 Wth the above crude but effectve objectve value (or the so-called ftness value n GA termnology) evaluaton model, we are ready to apply Step 2 of the two-stage algorthm to select N ( 1024) excellent nput vectors from Xˆ usng GA. he codng scheme we employed for all the vectors n Xˆ s rather straghtforward, because each component of the vector x s an nteger. We start from I( 5000) randomly selected vectors from Xˆ as our ntal populatons. he ftness value of each vector s calculated from F (x) based on the outputs of the two ANNs. Apply a GA wth the followng setup: smple roulette-wheel selecton scheme, p 0. 7, sngle-pont crossover scheme and p c m to these chromosomes. After the GA evolves for 20 teratons, we rank the fnal generaton of the I( 5000) chromosomes based on ther ftness values and pck the top N( 1024) 19

23 chromosomes to serve as the N( 1024) nput vectors needed n Step Applyng Step 3 Startng from the N( 1024) nput vectors obtaned n Step 2, we wll compute F (x) for each nput vector usng a more refned model than ANNs, that s a stochastc smulaton wth varous number of test wafers. he basc number of test wafers s L 0 =500, and n k s nk 1 nk determned by nk arg{ mn( L0k Ls L0k, 1 N n 10)}, where L s = From nk 1 to n k 1, use the stochastc smulaton wth L k L0 test wafers to estmate the k F (x) of the canddate N / k 1 x s; rank the canddate N / k 1 x s based on ther estmated F (x) and select the best N / k x s as the canddate soluton set for subphase Applyng Step 4 In ths step, we wll compute the objectve value of (4) for each of the N n k nput vectors obtaned n Step 3 usng the exact model that s a stochastc smulaton wth suffcently large number of test wafers (.e. L s =100000) that makes the estmated objectve value suffcently stable. hen the nput vector among N assocated wth the smallest F (x) s the good n k enough soluton that we seek. 4. est Results and Performance Evaluaton 4.1 est Results and Comparsons Our smulatons are based on the followng data collected from a practcal product of a renowned wafer foundry n awan. he product s made n 6-nches wafers. Each wafer conssts of 206 des. here are 10 bns n the wafers of ths product, and the values of ther means, j 1,...,10, are respectvely the followng 10 postve real numbers: 0.5, 0.5, 1.1, j 1.3, 0.8, 3.7, 3.5, 40, 45, and 13. he yeld rate of ths product s 46.6%. he mean of the 20

24 overklls that occurred n bn j s 0.03 B for j =1,,10, that s for all j. j he nput-varable space s X x [ W, b, j 1,...,10] W [1,206], b [1,206], j 1,...,10}. { mn j max mn j max We used the sgmod-type functon as our penalty functon P R r ) n (4),.e., j ( P( R r ) = 1 ( R r ) 1 e for R r, where ( ) s a normalzed coeffcent such that max {1,..., M }, and ( R r ) 0 max {1,..., M } V R P for R r. We have smulated three cases of dfferent r s, whch are 10, 40 and 80. Remark 2: he reason we use 6-nches wafer products s for easer dentfcaton of the bns and overklls n experments. In fact, our results can apply to any sze of wafer. Specfc data n the two-stage algorthm applyng to ths product are gven n the followng. In Step 1 of the frst stage approach, we have that (a) X s narrowed down ratonally however conservatvely to Xˆ { x [ Wmn, b j max, j 1,...,10] Wmn [50,206], b j max [1,6 j ], j 1,...,10}, and (b) M =16641 and L s = wafers. In Step 2, I =5000, N =1024, and the convergence crtera we employed for our GA s when the evolvng number of generatons exceed 20. It should be noted that all the test results shown n ths secton are smulated n a Pentum IV PC. In the second stage approach, we test the computaton tme and the goodness of the obtaned good enough soluton of our problem for varous k n order to choose the sutable one. We show the F ( x g ) (vertcal axs) of the good enough solutons, x g, obtaned and the correspondng CPU tme (horzontal axs) consumed by our algorthm wth k 2,3,4,5, 6 and 200 for the case of r =10 n Fgure 4. In general, smaller k corresponds to less CPU tme consumpton because of less smulaton replcatons. However, there s no guarantee that 21

25 larger k wll lead to smaller F x ). Nonetheless, for suffcently large k such as k =200, ( g the correspondng F x ) s the least and CPU tme consumpton s the longest among all the ( g tested k s as we expect. herefore, the choce of k s really problem dependent regardng how fast one ntends to obtan the soluton or how good one cares about the obtaned soluton. As can be observed n Fgure 4, the consumed CPU tme for k =2 n ths test s wthn 2 mnutes, t s sutable to choose k =2 for the sake of real-tme applcaton. herefore, the parameters n second stage of our algorthm are set as follows: k =2, L 0 =500, L = 2 n L0, n n 2 =8 and N = 1024 n 2 n1. Fgure 4: he F x ) obtaned and the correspondng CPU tme consumed by our algorthm ( g wth k 2,3,4,5, 6 and 200 for the case of r =10. able 1 shows the smulaton length and canddate soluton set n each subphase of second stage. In the last subphase, we use the stochastc smulaton wth smulaton length L s = to compute the F x ) of the N 8 canddate solutons. he x wth the smallest ( g n 2 F s (x) s the good enough vector of threshold values xg that we look for. 22

26 able 1: Number of canddate soluton and smulaton length n each subphase of second stage. subphase N n L n he good enough vector of threshold values and the average overkll percentage for three cases r =10, 40 and 80 we obtaned from the two-stage algorthm are shown n able 2. From ths table, we can observe that when r ncreases, the values of W ncreases as mn shown n row 2, and the values of leadng b j max, j =8 and 9, whch accounts for most of the retests, decrease as shown n rows 10 and 11, respectvely. hs ndcates that f we allow more retests, that s ncreasng r, we can set more strngent threshold values, that are ncreasng W and decreasng the leadng b j max, so as to save more overklls, that s mn decreasng the average overkll percentage, as ndcated n the last row of able 2. hs also demonstrates the conflctng nature between the two objectves reducng overklls and retests. We use 590 real test wafers, whose bns B j and overklls before retest o v j are known, to test the performance of the vector of threshold values obtaned by our algorthm for the three cases shown n able 2. he correspondng results of the par of the average overklls per wafer, V ( ), and the average retests per wafer, 1 R ( R ), for V 1 1 these 590 test wafers are shown n Fgure 5 as the ponts marked by,, wth the correspondng r shown on the top rght corner of the fgure. We also use 2000 randomly selected vectors of threshold values to test the same 590 test wafers; the resulted pars of V and R are shown as the ponts marked by n Fgure 5. 23

27 able 2: he good enough vector of threshold values and the average overkll percentage for three dfferent r s. Good enough vector x g r Wmn b1max b2 max b3max b4 max b5 max b6 max b7 max b8 max b9 max b10 max V % 1.36% 0.85% 0.23% R Fgure 5: he resulted pars of (V, R ) obtaned by our algorthm and the randomly generated vector of threshold values. V We have also used typcal GA and Smulated Annealng (SA) algorthm to solve (4) for the case of r =40. As ndcated at the begnnng of Secton 1, the global searchng technques are computatonally expensve n solvng (4). We stop the GA and SA when they consumed 30 24

28 tmes of the CPU tme consumed by the two-stage algorthm, and the objectve values of (4) they obtaned are stll 11.8% and 19.9% more than the fnal objectve value obtaned by the two-stage algorthm, respectvely. Usng the threshold values they obtaned to test the 590 wafers, the resulted (V, R ) pars from GA and SA are marked by and + n Fgure 5, respectvely. We found that usng two-stage algorthm, we can save 11.8% and 19.9% more overklls than usng the GA and SA for R 40, respectvely. In addton, both GA and SA do not generate the optmal soluton, because the best so far soluton they obtaned for one hour of CPU tme are stll far away from the optmal soluton of (4). We see that for R 40, the V resulted by the good enough vectors of threshold values obtaned by our algorthm s almost the mnmum compared wth the randomly selected vectors of threshold values. Smlar conclusons can be drawn for the cases of r =10 and 80. From Fgure 5, we can see that the results we obtaned for the cases of r =10, 40 and 80 are almost on the boundary of the regon resulted from the randomly generated vectors of threshold values; ths mplct boundary represents the (V, R ) pars resulted by the optmal vectors of threshold values. he above result mples that our algorthm not only controls the level of retests but also obtan a near optmal soluton. 4.2 Performance Evaluaton It should be nterestng to address how excellent the N selected vectors are among the varous types of nput-varable space X so as to demonstrate the valdty of our frst stage approach. Although there exsts n-depth analyss of the approxmaton errors for ANN to approxmate contnuous functons, the accuracy of approxmatng the nput and output relatonshps of a dscrete event smulated system s usually addressed usng emprcal results. hus, t s not surprsng that we do not get any analytcal result for the qualty of the N vectors selected n our frst stage approach. Snce the nput-varable space for test product s 25

29 X { x [ Wmn, b j max, j 1,...,10] Wmn [1,206], b j max [1,206], j 1,...,10}, the sze of the nput- varable space s 11 X 206. he methodology for our performance evaluaton s to smulate based on the Ordered Performance Curves (OPCs) (Lau & Ho, 1997) and the employed crude model. he Order Performance Curve (OPC) of all the ordered vectors x 1, x2,..., x X n X s determned by the spread of the order performance F [ 1], F[2],..., F[ X ], where F [] denotes F x ). Wthout loss of generalty, F [] s can be normalzed nto the range [0,1],.e., for 1,2,..., X, y F F ) /( F ). Meanwhle, the ordered X vectors, spaced ( [ ] [1] [ X ] F[1 ] equally, are also mapped nto the range [0,1] such that for 1,2,..., X, z( x ) z[ ] ( 1) /( X 1). here are fve broad categores of OPC models: () lots of good vectors, () lots of ntermedate but few good and bad vectors, () equally dstrbuted good, bad and ntermedate vectors, (v) lots of good and lots of bad but few ntermedate vectors, and (v) lots of bad vectors. Fgure 6 shows a graphcal expresson of these fve types of OPCs. More precsely, a standardzed OPC can be determned by a two-parameter smooth curve B ( z, ) = B ( z, ), where B ( z, ) s the Incomplete Beta functon of the two parameters (, ). In general, <1, >1 corresponds to the OPC of type (); >1, >1 corresponds to the OPC of type (); =1, =1 corresponds to the OPC of type (); <1, <1 corresponds to the OPC of type (v); >1, <1 corresponds to the OPC of type (v). As ndcated n Secton 1, we need not consder the types of X consstng of lots of good vectors n ths evaluaton, thus we take only the three OPC types (), () and (v) nto account. ( 26

30 Fgure 6: Fve types of standardzed OPCs. he roughness of the ANN model can be descrbed by addng a unform nose to the normalzed performances y s (Lau & Ho, 1997; Ho, 1999). hat means, the model of ANN can be descrbed by the nosy model y +, where the random nose representng a large modelng nose that s generated from a unform dstrbuton random varable. We assume varous magntudes of modelng nose of unform dstrbuton to represent the approxmaton errors caused by the proposed ANN based model and make the followng smple experments to compare the qualty of the N vectors selected by GA based on the ANN model wth those selected n random from the soluton space. We let U [-0.1,0.1] denote the unform dstrbuton of a random nose rangng from -0.1 to 0.1 to be added to the normalzed performance,.e. the normalzed objectve value, of the exact model. he normalzed performance for all solutons n a soluton space s equally-spaced rangng from 0 to 1 wth 0 as the top performance. We studed a total of 28 OPCs dstrbuted unformly among the three broadly generc types, (), () and (v), formed from the followng parameters: =1.0, 2.0, 4.0, 5.0 and =0.2, 0.4, 0.8, 1.0, 2.0, 4.0, 5.0. We carred out a Monte Carlo study for vast number of OPCs smlar to that n (Lau & Ho, 1997) for an assumed nose dstrbuton and pck the top N vectors usng GA. In all of our Monte-Carlo calculatons, we smulate realzatons of nosy OPCs. Consder three modelng nose dstrbutons, U [-0.01,0.01], U [-0.05,0.05] and U [-0.1,0.1], 27

31 6 the top 5% solutons n N, whch are selected by GA, are at least a top %, top %, and top % soluton n X wth probablty 0.95, respectvely. However, the top 5% solutons n N, whch are selected n random, s at best (.e. no modelng error) a top 5% soluton n X only. herefore, we have greatly mproved the qualty of the N vectors by replacng the exstng unformly selectng procedure. Remark 3: hough we do not nvestgate the actual order of the N vectors for the OPC types () and (v), our frst stage approach can stll be appled for problems wth of these two types of OPCs. hs s because even f the order of the obtaned N vectors of the two types of OPC may not be as good as those of the other three OPC types due to the sharp senstvty of the nose to the performance n these two types, however ther actual objectve values wll stll be good enough due to the exstence of lots of good vectors. hat means n both OPC types () and (v), there can be a bg dfference n the order of good vectors but the dfference n objectve values are very small. hus, no matter what types of OPC we are facng, our frst stage approach works the same way. 5. Conclusons o cope wth the computatonally ntractable stochastc smulaton optmzaton problems, we have proposed an ordnal optmzaton theory based two-stage algorthm to solve for a good enough soluton usng reasonable computatonal tme. o demonstrate the applcablty of the proposed algorthm, we have used t to solve for a vector of good enough threshold values to reduce overklls and retests n a wafer probe testng process of a wafer foundry. We have tested the performance of the soluton we obtaned usng the real data and found that the resultng average number of overklls and retests per wafer le almost on the boundary resulted from the optmal vector of threshold values of the consdered stochastc optmzaton problem. hs ndcates that the proposed algorthm wll not only control the tolerable level of 28

32 retests by takng the varous chp demand nto account but also provde a near optmal vector of threshold values. We have demonstrated the computatonal effcency of the proposed algorthm by comparng wth the genetc algorthm and the smulated annealng method and found that when the latter two methods consume more than 30 tmes of the CPU tmes consumed by the proposed algorthm, the best so far objectve values they obtaned are stll not better than that obtaned by the proposed algorthm. We have also justfed the performance of the proposed algorthm n a wafer probe testng process based on the ordnal optmzaton theory. References Ahmed, M.A. (2007). A modfcaton of the smulated annealng algorthm for dscrete stochastc optmzaton. Engneerng Optmzaton, 39(6), Aprl, J., Glover, F., Kelly, J.P. & Laguna, M. (2003). Practcal ntroducton to smulaton optmzaton. In: Proceedngs of the 2003 Wnter Smulaton Conference, vol.1 (pp.71-78). New Orleans, LA. Barnett,.S., Grady, M., Purdy, K. & Sngh, A.D. (2005). Explotng predcton defect clusterng for yeld and relablty. IEE Proceedngs-Computers and Dgtal echnques, 152(4), Battt, R. (1992). Frst and second order methods for learnng: Between steepest descent and Newton's method. Neural Computaton, 4(3), Blum, C. & Rol, A. (2003). Metaheurstcs n combnatoral optmzaton: overvew and conceptual comparson. ACM Computng Surveys, 35(4), Chen, C.-H., Wu, S.D. & Da, L. (1999). Ordnal comparson of heurstc algorthms usng stochastc optmzaton. IEEE ransactons on Robotcs and Automaton, 15(1), Chen, F.L., Ln, S.C., Doong, Y.Y. & Young, K.L. (2003). LOGIC product yeld analyss by wafer bn map pattern recognton supervsed neural network. In: Proceedngs of the