Process Monitoring versus Process Adjustment

Transcription

1 CHAPTER 1 Process Monioring versus Process Adjusmen I can be said ha he birh of modern saisical process conrol Ž SPC. ook place when Waler A. Shewhar, a physicis and saisician working for Bell Laboraories, developed he concep of a conrol char in he 1920s. Iniially, he applicaion of conrol chars was confined o he fabricaion of elephone ses and heir componens, bu evenually SPC echniques became popular in oher discree-par indusries, noably in meal machining. Afer some ime, SPC echniques were adoped in coninuous-process Ž chemical. indusries. The main idea behind SPC is o monior he sabiliy of one or more qualiy characerisics. 1 By a qualiy characerisic we mean some measurable physical aribue of a manufacured produc or par ha is of imporance for he producer or for he consumer. Noice ha he word monioring is somewha in conflic wih he word conrol, which has been used hisorically in his area. Tha is, process monioring usually has a more passive connoaion han he more acive process conrol. InSPC, a producion process is hough of running in eiher of wo muually exclusive saes: an in-conrol sae, and an ou-of-conrol sae. Graphical devices called conrol chars were developed by Shewhar o disinguish beween hese wo saes. As long as he char does no signal he exisence of an ou-of-conrol sae, he process is hough o be operaing in saisical conrol. Two sources of variabiliy are conemplaed in SPC, common-cause ariabiliy, variabiliy ha is inheren in he producion sysem and ha can only be modified by alering he exising producion process, and assignable cause ariabiliy, observed variabiliy ha can be raced o a paricular problem Ž e.g., human error, a problem wih a raw maerial, or a machine failure.. Assignable causes occur a unpredicable imes, and he goal of he char is 1 Since he purpose of his chaper is o conras SPC mehods wih engineering conrol mehods, only univariae conrol chars for he mean of a normally disribued process are discussed. Oher ypes of chars, such as chars for dispersion, for aribue daa, or mulivariae chars can be found in mos SPC books Ž e.g., Mongomery,

2 2 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT o deec hem as soon as possible. The diagnosis and correcive acions ha should accompany such deecion are lef o he process engineer or operaor in charge of he process and are no modeled. In conras, common-cause variabiliy is predicable wihin cerain limis. A process is in saisical conrol if assignable causes have been deeced and correced, so hese sources of variabiliy will no influence he process in he fuure and only common-cause variabiliy will remain presen. This makes he qualiy characerisic predicable in he sense ha he fracion of produc ha will fall wihin specificaions can be calculaed. By applying SPC chars coninuously, assignable causes are removed from he sysem and he qualiy characerisics are coninuously improved. The emphasis of his book is no, however, on process monioring ŽSPC chars. bu on process adjusmen using engineering process conrol Ž EPC. mehods. Our approach for moivaing he use of EPC echniques in a qualiy conrol applicaion is o undersand he problems of serially correlaed daa in radiional SPC schemes. One of he key assumpions behind he use of conrol chars is ha successive values of he qualiy characerisic as hey are observed hrough ime are no correlaed wih each oher. As we discuss in his chaper, modern manufacuring mehods and sensor echnologies ofen imply ha qualiy daa are serially correlaed in ime Ž auocorrelaed., and his can have a large impac on he performance of conrol chars. Firs we briefly inroduce basic SPC chars and discuss how he performance of some of hese chars can be assessed under ideal circumsances Ži.e., in he absence of auocorrelaion.. Formal definiions and saisical echniques are inroduced o demonsrae he auocorrelaion of a process and how i can be esimaed from ime-series daa. Then we discuss he performance of SPC conrol chars under he presence of auocorrelaion. The performance of SPC chars is badly affeced by auocorrelaion, and use of engineering process conrol mehods is proposed as an alernaive o compensae for process dynamics. SPC and EPC are no echniques in conflic wih each oher, and he majoriy of researchers in his field agree ha a combined SPC EPC sraegy is necessary in mos indusrial processes oday. In fac, one could define conrol as combining he ars of monioring and adjusmen. In he las secion of his chaper we discuss inegraed SPC EPC approaches and provide a guide o he echniques discussed elsewhere in he book. 1.1 SPC CHARTS: BRIEF OVERVIEW In classical SPC conrol chars, 2 he assumed in-conrol model is Y s q for s 1,2,... Ž Readers familiar wih SPC chars can skip o Secion 1.2.

3 SPC CHARTS: BRIEF OVERVIEW 3 a model someimes referred o as Shewhar s model. Here Y denoes he value of he qualiy characerisic ha was obained from he h sample. 3 The parameer is he mean of he qualiy characerisic, and he random variables 4 form wha is called a whie noise sequence, which means ha G 0 Ž 0, 2. and CovŽ, qj. s 0 for all ime and all lags j 0 where Cov denoes he covariance beween wo random variables separaed j periods or lags Ž and,inhis case.. The noaion Z Ž, 2. qj Z Z means ha he random variable Z is disribued wih consan mean Z and consan variance Z 2.Inoher words, he sequence of errors is uncorrelaed, 4 has zero mean, and has a variance ha is consan in ime. This implies ha Y Ž, 2., and ha CovŽ Y, Y. s 0 for j 0 Ž See Problem qj Shewhar Chars for Averages Several differen ypes of conrol chars were proposed by Shewhar. For he purposes of his chaper, we illusrae Shewhar chars wih he simple case of a char used o monior he sabiliy of he mean of a process, usually called an X char in he SPC lieraure, alhough we use he name Y char for consisency of noaion. In a Shewhar char, samples of size n pars are aken from he producion process wih a cerain frequency. If model Ž 1.1. is he correc descripion of he qualiy characerisic and he errors 4 are normally disribued Ž N. G1, 5 Ž 2 he averages are also normally disribued; ha is, Y N, rn. Ž 2 N,. Y.Tomonior he process, he averages are ploed in ime order on a Shewhar Y char wih limis a k. Y If and are unknown, hey are usually esimaed wih 6 Y R ˆ s Y and Y s d ' n where Y denoes he average of he averages of he samples and R denoes he average range of he samples. Boh of hese quaniies are compued 3 In many SPC books, X denoes he values of he qualiy characerisic. In his book we use X o denoe a conrollable facor or inpu of a process. The variable Y will denoe he process oupu, orqualiy characerisic of ineres. For his same reason we will alk abou a Y char insead of an X char. 4 Boh covariance and correlaion are measures of linear associaion beween wo random variables. In Secion 1.2 we define hese conceps for ime series. 5 A line over a variable denoes he average of he observed values of ha variable. A erage in his book is used o denoe wha some auhors refer o as sample mean, whereas mean in his book refers o a populaion parameer. 6 Throughou his book, a ha Ž. ˆ over a variable or parameer name denoes an esimae. 2

4 4 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT based on a reference se of daa ha was supposedly obained while he process was running in a sae of saisical conrol. The consan d2 depends only on he sample size and is a bias correcion facor for he esimae of Y. I can be read from ables found in mos SPC books. For 2 F n F 9, a good approximaion is d f ' n. Oher esimaes of are more efficien Ž 2 Y see, e.g., Mongomery, 2001., bu he range esimaor is sill very common in pracice. If no subgroups are formed Ž n s 1. and a char on he individual measuremens Yi is desired, hese are ploed on a char wih limis a Y krrd, where R is now he average of he mo ing ranges R s 2 i Yiy Y iy1. Inhis case, since he ranges are compued by arificial samples of size 2, he consan d2 used equals Shewhar s model Ž 1.1. allows us o consider he following wo ypes of process anomalies: hose ha are assignable o some cause and hose due o random or common causes Ž unconrollable noise.. As long as he sequence of random variables Y 4 G1 behaves according o model Ž 1.1., he only variaion presen in he process is random and unconrollable under he presen process condiions. A ime, his variabiliy is modeled by he random variable.inhis case he process is in saisical conrol. For a normally disribued process, assignable causes can make,, or he disribuion of Y vary. Chars for averages are ypically used in conjuncion wih chars ha monior. We hen have ha he design parameers of a Shewhar char are n, k, and he ime beween samples, h. Typically, n s 5 and k s 3 are used. The value of h is seleced direcly or indirecly on economic grounds, for example, based on he cos of sampling or measuring. Example 1.1: A Shewhar Char for A erages. Consider he fabricaion of discree meal pars in a CNC lahe machine. A single dimension of ineres is moniored wih a Y char. Figure 1.1 shows he char ploed using he Miniab sofware package. 7 Fify samples of size 5 each were colleced, and hese were used o obain he esimaes Y s 5.260, R s 2.108, and Y s Ž since d s for n s 5,. 2 giving upper and lower conrol limis Ž CL. of UCL s and LCL s All measuremens are in hundreds of an inch. No average falls ouside he limis; hus he process operaor concludes ha he process was in conrol while he daa were colleced. The conrol limis can hen be adoped o assess saisical conrol based on furher samples. Had any poins fallen ouside he limis, an invesigaion of he cause would need o be aken. Only when he problems associaed wih he occurrence of hese abnormal observaions can be found and correced 7 Ž. Miniab is an easy-o-use saisical sofware ool version 13 was used in his book. Daa can simply be enered in a column of is buil-in spreadshee. The user hen selecs from he op menu bar he opions Sa Conrol Chars and selecs he ype of char o plo. Miniab requess informaion abou where he daa are locaed on he spreadshee, he sample size, and so on.

5 SPC CHARTS: BRIEF OVERVIEW 5 Figure 1.1 Shewhar conrol char for averages. should he abnormal poins be deleed afer correcion of he problems. Ž and he conrol limis recompued. I is recommended ha he limis be recompued periodically wih recen informaion so ha process improvemens are refleced in he limis. For example, he process sandard deviaion will ypically be reduced wih ime afer use of an SPC char. Recompuing he limis will promoe furher improvemens in he process. I should be poined ou, however, ha a common difficuly in pracice is finding an assignable cause once an ou-ofconrol signal is deeced. Even if an assignable cause is found, managemen will no always solve he problem. This may lead o disenchanmen wih use of he char. Raional Subgroups An imporan concep in he applicaion of conrol chars relaes o how samples are aken from a producion process. Shewhar called his recommended sampling scheme he raional subgroup approach. For a char of averages, his consiss of sampling in such a way ha he wihin-sample variabiliy is due simply o common-cause or chance variabiliy, whereas he variabiliy beween samples, if any, should indicae assignable or special causes of rouble in he process. A ypical way of achieving hese goals is o ake he pars or observaions ha make up he sample as close ogeher in ime as possible, so ha we minimize our chances of observing an assignable cause while we collec he n observaions of a sample. In his way, he conrol limis widh is compued based on he wihin-sample variabiliy, which represens common-cause variabiliy. Furher discussion of his opic can be found in Alwan Ž and Mongomery Ž

6 6 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT EWMA Conrol Chars To monior he mean of a process, we can insead consider using he exponenially weighed moving average Ž EWMA. saisic: Zs Yq Ž 1 y. Zy1 for s 1, 2,... Ž 1.2. where is a weigh parameer such ha 0 1. Clearly, as approaches 1, more weigh is given o he mos recen average. As decreases, more weigh is given o older daa, and in he limi when s 0, all he Z s equal Z. 0 I can be shown Ž see Problem ha each observaion Y weigh Ž 1 y. j, ha is, y1 Ý yj receives a j yj Ž. Ž. Z s Y 1 y 1.3 js0 Thus he weighs given o older observaions decrease geomerically wih age, and he geomeric progression is he discree analogy of an exponenial funcion. For his reason, some auhors refer o his char as he geomerically weighed moving average char, bu he name EWMA is more common Ž 2 in pracice. If Y N, rn., applying he expeced value and variance operaors o equaion Ž 1.3., we ge Taking he limi as, 2 2 VarwZ x s 1 y Ž 1 y. Ž 1.4. n 2 y EwZx s 1 y 1 y VarwZ x s n 2 y EwZx s Ž. Ž. Wih hese resuls we can se up he conrol limis of he char in a manner analogous o ha used for Y chars. In an EWMA conrol char, he EWMA saisics Z are ploed ordered in ime on a char wih limis a ˆ k ' ( n 2 y where ˆ can be obained as for he Y char. Thus i can be seen ha he conrol limis of an EWMA can be obained from hose of an Y char by inroducing he correcion facor ' rž 2 y.. There is evidence in favor of always using he limis ˆ 2 ˆ k ( 1 y Ž 1 y. ' n 2 y

7 SPC CHARTS: BRIEF OVERVIEW 7 Table 1.1 EWMA Char Compuaions Y Z UCL LCL which provide a fas iniial response of he char o process upses ha occur early afer process sarup. Thus, as can be seen, he design parameers of an EWMA char are n, k,, and h. Example 1.2: EWMA Char for A erages. Consider he machining process daa in Example 1.1 and suppose ha an EWMA char is o be applied insead. The value s 0.2 is chosen. Using he iniial 50 samples of size 5 each, he esimaed parameers ˆ s 5.26 and ˆY s are obained. The EWMA recursive equaion is iniialized a Z0 s ˆ s Table 1.1 shows he deailed compuaions. As i can be seen, he conrol limis converge rapidly o heir long-run values. Figure 1.2 displays he corresponding EWMA char ploed using Miniab. Figure 1.2 EWMA conrol char for Example 1.2.

8 8 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Cumulaive Sum Conrol Chars A cumulaive sum Ž CUSUM. char is based on he concep of sequenially probabiliy raio ess Ž SPRTs.. A deailed descripion of he SPRT is ouside he scope of his book, 8 so only an inuiive explanaion of he use of he char is presened here. In he SPC lieraure, V mask CUSUM chars are also discussed, bu hese are no easily inerpreable and are difficul o design. A more modern approach, based on a abular CUSUM char, is presened here. The abular CUSUM defines wo one-sided cumulaive sums, where he firs one is SHŽ. s Yy Ž 0 q K. q SH Ž y 1. 9 wih iniial value S Ž. 0 s 0. The noaion Ž x. q s maxž 0, x. H means posiive par and 0 is he assumed in-conrol process mean or arge. This firs sum accumulaes deviaions from ha are greaer Ž i.e., higher. 0 han 0q K. The second sum, SLŽ. s Ž 0 y K. y Yq SLŽ y 1. wih iniial value S Ž. L 0 s 0, accumulaes deviaions from 0 ha are smaller-lower-han y K Ž see Figure The reference value K is usually 0 q q Figure 1.3 How a CUSUM char works. 8 For a comprehensive reamen of CUSUM chars, see he monograph by Hawkins and Olwell Ž Oher iniial values can be used o provide a fas iniial response o changes ha occur shorly afer sarup Ž see Lucas and Crosier,

9 SPC CHARTS: BRIEF OVERVIEW 9 recommended o be se a half he size of he smalles shif in he mean ha one wishes o deec quickly: Y K s 2 where is a muliple ha allows us o measure shif sizes in erms of he sandard deviaion of he saisic being ploed Žif individual measuremens are used, we use insead.. If S Ž i. H or S Ž i. H H his is evidence ha he process is ou of conrol wih respec o Shewhar s model. Inuiively, if several consecuive deviaions above he reference value are obained, he process is abnormally high, and his calls for an invesigaion. A similar argumen applies o consecuive deviaions in he opposie direcion. Typical values for H are four or five imes he magniude of Ž Y or imes if he sample size is 1.. The design parameers of a abular CUSUM char are: n, K, H, and h. Example 1.3: CUSUM Char for A erages. For he same machining process daa as used in Examples 1.1 and 1.2, we can plo a CUSUM char insead. Suppose ha we wish o deec changes in he mean of magniude equal o 1 sandard deviaion of he averages. If he hisorical process sandard deviaion is 0.9 and he process arge is assumed o be 0 s 5.25, we have ha K s 0.5Ž 0.9.' 5 s The limis are se equal o H, where we use H s 4 s 4Ž 0.9.' Y 5 s A deail of he compuaions involved is given in Table 1.2. The resuling char, ploed using Miniab, is shown in L Table 1.2 CUSUM Char Compuaions Ž. Ž. Y S S H L

10 10 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Figure 1.4 Example of a abular CUSUM char showing S Ž. i and ys Ž. i. H L Figure 1.4. Noice ha he CUSUM char signals an ou-of-conrol sae a sample 37. Such signals should be invesigaed Performance of Conrol Chars for Uncorrelaed Daa A classical SPC char can be hough of as a sequence of ess of hypohesis where he goal is o find evidence agains he hypohesized Shewhar model. Therefore, i is naural o hink of he probabiliies associaed wih ype I and ype II errors as performance measures of how well he char works. Consider he simple case of a Shewhar Y char. For his char, he probabiliy of a false alarm is 4 s PType I error4 s P one Y falls ouside limis process is in conrol 2 ' i Since Y N Ž, r n., assuming ha he in-conrol mean is some known value 0,wehave 5 Y y 0 LCL y 0 s 2 PY LCL s 04 s 2 P s 2 Ž yk ½. r' n r' n Ž. where denoes he sandard normal disribuion funcion. Thus he false alarm rae is deermined exclusively by he widh of he conrol limis. Figure 1.5 depics he false alarm probabiliy. Example 1.4: False Alarm Probabiliy of a Shewhar Char for A erages. Consider a Shewhar Y char in which 3 conrol limis are used. In his case we have ha k s 3, and herefore he false alarm probabiliy is s 2 Ž y3. s

11 SPC CHARTS: BRIEF OVERVIEW 11 Figure 1.5 Probabiliy of a ype I error, normal disribuion. Similarly o he siuaion in he false alarm case, we can define he probabiliy of no deecing a shif a a sampling ime: s Pype II error4 s P one Y falls inside limis process is ou of conrol 4 For compuing, weneed o describe exacly wha ype of disurbance is no being deeced. If we assume ha an assignable cause shifs he process mean from o s q, hen so s P LCL F Y F UCL s UCL y Ž 0q. LCL y Ž 0q. s y ž r' n / ž r' n / Ž '. y Ž yk y ' n. s k y n where he second erm is usually negligible. Noe ha is a funcion of he sample size, he process variance, and he size of he shif in he mean. Also, if s 0, we ge s 1 y 2 Ž yk. s 1 y Ž see Figure Figure 1.6 Probabiliy of a ype II error, normal disribuion.

12 12 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Example 1.5: Type II Error Probabiliy for a Shewhar Char for A erages. Consider again a Y conrol char in which samples of size 5 are aken and he widh of he conrol limis is 3 sandard deviaions of he qualiy characerisic. Therefore, we have ha n s 5 and k s 3. Suppose i is of ineres o deec shifs of size equal o 2 sandard deviaions. Then he probabiliy of a ype II error for such a char is s Ž 3 y 2' 5. y Ž y3 y 2' 5. s If we define he power of he char as 1 y power, i can be said ha in any conrol char we desire high power and low. However, o a process engineer or operaor, a discussion abou probabiliies of false alarms, or s and s, migh sound esoeric. In any case, is s low enough? To avoid hese issues, a much beer way of describing he performance of any conrol char is looking a is run lengh performance. The run lengh of a char is he number of samples prior o obser ing an ou-of-limis poin on he char. Since i is assumed ha he daa are generaed by an underlying probabilisic model, he run lengh is a random variable. Depending on he acual sae of he sysem Ž in conrol or ou of conrol., we can alk abou an in-conrol run lengh Ž RL. in and an ou of conrol run lengh Ž RL. ou. In he case of a Shewhar char for averages, i urns ou ha if we have enough hisorical process daa so ha we can consider 0 and o be known, hen RL geomericž. and RL geomericž 1 y. in The raionale is as follows. Consider firs he in-conrol case and compue, in sequence, he following probabiliies: Pou of limis deeced a firs sample process in conrol4 s Pou of limis deeced a second sample process in conrol4 s Ž 1 y. ou. Ž jy1. Pou of limis deeced a jh sample process in conrol4 s 1 y which is jus he probabiliy mass funcion of a geomeric random variable. Thus he in-conrol average run lengh Ž ARL. is in 1 RLy1 Ý ARL in s EwRL in x s RLŽ 1 y. s RLs1 Similarly Ž see Problem 1.13., for a geomerically disribued random variable wih parameer, iisknown ha is variance is given by ' 1 y r, sowe can define he sandard deviaion of he run lengh Ž SRL. as ' ( 1 y 1 y SRL in s VarwRL in x s s 2 ' in

13 SPC CHARTS: BRIEF OVERVIEW 13 Therefore, he sandard deviaion and he average of he run lengh disribuion will be very close if is small. By a similar argumen, he probabiliy mass funcion of he ou-of-conrol 4 jy1 run lengh disribuion is P RL s j s Ž 1 y., and herefore and 1 ARL ou s EwRL ou x s 1 y ' SRL ou s s' ARL ou Ž ARL ou y 1. 1 y Ž. so, again, SRL f ARL for large ARL small power. ou ou ou Example 1.6: A erage Run Lenghs, Shewhar Chars for A erages. Table 1.3 gives ARL values for Y chars ha use differen sample sizes and mulipliers of differen widhs. Noice ha he unis in he able are number of samples unil deecion. For he sample sizes mos common in pracice Ž n F 5,. a Shewhar Y char does no deec small Ž F 1. shifs in he mean rapidly compared o EWMA or CUSUM chars. Large shif sizes, on he order of 2, are deeced very quickly. This is no surprising since Shewhar chars were developed a a ime when large process upses were frequen. Narrowing he conrol limi widh resuls in more frequen false alarms and faser ou-of-conrol deecion. Noe ha o assess he performance of a single char, even wih known parameers, he ARL is misleading, due o he hick righ ail of he run lengh disribuion. For his reason, some auhors sugges looking a he median or a some percenile poins of his disribuion. Appendix 1C gives furher deails abou he use of ARLs as a performance indicaor in conrol chars. The run lengh performance of EWMA and CUSUM chars is very similar if he weigh in he EWMA is small, around 0.1 according o mos recommendaions. These chars deec small shifs in he mean much more Table 1.3 Average Run Lenghs, Shewhar Chars for Averages k s 3 k s 2.5 n s 2 n s 5 n s 10 n s 2 n s 5 n s

14 14 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Figure 1.7 Differen weighs given o he las 15 observaions of a process. The weighs for he EWMA char correspond o a value s 0.4. Ž. Ž. Ž. Figure 1.8 Ychar n s 5, k s 3 applied o a process wih a no auocorrelaion and b posiive auocorrelaion.

15 SPC CHARTS: BRIEF OVERVIEW 15 Figure 1.9 EWMA char Ž n s 5, k s 3, s 0.2. applied o a process wih Ž a. no auocorrelaion and Ž b. posiive auocorrelaion. rapidly han Y chars and should be used insead if small shif sizes in are suspeced in a process. The derivaion of he ARLs of hese chars is more involved and ouside he scope of his book Žsee Woodall, 1983, 1986; Crowder, The reason he EWMA and he CUSUM chars can deec small shifs quickly can be raced o he way ha daa are weighed Ž Huner, In a Shewhar Y char Ž see Figure 1.7., all weigh is given o he poin ploed mos recenly Ži.e., he decision wheher or no o sop a process is based only on he las Y observed.. In conras, EWMA and CUSUM chars give some weigh o all he previous daa. This suggess ha a small shif in he mean of he process will accumulae hrough he EWMA or CUSUM saisics and will rigger a signal sooner han relying on he las poin ploed. Evidenly, an Y char is a paricular case of an EWMA char if he EWMA weigh equals 1.0. Figures 1.8 and 1.9 show a process where daa are uncorrelaed and a process where daa are serially correlaed. Samples of size 5 were aken and Y and EWMA chars were ploed for he daa. If here is no auocorrelaion, he chars perform adequaely, bu for high posiive auocorrelaion a low lags here are a considerable number of poins ouside limis, making inerpreaion difficul. ŽAre hese rue ou-of-conrol signals, or are hey expeced

16 16 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT. given he auocorrelaion? We discuss hese issues in deail in Secion 1.3. Firs, some saisical definiions are needed o explain clearly wha we mean by posiive auocorrelaion a low lags of a sochasic process. 1.2 AUTOCORRELATION OF A PROCESS In Secion 1.1 i was poined ou ha classical conrol chars assume no correlaion beween successive observaions of he qualiy characerisic. In his secion we define in a more precise manner wha is mean by correlaion beween repeaedly observed measuremens of a single qualiy characerisic. We need o know how o esimae he serial correlaion, in case i exiss, and how o sudy is effec on classical SPC chars. To achieve hese goals, he concep of a sochasic process is firs necessary. A sochasic process YŽ., g I4 is a family of indexed random variables, where I is called he index se. Someimes we will refer o he mechanism generaing he sochasic process simply as he process, which can be undersood in he double sense of he underlying sochasic process ha he qualiy characerisic being modeled follows, or as he producion process iself, which in urn generaes he sochasic process Ž i.e., he qualiy characerisic.. In applicaions in his book, will relae o he discree poins of ime a which an observaion is obained by sampling Ži.e., he index se is I s...,y2, y1, 0, 1, 2, and he sochasic process is said o be a discree-ime sochasic process. If I s : y q 4,heprocessisaconinuous-ime sochasic process. For discree-ime sochasic processes, i is cusomary o denoe hem as Y 4 g I,ha is, a subscrip is used for discree-ime indices. Discree-ime processes can be idenified by describing he behavior of is h elemen. Thus we can wrie, for example, Ys q implying ha for he given process Ž Shewhar s model in his case. he equaion holds for all discree poins in ime. Inhis case, using noaion inroduced in Secion 1.1.1, he ime beween observaions h equals. A sochasic process can be hough of as a funcion of wo argumens: namely, Y, Ž w., g I, w g S 4, where S is he sample space of he random variables Y.I Ž. is imporan o poin ou ha for fixed g I, Y, Ž. is a random variable; for fixed w g S, YŽ, w. is a funcion of ime called a realizaion, orrajecory, of he process. A ime series is a se of observaions, or realizaion, of a discree-ime sochasic process. In basic probabiliy heory, his is analogous o he noion of a single observed value of a random variable Ž y. compared o he random variable iself Ž Y.. The se of all possible realizaions of a sochasic process is called he ensemble of he process, as depiced in Figure As in many oher areas of saisics, o perform valid saisical inferences from a sochasic process, we need some noion of how repeaable he underlying random experimen is under idenical condiions. For example,

17 AUTOCORRELATION OF A PROCESS 17 Figure 1.10 Single realizaion Ž darker line. compared o oher possible realizaions ha make up he ensemble of a single sochasic process. one such saisical inference could be o predic o where a qualiy characerisic will move in he near fuure. The only informaion available o us is a single realizaion of a sochasic process, he values recorded of he qualiy characerisic in he pas. If his process is such ha is random variables differ radically a differen poins of ime, no inference could be drawn from a single realizaion since he probabilisic properies of he process during he period of ime when he observaions were aken canno be generalized o oher periods of ime. For he class of models sudied in his book, he noion ha we need o make valid inferences is called saionariy. A sochasic process is sricly saionary if for any ineger k G 1, he join disribuion of Y, Y,...,Y 4 is idenical o he disribuion of 1 2 k Y, Y,...,Y 4 q q q, where ig I, iq g I. Thus he sochasic proper- 1 2 k ies of he process are unaffeced by changes in he ime origin. If in his definiion we look a he case k s 1, we noe ha sric saionariy implies ha he disribuion of Y is he same as he disribuion for Y q. Therefore, sric saionariy implies ha he disribuion of he random variables occurring a differen poins in ime is idenical Mean and Variance of a Saionary Process Since for a sricly saionary discree-ime process Y 4 sy he probabiliy densiy funcion fy Ž. is he same for all, ha is, fž Y. s fž Y.

18 18 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT we have ha H q s EwYx s yfž y. dy y H q w x Ž. y s Var Y s y f y dyy. Thus gives he long-run or asympoic average level of he process a every ime and 2 gives he long-run dispersion of he process a every ime. These long-run quaniies are obained over he ensemble of he sochasic process, which is he populaion from which we sample. For example, EY w x can be hough of as he level of he process a ime averaged over an infiniely large number of possible realizaions. 10 The corresponding poin esimaes are given by ˆ s Y s 1 N Ý y N s1 1 N 2 2 ˆ s Ý Ž y y Y. N s1 where N denoes he lengh of he ime series observed Auocovariance and Auocorrelaion The prefix auo means a reflexive ac upon oneself, hus auocovariance is he covariance ha a process has wih iself. The auocovariance funcion Ž. gives a measure of linear associaion Ž covariance., k beween wo vari- ables of he same process, Y and Y qk, ha are separaed k periods or lags, as a funcion of k. For any discree-ime process, we define s CovwY, Y x s E Ž Y y EwYx.Ž Y y EwY x., k qk qk qk for k s 0, 1, 2,... For a sricly saionary process, he mean is consan for all imes, and he auocovariance reduces o, ks ks Cov Y j, Yjqk s E Ž Yy.Ž Yqk y. for k s 0, 1, 2,... so he auocovariance depends only on he lag k. Also, noe ha 0 s wž. 2 x Ž. 2 E Y y s Var Y s. 10 See Appendix 1A a he end of his chaper for more echnical deails.

19 AUTOCORRELATION OF A PROCESS 19 I is usually beer o scale he auocovariance ino a uniless quaniy by dividing by he process variance. For a saionary process, k k s for k s 0, 1, 2,... 0 where y1 F F 1. Given ha s and s Ž k k yk k yk i.e., he auocorre- laion and auocovariance are e en funcions., i is common o plo hese wo funcions only for posiive lags. Boh he auocovariance and he auocorrelaion a lag k give a measure of he degree of linear associaion beween wo random variables of he same process ha are separaed k periods. When considering relaions beween random variables, i is useful o recall he following implicaions: 1. If Y and Y are independen, hey are uncorrelaed Ž qk i.e., k s 0 for all k.. 2. If Y and Y are correlaed Ž i.e., 0 for some k. qk k, Y and Yqk are dependen. The direcion of he implicaions is imporan; uncorrelaed random variables may or may no be independen. Correlaion is a measure of linear associaion, so here migh be some nonlinear associaion beween uncorrelaed variables. I should be poined ou ha he sandard error of he average Ž Y. is grealy affeced by auocorrelaion. Barle Ž showed ha ) N 0 k Ý N ž N/ ks1 s 1 q 2 1y Y Thus if he sochasic process is compleely uncorrelaed Ž s 0 for all k. k, he sandard error of he average Y is he usual r' N.Ashe Ž posiive. auocorrelaion increases, so does he sandard error and he poin esimae Y becomes less reliable. Poin esimaes of he auocovariance funcion are given by he sample auocovariance funcion, 11 compued as 1 Nyk c s s y y Y y y Y ks 0, 1, 2,... Ž 1.6. ˆ Ý Ž.Ž. k k qk N s1 Similarly, he sample auocorrelaion funcion is given by c k r s ˆ s k s 0, 1, 2,... k k c 0 k 11 If in he denominaor we use N y k, raher han N we ge an unbiased esimae bu wih larger mean square error; hus equaion Ž 1.6. is preferred. Since we defined a biased esimaor of 2 wih a denominaor of N, ifollows ha ˆ s ˆ 2. 0

20 20 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Example 1.7: Coninuous-Flow Chemical Process. Consider he daa provided by Box and Jenkins Ž represening he yields of 70 consecuive baches from a chemical process. 12 Figure 1.11 shows a ime-series plo of he daa and a plo of he sample auocorrelaion funcion of he daa, compued wih Miniab. Observe how he yield ime-series daa jump from bach o bach from one side of is mean Ž approximaely locaed a 50%. o he oher. This is an indicaion of negaive auocorrelaion a lag 1 Ž i.e., r 0issignifican. 1. This can be seen from he sample auocorrelaion plo, which indicaes a significan lag 1 negaive auocorrelaion esimaed a y0.39. Significan negaive auocorrelaion a low lags can indicae a process ha was subjec o succesive adjusmens, in such a way ha adjacen observaions lie on opposie sides of he mean. Example 1.8: A Discree-Par, Machining Process. As a second example, consider he daa in he file Fadal.x, which correspond o 40 measuremens of a dimension machined on aluminum pars processed on a Fadal compuer numerically conrolled Ž CNC. machine ool. The ime-series daa plo and he esimaed auocorrelaion funcion were obained wih Miniab and are shown in Figure The series is clearly nonsaionary, and his reflecs he auocorrelaions, which are significan for many lags. A deailed descripion and inerpreaion of sample auocorrelaion funcions is provided in Chaper 3. In his example, he auocorrelaion of he daa is due o ool wear of he cuing ool. Noice ha he drif in he process does no exacly follow a linear rend or ramp. 13 Example 1.8 illusraes ha no only in chemical processes, bu also in discree-par manufacuring, we migh find considerable auocorrelaion. In chemical processes and, in general, processes where maerial accumulaes, auocorrelaion in he daa observed resuls from he dynamic or inerial elemens in he producion process. In a discree-par manufacuring process, auocorrelaion exiss because of ool wear or because measuremens were aken on dimensions generaed from cuing operaions ha shared he same seup operaion or some oher process condiion ha is common o he pars being produced. Environmenal emperaure flucuaions can also induce auocorrelaion in dimensional daa of machined pars. 12 The daa can be found in he file BJ-F.x and correspond o series F in Box e al. Ž In fac, for his daa se he random walk model, Y s Y q Ž see Chaper 3. y1, provides a beer fi han he linear rend model, Y s a q bq Ž see Problem As discussed in Chaper 3, his is an insance of a nonsaionary process ha can be undersood as an ARŽ. 1 process wih parameer s 1.

21 AUTOCORRELATION OF A PROCESS 21

22 22 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT

23 EFFECT OF AUTOCORRELATION ON SPC CHART PERFORMANCE Need for Saionariy 2 Noe ha he poin esimaes Y, ˆ, c k, and rk are compued based on a single realizaion of he process Ži.e., hey are compued based on ime-series daa.. Inferences based on a single realizaion are possible because he process is saionary. 14 A differen form of saionariy is called weakly or co ariance saionariy. A process is weakly saionary if only is firs wo momens Žmean and covariance. are finie and independen of ime Žin such a case he covariance depends only on he lag.. If each Y is normally disribued and he process is weakly saionary, he process is sricly saionary. As i urns ou, he ype of sochasic models we sudy are fully characerized by heir firs wo momens, so when we refer o saionariy in laer chapers we will be referring o weak saionariy. Weak saionariy can be beer undersood from an engineering poin of view. If he mean of he process is consan, some process engineers would say ha he process is sable. Amore exac definiion of sabiliy is given in Chaper 2 in relaion o ransfer funcion models. As will be seen, sabiliy is a propery of a deerminisic dynamical sysem, whereas saionariy is a propery relaed o a sochasic process. So wha should we do if he process under sudy is nonsaionary? In Chaper 3 we describe differen ransformaions ha can be applied o a nonsaionary process o achieve saionariy. Then, 2, or k can be esimaed from he ransformed series. 1.3 EFFECT OF AUTOCORRELATION ON SPC CHART PERFORMANCE As menioned before, SPC conrol chars assume ha if in conrol, he process has a consan mean and is compleely uncorrelaed wprocess Ž 1.1.x. An imporan pracical quesion is o invesigae wha happens wih he performance of SPC chars as he process exhibis more and more serial correlaion. As one process engineer old his auhor, almos every producion process exhibis auocorrelaion. Posiive auocorrelaion a low lags is common because given he advances in sensor echnologies, measuremens are aken closer ogeher in ime. In discree-par manufacuring, his someimes implies ha every par is measured. Observaions ha were generaed close in ime will end o be similar; hence posiive correlaion a low lags will resul. The following wo examples illusrae he effec of auocorrelaion on he performance of SPC chars. 14 The precise condiion for hese ime averages o be consisen esimaes of he ensemble Ž. averages is called ergodiciy see Appendix 1A.

24 24 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Example 1.9. Suppose ha a process is described by Z s q Z q Ž 1 y. Z q Ž 1 y. Z q... 2 y1 y2 y3 Ys q Z ^ ` _ EWMA of pas Z s Ž 1.7. where 0 F F 1 and Y is he qualiy characerisic. There are wo exreme cases: if s 0, he process is jus Shewhar s process wequaion Ž 1.1.x, where he observaions are compleely uncorrelaed. If s 1, he process is Y s q Z s q q Z y1 which evidenly is highly correlaed 15 wih Yy1 s q Z y1. Ž. 16 Vander Weil 1996 considers his process as increases from 0 o 1. In he realizaions of Figure 1.13b and d, asusained shif in he mean of magniude 5 was induced a ime 100; ha is, he mean of he process changes according o s ½ if F q if where in his example s 5, 0s 100, and 0s 10. This ype of shif suddenly changes he mean of he process a ime 0.Ascan be seen from he figures, for s 0, deecion should be almos immediae for any SPC char, and false alarms should be infrequen. For G 0.8, he shif becomes indisinguishable from he auocorrelaion srucure. Such posiive auocorrelaion will no necessarily worsen he deecion capabiliies of he chars as measured by he ARL performance crierion Ž ou Goldsmih and Whifield, 1961; Lu and Reynolds, 1999, 2001., bu he number of false alarms will cerainly increase. A char ha gives frequen false alarms will soon be abandoned. Example The impac of auocorrelaion on SPC chars was sudied by Ž. Maragah and Woodall 1992, who consider he process Ys q Yy1 q Ž In his case, he process Zs Zy1 q is called a random walk. The auocorrelaion funcion of his process does no decay wih he lag since he process is nonsaionary. 16 Vander Weil considered ploing on SPC chars no he original observaions bu he forecas errors of an IMAŽ 1, 1. process, an approach we discuss in Chaper 3.

25 EFFECT OF AUTOCORRELATION ON SPC CHART PERFORMANCE 25

26 26 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Table 1.4 Number of Poins Ouside Limis Generaed by a Shewhar Char for Individuals Used o Monior 25 Observaions of an AR( 1) Process, Obained by Simulaion Average Sd. Dev. y y y y Ž. Source: Maragah and Woodall where y1 1isaparameer ha allows us o define he auocorrelaion in he Y 4 process. In Chaper 3 we show how his is a paricular insance of he ARIMA family of sochasic processes sudied by Box and Jenkins called an ARŽ. 1 process ŽMaragah and Woodall also considered oher ypes of sochasic processes.. If s 0inequaion Ž 1.8., we obain Shewhar s uncorrelaed process. I is shown in Chaper 3 ha he auocorrelaion funcion of his process is k s k,sohe process is saionary as long as 1. Noe ha for s 1 he process urns ou o be a random walk. In he case of posiive auocorrelaion Ž 0 1,. he movemen of he process is smooher han ha of an uncorrelaed process. In he case of negaive auocorrelaion Ž y1 0,. he process shows a sawooh paern, which compared o Shewhar s process is much more crumpled. Maragah and Woodall Ž compued by simulaion he average number of ou-of-conrol poins ha a Shewhar char for individuals will generae if 25 observaions of process Ž 1.8. wih y0.9 F F 0.9 are used o se he char limis. A char for individuals or a Y char is one in which no subgroups are formed. The conrol limis are compued using he moving-range esima- 17 or s MRrd, where MR s Y y Y ˆY 2 i i iy1. Table 1.4 shows some of heir resuls. Noe ha no shif in he mean was inroduced while simulaing he ARŽ. 1 process. From he able i can be seen ha he number of ou-of-conrol signals increases wih increasing posiive auocorrelaion. For s 0.9, for example, an average of abou four signals will occur in 25 samples. This is a much higher signal rae han he adverised average of one false alarm every 370 observaions for Shewhar chars. Negaive auocorrelaion reduces he number of signals, bu posiive auocorrelaion is much more common in 17 This esimaor was used because i is he mos common esimaor in conrol chars for individuals, bu as menioned below, i is no recommended for auocorrelaed daa.

27 DEMING S FUNNEL EXPERIMENT 27 pracice, as menioned before. Negaive auocorrelaion a low lags inflaes he variance and hence he conrol limis widh becomes oo large, so deecing acual shifs becomes more difficul. The eviden problem in Example 1.10 is ha he limis were compued based on a variance esimae which assumes ha he process is uncorrelaed. If 0 Ž posiive auocorrelaion., adjacen observaions will end o be similar and he moving-range esimaor will underesimae he variance of he process. This will resul in limis ha are oo narrow, producing many alarms compared o he uncorrelaed case. This resul is quie general: The wihin-sample ariance esimaor will underesimae he process ariance in a posii ely auocorrelaed process. If 0 Ž negaive auocorrelaion., he moving-range esimaor will overesimae he rue variance, resuling in limis ha are oo wide, so very few signals will be obained as opposed o he uncorrelaed case. Noe ha his will have an impac on he deecion capabiliies of real shifs Ži.e., wide limis will make he char ake much longer o deec shifs in he process mean DEMING S FUNNEL EXPERIMENT W. Edwards Deming Ž used o alk abou an experimen aimed a showing, by analogy, he dangers of adjusing a process ha is in a sae of saisical conrol. Based on his experimen and some remarks made by Deming in his wriings, some qualiy consulans have aken he exreme view ha a process should never be adjused and ha SPC chars are always sufficien. Given he influence of Deming in he qualiy area, i is relevan o analyze his funnel experimen in deail. MacGregor Ž analyzes his experimen and provides furher useful informaion. The experimen is conduced by mouning a funnel over a arge bull s-eye placed on a fla surface. Marbles are dropped consecuively hrough he funnel, and heir posiion wih respec o he arge is measured. The posiion of he funnel relaive o he arge can be adjused from drop o drop Žsee Figure Deming proposed four adjusmen rules ha mimic, in his analogy, four scenarios someimes found in pracice: Rule 1. Leave he funnel fixed, aimed a he arge, no adjusmen. Rule 2. A drop k Ž ks 1, 2, 3,.... he marble will come o res a poin Yk measured from he arge. Move he funnel a disance yyk from is las posiion. Rule 3. Move he funnel a disance yy k from he arge. Rule 4. Se he funnel for he nex drop Ž k q 1. righ over where he marble came o res a he preceding drop.

28 28 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Figure 1.14 arge. Funnel experimen seup. I is assumed ha iniially, he funnel is aimed a he The posiion of he funnel is he conrollable facor in a feedback mechanism, and he disance from he arge is similar o he deviaion of a qualiy characerisic from arge, a variable commonly moniored in SPC. Rules 2 o 4 conain a feedback acion in he sense ha previous measuremens are used o se he curren posiion of he funnel. Rule 1 corresponds o SPC. Rule 4 describes a siuaion where a process operaor is rying o make each par idenical o he preceding one o achieve consisency. In ha case, he process ends up chasing iself. Rules 2 and 3 are raher inuiive, recommending ha we adjus by as much as he las observed deviaion. Every person who has ried o shoo a a arge probably has used eiher of hese sraegies in a sequence of rials. Figures 1.15 o 1.17 illusrae he rajecories followed by he funnel as looked a from above he able under rules 2 o 4, respecively. There are some imporan assumpions ha Deming made explici wih his experimen. Firs, i is assumed ha he process producing deviaions from arge is in a sae of saisical conrol, which means ha Y 4 G1 obeys Shewhar s model. Furhermore, i is assumed ha he process is iniially on arge; ha is, i is assumed ha he process mean EY w x 1 equals he arge desired. Also, i is assumed ha i is possible o aim he funnel a he arge or a any oher objecive. Under he assumpions saed, he resuls are: Rule 1. This rule produces he minimum variance and a process ha is on arge. Rule 2. The process is on arge and sable bu has wice he variance as in rule 1. Rule 3. The process is nonsable and explodes in an oscillaory paern, wih oscillaions being wider and wider wih ime.

29 DEMING S FUNNEL EXPERIMENT 29 Figure 1.15 Sample rajecory of he posiion of he funnel under rule 2 for a process in a sae of saisical conrol. Ligher lines, firs hree observed deviaions from arge; darker lines, funnel rajecory. Figure 1.16 Sample rajecory of he posiion of he funnel under rule 3 for a process in a sae of saisical conrol. Ligher lines, firs hree observed deviaions from arge; darker lines, funnel rajecory. Rule 4. The process explodes, wih marbles moving farher away from Ž. he arge as ime passes he process is acually a random walk. The resuls above can be beer observed from Figure 1.18, which shows compuer simulaions ha illusrae he behavior of an in-conrol funnel process under he four adjusmen rules. Figure 1.19 shows he corresponding analogous simulaion for a ime-series process ha obeys Shewhar s

30 30 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Figure 1.17 Sample rajecory of he posiion of he funnel under rule 4 for a process in a sae of saisical conrol. Ligher lines, firs hree observed deviaions from arge; darker lines, funnel rajecory. Figure 1.18 Funnel simulaion of he four adjusmen rules for a process ha is in saisical conrol. The dos represen where he marbles sopped on he able.

31 DEMING S FUNNEL EXPERIMENT 31 Figure 1.19 Time-series simulaion of he four adjusmen rules for a process ha is in saisical conrol. model and is adjused by each of he rules. 18 The simulaion programs used o creae Figures 18 and 19 can be found in he Excel spreadshee Deming-funnel. xls. Clearly, afer observing he plos i is quie eviden ha a process ha is on arge and in a sae of saisical conrol should no be adjused. Grubbs Ž provided an adjusing mechanism for a process ha is in saisical conrol bu is iniially off arge. His adjusing mehod is described in Chaper 5. I is simple o see how under Deming s assumpions, rule 1 produces he bes resuls. Le he process be Y s q X y1 where Y is he de iaion from arge observed a drop, X he level of he conrollable facor a run Ž reseing of he funnel., and 4 represens a G 0 18 Ž The observaions in he funnel experimen are an insance of mulivariae bivariae, in his case. daa, given by he wo coordinaes of he marbles on he able, whereas he corresponding ime-series simulaions of Figure 1.19 are univariae since a scalar is observed a each ime insance. The effec of he four rules is he same regardless of he dimension of he vecor Y observed. Mulivariae processes are discussed in Chaper 9.

32 32 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT whie noise sequence. The adjusmens will be given by he differences X y X. Then o ge EY w x y1 s 0, ha is, no expeced deviaion from he arge, we simply se X s 0 for all, which implies ha no adjusmens are w x 2 made. Wih his we have Var Y s, which is he minimum possible variance we can achieve for his process, since i is assumed ha he variance we consider by inroducing in he model is unconrollable. If we follow any feedback conrol rule of he form Xs fž Y,Y y1,... w x 2 i follows ha regardless of he funcion f, his will imply ha Var Y. Thus we should no weak he process if i is on-arge and in saisical conrol. Agood quesion, however, is how much larger han 2 he variance of an on-arge and in-conrol process is under he acions of a given feedback conrol law. In Chapers 6 and 7 we describe how for cerain conrol rules and processes, he inflaion in variance is very moderae and always using an EPC scheme seems reasonable under some condiions. See also Problem 1.8, where a fifh adjusmen rule is proposed for Deming s funnel experimen; such a rule is less drasic han rules 2 o 4 and provides a reasonable increase in variance for he assurance of proecion agains sysemaic variabiliy of differen ypes. In general, if an unconrolled process Y 4 exhibis auocorrelaion Ž includ- ing he exreme case of nonsaionary behavior., a feedback conrol rule such as rules 2 and 3 migh prove beneficial. For example, suppose ha we le Ys q, where is he mean of he process. Then he process is nonsaionary since he mean is wandering and feedback is needed, because oherwise he process, lef unconrolled, will move away from he arge, depending on how moves wih ime. In erms of he funnel experimen, his is he case when he bull s-eye is moving; keeping he funnel fixed in such a siuaion is no he bes alernaive. Example Feedback adjusmen migh be beneficial no only for a nonsaionary process. Suppose ha Y s q a, where CovŽ a, a. o qk 0 for some k 0. Then he process is saionary since he mean and auocovariance are consan in ime, bu he auocorrelaion induces srucured variabiliy ha we can anicipae and compensae for by feedback adjusmen, a opic we discuss nex. 1.5 INTRODUCTION TO ENGINEERING PROCESS CONTROL By a conroller we mean a rule, funcion, or algorihm ha describes how he conrollable facor of a process Ž X. needs be adjused from observaion o observaion. In mechanical, chemical, and elecrical engineering applicaions, conrollers are usually implemened using commercially available sensors,

33 INTRODUCTION TO ENGINEERING PROCESS CONTROL 33 elecronic conrollers, and acuaors. In qualiy conrol applicaions, in conras, he conroller is usually implemened manually by an operaor Žperhaps dialing-in new operaing condiions ino a machine. once he daa become available. Someimes Ž see Chaper 7. he conroller runs on a small compuer and is suggesions are enered manually ino he equipmen by he process operaor. Auomaic closed-loop implemenaions are, of course, possible in qualiy conrol, bu i is preferable o keep an operaor on he loop wih auhoriy o change he seings recommended by he conroller Žas done, e.g., in semiconducor manufacuring; see Moyne e al., An engineering process conroller Ž e.g., a feedback conroller. akes advanage of he auocorrelaion srucure of he sysem o provide beer forecass some number of periods ahead, say Y ˆqk. The conrol acion X is chosen o ensure ha Yˆ qk is close o he arge T in some sense. There are issues relaed o he magniude of he adjusmens ha are imporan in pracice. For example, some conrol schemes may recommend nonzero conrol acions Ž adjusmens. even if he process is close o arge. Mos process operaors will be relucan o make miniscule adjusmens o a process oo frequenly. In oher processes, varying a conrollable facor over a wide range of values from observaion o observaion may be needed if ineres is only in he qualiy characerisic, bu his is no always pracical. Consider, for example, changing he emperaure in a furnace. Evidenly, frequen large emperaure adjusmens are no feasible since he furnace akes ime o cool down and warm up. If making an adjusmen is expensive or frequen adjusmens no feasible, i seems reasonable o wai unil here is a more significan deviaion from arge in order o adjus. These deadband policies can be shown o be opimal ŽBox and Jenkins, 1963; Box and Kramer, 1992; Crowder, 1992; Jensen and Vardeman, if besides quadraic off-arge coss, here is a fixed adjusmen cos, independen of he magniude of he adjusmen, or considerable adjusmen error. Some of hese cos issues are considered in deail in Chaper 5. Huner Ž illusraes wih an example when an engineering process conrol Ž EPC. scheme should be used and when an SPC scheme should be used. Consider he following sochasic model: Z s Z q y1 Y s Z q a Ž 1.9. where 4 and a 4 are wo whie noise sequences uncorrelaed wih each oher and Y denoes he deviaion from arge of he qualiy characerisic. The variable Z is no direcly observable and models he dynamic beha ior of he deviaion from he arge Ži.e., in his case, i models how he process mean changes wih ime.. We observe his dynamic behavior under he presence of measuremen error Ž given by he a s.. The parameer deer- mines how fas he process moves wih ime; ha is, i defines he process

34 34 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT dynamics. As shown in he following chapers, can be relaed o wha is called a firs-order dynamical sysem according o he relaion 19 y s expž T c / where is he ime beween samples. The variable Tc denoes he ime consan of he dynamical sysem, abou wo-hirds of he seady-sae or long-run level of Z. The shorer Tc is, he faser he process dynamics are Ži.e., he faser he qualiy characerisic reacs o changes in he conrollable facor.. Clearly, if T hen 0 and Y s qa c which is jus Shewhar s model. This means ha if sampling is slow relaive o he process dynamics, he process observed will be uncorrelaed and a SPC char will work adequaely. Alernaively, if T hen 0 and Z s Z q c y1 In his case, Ys Zq a is he sum of a random walk and measuremen noise. 20 Sampling rapidly resuls in an observed nonsaionary process, and an SPC char will be inappropriae. An adjusmen sraegy Ž EPC. based on predicions of Yql is preferred in his case. However, an imporan ime-series analysis resul is ha reducing he sampling rae of an nonsaionary process sill resuls in a nonsaionary process Ž MacGregor, Reducing he sampling frequency for SPC purposes is useful only if he original process is saionary. By a similar argumen, if in model Ž 1.9. he observaional noise Ž measuremen error. a 4 dominaes Z 4, an SPC char will be adequae again over an EPC sraegy. Figure 1.20 summarizes when each SPC and EPC apply beer. Some argumens have been raised agains he use of EPC schemes in qualiy conrol. In heir auomaic form, an auomaic process conroller Ž APC. may mask assignable causes ha need furher invesigaion. For example, i could be difficul o see if here is a problem wih hermal insulaion if a emperaure conroller is acing on a machine. Here an assignable cause Ž e.g., a broken insulaion seal. will no be fixed, and he operaion of he process wih he assignable cause will be more expensive han if he insulaion problem had been deeced and correced. In addiion, large process changes may no be conrollable by an EPC scheme, since he conroller may become sauraed before compensaing 19 Process dynamics conceps are explained in Chapers 2 and 4. As described in Chaper 4, model Ž 1.9. is a sae-space model known in he ime-series lieraure as he seady model. 20 A sum ha resuls in a correlaion paern idenical o ha of an IMAŽ 1, 1. process Žsee Chaper 3..

35 COMBINED EPC SPC APPROACHES 35 Ž. Figure 1.20 When SPC and EPC should be applied o a process. Adaped from Huner, compleely for he disurbance. These issues can be resolved by an inegraed EPC SPC approach in which he SPC scheme acs as a supervisor of he EPC scheme. This is represened in he origin of he diagram in Figure Noe ha he figure implies ha EPC may be needed or no, bu SPC is always necessary. The disincion of when o use EPC and SPC ools is no as clear-cu as Figure 1.20 implies. There are cases when EPC echniques can be used under condiions when SPC is recommended by he figure. Some of hese cases are discussed in Chapers 6 and COMBINED EPC SPC APPROACHES A considerable amoun of work has appeared in he lieraure during he 1990s abou mehods ha combine EPC and SPC schemes for he same process in an inegraed form. The raionale is ha large, abrup changes in a process sill need o be deeced and correced as soon as possible, since an EPC scheme will ypically have a hard ime compensaing agains hem. Wih his we have he advanages of closer conrol o arge wih a smaller variance even if here is auocorrelaion due o process or noise disurbances ŽEPC benefi.. Furhermore, a coninuous improvemen effor is pu ino place hrough he deecion and removal of assignable causes of variaion ha resul in large, unpredicable changes Ž SPC benefi..

36 36 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Figure 1.21 SPC EPC inegraion possibiliies: Ž a. SPC applied o he qualiy characerisic; Ž b. SPC applied o he process adjusmens. In an inegraed EPC SPC approach, i is no clear wha o monior wih an SPC char. There are a leas wo possibiliies, illusraed in Figure Monior he qualiy characerisic. Here he char signals if large deviaions from arge are observed. The oupu of a feedback-conrolled process may sill be correlaed Ž see, e.g., Chaper 5., and monioring for abrup changes in auocorrelaed daa becomes an imporan issue. wsee Basseville and Nikiforov Ž for a comprehensive reamen. x 2. Monior he adjusable facor. The raionale is ha a large change in he qualiy characerisic will resul in large adjusmens made, so he adjusmens should have informaion we can use for monioring he process. The adjusmens hemselves may be auocorrelaed, and he commens made above abou SPC and auocorrelaed daa also apply here. Furhermore, here will be a delay beween he observed qualiy characerisic and he compued adjusmen ha should be considered. The firs of hese approaches Ž monioring he qualiy characerisic. was sudied by Vander Weil e al. Ž and Tucker e al. Ž under he name algorihmic saisical process conrol and illusraed heir approach wih a bach polymerizaion example. Mongomery e al. Ž also considered monioring he qualiy characerisics. Applicaion of his approach o a coninuous polymerizaion process was repored by Capilla e al. Ž Some of hese applicaions are discussed in Chaper 5. Example 1.12: Monioring a Conrolled Process. Consider he daa in he file CNC.x, which relaes o a CNC lahe machine. Cylindrical meal pieces are produced in his operaion and he qualiy characerisic Ž Y. is he

37 COMBINED EPC SPC APPROACHES 37 Figure 1.22 EWMA conrol chars used o monior a conrolled process: Ž a. SPC char applied o he qualiy characerisic; Ž b. SPC char applied o he adjusmens. ouside diameer of he meal cylinders. The machine se poin Žaimed-a value. X is adjused according o he rule X y X s 0.3Ž T y Y. y1, where T is he arge diameer for he par and equals 15,010 unis, where 1 uni s inch. Sixy pars were processed. Figure 1.22 shows EWMA conrol chars applied o he qualiy characerisics and o he adjusmens. A value of s 0.2 was used in hese chars. Observe how each char deecs an abnormal process behavior around par 55. In his example, he adjusmens X y X 4 and he qualiy characerisic Y 4 are simply relaed Ž y1 here is a srong negaive correlaion beween hese wo processes, and his resuls in chars ha work almos idenically.. However, here migh be cases where he inpu oupu relaion is more complex, and hen i will make a difference if we monior he adjusmens or he qualiy characerisics. Of paricular imporance are inpu oupu delays, perhaps because measuremens are aken in an offline laboraory afer sampling. When delays are significan, an assignable cause will appear delayed in he adjusmens series wih respec o he qualiy characerisic series, and his needs o be aken ino accoun. Combinaions of he wo basic inegraed approaches are also possible. For example, Tsung and Shi Ž propose join monioring of he qualiy characerisic and he conrollable facor of a feedback-conrolled process using mulivariae SPC echniques. We can summarize by saying ha he goals of having an SPC char added o an EPC scheme include Ž Falin e al., he following: 1. Verify he adequacy of he adjusmen rule. 2. Help idenify he causes of changes in performance.

38 38 PROCESS MONITORING VERSUS PROCESS ADJUSTMENT Figure 1.23 Guide o he use of EPC and SPC echniques discussed in his book. We close his chaper wih an overview of he various univariae SPC and EPC echniques described in his book and when each should be used Ž Figure Addiional mulivariae adjusmen schemes are discussed in Chaper 9. PROBLEMS 1.1. Consider he machined pars daa of Example 1.8 Žsee he file Fadal. xls.. Using Miniab s regression opions, fi a linear rend model Ys a q b q and a random walk model Ys Yy1 q o he daa. Which model fis beer? Try fiing he quadraic model Y s a q b q c 2 q insead. Does his provide beer fi han he random walk model? 1.2. Consider again he daa in example 1.8 and he model Ys Yy1 q X q, where X is he value aimed a Ž he se poin. y1. Suppose ha he machine is adjused o he arge every period by seing i a Xs T y Y where T s is he arge. Draw a ime-series plo for he conrolled process Y 4.