An Introduction to Statistical Process Control

Transcription

1 An Inroducion o Saisical Process Conrol F. F. Gan Deparmen of Mahemaics Naional Universiy of Singapore Inroducion The basic idea in saisical process conrol (SPC) is o ake random samples of producs from a manufacuring line and examine he producs o ensure ha cerain crieria of qualiy are saisfied. If he producs sampled are found o be of inferior qualiy, hen he manufacuring process is checked o seek ou assignable causes of inferior qualiy o bring he process back o conrol. In he pas when producs were hand-made one a a ime, SPC was a redundan concep as all he producs made were checked for conformiy o cerain crieria of qualiy. An example of early manufacuring processes is coin molding whereby meled meal is poured ino a mold. When he meal hardens, he coins are removed and polished. Any defec found would immediaely be raced back o hese assignable causes: (i) he mold is defecive, (ii) he meal used is of inferior qualiy, (iii) he emperaure of he meled meal is oo high or oo low, (iv) he worker is no skillful enough, or ( v) oher causes ha could be found easily. The Indusrial Revoluion brough along machines ha can manufacure producs a breakneck speeds, so ha 1% inspecion is virually impossible (a leas in he pas) or very cosly. The procedure of aking random samples of producs from a manufacuring line a inervals and checking he qualiy is a reasonable alernaive. This procedure mus have been pracised for a long ime before Shewhar (1931) formally proposed a graphical procedure for monioring qualiy. The proposed procedure is now commonly known as he Shewhar conrol char. 8

2 Consider a manufacuring process producing ennis balls where he weigh of a ball is a key measure of qualiy. According o official ennis rules, he weigh of a ennis ball should be 2 ounces. In order o monior he qualiy, random samples of five balls each, for example, are aken from a manufacuring line a regular inervals. The weigh of each ball is measured and he sample mean of he measuremens is ploed agains he ime or he sample number. An example of such a graph is displayed in Figure 1. Three horizonal lines represening lower, upper conrol limis (LCL and UCL) and he arge weigh are also displayed. Each ploed poin gives an indicaion of wheher he process is in conrol or no. The process is considered o be in conrol if a ploed poin is wihin he wo char limis. Any poin ploed below he LCL or above he UCL is considered an ou-of-conrol poin and gives an indicaion ha he process could be ou-of-conrol. Any unusual paern like 6 or 7 consecuive poins above he arge value or a cyclical paern of poins also indicaes ha he process could be ou of conrol. The 3rd o 9h, 51h o 57h and 63rd o 68h poins are all above he arge value providing evidence ha he process could be ou of conrol a he early and also he laer sages. 2.6 w e g 2. h 1.98 x UCL.... ~ n, Onn,., ~ Oo Oo " oo I'N v o - LCL... I I I I I I I I I I I T r Figure 1. A Shew har Conrol Char for Monioring he Weighs of Tennis Balls Produced. If ou-of-conrol poins are found, he process should be checked immediaely for assignable causes and if found, should be removed o bring 9

3 he process back o conrol. Someimes, i is also likely ha no assignable cause can be found and ha he ou-of-conrol poins could be a resul of some inheren or naural variaions of he process. The qualiy conrol engineers should coninue o be on aler unil he nex few poins ploed are all wihin conrol or no unusual paern is found. I is pruden o increase he frequency of sampling a his aler sae. I is imporan o undersand ha a conrol char is merely a procedure for monioring he qualiy of producs from a manufacuring line, hereby keeping he manufacuring process in conrol. In oher words, a conrol char is only capable of removing assignable causes of variaions in he qualiy bu no he causes of random variaions inheren of a manufacuring process. For example, consider coin molding, he coins produced can only be as good as he mold. If coins wih sharper images are required, hen he enire mold has o be redesigned. Saisical analyses performed on manufacuring processes o seek ou opimal operaing condiions, hence reducing he variance in he qualiy are generally known as experimenal designs. The well known Taguchi mehod which is credied for much of he successes in Japanese indusries falls under he caegory of experimenal designs. The hird caegory of saisical qualiy conrol is accepance sampling. A cusomer upon receiving a lo of producs from a supplier, akes a random sample from he lo o examine he qualiy. A decision is hen made wheher o accep or rejec he lo based on he sample. A supplier who keeps receiving an unusually high proporion of rejeced los from cusomers would be forced o employ more sringen qualiy conrol effors in he facories, hereby improving he qualiy. Unlike conrol char procedures and experimenal designs, accepance sampling only has an indirec effec on improving he qualiy of producs. The oher ypes of conrol chars are inroduced in he nex secion. An applicaion of conrol char procedures o he cerificae of enilemen daa se is hen considered. Saisical Conrol Chars There are four main ypes of conrol chars commonly used in manufacuring lines. According o he populariy of usage, hey are (1) She- 1

4 whar, (2) cumulaive sum (CUSUM), (3) exponenially weighed moving average (EWMA) and (4) sraigh moving average (SMA) chars. The Shewhar char has already been inroduced in he previous secion. Le x 1, x 2,.. be a sequence of independen and idenically disribued sample means. If f-lo is he in-conrol process mean, hen he expeced value of X - f-lo is zero. Thus, wihou loss of generaliy, he arge mean is assumed o be zero. The upper-sided and lower-sided CUSUM chars proposed by Page (1954) are obained by ploing and S = max{o, S-1 + (x- k)} T = min{o, T-l+ (x + k)} agains he sample number for = 1, 2,..., respecively where he char parameer k is a suiably chosen posiive consan. The iniial saring values So and T are usually chosen o be zero. c.8... u.6 UCL s u.4 o o M ocp o.2 S Figure 2. An Upper-Sided Cumulaive Sum Conrol Char for Monioring he Weighs of Tennis Balls Produced. ~~ ~~ ~ ~. U' c ou- -v Oo u -.2- s u -.4- M -.6- LCL T -.8 I I I I I I I I I T I I I Figure 3. A Lower-Sided Cumulaive Sum Conrol Char for Monioring he Weighs of Tennis Balls Produced. 11

5 The upper-sided CUSUM char is inended o deec an upward shif in he mean and i issues an ou-of-conrol alarm a he firs for which S ;::: UCL. Similarly, he lower-sided CUSUM char is inended o deec a downward shif in he mean and i issues an ou-of-conrol alarm a he firs for which T ::; LCL. The upper-sided CUSUM char remains inacive (ha is, S = ) as long as X < k which means ha a CUSUM char wih a large value of k would no be sensiive o small shifs in he process mean. Thus, he value of k can be chosen such ha i is opimal in deecing a paricular shif in he mean which is usually aken o be he smalles shif ha is he leas olerable. The Shewhar char is a special case of he CUSUM char obained by seing boh LCL and UCL o zero. The upper-sided and lower-sided CUSUM chars consruced for he ennis balls' daa se are displayed in Figures 2 and 3 respecively. The lower-sided CUSUM char remains inacive mos of he ime indicaing ha here is no evidence of any downward shif in he process mean. In comparison, he upper-sided CUSUM char is more acive and an ouof-conrol alarm is issued a he 66h sample, indicaing ha he process could be ou of conrol. The EWMA char firs invesigaed by Robers (1959) is obained by ploing agains he sample number for= 1, 2,... where A is a smoohing consan such ha < A ::; 1. The saring value Q is usually chosen o be he in-conrol process mean. An ou-of-conrol alarm is issued a he firs for which Q ::; LCL or Q ;::: UCL. The consan A is he weigh given o he mos recen sample mean and 1 - A is he weigh given o he pas hisory. The Shewhar char is also a special case of he EWMA char wih A = 1 which means ha he enire weigh is given o he mos recen sample mean and no weigh is given o he pas hisory. The EWMA char for he ennis balls' daa se is displayed in Figure 4. The EWMA char ploed wih A =.154 shows a clearer rend of he process mean han he Shewhar char. The smoohness of he curve is deermined by he consan A. The EWMA char issues an ou-of-conrol alarm a he 64h sample mean providing evidence ha he process mean could have shifed upwards recenly. The SMA char is obained by ploing x 1 agains = 1, (x 1 + x 2 )/2 12

6 agains = 2,..., (x1 + i Xm-d/(m- 1) agains = rn- 1 and he average of he m mos recen sample means M = (i-(m-1) + i-(m-2) i-1 + i)/m agains for = m, m + 1,... An ou-of-conrol alarm is issued a he firs for which M :=; LCL or M 2:: UCL. The Shewhar char is also a special case of he SMA char wih m = 1. The SMA char wih m = 12 for he ennis balls' daa se is displayed in Figure 5. The smoohness of he curve of a SMA char is deermined by he consan m. The SMA char also issues an ou-of-conrol alarm a he 64h sample. A comparison of he EWMA and SMA chars shows ha boh are very similar in showing he rend of he mean E \V 2.5 M 2. A Q UCL od:po o ooo oooeo o (j. - ooo Oooo o o oooo ooo " no,.., nnooo o o vv v o (}.Jo ooooo ovo - LCL I I I I I I I I I I I I I Figure 4. An Exponenially Weighed Moving Average Conrol Char for Monioring he Weighs of Tennis Balls Produced s 2.5 M A 2. M UCL... oooooo o r:xy:;po oooo - oo o ooo nooooo oo Oo ooovvv o o v "'Vo - ooo LCL I I I I I I I I I I I I I Figure 5. A Sraigh Moving Average Conrol Char for Monioring he Weighs of Tennis Balls Produceds 13

7 vv e 1 g h w e 1 g h w e 1 g h w e 1 g h Shew har Conrol Char I I I I I I I I I I I I I Upper-Sided Cumulaive Sum Conrol Char I I I I I I I I I I I Exponenially Weighed Moving Average Conrol Char Sraigh Moving Average Conrol Char I I I I I I I I I I I Figure 6. Allocaions of Weighs o he Sample Means by he Various Conrol Chars for he Tennis Balls' Daa Se. 14

8 The weighs allocaed o he sample means by he various conrol chars for he ennis balls' daa se are displayed in Figure 6. Graphs similar o Figure 6 were firs given by Huner (1986). A Shewhar char pus all is weigh on he mos recen sample mean and compleely ignores he pas hisory and i is hus sensiive o large shifs in he process mean. The CUSUM char is he mos 'inelligen' in he sense ha when i becomes acive on encounering a sample mean greaer hank, i begins o pu equal weighs on he presen and all fuure sample means unil i becomes inacive again or issues an ou-of-conrol alarm. The EWMA char gives he larges weigh A o he mos recen sample mean, a weigh of A(1 -A) o he nex mos recen sample mean and so on according o he coefficien of x in he equaion Q = (1- A)Qo +A 2:::(1- A)-ixi. In oher words, he weigh decreases exponenially owards he pas. The SMA char based on m gives he same weigh o hem mos recen sample means. Unlike he CUSUM char, he number of sample means given he same weigh by he SMA char is always he same. Compared wih he Shewhar char, he CUSUM, EWMA and SMA chars are much more sensiive in deecing small and moderae shifs in he process mean and only slighly less sensiive in deecing large shifs. i=l Cerificae of Enilemen Example In an effor o minimize congesion on he roads, he Singapore governmen implemened a quoa sysem o conrol he growh of he vehicle populaion in May 199. A poenial buyer has o bid and secure a cerificae of enilemen (COE) before he buyer can purchase a car. The car is hen allowed o be on he roads for a period of 1 years. In order o use he car for anoher 1 years, anoher COE based on he prevailing quoa premium has o be purchased. The prevailing quoa premium is based on he sraigh moving average of he quoa premiums of he mos recen 12 monhs. The quoa premiums for vehicles wih engine capaciies in he range cc since May 199 ill Augus 1993 are lised in Table 1. A plo of he quoa premium agains he monh is displayed in Figure 7. 15

9 Q 3 u 25 a 2 p 15 r e m u m r~......?" Monh --- Exponenially Weighed Moving Average ---- Sraigh Moving Average Figure 7. The COE Quoa Premiums for Vehicles wih Engine Capaciies in he Range cc from May 199 o Augus Q u a p r e m 1 u m Monh Adjused Exponenially Weighed Moving Average ---- Adjused Sraigh Moving Average Figure 8. The COE Quoa Premiums for Vehicles wih Engine Capaciies in he Range cc from May 199 o Augus

10 Table 1. COE Quoa Premiums for Vehicles wih Engine Capaciies in he Range from May 199 o Augus c 2 u s u 15 M S Monh Figure 9. An Upper-Sided Cumulaive Sum Conrol Char for COE Quoa Premiums for Vehicles wih Engine Capaciies in he Range cc from May 199 o Augus

11 The variances of Q and M for large are given by.a.o-}/(2- A.) and a}/m respecively. Equaing hese wo variances yields m = (2- A.)/ A which provides a reasonable way of finding an EWMA char ha is 'compaible' o an SMA char based on m. The EWMA (based on A=.154) and SMA (based on m = 12) for he quoa premiums are also displayed in Figure 7. In he beginning, he process was in conrol for approximaely 19 monhs and hen wen compleely ou-of-conrol afer ha. Since an increase in he number of COE released each monh has no curbed he rise in he quoa premium, he main assignable cause for his ou-of-conrol sae has o be he srong economy prevailing locally. As long as his assignable cause says, he quoa premium will coninue o spiral up unil i sabilizes a a new value. Boh he EWMA and SMA show a very similar rend of he quoa premium and boh consisenly lag behind he quoa premium from he 13h monh onwards. The expecaion of Q is given by E(Q) = (1- A.)Qo +A. _'L(l-.A.)-iE(Xi) If he process mean is a consan, ha is E(X) = J.l.o and seing Q = p, i can be shown ha E(Q) = J.l.O However, if he process mean increases linearly wih, ha is E(X) = p, +!::l., hen i can be shown ha i=l for =, 1, 2,.... Thus, he EWMA underesimaes he rue process mean by he amoun (1-.A.)[l- (1-.A_)]!::l./.A.. For large, his bias is approximaely equal o (1-.A.)!::l./ A.. Relabeled he sample number such ha E( X ) = p +!::l. for =, 1, 2,... and E(X) = p for = -r, -r + 1,..., -1. If r ~ m- 1, he expecaion of he M is given by E(M) = J-Lo + ( + l)!::l./(2m) = J.l.o +!::l.- (!::l.- ( + l)!::l./(2m)), for=,1,2,...,m and E(M) = P,o +!::l.-!::l.(m- 1)/2, 18

12 for = m + 1, m + 2,... For ~ m + 1, he bias of M is hus given by ~( m - 1) /2 which is exacly he same as he asympoic bias of he EWMA, (1 -)..)~/A. The EWMA and SMA may be adjused such ha hey are unbiased esimaors of a process mean ha is increasing linearly: E(Q + (1- )..)[1- (1- )..)]~/A) = J-o + ~, =, 1, 2,..., E(M + ~- ( + 1)~/(2m)) = J-lo + ~, =, 1,..., m, and E(M- ~(m- 1)/2) = J-lo + ~, = m + 1, m + 2,... The quaniy ~ can be added o hese unbiased esimaors so ha hey can be used o predic he fuure process mean J-lo + ~( + 1). The quaniy~ which is he increase in he quoa premium per monh has o be esimaed. Based on he plo of quoa premiums in Figure 7, i is reasonable o make he assumpion ha he process mean a he 13h monh is J-lo and hen shifs linearly upwards as J-lo + ~. The quaniy ~ is esimaed here by fiing a leas-squares line o he quoa premiums from 13h o 44h monhs. The leas-squares line is given by Quoa Premium= , for = 13, 14,... and ~ = 58. These wo predicors are displayed in Figure 8 and hey provide much beer fi compared wih he unadjused curves. The prediced quoa premiums for Sepember 1993 using EWMA, SMA and regression procedures are shown in Table 2. The upper-sided CUSUM char wih k = consruced for he quoa premiums is displayed in Figure 9 and i indicaes ha he process has compleely gone ou of conrol. The CUSUM char may also be used o predic fuure quoa premiums. Assuming E(X) = J-lo + ~, = 1, 2,... For an upper-sided CUSUM char ha remains acive from ime wih S =, i can be shown ha s =I: xi -k, i=l 19

13 and E( S) = (Jo - k +!::.j2) +!::. 2 /2 = a + (32. By fiing a quadraic curve wih no inercep on he verical axis using he values of S for = 18, 19,..., 44, he leas-squares esimaes of a and (3 are given by & = 4499 and ~ = 219. Esimaes of Jo and!::. are given by j1 = and A = 439 from which he fuure mean can be esimaed. The prediced quoa premium for Sepember 1993 using he CUSUM procedure is also displayed in Table 2. As long as he mean coninues o increase linearly wih he main assignable causes remaining unchanged or varying lile, hese prediced values will be accurae. If he mean quoa premium sabilizes, hen hese models mus be adjused accordingly. The prediced values in Table 2 provide idea for realisic bid amouns for very serious car buyers who mus ge heir COEs by Sepember Table 2. Prediced Quoa Premiums for Sepember 1993 Obained Using EWMA, SMA, CUSUM and Regression Procedures EWMA SMA CUSUM Regression $26,472 $25,68 $25,237 $26,735 Conclusions The main objecive of SPC is on-line monioring of he qualiy of producs from a manufacuring line. The Shewhar char is he simples ype bu i is only sensiive in deecing large shifs in he process mean. The CUSUM, EWMA and SMA chars are much more sensiive in deecing small and moderae shifs in he process mean and only slighly less sensiive in deecing large shifs. A more horough reamen of SPC can be found in Duncan (1986) who has provided ables for deermining he char parameers of a Shew har char. Design procedures for deermining opimal char parameers of CUSUM and EWMA chars are given by Gan (1991) and Crowder (1989) respecively. Alhough he SMA has he advanage ha i is a quaniy ha is easily undersood bu he run 2

14 lengh (defined as he number of samples aken unil an ou-of-conrol alarm is issued) properies of he SMA char remains inracable. Unlike he CUSUM and EWMA chars, he SMA char canno be approximaed as a Markov Chain. In recogniion of Deming's effor in promoing qualiy conrol in Japan, he Deming Prize was insiued in 1951 by he Union of Japanese Scieniss and Engineers. The presigous award and he benefis derived from sysemaic qualiy conrol effor in a company have encouraged many companies o compee for he award. The use of conrol chars is lised as one of he crieria for he Deming Prize. Finally, i is imporan o undersand ha in order o reduce he variance of qualiy of producs, experimenal designs would have o be vigorously conduced o seek ou opimal operaing condiions. Conrol char procedures should hen be implemened o mainain such opimal operaing condiions. Acknowledgmen I would like o hank Ms Shrlinda See of he Auomobile Associaion of Singapore for providing me wih he COE daa se. References [1] S. V. Crowder, Design of Exponenially Weighed Moving Average Schemes, Journal of Qualiy Technology, 21 (1989), [2] F. F. Gan, An Opimal Design of CUSUM Qualiy Conrol Conrol Chars, Journal of Qualiy Technology, 23 (1991 ), [3] S. J. Huner, The Exponenially Weighed Moving Average, Journal of Qualiy Technology, 18 (1986), [4] E. S. Page, Coninuous Inspecion Schemes, Biomerika, 41 (1954), [5] S. W. Robers, Conrol Chars Based on Geomeric Moving Averages, Tecbnomerics, 1 (1959), [6] W. A. Shewhar, Economic Conrol of Qualiy of Manufacured Producs, Van Nosrand, New York (1931). 21