Exponentially Weighted Moving Average Control Charts with Time-Varying Control Limits and Fast Initial Response

Transcription

1 Exponenially Weighed Moving Average Conrol Chars wih Time-Varying Conrol Limis and Fas Iniial Response Sefan H. Seiner Dep. of Saisics and Acuarial Sciences Universiy of Waerloo Waerloo, Onario N2L 3G1 Canada Absrac The conrol limis of an exponenially weighed moving average (EWMA) conrol char should vary wih ime, approaching asympoic limis as ime increases. However, previous analyic analyses of EWMA chars consider only asympoic conrol limis. In his aricle, he run lengh properies of EWMAs wih ime-varying conrol limis are approximaed using nonhomogeneous Markov chains. Comparing he average run lenghs of EWMA wih ime-varying conrol limis and resuls previously obained for asympoic EWMA chars shows ha using ime-varying conrol limis is akin o he fas iniial response (FIR) feaure suggesed for Cumulaive Sum (CUSUM) chars. The ARL of he EWMA scheme wih ime-varying limis is subsanially more sensiive o early process shifs especially when he EWMA weigh is small. An addiional improvemen in FIR performance can be achieved by furher narrowing he conrol limis for he firs 20 observaions. The mehodology is illusraed assuming a normal process wih known sandard deviaion where we wish o deec shifs in he mean. Keywords: Average run lengh; Cumulaive Sum; CUSUM; Fas Iniial Response (FIR); EWMA; Non homogenous Markov Chain.

2 2 1. Inroducion EWMA conrol chars, and oher sequenial approaches like Cumulaive Sum (CUSUM) chars, are an alernaive o Shewhar conrol chars especially effecive in deecing small persisen process shifs (Mongomery, 1991). Firs inroduced by Robers (1959), EWMA chars have a fairly long hisory, bu only recenly have is properies been evaluaed analyically (Crowder 1987; Lucas and Saccucci 1990). The EWMA also is known o have opimal properies in some forecasing and conrol applicaions (Box, Jenkins, and MacGregor, 1974). In his aricle we focus on he qualiy monioring applicaions. For monioring he process mean, he EWMA conrol char consis of ploing: z = λ x + ( 1 λ )z 1, 0 < λ 1, (1) versus ime, where λ is a consan and he saring value z 0 is se equal o an esimae of he process mean, ofen given as x calculaed from previous daa. In his definiion x is he sample mean from ime period, z is he ploed es saisic and λ is he weigh assigned o he curren observaion. The definiion of he EWMA es saisic given in (1) can be adaped o monior any process parameer of ineres. By wriing ou he recursion in (1) he EWMA es saisic is shown o be an exponenially weighed average of all previous observaions. In qualiy monioring applicaions, ypical values for he weigh λ are beween 0.05 and 0.25, alhough larger values may be used in forecasing and conrol applicaions. In he limiing case, wih λ = 1, he EWMA char is he same as a Shewhar X conrol char. Using an EWMA char, he process is considered ou-of-conrol whenever he es saisic z falls ouside he range of he conrol limis. EWMA conrol limis are discussed in deail in he nex secion. As shown in Monogomery (1991), he conrol limis for EWMA chars should be ime-varying since he variance of he es saisic z depends on, because he effec of he

3 saring consan z 0 decreases as increases. However, all pas analyic sudy of he 3 properies of he EWMA char have used fixed (asympoic) conrol limis o make he analysis easier. This aricle presens mehodology for deermining he expeced value and sandard deviaion of he run lengh of he EWMA char wih ime-varying conrol limis. Numerical resuls are given for monioring he mean of a normal disribuion. No surprisingly, he resuls show ha EWMA chars wih ime-varying conrol limis has shorer average run lenghs (ARLs) han EWMA chars wih asympoic conrol limis for sar up qualiy problems. The effec for ou-of-conrol mean values is more pronounced han for he in-conrol case, especially for large process shifs. As a resul, EWMA conrol chars wih ime-varying conrol limis are appropriae in all siuaions where he iniial qualiy level is suspec. This is useful because processes are fairly likely differen from he arge value when a conrol scheme is iniiaed due o sar-up problems or because of ineffecive conrol acion afer he previous ou-of-conrol signal. In addiion, ofen afer a process change or adjusmen we wish o quickly confirm ha he change had he desired effec. Using ime-varying conrol limis has an effec similar o he fas iniial response (FIR) feaure recommended by Lucas and Crosier (1982) for CUSUM chars, since i helps deec problems wih he sar up qualiy. For CUSUMs, he FIR feaure subsanially decreases he ARL for an ou-of-conrol process while decreasing he ARL of an in-conrol process only slighly. For EWMA chars, Lucas and Saccucci (1990) suggesed he simulaneous use of wo one-sided EWMA chars wih iniial saes differen han zero as an implemenaion of he FIR feaure. One EWMA char moniors for increases in he process parameer, while he oher char moniors for decreases. Rhoads, Mongomery and Masrangelo (1996) adap he Lucas and Saccucci approach by allowing he one-sided EWMA o have ime-varying conrol limis as given by (2) and discussed in Secion 2. Rhoads e al. (1996) compare he run lengh properies deermined hrough simulaion.

4 Boh hese implemenaions of FIR-EWMA chars require monioring wo EWMA chars o 4 monior a process for wo-sided shifs. This aricle shows ha he use of ime-varying conrol limis makes a EWMA char more sensiive o sar up qualiy problems han he radiional asympoic limis. If addiional proecion o sar up qualiy problems is desired he furher narrowing of he ime-varying conrol limis according o an exponenial weighing scheme mimics he FIR feaure. The derivaion of ime-varying conrol limis for an EWMA is presened in Secion 2, and he effec of ime-varying conrol limis is illusraed for a simple example. Secion 3 uses numerical resuls o conras and compare EWMA s wih ime-varying conrol limis and EWMA s wih asympoic limis. Secion 4 inroduces a FIR feaure for wo-sided EWMA chars and shows ha his approach is superior o mehods suggesed previously by Lucas and Saccucci (1990) and Rhoads e al. (1996). In he Appendix, i is shown ha he run lengh properies of an EWMA char wih ime-varying conrol limis can be approximaed using a non-homogenous Markov chain. 2. EWMA Conrol Chars wih Time-Varying Conrol Limis From (1) he mean value and variance of z are easily derived (Mongomery, 1991). Assuming he x i`s are independen random variables wih mean and variance σ x 2 n, where n is he sample size used a each ime inerval o calculae x i, we ge µ z =, and σ 2 z = σ 2 x λ n 2 λ [ 1 ( 1 λ )2 ]. (2) Noice ha he variance of he EWMA es saisic z is a funcion of ime. This should be expeced since he number of observaions used o derive he EWMA es saisic varies wih ime and he influence of he iniial fixed value z 0 slowly decreases. Conrol limis for an EWMA conrol char are ypically derived based on ± L sigma limis, where L is usually equal o hree as in he design of Shewhar conrol char limis.

5 Thus, he ime-varying upper and lower EWMA conrol limis, UCL() and LCL() 5 respecively, are given by UCL LCL [ ] λ 1 ( 1 λ ) 2 () = + Lσ x ( 2 λ )n [ ] λ 1 ( 1 λ ) 2 () = Lσ x ( 2 λ )n and, (3) where, in applicaions, and σ x are ypically esimaed from preliminary daa as he sample mean and sample sandard deviaion. As increases he conrol limis UCL() and LCL() converge o he asympoic conrol limis, denoed as UCL and LCL, given by ± Lσ x λ ( 2 λ )n. The rae of convergence o his asympoic values depends criically on λ wih he convergence being much slower for small λ. To illusrae he effec of ime-varying limis consider he following example used by Lucas and Crosier (1982) o show he effec of he FIR feaure on a CUSUM char. For he example we shall assume = 0, σ x = 1 and L = 3. The raw daa is given by x in Table 1, and represens an iniial ou-of-conrol siuaion. The able also gives he EWMA values z derived from (1), and ime-varying conrol limis derived from (3) wih λ =.1. Table 1: Simple EWMA Example λ =.1 sample number x z UCL() Figure 1 shows he resuling EWMA chars for differen values of λ. The imevarying upper conrol limi UCL() is shown as a solid line, whereas he asympoic

6 conrol limi UCL is shown as a dashed line. Figure 1 shows only he upper conrol limis 6 o aid display; normally boh upper and lower conrol limis are shown. The number of observaions needed o generae an ou-of-conrol signal depends on boh he value of λ and wheher he ime-varying conrol limis are used. When λ equals.05,.1 or.25 an EWMA char wih ime-varying conrol limis signals afer only four observaions whereas using he asympoic limis a signal will no be generaed unil observaion seven for λ =.1 and λ =.25, or observaion nine for λ =.05. When λ =.5, he ime-varying conrol limi quickly converges o he asympoic value and hus has lile effec. When λ =.5 a signal occurs afer seven observaions using eiher UCL() or UCL as he conrol limi. 0.5 λ =.05 1 λ = z z z 0.5 λ = λ = z Figure 1: Plo of EWMA Conrol Chars wih Time-varying Conrol Limis dashed lines show he asympoic conrol limis solid lines show he ime-varying conrol limis generaed by (3) circles represen he EWMA values As can be seen in Figure 1, using asympoic conrol limis raher han he ime varying limis makes he EWMA char much less sensiive o process shifs in he firs few observaions. This could be a significan problem if a large shif occurs early, or if afer an ou-of-conrol condiion he process is no properly rese.

7 3. Run Lengh Properies of EWMA Chars wih Time-varying Conrol Limis 7 In his secion, he run lengh properies of EWMA chars wih ime-varying conrol limis, such as ARL, are compared wih he run lengh properies of EWMA chars wih asympoic conrol limis. As will be shown, while he process is in-conrol, he ARLs of EWMA conrol chars wih ime-varying conrol limis are nearly idenical o he ARLs of radiional EWMA chars wih asympoic conrol limis. However, when he iniial process level is ou-of-conrol he ARL of he wo chars may differ subsanially depending on he value of he EWMA weigh λ. I is imporan o quanify he effec of using ime-varying conrol limis since EWMA conrol chars are usually designed o have given average run lenghs (ARLs) under cerain operaing condiions. For an EWMA he design parameers include λ and L. However, since he ime-varying conrol limis converge o he consan asympoic values as ime increases, for process shifs ha occur laer in ime he wo chars will have similar run lengh properies. As a resul, EWMA conrol chars wih ime-varying conrol limis can be designed in he same manner as EWMA wih asympoic limis. See Crowder (1987) for guidelines. The run lengh properies of EWMA conrol chars wih asympoic conrol limis were deermined by Crowder (1987) using an inegral equaion approach. Unforunaely, his inegral equaion soluion approach is no applicable for EWMA chars wih imevarying conrol limis. However, he run lengh properies of he EWMA char wih imevarying conrol limis can be approximaed using a non-homogeneous discree Markov chain. Using a Markov chain he feasible sae space is approximaed hrough discreizaion and he probabiliy of moving from any one sae o any oher sae for each ime period is deermined. By using a greaer number of disinc saes he approximaion of he run lengh properies can be made more precise. A deailed explanaion of he soluion procedure is given in he Appendix.

8 The effec of ime-varying conrol limis on he ARL is illusraed in Figure 2. The 8 resuls were derived using L = 3.0 as he conrol limi consan, and wihou loss of generaliy assuming an in-conrol mean and sandard deviaion of zero and uniy respecively. In Figure 2, he horizonal axis gives he iniial rue process mean in σ X unis, he sandard deviaion of he sample mean. The resuls are given only for posiive shifs, bu since he problem is symmeric he same paern is observed for negaive shifs. These resuls are also abulaed in he Table A1 of he Appendix. ARL values for he asympoic case are aken from Crowder (1987), while ARL resuls for EWMA wih imevarying conrol limis are deermined using he mehodology presened in he Appendix. Figure 2 shows ha he effec of using ime-varying conrol limis on he ARL of he EWMA is subsanial when he process is no iniially in-conrol, especially when λ is small. The figure uses log(arl) o improve he visual comparison. As an example, from Table A1 assuming he iniial process mean value is 2.0 σ X unis greaer han he in-conrol value used o se up he EWMA char, hen for λ =.05 he ARL of he EWMA wih ime-varying conrol limis is 2.8 which is much shorer han he ARL of 6.0 required for an EWMA using asympoic conrol limis. The effec of he imevarying conrol limis, however, has very lile influence on he in-conrol run lengh as shown in Figure 2 and by he σ X = 0.0 row in Table A1. As such ime-varying conrol limis are recommended for all EWMA chars, since heir performance will be subsanially beer han asympoic limi EWMAs when he process is fairly likely o sar ou-of-conrol.

9 λ =.05 λ = ARL ARL 10 2 ime-varying conrol limis asympoic limis process mean ARL ARL 10 2 ime-varying conrol limis asympoic limis process mean λ =.25 λ = ime-varying conrol limis ime-varying conrol limis asympoic limis process mean asympoic limis process mean Figure 2: Plo of he ARLs for EWMA chars wih ime-varying and asympoic conrol limis Sandard deviaion values for he asympoic EWMA conrol chars are also given in Crowder (1987). Table A2 in he Appendix reproduces he Crowder resuls and gives he sandard deviaion values for he ime-varying case also calculaed using he ime nonhomogenous Markov chain mehodology presened in he Appendix. Table A2 shows ha he sandard deviaion of he run lenghs are nearly idenical for he asympoic EWMA and he EWMA wih ime-varying conrol limis. I is also of ineres o examine how he disribuion of he run lengh of an EWMA char changes when ime-varying conrol limis are adoped. The run lengh disribuion can be deermined using equaions (A3) given in he Appendix. Figures 3 and 4 show he run lengh disribuions for EWMAs wih ime-varying conrol limis and asympoic conrol limis when he iniial process is in-conrol and shifed one σ X uni respecively.

10 Probabiliy Densiy Asympoic Limis Time-varying Limis Run Lengh Figure 3: In-conrol Run Lengh Disribuion of EWMA wih Time-varying and Asympoic Conrol Limis, λ =.05, L = Probabiliy Densiy Asympoic Limis 0.02 Time-varying Limis Run Lengh Figure 4: Ou-of-conrol Run Lengh Disribuion of EWMA wih Time-varying and Asympoic Conrol Limis λ =.05, L = 2.587, iniial mean shif of one sandard deviaion uni Figure 3 shows an iniial spike in he run lengh probabiliy densiy for he EWMA wih ime-varying conrol limis, wih he wo probabiliy densiies nearly converging for long run lenghs. This greaer probabiliy of a shor run lengh is undesirable since he iniial process sae is in-conrol and we would like he run lengh o be very long. However, since he probabiliies involved are sill very small, his spike has a corresponding small influence on he ARL. The size of his iniial spike in he run lengh

11 probabiliy densiy funcion depends on L, wih smaller L leading o larger spikes. In 11 Figure 4, by conras, he bulk of he probabiliy densiy for he wo cases is quie differen, and he ARL under he ime-varying conrol limis will be subsanially shorer. Of course, given an iniial ou-of-conrol sae a shor ARL is desirable. Comparing he run lengh disribuion plos shown in Figure 3 and 4 wih similar plos for CUSUM and FIR CUSUM in Lucas and Crosier (1982) and for FIR-EWMA chars in Lucas and Saccucci (1990) suggess ha he effec of he ime-varying limis is similar o ha achieved wih he FIR feaure. The effec of he ime-varying limis appears less pronounced han he FIR-CUSUM which suggess ha an addiional narrowing of he ime-varying conrol limis for small values of may be appropriae o make he EWMA char even more sensiive o sar-up qualiy problems. 4. EWMA Conrol Chars wih Fas Iniial Response (FIR) EWMA chars wih ime-varying conrol limis were shown in he previous secion o have properies similar o he FIR feaure when compared wih asympoic EWMA. However, using ime-varying conrol limis is no he same as he FIR feaure for CUSUMs since he adjusmen of he conrol limis only correcs he conrol limis o ake ino accoun he ime dependen naure of he EWMA saisic given by (1). A few auhors have suggesed adapaions o he EWMA scheme o build in a rue FIR feaure. As discussed in he inroducion, o creae a wo-sided EWMA char ha reacs quickly Lucas and Saccucci (1990) suggesed he use of wo one-sided EWMA chars wih iniial saes differen han zero. Rhoads, Mongomery and Masrangelo (1996) adap he Lucas and Saccucci approach by allowing each one-sided char o have imevarying conrol limis. Boh hese mehods have he desired effec of making he char more sensiive o sar up qualiy problems, bu are raher awkward since hey require he simulaneous use of wo EWMA chars o accomplish he ask previously achieved wih jus one char.

12 Here a differen approach is suggesed ha reains he simpliciy of a single conrol 12 char. To give EWMA chars wih ime-varying conrol limis a FIR feaure, he conrol limis are narrowed furher for he firs few sample poins. This approach is easily implemened since he conrol limis are already ime-varying. Since he ime-varying conrol limis exponenially approach he asympoic limis i is reasonable o use an exponenially decreasing adjusmen o furher narrow he limis. Le FIR adj = 1 ( 1 f ) 1+a( 1) (4) Wih his seup he FIR adjusmen makes he conrol limis for he firs sample poin ( = 1) a proporion f of he original disance from he saring value. The effec of he FIR adjusmen decreases wih ime o ensure ha he long erm run lengh properies of he EWMA will be virually unchanged. A reasonable seup would be o se he adjusmen parameer a so ha he FIR adjusmen has very lile effec afer observaion 20, say ha he adjusmen FIR adj a observaion 20 is.99. This should be sufficien o allow he deecion of qualiy problems in he sarup. This idea implies ha we should se a = ( 2 log( f ) 1) 19. For example, using f = 0.5 yields a = 0.3. Using his adjusmen facor and (3), he FIR-EWMA conrol limis are: ( ( ) 1+a( 1) ) ± Lσ x 1 1 f [ ( ) 2 ] λ 1 1 λ ( 2 λ )n (5) The conrol limis given by (5) are ime-varying, hus he run lenghs properies of he proposed FIR-EWMA can also be deermined using he non-homogeneous Markov chain mehodology presened in he Appendix. Figure 5 shows he effec of using limis (5) wih f =.5, a =.3 on he example iniially discussed in Secion 2, and previously illusraed in Figure 1. In Figure 5, he advanage of he addiional narrowing of he conrol limis in deecing sar up qualiy

13 problems is clearly demonsraed. For all he differen values of λ he FIR-EWMA signals 13 in jus wo observaions. This is a subsanial improvemen over he run lenghs obained wih only he ime-varying conrol limis, especially for large values of λ λ =.05 λ = z z λ =.25 λ =.5 2 z z Figure 5: EWMA Conrol Chars wih Time-varying Conrol Limis dashed lines show he FIR ime-varying conrol limis from (4) solid lines show he ime-varying conrol limis generaed by (3) circles represen he EWMA values To explore he effec of differen levels of FIR Table 2 gives ARL resuls for differen levels of f. From hese resuls i is clear ha o derive a subsanial benefi from he FIR feaure he level off should be fairly small, say around f equal o 0.5. Using f equal o 0.5 corresponds o adjusing he ime-varying limis by a facor of one half for he firs ime period, as shown in Figure 5. In furher analysis of he FIR we use f equal o 0.5, his choice is also aracive because i mimics he 50% head sar ypically suggesed for FIR CUSUM chars.

14 Table 2: ARL Resuls for Differen FIR Proporions λ =.25 L =3 σ x f =0.4 f =0.5 f =0.6 f =0.7 f =0.8 f = λ =.10 L =3 σ x f =0.4 f =0.5 f =0.6 f =0.7 f =0.8 f = Table 3 compares he ARL of he Lucas and Saccucci (1990) FIR-EWMA, denoed L-FIR, he Rhoads e al. (1996) FIR-EWMA, denoed R-FIR, and a FIR-EWMA wih adjused ime varying conrol limis given by (4). The resuls for he L-FIR and he R-FIR are aken from simulaion resuls published in Rhoads e al. (1996), and he run lengh resuls for he proposed FIR-EWMA were approximaed using he mehodology described in he Appendix. For all he FIR-EWMAs he conrol limi muliple L has been adjused so ha, in-conrol, all mehods have approximaely he same ARL.

15 Table 3: Average Run Lengh Comparison of EWMAs wih FIR λ =0.25 λ =0.1 σ x L-FIR L=2.81 R-FIR L= 3.0 FIR L=3.07 L-FIR L=2.81 R-FIR L= 3.0 FIR L= σ x L-FIR L=2.62 λ =0.05 λ =0.03 R-FIR FIR L-FIR R-FIR L=2.72 L=2.69 L=2.44 L=2.54 FIR L= The resuls in Table 3 sugges ha proposed FIR-EWMA is superior o he previous approaches. For example, wih λ =.1 and a mean shif of one sandard deviaion uni, he proposed FIR-EWMA requires on average only 4.5 observaions o signal, while he Lucas and Saccucci FIR-EWMA, and he Rhoads e al. FIR-EWMA require 6.9 and 5.4 observaions respecively. The reducion in ou-of-conrol ARLs appears o be greaes when λ is no small. In addiion o he benefi of beer run lengh properies, he EWMAs wih ime-varying conrol limis also provide wo-sided proecion from sar up qualiy problem hrough only a single conrol char. This is a major advanage from an implemenaion perspecive. I should be noed ha he FIR-EWMA requires larger values of he conrol limi L han he radiional EWMA chars. As a resul, if he process shif does no occur near sarup he FIR-EWMA will acually have slighly longer ARLs han radiional EWMA chars.

16 Summary 16 This aricle derives he run lengh properies for EWMA conrol chars wih imevarying conrol limis. Since he variance of he EWMA es saisic is a funcion of ime, ime-varying conrol limis resul in improved process shif deecion capabiliies if he process is iniially ou-of-conrol, or if i goes ou-of-conrol quickly. The magniude of he benefi of using ime-varying conrol limis over radiional asympoic limis depends on he EWMA consan λ, and size of he iniial process shif. Resuls are presened ha quanify he difference for an EWMA designed o monior he process mean. In general, ime varying conrol limis are useful if λ is small, say less han.3. In siuaions where a he sar of process monioring here is good chance he process is ou-of-conrol furher narrowing of he ime-varying conrol limis is shown o provide an addiional fas iniial response benefi. Adjusing he conrol limis o sar a half he regular value and hen exponenially approach he regular ime-varying limis for 20 observaions is shown o perform beer han previously suggesed approaches o creae a FIR-EWMA. The proposed approach has he addiional benefi of reaining he benefi of he EWMA char ha allows he wo-sided deecion of problems wih a single char. Appendix In his appendix, approximaions for he disribuion, expeced value and variance of he run lengh of EWMA chars wih ime-varying conrol limis are derived. The soluion procedure uilizes a non-homogenous Markov chain wih g disinc saes. In he soluion he sae space beween he conrol limis is divided ino g-1 disinc discree saes, and he ou-of-conrol condiion corresponds o he gh sae. The differen saes are defined as s = s 1,s 2,K,s g 1 ( ) = ( LCL + w, LCL + 2w,..., UCL 2w, UCL w ), where w = ( UCL LCL) g and UCL and LCL are he asympoic conrol limis as given by seing = in (3). As g increases he approximaion improves. Assume ha he ransiion probabiliy marix for ime period is given by

17 17 P = p, p, K, p g p 21, K, p 2g M M p, K, p g1 gg = R, ( I R )1 0, K,0, 1 (A1) where I is he g by g ideniy marix, 1 is a g by 1 column vecor of ones, and p ij equals he ransiion probabiliy from sae s i o sae s j for ime period. The las row and column correspond o he absorbing sae ha represens an ou-of-conrol signal. The R marix equals he ransiion probabiliy marix wih he row and column ha correspond o he absorbing (ou-of-conrol) sae deleed. R will be used o derive he run lengh properies of he EWMA conrol char wih ime-varying conrol limis. Since he ime-varying conrol limis (3) asympoically approach consan values, he sae ransiion probabiliies p ij converge o probabiliies p ij and he marix R converges o he infinie ime ransiion marix R as. The values for p ij can be deermined by making some process assumpions. Assuming a normal model wih X i ~ ( ) and given he curren EWMA value, he disribuion of he fuure EWMA value 2 N,σ x z +1 is N λ + ( 1 λ )z,λ 2 2 ( σ x ). Thus, he infinie ime ransiion probabiliies are: p = Pr s ij j w 2 < z < s j + w 2, for p = Pr z > s + w ig g Pr z < s w 1 2, j = 1, 2,K,g 1 and (A2) where z ~ N( λµ + ( 1 λ )s i, λ 2 σ 2 ). These values can be easily calculaed o deermine P and R. The ime dependen ransiion marices R can be deermined from R by changing he ransiions probabiliies ha lead o an earlier signal. Transiions probabiliies in R from saring values (rows) ha are ouside he ime-varying conrol limis and o ending values (columns) ha resul in ou-of-conrol signals are se o zero. For each value of, he appropriae rows and columns are idenified by comparing he ime-varying conrol

18 limis wih he saes in he sae space. In oher words, o deermine R he firs f 1 () and 18 las f 2 () rows and columns of R are se o zero vecors, where f 1 () equals he larges ineger for which s f 1 w 2 LCL() and f 2 () is he smalles ineger for which s f 2 + w 2 UCL(). In an aemp o consisenly yield run lengh values less han he rue value any sae whose ransiion probabiliy is a all effeced by he changing conrol limi is se o zero. A sae s i is effeced if he ime-varying conrol limi is eiher closer o zero han s i or wihin w 2 of s i. Using his procedure esimaes for R 1, R 2, ec. are obained. Deermining he expeced run lengh and he variance of he run lengh can now proceed using he marices R. Leing RL equal he run lengh of he EWMA we have ( ) = I R i Pr RL Pr RL = 1 i=1 ( ) = R i R i i=1 1, and 1 for 1. i=1 (A3) Thus, ( ) = Pr( RL = ) ERL = R s 1. (A4) =1 =1 s=1 Similarly, he variance of he run lengh is ( ) = I + ( 2 + 1) R s 1 Var RL. (A5) =1 s=1 These expressions are gx1 vecors ha give he average run lengh and variance from any saring value or sae s i. The values ha correspond o he saring wih z 0 = X are easily found. Assuming ha he conrol limis are symmeric abou X he corresponding sae is s g 2. (A4) and (A5) give he momens of he run lengh in erms of infinie sums ha converge for large. These expression can be simplified in his case, since he conrol limis converge asympoically, and hus he ransiion probabiliy marices R also

19 converge o R as increases. Replacing all R marices for large values wih R, he 19 infinie sums (A4) and (A5) can be wrien as: ( ) = R s 1 ERL Var RL + R s I R max -1 =1 s=1 s=1 ( ) = 1 + ( 2 + 1) R s 1 ( ) 1 1, and (A6) max max + 1 =1 s=1 max +2 R s R 1 R s=1 max ( ) R s s=1 1 R ( ) 1 1 ( ) 2 1, (A7) where max equals he number of ime period for which differen ransiion probabiliy marices are used. For he compuaions, max was chosen based on λ and g so ha he marix R max is indisinguishable from R. In his way increasing max furher will have no influence on he soluion accuracy. The minimum value for max is derived by realizing ha if he ime-varying conrol limis a ime max differ from he asympoic limis by less han w 2 hen he marix R max is he same as R. Solving UCL UCL() w 2 and LCL LCL() w 2 for he minimum value yields max as he smalles ineger larger han log 12nw( 2 λ )σ λ n 2 λ 36λσ 2 ( ) w 2log( 1 λ ). For compuaional efficiency and accuracy, ERL ( ) and Var( RL) are deermined using Gaussian eliminaion raher han by finding he marix inverse direcly as suggesed by (A6) and (A7). In general, as g increases he ERL ( ) and Var( RL) values obained hrough (A6) and (A7) increase and more closely approximae he rue values. The values increase because he procedure always underesimaes he rue run lengh. The run lenghs are underesimaed for wo reasons; firs, he absorbing boundaries for R are narrower han he conrol limis since hey are se a LCL + w 2 and UCL w 2, and second for R he

20 absorbing probabiliies are conservaively calculaed since all saes even marginally 20 effeced by he conrol limi are assumed o lead o absorpion. The advanage of consisenly underesimaing he run lenghs of he EWMA are ha we can use he rae of increase o esimae he rue values. The values shown in he Tables A1, A2, and A3 were derived by esimaing he rue value ERL ( ) g= based on fiing he model ERL ( )= ERL ( ) g= + Bg+ Cg 2 derived using he resuls generaed wih g = 50, 100, and 150. Verificaion of his approach using simulaion suggess ha our resuls differ from he rue value by less han 1% excep for very large process shifs when he average run lengh is near uniy. For very large shifs, he values in he ransiion probabiliy marix R become smaller and calculaions required o derive ERL ( ) become more prone o rounding error. As a resul, for large shifs he ERL ( ) esimae may no increase as g increases. If his occurs, we use he larges obained ERL ( ) as an esimae of he rue ERL ( ) g=, and he esimae may be off by as much as 10%. A similar problem is also repored in Lucas and Crosier (1982). However, in our case, for comparison purposes, he resuls are adequae. Tables A1 and A2 give he deailed resuls required o generae Figures 1 and 2 in he ex. The iniial shif in he process mean is given σ X unis. Noe ha he value for σ x = 0.0 and λ =.05 is incorrecly given as in Crowder (1987), he correc value is given in Table A2.

21 Table A1: Average Run Lengh for Two-sided EWMA Chars Zero Sae Resuls, L = 3.0 Asympoic Conrol Limis Time-varying Conrol Limis σ x λ =.50 λ =.25 λ =.10 λ =.05 λ =.50 λ =.25 λ =.10 λ = Table A2: Sandard Deviaion of he Run Lengh for Two-sided EWMA Chars Zero Sae Resuls, L = 3.0 Asympoic Conrol Limis Time-varying Conrol Limis σ x λ =.50 λ =.25 λ =.10 λ =.05 λ =.50 λ =.25 λ =.10 λ = To provide more deails, Table A3 gives he ARL values for EWMAs wih imevarying conrol limis for some differen values of L.

22 Table A3: Average Run Lengh for Time-varying Conrol Limis EWMA Chars Zero Sae Resuls L = 3.5 L = 3.25 σ x λ =.50 λ =.25 λ =.10 λ =.05 λ =.50 λ =.25 λ =.10 λ = L = 3.0 L = 2.75 σ x λ =.50 λ =.25 λ =.10 λ =.05 λ =.50 λ =.25 λ =.10 λ = L = 2.5 L = 2.25 σ x λ =.50 λ =.25 λ =.10 λ =.05 λ =.50 λ =.25 λ =.10 λ = Resuls are derived for he wo-sided case bu he mehodology can be easily adaped for one-sided case EWMA chars defined as z = max( λx + ( 1 λ )z 1,z 0 ). In addiion, he examples provided assume he disribuion of he observed process parameer is normal. However, similar resuls are easily derived for oher underlying disribuions.

23 Acknowledgmens 23 This research was suppored, in par, by he Naural Sciences and Engineering Research Council of Canada and he Manufacuring Research Council of Onario. The auhor would also like o hank wo anonymous referees for heir helpful commens. References Box, G.E.P., Jenkins, G.M., and MacGregor, J.F. (1974), Some Recen Advances in Forecasing and Conrol, Applied Saisics, 23, Brook, D., Evans, D.A. (1972), An approach o he probabiliy disribuion of cusum run lengh, Biomerika, 59, Crowder, S.V. (1987), Run-Lengh Disribuions of EWMA Chars, Technomerics, 29, Lucas, J.M. and Crosier, R.B. (1982), "Fas Iniial Response for CUSUM Qualiy Conrol Schemes," Technomerics, 24, Lucas, J.M. and Saccucci, M.S. (1990), Exponenially Weighed Moving Average Conrol Schemes: Properies and Enhancemens, wih discussion, Technomerics, 32, Mongomery, D.C. (1991), Inroducion o Saisical Qualiy Conrol, Second Ediion, John Wiley and Sons, New York. Page, E. S. (1954), "Coninuous Inspecion Schemes," Biomerika, Vol. 41, Robers, S.W. (1959), Conrol Char Tess Based on Geomeric Moving Averages, Technomerics, 1, Rhoads, T.R., Mongomery, D.C. and Masrangelo, C.M. ( ) Fas Iniial Response Scheme for he Exponenially Weighed Moving Average Conrol Char, Qualiy Engineering, 9, Dr. Seiner is an Assisan Professor in he Deparmen of Saisics and Acuarial Sciences a he Universiy of Waerloo. He is a member of ASQ.