A Study on the False Alarm Rates of X, EWMA and CUSUM Control Charts when Parameters are Estimated

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1 11 Iteratioal Coferece o Circuits, System ad Simulatio IPCSIT vol.7 (11) (11) IACSIT Press, Sigapore A Study o the False Alarm Rates of X, EWMA ad CUSUM Cotrol Charts whe Parameters are Estimated Michael Boo Chog Khoo +, Si Yi Teh, Li Fe Ag ad Xi Wah Ng School of Mathematical Scieces, Uversiti Sais Malaysia, 118 Mide, Peag, Malaysia. Abstract. Nowadays, quality is a importat issue, especially i maufacturig idustries. Producig a quality product that meets customers requiremets is the mai obective of maufacturig idustries. A cotrol chart is a powerful tool i statistical process cotrol (SPC) which is widely used i maufacturig idustries. I order to costruct a cotrol chart, the estimates of the parameters mea, μ ad stadard deviatio, σ have to be computed first. The Shewhart X chart is slow i detectig small shifts i the process mea. Thus, usually the EWMA (expoetially weighted movig average) ad CUSUM (cumulative sum) charts are used i the detectio of small shifts. The aim of this study is to compare the performaces of the Shewhart X, EWMA ad CUSUM charts, i terms of their false alarm rates, whe parameters are estimated. A Mote Carlo simulatio is coducted usig the Statistical Aalysis Software (SAS) to compute the average ru legth (ARL) profiles of each of the charts. Overall, the results show that the false alarm rates whe the parameters are estimated is iversely proportioal to the umber of i-cotrol samples ad the size of each of these samples, used i the estimatio of the process parameters. Keywords: ARL; X chart; EWMA chart; CUSUM chart; false alarm rates; parameter estimatio 1. Itroductio Statistical process cotrol (SPC) is a collectio of problem-solvig tools that is useful i achievig process stability ad improvig capability through the reductio of variability [1]. A cotrol chart is a powerful tool i SPC. It is used i the moitorig of the quality of a process. A cotrol chart displays the quality characteristic of a process i a graphical way. It cosists of the lower cotrol limit (LCL), ceter lie (CL) ad the upper cotrol limit (UCL). The stability of a process ca be determied from a cotrol chart. If all the sample poits fall withi the cotrol limits without showig ay systematic behavior, it shows that the process is stable or i statistical cotrol []. It is sometimes difficult to compute the cotrol limits because the process parameters, such as μ ad σ are usually ukow. Therefore, these parameters are estimated from a prelimiary sample take from a icotrol process. The estimated sample mea ad sample variace should be ubiased of the populatio mea ad populatio variace, respectively. There are two distict phases i the implemetatio of a cotrol chart. I Phase I, parameters are estimated from the dataset give by the prelimiary samples, followed by computig the trial cotrol limits. After a Phase I cotrol chart is costructed, the process is checked to esure that it stays i-cotrol before cotrol chartig for a Phase II process is iitiated. I a Phase II process, a cotrol chart is used to moitor a future process. Studyig the false alarm rates is oe of the ways to evaluate the performace of a cotrol chart. A false alarm, a.k.a., a Type I error occurs whe a cotrol chart declares a process to be out-of-cotrol, whe i fact it is i-cotrol [3]. A efficiet cotrol chart is able to detect istataeously a process shift, stop samplig as quickly as possible whe a out-of-cotrol sigal is detected ad adopts essetial corrective actios to improve the process quality [4]. + Correspodig author. Tel.: + (6) ; fax: +(6) address: mkbc@usm.my 61

2 I this study, we are iterested to study the false alarm rates of the Shewhart X, EWMA ad CUSUM charts, whe parameters are estimated. The false alarm rates are computed usig a Mote Carlo simulatio. The remaider of this paper is orgaized as follows: Sectio discusses the ubiased estimators for estimatig the process mea ad variace of the Shewhart X, CUSUM ad EWMA cotrol charts. I Sectio 3, a simulatio study is carried out to compare the performaces of the Shewhart X, CUSUM ad EWMA charts, based o the false alarm rates computed. Some useful coclusios ad recommedatios of potetial future works are summarized i Sectio 4. Sectio 5 provides some ackowledgemets while Sectio 6 cotais the refereces that are used i this paper.. Ubiased Estimators for Estimatig the Process Mea ad Variace Most studies i evaluatig the performace of a cotrol chart assume that the i-cotrol process parameters are kow. However, there are may cases where the process parameters are ukow. Woodall ad Motgomery [5] studied the effects of parameter estimatio o the properties of cotrol charts. The most importat poit i parameter estimatio is selectig the most suitable estimator. We are cocered with estimatig the process average, μ ad the process stadard deviatio, σ from prelimiary samples or samples take from the process which is assumed to be i-cotrol ad follows a ormal distributio. Because of a high samplig ad ispectio costs whe the sample size is large, a appropriate sample size that is sufficietly large would be used i estimatig the process parameters. The performaces of the Shewhart X, EWMA ad CUSUM charts are compared, i terms of the false alarm rates whe parameters are estimated. The followig are some descriptios of these charts ad their respective ubiased estimators for estimatig the process mea ad variace..1. The Shewhart X Chart Suppose that a process follows a ormal distributio ad a sample of size, cosistig of measuremets X1, X,..., X is draw from this process, where the average of this sample is 1 X i i = 1 X =. (1) Let X1, X,..., X m be m sample averages computed from a i-cotrol process, the the best estimator of μ, i.e. the process grad average, X is computed as 1 m X m = 1 X =. () The process stadard deviatio, σ is aother ukow parameter i most of the applicatios of cotrol charts. It is a measuremet showig the spread of the data ad is estimated from either the stadard deviatios or rages of the m samples. Here, the estimatio of σ usig the sample rage method is used. Let R1, R,..., R m be the rages of the m samples, where R = Xmax Xmi is the differece betwee the largest sample observatio X max ad the smallest sample observatio X mi. The, the average sample rage, R is calculated by 1 m Rk m k = 1 R =. (3) The cotrol limits of the Shewhart X chart ca be computed as follows [1]: a LCL = X R, (4) d CL = X (5) ad a UCL = X + R, (6) d where a i Equatios (4) ad (6) are measured i multiples of stadard deviatio, whose value depeds o the desired i-cotrol average ru legth, ARL. The value of the costat d depeds o the sample size ad is give i most statistical quality cotrol textbooks. 6

3 The mai drawback of the Shewhart X cotrol chart [6] is that it oly cosiders the iformatio give i the last plotted poit ad igores all the iformatio i the sequece of poits. Thus, it is less effective i detectig small chages i the process. Page [7] proposed the CUSUM chart while Roberts [8] itroduced the EWMA chart to eable a quick detectio of small shifts. These charts complemet the Shewhart X chart whe the detectio of a small shift is of iterest. This is because the CUSUM ad EWMA charts take ito accout the iformatio cotai i the sequece of poits. Thus, they perform better i sigallig small chages compared to the Shewhart X chart... The CUSUM Chart The CUSUM chart, developed by Page [7] icorporates past iformatio ito each idividually plotted observatio. This icreases the sesitivity of the CUSUM chart i detectig small shifts i the process. The CUSUM chart plots the cumulative sums of deviatios of the sample values from a target value agaist time. There are two types of CUSUM charts, i.e., the tabular CUSUM ad the V-mask CUSUM charts. A CUSUM chart is a two-sided chart. It plots the cumulative sum of the sample average versus the sample umber. The tabular CUSUM employs two sample statistics ( C + ad C ), where oe is the oe-sided upper CUSUM that accumulates positive deviatios above the target ad the other is the oe-sided lower CUSUM that accumulates egative deviatios below the target [9]: C = mi{, C 1 + ( Y + k) } (7) ad + + C = max, C + ( Y k), (8) { 1 } δ X μ where k = is the chart s parameter, with k ad δ is the shift i the mea. Note that Y =. σ The lower ad upper cotrol limits of a CUSUM chart are as follows: LCL = H (9) ad UCL = H. (1) Similar to the Shewhart X chart, the ceter lie of the tabular CUSUM chart represets the target value, μ. If either C + or C exceeds the predetermied decisio iterval, H, the process is cosidered to be out-ofcotrol..3. The EWMA Chart The EWMA statistics [8] is defied as follows: Z =λ X + ( 1 λ ) Z 1, = 1,,..., m (11) Here, < λ 1 is the smoothig costat while Z = μ is the startig value of Z, whe the process is icotrol. The cotrol limits ad ceter lie of a EWMA chart are as follows: ( ) i 1 1 σ λ λ LCL =μ L, λ (1) CL = μ (13) ad ( ) i 1 1 σ λ λ UCL =μ + L λ. (14) The factor L cotrols the width of the cotrol limits. As i icreases, the term 1 ( 1 ) i λ approaches uity. Whe the parameters μ ad σ are estimated, the cotrol limits of the EWMA chart i Equatios (1) ad (14) become: 63

4 ad R λ LCL = X L d λ (15) R λ UCL = X + L. (16) d λ Note that the EWMA chart behaves like the Shewhart X chart whe λ = 1 because for this case, the EWMA statistics deped o oly the most recet sample. 3. A Compariso of the Performaces of the X, EWMA ad CUSUM Charts Whe Parameters are Estimated The mai obective of statistical process cotrol (SPC) is to detect as early as possible the presece of assigable causes of variatio that affects the quality of a process. This study compares the performaces of the Shewhart X, CUSUM ad EWMA charts, i terms of false alarm rates, whe parameters are estimated. Whe the process is i-cotrol, the false alarm rates should be sufficietly small or close to the target value. The i-cotrol ARLs of the Shewhart X, CUSUM ad EWMA charts are set as ARL = 5 ad their false alarm rates are compared, for a give combiatio of the umber of i-cotrol samples, m ad the size of each sample,. The umber of samples, m =, 5, 1 ad 1 ad the sample sizes = 3, 5, 1, 15,, 5, 3, 4, 5 ad 1 are cosidered. Table 1. False alarm rates for the Shewhart X, CUSUM ad EWMA charts for ARL =5. Shewhart X CUSUM EWMA k =.5 H = 4.36 λ =.15 L =.659 m Table 1 presets the false alarm rates for the Shewhart X, CUSUM ad EWMA charts whe parameters are estimated, based o ARL = 5. The false alarm rate correspodig to ARL = 5 whe parameters are kow is 1/5 =.4. The results for the Shewhart X chart i Table 1 show that whe m ad (or) icrease(s), the false alarm rates whe parameters are estimated approach.4, i.e., close to the value whe parameters are kow. For example, whe = 3 ad m varies from { 5 1 1}, the false alarm rates decrease from { }. However, whe m = 5 ad icreases from { }, the false alarm rates decrease from { }. We observe that whe either m or icreases, the false alarm rate decreases. I other words, the false alarm rate is iversely proportioal to m ad. From the false alarm rates for the CUSUM ad EWMA charts, we ca coclude that chages i m ad have the same effect o the Shewhart X, CUSUM ad EWMA charts. The false alarm rates of the Shewhart X, CUSUM ad EWMA charts whe parameters are estimated approach that of the case whe parameters are kow whe either or both m ad icrease. A iterestig fidig i Table 1 is that the effect of is egligible whe m is large. Note that the false alarm rates are close to oe aother whe m = 1, where they are almost the same as those for the case whe parameters are kow, i.e..4. We also otice that there is o sigificat differece amog the false 64

5 alarm rates whe 15. The Shewhart X, CUSUM ad EWMA charts show the same behaviour with respect to chages i m ad, whe parameters are estimated. I additio, we also foud from Table 1 that i cotrast to the Shewhart X chart, the CUSUM ad EWMA charts eed a larger m to give reliable estimates of their false alarm rates. Also ote that the false alarm rates of the CUSUM chart are slightly higher tha that of the EWMA chart. It is worth otig that similar treds are observed whe differet ARL values, such as 5, 1 ad are used. The results are ot displayed i this paper because of space limitatio. 4. Coclusios ad Recommedatios The fidigs i this study idicate that parameter estimatio is a importat issue i cotrol chartig. The choice of the values of m ad affects the false alarm rates whe parameters are estimated. We foud that the false alarm rates for the three charts cosidered are higher tha their omial values. If a large m, like m = 1 is selected, the false alarm rates of the charts approach that of the omial value, irrespective of the sample size,. However, meetig this requiremet is ot practical i most process moitorig situatios. It is very costly ad time cosumig to use a large m. A moderate umber of samples may be sufficiet to get a estimate that does ot differ much from the omial value. Thus, for estimatig appropriate cotrol limits, a moderately large m ad suffice. This study serves as a referece i determiig adequate umber of samples ad the sample size for the purpose of parameter estimatio. Future works related to this topic that is worthy of pursuig are as follows: To evaluate the performaces of cotrol charts i detectig shifts i the process variace whe parameters are estimated. To ivestigate the effects of parameter estimatio o multivariate EWMA ad multivariate CUSUM charts. To study the effects of parameter estimatio o attribute charts. 5. Ackowledgemets This work is fuded by the Uiversiti Sais Malaysia (USM) Icetive Grat, o. 11/PMATHS/88 ad supported by the Academic Staff Traiig Scheme (ASTS), USM. 6. Refereces [1] D. C. Motgomery. Statistical Quality Cotrol: A Moder Itroductio, 6th ed. Asia: Joh Wiley & Sos (Asia) Pte. Ltd., 9. [] J. S. Oaklad. Statistical Process Cotrol, 6th ed. Oxford, UK: Butterworth-Heiema, 8. [3] D. H. Besterfield. Quality Cotrol, 8th ed. Upper Saddle River, New Jersey: Pearso Educatio Ic., 9. [4] T. S. Bakir. A distibutio-free Shewhart quality cotrol chart based o siged-raks. Qual. Eg. 4, 16 (4): [5] W. H. Woodall, ad D. C. Motgomery. Research issues ad ideas i statistical process cotrol. J. Qual. Techol. 1999, 31: [6] W. A. Shewhart. Ecoomic Cotrol of Quality of Maufactured Products. Priceto, New Jersey: D. Va Nostrad. Reprited by the America Society for Quality Cotrol, Milwaukee, Wiscosi, [7] E. S. Page. Cotiuous ispectio schemes. Biometrika 1954, 41 (1-): [8] S. W. Roberts. Cotrol chart tests based o geometric movig averages. Techometrics 1959, 1: [9] F. F. Ga. A optimal desig of CUSUM quality cotrol charts. J. Qual. Techol. 1991, 3 (4):