nedumaran, g. & pignatiello, j. j. (1999) on constructing t² control charts for on-line process...

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On constructing T2control charts for on-line process

monitoringGUNABUSHANAM NEDUMARAN

a& JOSEPH J. PIGNATIELLO JR.

b

aOracle Corporation, 1710 Valley View Lane, Irving, TX, 75061, USA E-mail:

bFAMU-FSU College of Engineering, Florida State University, Florida A & M University,

Tallahassee, FL, 32310-6046, USA E-mail:

Version of record first published: 27 Jul 2007.

To cite this article:GUNABUSHANAM NEDUMARAN & JOSEPH J. PIGNATIELLO JR. (1999): On constructing T2control charts

on-line process monitoring, IIE Transactions, 31:6, 529-536

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IIE ransactions (1999) 31 529 536

On constructing ~ control charts for on-line process

GUNABUSHANAM N E D U M A R A N ' a n d JOSEPH J . PIGNATIELLO JRA2Oracle Corporation, 1710 Valley View Lane, Irving, TX 75061, U S AE-mail: gnedumar @us.oracle.com* FAM U - FSU College of Engineering. Florida State University, Florida A M University. Tallahassee, FL 32310-6046 U S AE-mail: [email protected]

Received May 1997 and accepted December 1997

In this paper we study the performance of X control limits on T~ control charts during on-line process monitoring. We makerecom men datio ns on the minimum number of subgroup s necessary for contr ol charts based o estimated parameters to performsimilar to control charts based on true parameters when th X contr ol limit is used. We discuss an exact procedure for constru ctingT~ ontrol charts using estimated parameters that perform similar to control charts based on true parameters ;egardless of thenumber of preliminary subgroups. Furthermore we suggest an implementation approach for using lilt s control limit on r2control charts un i1 the accumulation of recommended minimum number of subgroups.1. Introduction variate X control charts. He showed that when controlStatistical Process Control (SPC) charts are tools thatare used to monitor the state of a process by distin-guishing between common causes and special causes ofvariability. When several characteristics of a manufac-tured component are to be monitored simultaneously,multivariate Shewhart-type X o r T~ control charts canbe used [I]. As long as the points plotted on the X or T~control chart fall below the Upper Control Limit UCL)of the chart, the process is assumed to operate under astable system of common causes, and hence, in a state ofcontrol. When one or more points exceed the UCL, theprocess is deemed out of control due to one o r morespecial causes and an investigation is carried out todetect these special causes.When the in-control values of the process mean vectorand covariance matrix are known, these known parame-ter values are used to find the statistic plotted on a Xcontrol chart. The UCL of this chart is based on the chi-square distribution. However, in most cases these pa-rameter values are unknown, and hence are estimatedfrom some m initial subgroups of size n taken when theprocess is believed to be stable. When a future subgroupis drawn from the process, Hotelling's ~ statistic is cal-culated using parameter estimates and is plotted on a ~control chart . The U L of this chart is based on the Fdistribution.Quesenberry [2] has studied the effects of using esti-mated parameters to construct the control limits of uni-

limits are constructed using parameters estimated withda ta from a small number of subgroups, false alarm ratesand Average Run Lengths ARLs, ) of the control chartincreased from their nominal values that are based ontrue parameter values. Quesenbermry [2] further showedtha t m should be a t least 1 when n 5 in order for theestimated contro l limits to perform similar to true contro llimits with respect to the in-control run length distribu-tion and the ARL.

In this paper, we extend Quesenberry's [2] s tudy 1 0multivariate ~ control charts. I n particular, we considerthe issue of the minimum number of subgroups necessaryfor the control char t constructed using estimated param -eters to perform similar to the control chart constructc:dusing true p arameters d uring the on-line monitoring stage.Further , we suggest an implementation procedure so t haton-line mo nitoring with T~ control charts can begin at thecrucial st art- up stages of the process.This paper is organized as follows. In the next sectionwe describe the multivariate process model and reviewsome pertinent literature. In the following section, wepresent the results of som e simul;3tion experiments pe,r-formed along with o u r recomm endatio~l or the minimumnumber of subgroups necessary for the control cha-rtbased on parameter estimates to perform similar to thechart based on true parameters. We then describe an exactmethod and an implementation p~:ocedure or construct-ing control charts based on parameter estimates t ha tperform similar to the cha rt based on true param eters.

0740 8 7X O 1999 I I E


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Nedumaran and Pignatiello Jr2. Multivariate process model and literature reviewWe let X i j = X v l , X i , 2 , . . ,+) denote a p x vectorthat represents the p observatsons on the j th componentin the ith subgroup, = 1 , 2 , . . and = 1 ,2 , . , . Weassume that X i j s are independent and identically dis-tributed normal rando m variables with mean and co-variance matrix I when the process is in control. We leti denote the average vector for the i th subgroup, and welet Si denote the unbiased estimate of the covariancematrix for the ith sub group . Th at is,

and

When the process is in control and the in-control processparameter values are known, the statistic plotted on theX2 control chart for the ith subgroup is

When the process is in control this statistic has a chi-square distribution with p degrees of freedom [l]. It isplotted on a X 2 control cha rt with a UCL iven by

where is the )t h percentile point of chi-squaredis tr ibu t~on with p degrees of freedom and a is theprobability of a false alarm for each su bgrou p plotted o nthe x control chart.If the process parameter values are not known, datafrom m initial sub gro ups ar e collected when the process isbelieved to be in con trol. Then, pooling da ta from these msubgro ups and assum ing that the process was in control,unbiased estimates of the mean vector an d the covariancematrix are given by

respectively. A T~ ontrol chart is constructed and the minitial subgroups are tested retrospectively to ensure thatthe process was in control when these initial subgroupswere drawn.In this paper, we consider th e case where the process isbeing monitored on-line and where the in-control valuesof the process parameters are estimated from m initialsubgroups. When a fu ture subgroup i > m) is drawnfrom the process for on-line monitoring purposes, thestatistic plotted on the control c hart is

This statistic is plotted on a T~ ontrol chart . Alt 13 hasgiven the U L of this control chart as

whereCt(m', n , p ) = p ml I)(n 1)

(m n - m t - p + 1 ) 'F ,,,,s the 1 a th percentile point of the F distri bution with v l and v z degrees of freedom, and a is thespecified acceptable false alarm probability for eachsubgro up plotted on the T~ ontrol chart.If the process parameters are estimated from a rea-sonably large number of initial subgroups, the usualpractice for.constructing T~ ontrol cha rts is to use UCLzlinstead of the exact UCLp . However, according toMontgomery [ l ] we must be careful when followingthis practice. In this paper, we show tha t X2 control limitsissue a relatively large number of false alarms even whenthe process paramete rs are estimated fro m relatively largenumber of subgroups.Lowry and Montgomery [4] have compared the nu-merical values of UCL,2 and UCLp for several values ofm , n, and p, with a = 0.001. To select the minimum m forwhich UCLp can be replaced by the approximationUCL,,, they used the relative erro r between the twocontrol limits as the criterion. They have recommendedminimum rn values for several combinations of p and nfor which the relative erro r is less than 0.1. H owever, theydid not consider the average run length or the run lengthdistribution performance of the respective control charts.Quesenberry [2j has addressed the issue of the mini-mum number of initial subgroups necessary while usingunivariate control charts in order for the chart basedon estimated parameters to perform in a similar mannerto the chart based on true parameters during the-on-lineprocess mon itoring stage. Let B denote the event that theith future subgroup average i > m) plots outside thecontrol limits when the univariate normal process pa-rameters are estimated from m initial subgroups. Que-senberry [ ] showed that the events Bi n d Bj(i, > m ) a r epositively cbrrelated. This dependence causes both theA R L and the Standard Deviat ion of the Run Length(denoted by SDRL) to increase from their nominal val-ues, despite the increase in false alarm rates.

Quesenberry [2] used simulation to study the A R Lperformance of the univariate X chart for several valuesof m, with n = 5, to find the minimum value of m forwhich the estimated control limits perform like the true,but unknown, control limits. His simulation resultsshowed tha t for small m (i.e., m < 50) both the A RL andthe SDRL exceeded their nominal values. For instance,when m = 20 and n = 5 the A R L increased from 370.4(for true parameters) to 434, and the correspondingSDRL increased from 369.9 to 754 for the in-controlprocess. Both the A R L and the SDRL got reasonably


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Constructing 7 control chart sclose to their nominal values for the univariate X chart qwhen m was at least 100. ARL =o s tudy the efkct of m on the A-control Run Length 1 -j(RL) distribution for the univariate X chart, Quesenberry a n d[2] used Monte Carlo simulation to estimate Pr[RL 5 x ]for several values of x when the control limits are esti- SDRL =mated from m subgroups. The estimated probabilities fl G: ARL- I ,1 f lexceeded the nominal values for small m, and get rea- wheresonably close to the nominal values when m is at least100. For instance, when m = 20 and n = 5, Pr[RL 201 /?= 1 - Pr(Ai) = ~ r ( d LICL;,).increased from 0.0526 (for true parameters) to 0.073 forthe in-control process. Based on these two considerations,namely, the ARL and the in-control RL distribution,Quesenberry 121 concluded that m should be a t least 100for n = 5 in orde r for th e estimated X control char t limitsto perform like the tru e limits. F or other sub group sizes,he conjectured that m should be at least 400/ n - 1)based on the speculation that the degrees of freedom ofthe variance estimator should be approximately 400 forthis t o work.In the next section, we study the performance of T~control charts with X2 control limits. We compare theperformance of this cha rt with the ca se where th e processparameter values are known. We call the latter the trueparam eters case and refer to the characteristics of thischa rt as the nominal characteristics. We recommend aminimum m for which the control chart based on esti-mated performs similar to the control chartbased on true parameters. We further suggest a controlchart implementation program that can be used until theaccumulation of the recommended minimum number ofsubgroups.

3 Characterizing the run length distributionWe suppose tha t da ta on initial subgroups of size nfrom a stable N p p , Z ) distribution are availabk. heprocess mean vector and covariance matrix are esti-mated using (3). Future subgroups are drawn from theprocess at regular intervals for on-line monitoringpurposes. Suppose that there has been a step-change inthe process mean from the in-control value of p = CI, t op = p, where p, k. We let Ai denote the event thatvthe control chart statistic X of a future subgroupi > m xceeds UCL I. We let Y denote the run length,which is the number of points plotted on the chartuntil a point exceeds the UCL. I f the process parametervalues are known and the X 2 statistic (1) is plotted on acontrol chart with control limit 2), then the sequenceof trials that consist of determining if f exceeds UCLx2would be a sequence of Bernoulli trials. Consequently,the run length Y until event i occurs wouId have ageometric distribution with mean ARL and s tandarddeviation SDRL given by [ ]:

However, when param eter estimates from the m initialsubgroups are used this run length characterization isnot valid since q2 and q2are correlated for i j a n di, > m. T o verify the last state men t, we consider- -cov(Xi - X , X, h). I t can be shown by direct evalua-tion that

Let Bi i > m denote the event that q2 exceeds UCLZ.Events Bi a n d B, are not independent for i j i j > msince and q2 are correlated. Consequently, thesequence of trials comparing q2 with UCL? is not a se-quence of Bernoulli trials, and hence, the run length untilthe occurrence of event Bi does not follow a geomet~nicdistribution.s suggested by Quesenberry I: ],ne reasonable wayto decide when m is large enough for the control cha-rtbased on estimated parameters to behave essentiallysimilar to the control chart based on true parameters is todetermine when the run length distribution betweenevents Bi is close to tha t of Ai. urther, note from (6) tha tthe ARL and the SDRL are app~:oximatelyequal whenthe true process parameters are known and when /Isrelatively close to 1.To study the effect of m on t h ~n-control run lengthdistribution when n = 5, we simlilated 10 000 obser va-tions from t he run length distribution fo r several values ofm and p. We considered three .process dimensions ofp = 3, 6 and 10.We selected a = 0,0027, the value usuallyused in univariate X control charts.For each combination of p and m, we generated msubgroups from a stable multivariate normal distribution.The process parameters were estimated based on these min-control initial subgroups. Then, further subgroupswere generated from the same stable normal distributionand the T~ statistic was calculated for each subgroup.This statistic was plotted o n a co ,ntrol cha rt with UCL,,Iuntil a signal was issued. The number of subgroupsdrawn until a signal was given constituted one observa-tion from the run length distribution. This procedure wasreplicated 10 000 times and an empirical run length dis-tribution was found. These empirical run length distri-bution s are shown in Figs. 1-3 along with the nom inalrun length distribution (denoted by m = inf.). The curves


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532 Nedumaran nd Pignatiello Jr

Fig. 1 R u n length empirical distribution for p 3, 5 Fig. 3, R u n length empirical distribution for p 10 n 5a = 0.0027: r 0.0027.of m. Three process dimensions of p 3 6 and 10 wereconsidered. B oth in-control and out-of-control situationswere considered. T h e false alarm probability for eachsubgroup was set at 0.0027. For each combination of p mand shift magnitude I subgroups were generated froma stable multivariate normal distribution and the processparameters were estimated. Starting with subgroupm 1 the process mean was changed from ~ r o p pby introducing a shift of magnitude 2 where

Subgroups were generated from the out-of-control pro-cess and the ~ tatistic 4) was calculated for each sub-g roup using estimated parameters. This statistic wasFie. 2 Run length e m ~ i r i c a ldistribution or p 6 5 plotted o n a control ,-hart with ICLZ2until a signal wasa 0.0027. issued. Th e number of subgr oups until a signal was givenconstituted one observation from the run length distribution. This procedure was replicated 10 000 times andappear to have non-differentiable points due to the scal- the ARL and the S D R L were obtained. The results areing of the k-axis. showngin Tab les 1-3.To study further the effect of m when n 5, values of Several observ ations ca n be made from Figs. 1-3 andt h e A R L a n d theSDRLweres imulatedforsevera lva lues Tables 1-3. When the process parameyers are estimatedTnbtc I A R L and S D R L for p 3, 5 r 0.0027

A R L S D R L A R L S D R L A R L S D R L A R L S D R L A R L S D R L


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Constructing control charts 5Table 2 . A R L and S D R L for p 6 n 5 0 0027

0.0 0.5 1.0 1.5 2.0A R L S D R L A R L SDRL A R L S D R L A R L SDK L A R L S D R L

Table 3 AR L and SDRL for p 10, n 5 , a 0 0027

0.0 0.5 l O 1.5 2 0R L S D R L A R L S DR L A R L S DR L A R L S D R L A R L SDRL

from a small numbe r of sub group s, there is an increase inthe rate of false alarms for the in-control process and theA R L s are shorter for both in-control and out-of-controlprocesses. Also, for small values of m, the SDRLs arelarger than the A RLs , whereas they are approximatelyequal for the case where the true process parameters ar eknown. As m increases, both the run length distributionand the ARLs a pproa ch their nominal values.Following Quesenberry s [2] approach, we make thefollowing recommendations for the minimum value of mnecessary for the control chart based on estimated pa-rameters to behave similar to the control cha rt based o ntrue parameters. For p 3 and n 5, we recommendm 200. For p 6 and n 5 a minimum of m 400 srecommended. For p 10 and n 5 we recommend600.By contrast, the criterion used by Lowry andMontgomery [4] i.e., maintaining relative error of lessthan 0.1 between the x 2 control limit and Alt s controllimit, would lead to substantially lower recomm endationsfor the minimum m for the case where n 5 a n du 0.0027. In particular, their recommended minimumvalue of would be 40, 50 and 70 or p 3, 6 and 10respectively. It can be seen from Figs. 1-3 an d Tab les 1 3

that the Lowry and Montgomery [4] recommended min-imum rn would result in a large number of false alarmsand very short ARLs.I n general, we speculate that a m inimum m in the rangefrom 800p/3 n to 400p/ n 1 is necessary for thecontrol chart based on estimated parameters to behavelike the chart based on true parameters.To test this conjecture, we considered two subgroupsizes of n 4 and 10 when p 8 and a 0.0027. or (,hen 4 case , our recommended mir~ imumm is in the range700-1000, nd for the n 10 cast:, it is in the range 300-450. We selected m values in the middle of each recom-mended range for validation purposes. That is, for lhen 4 case we selected m 900, ind for the n 10 c;isewe selected m 400.Th e criterion used by Lowry andMontgomery [4] would lead to the following recommen-dations for minimum m for the case where u 0.0027.The Lowry and Montgomery [4] recommended m inimumwould be 70 and 30 for n 4 .and 10, respectively.For each case we found the in -co ntro l run length em-pirical distribution and the in-control a nd out-of-controlA R L and S D R L following the: procedures describedabove . The results presented in Ta.bIes 4 and 5 and Figs . 4


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534 Nedumaran and Pignut iello JrTnblc 4. Validation test: A R L and S D R L for p 8, n 4 I 0.0027

10 0 0 5 1O 1 5 2.0

A R L S D R L A R L SD RL A R L S RL A R L S RL A R L S DR L

TnbIe 5. Validation test: A RL and SDR L for p 8, n 10 0.0027

0 0 0 5 1 0 1 5 2 0A R L S D R L A R L S D R L A R L S D R L A R L S D R L RL S D R L

and 5 show that our recommended m n mum m providesadequate performance, whereas the Lowry and Mont-gomery [4] recommended minimum m would result in alarge number of false alarms.A number of additional computer runs were carriedout for different values of n and p to test our speculationon minimum m. The results (not reported here) suggestthat the speculation is reasonable for other values of nand p.

4 Chart implement tion during start up st geIn this section, we first describe an exact method forFig 4 Validation test: r u n length empirical distribution for constructing u L usin estimated parameters that per-p 8, n 4 0.0027. forms similar to the chart based on true parameters withrespect to specific percentile points of the in-control runlength distribution. Next, we describe how Alt s [3] con-trol limits can be used instead of the exact method.We suppose that data from m initial in-control sub-groups are available. W e let and x i, > m ) denotetwo future subgroups. Then,

and

In the exact approach, control limits are constructed forFig 5. Validation test: run length empirical distribution for a specified number k) of future subgroups. That is, wep 8 n 10 a 0.0027. establish control limits to be used for monitoring the


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Constructing control charts 5 3 5process for th e next k subgroups. The value of k may bechosen t o represent the daily or weekly production volume.The T~ tatistic plotted for th e ith su bgro up is given by

Let be the desired probability such that

We let U LE denote the exact upper control limit thatsatisfies (7). Then,

Th at is P~LT; U C L ~ ] 1 - I,where ~ ~~I I ~ X {n Xi P 3-l Xi F },

Here, T:,, is a %,,,,-type sta tis tic discussed by Siota ni[ 5 ] with Siotani's y (m 1)lmn and 6 l lmn. Thus,U LE is the upper (1 4)th percentile point of the dis-

ulated 10 000 observations fro m the run length distribu-tion for several values of m and p with n 5. Weconsidered three process dimensions of p 3, 6 and 10.We selected a 0.0027, the value usually used in uni-variate X ontrol charts. F or each combination of p andm, we generated m subgroups of size n 5 from a stablemultivariate norm al distribution. The process paramet.erswere estimated based on these rn in-control initial subgrou ps. Then, further subg roup s were generated from thesame stable normal distribution and the T* tatistic wascalculated for each subgroup. This statistic was plottedon a control chart with an uppe;i control l imit of UCLruntil a signal was issued. T he nu-mber of s ubgr oup s untila signal was given constituted one observation from therun length d istribution . This procedure was replicated10 000 t imes a nd an empirical run ength distribution wasfound. These empirical run 1t:ngth distributions areshown in Figs. 6 8 along with the nominal run lengthdistribution (denoted by m inf.).Figures 6-8 show th at when A:it's contro l limit is basedon 20 subgroups, the empirical run lengih distribution

tri bution ofFor the control char t based o n estimated co ntrol limitsto perfo rm in a similar man ner to the control char t basedon true parameters, is set equal to the run length dis-tribution of the true parameters ch art . Tha t is,

Pr[RL k] 1 1 a) ,where a is the probability of a false alarm for one sub-group.

Siotani [ 5 ] pointed o ut tha t the sampling distribution of?,,,,,-type statistic is extremely difficult to find.Sio ani 1:5] suggested a two-s age procedure for findingapproximations to the upper percenti le points of thedistribution of which was investigated fur ther bySeo an d Siotani [ 6 ] . However, this two-stage procedureinvolves extensive computations and gives only approxi-mate percentile points. As an alternative, we suggest animplementation procedure for using the simpler controllimit as given by Alt [3].Alt's control limit (5) is exact for any given futuresubgroup. However, it is not exact when several futuresubgroups are plotted on a chart with the same controllimit sinceit ignores the dependence between Xi fand (ft, x). Hence, one implementation metho d wouldbe to u pdate the co ntrol l imit after every future subgroupplotted on the chart . But, such an approach is notrecommended since it involves extensive computationsfor finding the control limit for each future subgroup.More importantly, such an approach will have a lowpower in detecting gradual mean shifts since the esti-mated process mean is updated after every subgroup.T o study the effect of on the in-control run lengthdistribution when Alt's control limit (5) is used, we sim-

Fig. 6 . Empirical run length distl-ibution for 3, n := 5r 0 0027

Fig. 7 Empirical run length dist:iibution for p 6 n = 5a 0.0027.


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536 Nedumaran and Pignatiello Jr

Fig. 8. Empirical run length distribution for p = 10 n = 5a = 0.0027.differs from the nominal run length distribution for arelatively large number of future subgroups, k > 150. It isessentially identical t o the nominal ru n length distributionfor a small number of fu ture subgroups, k 50. Further ,the empirical run length distribution approaches thenominal run length distribution for m 50.For the control cha rt with Alt s con trol limit t o per-form in a similar manner to the control chart based ontrue parameters, we propose the following implementa-tion approach. When m < 50, construct the control c hartfor small k, e.g., k 20 future subgroups, and update thecoutrol limit after every k future subgroups. When pro-cess parameters are estimated based on at least 50 sub-groups m 50) , the resulting control limit can then beused for a large number of future subgroups. In fact, thiscontrol limit can be used until the recommended mini-mum number of subgroups, i.e. 800p /3 (n to, 400p ln ) , is accumulated, after which a perm anent X2control limit can be established.The implementat ion approach that we have describedis based on the use of the T tatistic. W e showed that thefalse a la rm rates increase when standa rd X2 contro l limitsa r e used on T~ ontrol charts unless process parametersarc estimated from a relatively large number of sub-groups . O ur proposed im plementation approac h, which isbased on Alt s contro l limit, can help alleviate thisproblem. T he determination of Alt s control limit is onlyslightly inore difficult than that of the X control limit. Itshould be pointed out that an alternative to the entireimplementation approach described here is the use of Qcharts developed by Quesenberry [7].5. SummaryIn this paper we studied the performance of X2 controllimits on T~ ontrol charts. We made recommendationson the minimum number of subgroups necessary forcontrol charts based on estimated param eters to perform

similar to control charts based on true parameters whenX2 control limit s used. We discussed a n exact procedurefor constructing T~ control charts based on estimatedparameters that perform similar to control charts basedon true parameters regardless of the number of prelim-inary subgroups. Further, we presented an implementa-tion approach for using Alt s con trol limit on T ontrolcharts until the accumulation of recommended minimumnumber of subgroups. Finally, it should be noted that thefalse alarm rates discussed in this paper are the rates forthe entire charting program, an d not for individual chartsas is discussed by Quesenberry [2J.AcknowledgmentsThis material is based in part upon the work supportedby the Texas Advanced Research Program under GrantNO. 999903-120.References[ I ] Montgomery, D.C. 1 996) Introduction to Statistical Quality Con-rrol, John Wiley Sons, New York, NY, pp. 362-367.[2] Quesenberry, C.P. (1993) The effect of sample size on estimatedlimits for and X control charts. Journal of Quality Technology, 25,237-247.[3j Alt, F.B.1976) Small sample probability limits for the mean of amu1 ivariate normal process in ASQC Technical ConferenceTransactions, pp. 170-1 76.[ ] Lowry, C.A. and Montgomery, D.C. (1995) A review of multi-variate control charts. IIE Transactions, 27 800-810.[5] Siotani, M. (1959) The exact value of the generalized dist nces ofthe individual points in the multivariate normal sample. Annuls othe Institute of Starktical Mathematics, 10, 183-203.[6] Seo, T. and Siotani, S. 1993) Approximations to the upper per-cen tiles of T:,,-type statistics, in Statistical Science Data Anal-ysis, Matsusita, K . t al . , (eds), VSP Tokyo.[7] Quesenberry, C.P. (1 997) SPC Merho for Qualisy Improvement,John Wiley Sons, New York, NY.

BiographiesDr. Gunabushanam Nedumaran is a Senior Consultant with OracleCorporation. His research interests are in the area of applied statisticsand statistical process control. H e is an A S Q certified Quality Engineer.Dr Joseph J. Pignatiello, Jr is an Associate Professor in the Departmentof Industrial Engineering in the FAMU-FSU College of Engineering atFlorida State University and Florida A M University. Dr Pignati-ello s interests are in quality engineering and include statistical processcontrol, process capability, design and analysis of experiments, robustdesign and engineering statistics. Dr Pignatiello w o n the 1994 ShewellAward from ASQC s Chemical and Process Industries Division and the1994 Craig Award from ASQC s Automotive Division for his researchon process capability. In 1990 and 1991 he won awards from the EllisR. Ott Foundation for his research on SPC and the methods ofTaguchi. He serves on the editorial boards of IIE Transacrions, Journalof Quality Technology. Quality Engineering and the In fernat onalJournal of Industrial Engineering.Conrribured by the On-line Quaiily Engineering Deparrment.

nedumaran, g. & pignatiello, j. j. (1999) on constructing t² control charts for on-line process...

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