CUSUM Charts for Detecting Special Causes in Integrated...

Research Article

Published online 8 July 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qre.1045

CUSUM Charts for Detecting Special Causes inIntegrated Process ControlMarion R. Reynolds Jra∗† and Changsoon Parkb

This paper investigates control charts for detecting special causes in an ARIMA(0,1,1) process that is being adjustedautomatically after each observation using a minimum mean-squared error adjustment policy. It is assumed that theadjustment mechanism is designed to compensate for the inherent variation due to the ARIMA(0,1,1) process, but it isdesirable to detect and eliminate special causes that occur occasionally and produce additional process variation. It isassumed that these special causes can change the process mean, the process variance, the moving average parameter,or the effect of the adjustment mechanism. Expressions are derived for the process deviation from target for all of theseprocess parameter changes. Numerical results are presented for sustained shifts, transient shifts, and sustained drifts inthe process parameters. The objective is to find control charts or combinations of control charts that will be effective fordetecting special causes that result in any of these types of parameter changes in any or all of the parameters. CUSUMcharts designed for detecting specific parameter changes are considered. It is shown that combinations of CUSUM chartsthat include a CUSUM chart designed to detect mean shifts and a CUSUM chart of squared deviations from target givegood overall performance in detecting a wide range of process changes. Copyright © 2009 John Wiley & Sons, Ltd.

Keywords: automatic process control; average time to signal; engineering process control; monitoring; squared deviations fromtarget; statistical process control; steady state

1. Introduction

Two widely used approaches for monitoring and improving the quality of the output of a process are statistical process control(SPC) and automatic process control (APC). The basic approach of SPC is to reduce process variation by using control chartsto detect and facilitate the removal of special causes, while the basic approach of APC is to reduce process variation in an

autocorrelated process by using a control scheme to adjust the process so that it moves closer to target. In the recent years there hasbeen increasing interest in using these two approaches together to reduce process variation. In particular, control charts are used tomonitor a process under APC adjustment so that special causes can be detected and removed.

Standard SPC control charts designed for independent process observations may not work well for monitoring an autocorrelatedprocess under APC adjustment because the effect of a special cause on the process output will be modified by the adjustment action.A number of recent papers have investigated control charts that can be used for processes with autocorrelation and APC adjustment.See, for example, Box and Kramer1, Vander Wiel et al.2, Montgomery et al.3, Wardell et al.4, Vander Wiel5, Apley and Shi6, Tsung et al.7,Jiang and Tsui8, 9, Capilla et al.10, Shu et al.11, Tsung and Tsui12, Jiang13, Han and Tsung14, Runger et al.15, Apley and Chin16, andNembhard and Valverde-Ventura17. Most of these papers consider control charts designed to detect a particular process change thatis of interest. In many applications, however, it will not be possible to specify in advance the process change that will be produced byspecial causes, hence it may be necessary to use several control charts in combination to provide the ability to detect a wide rangeof possible process changes.

The basic objective of this paper is to investigate control charts and combinations of control charts for detecting special causes thatcan affect a process that is under APC adjustment. It is assumed that, in the absence of special causes, the process can be modeledas an ARIMA(0,1,1) process, and that a standard minimum mean-squared error (MMSE) adjustment policy is being used. In addition,it is assumed that special causes can occasionally occur and produce undesirable changes in any of the parameters associated withthe process or the adjustment mechanism. It is also assumed that these special causes occur relatively infrequently, and that it isdesirable to detect these special causes, so that they can potentially be removed (rather than to design an adjustment mechanism totry to compensate for them).

aDepartments of Statistics and Forestry, Virginia Tech, Blacksburg, VA 24061, U.S.A.bDepartment of Statistics, Chung-Ang University, Seoul 156-756, Korea∗Correspondence to: Marion R. Reynolds Jr, Departments of Statistics, Virginia Tech, Blacksburg, VA 24061, U.S.A.†E-mail: [email protected]

Contract/grant sponsor: Korean Research Foundation; contract/grant number: 2009-0073336

Copyright © 2009 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2010, 26 199--221

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M. R. REYNOLDS JR AND C. PARK

We consider a variety of different process parameter changes that could be produced by special causes, and derive expressionsfor the process deviation from target for these parameter changes. It is assumed here that it is necessary to detect many differenttypes of process changes, hence we consider some standard SPC Shewhart and CUSUM control charts, and also some CUSUM controlcharts specifically developed for SPC monitoring of the APC adjusted process.

2. The adjusted ARIMA(0,1,1) process without special causes

Suppose that a process is observed at times t =1, 2,. . ., where the time between consecutive observations is taken as the time unit.Assume that, in the absence of adjustment and special causes, the process deviation from target at time t, say Zt , is an ARIMA(0,1,1)process (Box et al.18) of the form

Zt =Zt−1 +�t −��t−1, t =1, 2, 3,. . . (1)

where �t ∼N(0,�2� ) and it is usually assumed that 0≤�<1. We assume that the ‘starting values’ are Z0 =0 and �0 =0, hence we can

write Equation (1) as

Zt = (1−�)t−1∑j=1

�j +�t , t =1, 2, 3,. . . (2)

As an example, consider Figure 1, which is a plot of 100 measurements (from Box and Luceno19) of the thickness of a very thinmetallic film made when no adjustment was applied in the manufacturing process. Box and Luceno19 report that this process can beadequately modeled as an ARIMA(0,1,1) process. For purposes of an example, we assume that the true parameters values are �=0.8and �� =11.2, corresponding to estimates obtained from the data. Figure 1 shows that the process is wandering from the target of80 in the absence of adjustment.

The MMSE forecast of Zt made at time t−1, say Zt , is an EWMA that can be written as:

Zt = (1−�)t−1∑j=1

�j (3)

We assume that the process is adjusted after every observation using the optimal MMSE adjustment policy with no constraint on theamount of adjustment. Let At be the total effect on the process at time t of all previous adjustments, and let at =At −At−1 be theeffect of the adjustment made at time t−1. We assume that the effect of any adjustment made at time t−1 takes effect at time t,hence there is only a one-period delay. Then the actual observed process (the actual deviation from target) at time t, say et , is

et =Zt +At (4)

The MMSE controller is a discrete integral controller that chooses the amount of adjustment that will produce the effect At =−Zt ,hence et from Equation (4) becomes et =Zt − Zt =�t . Thus, in Equation (3) we can replace �t with et and write

Zt = (1−�)t−1∑j=1

ej (5)

so this gives

at =−Zt + Zt−1 =−(1−�)�t−1 =−(1−�)et−1

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Figure 1. Unadjusted metallic film thickness. This figure is available in colour online at www.interscience.wiley.com/journal/qre

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Figure 2. Process deviation from target using MMSE adjustment when the process is in control. This figure is available in colour online atwww.interscience.wiley.com/journal/qre

Note that et =�t as long as the model and controller do not change, but this may not be true after a special cause changes theparameters of the model or the controller. Now it is et that is actually observed in practice, hence the computation of Zt and thedetermination of the amount of adjustment needed to produce at would be done in terms of et .

In the metallic film example, the film thickness can be adjusted using the metallic deposition rate. Figure 2 is a plot of the thicknessdeviation from the target of 80 (the value et) that would have resulted if the process had been effectively adjusted using the depositionrate (three-sigma control limits are shown for visual reference). We see that MMSE adjustment would have kept the thickness muchcloser to target (the deviation closer to 0).

3. The effects of special causes on the adjusted process

Most work on monitoring a process under feedback control assumes that a process change can be represented as some type of meanchange (see, for example, Hu and Roan20, Tsung and Tsui12, Runger et al.15, and Apley and Chin16). Here, we assume that a specialcause occurs between observations � and �+1, �∈{0, 1, 2,. . .}, and this special cause can change any or all of the four parameters inthe process (the mean, the variance, �, and the effect of adjustment) at any time after �. We assume that a parameter change canbe a sustained step shift or a linear drift that remains until detected by a control chart, or a transient step shift that lasts only for aspecific period of time (see Reynolds and Stoumbos21, 22 for discussion of sustained and transient changes). Some authors (e.g. Jianget al.23) refer to transient changes in terms of the effect on et , but here we use transient to refer to the effect on process parameters.

It will be convenient to represent parameter changes using multiplicative or additive factors along with the parameter, where weuse the parameter with a ‘∼’ to represent the factor. We will now derive expressions for et from the adjusted process at some timet =�+k after the change, where k =1, 2,. . . .

Consider first the situation in which the effect of a special cause is the addition of constants to the deviation from target, so thatthe actual observed deviation from target at time t =�+k is

e�+k =Z�+k +A�+k + �k��

These additive factors, which will be referred to as a mean change, could occur because of problems with the adjustment mechanism,problems with the Zt process, or problems with the measurement process that result in incorrect readings. Note that we may haveE(Z�+k +A�+k + �k��) �= �k�� because E(Z�+k +A�+k) �=0 as a result of past process changes.

The values of the additive factors �1, �2,. . . determine the type of mean change. For example, �k = � for k =1, 2,. . . correspondsto a sustained mean shift of size ��, while �k = � for k =1 and 2, and �k =0 for k =3, 4,. . . corresponds to a transient shift that lastsfor two time periods. A linear drift in the mean with drift rate r�� per unit time is represented as �k = rk for k =1, 2,. . . . If frequentmean changes of a certain type are expected, then it may be possible to design a controller to compensate for these changes. Here,we assume that mean changes of an unknown type are expected to occur relatively infrequently, hence the objective is to avoidincreased loss in the future by detecting these changes and eliminating the cause.

Consider next an increase in the variance of the process corresponding to an increase in the variance of �t . At time t =�+k, thevalue of �t is assumed to be �k��+k , hence the variance is now �2

k�2� . With Equation (1), this implies that Z�+1 =Z�+ �1��+1 −��

and

Z�+k =Z�+k−1 + �k��+k −��k−1��+k−1, k =2, 3,. . .


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Now consider a change in the parameter �, and assume that at time t =�+k the value of � is �k�. Using Equation (1) this implies that

Z�+k =Z�+k−1 +��+k − �k��+k−1, k =1, 2,. . .

hence there is a change in the underlying Zt process. An adaptive controller can be used when relatively frequent changes in � areexpected, but here we assume that it is desirable to detect these changes in � and attempt to remove the cause. When there is achange in � and/or the variance of �t , an easy inductive proof shows that at time �+k we have

Z�+k = Z�+k−1∑i=0

(1− �i+1�)(�i��+i)+ �k��+k (6)

where �0 =1.Finally, consider a possible change in the effect of the adjustment, and assume that at time t =�+k, the actual effect of the

attempted adjustment made at time �+k−1 is aka�+k . This change in the effect of the adjustment could be due to a problem withthe adjustment mechanism itself, or to a change in the process that changes the effect of a given adjustment.

Now consider the effect of parameter changes on the observed deviation from target after time �. If the special cause affects anyof the parameters, then using Equation (6) gives

e�+k = Z�+k +�∑

i=1ai +

k∑i=1

aia�+i + �k��

=[

Z�+k−1∑i=0

(1− �i+1�)(�i��+i)+ �k��+k

]− Z�−

k∑i=1

ai(1−�)e�+i−1 + �k��

=k−1∑i=0

(1− �i+1�)(�i��+i)−(1−�)k∑

i=1aie�+i−1 + �k��+k + �k��, k =1, 2,. . .

(7)

In evaluating the properties of e�+k it will be useful to have an expression that involves only the iid terms {��+i} and not the terms{e�+i}. The general expression for e�+k is complex, so we look at the special case of no change in the effect of adjustment (ai =1,i=1, 2,. . .). In this case

e�+1 = (1− �1)��+ �1��+1 + �1�� (8)

from Equation (7), and it is shown in the Appendix that

e�+k =k−1∑i=0

(1− �i+1)�k−i�i��+i + �k��+k +(

�k −(1−�)k−1∑i=1

�k−1−i�i

)��, k =2, 3,. . . (9)

The expressions for e�+k given by Equations (8) and (9) can be used to determine the effects of parameter changes on e�+k . We nextconsider the effect of specific parameter changes.

4. The effect of changes in the mean and/or variance

If there are changes in the mean and variance only (this corresponds to adding �k�� and changing ��+k to �k��+k at time �+k,k =1, 2,. . .), then we have e�+1 = �1��+1 + �1�� and

e�+k = �k��+k +[�k −

k−1∑i=1

�i(1−�)�k−1−i

]��, k =2, 3,. . . (10)

Thus, the variance changes affect e�+k only through the change at time �+k, while the mean changes affect e�+k through the changesat times �+1, �+2, . . ., �+k. Note that e�+1, e�+2,. . . are still independent after changes in the mean and/or variance.

If �k = �, k =1, 2,. . ., corresponding to a sustained shift in the mean, then

e�+k = �k��+k + ��k−1��

This shows the well-known result (see, for example, Vander Weil5 and Runger et al.15) that for �<1, the effect on e�+k of a sustainedmean shift is the largest immediately after it occurs, and then the effect dissipates as k →∞, due to the effect of the adjustmentprocess.

Figure 3 is a plot of ek for the metallic film example when a sustained mean shift of size �� =4�� is introduced starting afterobservation �=59. This mean shift was introduced by adding 4�� to each e�+k and then applying the standard MMSE policy.Figure 3 shows that the large 4�� shift initially produces large effects, but these effects dissipate quickly; after about observation 72the plot in Figure 3 is essentially the same as the in-control plot in Figure 2.

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Figure 3. Process deviation from target using MMSE adjustment when a 4�� mean shift occurs before observation 60. This figure is available in colour online atwww.interscience.wiley.com/journal/qre

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Figure 4. Process deviation from target using MMSE adjustment when a mean drift of rate 0.2�� starts before observation 60. This figure is available in colour onlineat www.interscience.wiley.com/journal/qre

If a special cause produces a sustained linear drift in the mean starting at time � with rate r�� per unit time, then �i = ri, i=1, 2,. . . , kin Equation (10), and we get

e�+k = �k��+k +r

[1−�k

1−�

]��

If �<1, then |E(e�+k)| increases with k and reaches a limit r�� / (1−�), hence this means that the controller considered here cannotcompensate for this special cause.

Figure 4 is a plot of ek for the metallic film example when there is a sustained mean drift with rate 0.2�� per unit time startingat �=59. This slow drift does not produce extremely large deviations from target, but there is a pattern of positive deviations thatpersists over time.

Suppose now that there is a transient shift in the mean of duration l time periods, so that �k = � �=0 for k =1, 2,. . . , l, and �k =0 fork > l. Then from Equation (10)

e�+k =⎧⎨⎩

�k��+k + ��k−1��, k =1, 2,. . . , l

�k��+k − ��k−l−1(1−�l)��, k = l+1, l+2,. . .

Note that the adjustment process must compensate for both the initial mean shift and the vanishing of the mean shift that occursafter l time periods, so the overall effect of the transient shift may actually be larger than the effect of the corresponding sustainedshift.

Figure 5 is a plot of ek for the same situation as in Figure 3, except that the shift is a transient mean shift that vanishes beforeobservation 70. Figure 5 has the same large positive deviations as Figure 3 when the shift starts, but Figure 5 also shows large negativedeviations starting at observation 70 when the adjustment process adjusts for the vanishing mean shift.


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Figure 5. Process deviation from target using MMSE adjustment when a 4�� transient mean shift occurs before observation 60 and lasts for 10 observations. This figureis available in colour online at www.interscience.wiley.com/journal/qre

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Figure 6. Process deviation from target using MMSE adjustment when a 50% increase in �� occurs before observation 60. This figure is available in colour online atwww.interscience.wiley.com/journal/qre

A sustained increase in the variance corresponds to �k = �>1, k =1, 2,. . . . Figure 6 is a plot of ek when there is a sustained varianceincrease of size �=1.50 starting at �=59. The increased variance of the deviations is clearly visible here.

5. The effect of changes in h

Consider the case in which � is the only parameter that changes, and �k = �, k =1, 2,. . . . Taking ak =1, �k =1, and �k =0, k =1, 2,. . .,in Equations (8) and (9) and simplifying gives

e�+k =��+k +(1− �)k−1∑i=0

�k−i��+i (11)

We also get the alternate expression

e�+k =��+k +�e�+k−1 − ��+k−1 (12)

Using Equation (11) it is easy to show that E(e�+k)=0, and

Var(e�+k)=[

1+(1− �)2�2 1−�2k

1−�2

]�2

�

If �<1 then

limk→∞

Var(e�+k)=[

1+ (1− �)2�2

1−�2

]�2

a

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Figure 7. Process deviation from target using MMSE adjustment when � shifts from 0.8 down to 0.4 before observation 60. This figure is available in colour online atwww.interscience.wiley.com/journal/qre

Equation (12) shows that e�+k has the form of an ARMA(1,1) process with parameters ‘�’=� and ‘�’= �� (see Box and Luceno19,p. 152), so the change in � produces a correlated sequence e�+1, e�+2,. . . with mean zero and increasing variance (assuming that�<1).

Figure 7 is a plot of ek for the metallic film example when there is a sustained decrease in � from 0.8 to 0.4 (this correspondsto �=0.5) starting at �=59. The most notable effect visible in Figure 7 is a relatively long run of positive deviations soon after thedecrease in �.

6. The effect of changes in adjustment

If a special cause changes only the effect of the adjustment, we take �k =1, �k =1, and �k =0, k =1, 2,. . . . It is shown in the Appendixthat e�+1 = (1−�)(1− a1)��+��+1 and

e�+k = (1−�)k−2∑i=0

((1− ai+1)

k∏j=i+2

(1− aj(1−�))

)��+i +(1−�)(1− ak)��+k−1 +��+k (13)

for k =2, 3,. . . . If ak = a, k =1, 2,. . ., then this simplifies to

e�+k = (1−�)(1− a)k−1∑i=0

(1− a(1−�))k−1−i��+i +��+k (14)

and from Equation (14) we can also get

e�+k =��+k +(1− a(1−�))e�+k−1 −��+k−1 (15)

Using Equation (14), it is easy to show that E(e�+k)=0, and

Var(e�+k)=[

1+(1−�)(1− a)2 1−{1− a(1−�)}2k

a(2− a(1−�))

]�2

�

If a>0 and 1− a(1−�)<1, then

limk→∞

Var(e�+k)=[

1+ (1−�)(1− a)2

a(2− a(1−�))

]�2

�

Equation (15) shows that e�+k has the form of an ARMA(1,1) process with parameters ‘�’=1− a(1−�) and ‘�’=�, hence a sustainedshift in the effect of adjustment produces a correlated sequence e�+1, e�+2,. . . with mean zero and increasing variance.

Figure 8 is a plot of ek for the metallic film example when there is a sustained three-fold increase in the effect of the adjustmentstarting at �=59 (the large value of a=3.0 was used here because the process is insensitive to over-adjustment or under-adjustment).Except for one large negative deviation, it is a bit difficult to see the effect of this over-adjustment in Figure 8.


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Figure 8. Process deviation from target using MMSE adjustment when the effect of adjustment increases by a factor of 3 before observation 60. This figure is availablein colour online at www.interscience.wiley.com/journal/qre

7. Control charts for detecting process changes

A Shewhart chart can be based on plotting et , with control limits at ±h��, where the traditional choice for the constant h is 3 (theseare the control limits shown in Figures 2–8). Some shorthand notation will be used here for control charts, with ‘S’ used for Shewhartcharts and ‘C’ for CUSUM charts. Call the traditional Shewhart chart the S� chart, with the subscript ‘�’ indicating that this chart is achart designed for detecting changes in the mean.

In developing CUSUM charts it will be convenient to write e�+k as

e�+k = �k��+k +b�+k

so that, apart from any change in the variance, b�+k represents the additional deviation from target that is a result of the specialcause that changes the other parameters.

Let H0 represent the hypothesis of no change in any of the parameters, H0 : �k =0, �k =1, �k =1, ak =1, k =1, 2,. . ., and let H1

represent a specific alternative hypothesis, H1 : �k = �∗k , �k = �∗

k , �k = �∗k , ak = a∗

k , k =1, 2,. . .., where a ‘∗’ is used to represent thesespecific out-of-control parameter values. Now �1,�2,. . . are assumed to be iid normal, so e1 / ��, e2 / ��,. . . are iid standard normal if H0is true. If H1 is true then

e�+k −b∗�+k

�∗k��

, k =1, 2,. . .

will also be iid standard normal, where b∗�+k is the value of b�+k corresponding to H1. Let � represent the joint density of iid standard

normal variables, and assume for the moment that the value of � is known. The log likelihood ratio statistic corresponding to H0 andH1 is

L�+k = lnf (e�+1,. . . , e�+k|H1)

f (e�+1,. . . , e�+k|H0)

= ln�((e�+1 −b∗

�+1) / �∗k��,. . . , (e�+k −b∗

�+k) / �∗k��)

�(e�+1 / ��,. . . , e�+k / ��)− ln

k∏i=1

�∗i

= − 1

2�2�

⎡⎣ k∑

i=1

(e�+i −b∗

�+i

�∗i

)2

−k∑

i=1e2�+i

⎤⎦−

k∑i=1

ln �∗i

=k∑

i=1c�,�+i

where c�,�+k is the increment or score for observation �+k given by

c�,�+k =− 1

2�2�

⎡⎣(

e�+k −b∗�+k

�∗k

)2

−e2�+k

⎤⎦− ln�∗

k (16)

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If there is no change in the variance (�∗k =1, k =1, 2,. . .), Equation (16) simplifies to

c�,�+k = 1

2�2�

b∗�+k(2e�+k −b∗

�+k) (17)

and if there is only a change in variance (b∗�+k =0, k =1, 2,. . .), Equation (16) simplifies to

c�,�+k = �∗2k −1

2�∗2k �2

�

e2�+k − ln �∗

k (18)

CUSUM charts can be constructed using the increments c�,�+k when � is known. For any given �=0, 1, 2,. . . define the CUSUMstatistic at time t =�+1, �+2,. . ., say C�,t , as

C�,t =max{C�,t−1, 0}+ c�,t (19)

where C�,� =0. This CUSUM chart would signal at time t if C�,t >h for some constant h, where h can be chosen to give specifiedin-control properties. CUSUM charts of this general form for detecting specific process changes are sometimes called CUSCORE charts.For additional discussions see, for example, Box and Ramirez24, Box and Luceno19, Luceno25, 26, Shu et al.11, Runger and Testik27,Han and Tsung14, Apley and Chin16, and Nembhard and Valverde-Ventura17. In practice, this CUSUM chart could be used in situationsin which there is some likely time that a process change will occur. Here, we are considering the case in which � is unknown, so thechart must be modified to account for the fact that � is unknown.

One option when � is unknown is to consider all values of C�,t for �=0, 1,. . . , t−1 and take the maximum. A potential computationalproblem occurs as t gets very large, so a simple option is to use a window of the last m points as possible values for �. Thus, signal ifthe statistic

Mt = maxm(t)≤�<t

C�,t (20)

exceeds an upper control limit h, where m(t)=0 for t =1, 2,. . . , m, and m(t)= t−m for t =m+1, m+2,. . .. Statistics of this type aresometimes called generalized likelihood ratio statistics (see, for example, Apley and Shi6, Runger and Testik27, and Han and Tsung14).

Another option is to use a CUSUM statistic of the form (19) and restart b∗�+k at any time that the CUSUM statistics becomes non-

positive (this is sometimes referred to as using reinitialization (Apley and Chin16)). In particular, define �(1) to be the first t for whichC0,t ≤0, �(2) be the first t for which C�(1),t ≤0, and, in general, �(i) is the first t for which C�(i−1),t ≤0. For a given value of t, definei(t)=max{i :�(i)< t}. Then signal at time t if the statistic

C�(i(t)),t =max{C�(i(t)),t−1, 0}+c�(i(t)),t (21)

exceeds a control limit h.Applying either of the CUSUM charts defined by Equations (20) and (21) requires that b∗

�+k be computed, and this can be done

using recursive relationships when � is specified. Previous expressions for b∗�+k were expressed in terms of {�i}, but it will be necessary

to give expressions in terms of {ei}, so that b∗�+k and the CUSUM statistics can be computed in terms of {ei}.

The construction of CUSUM charts using Equations (16), (17), or (18) requires that �∗k , �∗

k , �∗k , a∗

k , k =1, 2,. . . be specified. In practice,the exact form and size of a process change that will occur will rarely be known. Thus, as in the case of standard CUSUM charts for iidobservations, it is useful to think of these values as tuning parameters than can be used to tune the CUSUM charts to be sensitive tocertain sizes of process changes. For example, if �∗

k = �∗ then the choice of �∗ determines the size of the variance shift to which theCUSUM chart will be particularly sensitive.

8. CUSUM charts for the mean

The ‘standard’ CUSUM chart for detecting a change in the mean of e1, e2,. . . is designed to detect a sustained shift in E(et). Now thein-control mean of E(et) is 0, so to detect a sustained shift to E(e�+k)=�∗��, k =1, 2,. . ., the CUSUM increment from Equation (17)should be �∗(e�+k −�∗�� / 2) / ��. In most applications it would be desirable to detect both increases and decreases in the mean, hencea two-sided CUSUM chart would be carried out using two CUSUM statistics, one with �∗ >0 and the other with �∗ <0. We will callthis two-sided CUSUM chart the C� chart. In practice, the increment �∗(e�+k −�∗�� / 2) / �� is usually multiplied by �� / �∗ to give theincrement e�+k −�∗�� / 2. This gives the CUSUM chart for the mean used, for example, in Reynolds and Stoumbos21, 22 for the caseof independent observations (with no APC adjustment). Vander Wiel5 investigated the performance of the C� chart for the case of asustained shift in the mean when the process is under adjustment.

A sustained shift in the mean does not produce a sustained shift in E(e�+k), so it will be instructive to find the mean change thatproduces the sustained shift in E(e�+k) that the C� chart is designed to detect. From Equation (10) we see that b∗

�+1 = �∗1�� and

b∗�+k =

[�∗

k −k−1∑i=1

�∗i (1−�)�k−1−i

]��, k =2, 3,. . . (22)


20

7


Now E(e�+k)=b∗�+k��, and if �∗

1, �∗2,. . . satisfy �∗

1 =�∗ and

�∗k =�∗+

k−1∑i=1

�∗i (1−�)�k−1−i , k =2, 3,. . . (23)

then E(e�+k)=�∗��, k =1, 2,. . ., where �∗ is a constant. Thus, we see that the standard C� chart is designed to detect a mean changethat initially jumps to �∗�� and then Equation (23) shows that |�∗

k | continually increases (assuming that �<1).We next consider the question of whether a CUSUM chart designed specifically to detect a sustained mean shift can give significant

improvements in the performance for detecting these shifts. A sustained mean shift corresponds to �∗k = �∗, k =1, 2,. . ., hence using

this in Equation (22) gives

b∗�+k = �∗�k−1��, k =1, 2,. . .

and hence b∗�+k can be written as b∗

�+k =�b∗�+k−1, k =2, 3,. . ., with b∗

�+1 = �∗��. We see that, if �<1, then b∗�+k →0 as k →∞. We

will use the CUSUM chart based on Mt given by Equation (20) in a two-sided version to detect both increases and decreases in themean. We call this CUSUM chart for a sustained mean shift the C�s chart to distinguish it from the standard C� chart, where ‘s’ is usedto indicate a mean shift.

The standard C� chart is designed to detect a mean change that continually moves away from 0 (see Equation (23)), hence, whenthe actual mean change is a linear drift, it will be interesting to see how well this chart performs relative to a CUSUM chart designedspecifically for a linear drift. For a linear mean drift, we take �∗

k = r∗k, k =1, 2,. . . in Equation (22), and this gives

b∗�+k = r∗

(1−�k

1−�

)��, k =1, 2,. . .

and hence b∗�+k can be written as b∗

�+k = r∗��+�b∗�+k−1, k =2, 3,. . ., with b∗

�+1 = r∗��. We found that the CUSUM chart based onC�(i(t)),t in Equation (21) seems to have better performance for linear drifts than the CUSUM chart based on Mt , so we use C�(i(t)),t in atwo-sided version. Call this CUSUM chart for a sustained mean drift the C�d chart, where ‘d’ indicates drift.

9. A CUSUM chart for the variance

From Equation (18), the CUSUM increment for a sustained variance shift (�∗k = �∗) is

c�+k = �∗2 −1

2�∗2�2�

e2�+k − ln �∗ (24)

hence this increment does not depend on the unknown �. Here, we are primarily concerned with detecting variances increases, hencewe use a one-sided chart and call it the C� chart.

In the context of SPC monitoring, Reynolds and Stoumbos21, 22 investigated the combinations of CUSUM control charts fordetecting changes in the mean and variance of independent process observations, and one of the charts investigated in this work isthe C� chart. There, the CUSUM increment (24) was multiplied by 2�∗2�2

� / (�∗2 −1) to give the increment

e2�+k − 2 ln �∗

1−(1 / �∗2)�2

�

It was found that the combination of the C� and the C� charts provides very effective detection of both small and large changes inthe process mean or variance. The C� chart, while designed for detecting increases in the variance, is actually more effective than theC� chart for detecting large mean shifts when the C� chart is tuned to detect small mean shifts.

When monitoring an ARIMA(0,1,1) process under feedback adjustment, e1, e2,. . . are not necessarily independent after a change inthe process, hence the Reynolds and Stoumbos21, 22 results on the performance of various chart combinations do not apply directly.However, the C� chart based on squared deviations from target would seem to be a natural chart to use for monitoring an APCadjusted process. The objective of APC adjustment is usually to minimize the squared deviations from target, so any process changethat increases the squared deviations from target should show up on a chart based on these statistics. In particular, the results thathave been given here show that many process changes have the effect of increasing the variance of et .

10. A CUSUM chart for h

For a sustained shift in �(�∗k = �

∗), we obtain from Equation (11) that b∗

�+k satisfies

b∗�+k = �

∗�b∗

�+k−1 +(1− �∗

)�e�+k−1, k =2, 3,. . .

20

8



with b∗�+1 = (1− �

∗)�e�. This b∗

�+k can be used in Equation (17) to form the CUSUM increment. A two-sided CUSUM chart for � requires

two CUSUM statistics, one with �∗<1 and the other with �

∗>1. These two CUSUM statistics are not symmetric, hence two values of h

are required. Call this two-sided CUSUM chart the C� chart.

11. A CUSUM chart for adjustment

For a sustained shift in the effect of the adjustment (a∗k = a∗), we obtain from Equation (14) that b∗

�+k satisfies

b∗�+k =�b∗

�+k−1 +(1−�)(1− a∗)e�+k−1, k =2, 3,. . .

with b∗�+1 = (1−�)(1− a∗)e�. A two-sided CUSUM chart for the effect of adjustment requires two CUSUM statistics, one with a∗ <1 and

the other with a∗ >1. These two CUSUM statistics are not symmetric, hence two values of h are required. Call this two-sided CUSUMchart the Ca chart.

Note that several recent papers (Tsung, et al.7, Jiang and Tsui9, Tsung and Tsui12, Jiang13) have investigated control charts basedon a1, a2,. . ., but these charts are designed to detect changes in the mean through the effect of the mean change on the amount ofadjustment. In contrast, the Ca chart is specifically designed to detect changes in the adjustment process itself.

12. Performance measures

Define the average time to signal (ATS) to be the expected length of time from the start of process monitoring until a signal by acontrol chart (with the time between observations taken to be the time unit). Fair comparisons among different control charts in theSPC context can be made when all of the charts have the same in-control values of the ATS.

The effectiveness of a control chart in detecting a change in the process is usually evaluated using the expected time requiredfor detection. When �=0, the ATS can be used as a measure of the detection time, but in most situations it is likely that �>0. Here,we use the steady-state ATS (SSATS) which assumes that � is large enough for the process and control statistics to have reachedtheir steady-state distributions by time �. The SSATS also allows for the possibility that the process change can occur randomly in atime interval between samples. This definition of the SSATS is consistent with the definition in other papers, such as Reynolds andStoumbos21, 22.

The effectiveness of an APC adjustment mechanism is frequently evaluated using the expected quadratic loss E(ce2t ), where c is

a constant. If the optimal adjustment policy is used and there is no special cause, then E(ce2t )=E(c�2

t )=c�2� . It will be instructive to

consider the increase in the expected quadratic loss due to the occurrence of special causes. If the special cause occurs between times� and �+1 and is detected at observation �+T for some T ≥1, then the total additional quadratic loss incurred up to the time thatthe special cause is detected is

E

(T∑

i=1(ce2

�+i −c�2� )

)

We assume that the steady-state distributions have been reached by time �, and call this expectation the steady-state extra quadraticloss (SSEQL). Numerical results will be based on c=1 and �2

� =1 (in this case the expected quadratic loss is one unit per time unitwith no special causes).

For special causes that produce an effect on e�+k that dissipates as k increases, the SSATS may not be the most appropriateperformance measure because the effect may dissipate before it can be detected. However, the SSATS should provide a reasonablemeasure of the relative performance of different control charts, so we use it here to avoid introducing another performance measure.A value of the SSATS close to the in-control ATS can be interpreted to mean that it is unlikely that the special cause will be detectedbefore the effect dissipates.

In this paper, simulation with 1 000 000 runs was used to evaluate the performance measures for most control charts and controlchart combinations. Steady-state properties were modeled by using 100 initial in-control observations before the parameter changewas introduced. If a false alarm occurred in the initial in-control observations, then this sequence of observations was discarded anda new sequence was generated.

13. Parameter choices for the control chart comparisons

The control limits of the charts being compared were adjusted so that the in-control ATS is 370.4 (the value for the S� chart with h=3).When two or more charts are used in combination, the control limits were adjusted so that all of the charts have the same individualin-control ATS, and the in-control ATS of the combination is 370.4. The control limits of two-sided CUSUM charts were adjusted sothat the two one-sided charts have the same individual in-control ATS.

The tuning parameters of CUSUM charts determine the sensitivity to small versus large parameter shifts. We found that thetuning parameters did not have a large effect on the overall conclusions about the relative performance of different charts and chartcombinations, hence we present numerical results here for either one or two values of the tuning parameter for each chart.


20

9


We considered the C� chart tuned to detect a sustained shift in E(eu+k) of size either �∗ =0.5 or �∗ =2.0. The C�s chart had m=25and was tuned to detect a sustained mean shift of size either �∗ =0.5 or �∗ =2.0. The C�d chart was tuned to detect a sustainedmean drift with a rate of either 0.2 or 0.5. For the C� chart we used �∗ =1.5, corresponding to a 50% increase in the process standard

deviation. We tuned the C� and Ca charts for a 50% increase or decrease in the parameters, hence we used �∗ =0.5 and �

∗ =1.5 forthe C� chart, and a∗ =0.5 and a∗ =1.5 for the Ca chart. We present numerical results for two values of �, 0.4, and 0.8.

14. Initial comparisons of control charts for mean changes

We now investigate how the C�, C�s, and C�d charts perform in detecting the specific types of mean changes for which they weredesigned, as well as for other types of mean changes. Table I gives the SSATS values for these three charts when �=0.8. The columnslabeled [1] and [2] give results for the C� and C�s charts, respectively, when they are tuned to detect small shifts, while columns [3]and [4] give results when they are tuned to detect large shifts. Columns [5] and [6] give results for the C�d chart. Table II is the sameas Table I except that �=0.4.

The first row in Tables I and II is the in-control ATS (370.4), and the next set of rows corresponds to sustained mean shifts. Wefirst note that the SSATS is large for small sustained mean shifts (particularly when �=0.4), so this implies that there is a substantialprobability that the effect of this shift will dissipate before it is detected. This observation agrees with the conclusion from VanderWiel5 concerning the C� chart. From Tables I and II we see that the C�s chart, which is designed to detect the sustained mean shifts,is significantly better than the standard C� chart for many sustained shifts. Both of these charts are better than the C�d , which wasdesigned to detect mean drifts.

The next set of rows in Tables I and II correspond to mean drifts. When �=0.8, the C�d chart does not seem to be significantlybetter than the other two charts. However, when �=0.4, the C�d chart is significantly better, except for the C� chart with �∗ =0.5.The last two sets of rows in Tables I and II correspond to transient shifts. Here, the C�s chart has the best performance.

The results in Tables I and II show that the best chart to detect a specific mean change depends on the type and size of this meanchange, and the value of � has a significant effect on how quickly this change can be detected. Thus, in applications where thereis some knowledge of the most likely type of process change, a control chart can be selected based on this knowledge. However,here we are assuming that the type, size, and start of the mean change is unknown. It appears that the C�s chart has the best overallperformance, so this chart would be reasonable to recommend for general use. The C� chart is the standard CUSUM chart for mean

Table I. SSATS values for CUSUM control charts for the mean when �=0.8

C� C�s C� C�s C�d

�∗ =0.5 �∗ =0.5 �∗ =2.0 �∗ =2.0 �∗k =0.2k �∗

k =0.5kParameter values [1] [2] [3] [4] [5] [6]

In-control 370.4 370.4 370.4 370.4 370.4 370.4

Sustained 1.00 293.5 306.6 326.3 316.5 304.5 317.7Mean 2.00 123.1 104.4 144.5 119.9 141.7 141.0Shift 3.00 16.0 7.3 14.2 8.9 28.5 18.7(size) 4.00 2.4 1.3 1.0 0.9 4.1 2.0

5.00 1.6 0.8 0.6 0.6 2.1 1.2

Sustained 0.1 29.6 47.0 73.1 55.9 35.6 58.5Mean 0.2 13.6 15.0 18.6 16.1 13.2 16.4Drift 0.3 9.4 9.3 9.9 9.5 8.9 9.2(rate) 0.6 5.5 4.9 4.6 4.7 5.2 4.5

1.0 3.8 3.2 2.9 3.0 3.7 3.0

Transient 1.0 334.6 325.0 328.2 326.2 337.3 326.0Mean 2.0 241.5 150.2 150.6 147.3 242.5 165.1Shift 3.0 75.4 14.4 14.9 13.6 110.2 26.3l =4 4.0 6.3 1.3 0.9 0.9 24.2 2.2(size) 5.0 1.6 0.8 0.6 0.6 3.2 1.2

Transient 1.0 265.7 264.6 293.1 278.7 265.7 280.3Mean 2.0 62.2 38.7 65.8 48.3 68.8 63.8Shift 3.0 5.0 2.4 2.6 2.2 7.2 3.9l =16 4.0 2.2 1.3 0.8 0.9 2.8 1.6(size) 5.0 1.6 0.8 0.6 0.6 2.1 1.2

h 4.0052 2.1802 2.5174 5.1837 3.6868 4.1702

21

0



Table II. SSATS values for CUSUM control charts for the mean when �=0.4

C� C�s C� C�s C�d

�∗ =0.5 �∗ =0.5 �∗ =2.0 �∗ =2.0 �∗k =0.2k �∗

k =0.5kParameter values [1] [2] [3] [4] [5] [6]

in-control 370.4 370.4 370.4 370.4 370.4 370.4

Sustained 1.00 345.2 357.5 357.3 357.9 336.6 351.8Mean 2.00 316.9 289.4 294.8 291.2 314.3 317.1Shift 3.00 262.5 143.7 160.9 146.1 276.9 251.2(size) 4.00 183.7 32.4 43.9 33.7 227.7 165.7

5.00 96.1 3.1 5.3 3.3 169.5 85.7

Sustained 0.1 138.1 293.2 267.9 297.2 118.1 176.6Mean 0.2 49.2 175.3 138.1 182.0 44.8 64.1Drift 0.3 25.9 98.2 68.8 103.4 25.7 30.1(rate) 0.6 10.1 21.7 13.8 22.9 11.1 9.4

1.0 5.7 6.4 4.8 6.6 6.5 5.0

Transient 1.0 348.0 348.1 348.3 348.7 340.5 352.3Mean 2.0 324.0 234.3 244.6 236.7 327.9 312.0Shift 3.0 264.9 62.4 79.1 64.3 299.6 226.8l = 4 4.0 164.8 4.4 7.7 4.6 250.7 118.6(size) 5.0 60.5 0.6 0.7 0.6 183.8 39.3

Transient 1.0 339.2 347.3 346.6 348.1 333.9 343.0Mean 2.0 291.5 230.8 238.3 233.2 302.9 280.8Shift 3.0 206.7 59.7 73.9 61.8 247.5 178.8l = 16 4.0 105.9 4.6 7.4 4.9 175.7 80.5(size) 5.0 32.3 0.7 0.8 0.7 103.1 24.7

h 4.0052 1.5293 2.5174 4.4162 3.2686 4.3036

changes, so we include both the C�s and C� charts in the next set of comparisons where the process changes may involve any or allof the parameters in the process and adjustment model.

15. The setup for general comparisons of control charts

The next set of comparisons considers the S�, C�, C�s, C�, C�, and Ca charts used individually, the two-chart combinations of the C�and C� charts and the C�s and C� charts, and the four-chart combinations of the C�, C�, C�, and Ca charts and the C�s, C�, C�, and Ca

charts. Table III gives values of h for the combinations of charts for some values of the chart parameters �∗, �∗, �∗, �∗

, and a∗. Valuesof h are given for a range of values of � for use as an aid in designing these combinations of charts.

For the comparison of different charts or combinations of charts, Tables IV and V give SSATS values and SSEQL values, respectively,

for the case of �=0.8, and Tables VI and VII give values for the case of �=0.4. The values of �∗, �∗, �∗, �∗

, and a∗ used in TablesIV–IX are the values in the middle column of Table III for each chart combination. For the individual charts in Tables IV–IX, the valuesof �∗, �∗, �∗, �

∗, and a∗ are the same as the values used in the combinations.

Tables IV–VII contain results for sustained mean shifts, mean drifts, sustained shifts in variance (increases only), sustained shifts in �,and sustained shifts in effect of the adjustment. It was found that sustained shifts in � or in the effect of the adjustment are relativelydifficult to detect, hence quite large shifts were considered to see how large the shift would need to be to induce relatively quickdetection. In some cases this resulted in ��>1. For some of the parameter shifts the SSATS of a chart was larger than the in-controlATS, and in these cases a ‘∗’ was used in the table in place of the SSATS.

In practice, a special cause may produce a change in more than one parameter, hence it is important to consider the performanceof the charts in this situation. Tables VIII and IX contain results for combinations of charts when two or more parameters change atthe same time.

16. Some conclusions about detecting process changes

We have already seen in Tables I and II that detecting small mean shifts is difficult, and the same conclusion is obtained fromTables IV and VI when we consider combinations of charts. Although the SSATS is large for small mean shifts, the SSEQL is small inTables V and VII because the effect of mean shifts dissipates over time.


21

1


Tab

leII

I.V

alu

eso

fh

for

som

eco

mb

inat

ion

so

fch

arts

tog

ive

anin

-co

ntr

ol

ATS

of

370.

4fo

r�=

0.4,

0.5,

0.6,

0.7,

or

0.8

C�

and

C�

and

C�s

and

C�

and

C�

and

C�

C�s

and

C�

C�

and

Ca

C�

and

Ca

�∗o

r�∗ =

0.5

0.5

1.5

0.5

0.5

1.5

0.5

0.5

1.5

0.5

0.5

1.5

�∗=

1.25

1.5

2.5

1.25

1.5

2.5

1.25

1.5

2.5

1.25

1.5

2.5

�∗ =—

——

——

—0.

750.

50.

250.

750.

50.

251.

251.

51.

751.

251.

51.

75

a∗=

——

——

——

0.75

0.5

0.25

0.75

0.5

0.25

1.25

1.5

1.75

1.25

1.5

1.75

C�

or

C�s

4.56

944.

5877

5.52

622.

3185

2.31

815.

0782

4.94

835.

0246

6.05

162.

4351

2.44

885.

4701

�=

0.8

C�

3.03

203.

6684

3.95

372.

9751

3.58

943.

8306

3.38

264.

0981

4.36

283.

3941

4.08

744.

3731

C�

——

——

——

7.62

309.

1285

9.68

067.

6427

9.10

719.

7013

8.06

969.

6771

10.1

470

8.09

019.

6549

10.1

676

Ca

——

——

——

4.42

046.

5740

7.71

194.

4365

6.55

357.

7319

3.97

015.

8788

6.93

383.

9859

5.85

846.

9532

C�

or

C�s

4.56

944.

5877

5.52

622.

0095

2.00

664.

6439

4.92

285.

0125

6.03

882.

1028

2.11

684.

9937

�=

0.7

C�

3.03

203.

6684

3.95

372.

9580

3.55

973.

7890

3.36

014.

0864

4.35

023.

3660

4.07

084.

3627

C�

——

——

——

7.01

238.

7810

9.47

937.

0232

8.75

019.

5038

7.34

699.

3405

10.0

560

7.35

819.

3087

10.0

811

Ca

——

——

——

5.03

357.

2461

8.31

795.

0431

7.21

678.

3416

4.64

676.

6816

7.74

614.

6558

6.65

217.

7706

C�

or

C�s

4.56

944.

5877

5.52

621.

8200

1.81

514.

3179

4.89

704.

9789

6.01

131.

8983

1.90

954.

6377

�=

0.6

C�

3.03

203.

6684

3.95

372.

9460

3.53

523.

7415

3.33

614.

0531

4.45

693.

3342

4.03

154.

3284

C�

——

——

——

6.35

458.

2860

9.13

746.

3508

8.24

249.

1480

6.57

568.

7234

9.67

816.

5719

8.67

969.

6891

Ca

——

——

——

5.49

487.

6822

8.69

265.

4913

7.64

058.

7028

5.16

037.

2290

8.28

805.

1569

7.18

698.

2986

C�

or

C�s

4.56

944.

5877

5.52

621.

6937

1.68

724.

0712

4.88

104.

9552

5.97

421.

7631

1.77

164.

3679

�=

0.5

C�

3.03

203.

6684

3.95

372.

9383

3.51

573.

6888

3.32

144.

0299

4.28

683.

3140

3.99

954.

2788

C�

——

——

——

5.66

087.

6921

8.64

645.

6479

7.63

288.

6309

5.79

117.

9936

9.06

095.

7771

7.93

289.

0451

Ca

——

——

——

5.88

088.

0280

8.93

855.

8673

7.96

798.

9229

5.60

127.

6733

8.67

675.

5878

7.61

348.

6605

C�

or

C�s

4.56

944.

5877

5.52

621.

6063

1.59

943.

8816

4.89

874.

9660

5.96

911.

6744

1.68

144.

1768

�=

0.4

C�

3.03

203.

6684

3.95

372.

9346

3.50

643.

6378

3.33

774.

0405

4.28

163.

3286

4.00

844.

2574

C�

——

——

——

4.94

507.

0183

8.05

044.

9289

6.95

748.

0036

5.01

097.

2042

8.32

504.

9946

7.14

198.

2770

Ca

——

——

——

6.27

308.

3731

9.17

546.

2553

8.30

979.

1281

6.04

558.

1049

9.02

996.

0279

8.04

168.

9811

21

2



Table IV. SSATS values for control charts and control chart combinations when �=0.8

Chart or chart combination

C� C�sC� C�

C� C�s C� C�S� C� C�s C� C� Ca C� C� Ca Ca

Parameter values [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

In control 370.4 370.4 370.4 370.3 370.4 370.4 370.4 370.4 370.4 370.4

Sustained 1.00 352.3 293.5 306.4 337.3 321.3 295.7 306.5 316.0 303.8 304.7Mean 2.00 262.5 123.1 104.4 195.4 167.3 160.3 139.8 122.8 148.2 127.1Shift 3.00 101.0 16.0 7.3 33.5 29.6 30.3 17.7 9.6 20.2 11.4(size) 4.00 12.0 2.4 1.3 1.6 3.0 3.5 1.3 1.0 1.4 1.1

5.00 0.8 1.6 0.8 0.6 1.4 2.0 0.6 0.6 0.6 0.6

Sustained 0.1 158.4 29.6 47.0 102.7 55.9 37.6 33.0 51.6 35.6 42.7Mean 0.2 49.4 13.6 15.0 24.0 16.5 15.0 14.7 16.5 15.3 15.7Drift 0.3 21.0 9.4 9.3 12.2 9.9 10.0 10.0 9.9 10.2 9.9(rate) 0.6 6.5 5.5 4.9 5.3 5.1 5.7 5.4 5.1 5.3 5.1

1.0 3.4 3.8 3.2 3.2 3.4 4.0 3.4 3.3 3.4 3.3

Transient 2.0 251.7 241.5 150.2 168.5 202.4 269.1 182.3 156.5 183.1 162.8Mean shift 3.0 86.7 75.4 14.4 19.4 49.0 128.3 25.2 16.0 27.5 19.5l =4 4.0 8.5 6.3 1.3 1.0 4.6 19.7 1.3 1.1 1.5 1.2(size) 5.0 0.6 1.6 0.8 0.6 1.4 2.3 0.6 0.6 0.6 0.6


1.25 60.4 96.7 68.7 36.4 83.9 103.4 41.1 39.8 44.6 44.0Sustained 1.50 21.5 45.4 27.2 13.7 35.4 48.7 15.4 14.9 16.7 16.4Shift in 1.75 11.1 27.4 15.0 8.0 19.9 28.4 8.9 8.7 9.6 9.5Variance 2.00 7.0 18.8 9.9 5.6 13.1 18.9 6.2 6.0 6.7 6.5(size) 3.00 2.7 7.6 3.8 2.5 5.3 7.3 2.8 2.7 3.0 2.9

5.00 1.3 3.2 1.8 1.4 2.7 3.3 1.5 1.4 1.5 1.5

0 28.1 12.4 11.0 16.2 10.2 12.9 13.3 12.1 11.6 11.30.25 48.1 16.3 15.1 25.2 13.9 16.9 18.2 16.9 15.8 15.4

Sustained 0.50 99.2 24.1 24.1 48.5 22.3 25.1 28.2 28.0 25.0 24.5Shift in 0.75 234.0 48.1 55.1 141.8 54.7 52.4 59.1 68.2 56.0 56.4� 1.25 227.6 ∗ ∗ 171.2 229.8 71.5 270.2 247.3 92.9 92.1(size) 1.50 80.9 205.4 119.4 50.1 106.2 118.1 62.5 57.9 59.4 57.9

1.75 31.4 39.0 29.4 19.7 38.1 48.1 21.6 20.7 23.1 22.52.00 16.1 21.3 15.5 11.4 20.1 25.3 12.5 11.9 13.2 12.93.00 4.6 8.1 5.5 4.2 6.9 8.9 4.5 4.4 4.8 4.7

0 65.3 25.2 29.1 46.0 29.6 27.2 28.0 32.4 28.6 29.50.25 174.8 38.9 49.1 104.1 51.2 43.6 45.1 57.8 46.5 48.8

Sustained 0.50 294.2 68.3 92.3 222.1 101.7 80.7 83.1 114.8 86.7 92.5Shift in 0.75 354.8 140.3 183.1 330.9 209.9 178.2 173.2 223.3 186.1 199.4Adjustment 1.50 337.0 ∗ ∗ 303.0 ∗ 131.3 ∗ ∗ 174.2 171.3(size) 2.00 278.3 ∗ ∗ 215.3 253.0 68.2 352.8 326.9 93.3 92.9

3.00 170.7 ∗ ∗ 103.6 125.1 39.2 148.9 142.9 53.1 52.85.00 57.6 ∗ ∗ 29.9 40.8 22.5 36.6 35.7 26.5 26.4

Tables I, II, IV, and VI show that slow mean drifts are hard to detect, but fast drifts are easier to detect. The effect of a drift doesnot dissipate over time, so the SSEQL values in Tables V and VII are relatively high for all drifts. Thus, the type of mean change affectsboth the ability to detect the change and the consequence of not detecting the change.


21

3


Table V. SSEQL values for control charts and control chart combinations when �=0.8


C� C�sC� C�



In control 0 0 0 0 0 0 0 0 0 0


5.00 25.4 40.9 30.0 25.9 39.0 45.1 26.6 26.5 27.1 27.1


1.0 20.4 23.4 17.4 17.5 18.7 25.3 19.1 17.7 19.2 18.4




5.00 43.8 88.8 54.5 44.8 76.4 91.9 46.9 46.7 48.5 48.2

0 30.8 13.5 10.9 16.0 9.8 14.4 13.5 12.2 11.4 11.20.25 34.6 10.6 9.2 15.9 8.0 11.2 11.5 10.6 9.6 9.3

Sustained 0.50 36.7 7.6 7.3 16.0 6.5 8.0 8.9 8.8 7.6 7.4Shift in 0.75 24.1 4.3 4.9 13.9 4.8 4.7 5.4 6.2 5.0 5.0� 1.25 24.9 ∗ ∗ 18.7 25.4 7.9 29.7 27.2 10.3 10.2(size) 1.50 35.3 90.7 52.5 21.7 46.6 51.9 26.7 25.1 25.8 25.2

1.75 29.9 37.3 27.8 18.3 36.6 46.5 20.2 19.3 21.6 21.12.00 25.6 34.3 24.4 17.4 32.3 41.6 19.3 18.3 20.5 19.93.00 22.7 43.4 27.6 20.3 36.5 49.2 22.4 21.6 23.8 23.3

0 35.7 5.1 6.5 15.6 6.6 5.8 6.3 8.0 6.4 6.70.25 31.5 3.8 5.3 15.2 5.6 4.5 4.8 6.7 5.0 5.3

Sustained 0.50 14.3 2.5 3.6 10.3 4.1 3.1 3.2 4.7 3.4 3.7Shift in 0.75 3.1 1.1 1.5 2.9 1.7 1.4 1.4 1.8 1.5 1.6Adjustment 1.50 6.4 ∗ ∗ 5.8 ∗ 2.6 ∗ ∗ 3.4 3.3(size) 2.00 17.1 ∗ ∗ 13.3 15.8 4.3 21.9 20.2 5.8 5.8

3.00 31.7 ∗ ∗ 19.3 23.7 7.6 28.0 26.7 10.1 10.15.00 34.5 ∗ ∗ 18.1 25.7 14.5 22.4 21.8 16.5 16.5

Small increases in �� take a relatively long time to detect, and all increases in �� result in quite large SSEQL values. Thus, inmonitoring a process of this type, it is important to have effective detection of changes in variability. Changes in variability have thepotential to produce some of the largest losses, even when these variance changes are relatively small.

Small changes in � are hard to detect. The SSEQL may not be very high for small changes in �, particularly for decreases in �.

21

4



Table VI. SSATS values for control charts and control chart combinations when �=0.4


C� C�sC� C�



In control 370.4 370.4 370.4 370.3 370.4 370.4 370.4 370.4 370.4 370.4


5.00 7.5 96.1 3.1 7.6 135.6 145.5 11.7 4.0 15.1 5.0


1.0 11.9 5.7 6.4 6.2 6.2 5.6 5.8 6.2 5.8 5.6



1.25 60.5 96.7 58.4 36.4 95.2 89.4 41.1 39.1 44.0 43.1Sustained 1.50 21.5 45.4 20.9 13.8 42.6 38.4 15.4 14.5 16.5 15.9Shift in 1.75 11.1 27.4 10.9 8.0 24.2 21.5 8.9 8.3 9.6 9.1Variance 2.00 7.0 18.8 7.0 5.6 159 14.1 6.2 5.7 6.6 6.2(size) 3.00 2.6 7.6 2.8 2.5 6.0 5.5 2.8 2.6 2.9 2.7

5.00 1.3 3.2 1.4 1.4 2.9 2.7 1.5 1.4 1.5 1.4

0 175.3 48.0 74.6 93.1 32.7 36.0 56.3 81.8 38.4 38.50.25 234.6 65.3 109.1 143.5 46.4 52.5 79.5 122.9 54.3 54.9

Sustained 0.50 297.9 97.0 165.3 222.9 76.6 89.7 121.4 188.9 88.1 90.9Shift in 0.75 349.8 164.9 252.8 317.9 163.3 189.9 203.9 281.1 177.0 188.1� 1.25 349.8 ∗ ∗ 320.5 176.5 200.2 ∗ ∗ 222.1 209.7(size) 1.50 297.1 ∗ ∗ 233.9 79.0 90.7 384.1 293.9 99.9 97.1

1.75 231.9 ∗ ∗ 157.0 47.8 53.4 242.2 200.3 59.9 58.82.00 170.7 ∗ 323.3 103.4 35.2 38.5 148.9 129.4 43.6 43.03.00 43.6 ∗ 64.5 25.0 22.9 24.6 30.4 28.3 23.6 23.1

0 21.0 11.2 14.6 15.0 11.3 10.6 12.1 14.8 11.5 11.70.25 67.4 19.3 32.2 34.7 18.8 17.9 22.1 33.1 19.8 20.3

Sustained 0.50 186.0 40.0 81.3 99.5 36.5 36.6 48.1 88.8 39.6 41.5Shift in 0.75 318.9 103.2 193.8 256.5 98.3 109.0 131.4 220.0 106.8 114.7Adjustment 1.50 248.5 ∗ ∗ 166.4 53.4 52.6 256.1 213.4 64.2 63.2(size) 2.00 103.8 ∗ 213.5 51.0 23.7 21.0 65.5 113.6 26.1 25.9

3.00 13.3 ∗ 17.6 9.0 8.5 7.4 9.9 9.6 8.4 8.35.00 2.1 3.5 2.2 2.2 2.7 2.6 2.3 2.2 2.3 2.3


21

5


Table VII. SSEQL values for control charts and control chart combinations when �=0.4


C� C�sC� C�



In control 0 0 0 0 0 0 0 0 0 0


5.00 25.1 28.9 25.2 25.3 29.1 28.9 25.4 25.2 25.6 25.3

Sustained 0.1 9.0 3.8 8.1 8.4 7.3 7.0 4.3 8.1 4.7 7.3Mean 0.2 26.0 5.4 19.4 20.5 13.5 12.2 6.2 19.5 6.7 13.2Drift 0.3 38.7 6.3 24.4 24.6 13.6 121 7.2 23.6 7.7 13.2(rate) 0.6 43.9 9.5 21.1 18.4 13.5 11.7 10.5 18.5 11.0 12.7

1.0 31.3 14.1 16.2 15.7 15.4 13.7 14.4 15.6 14.4 14.0




5.00 43.7 88.9 45.3 44.8 81.3 77.4 46.9 44.8 48.3 46.2

0 31.6 8.7 13.4 16.6 5.8 9.4 10.1 14.7 6.8 6.80.25 24.0 6.7 11.2 14.6 4.7 5.3 8.1 12.6 5.5 5.6

Sustained 0.50 13.7 4.5 7.6 10.2 3.5 4.1 5.6 8.7 4.0 4.1Shift in 0.75 4.0 1.9 2.9 3.7 1.9 2.2 2.4 3.3 2.1 2.2� 1.25 4.1 ∗ ∗ 3.8 2.1 2.4 ∗ 4.4 2.6 2.5(size) 1.50 13.8 ∗ ∗ 11.0 3.8 4.3 ∗ 13.7 4.8 4.6

1.75 24.2 ∗ ∗ 16.5 5.1 5.7 25.6 21.1 6.4 6.32.00 31.6 ∗ 60.5 19.3 6.8 7.4 28.0 24.2 8.3 8.23.00 32.4 ∗ 48.1 18.7 17.7 19.0 22.8 21.1 17.9 17.5

0 28.7 11.2 15.4 15.4 10.7 9.3 12.1 15.5 10.9 10.80.25 36.2 8.4 14.8 15.8 7.9 7.1 9.7 15.2 8.2 8.4

Sustained 0.50 30.3 6.0 12.6 15.6 5.4 5.3 7.4 13.9 5.9 6.1Shift in 0.75 9.9 3.2 6.0 8.0 3.0 3.3 4.0 6.8 3.3 3.5Adjustment 1.50 21.9 ∗ ∗ 14.8 4.8 4.8 22.9 19.0 5.8 5.7(size) 2.00 36.4 ∗ 77.2 17.9 8.6 7.6 23.3 21.5 9.4 9.3

3.00 25.4 ∗ 37.4 16.6 19.0 15.5 19.2 18.3 16.7 16.45.00 23.9 183.4 29.2 29.1 69.1 55.0 33.4 29.8 35.8 32.6

21

6



Table VIII. SSATS and SSEQL values for the C� and C� and Call combinations when �=0.8 and multipleparameters shift simultaneously

SSATS SSEQL

C� C�s C� C�sC� C� C� C�

C� C�s C� C� C� C�s C� C�Parameter values C� C� Ca Ca C� C� Ca Ca

� � � a [1] [2] [3] [4] [5] [6] [7] [8]

0.00 1.00 1.00 1.00 370.4 370.4 370.4 370.4 0 0 0 0

1.00 1.50 1.00 1.00 13.0 12.3 14.2 13.8 19.4 18.5 20.9 20.31.00 2.00 1.00 1.00 5.5 5.3 5.9 5.8 20.3 19.6 21.6 21.13.00 1.50 1.00 1.00 3.1 2.7 3.4 3.0 20.0 19.0 20.9 20.13.00 2.00 1.00 1.00 2.4 2.2 2.6 2.4 23.6 22.8 24.6 24.0

0.00 1.50 1.00 0.00 11.6 11.2 12.3 12.0 17.9 17.1 19.0 18.40.00 1.50 1.00 2.00 14.5 14.1 15.4 15.3 20.3 19.8 21.7 21.50.00 1.50 0.50 1.00 10.3 9.6 10.7 10.2 18.1 16.7 18.7 17.90.00 1.50 0.50 0.00 7.4 6.9 7.4 7.2 18.3 16.9 18.6 17.80.00 1.50 0.50 2.00 13.6 12.8 14.5 14.0 18.9 17.9 20.2 19.5

0.00 2.00 1.00 0.00 5.8 5.5 6.1 6.0 20.1 19.2 21.4 20.80.00 2.00 1.00 2.00 6.0 5.9 6.5 6.4 20.5 20.0 21.9 21.70.00 2.00 0.50 1.00 5.4 5.1 5.7 5.5 20.5 19.4 21.6 20.90.00 2.00 0.50 0.00 4.7 4.5 4.9 4.7 21.3 20.1 22.3 21.60.00 2.00 0.50 2.00 6.0 5.7 6.4 6.2 20.1 19.3 21.4 20.8

1.00 1.00 1.00 0.00 16.6 18.7 17.0 17.2 9.7 10.1 9.7 9.51.00 1.00 1.00 2.00 339.8 313.0 95.0 94.3 22.6 20.9 7.5 7.41.00 1.00 0.50 1.00 25.0 24.3 22.1 21.5 10.1 9.6 8.8 8.61.00 1.00 0.50 0.00 10.4 9.9 9.8 9.6 13.1 12.0 11.8 11.51.00 1.00 0.50 2.00 86.9 91.1 77.7 76.0 6.6 6.8 6.0 5.9

1.00 1.50 1.00 0.00 8.9 8.4 9.4 9.1 17.9 16.8 18.8 18.11.00 1.50 1.00 2.00 13.1 12.7 14.2 14.0 20.1 19.5 21.6 21.31.00 1.50 0.50 1.00 9.4 8.7 9.7 9.3 18.2 16.8 18.9 18.01.00 1.50 0.50 0.00 6.6 6.2 6.7 6.4 18.6 17.1 19.0 18.21.00 1.50 0.50 2.00 12.6 11.7 13.4 12.9 18.9 17.8 20.1 19.4

1.00 2.00 1.00 0.00 5.0 4.8 5.3 5.2 20.6 19.6 21.7 21.11.00 2.00 1.00 2.00 5.5 5.4 6.0 5.9 20.6 20.0 22.0 21.71.00 2.00 0.50 1.00 5.1 4.8 5.3 5.2 20.7 19.6 21.8 21.11.00 2.00 0.50 0.00 4.4 4.2 4.6 4.4 21.6 20.4 22.6 21.91.00 2.00 0.50 2.00 5.6 5.3 6.0 5.8 20.3 19.4 21.5 20.9

3.00 1.00 1.00 0.00 1.8 1.6 1.9 1.8 18.6 17.7 19.6 18.93.00 1.00 1.00 2.00 119.7 61.2 44.3 28.0 20.1 16.3 15.7 14.63.00 1.00 0.50 1.00 7.9 6.1 7.3 6.1 16.1 15.1 16.3 15.73.00 1.00 0.50 0.00 3.1 2.7 3.1 2.8 18.2 17.2 19.0 18.33.00 1.00 0.50 2.00 36.5 29.6 34.6 27.9 13.8 13.3 14.0 13.5

3.00 1.50 1.00 0.00 2.0 1.9 2.1 2.0 21.3 20.4 22.3 21.73.00 1.50 1.00 2.00 4.9 4.1 5.6 4.9 19.9 18.8 21.1 20.23.00 1.50 0.50 1.00 4.2 3.7 4.5 4.1 20.1 19.0 21.0 20.23.00 1.50 0.50 0.00 3.0 2.7 3.1 2.9 21.2 20.2 22.2 21.53.00 1.50 0.50 2.00 6.0 5.3 6.6 6.1 19.8 18.8 20.9 20.1

3.00 2.00 1.00 0.00 2.0 1.9 2.1 2.0 24.9 24.1 25.9 25.43.00 2.00 1.00 2.00 2.8 2.6 3.1 2.9 23.0 22.3 24.0 23.53.00 2.00 0.50 1.00 2.9 2.8 3.2 3.0 23.3 22.5 24.4 23.83.00 2.00 0.50 0.00 2.5 2.4 2.7 2.6 24.5 23.7 25.7 25.13.00 2.00 0.50 2.00 3.3 3.1 3.6 3.5 22.6 21.9 23.7 23.1


21

7


Table IX. SSATS and SSEQL values for the C� and C� and Call combinations when �=0.4 and multipleparameters shift simultaneously

SSATS SSEQL

C� C�s C� C�sC� C� C� C�

C� C�s C� C� C� C�s C� C�Parameter values C� C� Ca Ca C� C� Ca Ca

� � � a [1] [2] [3] [4] [5] [6] [7] [8]

0.00 1.00 1.00 1.00 370.4 370.4 370.4 370.4 0 0 0 0

1.00 1.50 1.00 1.00 14.4 13.5 15.5 14.8 19.8 18.6 21.2 20.31.00 2.00 1.00 1.00 5.9 5.4 6.2 5.9 20.2 18.8 21.4 20.23.00 1.50 1.00 1.00 7.5 6.4 8.4 7.4 20.1 18.6 21.2 19.93.00 2.00 1.00 1.00 3.6 3.2 3.9 3.5 22.3 21.0 23.3 22.0

0.00 1.50 1.00 0.00 7.4 7.2 7.4 7.2 18.4 17.7 18.8 17.80.00 1.50 1.00 2.00 10.3 9.9 10.2 10.0 20.0 18.9 20.0 19.50.00 1.50 0.50 1.00 13.9 13.2 14.8 14.2 19.1 18.2 20.3 19.50.00 1.50 0.50 0.00 6.2 6.0 6.2 6.0 19.1 18.1 19.5 18.30.00 1.50 0.50 2.00 12.7 12.0 12.9 12.6 20.1 19.0 20.5 19.9

0.00 2.00 1.00 0.00 4.7 4.4 4.9 4.6 21.3 19.6 22.4 20.70.00 2.00 1.00 2.00 5.2 4.9 5.4 5.2 21.4 20.1 22.5 21.60.00 2.00 0.50 1.00 6.0 5.6 6.4 6.0 20.1 18.7 21.3 20.00.00 2.00 0.50 0.00 4.2 4.0 4.3 4.1 22.2 20.4 23.3 21.50.00 2.00 0.50 2.00 5.7 5.4 6.0 5.8 20.8 19.5 21.9 21.0

1.00 1.00 1.00 0.00 10.4 12.6 9.9 9.9 13.1 15.7 12.2 11.91.00 1.00 1.00 2.00 63.3 58.7 25.5 25.3 23.6 21.7 10.2 10.01.00 1.00 0.50 1.00 117.2 181.4 84.8 87.2 6.6 9.5 5.0 5.11.00 1.00 0.50 0.00 8.6 9.7 8.1 8.0 14.7 16.2 13.6 13.21.00 1.00 0.50 2.00 145.2 128.2 43.7 43.2 24.5 21.6 8.1 8.0

1.00 1.50 1.00 0.00 6.6 6.5 6.7 6.5 18.6 17.7 19.1 18.01.00 1.50 1.00 2.00 9.8 9.3 9.8 9.6 20.0 18.9 20.1 19.61.00 1.50 0.50 1.00 13.1 12.3 13.9 13.2 19.1 18.1 20.3 19.41.00 1.50 0.50 0.00 5.7 5.5 5.7 5.5 19.3 18.2 19.8 18.51.00 1.50 0.50 2.00 12.0 11.4 12.3 12.0 20.0 18.9 20.5 19.9

1.00 2.00 1.00 0.00 4.4 4.1 4.5 4.3 21.6 19.8 22.6 21.01.00 2.00 1.00 2.00 4.9 4.7 5.2 5.0 21.4 20.2 22.5 21.61.00 2.00 0.50 1.00 5.7 5.2 6.0 5.7 20.2 18.8 21.4 20.11.00 2.00 0.50 0.00 4.0 3.7 4.1 3.9 22.4 20.6 23.5 21.71.00 2.00 0.50 2.00 5.4 5.1 5.7 5.5 20.8 19.6 21.9 21.0

3.00 1.00 1.00 0.00 3.1 3.2 3.1 2.8 18.2 17.0 19.0 17.23.00 1.00 1.00 2.00 39.0 33.3 18.2 16.8 22.8 20.7 15.5 14.93.00 1.00 0.50 1.00 65.7 80.7 50.0 41.9 13.0 13.5 12.4 11.83.00 1.00 0.50 0.00 3.4 3.4 3.4 3.1 18.6 17.4 19.2 17.53.00 1.00 0.50 2.00 90.8 72.7 30.5 27.7 23.7 20.7 14.0 13.5

3.00 1.50 1.00 0.00 3.0 2.7 3.1 2.8 21.2 19.6 22.1 20.43.00 1.50 1.00 2.00 6.1 5.5 6.4 6.0 20.8 19.6 21.5 20.63.00 1.50 0.50 1.00 7.2 6.3 7.9 7.0 19.9 18.6 21.0 19.63.00 1.50 0.50 0.00 3.0 2.8 3.1 2.9 21.6 20.0 22.5 20.83.00 1.50 0.50 2.00 7.2 6.5 7.7 7.2 20.6 19.4 21.4 20.5

3.00 2.00 1.00 0.00 2.5 2.3 2.7 2.4 24.5 22.7 25.6 23.83.00 2.00 1.00 2.00 3.4 3.1 3.6 3.4 23.2 22.0 24.2 23.23.00 2.00 0.50 1.00 3.6 3.2 3.9 3.5 22.4 21.0 23.3 22.03.00 2.00 0.50 0.00 2.6 2.3 2.7 2.4 24.9 23.1 26.0 24.23.00 2.00 0.50 2.00 3.7 3.3 3.9 3.7 22.7 21.5 23.7 22.7

21

8



All but very large changes in adjustment are hard to detect, and most changes in adjustment do not result in large values of theSSEQL. Our results agree with Box and Luceno19, who note that the amount of adjustment can deviate somewhat from the optimalamount with little effect.

The difficulty of detecting a process change and the associated loss depends on the value of �. For example, a shift in � is muchharder to detect (SSATS larger) when �=0.4 than when �=0.8, but the SSEQL is larger at �=0.8. In addition, a shift in adjustment isharder to detect (SSATS larger) when �=0.8 than when �=0.4, but the SSEQL is larger at �=0.4. Apparently, detection is relativelyfast when there is a large change in the process, and a large change in the process is associated with a large SSEQL.

17. Conclusions about the relative performance of the charts

It is interesting to note in Tables IV–VII that the C� chart has a better performance than the C� chart for large sustained shifts in themean. The same is true for transient shifts, and for drifts when �=0.4. This is consistent with the conclusions from Reynolds andStoumbos21, 22 in the case of independent observations. However, the C� chart does not seem to be better than the C�s chart formean shifts.

Comparing the two two-chart combinations in columns [7] and [8] of Tables IV–VII shows that the C�s and C� chart combinationis better than the C� and C� chart combination in most cases. Similarly, comparing the two four-chart combinations in columns [9]and [10] shows that the C�s, C�, C�, and Ca chart combination is better than the C�, C�, C�, and Ca chart combination in most cases.

Comparing the two-chart combinations in columns [7] and [8] with the four-chart combinations in columns [9] and [10] shows thatthe two-chart combinations have approximately the same performance as the four-chart combinations, except for small changes in �or in adjustment, where the four-chart combinations are better. Thus, adding the C� and Ca charts to the C� and C� charts or to theC�s and C� charts to form a four-chart combination gives additional protection against changes in � or adjustment. However, addingthe C� and Ca charts produces a slight deterioration in the performance for detecting some other types of process changes.

Tables VIII and IX contain results for the arguably more realistic situation in which a special cause produces changes in morethan one parameter. We see that the combinations with the C�s chart perform slightly better than the combinations with the C�chart in almost all cases. The performance of the four-chart combinations is slightly worse than the performance of the two-chartcombinations in many cases, but is significantly better in a few cases. The four-chart combinations seem to be significantly better inthe cases with a=2.0 and no increase in the variance. It is not too surprising that the two-chart combinations frequently perform aswell as the four-chart combinations, as there is some confounding of the effects of the different parameter changes (see also Rungeret al.15 for additional discussion of confounding of effects).

18. Overall conclusions and comments

This paper has investigated the problem of detecting changes in an ARIMA(0,1,1) process under MMSE control. It is assumed that thecontroller is designed to compensate for the inherent variation due to the ARIMA(0,1,1) process, but it is desirable to detect specialcauses that occur occasionally and produce additional process variation. The objective in detecting these special causes is to try toeliminate them, rather than to compensate for them through the controller. It is assumed that special causes can produce manytypes of process changes, including changes in the mean, the variance, �, and the effect of adjustment. In addition to looking at theperformance of individual control charts, the performance of combinations of control charts has been investigated.

The C�s chart, which is designed specifically to detect mean shifts, has better performance for detecting mean shifts than thestandard C� chart. Except for mean drifts, the C�s chart also seems to have better performance for other process changes.

In the case in which the process observations are independent, Reynolds and Stoumbos21,22 found that the C� and C� chartcombination provides excellent performance in detecting changes in the mean or variance. The results presented here show that theperformance of the C�s and C� chart combination is slightly better than the performance of the C� and C� chart combination, and theC�s and C� chart combination provides good overall performance for detecting general process changes. Thus, it would be reasonableto use the C�s and C� chart combination in many applications. If it is important to detect small changes in � or in adjustment, thenthe four-chart combination consisting of the C�s, C�, C�, and Ca charts would be a reasonable choice.

It appears that detecting most small parameter changes in this type of process is relatively difficult in the sense that the expectedtime required to detect these changes is large. Some small process changes do not result in large loss even if they are not detectedquickly, but it may still be important to detect these process changes because, with detection, it may be possible to prevent the futureoccurrence of these changes.

The results presented here show that even a small increase in the process variance produces a large increase in loss, hence detectionof variance increases seems to be particularly important when monitoring a process of this type. So far, detection of variance increasesdoes seem to have received much attention in the literature on monitoring processes under APC adjustment, so there is an opportunityfor additional research on this topic.

References1. Box GEP, Kramer T. Statistical process monitoring and feedback adjustment: A discussion. Technometrics 1992; 34:251--267.2. Vander Wiel S, Tucker WT, Faltin FW, Doganaksoy N. Algorithmic statistical process control: concepts and an application. Technometrics 1992;

34:286--297.


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9


3. Montgomery DC, Keats BJ, Runger GC, Messina WS. Integrating statistical process control and engineering process control. Journal of QualityTechnology 1994; 26:79--87.

4. Wardell DG, Moskowitz H, Plant RD. Run length distributions of special-cause control charts for correlated processes. Technometrics 1994; 36:3--17.5. Vander WS. Monitoring processes that wonder using integrated moving average models. Technometrics 1996; 38:139--151.6. Apley DW, Shi J. The GLRT for statistical process control of autocorrelated processes. IIE Transactions 1999; 31:1123--1134.7. Tsung F, Shi J, Wu CFJ. Joint monitoring of PID-controlled processes. Journal of Quality Technology 1999; 31:275--285.8. Jiang W, Tsui KL. An economic model for integrated APC and SPC control charts. IIE Transactions 2000; 32:505--513.9. Jiang W, Tsui KL. SPC monitoring of MMSE- and PI-controlled processes. Journal of Quality Technology 2002; 34:384--398.

10. Capilla C, Ferrer A, Romero R, Hualda A. Integration of statistical and engineering process control in a continuous polymerization process.Technometrics 1999; 41:14--28.

11. Shu L, Apley DW, Tsung F. Autocorrelated process monitoring using triggered cuscore charts. Quality and Reliability Engineering International2002; 18:411--421.

12. Tsung F, Tsui KL. A mean-shift pattern study on integration of SPC and APC for process monitoring. IIE Transactions 2003; 35:231--242.13. Jiang W. A joint monitoring scheme for automatically controlled processes. IIE Transactions 2004; 36:1201--1210.14. Han D, Tsung F. A reference-free cuscore chart for dynamic mean change detection and a unified framework for charting performance comparison.

Journal of the American Statistical Association 2006; 101:368--386.15. Runger GC, Testik MC, Tsung F. Relationships among control charts used with feedback control. Quality and Reliability Engineering International

2006; 22:877--887.16. Apley DW, Chin C. An optimal filter design approach to statistical process control. Journal of Quality Technology 2007; 39:93--117.17. Nembhard HB, Valverde-Ventura R. Cuscore statistics to monitor a non-stationary system. Quality and Reliability Engineering International 2007;

23:303--325.18. Box GEP, Jenkins GM, Reinsel GC. Time Series Analysis: Forecasting and Control (3rd edn). Prentice-Hall: Upper Saddle River, NJ, 1994.19. Box GEP, Luceno A. Statistical Control by Monitoring and Feedback Adjustment. Wiley: New York, 1997.20. Hu SJ, Roan C. Change patterns in time series-based control charts. Journal of Quality Technology 1996; 28:302--312.21. Reynolds MR Jr, Stoumbos ZG. Control charts and the efficient allocation of sampling resources. Technometrics 2004; 46:200--214.22. Reynolds MR Jr, Stoumbos ZG. Should observations be grouped for effective process monitoring? Journal of Quality Technology 2004; 36:343--366.23. Jiang W, Wu H, Tsung F, Nair VN, Tsui KL. Proportional integral derivative charts for process monitoring. Technometrics 2002; 44:205--214.24. Box GEP, Ramirez J. Cumulative score charts. Quality and Reliability Engineering International 1992; 8:17--27.25. Luceno A. Average run lengths and run length probability distributions for cuscore charts to control normal mean. Computational Statistics and

Data Analysis 1999; 32:177--195.26. Luceno A. CUSCORE charts to detect level shifts in autocorrelated noise. Quality Technology and Quantitative Management 2004; 1:27--45.27. Runger GC, Testik MC. Control charts for monitoring fault signatures: cuscore versus GLR. Quality and Reliability Engineering International 2003;

19:387--396.

Appendix

For the inductive proof of (9) assume that (8) and (9) hold for e�+j for j=1, 2,. . . , k and that �0 =1. Then

(1−�)k∑

j=0e�+j = (1−�)e�+(1−�)((1− �1)��+ �1��+1 + �1��)

+(1−�)k∑

j=2

[j−1∑i=0

(1− �i+1)�j−i�i��+i + �j��+j +(

�j −(1−�)j−1∑i=1

�j−1−i�i

)��

]

= (1−�)e�+k−1∑i=0

(1− �i+1)�(1−�k−i)�i��+i +(1−�)k∑

j=1�j��+j +(1−�)

(k∑

j=1�j −

k−1∑i=1

(1−�k−i)�i

)��

=k−1∑i=0

(1− �i+1�)�i��+i −k−1∑i=0

(1− �i+1)�k+1−i�i��+i +(1−�)�k��+k +(1−�)

(k∑

i=1�k−i�i

)��

Using (5), (6), and (7) gives

e�+k+1 = Z�+k+1 − Z�−(1−�)k∑

j=0e�+j + �k+1��

=k∑

i=0(1− �i+1�)(�i��+i)+ �k+1��+k+1 −

k−1∑i=0

(1− �i+1�)�i��+i

+k−1∑i=0

(1− �i+1)�k+1−i�i��+i −(1−�)�k��+k −(1−�)

(k∑

i=1�k−i�i

)��+ �k+1��

=k∑

i=0(1− �i+1)�k+1−i�i��+i + �k+1��+k+1 +

(�k+1 −(1−�)

k∑i=1

�k−i�i

)��

and hence (9) holds for e�+k+1.

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0



For the inductive proof of (13), suppose that (13) holds for e�+k . From (7) we get

e�+k = (1−�)k−1∑i=0

��+i −(1−�)k∑

i=1aie�+i−1 +��+k

Then

e�+k+1 = (1−�)k−1∑i=0

��+i −(1−�)k∑

i=1aie�+i−1 +��+k +(1−�)��+k −(1−�)ak+1e�+k −��+k +��+k+1

= (1−(1−�)ak+1)e�+k −��+k +��+k+1

Then substituting (13) for e�+k above and simplifying gives (13) for e�+k+1.

Authors’ biographies

Marion R. Reynolds Jr is a professor in the Departments of Statistics and Forestry at Virginia Polytechnic Institute and State University.His research interests include statistical process control and applications of statistics to natural resource problems. He is a member ofASA, INFORMS, and IIE, and is on the editorial board of Journal of Quality Technology and IIE Transactions.

Changsoon Park is a full professor at Chung-Ang University in Korea. He received his PhD in Statistics from Virginia Tech in 1984.He began his professional career in the department of Statistics at Chung-Ang University. He has visited Virginia Tech during 1990–1991, 1998 as a visiting associate professor and professor, respectively, and Purdue university during 2006 as a visiting professor. Hisprimary research interests are Statistical process control, Robust design of experiments, and protein function/structure prediction inBioinformatics field. He is a member of the American Statistical Association, the American Society for Quality, and a lifetime memberof the Korean Statistical Society and the Korean Society for Quality Management.


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