on the performance of vsi shewhart control chart for

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On the performance of VSI Shewhart control chart formonitoring the coefficient of variation in the presence of

measurement errorsHuu Du Nguyen, Quoc Thong Nguyen, Kim Phuc Tran, Dang Phuc Ho

To cite this version:Huu Du Nguyen, Quoc Thong Nguyen, Kim Phuc Tran, Dang Phuc Ho. On the performance of VSIShewhart control chart for monitoring the coefficient of variation in the presence of measurement errors.International Journal of Advanced Manufacturing Technology, Springer Verlag, 2019, 10.1007/s00170-019-03352-7. hal-02005536

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On the Performance of VSI Shewhart control chartfor monitoring the Coefficient of Variation in thePresence of Measurement Errors

Huu Du Nguyen · Quoc Thong Nguyen ·Kim Phuc Tran · Dang Phuc Ho

Received: date / Accepted: date

Abstract In this paper, we propose a variable sampling interval Shewhartcontrol chart to monitor the coefficient of variation (CV) squared, denoted byVSI SH-γ2. The new model overcomes the ARL-biased (average run length)property of the control chart monitoring the CV in a previous study by design-ing two one-sided charts rather than one two-sided chart. Moreover, the effectof measurement error on the performance of the VSI SH-γ2 control chart isinvestigated. The incorrect formula for the distribution of the CV in the pres-ence of measurement error in a former study is fixed. Numerical simulationsshow that the precision errors and accuracy errors do have negative influenceson the VSI SH-γ2 chart. An appropriate strategy based on the obtained resultsis suggested to reduce these negative effects.

Keywords VSI control chart · Coefficient of Variation · Measurement Errors

H. D. NguyenDivision of Artificial Intelligence, Dong A University, Danang, VietnamE-mail: [email protected]

Q. T. NguyenUniversite de Bretagne Sud, Laboratoire de Mathematiques de Bretagne Atlantique, UMRCNRS 6205, Campus de Tohannic, Vannes, FranceE-mail: [email protected]

K. P. TranEcole Nationale Superieure des Arts et Industries Textiles, GEMTEX Laboratory, 59056Roubaix, FranceE-mail: [email protected]

P. Ho DangDepartment of Probability and Mathematical Statistics, Institute of Mathematics, Hanoi,VietnamE-mail: [email protected]

2 Huu Du Nguyen et al.

1 Introduction

Coefficient of variation (CV) is an important measure of dispersion of a prob-ability distribution. It is widely used in a large number of domains such as an-alytical chemistry (e.g. expressing the precision and repeatability of an assay),engineering or physics (e.g. doing quality assurance studies) and economics(e.g. determining the volatility of a security). Monitoring this characteristic ofa variable has attracted the attention of many researchers. The first Shewhartcontrol chart monitoring the CV was proposed in [11]. In [10], Hong et al.designed an exponential weighted moving average (EWMA) to monitor theCV using the intensive simulations for calculating the ARL. The EWMA wasalso used to monitor the CV squared in [7], where the ARL was computed bya Markov chain method. A synthetic CV control chart was next introduced in[2] while a Shewhart type chart with supplementary Run Rules to track theCV was developed in [3]. Castagliola [6] proposed a one-sided Shewhart CVchart for short production runs. In [29], the authors studied a side-sensitivegroup runs (SSGR) chart. The CUSUM CV chart designed in [25] outperformsthe others in detecting the process shifts. Recently, the variable parametersCV chart has been suggested in [28]. In addition to developing new types ofcontrol chart, adaptive strategies enhancing the performance of the controlchart were also developed. Some authors have explored the variable samplinginterval (VSI) and variable sample size (VSS) control charts, see [4,5,17,26];while [12] suggested combining VSI type with VSS type to create the VSSICV control chart. An overview of control charts monitoring the CV has beendiscussed in [27].

For the sake of simplicity, most of aforementioned studies omitted the in-fluence of measurement error on the control charts. However, no matter howimproved the measurement method is, the measurement error is inevitable.This fact is a motivation for a large number of authors to study the conse-quence of measurement error on the performance of various control charts [1,9,14,16,18,21,24,22]. Among control charts monitoring the CV, the measure-ment error has been firstly considered by [27]. However, the authors used anassumption that the ratio γ∗1/γ

∗0 (explained in Section 5) are independent from

the presence of measurement error. This assumption seems to be unrealisticsince the measurement errors might change the relation between the in-controland out-of-control value of the CV. Very recently, Tran et al. [23] have relaxedthis assumption by letting γ∗1/γ

∗0 to vary with measurement error. In this pa-

per, we use the same measurement error model as in [23] to study the effect ofmeasurement error on the variable sampling interval (VSI) Shewhart controlchart monitoring the CV squared (denoted by VSI SH-γ2 chart).

In literature, a two-sided VSI SH-γ control chart has been already designedin [4]. However, since the distribution of the CV is skewed, the two-sided VSISH-γ chart is ARL-biased, i.e., some out-of-control average run length valuesare larger than the in-control value ARL0. That was the reason why in [4],the authors have willingly omitted the decreasing shift case. We avoid thissituation by designing two one-sided VSI SH-γ2 control charts instead of one

Title Suppressed Due to Excessive Length 3

two-sided chart. A number of considerable contributions from this study canbe mentioned, involving (1) we overcome the ARL-biased property of the two-sided VSI SH-γ chart, (2) we investigate the effect of measurement error onthe performance of the proposed charts, and (3) we suggest a method to lowerthe negative effect of measurement error based on the obtained results.

The paper is organized as follows. Section 2 presents the design and theimplementation of two one-sided VSI SH-γ2 control charts. The performanceof these two VSI SH-γ2 charts is shown in Section 3 along with a direct com-parison with the performance of the two-sided VSI SH-γ chart. Section 4 isdevoted to the linear covariate error model for the CV. The design and theperformance of VSI SH-γ2 control charts in the presence of measurement er-rors are presented in Section 5. Section 6 illustrates two examples of using VSISH-γ2 control chart. Some concluding remarks are given in Section 7.

2 Implementation of the one-sided VSI SH-γ2 control charts

As mentioned before, the two-sided VSI SH-γ control chart investigated in [4]is inefficient in detecting decreasing CV shifts. We overcome this drawback ofthe two-sided VSI SH-γ control chart by designing two one-sided VSI SH-γ2

control charts. That is to say, we propose separately an upward control chartfor detecting increasing shifts and a downward chart for detecting decreasingshifts.

Suppose that at the ith step of sampling, a subset of independent samplesXi,1, ..., Xi,n is taken, where Xi,j follows a normal distribution with param-eters (µi, σi). In addition, assume that the values of µi and σi are able tochange from sample to sample, but the CV remains stable and be equal to apredefined in-control value, i.e, γi = µi

σi= γ0 ∀i. From this subset of samples,

the sample statistic γ2i is defined as γ2

i =(Si

Xi

)2

, where Xi and Si are the

mean and the standard deviation of the samples Xi,1, ..., Xi,n. The valueof γ2

i is then plotted in the VSI SH-γ2 control charts to monitor the process.The chart parameters include the warning limit and the control limit, i.e.,(UCL+, UWL+) for upward chart and (LCL−, LWL−) for downward chart.

It is customary that the control interval in the VSI chart is divided intothree regions; see Figure 1 as an example for the case of upward chart. If themonitored statistic γi

2 drops into central region, i.e. 0 < γi2 < UWL+ (upward

chart) or LWL− < γi2 (downward chart), the process is definitely believed to

be in-control, then the next sample can be taken after a longer time, say hL. Ifγi

2 drops into the warning region, i.e. γi2 ∈ [UWL+, UCL+] (upward chart) or

γi2 ∈ [LCL−, LWL−] (downward chart), the process is still considered to be

in-control but it seems to be gradually shifted towards out-of-control condition.The next sample should be taken after a shorter time, say hS , to quickly detectassignable causes if they do exist. Finally, the process is considered to be out-of-control if γi

2 falls in out-of-control region, i.e., γi2 > UCL+ (upward chart)

or γi2 < LCL− (downward chart).


0

UWL+

UCL+

Out-of-control region

Central region

Next sampling interval: hL

Warning regionNext sampling interval: hS

Fig. 1: Three regions and principle of the upward VSI SH-γ2 control chart.

Because of the variance of sampling interval, the efficiency of the VSI typechart is usually evaluated by the average time to signal ATS. Once the controlchart parameters are defined, the ATS is numerically calculated for a partic-ular shift from an in-control value γ0 to an out-of-control value γ1 = τγ0,where τ is the shift size. The average sampling interval (ASI) is also consid-ered to make a fair comparison with other control charts. The formulae andcalculation of ATS and ASI are given in Appendix A. With the constraint onthe in-control value of ATS and ASI (denoted by ATS0 and ASI0), the chartparameters (UCL+, UWL+) or (LCL−, LWL−) of the properly designed VSISH-γ2 charts should be a solution of the following equations:

– for downward chart,ATS(n, hS , hL, LCL

−, LWL−, γ0, τ = 1) = ATS0

ASI(n, hS , hL, LCL−, LWL−, γ0, τ = 1) = ASI0

; (1)

– for upward chart,ATS(n, hS , hL, UCL

+, UWL+, γ0, τ = 1) = ATS0

ASI(n, hS , hL, UCL+, UWL+, γ0, τ = 1) = ASI0

. (2)

3 Performance of the one-sided VSI SH-γ2 control charts withoutthe presence of measurement errors

According to the discussion in [20], most of the improvement in detectionpower of VSI type chart compared to fixed sampling interval (FSI) type chartcan be obtained by using only two sampling intervals. Using only two samplingintervals also reduces the complexity of VSI scheme. We therefore assume thatthe sampling interval in VSI Shewhart chart is only pick from a set of two val-ues. For the sake of comparison with the two-sided chart designed in [4], ASI0is set to 1, ATS0 is set to 370.4 and the following combinations of two sam-pling intervals (hS , hL) are applied: (0.5, 1.5), (0.3, 1.7), (0.1, 1.1), (0.1, 1.3),


(0.1, 1.5), (0.1, 1.9) and (0.1, 4.0). The experiment is implemented under theassumption that the shift size τ is deterministic, i. e. τ ∈ 0.5, 0.8, 0.9, 0.95for downward chart and τ ∈ 1.05, 1.1, 1.2, 1.5 for upward chart. Table 3 pro-vides values of ATS1 for both FSI SH-γ2 control charts (i.e. hS = hL = 1 inthe column titled FSI) and VSI SH-γ2 control charts. Some advantages of theone-sided VSI SH-γ2 control charts can be drawn from the numerical result asfollows.

– Firstly, the VSI type chart is much better than the FSI type chart. This isshown by the higher values of ATS1 in column titled FSI compared to theATS1 values in the same rows.

– Secondly, the decreasing process shifts, which were omitted in [4]), aredetected efficiently by using the downward VSI SH-γ2 chart. For example,with n = 5, (hS , hL) = (0.5, 1.5) and γ0 = 0.1, we have ATS1 = 13.80when τ = 0.5.

– Finally, the upward VSI SH-γ2 chart perform better than the two-sidedVSI SH-γ chart in detecting the increasing process shifts. Indeed, the ATS1

values of the upward VSI SH-γ2 chart are smaller than the ones reportedin [4] for whatever the values of n, γ0 and τ . For instance, if γ0 = 0.05, n =5, τ = 1.5 and (hS , hL) = (0.3, 1.7), the value of ATS1 using upward VSISH-γ2 chart is 4.6 while the corresponding value using two-sided VSI SH-γchart is ATS1 = 6.0.

Thus, our one-sided proposed charts not only overcome the ARL-biased prop-erty but also outperform the two-sided SH-γ control chart in detecting processshifts. It should be consider that in literature, there are other advanced con-trol charts monitoring the CV with higher performance compared to VSI typechart. However, one should consider that the parameters in these advancedcharts need to be optimized. Therefore, the Shewhart charts and its strate-gies versions are well-know to be easier to implement in practice. In the nextsections, we will investigate the effect of measurement error on the VSI SH-γ2

control charts.

PLEASE INSERT TABLE 3 HERE

4 Linear covariate error model for the coefficient of variation

In this Section, we briefly provide the linear covariate error model for thesample CV, which has been recently discussed in [23]. Suppose that a set ofindependent samples Xi,1, Xi,2, . . . , Xi,n is selected at times i, i = 1, 2, ...and Xi,j follows a normal distribution with mean µ0 +aσ0 and standard devi-ation bσ0. The constants a and b are introduced to describe the process shift.The values a = 0 and b = 1 are corresponding to the in-control conditionwhile the values a 6= 0 or b 6= 1 are corresponding to the out-of-control con-dition. Due to problem of measurement error, the true value of Xi,j is notobservable. Instead, we can only assessed this true value via m observations


X∗i,j,1, X

∗i,j,2, . . . , X

∗i,j,m, m > 1. As suggested in [13], we use the following

linearly covariate error model:

X∗i,j,k = A+BXi,j + εi,j,k,

where X∗i,j,k is the actually observed value in kth measurement of the item

j at the i sampling; A and B are constant parameters of the model; εi,j,kis a normal (0, σM ) random error independent of Xi,j which represents themeasurement inaccuracy.

In general, mean of multiple measurements represents the true value of theinterest. Let X∗

i,j be the mean of m observed quantities of the same item j at

the ith sampling, then

X∗i,j =

1

m

m∑k=1

X∗i,j,k = A+BXi,j +

1

m

m∑k=1

εi,j,k.

Thus, X∗i,j also follows the normal distribution with parameters (µ∗, σ∗2) in

which µ∗ = A+B(µ0 + aσ0)

σ∗2 = B2b2σ20 +

σ2M

m

.

Hence, the coefficient of variation of the measured quantity X∗i,j is

γ∗ =

√B2b2σ2

0 +σ2M

m

A+B(µ0 + aσ0)=

√B2b2 + η2

m

θ +B(1 + aγ0)× γ0, (3)

where η = σM

σ0, γ0 = σ0

µ0, θ = A

µ0. By these notations, γ0 is the in-control value

of CV, η is the precision error ratio and θ stands for the accuracy error. Itis important to consider that the measurement error also alters the relationγ1 = τγ0 between in-control and out-of-control value of the CV (τ representsthe true shift size). Indeed, without measurement error, the out-of-controlvalue γ1 of the CV is

γ1 =bσ0

µ0 + aσ0=

b

1 + aγ0γ0.

Therefore, it is easy to see that 1+aγ0 = b/τ . Based on this equation and (3),the in-control value of the CV (corresponding to a = 0 and b = 1) is

γ∗0 =

√B2 + η2

m

θ +B× γ0, (4)

while the out-of-control value of the CV is

γ∗1 =

√B2b2 + η2

m

θ +Bb/τ× γ0. (5)


Equations (4)-(5) show that in general γ∗1 6= τγ∗0 . That is to say, the relationγ∗1 = τγ∗0 using in [27] (Eqs. (8) and (9)) is no longer held in general situations(it only holds for the case θ = 0 and b = 1). .

Let ¯X∗i and S∗

i denote the sample mean and the sample standard deviationof X∗

1,j , . . . , X∗n,j , i.e.,

¯X∗i =

1

n

n∑j=1

X∗i,j and S∗

i =

√√√√ 1

n− 1

n∑j=1

(X∗i,j − ¯X∗

i )2.

Then, the sample coefficient of variation γ∗i is defined by

γ∗i =S∗i

¯X∗i

.

In order to monitor the CV squared, its distribution is needed. In this paper,we apply an approximation for the cumulative distribution function (c.d.f)and the inverse distribution function (i.d.f) of the CV squared suggested in[7]; that is

Fγ2(x|n, γ) = 1− FF(n

x

∣∣∣ 1, n− 1,n

γ2

), (6)

andF−1γ2 (p|n, γ) =

n

F−1F

(1− p

∣∣∣1, n− 1, nγ2

) , (7)

where FF (.) and F−1F (.) is the c.d.f and i.d.f of the non-central F distribution.

Changing the role of Xi in [7] by X∗i results in the following approximation

of the c.d.f and i.d.f of γ∗2 as:

Fγ∗2(x|n, γ∗) = 1− FF(n

x

∣∣∣ 1, n− 1,n

γ∗2

), (8)

andF−1γ∗2(p|n, γ∗) =

n

F−1F

(1− p

∣∣∣1, n− 1, nγ∗2

) , (9)

with γ∗2 is defined in (3).

5 The effect of measurement error on the VSI SH-γ2 control charts

The implementation of VSI SH-γ2 control charts under the presence of mea-surement error is similar to the implementation of VSI SH-γ2 without takinginto account measurement error in Section 2. The control limit and warn-ing limit parameters, (UCL+, UWL+) or (LCL−, LWL−), are the solutionof equations corresponding to two constrains on the in-control value of ATS0

and ASI0 as in (1)-(2). The difference is that the ATS and the ASI should becalculated by using the distribution of γ∗2 rather than the distribution of γ2

as pointed out in Appendix A.


Similar to Section 3, the in-control values of ATS0 and ASI0 are set to370.4 and 1, respectively. The following values of parameters are applied:

– the sample sizes n ∈ 5, 15,– the shift sizes τ ∈ 0.5, 0.8, 0.9, 0.95, 1.05, 1.1, 1.2, 1.5,– the coefficient of variation values γ0 ∈ 0.05, 0.1, 0.2,– the combinations of sampling interval (hS , hL) = (0.5, 1.5), (0.3, 1.7),

(0.1, 1.3), (0.1, 1.5), (0.1, 1.9), (0.1, 4.0).

We also consider other values for parameters representing the measurementerror, which are η ∈ 0.0, 0.28, 0.5, 1.0, θ ∈ 0.0, 0.03, 0.05, B ∈ 2, 3, 5and m ∈ 1, 5, 10. These parameters are taken into account for a desire ofcovering a large range of possible scenarios of parameters. For example, n = 5is corresponding to moderate sample size while n = 15 is for large sample size;τ = 0.5 or τ = 1.5 represent the large shift sizes, while τ = 0.95 or τ = 1.05represent small shift sizes. The specific value of η = 0.28 is motivated byassuming an acceptable value for the signal-to-noise ratio (see AIAG guidelines[15]), while the value of η = 1.0 is to describe a large accuracy error. The valuesof (hS , hL) follows those suggested in [19]. In our study, the numerical resultshave also been obtained for a number of other combinations of parameters,but they are not presented here and are available upon request from authors.

Without loss of generality, we suppose that b = 1. Tables 4-7 present thecontrol limit and warning limit for both upward and downward charts. It isconsidered from these tables that the control limit does not depend on thesampling interval and the shift size. This result can be proven as follows. SinceASI0 = 1, equation (16) leads to hSpS0 + hLpL0 = 1 − q0, in which pS0, pL0

and q0 are the corresponding in-control values of the probabilities pS , pL andq. Substituting this result into (17), we obtain

– for upward chart,

1

ATS0= q0 = 1− Fγ∗2(UCL+|n, γ∗0),

– for downward chart,

1

ATS0= q0 = Fγ∗2(LCL−|n, γ∗0).

That means UCL+ and LCL− do not depend on (hS , hL) and τ .

PLEASE INSERT TABLE 4 HEREPLEASE INSERT TABLE 5 HEREPLEASE INSERT TABLE 6 HEREPLEASE INSERT TABLE 7 HERE

Tables 8-12 present the corresponding values of ATS1 based on the valuesof control limit and warning limit in the Tables 4-7. The ATS1 value of theShewhart-γ2 control chart under the presence of measurement error basedon corrected formula is calculated again and presented in the second column


of the Tables 8-12 (FSI columns). As expected, the VSI feature in generalenhance remarkably the efficiency of Shewhart-γ2 regardless the presence ofmeasurement error. The values of ATS1 using FSI SH-γ2 control chart arealways larger than those the values of ATS1 using VSI SH-γ2 control chartwith the same parameters. For example, with n = 5, τ = 0.8, η = 0.28, θ =0.05, B = 1,m = 1 and γ0 = 0.05 in Table 8, we have ATS1 = 162.03 in FSIchart, larger than any value of ATS1 in the same row.

The results in Tables 8-10 show that both the precision and accuracy er-rors have negative influence on the chart performance but with different level.While the increase of θ leads to the significant increase of ATS1 (Table 10),the increase of ATS1 is insignificant as η increases (Table 8). For example,with n = 5, B = 1,m = 1, γ0 = 0.1, θ = 0.05, τ = 1.1 and (hS , hL) = (0.1, 1.5),we have ATS1 = 98.84 when η = 0.2 and ATS1 = 99.54 when η = 1 (Ta-ble 8). With the same value of n,B,m, γ0, τ , (hS , hL) and η = 0.28, we haveATS1 = 92.88 when θ = 0 and ATS1 = 98.86 when θ = 0.05 (Table 10).

PLEASE INSERT TABLE 8 HEREPLEASE INSERT TABLE 9 HEREPLEASE INSERT TABLE 10 HERE

In contrast to θ and η, the larger the value of constant coefficient B in linearcovariate error model, the smaller the value of ATS1. That is to say, theincrease of B enhances slightly the chart performance. For instance, we obtainATS1 = 5.19 when B = 2 and ATS1 = 5.13 when B = 5 with the samen = 5, η = 0.028, θ = 0.01,m = 1, τ = 1.5, (hS , hL) = (0.5, 1.5) in Table 11.


To deal with the problem of measurement error, a common approach is to usemultiple measures for the same item. However, the obtained result in Table 12show that this method is inefficient in enhancing the efficiency of VSI SH-γ2

control chart. As the value of m increases from 1 to 10, the ATS1 remainsunchanged or reduces negligibly. For instance, with n = 5, τ = 0.8, (hS , hL) =(0.1, 1.1), η = 0.28, θ = 0.05, B = 1, γ0 = 0.05, we have ATS1 = 146.50 whenm = 1 and ATS1 = 146.49 when m = 10.


In many practical situations, it is difficult for practitioners to determine aspecific shift size. Considering the shift size as a random variable and selectinga probability distribution is suggested to model it. Several potential statisticaldistributions is provided in [8]. In the case the quality practitioners do not haveany reference for the shift size, the uniform distribution can be used which givesan equal weight to each shift size included within the interval Ω = [a, b]. Theperformance of VSI SH-γ2 now is measured by expected average time to sign,

EATS =

∫ b

a

ATS × f(τ)dτ,

where f(τ) = 1b−a over the range [a, b]. The chart parameters is now the

solution of following equations:


– for downward chart,EATS(n, hS , hL, LCL

−, LWL−, γ0, τ = 1) = ATS0

ASI(n, hS , hL, LCL−, LWL−, γ0, τ = 1) = ASI0;

– for upward chart,EATS(n, hS , hL, UCL

+, UWL+, γ0, τ = 1) = ATS0

ASI(n, hS , hL, UCL+, UWL+, γ0, τ = 1) = ASI0.

The value of EATS is computed based on these chart parameters. We considertwo situations, [a, b] = [0.5, 1) corresponding to the decreasing case, denotedby (D) and [a, b] = (1, 1.5] corresponding to the increasing case, denoted by(I). Tables 13-16 show the impact of parameters η, θ, B and m on the overallperformance of VSI SH-γ2 control charts. In general, it can be seen that theEATS values in these tables show a similar tendency as for the deterministicshift size results discussed above. That is: (1) the measurement errors (repre-sented by η and θ) have a negative effect on the power of the VSI SH-γ2 chart,(2) the coefficient B in the linear covariate model can reduce the negative effectof measurement errors, and (3) taking more measurements per item in eachsample is not an efficient way to compensate for the effect of measurementerrors. Moreover, some more conclusion can be drawn as follows.

– EATS values are always larger for the decreasing case compared to theincreasing case. For example, with n = 15, θ = 0.05, η = 0.2,m = 1, B =1, γ0 = 0.05, (hS , hL) = (0.5, 1.5) we have EATS1 = 40.57 for the de-creasing case and EATS1 = 12.65 for the increasing case (Table 13). Thatmeans it is easier for VSI SH-γ2 to detect upward shifts than the down-ward shifts. This can be explained by the asymmetric property of the CVdistribution.

– The value of (hS , hL) = (0.1, 4.0), corresponding to the widest intervalamong the intervals suggested by [19], results in the smallest values ofEATS. This couple is then recommended to reduce the effect of measure-ment errors on the chart performance.

– The control chart sensitivity is improved remarkably with the sample size.As n increases from 5 to 15, the values of EATS reduce by more thanhalf. For example, with θ = 0.03, η = 0.28,m = 1, B = 1, γ0 = 0.05 and(hS , hL) = (0.1, 1.5), we have EATS1 = 22.46 when n = 5 and EATS1 =10.65 when n = 15 (Increasing case in Table 14). Therefore, a larger samplesize is a very effective way to enhance the chart efficiency.

PLEASE INSERT TABLE 13 HEREPLEASE INSERT TABLE 14 HEREPLEASE INSERT TABLE 15 HEREPLEASE INSERT TABLE 16 HERE


6 Application in real industrial scenarios

The use of control charts is widespread in many industrial processes wherethe assignable causes and out-of-control conditions should be detected as soonas possible to ensure the product quality and reduce waste. In practice, thedata are collected by measuring samples taken during production. The interestcharacteristic (which is the CV squared in our study) is calculated directly fromthese observed data, which commonly contain measurement error. That is tosay, the obtained value of the CV is under the presence of measurement error(denoted by γ∗2). As a result, the characteristic we monitor in practice shouldbe γ∗2, but not the true value γ2 (we do not observe this exact value dueto measurement error). However, by monitoring γ∗2 using the linear covariateerror model, we can still detect the change in the true value of the CV. Animportant issue related to choosing the proper parameters of the model canbe arisen in practice. This is equivalent to how to determine the parametersin the linear covariate regression model (using in a measurement system). Anumber of techniques to deal with this problem are provided in [15] (SectionB).

6.1 Example 1: Sintering process

In this Section, we illustrate the implementation of VSI SH-γ2 control chartin the presence of measurement errors. We consider a simulated set of datafrom a sintering process in an Italian company that manufactures sinteredmechanical parts. This context is similar to the scenario presented in [7]. Theprocess manufactures parts have to ensure a pressure test drop time Tpd from2 bar to 1.5 bar more than 30 sec, which is a quality characteristic related tothe pore shrinkage. It is stated that the preliminary regression study relatingTpd to the quantity QC of molten copper has demonstrated the presence of aconstant proportionality σpd = γpd × µpd between the standard deviation ofthe pressure drop time and its mean. The quality practitioner then decidedto monitor the coefficient of variation γpd = σpd/µpd to detect changes in theprocess variability.

From a Phase I dataset, an estimated γ0 = 0.01 is computed based on rootmean square computation. The related parameters of the linear covariate errormodel are supposed to be η = 0.28, θ = 0, B = 1, and m = 1. A dataset of 20observations with sample size n = 5 is measured for phase II. For each samplei = 1, 2, ..., 20, the sample mean X∗

i , the sample standard deviation S∗i and

the sample CV squared γ∗2i =

(X∗

i

S∗i

)2

have been computed and presented in

Table 1. In this simulated data, the process is assumed to run in-control upto sample #9. Then, between samples #10 and #12, we have simulated theoccurrence of a cause shifting γpd from γpd = 0.01 to γpd = 0.011, i.e. τ = 1.1.

In this example, the downward and the upward control charts are imple-mented with hS = 0.1 and hL = 4.0. Based on the numerical procedure inSection 5 with ATS0 = 370.4, for the downward VSI SH-γ2 chart, the control


Subgroup i X∗i S∗

i γ∗i γ∗2i

1 595.7 4.729 0.0079 0.000062 602.6 7.215 0.0120 0.000143 603.7 7.642 0.0127 0.000164 603.5 4.520 0.0075 0.000065 597.5 4.856 0.0081 0.000076 597.4 6.130 0.0103 0.000117 603.0 3.658 0.0061 0.000048 602.4 8.528 0.0142 0.000209 592.1 9.307 0.0157 0.0002510 604.3 14.201 0.0235 0.0005511 596.4 13.092 0.0220 0.0004812 602.8 12.607 0.0209 0.0004413 602.7 4.420 0.0073 0.0000514 605.0 5.940 0.0098 0.0001015 597.0 4.453 0.0075 0.0000616 599.5 4.331 0.0072 0.0000517 601.1 9.291 0.0155 0.0002418 604.6 2.070 0.0034 0.0000119 598.6 6.086 0.0102 0.0001020 597.3 5.208 0.0087 0.00008

Table 1: Illustrative example of Phase II dataset from a sintering process.

and warning limits with measurement error are LCL− = 0.0000040623 andLWL− = 0.00015128, respectively. The downward chart is illustrated in Fig-ure 2. It can be seen from Table 1 and Figure 2 that most of the monitoringvalues are above the warning limit, and no value is below the control limit.This is reasonable since there is no decrease shift in the process.

For the upward case, the control limit and warning limit are equal toUCL+ = 0.00043826 and UWL+ = 0.000048914, respectively. The upwardchart is illustrated in Figure 3. Most of the sample CV values are in the warn-ing region, the samples #10, #11 and #12 are detected as out-of-control.As we know, the change of the process is simulated from sample #10, whichmeans the chart detects the true shift at time 4.8 (time unit). In this particularexample, the time required for the VSI SH-γ2 chart to detect the process shiftis much less than the time that would have been necessary with the Shewhartchart (10 time unit).

6.2 Example 2: Die casting hot chamber process

Let us consider another scenario from a die casting hot chamber process intro-duced in [4]. A set of data has been simulated based on a context given by aTunisian company manufacturing zinc alloy (ZAMAK) parts for the sanitary


0

0.0001

0.0002

0.0003

0.0004

0.0005

0 5 10 15 20 25

t

γ2 i

LCL−

LWL−

Fig. 2: The downward VSI SH-γ2 control chart with hS = 0.1 and hL = 4.0 inthe presence of measurement error for the industrial data-set in Table 1.

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 1 2 3 4 5 6 7 8 9

t

γ2 i

UCL+

UWL+

Fig. 3: The upward VSI SH-γ2 control chart with hS = 0.1 and hL = 4.0 inthe presence of measurement error for the industrial data-set in Table 1.

sector. The weight (X, in grams) of scrap zinc alloy material to be removedbetween the molding process and the continuous plating surface treatment isthe character of interest. The presence of a constant proportionality σ = γ×µbetween the standard deviation σ and the mean µ of the weight of scrap alloyhas been illustrated by a preliminary regression study. With the regressionstudy, the in-control CV γ0 has been estimated to 0.01.

According to the process engineer, the most important special cause thatleads to an anomalous increase in σ is due to the shift from the nominal value


of the injection pressure of the zinc alloy into the die. In fact, the injectionpressure holds the molten metal into the die during solidification. As a re-sult, this variation can lead to an uncontrolled item solidification leading toexcessive scrap material.

Subgroup i X∗i S∗

i γ∗i γ∗2i

1 449.0 5.491 0.0122 0.000152 453.0 4.354 0.0096 0.000093 451.5 7.137 0.0158 0.000254 455.2 4.888 0.0107 0.000125 447.0 7.660 0.0171 0.000296 446.3 2.629 0.0059 0.000037 445.3 6.016 0.0135 0.000188 451.5 3.324 0.0074 0.000059 451.4 2.311 0.0051 0.0000310 448.3 5.782 0.0129 0.0001711 449.7 7.656 0.0170 0.0002912 447.7 3.406 0.0076 0.0000613 454.0 8.420 0.0185 0.0003414 451.0 4.885 0.0108 0.0001215 452.3 3.989 0.0088 0.0000816 450.7 8.315 0.0184 0.0003417 446.5 3.645 0.0082 0.0000718 450.2 9.553 0.0212 0.0004519 449.3 10.131 0.0225 0.0005120 449.2 4.186 0.0093 0.0000921 452.2 4.788 0.0106 0.0001122 448.7 3.890 0.0087 0.0000823 449.7 8.613 0.0192 0.0003724 450.1 7.376 0.0164 0.0002725 449.8 5.475 0.0122 0.0001526 451.9 4.399 0.0097 0.0000927 450.6 4.310 0.0096 0.0000928 453.4 3.627 0.0080 0.0000629 450.5 4.806 0.0107 0.0001130 450.9 4.358 0.0097 0.00009

Table 2: Illustrative example of Phase II dataset from a zinc alloy die castinghot chamber process.

Similar to the first example, the data is simulated under the presence ofmeasurement error with η = 0.28, θ = 0, B = 1, and m = 1 for the parametersof the linear covariate error mode. The dataset of 30 samples of size n = 5 ismeasured for phase II. For each sample i = 1, 2, ..., 30, the sample mean X∗

i ,


the sample standard deviation S∗i of scrap alloy and the sample CV squared

γ∗2i have been computed (see Table 2). In this simulated data, the process is

assumed to run in-control up to sample #17. Then, between samples #18 and#19, we have simulated an increase of 20% of CV (i.e. τ = 1.2).

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 5 10 15 20 25 30 35 40 45

t

γ2 i

LCL−

LWL−

Fig. 4: The downward VSI SH-γ2 control chart with hS = 0.1 and hL = 4.0 inthe presence of measurement error for the industrial data-set in Table 2.

We implement the downward and the upward control charts with hS =0.1, hL = 4.0 and ATS0 = 370.4. Therefore, the control and warning limitswith measurement error are the same as the previous example, i.e. LCL− =0.0000040623, LWL− = 0.00015128 for the downward chart and UCL+ =0.00043826, UWL+ = 0.000048914 for the upward chart. Figure 4 shows theoperation of the downward chart. As expected, there is no detection from thisfigure since only the upward shift is simulated.

The operation of upward VSI SH-γ2 chart is illustrated in Figure 5. Thecontrol chart triggers a signal by plotting points #18 and#19 above the controllimit UCL+ = 0.00043826. This figure confirms that these two batches areout-of-control. After the corrective actions, samples collected from successivebatches do not show any anomaly. The upward VSI SH-γ2 chart requires only9.5 (time unit) to detect the change in the process, while the Shewhart chartneeds 18 samples (i.e. 18 time unit).

7 Concluding remarks

We have investigated in this paper two one-sided VSI Shewhart control chartsfor monitoring the coefficient of variation squared. In comparison with thetwo-sided chart suggested in [4], using two-sided charts leads to better per-formance in detecting both downward shifts and upward shifts. The effect of


0

0.0001

0.0002

0.0003

0.0004

0.0005

0 2 4 6 8 10

t

γ2 i

UCL+

UWL+

Fig. 5: The upward VSI SH-γ2 control chart with hS = 0.1 and hL = 4.0 inthe presence of measurement error for the industrial data-set in Table 2.

measurement error on the performance of VSI SH−γ2 control chart using alinear covariate error model is also discovered. The numerical results show thatthe measurement error has negative influences on the proposed control charts:The larger the value of precision errors and especially accuracy errors, thelarger the value of average time to signal. Studying the effect of each param-eters on the chart performance suggests us some efficient strategies to reducethese negative impacts as well as enhance the efficiency of the proposed charts,involving increasing the sample size n or choosing the couple of sampling inter-val (hS , hL) with width as large as possible (which is (0.1, 4.0) in this study).It is also worthy to consider that increasing the number of multiple measure-ment per item is not a good strategy since it reduces insignificantly the chartefficiency in the presence of measurement error.

Acknowledgements The authors thank the anonymous referees for their insightful andvaluable suggestions which helped to improve the quality of the final manuscript.

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A Appendix

Let pS , pL and q be the probability that a monitored sample point drops into the centralregion, the warning region and the out-of-control region, respectively. According to thesubdivision of control interval, the formulae to calculate pS , pL and q in VSI SH-γ2 withoutconsidering the measurement errors are as follows.

– For downward chart,

pL = P (γ2 ≥ LWL−) = 1− Fγ2 (LWL− | n, γ2), (10)

pS = P (LCL− 6 γ2 6 LWL−) (11)

= Fγ2 (LWL−|n, γ2)− Fγ2 (LCL−|n, γ2),

q = P (γ2 < LCL−) = 1− pS − pL. (12)

– For upward chart,

pL = P (γ2 6 UWL+) = Fγ2 (UWL+|n, γ2), (13)

pS = P (UWL+ 6 γ2 6 UCL+) (14)

= Fγ2 (UCL+|n, γ2)− Fγ2 (UWL+|n, γ2),

q = P (γ2 > UCL+) = 1− pS − pL. (15)

The c.d.f Fγ2 (.|n, γ2) in this case is defined in (6).

In the VSI SH-γ2 control charts considering the presence of measurement errors, theformulae for pS , pL and q are the same in equations (10)-(15) for both charts, but thedistribution Fγ2 (.|n, γ2) in these equations are replaced by Fγ∗2 (.|n, γ∗2), defined in (8).

From its definition, the ASI is calculated by

ASI = E(h) =hSpS + hLpL

1− q . (16)

The formula of ATS is given by [20] and adopted by [4] as

ATS =hSpS + hLpL

q(1− q) . (17)


FSI VSI(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

τ γ0 n = 50.5 .05 26.91 13.71 8.43 12.90 6.12 4.23 3.16 2.71

0.1 27.07 13.80 8.49 13.03 6.19 4.28 3.19 2.730.2 27.72 14.15 8.73 13.54 6.48 4.47 3.30 2.79

0.8 .05 156.19 119.29 104.53 140.34 120.23 106.93 89.77 56.890.1 156.67 119.79 105.03 140.85 120.75 107.45 90.28 57.300.2 158.56 121.73 107.00 142.84 122.80 109.50 92.27 58.96

0.9 .05 246.01 218.26 207.16 236.04 221.75 211.22 196.06 159.370.1 246.42 218.74 207.66 236.47 222.22 211.71 196.59 159.940.2 247.99 220.57 209.61 238.16 224.05 213.64 198.64 162.19

0.95 .05 303.41 286.87 280.25 297.87 289.57 283.20 273.64 248.280.1 303.67 287.18 280.59 298.15 289.87 283.52 274.00 248.710.2 304.66 288.40 281.90 299.22 291.06 284.80 275.39 250.41

1.05 .05 189.91 180.56 176.82 183.99 179.04 176.29 173.08 167.520.1 190.48 181.15 177.41 184.57 179.63 176.88 173.68 168.130.2 192.85 183.56 179.85 186.98 182.07 179.33 176.14 170.60

1.1 .05 107.13 97.10 93.09 100.21 95.00 92.23 89.08 83.820.1 107.73 97.69 93.68 100.81 95.59 92.81 89.66 84.390.2 110.21 100.12 96.09 103.28 98.03 95.24 92.06 86.73

1.2 .05 42.63 35.46 32.59 36.91 33.37 31.62 29.72 26.740.1 43.04 35.83 32.95 37.30 33.74 31.98 30.07 27.060.2 44.75 37.39 34.45 38.93 35.28 33.47 31.51 28.40

1.5 .05 8.07 5.62 4.64 5.53 4.51 4.08 3.65 3.060.1 8.20 5.72 4.72 5.63 4.60 4.17 3.73 3.120.2 8.75 6.14 5.10 6.08 4.99 4.52 4.06 3.40

FSI VSI(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

τ γ0 n = 150.5 .05 1.70 0.85 0.51 0.19 0.17 0.17 0.17 0.17

0.1 1.72 0.86 0.52 0.19 0.17 0.17 0.17 0.170.2 1.78 0.89 0.53 0.20 0.18 0.18 0.18 0.18

0.8 .05 38.61 23.48 17.43 27.13 19.02 15.19 11.38 6.550.1 39.02 23.79 17.69 27.51 19.33 15.46 11.60 6.670.2 40.66 24.99 18.72 29.00 20.56 16.52 12.45 7.17

0.9 .05 121.77 95.85 85.48 108.64 94.68 86.02 75.11 53.880.1 122.53 96.59 86.21 109.41 95.45 86.77 75.83 54.500.2 125.46 99.45 89.05 112.41 98.42 89.69 78.65 56.94

0.95 0.05 213.58 190.95 181.90 203.92 192.08 183.98 172.85 147.230.1 214.28 191.72 182.69 204.66 192.86 184.78 173.66 148.080.2 216.98 194.67 185.74 207.50 195.84 187.83 176.81 151.35

1.05 0.05 135.94 122.83 117.59 128.07 121.19 117.20 112.34 103.030.1 136.80 123.70 118.45 128.95 122.07 118.08 113.21 103.880.2 140.23 127.13 121.89 132.41 125.53 121.53 116.65 107.25

1.1 .05 59.41 48.74 44.48 51.95 46.49 43.57 40.21 34.290.1 60.08 49.36 45.08 52.60 47.11 44.17 40.79 34.820.2 62.77 51.85 47.48 55.21 49.60 46.59 43.10 36.93

1.2 .05 16.93 11.81 9.76 12.35 9.97 8.87 7.72 5.950.1 17.23 12.05 9.98 12.61 10.20 9.08 7.90 6.100.2 18.46 13.03 10.85 13.68 11.13 9.93 8.68 6.73

1.5 .05 2.55 1.41 0.95 1.06 0.71 0.60 0.50 0.380.1 2.60 1.44 0.98 1.09 0.74 0.62 0.51 0.390.2 2.83 1.58 1.08 1.23 0.84 0.70 0.58 0.44

Table 3: The out-of-control ARL1 of the FSI SH-γ chart and the out-of-controlATS1 of the VSI SH-γ2 charts for γ0 = 0.05, 0.1, 0.2, n = 5, 15, τ =0.5, 0.8, 0.9, 0.95, 1.05, 1.1, 1.2, 1.5. The first row shows the values of (hS , hL).


n = 5Downward CV chart

γ0 η LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 0.0 0.0001 0.0019 0.0019 0.0006 0.0011 0.0014 0.0019 0.00320.2 0.0001 0.0020 0.0020 0.0006 0.0011 0.0015 0.0020 0.00330.5 0.0001 0.0024 0.0024 0.0008 0.0014 0.0018 0.0024 0.00401.0 0.0002 0.0038 0.0038 0.0012 0.0022 0.0029 0.0038 0.0064

0.1 0.0 0.0003 0.0076 0.0076 0.0024 0.0044 0.0057 0.0076 0.01280.2 0.0004 0.0079 0.0079 0.0025 0.0046 0.0059 0.0079 0.01330.5 0.0004 0.0095 0.0095 0.0030 0.0055 0.0071 0.0095 0.01601.0 0.0007 0.0153 0.0153 0.0049 0.0087 0.0114 0.0153 0.0257

0.2 0.0 0.0013 0.0306 0.0306 0.0097 0.0174 0.0228 0.0306 0.05190.2 0.0014 0.0318 0.0318 0.0100 0.0181 0.0237 0.0318 0.05400.5 0.0017 0.0383 0.0383 0.0120 0.0217 0.0284 0.0383 0.06511.0 0.0026 0.0613 0.0613 0.0191 0.0345 0.0454 0.0613 0.1057

Upward CV chartγ0 η UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 0.0 0.0093 0.0019 0.0019 0.0044 0.0030 0.0025 0.0019 0.0010

0.2 0.0096 0.0020 0.0020 0.0046 0.0032 0.0026 0.0020 0.00110.5 0.0116 0.0024 0.0024 0.0055 0.0038 0.0031 0.0024 0.00131.0 0.0186 0.0038 0.0038 0.0088 0.0061 0.0050 0.0038 0.0021

0.1 0.0 0.0378 0.0076 0.0076 0.0177 0.0122 0.0099 0.0076 0.00410.2 0.0393 0.0079 0.0079 0.0184 0.0127 0.0103 0.0079 0.00430.5 0.0475 0.0095 0.0095 0.0221 0.0153 0.0124 0.0095 0.00511.0 0.0774 0.0152 0.0152 0.0356 0.0245 0.0199 0.0152 0.0082

0.2 0.0 0.1629 0.0304 0.0304 0.0725 0.0495 0.0400 0.0304 0.01630.2 0.1701 0.0317 0.0317 0.0755 0.0515 0.0416 0.0317 0.01700.5 0.2090 0.0381 0.0381 0.0914 0.0621 0.0501 0.0381 0.02041.0 0.3628 0.0610 0.0610 0.1500 0.1006 0.0807 0.0610 0.0324


γ0 η LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 0.0 0.0006 0.0022 0.0022 0.0013 0.0017 0.0019 0.0022 0.00280.2 0.0006 0.0023 0.0023 0.0013 0.0017 0.0019 0.0023 0.00300.5 0.0007 0.0027 0.0027 0.0016 0.0021 0.0023 0.0027 0.00351.0 0.0012 0.0043 0.0043 0.0025 0.0033 0.0037 0.0043 0.0057

0.1 0.0 0.0023 0.0087 0.0087 0.0051 0.0066 0.0075 0.0087 0.01140.2 0.0024 0.0090 0.0090 0.0053 0.0069 0.0078 0.0090 0.01180.5 0.0029 0.0108 0.0108 0.0063 0.0082 0.0094 0.0108 0.01421.0 0.0047 0.0173 0.0173 0.0101 0.0132 0.0150 0.0173 0.0228

0.2 0.0 0.0092 0.0346 0.0346 0.0201 0.0262 0.0299 0.0346 0.04590.2 0.0096 0.0360 0.0360 0.0209 0.0273 0.0311 0.0360 0.04770.5 0.0114 0.0433 0.0433 0.0250 0.0327 0.0373 0.0433 0.05751.0 0.0180 0.0693 0.0693 0.0397 0.0521 0.0595 0.0693 0.0927

Upward CV chartγ0 η UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 0.0 0.0054 0.0022 0.0022 0.0034 0.0028 0.0025 0.0022 0.0016

0.2 0.0056 0.0022 0.0022 0.0035 0.0029 0.0026 0.0022 0.00170.5 0.0067 0.0027 0.0027 0.0043 0.0035 0.0031 0.0027 0.00201.0 0.0108 0.0043 0.0043 0.0068 0.0055 0.0050 0.0043 0.0032

0.1 0.0 0.0218 0.0086 0.0086 0.0137 0.0111 0.0099 0.0086 0.00640.2 0.0227 0.0090 0.0090 0.0142 0.0115 0.0103 0.0090 0.00670.5 0.0273 0.0108 0.0108 0.0171 0.0139 0.0124 0.0108 0.00801.0 0.0441 0.0173 0.0173 0.0274 0.0222 0.0199 0.0173 0.0128

0.2 0.0 0.0904 0.0345 0.0345 0.0554 0.0447 0.0398 0.0345 0.02550.2 0.0942 0.0359 0.0359 0.0576 0.0465 0.0414 0.0359 0.02650.5 0.1144 0.0432 0.0432 0.0696 0.0560 0.0499 0.0432 0.03181.0 0.1900 0.0691 0.0691 0.1129 0.0902 0.0801 0.0691 0.0505

Table 4: Control limits (boldfaced) and warning limits of VSI SH-γ2 control charts in thepresence of measurement error for different values of γ0, η, n and fixed θ = 0.05, B = 1,m =1.



γ0 θ LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 0.0 0.0001 0.0023 0.0023 0.0007 0.0013 0.0017 0.0023 0.00380.03 0.0001 0.0021 0.0021 0.0007 0.0012 0.0016 0.0021 0.00360.05 0.0001 0.0021 0.0021 0.0007 0.0012 0.0015 0.0021 0.0034

0.1 0.0 0.0004 0.0091 0.0091 0.0029 0.0052 0.0068 0.0091 0.01520.03 0.0004 0.0086 0.0086 0.0027 0.0049 0.0064 0.0086 0.01430.05 0.0004 0.0082 0.0082 0.0026 0.0047 0.0062 0.0082 0.0138

0.2 0.0 0.0016 0.0364 0.0364 0.0114 0.0207 0.0271 0.0364 0.06190.03 0.0015 0.0343 0.0343 0.0108 0.0195 0.0255 0.0343 0.05830.05 0.0014 0.0330 0.0330 0.0104 0.0188 0.0246 0.0330 0.0560

Upward CV chartγ0 θ UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 0.0 0.0110 0.0023 0.0023 0.0052 0.0036 0.0029 0.0023 0.0012

0.03 0.0104 0.0021 0.0021 0.0049 0.0034 0.0028 0.0021 0.00120.05 0.0100 0.0020 0.0020 0.0047 0.0033 0.0027 0.0020 0.0011

0.1 0.0 0.0451 0.0090 0.0090 0.0210 0.0145 0.0118 0.0090 0.00490.03 0.0424 0.0085 0.0085 0.0198 0.0137 0.0111 0.0085 0.00460.05 0.0408 0.0082 0.0082 0.0191 0.0132 0.0107 0.0082 0.0044

0.2 0.0 0.1975 0.0362 0.0362 0.0867 0.0590 0.0476 0.0362 0.01940.03 0.1848 0.0341 0.0341 0.0815 0.0556 0.0449 0.0341 0.01830.05 0.1771 0.0328 0.0328 0.0784 0.0534 0.0431 0.0328 0.0176


γ0 θ LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 0.0 0.0007 0.0026 0.0026 0.0015 0.0020 0.0022 0.0026 0.00340.03 0.0007 0.0024 0.0024 0.0014 0.0018 0.0021 0.0024 0.00320.05 0.0006 0.0023 0.0023 0.0014 0.0018 0.0020 0.0023 0.0031

0.1 0.0 0.0028 0.0103 0.0103 0.0060 0.0078 0.0089 0.0103 0.01350.03 0.0026 0.0097 0.0097 0.0057 0.0074 0.0084 0.0097 0.01280.05 0.0025 0.0093 0.0093 0.0055 0.0071 0.0081 0.0093 0.0123

0.2 0.0 0.0109 0.0412 0.0412 0.0238 0.0312 0.0355 0.0412 0.05460.03 0.0103 0.0388 0.0388 0.0225 0.0294 0.0335 0.0388 0.05150.05 0.0099 0.0374 0.0374 0.0216 0.0283 0.0322 0.0374 0.0495

Upward CV chartγ0 θ UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 0.0 0.0064 0.0026 0.0026 0.0040 0.0033 0.0029 0.0026 0.0019

0.03 0.0060 0.0024 0.0024 0.0038 0.0031 0.0028 0.0024 0.00180.05 0.0058 0.0023 0.0023 0.0037 0.0030 0.0027 0.0023 0.0017

0.1 0.0 0.0259 0.0103 0.0103 0.0162 0.0132 0.0118 0.0103 0.00760.03 0.0244 0.0097 0.0097 0.0153 0.0124 0.0111 0.0097 0.00720.05 0.0235 0.0093 0.0093 0.0147 0.0120 0.0107 0.0093 0.0069

0.2 0.0 0.1085 0.0411 0.0411 0.0661 0.0532 0.0474 0.0411 0.03020.03 0.1019 0.0387 0.0387 0.0622 0.0501 0.0447 0.0387 0.02850.05 0.0978 0.0373 0.0373 0.0598 0.0482 0.0430 0.0373 0.0274

Table 5: Control limits (boldfaced) and warning limits of VSI SH-γ2 control charts in thepresence of measurement error for different values of γ0, θ, n and fixed η = 0.28, B = 1,m =1.



γ0 B LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 2 0.0001 0.0021 0.0021 0.0007 0.0012 0.0016 0.0021 0.00353 0.0001 0.0021 0.0021 0.0007 0.0012 0.0016 0.0021 0.00355 0.0001 0.0021 0.0021 0.0007 0.0012 0.0016 0.0021 0.0035

0.1 2 0.0004 0.0085 0.0085 0.0027 0.0049 0.0064 0.0085 0.01423 0.0004 0.0084 0.0084 0.0027 0.0048 0.0063 0.0084 0.01415 0.0004 0.0084 0.0084 0.0027 0.0048 0.0063 0.0084 0.0141

0.2 2 0.0015 0.0341 0.0341 0.0107 0.0194 0.0253 0.0341 0.05783 0.0015 0.0338 0.0338 0.0106 0.0192 0.0251 0.0338 0.05745 0.0015 0.0337 0.0337 0.0106 0.0192 0.0251 0.0337 0.0572

Upward CV chartγ0 B UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 2 0.0103 0.0021 0.0021 0.0049 0.0034 0.0028 0.0021 0.0011

3 0.0102 0.0021 0.0021 0.0048 0.0034 0.0027 0.0021 0.00115 0.0102 0.0021 0.0021 0.0048 0.0034 0.0027 0.0021 0.0011

0.1 2 0.0421 0.0085 0.0085 0.0197 0.0136 0.0110 0.0085 0.00463 0.0418 0.0084 0.0084 0.0195 0.0135 0.0110 0.0084 0.00455 0.0417 0.0084 0.0084 0.0195 0.0135 0.0109 0.0084 0.0045

0.2 2 0.1834 0.0339 0.0339 0.0810 0.0552 0.0445 0.0339 0.01813 0.1819 0.0336 0.0336 0.0803 0.0548 0.0442 0.0336 0.01805 0.1813 0.0335 0.0335 0.0801 0.0546 0.0441 0.0335 0.0180


γ0 B LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 2 0.0007 0.0024 0.0024 0.0014 0.0018 0.0021 0.0024 0.00323 0.0006 0.0024 0.0024 0.0014 0.0018 0.0021 0.0024 0.00315 0.0006 0.0024 0.0024 0.0014 0.0018 0.0021 0.0024 0.0031

0.1 2 0.0026 0.0096 0.0096 0.0056 0.0073 0.0083 0.0096 0.01273 0.0026 0.0096 0.0096 0.0056 0.0073 0.0083 0.0096 0.01265 0.0026 0.0095 0.0095 0.0056 0.0073 0.0082 0.0095 0.0125

0.2 2 0.0102 0.0386 0.0386 0.0223 0.0292 0.0332 0.0386 0.05113 0.0101 0.0383 0.0383 0.0221 0.0290 0.0330 0.0383 0.05075 0.0101 0.0382 0.0382 0.0221 0.0289 0.0329 0.0382 0.0506

Upward CV chartγ0 B UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 2 0.0060 0.0024 0.0024 0.0038 0.0031 0.0028 0.0024 0.0018

3 0.0060 0.0024 0.0024 0.0038 0.0031 0.0027 0.0024 0.00185 0.0059 0.0024 0.0024 0.0037 0.0030 0.0027 0.0024 0.0018

0.1 2 0.0243 0.0096 0.0096 0.0152 0.0123 0.0110 0.0096 0.00713 0.0241 0.0095 0.0095 0.0151 0.0123 0.0110 0.0095 0.00715 0.0240 0.0095 0.0095 0.0150 0.0122 0.0109 0.0095 0.0070

0.2 2 0.1011 0.0385 0.0385 0.0618 0.0498 0.0444 0.0385 0.02833 0.1004 0.0382 0.0382 0.0613 0.0494 0.0440 0.0382 0.02815 0.1000 0.0381 0.0381 0.0611 0.0493 0.0439 0.0381 0.0280

Table 6: Control limits (boldfaced) and warning limits of VSI SH-γ2 control charts inthe presence of measurement error for different values of γ0, B, n and fixed η = 0.28, θ =0.01,m = 1.



γ0 m LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 1 0.0001 0.0021 0.0021 0.0007 0.0012 0.0015 0.0021 0.00345 0.0001 0.0019 0.0019 0.0006 0.0011 0.0015 0.0019 0.003210 0.0001 0.0019 0.0019 0.0006 0.0011 0.0014 0.0019 0.0032

0.1 1 0.0004 0.0082 0.0082 0.0026 0.0047 0.0062 0.0082 0.01385 0.0003 0.0078 0.0078 0.0025 0.0044 0.0058 0.0078 0.013010 0.0003 0.0077 0.0077 0.0025 0.0044 0.0058 0.0077 0.0129

0.2 1 0.0014 0.0330 0.0330 0.0104 0.0188 0.0246 0.0330 0.05605 0.0014 0.0311 0.0311 0.0098 0.0177 0.0231 0.0311 0.052710 0.0013 0.0308 0.0308 0.0097 0.0175 0.0230 0.0308 0.0523

Upward CV chartγ0 m UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 1 0.0100 0.0020 0.0020 0.0047 0.0033 0.0027 0.0020 0.0011

5 0.0094 0.0019 0.0019 0.0045 0.0031 0.0025 0.0019 0.001010 0.0093 0.0019 0.0019 0.0044 0.0031 0.0025 0.0019 0.0010

0.1 1 0.0408 0.0082 0.0082 0.0191 0.0132 0.0107 0.0082 0.00445 0.0384 0.0077 0.0077 0.0179 0.0124 0.0101 0.0077 0.004210 0.0381 0.0077 0.0077 0.0178 0.0123 0.0100 0.0077 0.0041

0.2 1 0.1771 0.0328 0.0328 0.0784 0.0534 0.0431 0.0328 0.01765 0.1657 0.0309 0.0309 0.0736 0.0503 0.0406 0.0309 0.016610 0.1643 0.0307 0.0307 0.0731 0.0499 0.0403 0.0307 0.0164


γ0 m LCL LWL(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

0.05 1 0.0006 0.0023 0.0023 0.0014 0.0018 0.0020 0.0023 0.00315 0.0006 0.0022 0.0022 0.0013 0.0017 0.0019 0.0022 0.002910 0.0006 0.0022 0.0022 0.0013 0.0017 0.0019 0.0022 0.0029

0.1 1 0.0025 0.0093 0.0093 0.0055 0.0071 0.0081 0.0093 0.01235 0.0024 0.0088 0.0088 0.0051 0.0067 0.0076 0.0088 0.011610 0.0024 0.0087 0.0087 0.0051 0.0066 0.0075 0.0087 0.0115

0.2 1 0.0099 0.0374 0.0374 0.0216 0.0283 0.0322 0.0374 0.04955 0.0093 0.0352 0.0352 0.0204 0.0266 0.0303 0.0352 0.046610 0.0093 0.0349 0.0349 0.0202 0.0264 0.0301 0.0349 0.0462

Upward CV chartγ0 m UCL UWL

(0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)0.05 1 0.0058 0.0023 0.0023 0.0037 0.0030 0.0027 0.0023 0.0017

5 0.0055 0.0022 0.0022 0.0035 0.0028 0.0025 0.0022 0.001610 0.0054 0.0022 0.0022 0.0034 0.0028 0.0025 0.0022 0.0016

0.1 1 0.0235 0.0093 0.0093 0.0147 0.0120 0.0107 0.0093 0.00695 0.0221 0.0088 0.0088 0.0139 0.0113 0.0101 0.0088 0.006510 0.0219 0.0087 0.0087 0.0138 0.0112 0.0100 0.0087 0.0064

0.2 1 0.0978 0.0373 0.0373 0.0598 0.0482 0.0430 0.0373 0.02745 0.0919 0.0351 0.0351 0.0563 0.0454 0.0405 0.0351 0.025810 0.0911 0.0348 0.0348 0.0558 0.0450 0.0401 0.0348 0.0257

Table 7: Control limits (boldfaced) and warning limits of VSI SH-γ2 control charts inthe presence of measurement error for different values of γ0,m, n and fixed η = 0.28, θ =0.01, B = 1.

24 Huu Du Nguyen et al.F

SI

(0.5

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29.5

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(15.0

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8)

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6)

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9)

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146.9

5,

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7)

(126.5

6,

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5,

113.7

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2)

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9,

37.4

6,

39.0

0)

(35.2

3,

35.5

9,

37.0

8)

(33.2

1,

33.5

6,

35.0

0)

(30.0

0,

30.3

3,

31.6

8)

1.5

(9.2

5,

9.3

9,

9.9

6)

(6.5

4,

6.6

5,

7.1

0)

(5.4

5,

5.5

5,

5.9

5)

(6.5

0,

6.6

1,

7.0

9)

(5.3

5,

5.4

5,

5.8

7)

(4.8

6,

4.9

5,

5.3

3)

(4.3

7,

4.4

5,

4.8

0)

(3.6

7,

3.7

4,

4.0

5)

τη

=0.5

0.5

(29.3

6,

29.5

6,

30.3

5)

(15.0

5,

15.1

6,

15.6

0)

(9.3

3,

9.4

0,

9.7

0)

(14.8

4,

15.0

0,

15.6

4)

(7.2

4,

7.3

4,

7.7

2)

(4.9

7,

5.0

4,

5.3

0)

(3.6

1,

3.6

4,

3.8

0)

(2.9

7,

2.9

9,

3.0

7)

0.8

(162.0

6,

162.6

1,

164.7

4)

(125.3

4,

125.9

0,

128.1

3)

(110.6

5,

111.2

2,

113.4

9)

(146.5

3,

147.1

1,

149.3

7)

(126.6

0,

127.2

0,

129.5

4)

(113.3

0,

113.8

9,

116.2

4)

(95.9

6,

96.5

4,

98.8

4)

(62.0

2,

62.5

1,

64.4

7)

0.9

(250.6

2,

251.0

7,

252.8

0)

(223.6

5,

224.1

7,

226.2

1)

(212.8

6,

213.4

1,

215.5

7)

(240.9

9,

241.4

7,

243.3

3)

(227.1

3,

227.6

5,

229.6

8)

(216.8

8,

217.4

3,

219.5

7)

(202.0

7,

202.6

5,

204.9

3)

(165.9

0,

166.5

5,

169.0

7)

0.9

5(3

06.2

4,

306.5

2,

307.6

0)

(290.3

4,

290.6

8,

292.0

0)

(283.9

7,

284.3

4,

285.7

7)

(300.9

3,

301.2

3,

302.4

0)

(292.9

5,

293.2

9,

294.5

8)

(286.8

2,

287.1

8,

288.5

7)

(277.6

1,

278.0

0,

279.5

3)

(253.0

7,

253.5

5,

255.4

1)

1.0

5(1

95.8

9,

196.5

4,

199.1

9)

(186.7

1,

187.3

7,

190.0

8)

(183.0

4,

183.7

0,

186.4

4)

(190.1

0,

190.7

6,

193.4

6)

(185.2

4,

185.9

0,

188.6

4)

(182.5

3,

183.2

0,

185.9

4)

(179.3

6,

180.0

4,

182.8

0)

(173.8

7,

174.5

5,

177.3

3)

1.1

(113.3

0,

113.9

9,

116.8

4)

(103.2

0,

103.8

7,

106.6

9)

(99.1

5,

99.8

3,

102.6

2)

(106.3

8,

107.0

7,

109.9

2)

(101.1

2,

101.8

1,

104.6

3)

(98.3

1,

98.9

9,

101.7

9)

(95.1

1,

95.7

8,

98.5

6)

(89.7

4,

90.4

0,

93.1

3)

1.2

(46.7

4,

47.2

3,

49.2

8)

(39.2

4,

39.6

8,

41.5

7)

(36.2

3,

36.6

7,

38.4

8)

(40.8

5,

41.3

1,

43.2

8)

(37.1

1,

37.5

6,

39.4

3)

(35.2

5,

35.6

9,

37.5

0)

(33.2

3,

33.6

5,

35.4

0)

(30.0

2,

30.4

2,

32.0

6)

1.5

(9.2

6,

9.4

2,

10.1

3)

(6.5

5,

6.6

7,

7.2

3)

(5.4

6,

5.5

8,

6.0

6)

(6.5

1,

6.6

4,

7.2

2)

(5.3

6,

5.4

8,

5.9

8)

(4.8

7,

4.9

8,

5.4

4)

(4.3

7,

4.4

8,

4.9

0)

(3.6

8,

3.7

6,

4.1

3)

τη

=1

0.5

(29.4

0,

29.7

2,

30.9

8)

(15.0

8,

15.2

5,

15.9

5)

(9.3

4,

9.4

6,

9.9

4)

(14.8

7,

15.1

3,

16.1

5)

(7.2

6,

7.4

1,

8.0

3)

(4.9

9,

5.0

9,

5.5

1)

(3.6

1,

3.6

7,

3.9

3)

(2.9

7,

3.0

1,

3.1

4)

0.8

(162.1

7,

163.0

4,

166.3

9)

(125.4

5,

126.3

6,

129.8

7)

(110.7

6,

111.6

8,

115.2

6)

(146.6

5,

147.5

7,

151.1

1)

(126.7

2,

127.6

8,

131.3

5)

(113.4

2,

114.3

7,

118.0

7)

(96.0

7,

97.0

1,

100.6

5)

(62.1

2,

62.9

0,

66.0

4)

0.9

(250.7

1,

251.4

2,

254.1

2)

(223.7

5,

224.5

9,

227.7

7)

(212.9

7,

213.8

5,

217.2

3)

(241.0

8,

241.8

5,

244.7

5)

(227.2

3,

228.0

6,

231.2

3)

(216.9

9,

217.8

6,

221.2

1)

(202.1

8,

203.1

2,

206.6

9)

(166.0

3,

167.0

6,

171.0

4)

0.9

5(3

06.3

0,

306.7

4,

308.4

2)

(290.4

0,

290.9

5,

293.0

2)

(284.0

5,

284.6

3,

286.8

5)

(300.9

9,

301.4

7,

303.2

8)

(293.0

2,

293.5

5,

295.5

7)

(286.9

0,

287.4

7,

289.6

3)

(277.6

9,

278.3

1,

280.6

9)

(253.1

7,

253.9

2,

256.8

5)

1.0

5(1

96.0

2,

197.0

5,

201.4

0)

(186.8

4,

187.9

0,

192.3

5)

(183.1

7,

184.2

4,

188.7

2)

(190.2

3,

191.2

9,

195.7

2)

(185.3

7,

186.4

4,

190.9

2)

(182.6

6,

183.7

4,

188.2

3)

(179.5

0,

180.5

8,

185.1

0)

(174.0

0,

175.1

0,

179.6

4)

1.1

(113.4

4,

114.5

4,

119.2

5)

(103.3

3,

104.4

2,

109.0

5)

(99.2

9,

100.3

8,

104.9

7)

(106.5

2,

107.6

3,

112.3

3)

(101.2

6,

102.3

6,

107.0

1)

(98.4

5,

99.5

4,

104.1

5)

(95.2

5,

96.3

3,

100.8

9)

(89.8

8,

90.9

4,

95.4

1)

1.2

(46.8

4,

47.6

3,

51.0

4)

(39.3

2,

40.0

5,

43.1

8)

(36.3

2,

37.0

2,

40.0

4)

(40.9

4,

41.6

9,

44.9

5)

(37.2

0,

37.9

2,

41.0

2)

(35.3

4,

36.0

4,

39.0

5)

(33.3

1,

33.9

9,

36.8

9)

(30.1

0,

30.7

3,

33.4

5)

1.5

(9.2

9,

9.5

6,

10.7

4)

(6.5

7,

6.7

8,

7.7

1)

(5.4

8,

5.6

7,

6.4

9)

(6.5

3,

6.7

5,

7.7

3)

(5.3

8,

5.5

8,

6.4

2)

(4.8

9,

5.0

7,

5.8

5)

(4.3

9,

4.5

6,

5.2

8)

(3.6

9,

3.8

4,

4.4

6)

Table 8: The out-of-control ARL1 of the FSI SH-γ chart and the out-of-control ATS1 ofthe VSI SH-γ2 charts in the presence of measurement errors for different values of η, τ, n,fixed θ = 0.05, B = 1,m = 1, γ0 = 0.05 (left side), γ0 = 0.1 (middle), γ0 = 0.2 (right side)and sample size n = 5; the first row shows the values of (hS , hL).

Title Suppressed Due to Excessive Length 25F

SI

(0.5

,1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

η=

00.5

(1.8

6,

1.8

8,

1.9

4)

(0.9

3,

0.9

4,

0.9

7)

(0.5

6,

0.5

6,

0.5

8)

(0.2

2,

0.2

2,

0.2

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

0.8

(42.1

9,

42.5

9,

44.1

7)

(26.1

2,

26.4

2,

27.6

1)

(19.6

9,

19.9

5,

20.9

8)

(30.4

0,

30.7

6,

32.2

2)

(21.7

2,

22.0

3,

23.2

6)

(17.5

3,

17.8

0,

18.8

7)

(13.2

7,

13.4

8,

14.3

6)

(7.6

5,

7.7

7,

8.2

9)

0.9

(127.8

4,

128.5

3,

131.2

3)

(101.7

7,

102.4

6,

105.1

2)

(91.3

5,

92.0

3,

94.6

7)

(114.8

4,

115.5

5,

118.3

1)

(100.8

3,

101.5

4,

104.3

0)

(92.0

6,

92.7

5,

95.4

7)

(80.9

2,

81.6

0,

84.2

3)

(58.9

1,

59.5

0,

61.8

2)

0.9

5(2

19.0

1,

219.6

4,

222.0

6)

(196.8

7,

197.5

6,

200.2

1)

(188.0

2,

188.7

3,

191.4

7)

(209.6

3,

210.2

9,

212.8

3)

(198.0

7,

198.7

6,

201.4

3)

(190.1

2,

190.8

3,

193.5

8)

(179.1

6,

179.9

0,

182.7

4)

(153.7

8,

154.5

5,

157.5

2)

1.0

5(1

42.2

5,

143.0

3,

146.1

4)

(129.1

7,

129.9

6,

133.0

9)

(123.9

4,

124.7

3,

127.8

7)

(134.4

7,

135.2

6,

138.4

1)

(127.6

0,

128.3

9,

131.5

5)

(123.6

0,

124.3

9,

127.5

5)

(118.7

1,

119.5

0,

122.6

4)

(109.3

0,

110.0

8,

113.1

8)

1.1

(64.2

8,

64.9

1,

67.4

5)

(53.2

5,

53.8

4,

56.2

0)

(48.8

4,

49.4

1,

51.7

0)

(56.6

8,

57.2

9,

59.7

6)

(51.0

2,

51.6

1,

53.9

8)

(47.9

7,

48.5

4,

50.8

4)

(44.4

3,

44.9

8,

47.2

0)

(38.1

5,

38.6

6,

40.6

9)

1.2

(19.0

8,

19.3

7,

20.5

9)

(13.5

2,

13.7

6,

14.7

4)

(11.3

0,

11.5

2,

12.4

0)

(14.2

2,

14.4

8,

15.5

6)

(11.6

1,

11.8

4,

12.7

8)

(10.3

8,

10.5

9,

11.4

6)

(9.0

8,

9.2

7,

10.0

6)

(7.0

6,

7.2

2,

7.8

6)

1.5

(2.9

0,

2.9

6,

3.1

9)

(1.6

2,

1.6

6,

1.8

1)

(1.1

1,

1.1

4,

1.2

5)

(1.2

8,

1.3

1,

1.4

6)

(0.8

7,

0.9

0,

1.0

0)

(0.7

3,

0.7

5,

0.8

4)

(0.6

0,

0.6

2,

0.7

0)

(0.4

6,

0.4

7,

0.5

2)

τη

=0.2

0.5

(1.8

6,

1.8

8,

1.9

4)

(0.9

3,

0.9

4,

0.9

7)

(0.5

6,

0.5

6,

0.5

8)

(0.2

2,

0.2

2,

0.2

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

0.8

(42.2

0,

42.6

1,

44.2

6)

(26.1

3,

26.4

4,

27.6

7)

(19.7

0,

19.9

7,

21.0

4)

(30.4

0,

30.7

8,

32.3

0)

(21.7

3,

22.0

5,

23.3

3)

(17.5

3,

17.8

1,

18.9

3)

(13.2

7,

13.4

9,

14.4

0)

(7.6

5,

7.7

8,

8.3

2)

0.9

(127.8

5,

128.5

7,

131.3

7)

(101.7

8,

102.4

9,

105.2

6)

(91.3

6,

92.0

6,

94.8

1)

(114.8

5,

115.5

9,

118.4

6)

(100.8

4,

101.5

8,

104.4

4)

(92.0

7,

92.7

9,

95.6

2)

(80.9

3,

81.6

3,

84.3

7)

(58.9

1,

59.5

3,

61.9

4)

0.9

5(2

19.0

2,

219.6

7,

222.1

8)

(196.8

8,

197.6

0,

200.3

5)

(188.0

3,

188.7

7,

191.6

1)

(209.6

4,

210.3

3,

212.9

6)

(198.0

8,

198.8

0,

201.5

7)

(190.1

3,

190.8

7,

193.7

2)

(179.1

7,

179.9

4,

182.8

8)

(153.7

9,

154.5

9,

157.6

7)

1.0

5(1

42.2

6,

143.0

7,

146.3

1)

(129.1

8,

130.0

0,

133.2

5)

(123.9

5,

124.7

7,

128.0

3)

(134.4

8,

135.3

0,

138.5

8)

(127.6

1,

128.4

4,

131.7

2)

(123.6

1,

124.4

3,

127.7

1)

(118.7

2,

119.5

4,

122.8

1)

(109.3

1,

110.1

3,

113.3

5)

1.1

(64.2

9,

64.9

5,

67.5

8)

(53.2

6,

53.8

7,

56.3

2)

(48.8

5,

49.4

4,

51.8

2)

(56.6

9,

57.3

3,

59.9

0)

(51.0

3,

51.6

4,

54.1

0)

(47.9

7,

48.5

7,

50.9

6)

(44.4

4,

45.0

1,

47.3

2)

(38.1

6,

38.6

8,

40.8

0)

1.2

(19.0

8,

19.3

9,

20.6

6)

(13.5

2,

13.7

7,

14.7

9)

(11.3

0,

11.5

3,

12.4

5)

(14.2

2,

14.5

0,

15.6

1)

(11.6

1,

11.8

5,

12.8

3)

(10.3

8,

10.6

0,

11.5

1)

(9.0

8,

9.2

8,

10.1

0)

(7.0

6,

7.2

3,

7.9

0)

1.5

(2.9

0,

2.9

6,

3.2

1)

(1.6

2,

1.6

6,

1.8

1)

(1.1

1,

1.1

4,

1.2

6)

(1.2

8,

1.3

1,

1.4

7)

(0.8

7,

0.9

0,

1.0

1)

(0.7

3,

0.7

5,

0.8

5)

(0.6

0,

0.6

2,

0.7

0)

(0.4

6,

0.4

7,

0.5

3)

τη

=0.5

0.5

(1.8

6,

1.8

8,

1.9

6)

(0.9

3,

0.9

4,

0.9

8)

(0.5

6,

0.5

6,

0.5

9)

(0.2

2,

0.2

2,

0.2

4)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

0.8

(42.2

3,

42.7

3,

44.6

9)

(26.1

5,

26.5

2,

28.0

0)

(19.7

1,

20.0

4,

21.3

3)

(30.4

3,

30.8

9,

32.7

0)

(21.7

5,

22.1

4,

23.6

7)

(17.5

5,

17.8

9,

19.2

3)

(13.2

8,

13.5

6,

14.6

5)

(7.6

6,

7.8

2,

8.4

7)

0.9

(127.8

9,

128.7

6,

132.1

0)

(101.8

3,

102.6

8,

105.9

8)

(91.4

0,

92.2

5,

95.5

3)

(114.9

0,

115.7

9,

119.2

0)

(100.8

9,

101.7

7,

105.1

9)

(92.1

1,

92.9

8,

96.3

6)

(80.9

8,

81.8

2,

85.0

9)

(58.9

6,

59.6

9,

62.5

8)

0.9

5(2

19.0

6,

219.8

5,

222.8

3)

(196.9

3,

197.7

9,

201.0

6)

(188.0

8,

188.9

6,

192.3

5)

(209.6

9,

210.5

1,

213.6

5)

(198.1

3,

198.9

9,

202.2

9)

(190.1

8,

191.0

7,

194.4

6)

(179.2

2,

180.1

4,

183.6

5)

(153.8

4,

154.8

0,

158.4

8)

1.0

5(1

42.3

1,

143.2

9,

147.1

7)

(129.2

4,

130.2

2,

134.1

2)

(124.0

1,

124.9

9,

128.9

0)

(134.5

3,

135.5

2,

139.4

5)

(127.6

6,

128.6

6,

132.6

0)

(123.6

6,

124.6

6,

128.5

9)

(118.7

8,

119.7

7,

123.6

8)

(109.3

7,

110.3

4,

114.2

0)

1.1

(64.3

3,

65.1

2,

68.3

0)

(53.3

0,

54.0

4,

56.9

9)

(48.8

9,

49.6

0,

52.4

6)

(56.7

3,

57.5

0,

60.5

9)

(51.0

7,

51.8

1,

54.7

7)

(48.0

1,

48.7

3,

51.6

1)

(44.4

8,

45.1

7,

47.9

4)

(38.1

9,

38.8

3,

41.3

7)

1.2

(19.1

0,

19.4

7,

21.0

1)

(13.5

4,

13.8

4,

15.0

7)

(11.3

2,

11.5

9,

12.7

0)

(14.2

4,

14.5

7,

15.9

2)

(11.6

2,

11.9

1,

13.1

0)

(10.3

9,

10.6

6,

11.7

6)

(9.0

9,

9.3

3,

10.3

3)

(7.0

7,

7.2

7,

8.0

9)

1.5

(2.9

0,

2.9

8,

3.2

7)

(1.6

3,

1.6

7,

1.8

6)

(1.1

2,

1.1

5,

1.2

9)

(1.2

8,

1.3

2,

1.5

1)

(0.8

7,

0.9

1,

1.0

4)

(0.7

3,

0.7

6,

0.8

7)

(0.6

1,

0.6

3,

0.7

2)

(0.4

6,

0.4

7,

0.5

4)

τη

=1

0.5

(1.8

7,

1.9

0,

2.0

3)

(0.9

3,

0.9

5,

1.0

1)

(0.5

6,

0.5

7,

0.6

1)

(0.2

2,

0.2

3,

0.2

5)

(0.1

9,

0.1

9,

0.2

1)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

0.8

(42.3

3,

43.1

2,

46.2

2)

(26.2

2,

26.8

2,

29.1

7)

(19.7

8,

20.3

0,

22.3

5)

(30.5

2,

31.2

5,

34.1

2)

(21.8

3,

22.4

4,

24.8

8)

(17.6

2,

18.1

5,

20.2

9)

(13.3

4,

13.7

7,

15.5

3)

(7.6

9,

7.9

5,

9.0

0)

0.9

(128.0

7,

129.4

4,

134.6

2)

(102.0

0,

103.3

6,

108.5

0)

(91.5

7,

92.9

2,

98.0

5)

(115.0

8,

116.4

9,

121.8

0)

(101.0

7,

102.4

7,

107.7

9)

(92.2

9,

93.6

7,

98.9

4)

(81.1

5,

82.4

8,

87.6

0)

(59.1

0,

60.2

8,

64.8

2)

0.9

5(2

19.2

2,

220.4

6,

225.0

5)

(197.1

0,

198.4

6,

203.5

1)

(188.2

6,

189.6

6,

194.9

0)

(209.8

5,

211.1

6,

215.9

9)

(198.3

0,

199.6

7,

204.7

6)

(190.3

6,

191.7

7,

197.0

1)

(179.4

1,

180.8

6,

186.2

8)

(154.0

4,

155.5

5,

161.2

7)

1.0

5(1

42.5

1,

144.0

7,

150.2

3)

(129.4

3,

131.0

1,

137.1

9)

(124.2

0,

125.7

8,

131.9

8)

(134.7

3,

136.3

2,

142.5

5)

(127.8

6,

129.4

5,

135.7

0)

(123.8

6,

125.4

5,

131.6

8)

(118.9

7,

120.5

5,

126.7

6)

(109.5

6,

111.1

2,

117.2

4)

1.1

(64.4

9,

65.7

6,

70.8

4)

(53.4

5,

54.6

2,

59.3

5)

(49.0

3,

50.1

7,

54.7

6)

(56.8

8,

58.1

2,

63.0

7)

(51.2

2,

52.4

0,

57.1

5)

(48.1

6,

49.3

0,

53.9

2)

(44.6

2,

45.7

2,

50.1

6)

(38.3

2,

39.3

3,

43.4

1)

1.2

(19.1

7,

19.7

8,

22.2

7)

(13.6

0,

14.0

8,

16.0

9)

(11.3

7,

11.8

1,

13.6

2)

(14.3

1,

14.8

4,

17.0

3)

(11.6

8,

12.1

5,

14.0

8)

(10.4

5,

10.8

8,

12.6

7)

(9.1

4,

9.5

3,

11.1

6)

(7.1

1,

7.4

3,

8.7

7)

1.5

(2.9

2,

3.0

3,

3.5

3)

(1.6

4,

1.7

1,

2.0

2)

(1.1

2,

1.1

8,

1.4

1)

(1.2

9,

1.3

6,

1.6

7)

(0.8

8,

0.9

3,

1.1

6)

(0.7

4,

0.7

8,

0.9

7)

(0.6

1,

0.6

5,

0.8

1)

(0.4

6,

0.4

9,

0.6

0)

Table 9: The out-of-control ARL1 of the FSI SH-γ chart and the out-of-control ATS1 ofthe VSI SH-γ2 charts in the presence of measurement errors for different values of η, τ, n,fixed θ = 0.05, B = 1,m = 1, γ0 = 0.05 (left side), γ0 = 0.1 (middle), γ0 = 0.2 (right side)and sample size n = 15; the first row shows the values of (hS , hL).

26 Huu Du Nguyen et al.n

=5

FS

I(0

.5,

1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

θ=

00.5

(26.9

1,

27.0

9,

27.7

8)

(13.7

1,

13.8

1,

14.1

9)

(8.4

4,

8.5

0,

8.7

5)

(12.9

1,

13.0

4,

13.5

9)

(6.1

2,

6.2

0,

6.5

1)

(4.2

3,

4.2

8,

4.4

9)

(3.1

6,

3.1

9,

3.3

1)

(2.7

1,

2.7

3,

2.8

0)

0.8

(156.2

0,

156.7

2,

158.7

5)

(119.3

0,

119.8

4,

121.9

3)

(104.5

5,

105.0

8,

107.2

0)

(140.3

5,

140.9

0,

143.0

4)

(120.2

4,

120.8

0,

123.0

1)

(106.9

4,

107.5

0,

109.7

1)

(89.7

9,

90.3

3,

92.4

7)

(56.9

0,

57.3

5,

59.1

3)

0.9

(246.0

3,

246.4

6,

248.1

5)

(218.2

8,

218.7

8,

220.7

6)

(207.1

8,

207.7

1,

209.8

1)

(236.0

5,

236.5

1,

238.3

3)

(221.7

6,

222.2

7,

224.2

4)

(211.2

3,

211.7

6,

213.8

4)

(196.0

8,

196.6

4,

198.8

5)

(159.3

9,

160.0

0,

162.4

2)

0.9

5(3

03.4

2,

303.6

9,

304.7

6)

(286.8

8,

287.2

2,

288.5

2)

(280.2

6,

280.6

2,

282.0

3)

(297.8

8,

298.1

7,

299.3

3)

(289.5

8,

289.9

0,

291.1

8)

(283.2

1,

283.5

6,

284.9

3)

(273.6

5,

274.0

3,

275.5

4)

(248.2

9,

248.7

6,

250.5

8)

1.0

5(1

89.9

2,

190.5

4,

193.1

0)

(180.5

7,

181.2

1,

183.8

2)

(176.8

3,

177.4

8,

180.1

1)

(184.0

0,

184.6

3,

187.2

4)

(179.0

5,

179.6

9,

182.3

3)

(176.3

0,

176.9

5,

179.5

9)

(173.1

0,

173.7

4,

176.4

0)

(167.5

4,

168.1

9,

170.8

6)

1.1

(107.1

5,

107.7

9,

110.4

7)

(97.1

2,

97.7

5,

100.3

9)

(93.1

1,

93.7

4,

96.3

5)

(100.2

2,

100.8

7,

103.5

4)

(95.0

2,

95.6

5,

98.3

0)

(92.2

4,

92.8

8,

95.5

0)

(89.1

0,

89.7

2,

92.3

2)

(83.8

4,

84.4

5,

86.9

9)

1.2

(42.6

4,

43.0

8,

44.9

3)

(35.4

7,

35.8

7,

37.5

6)

(32.6

0,

32.9

9,

34.6

1)

(36.9

3,

37.3

5,

39.1

1)

(33.3

8,

33.7

8,

35.4

5)

(31.6

3,

32.0

2,

33.6

3)

(29.7

3,

30.1

1,

31.6

6)

(26.7

4,

27.0

9,

28.5

4)

1.5

(8.0

7,

8.2

1,

8.8

1)

(5.6

2,

5.7

3,

6.1

9)

(4.6

4,

4.7

3,

5.1

4)

(5.5

3,

5.6

5,

6.1

3)

(4.5

1,

4.6

1,

5.0

3)

(4.0

8,

4.1

7,

4.5

6)

(3.6

6,

3.7

4,

4.0

9)

(3.0

6,

3.1

3,

3.4

3)

τθ

=0.0

30.5

(28.3

7,

28.5

4,

29.2

3)

(14.5

1,

14.6

0,

14.9

8)

(8.9

6,

9.0

3,

9.2

8)

(14.0

5,

14.1

9,

14.7

4)

(6.7

8,

6.8

6,

7.1

8)

(4.6

6,

4.7

2,

4.9

3)

(3.4

2,

3.4

5,

3.5

8)

(2.8

6,

2.8

8,

2.9

5)

0.8

(159.7

4,

160.2

3,

162.1

5)

(122.9

4,

123.4

4,

125.4

4)

(108.2

1,

108.7

3,

110.7

5)

(144.0

8,

144.6

0,

146.6

2)

(124.0

7,

124.6

1,

126.7

1)

(110.7

7,

111.3

0,

113.4

0)

(93.4

9,

94.0

1,

96.0

7)

(59.9

6,

60.3

9,

62.1

2)

0.9

(248.8

1,

249.2

2,

250.7

9)

(221.5

3,

222.0

0,

223.8

5)

(210.6

2,

211.1

2,

213.0

7)

(239.0

4,

239.4

8,

241.1

6)

(225.0

1,

225.4

8,

227.3

2)

(214.6

5,

215.1

5,

217.0

9)

(199.7

0,

200.2

3,

202.3

0)

(163.3

2,

163.9

0,

166.1

8)

0.9

5(3

05.1

3,

305.3

9,

306.3

7)

(288.9

8,

289.2

9,

290.5

0)

(282.5

1,

282.8

5,

284.1

5)

(299.7

3,

300.0

1,

301.0

7)

(291.6

3,

291.9

3,

293.1

1)

(285.4

0,

285.7

3,

287.0

0)

(276.0

5,

276.4

1,

277.8

0)

(251.1

9,

251.6

2,

253.3

2)

1.0

5(1

93.5

3,

194.1

1,

196.4

9)

(184.2

8,

184.8

8,

187.3

1)

(180.5

8,

181.1

8,

183.6

4)

(187.6

9,

188.2

8,

190.7

1)

(182.7

9,

183.3

9,

185.8

5)

(180.0

6,

180.6

7,

183.1

3)

(176.8

8,

177.4

9,

179.9

7)

(171.3

6,

171.9

7,

174.4

7)

1.1

(110.8

5,

111.4

6,

113.9

9)

(100.7

7,

101.3

7,

103.8

7)

(96.7

4,

97.3

4,

99.8

2)

(103.9

3,

104.5

4,

107.0

7)

(98.6

9,

99.2

9,

101.8

0)

(95.8

9,

96.4

9,

98.9

8)

(92.7

1,

93.3

1,

95.7

7)

(87.3

8,

87.9

7,

90.3

8)

1.2

(45.0

8,

45.5

1,

47.3

1)

(37.7

1,

38.1

0,

39.7

5)

(34.7

6,

35.1

4,

36.7

2)

(39.2

6,

39.6

7,

41.3

8)

(35.6

0,

35.9

9,

37.6

2)

(33.7

9,

34.1

6,

35.7

4)

(31.8

1,

32.1

8,

33.7

0)

(28.6

9,

29.0

3,

30.4

6)

1.5

(8.7

7,

8.9

1,

9.5

1)

(6.1

6,

6.2

7,

6.7

4)

(5.1

2,

5.2

2,

5.6

3)

(6.1

0,

6.2

2,

6.7

1)

(5.0

1,

5.1

1,

5.5

3)

(4.5

4,

4.6

3,

5.0

3)

(4.0

8,

4.1

6,

4.5

2)

(3.4

2,

3.4

9,

3.8

0)

τθ

=0.0

50.5

(29.3

5,

29.5

3,

30.2

1)

(15.0

5,

15.1

4,

15.5

2)

(9.3

3,

9.3

9,

9.6

5)

(14.8

3,

14.9

7,

15.5

2)

(7.2

4,

7.3

2,

7.6

5)

(4.9

7,

5.0

3,

5.2

5)

(3.6

0,

3.6

4,

3.7

7)

(2.9

7,

2.9

9,

3.0

6)

0.8

(162.0

4,

162.5

1,

164.3

6)

(125.3

1,

125.8

0,

127.7

3)

(110.6

2,

111.1

2,

113.0

8)

(146.5

0,

147.0

0,

148.9

6)

(126.5

8,

127.0

9,

129.1

2)

(113.2

7,

113.7

8,

115.8

2)

(95.9

3,

96.4

3,

98.4

3)

(61.9

9,

62.4

2,

64.1

2)

0.9

(250.6

0,

250.9

9,

252.4

9)

(223.6

2,

224.0

8,

225.8

4)

(212.8

3,

213.3

1,

215.1

8)

(240.9

6,

241.3

8,

242.9

9)

(227.1

0,

227.5

5,

229.3

1)

(216.8

5,

217.3

3,

219.1

8)

(202.0

4,

202.5

5,

204.5

2)

(165.8

8,

166.4

3,

168.6

1)

0.9

5(3

06.2

3,

306.4

7,

307.4

1)

(290.3

2,

290.6

2,

291.7

7)

(283.9

6,

284.2

7,

285.5

1)

(300.9

2,

301.1

8,

302.1

9)

(292.9

4,

293.2

3,

294.3

5)

(286.8

1,

287.1

2,

288.3

2)

(277.5

9,

277.9

3,

279.2

5)

(253.0

5,

253.4

6,

255.0

7)

1.0

5(1

95.8

7,

196.4

2,

198.6

9)

(186.6

8,

187.2

5,

189.5

8)

(183.0

1,

183.5

8,

185.9

3)

(190.0

7,

190.6

4,

192.9

6)

(185.2

1,

185.7

8,

188.1

3)

(182.5

0,

183.0

7,

185.4

3)

(179.3

3,

179.9

1,

182.2

9)

(173.8

4,

174.4

2,

176.8

1)

1.1

(113.2

7,

113.8

6,

116.3

0)

(103.1

7,

103.7

5,

106.1

6)

(99.1

2,

99.7

1,

102.1

0)

(106.3

5,

106.9

4,

109.3

9)

(101.0

9,

101.6

8,

104.1

0)

(98.2

8,

98.8

6,

101.2

7)

(95.0

8,

95.6

6,

98.0

4)

(89.7

1,

90.2

8,

92.6

2)

1.2

(46.7

2,

47.1

4,

48.9

0)

(39.2

1,

39.6

0,

41.2

1)

(36.2

1,

36.5

9,

38.1

4)

(40.8

3,

41.2

3,

42.9

1)

(37.0

9,

37.4

8,

39.0

8)

(35.2

4,

35.6

1,

37.1

6)

(33.2

1,

33.5

7,

35.0

7)

(30.0

0,

30.3

4,

31.7

5)

1.5

(9.2

5,

9.3

9,

9.9

9)

(6.5

4,

6.6

5,

7.1

2)

(5.4

5,

5.5

5,

5.9

7)

(6.5

0,

6.6

2,

7.1

1)

(5.3

5,

5.4

6,

5.8

9)

(4.8

6,

4.9

6,

5.3

5)

(4.3

7,

4.4

6,

4.8

2)

(3.6

7,

3.7

5,

4.0

6)

n=

15

FS

I(0

.5,

1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

θ=

00.5

(1.7

0,

1.7

2,

1.7

8)

(0.8

5,

0.8

6,

0.8

9)

(0.5

1,

0.5

2,

0.5

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

0.8

(38.6

2,

39.0

7,

40.8

3)

(23.4

9,

23.8

2,

25.1

1)

(17.4

4,

17.7

2,

18.8

3)

(27.1

4,

27.5

5,

29.1

5)

(19.0

3,

19.3

6,

20.6

9)

(15.2

0,

15.4

8,

16.6

3)

(11.3

9,

11.6

2,

12.5

4)

(6.5

6,

6.6

9,

7.2

2)

0.9

(121.7

9,

122.6

1,

125.7

6)

(95.8

7,

96.6

6,

99.7

5)

(85.5

0,

86.2

9,

89.3

4)

(108.6

6,

109.4

9,

112.7

2)

(94.7

0,

95.5

3,

98.7

3)

(86.0

4,

86.8

5,

89.9

9)

(75.1

3,

75.9

1,

78.9

4)

(53.9

0,

54.5

7,

57.1

9)

0.9

5(2

13.6

0,

214.3

6,

217.2

5)

(190.9

7,

191.8

0,

194.9

6)

(181.9

2,

182.7

7,

186.0

5)

(203.9

4,

204.7

4,

207.7

8)

(192.1

0,

192.9

4,

196.1

4)

(184.0

0,

184.8

6,

188.1

4)

(172.8

7,

173.7

5,

177.1

3)

(147.2

5,

148.1

7,

151.6

9)

1.0

5(1

35.9

6,

136.8

9,

140.5

8)

(122.8

5,

123.7

9,

127.4

8)

(117.6

1,

118.5

4,

122.2

4)

(128.1

0,

129.0

4,

132.7

7)

(121.2

2,

122.1

6,

125.8

9)

(117.2

3,

118.1

7,

121.8

9)

(112.3

7,

113.3

0,

117.0

0)

(103.0

5,

103.9

7,

107.6

0)

1.1

(59.4

2,

60.1

5,

63.0

6)

(48.7

6,

49.4

3,

52.1

1)

(44.5

0,

45.1

4,

47.7

3)

(51.9

6,

52.6

6,

55.4

8)

(46.5

1,

47.1

8,

49.8

6)

(43.5

9,

44.2

4,

46.8

4)

(40.2

3,

40.8

5,

43.3

5)

(34.3

1,

34.8

7,

37.1

5)

1.2

(16.9

4,

17.2

6,

18.6

0)

(11.8

2,

12.0

7,

13.1

3)

(9.7

7,

10.0

0,

10.9

5)

(12.3

5,

12.6

4,

13.8

0)

(9.9

8,

10.2

2,

11.2

3)

(8.8

7,

9.1

0,

10.0

3)

(7.7

2,

7.9

2,

8.7

6)

(5.9

5,

6.1

2,

6.8

0)

1.5

(2.5

5,

2.6

1,

2.8

5)

(1.4

1,

1.4

4,

1.5

9)

(0.9

5,

0.9

8,

1.0

9)

(1.0

6,

1.1

0,

1.2

4)

(0.7

2,

0.7

4,

0.8

5)

(0.6

0,

0.6

2,

0.7

1)

(0.5

0,

0.5

1,

0.5

9)

(0.3

8,

0.3

9,

0.4

4)

τθ

=0.0

30.5

(1.8

0,

1.8

1,

1.8

8)

(0.9

0,

0.9

1,

0.9

4)

(0.5

4,

0.5

4,

0.5

6)

(0.2

1,

0.2

1,

0.2

2)

(0.1

8,

0.1

8,

0.1

9)

(0.1

8,

0.1

8,

0.1

9)

(0.1

8,

0.1

8,

0.1

9)

(0.1

8,

0.1

8,

0.1

9)

0.8

(40.7

7,

41.2

0,

42.9

3)

(25.0

6,

25.3

9,

26.6

7)

(18.7

8,

19.0

6,

20.1

7)

(29.0

9,

29.4

9,

31.0

7)

(20.6

4,

20.9

7,

22.3

0)

(16.5

8,

16.8

7,

18.0

3)

(12.5

0,

12.7

4,

13.6

7)

(7.2

0,

7.3

3,

7.8

9)

0.9

(125.4

6,

126.2

3,

129.2

3)

(99.4

4,

100.2

0,

103.1

5)

(89.0

4,

89.7

9,

92.7

2)

(112.4

1,

113.2

0,

116.2

7)

(98.4

1,

99.2

0,

102.2

6)

(89.6

8,

90.4

5,

93.4

7)

(78.6

3,

79.3

8,

82.2

9)

(56.9

2,

57.5

6,

60.1

1)

0.9

5(2

16.9

0,

217.6

1,

220.3

2)

(194.5

7,

195.3

4,

198.3

1)

(185.6

3,

186.4

3,

189.5

1)

(207.4

1,

208.1

6,

211.0

1)

(195.7

4,

196.5

2,

199.5

2)

(187.7

3,

188.5

3,

191.6

1)

(176.7

0,

177.5

3,

180.7

1)

(151.2

2,

152.0

8,

155.4

0)

1.0

5(1

39.7

8,

140.6

6,

144.1

4)

(126.6

9,

127.5

7,

131.0

7)

(121.4

5,

122.3

3,

125.8

4)

(131.9

7,

132.8

5,

136.3

8)

(125.0

9,

125.9

8,

129.5

1)

(121.0

9,

121.9

8,

125.5

1)

(116.2

1,

117.1

0,

120.6

1)

(106.8

4,

107.7

1,

111.1

7)

1.1

(62.3

5,

63.0

5,

65.8

5)

(51.4

7,

52.1

1,

54.7

1)

(47.1

1,

47.7

4,

50.2

5)

(54.8

0,

55.4

8,

58.2

1)

(49.2

2,

49.8

7,

52.4

8)

(46.2

2,

46.8

5,

49.3

8)

(42.7

6,

43.3

6,

45.7

9)

(36.6

1,

37.1

7,

39.3

9)

1.2

(18.2

1,

18.5

4,

19.8

6)

(12.8

3,

13.0

9,

14.1

5)

(10.6

8,

10.9

1,

11.8

6)

(13.4

7,

13.7

5,

14.9

1)

(10.9

4,

11.1

9,

12.2

1)

(9.7

6,

9.9

9,

10.9

3)

(8.5

2,

8.7

3,

9.5

8)

(6.6

1,

6.7

8,

7.4

7)

1.5

(2.7

6,

2.8

2,

3.0

7)

(1.5

4,

1.5

7,

1.7

3)

(1.0

5,

1.0

7,

1.1

9)

(1.1

9,

1.2

2,

1.3

8)

(0.8

1,

0.8

3,

0.9

5)

(0.6

7,

0.7

0,

0.7

9)

(0.5

6,

0.5

8,

0.6

6)

(0.4

2,

0.4

4,

0.4

9)

τθ

=0.0

50.5

(1.8

6,

1.8

8,

1.9

5)

(0.9

3,

0.9

4,

0.9

7)

(0.5

6,

0.5

6,

0.5

8)

(0.2

2,

0.2

2,

0.2

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

0.8

(42.2

0,

42.6

4,

44.3

4)

(26.1

3,

26.4

5,

27.7

3)

(19.7

0,

19.9

8,

21.0

9)

(30.4

1,

30.8

0,

32.3

7)

(21.7

3,

22.0

7,

23.3

9)

(17.5

4,

17.8

2,

18.9

8)

(13.2

7,

13.5

1,

14.4

5)

(7.6

5,

7.7

9,

8.3

5)

0.9

(127.8

5,

128.6

0,

131.5

0)

(101.7

9,

102.5

3,

105.3

9)

(91.3

6,

92.1

0,

94.9

5)

(114.8

6,

115.6

2,

118.5

9)

(100.8

5,

101.6

1,

104.5

8)

(92.0

7,

92.8

3,

95.7

5)

(80.9

4,

81.6

7,

84.5

0)

(58.9

2,

59.5

6,

62.0

6)

0.9

5(2

19.0

3,

219.7

1,

222.3

0)

(196.8

9,

197.6

3,

200.4

8)

(188.0

4,

188.8

0,

191.7

5)

(209.6

5,

210.3

6,

213.0

9)

(198.0

9,

198.8

3,

201.7

0)

(190.1

4,

190.9

1,

193.8

6)

(179.1

8,

179.9

7,

183.0

2)

(153.8

0,

154.6

2,

157.8

2)

1.0

5(1

42.2

7,

143.1

1,

146.4

6)

(129.1

9,

130.0

4,

133.4

1)

(123.9

6,

124.8

1,

128.1

9)

(134.4

9,

135.3

4,

138.7

4)

(127.6

2,

128.4

8,

131.8

8)

(123.6

2,

124.4

7,

127.8

7)

(118.7

3,

119.5

8,

122.9

7)

(109.3

2,

110.1

6,

113.5

0)

1.1

(64.3

0,

64.9

8,

67.7

1)

(53.2

7,

53.9

0,

56.4

4)

(48.8

6,

49.4

7,

51.9

4)

(56.7

0,

57.3

6,

60.0

2)

(51.0

3,

51.6

7,

54.2

2)

(47.9

8,

48.6

0,

51.0

8)

(44.4

5,

45.0

4,

47.4

3)

(38.1

6,

38.7

1,

40.9

0)

1.2

(19.0

8,

19.4

1,

20.7

2)

(13.5

3,

13.7

9,

14.8

5)

(11.3

0,

11.5

4,

12.4

9)

(14.2

3,

14.5

1,

15.6

7)

(11.6

1,

11.8

6,

12.8

8)

(10.3

8,

10.6

1,

11.5

5)

(9.0

8,

9.2

9,

10.1

4)

(7.0

6,

7.2

3,

7.9

3)

1.5

(2.9

0,

2.9

6,

3.2

2)

(1.6

2,

1.6

6,

1.8

2)

(1.1

1,

1.1

4,

1.2

6)

(1.2

8,

1.3

2,

1.4

8)

(0.8

7,

0.9

0,

1.0

2)

(0.7

3,

0.7

5,

0.8

5)

(0.6

0,

0.6

2,

0.7

1)

(0.4

6,

0.4

7,

0.5

3)

Table 10: The out-of-control ARL1 of the FSI SH-γ chart and the out-of-control ATS1 ofthe VSI SH-γ2 charts in the presence of measurement errors for different values of θ, τ, n,fixed η = 0.28, B = 1,m = 1, γ0 = 0.05 (left side), γ0 = 0.1 (middle) and γ0 = 0.2 (rightside); the first row shows the values of (hS , hL).

Title Suppressed Due to Excessive Length 27n

=5

FS

I(0

.5,

1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

B=

20.5

(27.1

5,

27.3

1,

27.9

7)

(13.8

4,

13.9

3,

14.2

9)

(8.5

2,

8.5

8,

8.8

2)

(13.0

9,

13.2

2,

13.7

4)

(6.2

2,

6.3

0,

6.6

0)

(4.3

0,

4.3

5,

4.5

4)

(3.2

0,

3.2

3,

3.3

5)

(2.7

3,

2.7

5,

2.8

2)

0.8

(156.7

9,

157.2

8,

159.1

8)

(119.9

1,

120.4

1,

122.3

7)

(105.1

5,

105.6

6,

107.6

5)

(140.9

7,

141.4

8,

143.4

9)

(120.8

8,

121.4

0,

123.4

8)

(107.5

7,

108.1

0,

110.1

7)

(90.4

0,

90.9

1,

92.9

2)

(57.4

0,

57.8

2,

59.5

0)

0.9

(246.4

9,

246.9

0,

248.4

8)

(218.8

2,

219.2

9,

221.1

4)

(207.7

5,

208.2

5,

210.2

1)

(236.5

5,

236.9

8,

238.6

8)

(222.3

0,

222.7

8,

224.6

2)

(211.8

0,

212.3

0,

214.2

5)

(196.6

8,

197.2

1,

199.2

8)

(160.0

4,

160.6

2,

162.8

8)

0.9

5(3

03.7

0,

303.9

6,

304.9

6)

(287.2

3,

287.5

4,

288.7

7)

(280.6

4,

280.9

8,

282.2

9)

(298.1

9,

298.4

7,

299.5

4)

(289.9

2,

290.2

3,

291.4

2)

(283.5

7,

283.9

0,

285.1

8)

(274.0

5,

274.4

1,

275.8

1)

(248.7

7,

249.2

1,

250.9

2)

1.0

5(1

90.5

2,

191.1

0,

193.4

9)

(181.1

9,

181.7

8,

184.2

2)

(177.4

6,

178.0

6,

180.5

2)

(184.6

2,

185.2

1,

187.6

4)

(179.6

7,

180.2

7,

182.7

3)

(176.9

3,

177.5

3,

180.0

0)

(173.7

2,

174.3

3,

176.8

1)

(168.1

7,

168.7

8,

171.2

8)

1.1

(107.7

6,

108.3

6,

110.8

7)

(97.7

2,

98.3

2,

100.7

8)

(93.7

1,

94.3

0,

96.7

4)

(100.8

3,

101.4

4,

103.9

4)

(95.6

2,

96.2

2,

98.6

9)

(92.8

4,

93.4

4,

95.8

9)

(89.6

9,

90.2

8,

92.7

0)

(84.4

2,

85.0

0,

87.3

7)

1.2

(43.0

4,

43.4

5,

45.1

9)

(35.8

3,

36.2

1,

37.8

0)

(32.9

5,

33.3

2,

34.8

4)

(37.3

1,

37.7

0,

39.3

5)

(33.7

4,

34.1

2,

35.6

9)

(31.9

8,

32.3

5,

33.8

6)

(30.0

7,

30.4

2,

31.8

8)

(27.0

6,

27.3

9,

28.7

5)

1.5

(8.1

9,

8.3

2,

8.8

8)

(5.7

1,

5.8

1,

6.2

4)

(4.7

1,

4.8

1,

5.1

9)

(5.6

2,

5.7

3,

6.1

9)

(4.5

9,

4.6

9,

5.0

8)

(4.1

6,

4.2

4,

4.6

1)

(3.7

2,

3.8

0,

4.1

3)

(3.1

2,

3.1

8,

3.4

7)

τB

=3

0.5

(27.0

7,

27.2

3,

27.8

8)

(13.8

0,

13.8

9,

14.2

4)

(8.4

9,

8.5

5,

8.7

9)

(13.0

3,

13.1

6,

13.6

7)

(6.1

9,

6.2

6,

6.5

5)

(4.2

8,

4.3

2,

4.5

2)

(3.1

9,

3.2

1,

3.3

3)

(2.7

3,

2.7

4,

2.8

1)

0.8

(156.5

9,

157.0

7,

158.9

6)

(119.7

0,

120.2

0,

122.1

5)

(104.9

5,

105.4

5,

107.4

2)

(140.7

6,

141.2

7,

143.2

6)

(120.6

6,

121.1

8,

123.2

4)

(107.3

6,

107.8

8,

109.9

4)

(90.1

9,

90.7

0,

92.6

9)

(57.2

3,

57.6

5,

59.3

1)

0.9

(246.3

3,

246.7

4,

248.3

1)

(218.6

3,

219.1

1,

220.9

4)

(207.5

5,

208.0

5,

210.0

0)

(236.3

8,

236.8

1,

238.5

0)

(222.1

2,

222.5

9,

224.4

2)

(211.6

1,

212.1

0,

214.0

3)

(196.4

8,

197.0

0,

199.0

6)

(159.8

2,

160.3

9,

162.6

4)

0.9

5(3

03.6

1,

303.8

6,

304.8

5)

(287.1

1,

287.4

2,

288.6

4)

(280.5

1,

280.8

5,

282.1

5)

(298.0

8,

298.3

6,

299.4

3)

(289.8

0,

290.1

1,

291.2

9)

(283.4

5,

283.7

8,

285.0

5)

(273.9

1,

274.2

7,

275.6

7)

(248.6

1,

249.0

4,

250.7

4)

1.0

5(1

90.3

2,

190.8

9,

193.2

6)

(180.9

8,

181.5

7,

183.9

9)

(177.2

4,

177.8

4,

180.2

8)

(184.4

1,

184.9

9,

187.4

0)

(179.4

6,

180.0

6,

182.4

9)

(176.7

1,

177.3

1,

179.7

6)

(173.5

1,

174.1

1,

176.5

7)

(167.9

5,

168.5

6,

171.0

4)

1.1

(107.5

5,

108.1

5,

110.6

3)

(97.5

2,

98.1

1,

100.5

4)

(93.5

0,

94.0

9,

96.5

1)

(100.6

2,

101.2

2,

103.7

0)

(95.4

1,

96.0

1,

98.4

6)

(92.6

4,

93.2

3,

95.6

6)

(89.4

9,

90.0

7,

92.4

7)

(84.2

2,

84.7

9,

87.1

4)

1.2

(42.9

0,

43.3

1,

45.0

3)

(35.7

1,

36.0

8,

37.6

5)

(32.8

3,

33.1

9,

34.7

0)

(37.1

7,

37.5

7,

39.2

0)

(33.6

2,

33.9

9,

35.5

4)

(31.8

6,

32.2

2,

33.7

2)

(29.9

5,

30.3

0,

31.7

5)

(26.9

5,

27.2

8,

28.6

3)

1.5

(8.1

5,

8.2

8,

8.8

3)

(5.6

8,

5.7

8,

6.2

1)

(4.6

9,

4.7

8,

5.1

6)

(5.5

9,

5.7

0,

6.1

5)

(4.5

7,

4.6

6,

5.0

5)

(4.1

3,

4.2

2,

4.5

8)

(3.7

0,

3.7

8,

4.1

1)

(3.1

0,

3.1

6,

3.4

4)

τB

=5

0.5

(27.0

0,

27.1

7,

27.8

2)

(13.7

6,

13.8

5,

14.2

1)

(8.4

7,

8.5

3,

8.7

6)

(12.9

8,

13.1

1,

13.6

1)

(6.1

6,

6.2

3,

6.5

2)

(4.2

6,

4.3

0,

4.5

0)

(3.1

7,

3.2

0,

3.3

2)

(2.7

2,

2.7

4,

2.8

0)

0.8

(156.4

3,

156.9

1,

158.8

0)

(119.5

4,

120.0

3,

121.9

8)

(104.7

8,

105.2

8,

107.2

5)

(140.5

9,

141.1

0,

143.0

9)

(120.4

9,

121.0

1,

123.0

6)

(107.1

9,

107.7

1,

109.7

5)

(90.0

2,

90.5

3,

92.5

2)

(57.1

0,

57.5

1,

59.1

6)

0.9

(246.2

0,

246.6

1,

248.1

7)

(218.4

9,

218.9

6,

220.7

9)

(207.4

0,

207.9

0,

209.8

4)

(236.2

4,

236.6

7,

238.3

6)

(221.9

7,

222.4

4,

224.2

7)

(211.4

5,

211.9

5,

213.8

7)

(196.3

1,

196.8

3,

198.8

8)

(159.6

4,

160.2

1,

162.4

5)

0.9

5(3

03.5

3,

303.7

8,

304.7

7)

(287.0

1,

287.3

3,

288.5

4)

(280.4

1,

280.7

4,

282.0

5)

(298.0

0,

298.2

7,

299.3

4)

(289.7

1,

290.0

1,

291.2

0)

(283.3

5,

283.6

7,

284.9

4)

(273.8

0,

274.1

6,

275.5

5)

(248.4

8,

248.9

1,

250.6

0)

1.0

5(1

90.1

5,

190.7

3,

193.0

9)

(180.8

1,

181.4

0,

183.8

1)

(177.0

7,

177.6

7,

180.1

0)

(184.2

4,

184.8

2,

187.2

3)

(179.2

9,

179.8

8,

182.3

2)

(176.5

4,

177.1

4,

179.5

8)

(173.3

4,

173.9

4,

176.3

9)

(167.7

8,

168.3

9,

170.8

6)

1.1

(107.3

8,

107.9

8,

110.4

6)

(97.3

5,

97.9

4,

100.3

7)

(93.3

4,

93.9

2,

96.3

4)

(100.4

6,

101.0

6,

103.5

3)

(95.2

5,

95.8

4,

98.2

8)

(92.4

7,

93.0

6,

95.4

8)

(89.3

3,

89.9

1,

92.3

0)

(84.0

6,

84.6

3,

86.9

7)

1.2

(42.7

9,

43.2

0,

44.9

1)

(35.6

1,

35.9

8,

37.5

4)

(32.7

4,

33.1

0,

34.6

0)

(37.0

7,

37.4

6,

39.0

9)

(33.5

2,

33.8

9,

35.4

3)

(31.7

6,

32.1

2,

33.6

2)

(29.8

6,

30.2

1,

31.6

5)

(26.8

6,

27.1

9,

28.5

3)

1.5

(8.1

2,

8.2

5,

8.8

0)

(5.6

5,

5.7

5,

6.1

8)

(4.6

7,

4.7

6,

5.1

3)

(5.5

7,

5.6

7,

6.1

2)

(4.5

4,

4.6

4,

5.0

2)

(4.1

1,

4.2

0,

4.5

5)

(3.6

8,

3.7

6,

4.0

9)

(3.0

8,

3.1

5,

3.4

3)

n=

15

FS

I(0

.5,

1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

B=

20.5

(1.7

2,

1.7

3,

1.7

9)

(0.8

6,

0.8

7,

0.9

0)

(0.5

2,

0.5

2,

0.5

4)

(0.1

9,

0.2

0,

0.2

1)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

0.8

(38.9

7,

39.3

9,

41.0

5)

(23.7

5,

24.0

5,

25.2

8)

(17.6

6,

17.9

2,

18.9

7)

(27.4

6,

27.8

4,

29.3

5)

(19.2

9,

19.6

0,

20.8

6)

(15.4

2,

15.6

9,

16.7

8)

(11.5

7,

11.7

8,

12.6

6)

(6.6

6,

6.7

8,

7.2

9)

0.9

(122.3

9,

123.1

6,

126.1

2)

(96.4

6,

97.2

0,

100.1

0)

(86.0

8,

86.8

2,

89.6

9)

(109.2

8,

110.0

6,

113.0

9)

(95.3

1,

96.0

8,

99.0

9)

(86.6

3,

87.3

9,

90.3

5)

(75.7

0,

76.4

3,

79.2

8)

(54.3

9,

55.0

2,

57.4

9)

0.9

5(2

14.1

4,

214.8

5,

217.5

7)

(191.5

6,

192.3

4,

195.3

1)

(182.5

3,

183.3

3,

186.4

0)

(204.5

1,

205.2

6,

208.1

2)

(192.7

0,

193.4

9,

196.4

9)

(184.6

2,

185.4

2,

188.5

0)

(173.5

0,

174.3

3,

177.5

0)

(147.9

1,

148.7

6,

152.0

7)

1.0

5(1

36.5

9,

137.4

6,

140.9

2)

(123.4

8,

124.3

6,

127.8

2)

(118.2

4,

119.1

1,

122.5

8)

(128.7

3,

129.6

1,

133.1

1)

(121.8

5,

122.7

3,

126.2

3)

(117.8

6,

118.7

4,

122.2

3)

(113.0

0,

113.8

7,

117.3

4)

(103.6

7,

104.5

3,

107.9

4)

1.1

(59.9

0,

60.5

8,

63.3

1)

(49.2

0,

49.8

3,

52.3

5)

(44.9

2,

45.5

2,

47.9

6)

(52.4

2,

53.0

8,

55.7

3)

(46.9

5,

47.5

8,

50.1

0)

(44.0

2,

44.6

2,

47.0

8)

(40.6

4,

41.2

2,

43.5

7)

(34.6

8,

35.2

1,

37.3

6)

1.2

(17.1

4,

17.4

5,

18.7

1)

(11.9

8,

12.2

2,

13.2

2)

(9.9

1,

10.1

3,

11.0

3)

(12.5

3,

12.8

0,

13.8

9)

(10.1

3,

10.3

6,

11.3

1)

(9.0

1,

9.2

3,

10.1

1)

(7.8

5,

8.0

4,

8.8

3)

(6.0

6,

6.2

1,

6.8

6)

1.5

(2.5

8,

2.6

4,

2.8

7)

(1.4

3,

1.4

6,

1.6

0)

(0.9

7,

0.9

9,

1.1

0)

(1.0

8,

1.1

1,

1.2

5)

(0.7

3,

0.7

5,

0.8

6)

(0.6

1,

0.6

3,

0.7

2)

(0.5

1,

0.5

2,

0.5

9)

(0.3

9,

0.4

0,

0.4

5)

τB

=3

0.5

(1.7

1,

1.7

3,

1.7

9)

(0.8

6,

0.8

6,

0.8

9)

(0.5

1,

0.5

2,

0.5

4)

(0.1

9,

0.2

0,

0.2

1)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

0.8

(38.8

5,

39.2

7,

40.9

1)

(23.6

6,

23.9

6,

25.1

7)

(17.5

8,

17.8

4,

18.8

8)

(27.3

5,

27.7

3,

29.2

3)

(19.2

0,

19.5

1,

20.7

5)

(15.3

4,

15.6

1,

16.6

9)

(11.5

1,

11.7

2,

12.5

8)

(6.6

2,

6.7

5,

7.2

5)

0.9

(122.1

9,

122.9

5,

125.8

8)

(96.2

5,

96.9

9,

99.8

7)

(85.8

8,

86.6

1,

89.4

6)

(109.0

6,

109.8

4,

112.8

4)

(95.1

0,

95.8

7,

98.8

5)

(86.4

3,

87.1

8,

90.1

2)

(75.5

1,

76.2

3,

79.0

6)

(54.2

2,

54.8

4,

57.3

0)

0.9

5(2

13.9

5,

214.6

6,

217.3

6)

(191.3

6,

192.1

3,

195.0

8)

(182.3

2,

183.1

1,

186.1

7)

(204.3

1,

205.0

6,

207.9

0)

(192.5

0,

193.2

7,

196.2

5)

(184.4

0,

185.2

0,

188.2

6)

(173.2

8,

174.1

0,

177.2

5)

(147.6

8,

148.5

3,

151.8

1)

1.0

5(1

36.3

7,

137.2

4,

140.6

7)

(123.2

6,

124.1

3,

127.5

7)

(118.0

2,

118.8

9,

122.3

3)

(128.5

1,

129.3

9,

132.8

6)

(121.6

3,

122.5

1,

125.9

8)

(117.6

4,

118.5

1,

121.9

8)

(112.7

8,

113.6

5,

117.0

9)

(103.4

6,

104.3

1,

107.6

9)

1.1

(59.7

3,

60.4

1,

63.1

2)

(49.0

5,

49.6

7,

52.1

6)

(44.7

7,

45.3

7,

47.7

8)

(52.2

6,

52.9

2,

55.5

4)

(46.7

9,

47.4

2,

49.9

2)

(43.8

7,

44.4

7,

46.9

0)

(40.5

0,

41.0

8,

43.4

0)

(34.5

5,

35.0

8,

37.2

0)

1.2

(17.0

7,

17.3

7,

18.6

2)

(11.9

2,

12.1

6,

13.1

5)

(9.8

6,

10.0

8,

10.9

6)

(12.4

7,

12.7

3,

13.8

2)

(10.0

8,

10.3

1,

11.2

5)

(8.9

6,

9.1

8,

10.0

4)

(7.8

0,

7.9

9,

8.7

7)

(6.0

2,

6.1

8,

6.8

1)

1.5

(2.5

7,

2.6

3,

2.8

5)

(1.4

2,

1.4

6,

1.6

0)

(0.9

6,

0.9

9,

1.0

9)

(1.0

7,

1.1

1,

1.2

5)

(0.7

2,

0.7

5,

0.8

5)

(0.6

1,

0.6

3,

0.7

1)

(0.5

0,

0.5

2,

0.5

9)

(0.3

8,

0.3

9,

0.4

4)

τB

=5

0.5

(1.7

1,

1.7

2,

1.7

8)

(0.8

5,

0.8

6,

0.8

9)

(0.5

1,

0.5

2,

0.5

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

(0.1

7,

0.1

7,

0.1

8)

0.8

(38.7

5,

39.1

7,

40.8

0)

(23.5

9,

23.8

9,

25.1

0)

(17.5

2,

17.7

8,

18.8

1)

(27.2

6,

27.6

4,

29.1

3)

(19.1

2,

19.4

3,

20.6

7)

(15.2

8,

15.5

5,

16.6

2)

(11.4

6,

11.6

7,

12.5

3)

(6.5

9,

6.7

2,

7.2

2)

0.9

(122.0

2,

122.7

8,

125.7

1)

(96.0

9,

96.8

3,

99.7

0)

(85.7

2,

86.4

5,

89.2

9)

(108.8

9,

109.6

7,

112.6

6)

(94.9

3,

95.7

0,

98.6

7)

(86.2

6,

87.0

1,

89.9

4)

(75.3

5,

76.0

7,

78.8

9)

(54.0

9,

54.7

1,

57.1

5)

0.9

5(2

13.8

0,

214.5

1,

217.2

0)

(191.1

9,

191.9

6,

194.9

1)

(182.1

5,

182.9

4,

185.9

9)

(204.1

6,

204.9

0,

207.7

3)

(192.3

3,

193.1

1,

196.0

8)

(184.2

3,

185.0

3,

188.0

8)

(173.1

1,

173.9

2,

177.0

7)

(147.5

0,

148.3

5,

151.6

2)

1.0

5(1

36.2

0,

137.0

6,

140.4

8)

(123.0

9,

123.9

5,

127.3

8)

(117.8

5,

118.7

1,

122.1

4)

(128.3

4,

129.2

1,

132.6

7)

(121.4

6,

122.3

3,

125.7

9)

(117.4

7,

118.3

4,

121.7

9)

(112.6

0,

113.4

7,

116.9

0)

(103.2

9,

104.1

4,

107.5

1)

1.1

(59.6

0,

60.2

7,

62.9

7)

(48.9

3,

49.5

4,

52.0

3)

(44.6

5,

45.2

5,

47.6

5)

(52.1

4,

52.7

9,

55.4

0)

(46.6

7,

47.2

9,

49.7

9)

(43.7

5,

44.3

5,

46.7

7)

(40.3

8,

40.9

6,

43.2

8)

(34.4

5,

34.9

7,

37.0

8)

1.2

(17.0

1,

17.3

2,

18.5

5)

(11.8

8,

12.1

2,

13.1

0)

(9.8

2,

10.0

4,

10.9

2)

(12.4

2,

12.6

8,

13.7

6)

(10.0

3,

10.2

6,

11.2

0)

(8.9

3,

9.1

4,

10.0

0)

(7.7

7,

7.9

6,

8.7

3)

(5.9

9,

6.1

5,

6.7

8)

1.5

(2.5

6,

2.6

2,

2.8

4)

(1.4

2,

1.4

5,

1.5

9)

(0.9

6,

0.9

8,

1.0

9)

(1.0

7,

1.1

0,

1.2

4)

(0.7

2,

0.7

4,

0.8

4)

(0.6

0,

0.6

2,

0.7

1)

(0.5

0,

0.5

2,

0.5

8)

(0.3

8,

0.3

9,

0.4

4)

Table 11: The out-of-control ARL1 of the FSI SH-γ chart and the out-of-control ATS1 ofthe VSI SH-γ2 charts in the presence of measurement errors for different values of B, τ, n,fixed η = 0.28, θ = 0.01,m = 1, γ0 = 0.05 (left side), γ0 = 0.1 (middle) and γ0 = 0.2 (rightside); the first row shows the values of (hS , hL).

28 Huu Du Nguyen et al.n

=5

FS

I(0

.5,

1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

m=

10.5

(29.3

5,

29.5

3,

30.2

1)

(15.0

5,

15.1

4,

15.5

2)

(9.3

3,

9.3

9,

9.6

5)

(14.8

3,

14.9

7,

15.5

2)

(7.2

4,

7.3

2,

7.6

5)

(4.9

7,

5.0

3,

5.2

5)

(3.6

0,

3.6

4,

3.7

7)

(2.9

7,

2.9

9,

3.0

6)

0.8

(162.0

4,

162.5

1,

164.3

6)

(125.3

1,

125.8

0,

127.7

3)

(110.6

2,

111.1

2,

113.0

8)

(146.5

0,

147.0

0,

148.9

6)

(126.5

8,

127.0

9,

129.1

2)

(113.2

7,

113.7

8,

115.8

2)

(95.9

3,

96.4

3,

98.4

3)

(61.9

9,

62.4

2,

64.1

2)

0.9

(250.6

0,

250.9

9,

252.4

9)

(223.6

2,

224.0

8,

225.8

4)

(212.8

3,

213.3

1,

215.1

8)

(240.9

6,

241.3

8,

242.9

9)

(227.1

0,

227.5

5,

229.3

1)

(216.8

5,

217.3

3,

219.1

8)

(202.0

4,

202.5

5,

204.5

2)

(165.8

8,

166.4

3,

168.6

1)

0.9

5(3

06.2

3,

306.4

7,

307.4

1)

(290.3

2,

290.6

2,

291.7

7)

(283.9

6,

284.2

7,

285.5

1)

(300.9

2,

301.1

8,

302.1

9)

(292.9

4,

293.2

3,

294.3

5)

(286.8

1,

287.1

2,

288.3

2)

(277.5

9,

277.9

3,

279.2

5)

(253.0

5,

253.4

6,

255.0

7)

1.0

5(1

95.8

7,

196.4

2,

198.6

9)

(186.6

8,

187.2

5,

189.5

8)

(183.0

1,

183.5

8,

185.9

3)

(190.0

7,

190.6

4,

192.9

6)

(185.2

1,

185.7

8,

188.1

3)

(182.5

0,

183.0

7,

185.4

3)

(179.3

3,

179.9

1,

182.2

9)

(173.8

4,

174.4

2,

176.8

1)

1.1

(113.2

7,

113.8

6,

116.3

0)

(103.1

7,

103.7

5,

106.1

6)

(99.1

2,

99.7

1,

102.1

0)

(106.3

5,

106.9

4,

109.3

9)

(101.0

9,

101.6

8,

104.1

0)

(98.2

8,

98.8

6,

101.2

7)

(95.0

8,

95.6

6,

98.0

4)

(89.7

1,

90.2

8,

92.6

2)

1.2

(46.7

2,

47.1

4,

48.9

0)

(39.2

1,

39.6

0,

41.2

1)

(36.2

1,

36.5

9,

38.1

4)

(40.8

3,

41.2

3,

42.9

1)

(37.0

9,

37.4

8,

39.0

8)

(35.2

4,

35.6

1,

37.1

6)

(33.2

1,

33.5

7,

35.0

7)

(30.0

0,

30.3

4,

31.7

5)

1.5

(9.2

5,

9.3

9,

9.9

9)

(6.5

4,

6.6

5,

7.1

2)

(5.4

5,

5.5

5,

5.9

7)

(6.5

0,

6.6

2,

7.1

1)

(5.3

5,

5.4

6,

5.8

9)

(4.8

6,

4.9

6,

5.3

5)

(4.3

7,

4.4

6,

4.8

2)

(3.6

7,

3.7

5,

4.0

6)

τm

=5

0.5

(29.3

5,

29.5

1,

30.1

6)

(15.0

5,

15.1

4,

15.4

9)

(9.3

2,

9.3

9,

9.6

3)

(14.8

3,

14.9

6,

15.4

8)

(7.2

3,

7.3

1,

7.6

2)

(4.9

7,

5.0

2,

5.2

3)

(3.6

0,

3.6

3,

3.7

6)

(2.9

7,

2.9

8,

3.0

5)

0.8

(162.0

3,

162.4

7,

164.2

2)

(125.3

0,

125.7

6,

127.5

8)

(110.6

1,

111.0

8,

112.9

3)

(146.4

9,

146.9

6,

148.8

1)

(126.5

7,

127.0

5,

128.9

6)

(113.2

6,

113.7

4,

115.6

6)

(95.9

2,

96.4

0,

98.2

7)

(61.9

9,

62.3

9,

63.9

8)

0.9

(250.5

9,

250.9

6,

252.3

7)

(223.6

1,

224.0

4,

225.7

1)

(212.8

2,

213.2

7,

215.0

4)

(240.9

6,

241.3

5,

242.8

7)

(227.0

9,

227.5

2,

229.1

8)

(216.8

4,

217.2

9,

219.0

4)

(202.0

3,

202.5

1,

204.3

7)

(165.8

6,

166.3

9,

168.4

4)

0.9

5(3

06.2

3,

306.4

5,

307.3

4)

(290.3

1,

290.5

9,

291.6

8)

(283.9

5,

284.2

5,

285.4

2)

(300.9

1,

301.1

6,

302.1

1)

(292.9

3,

293.2

1,

294.2

7)

(286.8

0,

287.0

9,

288.2

3)

(277.5

8,

277.9

0,

279.1

5)

(253.0

4,

253.4

3,

254.9

5)

1.0

5(1

95.8

5,

196.3

7,

198.5

1)

(186.6

7,

187.2

0,

189.3

9)

(183.0

0,

183.5

4,

185.7

5)

(190.0

6,

190.5

9,

192.7

7)

(185.2

0,

185.7

4,

187.9

4)

(182.4

9,

183.0

3,

185.2

5)

(179.3

2,

179.8

7,

182.1

0)

(173.8

3,

174.3

8,

176.6

3)

1.1

(113.2

6,

113.8

1,

116.1

1)

(103.1

5,

103.7

0,

105.9

7)

(99.1

1,

99.6

6,

101.9

1)

(106.3

4,

106.9

0,

109.1

9)

(101.0

8,

101.6

3,

103.9

1)

(98.2

7,

98.8

2,

101.0

8)

(95.0

7,

95.6

2,

97.8

5)

(89.7

0,

90.2

4,

92.4

3)

1.2

(46.7

1,

47.1

1,

48.7

5)

(39.2

1,

39.5

7,

41.0

9)

(36.2

1,

36.5

6,

38.0

2)

(40.8

2,

41.2

0,

42.7

7)

(37.0

9,

37.4

5,

38.9

5)

(35.2

3,

35.5

8,

37.0

4)

(33.2

0,

33.5

4,

34.9

5)

(30.0

0,

30.3

2,

31.6

4)

1.5

(9.2

5,

9.3

8,

9.9

4)

(6.5

4,

6.6

4,

7.0

8)

(5.4

5,

5.5

5,

5.9

4)

(6.5

0,

6.6

1,

7.0

7)

(5.3

5,

5.4

5,

5.8

5)

(4.8

6,

4.9

5,

5.3

2)

(4.3

7,

4.4

5,

4.7

9)

(3.6

7,

3.7

4,

4.0

4)

τm

=10

0.5

(29.3

5,

29.5

1,

30.1

5)

(15.0

5,

15.1

3,

15.4

9)

(9.3

2,

9.3

8,

9.6

2)

(14.8

3,

14.9

6,

15.4

7)

(7.2

3,

7.3

1,

7.6

2)

(4.9

7,

5.0

2,

5.2

3)

(3.6

0,

3.6

3,

3.7

6)

(2.9

7,

2.9

8,

3.0

5)

0.8

(162.0

3,

162.4

7,

164.2

0)

(125.3

0,

125.7

6,

127.5

6)

(110.6

1,

111.0

7,

112.9

1)

(146.4

9,

146.9

6,

148.7

9)

(126.5

7,

127.0

5,

128.9

4)

(113.2

6,

113.7

4,

115.6

4)

(95.9

2,

96.3

9,

98.2

5)

(61.9

9,

62.3

8,

63.9

7)

0.9

(250.5

9,

250.9

5,

252.3

6)

(223.6

1,

224.0

4,

225.6

9)

(212.8

2,

213.2

7,

215.0

2)

(240.9

6,

241.3

4,

242.8

5)

(227.0

9,

227.5

1,

229.1

6)

(216.8

4,

217.2

9,

219.0

2)

(202.0

3,

202.5

0,

204.3

5)

(165.8

6,

166.3

8,

168.4

2)

0.9

5(3

06.2

3,

306.4

5,

307.3

3)

(290.3

1,

290.5

9,

291.6

7)

(283.9

5,

284.2

5,

285.4

0)

(300.9

1,

301.1

6,

302.1

0)

(292.9

3,

293.2

0,

294.2

5)

(286.8

0,

287.0

9,

288.2

2)

(277.5

8,

277.9

0,

279.1

4)

(253.0

4,

253.4

2,

254.9

3)

1.0

5(1

95.8

5,

196.3

7,

198.4

9)

(186.6

7,

187.2

0,

189.3

7)

(182.9

9,

183.5

3,

185.7

2)

(190.0

6,

190.5

9,

192.7

5)

(185.2

0,

185.7

3,

187.9

2)

(182.4

8,

183.0

2,

185.2

2)

(179.3

2,

179.8

6,

182.0

7)

(173.8

2,

174.3

7,

176.6

0)

1.1

(113.2

6,

113.8

1,

116.0

9)

(103.1

5,

103.7

0,

105.9

4)

(99.1

1,

99.6

5,

101.8

9)

(106.3

4,

106.8

9,

109.1

7)

(101.0

8,

101.6

3,

103.8

9)

(98.2

7,

98.8

1,

101.0

5)

(95.0

7,

95.6

1,

97.8

3)

(89.7

0,

90.2

3,

92.4

1)

1.2

(46.7

1,

47.1

0,

48.7

4)

(39.2

1,

39.5

7,

41.0

7)

(36.2

1,

36.5

5,

38.0

0)

(40.8

2,

41.1

9,

42.7

6)

(37.0

9,

37.4

4,

38.9

3)

(35.2

3,

35.5

8,

37.0

2)

(33.2

0,

33.5

4,

34.9

4)

(30.0

0,

30.3

1,

31.6

2)

1.5

(9.2

5,

9.3

8,

9.9

4)

(6.5

4,

6.6

4,

7.0

8)

(5.4

5,

5.5

5,

5.9

3)

(6.5

0,

6.6

1,

7.0

7)

(5.3

5,

5.4

5,

5.8

5)

(4.8

6,

4.9

5,

5.3

2)

(4.3

7,

4.4

5,

4.7

9)

(3.6

7,

3.7

4,

4.0

4)

n=

15

FS

I(0

.5,

1.5

)(0

.3,

1.7

)(0

.1,

1.1

)(0

.1,

1.3

)(0

.1,

1.5

)(0

.1,

1.9

)(0

.1,

4.0

)τ

m=

10.5

(1.8

6,

1.8

8,

1.9

5)

(0.9

3,

0.9

4,

0.9

7)

(0.5

6,

0.5

6,

0.5

8)

(0.2

2,

0.2

2,

0.2

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

0.8

(42.2

0,

42.6

4,

44.3

4)

(26.1

3,

26.4

5,

27.7

3)

(19.7

0,

19.9

8,

21.0

9)

(30.4

1,

30.8

0,

32.3

7)

(21.7

3,

22.0

7,

23.3

9)

(17.5

4,

17.8

2,

18.9

8)

(13.2

7,

13.5

1,

14.4

5)

(7.6

5,

7.7

9,

8.3

5)

0.9

(127.8

5,

128.6

0,

131.5

0)

(101.7

9,

102.5

3,

105.3

9)

(91.3

6,

92.1

0,

94.9

5)

(114.8

6,

115.6

2,

118.5

9)

(100.8

5,

101.6

1,

104.5

8)

(92.0

7,

92.8

3,

95.7

5)

(80.9

4,

81.6

7,

84.5

0)

(58.9

2,

59.5

6,

62.0

6)

0.9

5(2

19.0

3,

219.7

1,

222.3

0)

(196.8

9,

197.6

3,

200.4

8)

(188.0

4,

188.8

0,

191.7

5)

(209.6

5,

210.3

6,

213.0

9)

(198.0

9,

198.8

3,

201.7

0)

(190.1

4,

190.9

1,

193.8

6)

(179.1

8,

179.9

7,

183.0

2)

(153.8

0,

154.6

2,

157.8

2)

1.0

5(1

42.2

7,

143.1

1,

146.4

6)

(129.1

9,

130.0

4,

133.4

1)

(123.9

6,

124.8

1,

128.1

9)

(134.4

9,

135.3

4,

138.7

4)

(127.6

2,

128.4

8,

131.8

8)

(123.6

2,

124.4

7,

127.8

7)

(118.7

3,

119.5

8,

122.9

7)

(109.3

2,

110.1

6,

113.5

0)

1.1

(64.3

0,

64.9

8,

67.7

1)

(53.2

7,

53.9

0,

56.4

4)

(48.8

6,

49.4

7,

51.9

4)

(56.7

0,

57.3

6,

60.0

2)

(51.0

3,

51.6

7,

54.2

2)

(47.9

8,

48.6

0,

51.0

8)

(44.4

5,

45.0

4,

47.4

3)

(38.1

6,

38.7

1,

40.9

0)

1.2

(19.0

8,

19.4

1,

20.7

2)

(13.5

3,

13.7

9,

14.8

5)

(11.3

0,

11.5

4,

12.4

9)

(14.2

3,

14.5

1,

15.6

7)

(11.6

1,

11.8

6,

12.8

8)

(10.3

8,

10.6

1,

11.5

5)

(9.0

8,

9.2

9,

10.1

4)

(7.0

6,

7.2

3,

7.9

3)

1.5

(2.9

0,

2.9

6,

3.2

2)

(1.6

2,

1.6

6,

1.8

2)

(1.1

1,

1.1

4,

1.2

6)

(1.2

8,

1.3

2,

1.4

8)

(0.8

7,

0.9

0,

1.0

2)

(0.7

3,

0.7

5,

0.8

5)

(0.6

0,

0.6

2,

0.7

1)

(0.4

6,

0.4

7,

0.5

3)

τm

=5

0.5

(1.8

6,

1.8

8,

1.9

4)

(0.9

3,

0.9

4,

0.9

7)

(0.5

6,

0.5

6,

0.5

8)

(0.2

2,

0.2

2,

0.2

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

0.8

(42.1

9,

42.6

0,

44.2

1)

(26.1

2,

26.4

3,

27.6

3)

(19.6

9,

19.9

6,

21.0

0)

(30.4

0,

30.7

7,

32.2

5)

(21.7

3,

22.0

4,

23.2

9)

(17.5

3,

17.8

0,

18.8

9)

(13.2

7,

13.4

9,

14.3

8)

(7.6

5,

7.7

8,

8.3

1)

0.9

(127.8

4,

128.5

5,

131.2

8)

(101.7

8,

102.4

7,

105.1

7)

(91.3

5,

92.0

4,

94.7

3)

(114.8

4,

115.5

6,

118.3

7)

(100.8

3,

101.5

5,

104.3

5)

(92.0

6,

92.7

7,

95.5

3)

(80.9

3,

81.6

1,

84.2

8)

(58.9

1,

59.5

1,

61.8

6)

0.9

5(2

19.0

1,

219.6

5,

222.1

0)

(196.8

8,

197.5

8,

200.2

6)

(188.0

2,

188.7

4,

191.5

3)

(209.6

3,

210.3

1,

212.8

8)

(198.0

7,

198.7

8,

201.4

9)

(190.1

3,

190.8

5,

193.6

4)

(179.1

7,

179.9

1,

182.7

9)

(153.7

8,

154.5

6,

157.5

8)

1.0

5(1

42.2

5,

143.0

5,

146.2

1)

(129.1

8,

129.9

8,

133.1

5)

(123.9

4,

124.7

5,

127.9

3)

(134.4

7,

135.2

8,

138.4

8)

(127.6

0,

128.4

1,

131.6

2)

(123.6

0,

124.4

1,

127.6

1)

(118.7

1,

119.5

2,

122.7

1)

(109.3

1,

110.1

0,

113.2

5)

1.1

(64.2

8,

64.9

3,

67.5

0)

(53.2

6,

53.8

5,

56.2

5)

(48.8

5,

49.4

2,

51.7

4)

(56.6

8,

57.3

1,

59.8

2)

(51.0

2,

51.6

2,

54.0

3)

(47.9

7,

48.5

5,

50.8

9)

(44.4

4,

45.0

0,

47.2

4)

(38.1

5,

38.6

7,

40.7

3)

1.2

(19.0

8,

19.3

8,

20.6

2)

(13.5

2,

13.7

7,

14.7

6)

(11.3

0,

11.5

2,

12.4

2)

(14.2

2,

14.4

9,

15.5

8)

(11.6

1,

11.8

4,

12.8

0)

(10.3

8,

10.5

9,

11.4

8)

(9.0

8,

9.2

7,

10.0

8)

(7.0

6,

7.2

2,

7.8

8)

1.5

(2.9

0,

2.9

6,

3.2

0)

(1.6

2,

1.6

6,

1.8

1)

(1.1

1,

1.1

4,

1.2

5)

(1.2

8,

1.3

1,

1.4

6)

(0.8

7,

0.9

0,

1.0

1)

(0.7

3,

0.7

5,

0.8

4)

(0.6

0,

0.6

2,

0.7

0)

(0.4

6,

0.4

7,

0.5

2)

τm

=10

0.5

(1.8

6,

1.8

8,

1.9

4)

(0.9

3,

0.9

4,

0.9

7)

(0.5

6,

0.5

6,

0.5

8)

(0.2

2,

0.2

2,

0.2

3)

(0.1

9,

0.1

9,

0.2

0)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

(0.1

9,

0.1

9,

0.1

9)

0.8

(42.1

9,

42.6

0,

44.1

9)

(26.1

2,

26.4

2,

27.6

2)

(19.6

9,

19.9

5,

20.9

9)

(30.4

0,

30.7

7,

32.2

3)

(21.7

3,

22.0

4,

23.2

7)

(17.5

3,

17.8

0,

18.8

8)

(13.2

7,

13.4

9,

14.3

7)

(7.6

5,

7.7

8,

8.3

0)

0.9

(127.8

4,

128.5

4,

131.2

5)

(101.7

7,

102.4

6,

105.1

4)

(91.3

5,

92.0

3,

94.7

0)

(114.8

4,

115.5

6,

118.3

4)

(100.8

3,

101.5

5,

104.3

2)

(92.0

6,

92.7

6,

95.5

0)

(80.9

2,

81.6

0,

84.2

6)

(58.9

1,

59.5

0,

61.8

4)

0.9

5(2

19.0

1,

219.6

5,

222.0

8)

(196.8

8,

197.5

7,

200.2

4)

(188.0

2,

188.7

4,

191.5

0)

(209.6

3,

210.3

0,

212.8

6)

(198.0

7,

198.7

7,

201.4

6)

(190.1

2,

190.8

4,

193.6

1)

(179.1

7,

179.9

1,

182.7

6)

(153.7

8,

154.5

5,

157.5

5)

1.0

5(1

42.2

5,

143.0

4,

146.1

7)

(129.1

7,

129.9

7,

133.1

2)

(123.9

4,

124.7

4,

127.9

0)

(134.4

7,

135.2

7,

138.4

4)

(127.6

0,

128.4

0,

131.5

8)

(123.6

0,

124.4

0,

127.5

8)

(118.7

1,

119.5

1,

122.6

8)

(109.3

0,

110.0

9,

113.2

1)

1.1

(64.2

8,

64.9

2,

67.4

7)

(53.2

6,

53.8

5,

56.2

2)

(48.8

4,

49.4

2,

51.7

2)

(56.6

8,

57.3

0,

59.7

9)

(51.0

2,

51.6

1,

54.0

0)

(47.9

7,

48.5

5,

50.8

6)

(44.4

3,

44.9

9,

47.2

2)

(38.1

5,

38.6

6,

40.7

1)

1.2

(19.0

8,

19.3

8,

20.6

1)

(13.5

2,

13.7

6,

14.7

5)

(11.3

0,

11.5

2,

12.4

1)

(14.2

2,

14.4

9,

15.5

7)

(11.6

1,

11.8

4,

12.7

9)

(10.3

8,

10.5

9,

11.4

7)

(9.0

8,

9.2

7,

10.0

7)

(7.0

6,

7.2

2,

7.8

7)

1.5

(2.9

0,

2.9

6,

3.2

0)

(1.6

2,

1.6

6,

1.8

1)

(1.1

1,

1.1

4,

1.2

5)

(1.2

8,

1.3

1,

1.4

6)

(0.8

7,

0.9

0,

1.0

1)

(0.7

3,

0.7

5,

0.8

4)

(0.6

0,

0.6

2,

0.7

0)

(0.4

6,

0.4

7,

0.5

2)

Table 12: The out-of-control ARL1 of the FSI SH-γ chart and the out-of-control ATS1 ofthe VSI SH-γ2 charts in the presence of measurement errors for different values of m, τ, n,fixed η = 0.28, θ = 0.05, B = 1, γ0 = 0.05 (left side), γ0 = 0.1 (middle) and γ0 = 0.2 (rightside); the first row shows the values of (hS , hL).


n = 5Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

η = 0(D) (107.87,108.17,109.35) (96.98,97.27,98.42) (120.54,120.87,122.15) (105.52,105.84,107.07) (96.70,97.01,98.20) (86.09,86.38,87.50) (66.51,66.75,67.71)(I) (25.26,25.43,26.15) (23.72,23.88,24.57) (25.80,25.98,26.72) (24.00,24.17,24.86) (23.12,23.29,23.96) (22.18,22.34,22.99) (20.70,20.84,21.45)

η = 0.2(D) (107.87,108.18,109.41) (96.98,97.29,98.49) (120.54,120.88,122.21) (105.53,105.85,107.14) (96.71,97.02,98.26) (86.10,86.39,87.56) (66.52,66.77,67.76)(I) (25.26,25.44,26.19) (23.72,23.89,24.61) (25.80,25.99,26.76) (24.00,24.18,24.90) (23.13,23.30,24.00) (22.18,22.35,23.02) (20.70,20.85,21.48)

η = 0.5(D) (107.89,108.27,109.74) (97.00,97.37,98.80) (120.57,120.97,122.56) (105.55,105.94,107.48) (96.73,97.11,98.59) (86.12,86.47,87.87) (66.53,66.83,68.03)(I) (25.27,25.49,26.40) (23.73,23.94,24.81) (25.81,26.04,26.98) (24.01,24.22,25.11) (23.14,23.34,24.19) (22.19,22.39,23.21) (20.71,20.89,21.66)

η = 1(D) (107.97,108.56,110.89) (97.08,97.66,99.92) (120.65,121.30,123.80) (105.63,106.25,108.68) (96.80,97.40,99.75) (86.19,86.75,88.96) (66.59,67.07,68.98)(I) (25.31,25.67,27.19) (23.77,24.11,25.55) (25.86,26.22,27.79) (24.05,24.39,25.86) (23.18,23.51,24.92) (22.23,22.55,23.91) (20.74,21.04,22.31)

n = 15Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

η = 0(D) (40.57,40.82,41.81) (36.09,36.32,37.25) (44.06,44.34,45.40) (38.41,38.66,39.64) (35.30,35.53,36.46) (31.61,31.83,32.70) (24.78,24.97,25.71)(I) (12.65,12.78,13.31) (11.46,11.59,12.08) (12.98,13.12,13.67) (11.66,11.79,12.29) (11.01,11.13,11.61) (10.28,10.39,10.85) (9.04,9.14,9.55)

η = 0.2(D) (40.57,40.83,41.86) (36.09,36.34,37.30) (44.07,44.35,45.45) (38.42,38.68,39.70) (35.30,35.55,36.51) (31.61,31.84,32.74) (24.78,24.98,25.75)(I) (12.65,12.79,13.34) (11.47,11.59,12.10) (12.98,13.12,13.70) (11.66,11.79,12.32) (11.01,11.13,11.63) (10.28,10.40,10.87) (9.04,9.15,9.57)

η = 0.5(D) (40.59,40.90,42.13) (36.11,36.40,37.55) (44.09,44.43,45.74) (38.44,38.75,39.96) (35.32,35.61,36.77) (31.63,31.90,32.98) (24.80,25.03,25.95)(I) (12.66,12.82,13.49) (11.47,11.63,12.24) (12.99,13.16,13.86) (11.67,11.83,12.46) (11.02,11.17,11.77) (10.29,10.43,11.00) (9.05,9.18,9.69)

η = 1(D) (40.65,41.15,43.07) (36.17,36.64,38.44) (44.16,44.69,46.76) (38.50,38.99,40.90) (35.38,35.85,37.66) (31.68,32.12,33.81) (24.84,25.22,26.67)(I) (12.69,12.95,14.03) (11.50,11.75,12.75) (13.03,13.30,14.42) (11.70,11.95,12.98) (11.05,11.29,12.26) (10.32,10.54,11.46) (9.08,9.28,10.10)

Table 13: The effect of η on the overall performance of the VSI SH-γ2 control charts inthe presence of measurement errors for B = 1,m = 1, θ = 0.05, n = 5, 15, γ0 = 0.05 (leftside), γ0 = 0.1 (middle) and γ0 = 0.2 (right side).

n = 5Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

θ = 0(D) (103.86,104.21,105.58) (93.12,93.46,94.79) (116.13,116.51,118.00) (101.33,101.69,103.13) (92.71,93.06,94.43) (82.38,82.71,84.00) (63.41,63.69,64.79)(I) (23.49,23.68,24.50) (22.03,22.22,22.99) (23.98,24.18,25.02) (22.28,22.47,23.26) (21.46,21.64,22.40) (20.58,20.75,21.48) (19.18,19.35,20.03)

θ = 0.03(D) (106.28,106.62,107.93) (95.45,95.78,97.05) (118.80,119.16,120.58) (103.86,104.21,105.58) (95.12,95.46,96.77) (84.62,84.94,86.18) (65.28,65.55,66.61)(I) (24.55,24.74,25.53) (23.04,23.22,23.98) (25.07,25.27,26.09) (23.31,23.50,24.26) (22.46,22.64,23.38) (21.54,21.71,22.42) (20.09,20.25,20.92)

θ = 0.05(D) (107.87,108.20,109.47) (96.99,97.30,98.54) (120.55,120.90,122.28) (105.53,105.87,107.20) (96.71,97.04,98.32) (86.10,86.41,87.62) (66.52,66.78,67.81)(I) (25.26,25.45,26.23) (23.72,23.90,24.64) (25.80,26.00,26.80) (24.00,24.18,24.94) (23.13,23.30,24.03) (22.19,22.35,23.06) (20.70,20.86,21.51)

n = 15Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

θ = 0(D) (38.37,38.66,39.78) (34.04,34.31,35.36) (41.69,42.00,43.21) (36.24,36.53,37.64) (33.25,33.52,34.57) (29.71,29.96,30.94) (23.18,23.39,24.23)(I) (11.68,11.83,12.42) (10.56,10.70,11.25) (11.97,12.12,12.74) (10.73,10.87,11.44) (10.12,10.26,10.79) (9.44,9.57,10.07) (8.29,8.40,8.85)

θ = 0.03(D) (39.70,39.97,41.06) (35.27,35.53,36.55) (43.12,43.42,44.59) (37.55,37.83,38.90) (34.48,34.74,35.77) (30.85,31.10,32.05) (24.14,24.35,25.16)(I) (12.26,12.40,12.98) (11.10,11.24,11.77) (12.58,12.73,13.33) (11.29,11.43,11.98) (10.65,10.78,11.31) (9.95,10.07,10.56) (8.74,8.85,9.29)

θ = 0.05(D) (40.57,40.85,41.91) (36.09,36.35,37.35) (44.07,44.36,45.51) (38.42,38.69,39.74) (35.30,35.56,36.56) (31.61,31.85,32.79) (24.79,24.99,25.78)(I) (12.65,12.79,13.37) (11.47,11.60,12.13) (12.98,13.13,13.73) (11.66,11.80,12.34) (11.01,11.14,11.66) (10.28,10.41,10.89) (9.04,9.15,9.59)

Table 14: The effect of θ on the overall performance of the VSI SH-γ2 control charts inthe presence of measurement errors for B = 1,m = 1, η = 0.28, n = 5, 15, γ0 = 0.05 (leftside), γ0 = 0.1 (middle) and γ0 = 0.2 (right side).


n = 5Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

B = 1(D) (104.67,105.02,106.37) (93.90,94.23,95.55) (117.03,117.40,118.87) (102.18,102.54,103.95) (93.52,93.86,95.22) (83.13,83.45,84.73) (64.04,64.31,65.40)(I) (23.84,24.03,24.84) (22.37,22.55,23.32) (24.34,24.54,25.37) (22.63,22.81,23.59) (21.79,21.97,22.73) (20.90,21.07,21.79) (19.48,19.65,20.32)

B = 3(D) (104.12,104.45,105.73) (93.37,93.69,94.93) (116.42,116.78,118.17) (101.61,101.94,103.28) (92.97,93.29,94.58) (82.62,82.93,84.13) (63.61,63.87,64.90)(I) (23.60,23.78,24.54) (22.14,22.31,23.03) (24.10,24.28,25.06) (22.39,22.57,23.30) (21.57,21.74,22.44) (20.68,20.84,21.52) (19.28,19.43,20.07)

B = 5(D) (104.01,104.34,105.61) (93.27,93.58,94.82) (116.30,116.66,118.04) (101.49,101.83,103.16) (92.86,93.18,94.46) (82.52,82.83,84.03) (63.53,63.79,64.81)(I) (23.55,23.74,24.49) (22.10,22.27,22.98) (24.05,24.23,25.01) (22.35,22.52,23.25) (21.52,21.69,22.39) (20.64,20.80,21.47) (19.24,19.39,20.02)

n = 15Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

B = 1(D) (38.81,39.10,40.21) (34.45,34.72,35.76) (42.17,42.47,43.67) (36.68,36.96,38.06) (33.66,33.93,34.97) (30.09,30.34,31.31) (23.50,23.71,24.54)(I) (11.87,12.02,12.61) (10.74,10.88,11.42) (12.17,12.32,12.94) (10.92,11.06,11.62) (10.30,10.43,10.96) (9.61,9.74,10.24) (8.44,8.55,9.00)

B = 3(D) (38.51,38.78,39.83) (34.17,34.42,35.41) (41.84,42.13,43.26) (36.38,36.65,37.69) (33.38,33.63,34.62) (29.83,30.06,30.98) (23.28,23.48,24.26)(I) (11.74,11.88,12.43) (10.62,10.75,11.26) (12.03,12.18,12.75) (10.79,10.92,11.45) (10.18,10.30,10.80) (9.50,9.61,10.08) (8.34,8.44,8.86)

B = 5(D) (38.45,38.72,39.76) (34.12,34.37,35.34) (41.78,42.07,43.19) (36.33,36.59,37.63) (33.32,33.58,34.56) (29.78,30.01,30.93) (23.24,23.44,24.22)(I) (11.72,11.85,12.40) (10.60,10.72,11.23) (12.01,12.15,12.72) (10.77,10.90,11.42) (10.16,10.28,10.77) (9.48,9.59,10.06) (8.32,8.42,8.84)

Table 15: The effect of B on the overall performance of the VSI SH-γ2 control charts inthe presence of measurement errors for θ = 0.01,m = 1, η = 0.28, n = 5, 15, γ0 = 0.05(left side), γ0 = 0.1 (middle) and γ0 = 0.2 (right side).

n = 5Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

m = 1(D) (107.87,108.20,109.47) (96.99,97.30,98.54) (120.55,120.90,122.28) (105.53,105.87,107.20) (96.71,97.04,98.32) (86.10,86.41,87.62) (66.52,66.78,67.81)(I) (25.26,25.45,26.23) (23.72,23.90,24.64) (25.80,26.00,26.80) (24.00,24.18,24.94) (23.13,23.30,24.03) (22.19,22.35,23.06) (20.70,20.86,21.51)

m = 5(D) (107.87,108.17,109.37) (96.98,97.28,98.45) (120.54,120.87,122.17) (105.52,105.84,107.10) (96.70,97.01,98.22) (86.10,86.38,87.52) (66.51,66.76,67.73)(I) (25.26,25.43,26.17) (23.72,23.89,24.58) (25.80,25.98,26.74) (24.00,24.17,24.88) (23.12,23.29,23.97) (22.18,22.34,23.00) (20.70,20.85,21.46)

m = 10(D) (107.87,108.17,109.36) (96.98,97.28,98.44) (120.54,120.87,122.16) (105.52,105.84,107.09) (96.70,97.01,98.21) (86.09,86.38,87.51) (66.51,66.76,67.72)(I) (25.26,25.43,26.16) (23.72,23.89,24.58) (25.80,25.98,26.73) (24.00,24.17,24.87) (23.12,23.29,23.97) (22.18,22.34,22.99) (20.70,20.84,21.45)

n = 15Ω (0.5, 1.5) (0.3, 1.7) (0.1, 1.1) (0.1, 1.3) (0.1, 1.5) (0.1, 1.9) (0.1, 4.0)

m = 1(D) (40.57,40.85,41.91) (36.09,36.35,37.35) (44.07,44.36,45.51) (38.42,38.69,39.74) (35.30,35.56,36.56) (31.61,31.85,32.79) (24.79,24.99,25.78)(I) (12.65,12.79,13.37) (11.47,11.60,12.13) (12.98,13.13,13.73) (11.66,11.80,12.34) (11.01,11.14,11.66) (10.28,10.41,10.89) (9.04,9.15,9.59)

m = 5(D) (40.57,40.83,41.83) (36.09,36.33,37.27) (44.07,44.34,45.42) (38.42,38.67,39.66) (35.30,35.54,36.48) (31.61,31.83,32.71) (24.78,24.97,25.72)(I) (12.65,12.78,13.32) (11.46,11.59,12.09) (12.98,13.12,13.68) (11.66,11.79,12.30) (11.01,11.13,11.62) (10.28,10.40,10.86) (9.04,9.15,9.56)

m = 10(D) (40.57,40.82,41.82) (36.09,36.33,37.26) (44.07,44.34,45.41) (38.41,38.67,39.65) (35.30,35.54,36.47) (31.61,31.83,32.70) (24.78,24.97,25.72)(I) (12.65,12.78,13.32) (11.46,11.59,12.08) (12.98,13.12,13.68) (11.66,11.79,12.30) (11.01,11.13,11.61) (10.28,10.39,10.85) (9.04,9.14,9.55)

Table 16: The effect of m on the overall performance of the VSI SH-γ2 control charts inthe presence of measurement errors for B = 1, θ = 0.05, η = 0.28, n = 5, 15, γ0 = 0.05(left side), γ0 = 0.1 (middle) and γ0 = 0.2 (right side).

on the performance of vsi shewhart control chart for

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