quality and control control charts for monitoring process

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Research Article

(wileyonlinelibrary.com) DOI: 10.1002/qre.1208

A Comparison Study on Effectiveness and

Robustness of Control Charts for MonitoringProcess Mean and VarianceYanjing Ou,a Di Wen,b Zhang Wua and Michael B. C. Khooc

This article compares the effectiveness and robustness of nine typical control charts for monitoring both process meanand variance, including the most effective optimal and adaptive sequential probability ratio test (SPRT) charts. The nine

charts are categorized into three types (the X type, CUSUM type and SPRT type) and three versions (the basic version,optimal version and adaptive version). While the charting parameters of the basic charts are determined by commonwisdoms, the parameters of the optimal and adaptive charts are designed optimally in order to minimize an indexaverage extra quadratic loss for the best overall performance. Moreover, the probability distributions of the mean

shift and standard deviation shift are studied explicitly as the influential factors in a factorial experiment. Themain findings obtained in this study include: (1) From an overall viewpoint, the SPRT-type chart is more effective than

the CUSUM-type chart and X-type chart by 15 and 73%, respectively; (2) in general, the adaptive chart outperformsthe optimal chart and basic chart by 16 and 97%, respectively; (3) the optimal CUSUM chart is the most effective fixedsample size and sampling interval chart and the optimal SPRT chart is the best choice among the adaptive charts;

and (4) the optimal sample sizes of both the X charts and the CUSUM charts are always equal to one. Furthermore,this article provides several design tables which contain the optimal parameter values and performance indices of 54charts under different specifications. Copyright 2011 John Wiley & Sons, Ltd.

Keywords: extra quadratic loss; quality control; statistical process control; control charts; adaptive charts

1. Introduction

Since it was introduced as a statistical process control (SPC) tool to monitor processes and ensure quality, the control chart has

been increasingly adopted in modern industries and non-manufacturing sectors1--5. When dealing with a quality characteristic

x which is a variable, it is usually necessary to monitor both the mean and variability 6 . The combination of a Shewhart X

chart and a R chart (or an S chart) has been employed widely for this purpose. As online measurement and distributed computing

systems become a norm in todays SPC applications, more sophisticated CUSUM, EWMA and sequential probability ratio test

(SPRT) schemes have also been developed to detect mean shift and standard deviation shift . This includes the omnibus

EWMA chart7 , the MaxEWMA chart8, the WLC chart9, the ABS CUSUM chart10, and some combined CUSUM or EWMA schemes

which is composed of several individual CUSUM or EWMA charts 11, 12.

As more and more new charts have been developed over the decades, it is essential to conduct a systematic study that

evaluates and compares the performance of different charts in a quantitative and analytical manner under various conditions, so

that the SPC users not only roughly know which charts are better than the others, but are also aware of the degree of difference

among the performance of the charts. This kind of information will greatly facilitate the SPC users to select a suitable chart fortheir applications. Usually, a user may not be interested to adopt a complicated chart just for a performance improvement of 1

to 2%. But if the achievable improvement resulting from an advanced chart is more than, say, 20%, one would be willing to put

in effort and spend resources to adopt the advanced chart. The probability distributions of mean shift and standard deviation

shift will be different for different processes. Their influence on the chart performance should also be investigated carefully.

Some comparison studies of chart performance are available in the literature, such as the comparisons of the multivariate EWMA

charts13, the adaptive X, CUSUM charts and a special SPRT chart14, the economic control charts15, the Shewhart and CUSUM

aSchool of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, SingaporebWee Kim Wee School of Communication and Information, Nanyang Technological University, Singapore 639798, SingaporecSchool of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia 11800, MalaysiaCorrespondence to: Yanjing Ou, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore.

E-mail: [email protected]

Copyright 2011 John Wiley & Sons, Ltd.

Published online 22 June 2011 in Wiley Online Library

Qual. Reliab. Engng. Int. 201 , 22 8 3--17


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Y. OU ET AL.

charts for monitoring process mean and variance12 , the non-parametric charts for detecting defect shifts16, the profile monitoring

approaches17, the GLR chart18 and the CUSUM charts for detecting mean shifts19 or both mean and standard deviation shifts10.

However, few of the comparative studies have covered all major types and versions of the control charts for monitoring process

mean and variance. Specifically, seldom are the very effective SPRT charts and adaptive charts included. Many of these studies are

conducted in a qualitative or descriptive manner. Such qualitative approaches are unable to tell the degree of the superiority of

one chart over another or to rank the charts performance properly when several charts are involved in the comparison. Moreover,

almost none of these comparisons have either considered the robustness of chart performance or the probability distributions

of the mean shift and standard deviation shift . In most of the studies, the charting parameters have not been optimized,that is, a chart used for comparison may not be the best one of its type and version.

This article studies the overall performance of nine typical control charts for monitoring process mean and variance in a

quantitative and analytical manner. It includes the X-type, CUSUM-type and SPRT-type charts, and three versions (the basic

version, optimal version and adaptive version) for each type. The optimal and adaptive charts will be designed by an optimization

procedure in which the objective function to be minimized is the average extra quadratic loss (AEQL), which is a measure of the

overall performance of the charts. A factorial experiment will be used to compare the nine charts under different specifications

on false alarm rate, mean shift range and standard deviation shift range. Moreover, the probability distributions of the the mean

shift and standard deviation shift will be handled explicitly as one input factor. The results and findings obtained in this

study, together with a set of design tables, will provide the SPC users with useful aids to select or design a suitable chart for

their application.

In this study, the quality characteristic x is assumed to follow an identical and independent normal distribution with known

in-control 0 and standard deviation 0 (0 and 0 may be estimated from the field records or historical data). When process

shift occurs, the mean and standard deviation of x will change.

=0+0, =0. (1)

Here, and are the mean shift and standard deviation shift, respectively, in terms of0. When a process is in control, =0

and =1. For convenience of discussion, x will be converted to z which follows a standard normal distribution when the process

is in control.

z=x00

. (2)

Since control charts are often used to detect deterioration in product quality in most of the applications 20, 21, this article only

handles the increasing variance shift.

The remainder of this article proceeds as follows. The nine control charts will be introduced in the next section, followed by

a discussion of the general methodologies of the comparative study. The chart performance will be first compared for a general

case, and then further studied in a factorial experiment. Subsequently, the design tables are introduced and explained, and an

illustrative example is presented. The discussions and conclusions are drawn in the last section.

2. Nine control charts

It is impossible to include all the control charts with different types and versions in any comparative study. In this study, nine

typical and representative charts will be investigated. They are categorized into three types ( X, CUSUM, SPRT) and three versions

(basic, optimal, adaptive).

Basic Optimal Adaptive

X Basic X Optimal X VSSI X

CUSUM Basic CUSUM Optimal CUSUM VSSI CUSUM

SPRT Basic SPRT Optimal SPRT VSI SPRT

Each type or version has a distinctive level of performance complexity and a different way of implementation. The X-type

charts22 decide process status based on the observations in the latest sample. The CUSUM-type charts23 consider the data

in all sample points by checking the cumulative sum. The SPRT charts are developed based on the SPRT24. A typical model

of the SPRT chart proposed by Stoumbos and Reynolds 25 will be adopted in this article. This SPRT chart allows the sampling

rate to vary based on the data observed at the current sample. Its statistical properties are evaluated based on an assump-

tion that the time required to obtain an individual observation is short enough to be neglected, relative to the sampling

interval.

The EWMA-type charts are not included as their performance is quite similar to that of the CUSUM-type charts26.

The basic version of each type of charts serves as the baseline for the comparison and its charting parameters are determined

loosely according to common wisdoms and conventions. On the contrary, the parameters of the optimal and adaptive charts will

be determined analytically aiming at the best overall performance. In view of implementation, the basic and optimal charts use

fixed sample size and sampling interval (FSSI). But the adaptive charts (such as the variable sampling intervals (VSI) chart and

Copyright 2011 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 201 , 22 8 3--17


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variable sample sizes and sampling intervals (VSSI) chart) vary the sampling rate as a function of the observed data. They take

samples at a higher rate when there is an indication of a change in the process. In this study, while the adaptive versions of

both the X and CUSUM charts adopt the VSSI model (i.e. adapting both sample sizes and sampling intervals), the adaptive SPRT

chart will use the VSI model (adapting sampling intervals only), as the SPRT chart has already incorporated the adaptive sample

size feature intrinsically. In fact, Ou and Wu27 has found that the VSSI SPRT chart is only slightly more effective than the VSI SPRT

chart.

Brief introductions to the design and implementation of each of the nine charts are given as follows:

(1) The basic X chart: This chart detects the two-sided mean shift and increasing variance shift by checking the absolute value

of the sample mean |z| of z (the standardized x in Equation (2)). It has three parameters: the sample size n, sampling

interval d, and control limit H. When |z|>H, the process is thought to be out of control. According to the common practice

in SPC, the sample size n usually takes a value between four and six in order to make the X chart reasonably sensitive to

both small and large process shifts. In this study, n is set to five for the basic X chart. The sampling interval d depends on

the allowable inspection rate, and the control limit H is determined by the specified false alarm rate.

(2) The optimal X chart: This chart has the same three parameters n, d and H as the basic X chart. However, the sample size n,

as an independent design variable, will be determined by an optimization design. The other two parameters d and H are

dependant design variables and are determined by the two constraints on inspection rate and false alarm rate.

(3) The VSSI X chart: This chart has six parameters: the sample size n1 and n2 (n1d2),

warning limit w and control limit H (w


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d is determined by the constraint on inspection rate which is equal to ASN / d. Finally, the two control limits g and H are

adjusted simultaneously so that the constraints on false alarm rate and ASN are satisfied.

(8) The optimal SPRT chart30: Like the basic SPRT chart, this chart has four parameters d, k, g and H. The sampling interval d

and reference parameter k will be optimized. The two control limits g and H are again adjusted simultaneously in order

to satisfy the constraints on ASN and false alarm rate. The optimal SPRT chart is implemented exactly in the same manner

as the basic SPRT chart.

(9) The VSI SPRT chart27: This chart has six parameters: the sampling interval d1 and d2 (d1>d2), reference parameter k,

warning limit w, and lower and upper control limits g and H. The independent variables d1, d2, k and w will be determinedoptimally. The dependent variables g and H are determined so that the constraints are satisfied. During implementation, if

the statistic |z| of a sample is smaller than w, the longer sampling interval d1 is adopted for the next sample; otherwise,

the shorter sampling interval d2 is to be used.

3. Methodologies of the investigation

3.1. Specifications

To carry out the design of a control chart, all or part of the following four specifications have to be determined:

(1) The allowable inspection rate (R). It is equal to the ratio between the average sample size and the average sampling

interval. The value of R depends on the available resources such as manpower and instrument. Usually, only the in-control

(or long run) value of R is considered, because a process often runs in an in-control condition for a long period and only

occasionally falls into an out-of-control status for a short time period. The inspection rate in the short out-of-control period

has little influence on the long run value of R and is of much less concern31 .

(2) The minimum allowable value () of the in-control ATS0 . The value of is decided based on the requirement on false alarm

rate.

(3) The maximum value (,max) of the mean shift . The corresponding (,max) of the standard deviation shift is usually

set as (,max+1)12. The values of ,max and ,max may be chosen based on the knowledge about a process (e.g. the

maximum possible shift in a process) or on the shift domain, the users are interested to investigate.

(4) The minimum allowable value (dmin) of sampling interval d. The value ofdmin is determined by some practical considerations,

such as the amount of inspection that can be finished in a short time period or the assumption for an SPRT chart that d

must be significantly larger than the time for taking an observation.

In this study, the time unit is made equal to the time period in which one unit (or one product) can be inspected. For example,

if the available resource allows five units to be inspected per hour, the time unit is 12 (=60

5 ) min. This setting has a benefit thatthe inspection rate R always equals one. It only influences the scaling of the sampling interval d, but has no effect on the results

of comparison. In addition, the minimum sampling interval dmin is fixed at 0.3, in terms of the standard time unit. As R and dminare fixed, only the remaining specifications , ,max and ,max have to be determined.

3.2. Performance measures

The in-control and out-of-control performance of a control chart is usually measured by the average time to signal (ATS). The

in-control ATS0 must be large enough so that false alarm occurs infrequently. On the other hand, the out-of-control ATS should

be short enough in order to detect the process shifts quickly. In this study, out-of-control ATS is computed under the steady-state

mode. It assumes that the process has reached its stationary distribution at the time when the process shift occurs. While the ATS

values of the basic and optimal X charts are calculated by simple formulae, the ATS values of other seven charts are determined

by Markov chain procedures. For example, for a VSSI CUSUM chart, the zone between zero and control limit H is divided into m

in-control transient states (m is set to 100 in this study). The width of the interval of each state is equal to H /(m0.5), and

the center Oi of state i is equal to i (i=0,1,. . ., m1). The ATS of a VSSI CUSUM chart is calculated by:

ATSm1= (ImmRmm)1 Dm1

where ATS and D are the ATS vector and sampling interval vector, respectively, I is an identify matrix and R is the transition

matrix. If the center Oi of state i is larger than the warning limit w, the corresponding element di in the sampling interval vector

D is equal to the shorter sampling interval d2 and the sample size used to calculate the transition probability pij (from state i to

state j in matrix R) is equal to the larger sample size n2. Conversely, if Oi is smaller than w, di is equal to the longer sampling

interval d1 and the sample size takes the smaller one n1.

Since it is difficult to predict the actual magnitudes and types of process shifts for most of the applications, many

researchers12, 26, 32 pointed out that a control chart must have an excellent overall performance in a broad shift domain, that is,

being effective for detecting both small and large process shifts of different combinations of and .

In this article, the overall effectiveness of the control charts will be measured by the AEQL and average ratio of ATS (ARATS)33 .

The index AEQL stems from the quadratic loss function

34

.



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AEQL=1

,max (,max1)

,max0

,max1

(2+21)ATS(,) d d. (5)

It is actually a weighted average ATS over the shift domain of (0


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Table I. The nine control charts in the general case (=370,,max=4.5,,max=5.5)

Parameters

Chart n or n1 n2 d or d1 d2 k g w H ASN AEQL PCI ARATS

Basic x 5 5.0000 1.1046 70.8180 3.6878 (9) 3.5827 (9)

Optimal x 1 1.0000 2.9997 26.0932 1.3588 (6) 1.4458 (6)

VSSI x 1 1 1.1419 0.3000 1.3735 2.9990 21.9421 1.1426 (4) 1.1885 (4)Basic CUSUM 1 1.0000 1.0000 3.8811 30.3176 1.5788 (7) 1.6450 (8)

Optimal CUSUM 1 1.0000 1.6500 1.4878 24.9708 1.3003 (5) 1.3561 (5)

VSSI CUSUM 1 1 1.1419 0.3000 1.4081 0.1148 1.9338 20.8559 1.0861 (3) 1.1026 (3)

Basic SPRT 3.0000 1.2000 0.8671 2.3913 2.9986 31.8586 1.6590 (8) 1.6052 (7)

Optimal SPRT 1.1500 1.4338 0.2102 1.8694 1.1497 19.4424 1.0125 (2) 1.0170 (2)

VSI SPRT 1.2500 0.8081 1.4418 0.1904 1.1500 1.8536 1.1527 19.2033 1.0000 (1) 1.0000 (1)

than others. It is observed that the differences among the ATS values of the nine charts are very substantial. Specifically, the

SPRT-type charts significantly outperform the CUSUM-type charts and the latter are substantially more effective than the X-type

charts. On the other hand, within each type of charts, the adaptive version generally outperforms the optimal version which in

turn surpasses the basic version. It is noted that the VSI SPRT chart always produces the lowest, or nearly the lowest, ATS across

the entire shifts domain compared with other charts. However, the superiority of the VSI SPRT chart over the optimal SPRT chartis generally minor.

The values of the two overall performance indices AEQL (Equation (5)) and ARATS (Equation (6)) are shown in Table I, together

with another index, Performance Comparison Index (PCI). The index PCI is the ratio between the AEQL of a chart and the AEQL of

the best chart under the same conditions (i.e. the same specified values of , ,max and ,max, as well as the same probability

distributions of and that will be discussed shortly). This index PCI facilitates the performance comparison and ranking based

on AEQL. Under a given condition, the chart with the lowest AEQL has a PCI value equal to one, and the PCI values of all other

charts are larger than one. Since the VSI SPRT chart has the smallest AEQL in this general case, it stands as the best chart and

PCI=AEQL

AEQLbest=

AEQL

AEQLVSI SPRT. (11)

The VSI SPRT chart is also used, in this general case, as the benchmark for calculating ARATS in Equation (6). These three indices,

AEQL, ARATS, and PCI, provide fairly comprehensive information regarding the comparison of the overall performance of the

charts.

In Table I, the numbers within the parentheses under columns PCI and ARATS indicate the performance ranking of the charts,

with (1) for the most effective chart and (9) for the least effective one. The ranking based on PCI is as follows:

VSI SPRT,optimal SPRT, VSSI CUSUM, VSSI X, optimal CUSUM, optimal X, basic CUSUM, basic SPRT, basic X chart. (12)

It is interesting to note that the ranking based on ARATS is almost the same except one exchange of the ranking positions

between the basic CUSUM and basic SPRT charts. Since both PCI and ARATS usually deliver similar results about the relative

performance of the charts, only PCI will be pursued in the following discussions.

From the chart rankings, it is also noted that, without the optimization design, a basic SPRT chart or a basic CUSUM chart

may be less effective than an optimal X chart.

A joint probability distribution of and must exist in any process. However, as aforementioned, it is usually unknown and

difficult to be estimated. As a result, one may have to design the control charts based on an assumed uniform distribution over

the shift domain. A concern arises on how well these charts will work if the actual distribution of and is quite different

from a uniform one. A test has been carried out for this general case. The control charts designed based on the assumption of

uniform distribution (as listed in Table I) are now applied to three different circumstances. In each circumstance, each of and

follows a beta probability distribution.

f () =(a+b)

(a)(b)a1 (,max)

b1

(,max)a+b1,

f () =(a+b)

(a)(b)

(1)a1 (,max)

b1

(,max1)a+b1.

(13)

These beta distributions serve as the representatives of different types of non-uniform marginal probability distributions of and (Figure 1). The skewness of a beta distribution depends on the parameters a and b. Here a and b are the parameters

for distribution f (), whereas a and b are for f (). If (ab), the probability

distribution is skewed to the left (Figure 1(c)). This arises when most of the shifts cluster to the upper end. Finally, if ( a=

b), the



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Table II. ATS values of the nine control charts in the general case (=370,,max=4.5,,max=5.5)

Chart 0 0.9 1.8 2.7 3.6 4.5

1.0 Basic x 370.0 12.95 2.819 2.501 2.500 2.500

Optimal x 370.0 55.28 8.186 2.116 0.878 0.572

VSSI x 369.7 48.39 5.293 1.237 0.678 0.572Basic CUSUM 369.2 22.23 4.363 2.099 1.354 0.973

Optimal CUSUM 369.7 40.05 4.909 1.613 0.847 0.583

VSSI CUSUM 369.4 27.74 2.690 1.029 0.694 0.587

Basic SPRT 369.9 15.85 1.943 1.529 1.502 1.500

Optimal SPRT 369.9 27.23 2.192 0.800 0.605 0.577

VSI SPRT 369.9 25.91 2.009 0.779 0.615 0.593

1.9 Basic x 23.32 9.577 3.799 2.656 2.508 2.500

Optimal x 8.242 5.967 3.208 1.780 1.102 0.774

VSSI x 6.770 4.815 2.497 1.368 0.887 0.684

Basic CUSUM 7.034 5.517 3.405 2.142 1.458 1.066

Optimal CUSUM 7.054 5.195 2.881 1.661 1.070 0.773

VSSI CUSUM 5.218 3.838 2.138 1.275 0.881 0.697

Basic SPRT 3.889 3.108 2.183 1.756 1.590 1.529Optimal SPRT 4.796 3.487 1.905 1.132 0.801 0.660

VSI SPRT 4.540 3.308 1.826 1.106 0.798 0.668

2.8 Basic x 10.74 7.708 4.436 3.056 2.618 2.517

Optimal x 3.021 2.741 2.150 1.591 1.183 0.913

VSSI x 2.420 2.200 1.738 1.308 1.003 0.809

Basic CUSUM 3.304 3.056 2.503 1.935 1.485 1.161

Optimal CUSUM 2.789 2.551 2.038 1.538 1.164 0.910

VSSI CUSUM 2.142 1.976 1.614 1.260 0.995 0.817

Basic SPRT 2.243 2.161 1.981 1.807 1.680 1.599

Optimal SPRT 1.938 1.790 1.469 1.158 0.929 0.779

VSI SPRT 1.866 1.728 1.430 1.139 0.924 0.783

3.7 Basic x 7.412 6.371 4.618 3.429 2.839 2.601

Optimal x 1.895 1.819 1.627 1.389 1.162 0.973VSSI x 1.552 1.495 1.351 1.173 1.006 0.869

Basic CUSUM 2.238 2.158 1.951 1.686 1.422 1.193

Optimal CUSUM 1.818 1.751 1.578 1.360 1.149 0.970

VSSI CUSUM 1.469 1.422 1.302 1.150 1.001 0.874

Basic SPRT 1.918 1.896 1.839 1.767 1.697 1.637

Optimal SPRT 1.347 1.307 1.203 1.073 0.945 0.837

VSI SPRT 1.317 1.280 1.183 1.061 0.942 0.840

4.6 Basic x 5.956 5.496 4.529 3.649 3.065 2.742

Optimal x 1.444 1.415 1.333 1.219 1.093 0.972

VSSI x 1.217 1.195 1.136 1.053 0.962 0.874

Basic CUSUM 1.742 1.708 1.612 1.476 1.323 1.171

Optimal CUSUM 1.412 1.385 1.310 1.204 1.085 0.969

VSSI CUSUM 1.189 1.170 1.118 1.044 0.961 0.879Basic SPRT a1.788 1.779 1.755 1.721 1.682 1.644

Optimal SPRT 1.108 1.092 1.048 0.985 0.915 0.846

VSI SPRT 1.094 1.080 1.039 0.980 0.914 0.849

5.5 Basic x 5.153 4.914 4.348 3.729 3.229 2.89

Optimal x 1.208 1.194 1.153 1.092 1.019 0.941

VSSI x 1.046 1.035 1.006 0.962 0.910 0.855

Basic CUSUM 1.460 1.442 1.391 1.314 1.222 1.123

Optimal CUSUM 1.194 1.180 1.142 1.084 1.014 0.940

VSSI CUSUM 1.038 1.028 1.002 0.962 0.913 0.861

Basic SPRT 1.719 1.714 1.702 1.684 1.662 1.638

Optimal SPRT 0.981 0.973 0.950 0.917 0.876 0.832

VSI SPRT 0.976 0.968 0.947 0.916 0.877 0.836




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Figure 1. Three sets of Beta probability distributions of (,): (a) Set 1; (b) Set 2; and (c) Set 3

probability distribution is symmetrical (Figure 1(b)). In this article, three different circumstances or different combinations of the

values of (a, b, a and b) for the marginal probability distributions of and will be studied. Table III lists these three

circumstances together with the uniform distribution one.

With these given probability distributions f () and f (), the AEQL produced by the control charts is calculated by thefollowing formula:

AEQL=

,max0

,max1

(2+21)ATS(,)f ()f () d d. (14)

It is similar to Equation (5) except that the uniform density function (1/(,max(,max1))) in Equation (5) is replaced by

f ()f (). Now each of the nine control charts that are designed based on the uniform assumption is applied to three processes

with different f () and f () as shown in Figure 1. That is, the AEQL value will be calculated by Equation (14). The resultant

AEQL values under the three beta distributions, as well as the AEQL values under the uniform distribution, are displayed in the

top half of Table III. As expected, the AEQL values produced by all nine charts become smaller when the probability distributions

f () and f () are skewed to the right, and they become larger when f () and f () are skewed to the left. The bottom

half of Table III shows the PCI (the ratio of AEQL/ AEQLbest) values for each circumstance. For example, each PCI value under the

column skew to left is obtained by dividing each AEQL value under that column by the AEQL value of the best chart (i.e. the

VSI SPRT chart) that also appears in the same column. It can be seen that the VSI SPRT chart outdoes all other charts in terms of



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Table III. Effect of the probability distributions of and (=370,,max=4.5,,max=5.5)

AEQL

Beta

Skew to right Symmetrical Skew to left

(a=2, b=4) (a=b=3) (a=4, b=2)

Uniform (a=2, b=4) (a=b=3) (a=4, b=2)

Basic x 70.8180 46.5188 62.4804 84.3568

Optimal x 26.0932 21.7717 24.8483 30.6235

VSSI x 21.9421 17.5720 20.6543 26.3897

Basic CUSUM 30.3176 23.3255 29.6059 37.1621

optimal CUSUM 24.9708 19.9381 23.9713 30.1687

VSSI CUSUM 20.8559 15.4960 19.8134 26.1067

Basic SPRT 31.8586 18.0428 28.9390 43.2999

Optimal SPRT 19.4424 14.1193 18.3339 24.6068

VSI SPRT 19.2033 13.6797 18.0570 24.4740

PCI

BetaUniform Skew to right Symmetrical Skew to left Average

Basic x 3.6878 (9) 3.4006 (9) 3.4602 (9) 3.4468 (9) 3.4988 (9)

Optimal x 1.3588 (6) 1.5915 (7) 1.3761 (6) 1.2513 (6) 1.3944 (6)

VSSI x 1.1426 (4) 1.2845 (4) 1.1438 (4) 1.0783 (4) 1.1623 (4)

Basic CUSUM 1.5788 (7) 1.7051 (8) 1.6396 (8) 1.5184 (7) 1.6105 (8)

Optimal CUSUM 1.3003 (5) 1.4575 (6) 1.3275 (5) 1.2327 (5) 1.3295 (5)

VSSI CUSUM 1.0861 (3) 1.1328 (3) 1.0973 (3) 1.0667 (3) 1.0957 (3)

Basic SPRT 1.6590 (8) 1.3190 (5) 1.6027 (7) 1.7692 (8) 1.5875 (7)

Optimal SPRT 1.0125 (2) 1.0321 (2) 1.0153 (2) 1.0054 (2) 1.0163 (2)

VSI SPRT 1.0000 (1) 1.0000 (1) 1.0000 (1) 1.0000 (1) 1.0000 (1)

AEQL under all circumstances. The optimal SPRT chart is also very competitive and always has an AEQL value very close to that

of the VSI SPRT chart.

The numbers inside the parentheses in the bottom half of Table III indicate the rankings of the charts based on PCI in each

column. It is found that the rankings in each column are similar to the ranking under the uniform assumption with no or only

one or two exchange of chart ranking positions. This implies that, if a chart is designed using a uniform assumption, the relative

performance of this chart (or its ranking position) will be nearly the same even if f () and f () are non-uniform.

The above discussions show that the probability distributions of and have some, but quite limited, influence on the

relative performance of the control charts. Usually, if a chart that is designed by using uniform f () and f () has a better (or

worse) performance compared with other charts, it also has better (or worse) performance under different non-uniform f ()

and f ().

5. A factorial experiment

Next, a factorial experiment is carried out in order to further study and compare the performance of the nine charts. Three

factors will be considered including (1) ; (2) ,max (,max=,max+1); and (3) the parameters (a, b,a, b) of the probability

distributions. Each of these factors will be studied in two levels:

Factor Level 1 Level 2

1 200 600

2 ,max 3 6

,max 4 7

3 f () a=2, b=4, skew to right a=4, b=2, skew to left

f () a=2, b=4, skew to right a=4, b=2, skew to left


1



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Table IV. PCI values of the charts in a factorial experiment

Case 1 2 3 4 5 6 7 8

=200 =600

(,max=3,,max=4) (,max=6,,max=7) (,max=3,,max=4) (,max=6,,max=7)

a=2 a=4 a=2 a=4 a=2 a=4 a=2 a=4

b=4 b=2 b=4 b=2 b=4 b=2 b=4 b=2

a=2 a=4 a=2 a=4 a=2 a=4 a=2 a=4

Chart b=4 b=2 b=4 b=2 b=4 b=2 b=4 b=2 PCI SPCI

Basic x 2.9824 3.1536 3.3807 3.6203 3.5811 3.3785 3.6288 3.7083 3.4292 (9) 0.2557

Optimal x 1.6354 1.3380 1.2949 1.1322 2.1550 1.5257 1.4881 1.2119 1.4727 (6) 0.3227

VSSI x 1.3276 1.1269 1.1034 1.0429 1.6733 1.2178 1.2020 1.0718 1.2207 (4) 0.2046

Basic CUSUM 1.5474 1.5145 1.4452 1.2924 1.9008 1.8540 1.7545 1.5143 1.6029 (8) 0.2121

Optimal CUSUM 1.4506 1.3019 1.2524 1.1262 1.7548 1.4541 1.3945 1.2019 1.3670 (5) 0.1958

VSSI CUSUM 1.1259 1.0970 1.0613 1.0511 1.2115 1.1445 1.0992 1.0743 1.1081 (3) 0.0523

Basic SPRT 1.1103 1.4995 1.5471 1.9944 1.0936 1.4708 1.5098 1.9843 1.5262 (7) 0.3367

Optimal SPRT 1.0000 1.0205 1.0066 1.0174 1.0324 1.0044 1.0000 1.0194 1.0126 (2) 0.0116

VSI SPRT 1.0013 1.0000 1.0000 1.0000 1.0000 1.0000 1.0042 1.0000 1.0007 (1) 0.0015

Figure 2. The PCI values for nine charts under different cases: (a) X chart; (b) CUSUM chart; and (c) SPRT chart

It results in eight different cases or combinations of the three factors. The general case discussed in the last section is nearly

at the center of this experiment space.

The data (the PCI values) obtained from this experiment are displayed in Table IV. As before the PCI values are calculated to

indicate the AEQL values of the charts relative to the AEQL value of the best chart, for each case or under each column. For example,

in case 2, the AEQL value of each chart is first determined under (=200,,max=3,,max=4, (a=4, b=2), (a=4, b=2)). Since

the VSI SPRT chart has the smallest AEQL in this case, then the PCI value of each chart is obtained by dividing its AEQL by the

AEQL of the VSI SPRT chart. In this factorial experiment, the VSI SPRT chart usually outdoes all other charts, in terms of AEQL

under many cases. However, the optimal SPRT chart is also very competitive. Especially in cases 1 and 7, the AEQL of the optimal

SPRT is slightly smaller than that of the VSI SPRT chart. In these two cases, PCI=AEQL / AEQLoptimal SPRT, as the best chart here

is the optimal SPRT chart rather than the VSI SPRT chart.

The rightmost two columns in Table IV display the sample mean PCI and sample standard deviation SPCI of the PCI values of

each chart over the eight cases. The sample mean PCI is the most holistic measure of the effectiveness of a chart, considering

not only different specifications , ,max and ,max, but also various probability distributions f () and f (). According to

PCI, the charts are ranked as follows:

VSI SPRT, optimal SPRT, VSSI CUSUM, VSSI X, optimal CUSUM, optimal X, basic SPRT, basic CUSUM, basic X chart. (15)

The sample standard deviation SPCI of PCI is a measure of the variability of the PCI values of a chart under different cases. If

a chart has a smaller SPCI, its PCI value may vary less under different situations, and this chart is more robust in performance.

For example, the SPCI of the VSI SPRT chart is very small, because the PCI values of this chart are always equal or very close to

one. On the contrary, the basic SPRT chart has a very large SPCI, because its PCI value is relatively small for some cases (e.g. 1.0936



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Figure 3. Overall comparison of the nine charts in terms of PCI

in case 5) and becomes very large for other cases (e.g. 1.9944 in case 4). This reveals that the degree of the inferiority of the

basic SPRT chart to the best chart alters severely under different conditions.

Figure 2 also shows the effectiveness and stability of the nine charts under different cases in terms of PCI. Within each type

of control charts ( X, CUSUM and SPRT), the VSSI version always outperforms the optimal version, whereas the optimal version

always outdoes the basic version. In addition, it is obvious that the PCI values of the basic SPRT chart and optimal X chart are

least robust under different conditions.

Figure 3 illustrates the PCI values of the nine charts. It clearly shows that the SPRT-type charts outperform the CUSUM-type

charts and the latter outdo the X-type charts. The grand average PCISPRT of the SPRT-type charts (i.e. the average of the PCI

values of the basic SPRT, optimal SPRT and VSI SPRT charts) is equal to 1.180. Similarly, the grand average PCICUSUM is 1.359 for

the CUSUM-type charts and PCIX is 2.041 for theX-type charts. This indicates that, from an overall viewpoint, the SPRT chart is

more effective than the CUSUM and X charts by 15 and 73%, respectively. Within each type, the adaptive version outperforms

the optimal version and the latter surpasses the basic version. The grand averages of the PCI values for the adaptive, optimal

and basic versions are PCIadaptive=1.110, PCIoptimal=1.284 and PCIbasic=2.186. This reveals that the adaptive chart outperforms

the optimal chart and basic chart by 16 and 97%, respectively.

6. Design tables

The earlier discussions highlight the importance of properly selecting and designing a control chart for an SPC application. Thissection provides three design tables (Tables VVII) for three specified values of (=200, 370, 600). In each table, ,max and

,max are specified at three levels (,max=3, 4.5,6,,max=4, 5.5, 7). The users can select a chart for which the tabulated values

of , ,max and ,max are closest to the desirable values for their application. These three design tables should cover many

applications in SPC practice.

For each set of specified , ,max and ,max, the charting parameters of the optimal and adaptive versions of the X, CUSUM and

SPRT charts are listed. The basic versions are not recommended, because they are substantially less effective than the optimal versions.

As aforementioned, the sampling intervals, as well as the specification , in the design tables are expressed in terms of a

standard time unit equal to the time period in which one product is inspected. This setting will be further explained in the next

section of the example.

It is found with surprise that the optimal sample sizes (including n1 and n2) of all charts in all three tables are always equal to

one for detecting both and , which is quite different from some conventional settings, such as ( n=4) for X charts. Moreover,

the fact that the two sample sizes n1 and n2 are always the same for the VSSI version reveals that a VSI chart is actually as

effective as a VSSI chart for detecting both and .


3



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Table V. Design table (=200)

,max ,max Chart n1 n2 d1 d2 k g w H ASN PCI R PCIS PCIL PCIAVE

3 4 Optimal x 1 1.0000 2.8070 1.6354 1.4731 1.3380 1.4822

VSSI x 1 1 1.2500 0.3000 0.0000 1.1122 2.8063 1.3276 1.2041 1.1269 1.2195

Optimal CUSUM 1 1.0000 1.4419 1.5942 1.4506 1.3863 1.3019 1.3796

VSSI CUSUM 1 1 1.1919 0.3000 1.2500 0.2030 2.0411 1.1259 1.1200 1.0970 1.1143

Optimal SPRT 1.3081 1.3081 0.0428 1.8835 1.3087 1.0000 1.0119 1.0205 1.0108

VSI SPRT 1.3000 0.7581 1.3419 0.1958 1.1581 1.7945 1.1984 1.0013 1.0000 1.0000 1.0004

4.5 5.5 Optimal x 1 1.0000 2.8070 1.4302 1.2895 1.1991 1.3063

VSSI x 1 1 1.1419 0.3000 1.3682 2.8061 1.1887 1.1050 1.0620 1.1186

Optimal CUSUM 1 1.0000 1.6000 1.3313 1.34264 1.2570 1.1869 1.2622

VSSI CUSUM 1 1 1.1419 0.3000 1.3500 0.1733 1.7834 1.0914 1.0791 1.0604 1.0770

Optimal SPRT 1.1500 1.5419 0.0424 1.4199 1.1497 1.0275 1.0177 1.0118 1.0190

VSI SPRT 1.2000 0.6500 1.5500 0.1038 1.3000 1.4075 1.1270 1.0000 1.0000 1.0000 1.0000

6 7 Optimal x 1 1.0000 2.8070 1.2949 1.1907 1.1322 1.2060

VSSI x 1 1 1.1419 0.3000 1.3682 2.8061 1.1034 1.0606 1.0429 1.0690

Optimal CUSUM 1 1.0000 1.7581 1.1153 1.2524 1.1758 1.1262 1.1848

VSSI CUSUM 1 1 1.1419 0.3000 1.4000 0.1130 1.6738 1.0613 1.0581 1.0511 1.0569

Optimal SPRT 1.1500 1.7081 0.1656 1.1770 1.1492 1.0066 1.0118 1.0174 1.0119

VSI SPRT 1.1419 0.9081 1.5500 0.2581 1.1081 1.4002 1.0873 1.0000 1.0000 1.0000 1.0000

Table VI. Design table (=370)


3 4 Optimal x 1 1.0000 2.9997 1.9065 1.6392 1.4423 1.6627

VSSI x 1 1 1.3000 0.3000 1.0346 2.9983 1.5029 1.2961 1.1798 1.3263

Optimal CUSUM 1 1.0000 1.4419 1.8604 1.6119 1.5089 1.3922 1.5043

VSSI CUSUM 1 1 1.1919 0.3000 1.3081 0.1194 2.1964 1.1896 1.1525 1.1123 1.1515

Optimal SPRT 1.3581 1.2581 0.0730 2.3552 1.3593 1.0000 1.0149 1.0266 1.0138

VSI SPRT 1.3500 0.7000 1.3081 0.2314 1.1338 2.1848 1.2164 1.0070 1.0000 1.0000 1.0023

4.5 5.5 Optimal x 1 1.0000 2.9997 1.5915 1.3761 1.2513 1.4063VSSI x 1 1 1.1419 0.3000 1.3735 2.9990 1.2845 1.1438 1.0783 1.1689

Optimal CUSUM 1 1.0000 1.6500 1.4878 1.4575 1.3275 1.2327 1.3392

VSSI CUSUM 1 1 1.1419 0.3000 1.4081 0.1148 1.9338 1.1328 1.0973 1.0667 1.0989

Optimal SPRT 1.1500 1.4338 0.2102 1.8694 1.1497 1.0321 1.0153 1.0054 1.0176

VSI SPRT 1.2500 0.8081 1.4419 0.1904 1.1500 1.8536 1.1527 1.0000 1.0000 1.0000 1.0000

6 7 Optimal x 1 1.0000 2.9997 1.4022 1.2553 1.1753 1.2776

VSSI x 1 1 1.1419 0.3000 1.3735 2.9990 1.1585 1.0880 1.0577 1.1014

Optimal CUSUM 1 1.0000 1.7581 1.3333 1.3346 1.2335 1.1673 1.2451

VSSI CUSUM 1 1 1.1419 0.3000 1.4081 0.1148 1.9338 1.0851 1.0760 1.0623 1.0745

Optimal SPRT 1.1500 1.6081 0.0217 1.5525 1.1492 1.0083 1.0141 1.0174 1.0133

VSI SPRT 1.1419 0.4919 1.6500 0.1696 1.4743 1.4854 1.0850 1.0000 1.0000 1.0000 1.0000

The rightmost four columns in each design table enumerate the values of PCIR , PCIS , PCIL and PCIAVE for each chart. Here,

PCIR carries the PCI value of the chart when the real probability distributions of f () and f () are the beta distributions

that are skewed to the right. Similarly, PCIS or PCIL is related to the beta distributions that are symmetrical or skewed to the

left. Finally, PCIAVE is the average of PCIR, PCIS and PCIL. Usually, the users do not know the accurate distributions of and .

However, they may have some general and rough idea about them. For example, they may be able to tell that most of the mean

shifts and standard deviation shifts cluster to the lower end, that is, having f () and f () skewed to the right. Under such

a circumstance, they may make use of the data under the column of PCIR . On the other hand, if the users have no idea at all

about f () and f (), they can refer to PCIAVE which takes all different distributions into consideration.

In addition to detection effectiveness, simplicity in implementation is also an important consideration when selecting a control

chart. The adaptive charts (the VSSI X, VSSI CUSUM, optimal SPRT and VSI SPRT charts) generally have higher detection effectiveness

than the FSSI charts (the optimal X and optimal CUSUM charts), because these adaptive charts allow the sampling rate to vary

at each sampling point. However, it will in turn increase the difficulty to run the charts. This operational complexity cannot be

overcome just by means of a computer, because changes in sample sizes and sampling intervals must be carried out by the



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Table VII. Design table (=600)


3 4 Optimal x 1 1.0000 3.1440 2.1550 1.7850 1.5257 1.8219

VSSI x 1 1 1.3000 0.3000 1.0362 3.1435 1.6733 1.3785 1.2178 1.4232

Optimal CUSUM 1 1.0000 1.4919 1.9569 1.7548 1.6065 1.4541 1.6051

VSSI CUSUM 1 1 1.2419 0.3000 1.2500 0.1178 2.6551 1.2115 1.1875 1.1445 1.1812

Optimal SPRT 1.2581 1.2581 0.2337 2.6093 1.2639 1.0324 1.0157 1.0044 1.0175

VSI SPRT 1.3581 0.8081 1.2581 0.2209 1.1757 2.6104 1.2711 1.0000 1.0000 1.0000 1.0000

4.5 5.5 Optimal x 1 1.0000 3.1440 1.7512 1.4608 1.3019 1.5047

VSSI x 1 1 1.1919 0.3000 1.2389 3.1433 1.3715 1.1852 1.1037 1.2202

Optimal CUSUM 1 1.0000 1.6500 1.6633 1.5631 1.3964 1.2787 1.4127

VSSI CUSUM 1 1 1.1419 0.3000 1.4000 0.1291 2.1722 1.1711 1.1216 1.0807 1.1244

Optimal SPRT 1.2000 1.3919 0.1536 2.1893 1.1990 1.0221 1.0157 1.0106 1.0161

VSI SPRT 1.2500 0.6000 1.4419 0.1332 1.3081 2.0677 1.1777 1.0000 1.0000 1.0000 1.0000

6 7 Optimal x 1 1.0000 3.1440 1.4881 1.3057 1.2119 1.3352

VSSI x 1 1 1.1419 0.3000 1.3759 3.1434 1.2020 1.1089 1.0718 1.1276

Optimal CUSUM 1 1.0000 1.7581 1.4989 1.3945 1.2771 1.2019 1.2912

VSSI CUSUM 1 1 1.1419 0.3000 1.4000 0.1291 2.1722 1.0992 1.0896 1.0743 1.0877

Optimal SPRT 1.1500 1.5081 0.1208 1.9201 1.1492 1.0000 1.0132 1.0194 1.0109

VSI SPRT 1.1419 0.4919 1.7081 0.1323 1.4743 1.5717 1.0809 1.0042 1.0000 1.0000 1.0014

operators or an automatic inspection system. Among the four adaptive charts, the implementation of the optimal SPRT chart is

relatively easier, because this chart uses a fixed sampling interval and only adapts the sample size. On the other hand, the FSSI

charts use FSSI; therefore, they are simpler for operation. The optimal X chart is especially easy for implementation. However,

if a computer-aided SPC system is in use, there is almost no difference in the difficulty of running the two FSSI charts (i.e. the

optimal X and optimal CUSUM charts).

The four types of PCI in the design tables facilitate the users to consider both effectiveness of performance and simplicity in

operation when selecting a chart. If the PCI value shows that the improvement gained by a complicated chart in effectiveness

is insignificant, it may not be worthwhile to replace a simpler chart by this complicated one. On the contrary, if the achievable

improvement in terms of PCI is substantial, the users may believe that the gain in effectiveness is sufficient to outweigh the

increase in complexity.

As a general guideline, the optimal CUSUM chart and optimal SPRT chart are recommended for most of the SPC applications.The optimal CUSUM chart is the most effective and robust FSSI chart and can be implemented as easily as the optimal X chart

when an on-site computer is available. This chart should be selected when the users prefer to using FSSI. The optimal SPRT chart

is the best adaptive chart and should be selected when detection speed is critical and the users would like to adopt the adaptive

feature for better performance. This chart is almost as effective as the VSI SPRT chart and is considerably more effective than the

VSSI X and VSSI CUSUM charts. Meanwhile, the optimal SPRT chart enjoys the ease of using a fixed sampling interval. In all other

three adaptive charts (i.e. the VSSI X, VSSI CUSUM and VSI SPRT charts), both sample sizes and sampling intervals will have to be

changed during the operation.

7. An example

This example involves a polymerization process in a silicon-filler manufacturing factory36. The density x of the silicon filler is a

key quality characteristic. The probability distribution of x is approximately normal and the in-control standard deviation 0 is

estimated to be 0.001g/cm3. The in-control mean 0 can be easily adjusted to the nominal value of 1.145g/cm3. In order tostandardize the design and operation, the density x is converted to z, conforming to a standard normal distribution when the

process is in control.

z=x1.145

0.001.

The inspection rate is 1 unit/h and the in-control ATS0 is specified as 350 h by the Quality Assurance (QA) engineer. Thus, the

standard time unit is equal to 1 h or 60 min, and the tabulated value for in the design tables should be 350. The QA engineer

is interested to study the chart performance in a mean shift range (0


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Y. OU ET AL.

decides to adopt the optimal SPRT chart for monitoring the density x. The parameters of the control charts can be obtained

from Table VI.

It is noted that the sampling interval for the optimal SPRT chart is d=1.150060=69 minutes. In order to facilitate the

implementation, d is rounded off to 70 min. This adjustment will change the value of ATS0 to 375 (1.35% bigger than the original

value of 370) and decrease the value of r0 to 0.9855 (1.45% smaller). Both changes are minor and tolerable.

The values of AEQLVSSIx/ AEQLoptimal SPRT and AEQLVSSI CUSUM/ AEQLoptimal SPRT are equal to 1.1487 and 1.0799, respec-

tively. This means that the average speed of the optimal SPRT chart for detecting out-of-control cases is higher than that of the

VSSI X and VSSI CUSUM charts by 14.87 and 7.99%, respectively. Meanwhile, the optimal SPRT chart is easier for implementationas it uses fixed sampling interval.

8. Discussions and conclusions

Promptly detecting both mean shift and standard deviation shift is the most challenging task in SPC for variables. This article

carries out a systematic and analytic comparison among three main types of control charts ( X, CUSUM and SPRT charts), each

with three versions (basic, optimal and adaptive versions). The comparisons are conducted under various conditions, considering

not only different in-control ATS0 , mean shift ranges and standard deviation shift ranges, but also various probability distributions

of and .

This study ranks the charts in line of both types and versions according to detection effectiveness. The main findings include

that the SPRT-type chart is more effective than the CUSUM-type chart and X-type chart by 15 and 73%, respectively; the adaptive

chart outperforms the optimal chart and basic chart by 16 and 97%, respectively; the optimal CUSUM chart is the most effectiveFSSI chart and the optimal SPRT chart is the best choice among the adaptive charts; the optimal sample sizes (including n1 and

n2) of all charts are always equal to one for detecting both and . Such information will facilitate the users to design or

select a chart based on both superiority in performance and simplicity in implementation.

Both the optimal and adaptive charts are designed by an optimization procedure using AEQL as the objective function. The

optimization design of the control charts is strongly recommended, as it can significantly improve the overall performance of the

charts and meanwhile does not increase the difficulty in implementation.

This study also provides several design tables containing 54 charts for different design specifications. These tables will aid the

users to select a chart by considering both performance and simplicity in implementation, as well as the probability distributions

of and .

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Authors biographies

Yanjing Ou is a PhD student in the School of Mechanical and Aerospace Engineering of Nanyang Technological University,Singapore. She received her Bachelor degree and Co-Bachelor degree at the Wuhan Polytechnic University in 2007 and receivedher Master degree at the City University of Hong Kong in 2008. Her current research work focuses on control chart.

Di Wen is a Master student in the Wee Kim Wee School of Communication and Information of Nanyang Technological University,Singapore. He received his Bachelor degree in the Renmin University of China.

Zhang Wu is an Associate Professor in the School of Mechanical and Aerospace Engineering of Nanyang Technological University,Singapore. He obtained his PhD in 1988 and MEng in 1984 from the McMaster University, Canada, and obtained his BEng in1982 from the Huazhong University of Science and Technology, China. His current research interests include the development ofalgorithms in quality control, reliability, nonlinear optimization and geometrical tolerance. He is a senior member of the AmericanSociety for Quality.

Michael B. C. Khoo is a Professor in the School of Mathematical Sciences, Universiti Sains Malaysia (USM). He received his PhD in

Applied Statistics in 2001 from the Universiti Sains Malaysia. His research interests include statistical process control and reliability

analysis. He is a member of the American Society for Quality and serves as a member of the editorial board of Quality Engineering.

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