Upload
hacong
View
217
Download
2
Embed Size (px)
Citation preview
A New Monitoring Design for Uni-Variate Statistical Quality Control Charts
Mohammad Saber Fallah Nezhad, Ph.D.
Assistant Professor of Industrial Engineering, Yazd University, Yazd, Iran
Email: [email protected]
Seyed Taghi Akhavan Niaki, Ph.D. 1
Professor of Industrial Engineering, Sharif University of Technology
P.O. Box 11155-9414, Azadi Ave., Tehran, Iran 1458889694
Phone: (+9821) 66165740, Fax: (+9821) 66022702, Email: [email protected]
Abstract
In this research, an iterative approach is employed to analyze and classify the states of uni-
variate quality control systems. To do this, a measure (called the belief that process is in-
control) is first defined and then an equation is developed to update the belief recursively by
taking new observations on the quality characteristic under consideration. Finally, the upper
and the lower control limits on the belief are derived such that when the updated belief falls
outside the control limits an out-of-control alarm is received. In order to understand the
proposed methodology and to evaluate its performance, some numerical examples are
provided by means of simulation. In these examples, the in and out-of-control average run
lengths (ARL) of the proposed method are compared to the corresponding ARL's of the
optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM [11] methods within
different scenarios of the process mean shifts. The simulation results show that the proposed
methodology performs better than other charts for all of the examined shift scenarios. In
addition, for an autocorrelated AR(1) process, the performance of the proposed control chart
compared to the other existing residual-based control charts turns out to be promising.
1 Corresponding Author
2
Keywords: Statistical Quality Control; Process Monitoring; CUSUM Chart; EWMA Chart;
Average Run Length
1. Introduction and literature review
Traditional statistical process control (SPC) methods provide a group of statistical tests of a
general hypothesis which maintains that the mean value of the quality characteristic of a
process, or process mean in short, is consistently on its target level. A variety of graphical
tools, such as Shewhart, cumulative sum (CUSUM), and exponentially weighted moving
average (EWMA) charts, has been developed to monitor a process mean.
Shewhart charts, first introduced by W.A. Shewhart [24], plot either the individual
process measure or the average value of a small sample group (usually not more than five
samples) against the target level as well as the control limits. Under the assumption that the
plotted data are normally distributed around the process target value when the process is
within statistical control, the possibility of observing a point that is out of the three-sigma
control limits is less than 2.7 in a thousand. Therefore, spotting a point out-of-control-
limits leads to an out-of-control alarm which in turn calls for investigating the process.
Although it has been the common belief for many decades that a Shewhart chart is
not the most effective tool for some common process errors, such as small shifts in the
process mean, recent research has shown that the difference between Shewhart and
CUSUM charts is not that significant. For instance, Nenes and Tagaras [20] compared the
economic performance of CUSUM and Shewhart schemes in monitoring the process mean.
The results of their study showed that the economic advantage of using a CUSUM chart
over the simpler Shewhart scheme is substantial only when the sample size is one or it is
constrained to low values.
It is essential that the process mean be consistently maintained at its target level;
however, random process errors, or random “shocks”, could shift the process mean to an
3
unknown level. Furthermore, while a control chart is required to detect such shifts as soon
as possible, it is also desired that it does not signal too many false alarms when the process
mean is on target. These criteria are usually defined in terms of the average run length
(ARL) of the control chart for in-control and out-of-control operations, i.e., in-control ARL
0( )ARL and out-of-control ARL 1( )ARL , respectively.
At any sampling instant t , the EWMA control chart for the process mean uses the
control statistic 1(1 )t t tY X Y , where tX is the sample mean, 0Y is the in-control
process mean ( 0 ), and is the smoothing constant or the chart parameter. The chart
signals a change in the process level when tY exceeds control limits that are expressed by
means of multiples of the asymptotic standard deviation of tY . In other words, the lower
and the upper control limits (LCL and UCL) of the charts are given by
0 0 and Y YLCL L UCL L , where 0 (2 )Y n
is the asymptotic
standard deviation of tY , 0 is the in-control process standard deviation, and L is the
control limit parameter. We note that since the asymptotic standard deviation is not used in
the initial samples, one needs to provide the exact standard deviation of the EWMA chart at
these sampling epochs as 20 1 (1 )
(2 )t
tY n
.
The EWMA chart parameter is chosen to impose some desired properties on the
chart. In industrial applications, the commonly used value of is within the range of 0.1
to 0.5. Besides, the control limit parameter is chosen to achieve a desired probability of
type-I error (or desired 0ARL ). Moreover, for an EWMA chart, the literature suggests
detecting shifts of one-half to one standard deviation of the process mean [27].
4
The CUSUM chart uses one statistic for detecting a positive shift and another to
identify a negative shift in the process mean. The statistics for detecting positive and
negative shifts are 10,t t tY Max Y Z k and 10,t t tY Max Y Z k
respectively, where the starting values are 0 0 0Y Y , t tZ X is the sample mean and
k is the reference value or the chart parameter. The value of k is often chosen about
halfway between the target 0 and the out-of-control value of the process mean 1 that has
to be detected quickly [18]. If either tY or tY exceeds the decision interval H , the process
is considered to be out-of-control. In addition to the reference value, the decision interval is
chosen so that to give the CUSUM chart some good properties. The value of H is often
chosen as a multiple (five times) of the process standard deviation [18].
CUSUM charts were first introduced by Page [21]. When a shift in the process
mean is to be detected and the size of the shift is known, then a CUSUM control chart is
the most efficient method according to its ARL properties [2]. The CUSUM procedure can
be seen as equivalent to applying a sequential likelihood ratio test to a shift in the process
mean [1 & 9]. A CUSUM chart monitors the accumulated process deviation after the
process is determined to be in an out-of-control state. The parameters of a CUSUM chart
can be assigned such that it turns to be the optimal likelihood ratio test on a particular shift
size.
The EWMA [7 & 13] chart can be seen as a variation of the CUSUM [28] control
scheme in the sense that they both have been used to improve the detection of small
process shifts. Based on the notion that the most recent observed process deviation can
have more information on process errors than the previous ones, different weights may be
assigned to data according to their recorded times. An EWMA scheme lets the weights
decrease exponentially with the age of data, while a CUSUM scheme keeps the same value
5
for the weights. A Shewhart scheme, in contrast, assigns the total weight to the most recent
observation and zero to others. Usually, an EWMA chart can be designed to have
similar ARL properties to a CUSUM chart's through simulation studies [6].
Vargas et al. [27] presented a comparative study of the performance of CUSUM
and EWMA control charts by simulation. The objective of their research was to verify when
CUSUM and EWMA control charts provide the best control region to detect small changes
in the process mean. They observed that the CUSUM control chart practically did not
signal out-of-control points for the levels of variation between 1 standard deviation,
whereas the EWMA control chart was more efficient. Among the parameters of EWMA
control chart, the ones with 0.1 and 0.05 detected a larger number of altered positions.
While both the optimal EWMA and CUSUM control charts are based on a given
reference value, say δ, Han and Tsung [11] proposed a generalized EWMA control chart
(GEWMA) which does not depend on δ. They theoretically compared the performance of
GEWMA control chart with the optimal EWMA, CUSUM, Shewhart-EWMA (a combination
of Shewhart and Optimal EWMA), and the generalized likelihood ratio (GLR) control
charts. The results of the comparison, which considered the in-control average run length
approaching infinity, showed that the GEWMA control chart was better than the optimal
EWMA control chart in detecting a mean shift of any size. For the mean shifts not in the
interval (0.7842δ, 1.3798δ), the GEWMA chart performed better than the CUSUM control
chart. Furthermore, the GLR control chart had the best performance in detecting mean
shifts among the five control charts except for detecting a particular mean shift δ, when in-
control average run length approaches infinity.
Serel and Moskowitz [23] proposed an EWMA control chart to jointly monitor the
mean and variance of a process. In this research, an EWMA cost minimization model was
presented to design the joint control scheme based on purely economic or economic-
6
statistical criteria. Through a computational study, they showed that the optimal sample
sizes decrease as the magnitude of shifts in mean and/or variance is increased.
Recently developed quality control schemes are mostly based on sequential
analysis, which requires to analyze the data at hand in order to determine the necessary
number of additional observations at the next stage. For example, Wu and Shamsuzzaman
[30] proposed a scenario for continuously improving the &X S control charts, where the
information collected from out-of-control cases in a manufacturing process was used to
update the chart parameters. Moreover, Zhang et al. [32] employed a sequential sampling
scheme in phase one of an exponential control chart to monitor the time between events.
Adaptive charts may either be based on the Bayes’ rule for continuously updating
the knowledge about the state of the process, or not [26]. Over the years, to simultaneously
monitor the process mean and variance, numerous researchers have proposed different
statistically designed adaptive charts. Examples of these research efforts are the X control
chart of Lin and Chou [14], the joint X and R control charts of Costa and Rahim [5], and
the CUSUM chart of Wu et al. [31]. However, the performance criterion that is
increasingly used to measure the effectiveness of these adaptive charts, especially the
Bayesian ones, is the minimization of total expected quality-related costs.
Nenes and Tagaras [19] studied a model for the economic design of an
adaptive X chart for short production runs that were subject to the occurrence of assignable
causes. These causes may lead to either an increase or a decrease in the mean of the quality
characteristic. At each sampling instance, the probabilities that the process operates under
the effect of an assignable cause were updated using the Bayes’ theorem. All three chart-
parameters i.e., the time until the next sampling instance, the sample size, and the control
limits were adaptive and depended on these probabilities.
7
Chun and Rinks [4] applied Bayesian analysis to a single sampling plan in which
the defective-proportion was assumed a random variable that follows a Beta distribution.
Furthermore, Wu [29] applied Bayesian analysis to assess the process capability index in a
sequential manner based on subsamples. Sachs et al. [22] used a sequential Bayesian
statistical approach of feedback control in which the process was first divided into in-
control and out-of-control states. The out-of-control and in-control probability functions
were then defined and the certainty of the shift was updated with each addition of new
data. This sequential approach reflects the latest evidence supporting or discounting the
occurrence of a disturbance. They assumed a prior probability distribution for the shift and
obtained an expression that highlights the sequential nature of updating the shift
distribution.
Marcellus [17] presented a Bayesian analogue of the Shewhart X chart and
compared it with CUSUM charts. He showed the advantage of changing from Shewhart or
CUSUM chart to Bayesian monitoring in situations where the required information about
the process structure is obtained. Although implementing the Bayesian chart requires more
detailed knowledge of the process structure than the best-known types of charts, acquiring
this information can yield tangible benefits. For this case, a Bayesian monitoring system
was defined for a standard production process model introduced by Duncan [8] and it was
shown that the monitoring system to be equivalent to an adaptive Shewhart monitoring
scheme.
Some efforts have been made to take a fuzzy approach to control charting. The
major contribution of fuzzy set theory lies in its capability of representing vague data.
Fuzzy logic offers a systematic base to deal with situations that are ambiguous or not well
defined. A number of papers on fuzzy control charts use defuzziffication methods in the
early steps of their algorithms that make their approach similar to the conventional ways of
8
control charting. However, Gülbay and Kahramana [10] proposed a new alternative
approach of “direct fuzzy approach (DFA)” to control charting method. In contrast to the
existing fuzzy control charts, the proposed approach does not require taking the
defuzziffication step. This prevents the loss of information included in the samples and
directly compares the linguistic data in fuzzy space without making any transformation. At
the end, they provide some numerical examples to illustrate the performance of their
method and to interpret the results.
In this research, a recursive equation is first defined to update a statistic (called
belief) in each iteration of the data gathering process. Then, similar to the well-known
CUSUM and EWMA methods, thresholds are derived for the updated values of the beliefs.
When the updated belief is out of the derived threshold range, an out-of-control signal is
issued.
The rest of the paper is organized as follows: in section 2, the belief and the
recursive method of its improvement are first defined. Then, we design the new univariate
control charting method. Section 3 contains the results of some simulation experiments in
which the performance of the proposed methodology is compared to the performance of
the optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM methods. This section
also contains the results of a comparative study on an autocorrelated AR(1) process. The
performance of the proposed scheme is further evaluated in a case study in section 4.
Finally, the conclusions and recommendation for future research appear in section 5.
2. Belief and the approach to its improvement
For the sake of simplicity, assume only one single observation ( 1n ) is gathered on the
quality characteristic of interest in each iteration of the data gathering process. At the kth
iteration, let 1 2( , ,..., )k kO x x x be a vector of observations on the quality characteristic of
9
the current and the previous 1k iterations. After taking a new observation, kx ,
let 1( , )k kB x O define the belief in the process to be in an in-control state. In this iteration,
our aim is to improve this belief based on the observation vector 1kO and the new
observation kx .
Assuming that the quality characteristic of interest follows a normal distribution
with mean 0 and variance 20 and letting 1 1 2( ) ( , )k k kB O B x O be the prior belief in an
in-control state, in order to update the posterior belief 1( , )k kB x O , we define
0
0
0
0
11
1 1
,
1
k
k
x
kk k k x
k k
B O eB x O B O
B O e B O
(1)
Then, by defining the statistic
1
1
,
1 , 1k k k
kk k k
B x O B OZ
B x O B O
(2)
the recursive equation will be
0 0
0 011
11
k kx x
kk k
k
B OZ e e Z
B O
(3)
Hence,
00 0 1 0 1
0 0 0 01 2 .......
k
ik k k i
x kx x x
k k kZ e Z e Z e
(4)
In other words,
0
01
10 0
0,
k
i kii
ki
x kx
Ln Z N k
(5)
10
For initial values of 0Z and 0( )B O , note that for 1k equation (4) shows that
1
01 01
0 01
ii
xx
Z e e
. In this case, by equation (3) we have 1 0
01 0
x
Z e Z
which means
that 0 1Z . Hence 0( ) 0.5B O .
Now we define the upper and the lower control limits (UCL and LCL) for kLn Z
as
and k kLn Z Ln ZUCL c k LCL c k (6)
Where c is a multiple of the standard deviation of kLn Z and is determined such that for
a given probability of type-I error, , we have
1kP c k Ln z c k (7)
Since we need to determine threshold values for recursive statistic 1,k kB x O and
since computing kLn Z for small values of kZ is difficult in terms of computer
limitations, we may instead determine the values of 1,k kB x O and compare them to their
lower and upper control limits as derived bellow.
Substituting equation (2) for kZ in equation (7) results in
1
1
,1
1 ,k k
k k
B x OP c k Ln c k
B x O
(8)
Or
1
1
,1
1 ,k kc k c k
k k
B x OP e e
B x O
(9)
This means that
1
11 1 1
1 ,c k c k
k k
P e eB x O
(10)
11
which leads to a 100 (1 ) % confidence interval for 1,k kB x O as given in equation
(11).
1, 11 1
c k c k
k kc k c k
e eP B x O
e e
(11)
Furthermore, since in the initial stages of the data gathering process the false alarm rate
may be high, the parameter l is also introduced in equation (11) for the upper and the
lower control limits (UCL and LCL) of 1,k kB x O to become
1 1, , and 1 1k k k k
c k l c k l
B x O B x Oc k l c k l
e eUCL LCL
e e
(12)
in which the exponential terms are computed easier than the logarithmic terms needed in
equation (6).
It is worth noting that the values of c and l should be determined such that to
ascertain reasonable properties for the proposed control charting method. In other words,
for a desired in-control average run length, small values of out-of-control average run
length in different scenarios of the process mean shifts are in order (the probability of both
type-I and type-II error must be small).
Following Han and Tsung [11] who compared the abilities of their proposed
GEWMA control charts to the performance of the optimal EWMA, Shewhart EWMA, GLR,
and CUSUM, in the next section we perform some simulation experiments in which the
performance of the proposed methodology in terms of both in-control and out-of-control
average run lengths criterion is compared with other control charts. Furthermore, in
situations in which the collected data on the quality characteristic are auto-correlated, the
performance of the proposed procedure is compared with the residual-based EWMA chart
(Lu and Reynolds [15]), residual-based CUSUM chart (Lu and Reunolds [16]) and
12
triggered CUSCORE chart (Shu et al. [25]) for different values of the autocorrelation
coefficient in an AR(1) process.
3. Simulation experiments
Simulation experiments are performed for two classes of independent standard normal and
autocorrelated AR(1) observations.
3.1 Independent standard normal process
Suppose that the quality characteristic of interest in different stages of the data gathering
process are identically and independently distributed (IID) standard normal random
variables. In order to simulate this process, pairs of independent uniform random variates
( 1( , ) ; 2 1 ; 1, 2,3,...i iR R i k k ) are first generated and then
12 ( ) cos(2 )k i ix Ln R R is employed to generate a standard normal observation in
the kth iteration of the data gathering process [3]. In the next step, using equation (1) the
belief ( ( )kB O ) is updated in that iteration. When ( )kB O is out of the interval
,
1 1
c k l c k l
c k l c k l
e e
e e
, then an out-of-control signal is observed.
Because the two-sided CUSUM [12 & 20], GEWMA, EWMA, and Shewhart EWMA
charts [12] also incorporate past information, it is natural to compare their performance
with the proposed control chart to see whether the new chart performs better in terms of
out-of-control average run lengths (ARL1). Furthermore, the optimal EWMA, Shewhart
EWMA, GEWMA, GLR, and CUSUM control charts are based on a given reference value,
which for the CUSUM chart is the magnitude of a shift in the process mean that should be
13
quickly detected. Similar to Han and Tsung [11], in this research a reference value of 1 for
the optimal EWMA and CUSUM chart will be used in the simulation experiments.
Results based on 10000 independent replications, each representing a series of
observations that ends with a signal (i.e., in each replication we generate standard normal
deviates until the defined statistic goes out of the derived control interval) are summarized
in Table (1). The simulation results are also given for various values of the process mean
shift defined as multiples of the process standard deviation. The values in the “SD”
columns of Table (1) are the standard deviations of the run lengths. The parameters of
different control charts are given in the last row of Table (1). The reference value for the
optimal EWMA and CUSUM is taken to be 1. For the proposed method we pick c=1.5 and
0l to ascertain an accessible value of 0.ARL
The results of Table (1) show that while the GEWMA control chart maintains a
relative advantage over the optimal EWMA control chart (except in detecting the shifts of
around 0.5 to 1.25 ), Shewhart EWMA control charts (except in detecting the shifts of
around 0.5 to 1.25 ), the CUSUM chart (except in detecting the shifts of 0.75 to
1.25 ), and the GLR control chart (except in detecting the shifts of less than 0.25 ), the
performance of the proposed methodology is the best for shifts of different magnitudes in
the process mean. Moreover, the standard deviations of the run lengths obtained by the
proposed method are generally less than corresponding values of the other methods. We
also note that the in-control average run length of the proposed control chart (548.00) is
larger than the corresponding values in the other control charts. In other words, not only is
the probability of type-I but also the probability of type-II error associated with the
proposed method is less than their corresponding values in the other two methods
(according to 1 0
1 1 and
1ARL ARL
).
14
Insert Table (1) about here
3.2 Auto-correlated AR(1) process
The usual assumption of using a control chart to monitor a process is that the observations
from the process output are independent. However, for many processes this assumption
does not hold and the observations are autocorrelated. This autocorrelation can have a
significant effect on the performance of the control chart. In this section, we assume that
the observations on the quality characteristic in different stages of the data gathering
process can be modeled as an AR(1) process plus a normally distributed random error term.
A process{ }ky is said to be AR(1) if it is generated by
0 1 0k k ky y (13)
where is the autocorrelation coefficient satisfying 1,1 and k is a sequence of IID
normal error term, i.e., 20, ; k N k Z . In this process, the variance of the
observations is 2
21kVar y
and the residuals are defined as
0 1 0 ; 1, 2,...k k ky y k (14)
Since the residuals are IID random variables, in order to update the posterior belief
( 1( , )k kB y O ) let define
0 1 0
0 1 0
1 11
1 1 1 1
,
1 1
k kk
k k k
y y
k kk k y y
k k k k
B O e B O eB y O
B O e B O B O e B O
(15)
Then, a 100 (1 )% confidence interval for 1,k kB y O is easily determined using
Equation (16).
15
1, 11 1
c k l c k l
k kc k l c k l
e eP B y O
e e
(16)
Hence, when ( )kB O is out of the interval
,1 1
c k l c k l
c k l c k l
e e
e e
, an out-of-control
signal is observed.
In this section the performance of the proposed method in terms of both 0ARL and
1ARL is compared to the performance of the residual-based EWMA chart [15], residual-
based CUSUM chart [16] and one-sided CUSCORE chart [25] for selected auto-correlation
coefficients of 0.1, 0.5 and 0.90.
For 0.5 , based on 10000 independent replications, the simulation results are
summarized in Table (2). The results in Tables (3) and (4) correspond to 0.9 and
0.1 , respectively.
Insert Table (2) about here
Insert Table (3) about here
Insert Table (4) about here
The results of Table (2) show that for a moderate-level of autocorrelation, the
proposed method performs better than the other charts for all shifts of less than 2 . For
larger shifts than that the CUSCORE chart is the best. Furthermore, the standard deviation
of the out-of-control run length of the proposed method is the least among all the
competing methods.
For a highly correlated process, the results of Table (3) show that the residual-based
EWMA chart is the best for shifts of less than 0.1 . However, while for the shifts between
16
0.1 and 0.25 the CUSCORE chart enjoys the best performance, for shifts larger than
0.25 the proposed chart is the best. Moreover, standard deviation of the out-of-control
run length of the CUSCORE chart is the least for shifts of less than 0.75 . Nonetheless,
for shifts larger than that the standard deviation of the out-of-control run length is the least
for the proposed procedure.
For a low-level correlated process, the results in Table (4) show that the proposed
chart performs as well as the other methods with the least standard deviation for the out-of-
control run length.
4. A case study
Consider an in-control standard normal process ( 20 00 and 1 ), and suppose that at a
certain time, the process shifts to the mean 1 0.1 . We collect 20 observations as given in
Table (5), and compare the performance of the proposed method against the standard
Shewhart, CUSUM, and EWMA charts.
Figure (1) shows the charts of all competing methods, which are obtained using the
Minitab computer software. We note while none of the conventional charts is able to detect
a small shift of 0.1 in the process mean, the proposed method shows out-of-control signals
at observations 15, 16, 17 and 18. This means that the proposed chart is more capable in
detecting the mean shift than the conventional charts and is likely to be more practical.
The most important finding of this case study is the high sensitivity of the proposed
method to the shifts in the process mean for large and small number of observations.
Insert Figure (1) about here
17
5- Conclusions and recommendation for future research
In this paper, an iterative approach was employed to analyze and classify the states of uni-
variate quality control systems. This approach starts out with defining a measure called
belief, and subsequently the beliefs in the system to be in-control are updated by taking
new observations on the quality characteristic under study. Then by means of a control
charting method when the updated beliefs are out of the control limits, the process is
determined to be in an out-of-control state.
In the simulation experiments of an independent standard normal process, we
concluded that applying the proposed method to the process mean monitoring would
improve the performance of the charts and would result in reduced probability of both
type-I and type-II errors. Moreover, the results of another simulation study of an AR(1)
process revealed that the proposed control chart performs at least as well as the other
residual-based control charts.
In this research, the performance of the proposed method was compared against the
other existing methods based on simulated observations of a standard normal process. In
case where the observations are taken from a general normal or even a non-normal process,
the results of the comparison study remain to be shown in future research. Furthermore,
determining the control threshold of the beliefs for different values of autocorrelation
coefficient in different autocorrelated processes is an interesting subject for future research.
6. Acknowledgment
The authors would like to thank the referees for their valuable comments and suggestions
that improved the presentation of this paper.
18
7. References
[1]. Barnard, G.A., (1959), “Control charts and stochastic processes.” Journal of Royal
Statistical Society, Series B, 21, 239–271.
[2]. Basseville, M., and Nikiforov, I.V., (1993), “Detection of abrupt changes: Theory and
application.” Prentice Hall Inc., Upper Saddle River, NJ, USA.
[3]. Box, G.E.P., and Muler, M.E., (1958), “A note on the generation of random normal
deviates.” Annals of Mathematical Statistics, 29, 610-611.
[4]. Chun, Y.H. and Rinks, D.B., (1998), “Three types of producer's and consumer's risks in
the single sampling plan.” Journal of Quality Technology, 30, 254-268.
[5]. Costa, A.F.B., and Rahim, M.A., (2004), “Joint X and R charts with two-stage
samplings.” Quality and Reliability Engineering International, 20, 699-708.
[6]. Crowder, S.V., (1986), “Kalman filtering and statistical process control.” Ph.D. thesis,
Statistics, Iowa State University.
[7]. Crowder, S.V., (1987), “A simple method for studying run-length distributions of
exponentially weighted moving average charts.” Technometrics, 29, 401-407.
[8]. Duncan A.J., (1956), “The economic design of X charts used to maintain current
control of a process.” Journal of the American Statistical Association, 51, 228-242.
[9]. Ewan, W.D. and Kemp, K.W., (1960), “Sampling inspection of continuous processes
with no autocorrelation between successive results.” Biometrika, 47, 363-380.
[10]. Gülbay, M., and Kahramana, C., (2007), “An alternative approach to fuzzy control
charts: Direct fuzzy approach.” Information Sciences, 177, 1463-1480.
[11]. Han, D. and Tsung, F., (2004), “A generalized EWMA control chart and its
comparison with the optimal EWMA, CUSUM and GLR schemes.” Annals of Statistics,
32, 316-339.
19
[12]. Hawkins D.M. and Olwell, D.H., (1998), “Cumulative sum charts and charting for
quality improvement.” Springer, New York.
[13]. Hunter, S.J., (1986), “The exponentially weighted moving average.” Journal of
Quality Technology, 18, 203-210.
[14]. Lin, Y.-C., and Chou, C.-Y., (2005), “Adaptive X control charts with sampling at
fixed times.” Quality and Reliability Engineering International, 21, 163-175.
[15]. Lu, C.W. and Reynolds Jr., M.R., (1999), “EWMA control charts for monitoring
the mean of autocorrelated processes.” Journal of Quality Technology, 31, 166-188.
[16]. Lu, C.W. and Reynolds Jr., M.R., (2001), “CUSUM charts for monitoring an
autocorrelated process.” Journal of Quality Technology, 33, 316-334.
[17]. Marcellus, R.L., (2008), “Bayesian statistical process control.” Quality
Engineering, 20, 113-127.
[18]. Montgomery, D. (2005), “Introduction to statistical quality control.” 5th Ed., John
Wiley and Sons Inc., New York.
[19]. Nenes, G., and Tagaras, G., (2007), “The economically designed two-sided
Bayesian X control chart.” European Journal of Operational Research, 183, 263-277.
[20]. Nenes, G., and Tagaras, G., (2008), “An Economic Comparison of CUSUM and
Shewhart Charts,” IIE Transactions, 40, 133-146.
[21]. Page, E.S., (1954), “Continuous inspection schemes.” Biometrika, 14, 100-115.
[22]. Sachs, E., Hu, A., and Ingolfsson, A., (1995), “Run by run process control:
Combining SPC and feedback control.” IEEE Transactions on Semiconductor
Manufacturing, 8, 26-43.
[23]. Serel, D.A., and Moskowitz, H., (2008), “Joint economic design of EWMA control
charts for mean and variance.” European Journal of Operational Research, 184, 157-168.
20
[24]. Shewhart, W.A.T., (1931), “Economic control of quality of manufactured product.”
Van Nostrand, New York, USA.
[25]. Shu, L., Apley, D.W., and Tsung, F., (2002), “Autocorrelated process monitoring
using triggered CUSCORE charts.” Quality and Reliability Engineering International, 18,
411-421.
[26]. Tagaras, G., (1998), “A survey of recent developments in the design of adaptive
control charts.” Journal of Quality Technology, 30, 212-231.
[27]. Vargas, V.C.C., Lopes, L.F.D, and Souza, A.M, (2004), “Comparative study of the
performance of the CuSum and EWMA control charts.” Computers and Industrial
Engineering, 46, 707-724.
[28]. Woodall, W.H., (1986), “The design of CUSUM quality control charts.” Journal of
Quality Technology, 18, 99-101.
[29]. Wu, C.W., (2006), “Assessing process capability based on Bayesian approach with
subsamples.” European Journal of Operational Research; 184, 207-228.
[30]. Wu, Z. and Shamsuzzaman, M., (2006), “The updatable &X S control charts.”
Journal of Intelligent Manufacturing, 17, 243-250.
[31]. Wu, Z., Zhang, S., and Wang, P., (2006), “A CUSUM scheme with variable sample
sizes and sampling intervals for monitoring the process mean and variance.” Quality and
Reliability Engineering International, 23, 157-170.
[32]. Zhang, C.W., Xie, M., and Goh, T.N., (2006), “Design of exponential control
charts using a sampling sequential scheme.” IIE Transactions, 38, 1105-1116.
21
Table (1): The results of 0ARL and 1ARL study for IID N(0,1) observations
SD CUSUM SD GLR SD GEWMA SD Shewhart
SD Optimal
SD Proposed Method
Shifts EWMA EWMA
4.36 434.00 4.35 439.00 4.24 438.00 4.28 430.00 4.34 437.00 21.370 548.00 0.00
3.23 326.00 2.67 295.00 2.75 304.00 2.85 294.00 4.30 297.00 0.880 63.50 0.10
1.23 132.00 0.80 108.00 0.79 105.00 1.02 109.00 1.02 110.00 0.230 23.01 0.25
0.30 37.20 0.23 36.20 0.23 34.90 0.25 32.40 0.25 32.40 0.070 8.55 0.50
0.11 16.70 0.11 18.10 0.10 17.40 0.10 15.70 0.10 15.70 0.030 4.66 0.75
0.05 10.30 0.06 11.10 0.06 10.70 0.05 9.92 0.05 9.95 0.020 3.15 1.00
0.03 7.34 0.04 7.58 0.04 7.36 0.03 7.19 0.03 7.24 0.010 2.51 1.25
0.02 5.70 0.03 5.59 0.03 5.41 0.02 5.67 0.02 5.37 0.015 1.93 1.50
0.01 3.98 0.02 3.54 0.02 3.41 0.01 3.91 0.01 4.03 0.009 1.40 2.00
0.01 2.55 0.01 1.91 0.01 1.85 0.01 2.29 0.01 2.63 0.007 1.04 3.00
4.94H
3.45L
3.29L
0.128
2.82
3.9
C
L
0.128
2.82L
1.5
0
c
l
Parameters
22
Table (2): The results of 0ARL and 1ARL study for AR(1) with 0.5
SD Residual-
based CUSUM
SD CUSCORE SD
Residual-based
EWMASD Proposed Method
Shifts
4.26 430.00 4.35 420.00 4.17 421.00 17.37 451.00 0.00
2.83 301.00 2.67 295.00 2.59 257.00 1.24 93.50 0.10
1.73 185.201 1.30 144.532 1.22 134.47 0.28 32.01 0.25
0.80 89.979 0.5 67.282 0.55 67.328 0.1 15.55 0.50
0.41 48.2835 0.26 36.677 0.30 36.8727 0.06 10.66 0.75
0.23 29.3704 0.15 26.22 0.16 23.682 0.03 7.27 1.00
0.14 19.8704 0.1 17.944 0.11 17.4242 0.02 5.51 1.25
0.09 14.2967 0.07 14.493 0.07 13.1241 0.015 4.93 1.50
0.05 8.98 0.04 3.9 0.04 4.93 0.008 3.60 2.00
0.02 4.05 0.02 1.8 0.02 3.32 0.005 2.4 3.00
4.25
0.5
H
k
3.5
0.25
0.5r
L
k
0.1
2.51L
1.2
20
c
l
Parameters
23
Table (3): The results of 0ARL and 1ARL study for AR(1) with 0.9
SD Residual-
based CUSUM
SD CUSCORE SD
Residual-based
EWMA SD Proposed Method
Shifts
4.19 426.00 4.05 443.00 4.25 418.00 17.37 451.00 0.00
3.83 391.00 3.5 393.00 3.97 386.00 11.24 394.63 0.10
3.48 358.11 3.22 285.02 3.34 330.35 6.77 291.24 0.25
3.01 301.27 2.55 201.48 2.69 268.02 3.34 178.13 0.50
2.49 256.89 2.09 146.79 2.12 211.28 1.90 127.09 0.75
2.09 217.42 1.70 115.40 1.69 177.96 1.22 91.84 1.00
1.79 183.75 1.35 88.62 1.31 144.45 0.85 72.14 1.25
1.52 160.25 1.10 72.97 1.11 116.64 0.63 60.18 1.50
1.13 118.98 0.47 82.9 0.72 85.93 0.41 38.1 2.00
0.63 63.05 0.26 55.8 0.39 51.32 0.23 28.2 3.00
4.25
0.5
H
k
1.45
0.05
0.1
L
k
r
1.2
20
c
l
Parameters
0.1
2.51L
24
Table (4): The results of 0ARL and 1ARL study for AR(1) with 0.1
SD Residual-
based CUSUM
SD CUSCORE SD
Residual-based
EWMA SD Proposed Method
Shifts
4.17 428.00 4.05 445.00 4.37 425.00 17.37 451.00 0.00
3.83 391.00 2.22 232.00 1.87 189.4.00 1.54 103.63 0.10
0.97 101.25 0.40 100.30 0.62 69.01 0.36 40.04 0.25
0.30 35.33 0.14 33.63 0.21 27.51 0.11 17.68 0.50
0.12 17.22 0.08 16.84 0.08 15.83 0.05 10.87 0.75
0.06 10.44 0.05 10.54 0.05 10.57 0.04 8.29 1.00
0.04 7.49 0.03 7.64 0.03 7.70 0.03 6.30 1.25
0.03 5.74 0.03 5.93 0.03 6.43 0.02 5.25 1.50
0.01 3.98 0.01 4.15 0.01 4.39 0.01 4 2.00
0.01 2.54 0.01 2.67 0.01 2.84 0.01 2.67 3.00
4.25
0.5
H
k
4.2
0.45
0.9
L
k
r
0.1
2.51L
1.2
20
c
l
Parameters
25
Table (5): The process data
No. Obs. No. Obs.
1 -0.4037 11 -0.9202
2 1.6336 12 0.4118
3 -0.2238 13 0.7242
4 0.7271 14 1.6376
5 -1.3756 15 1.8857
6 -0.7735 16 0.6630
7 1.8541 17 0.4673
8 0.5053 18 1.8906
9 -0.1224 19 -2.5656
10 0.5932 20 -0.5820
Chart1: Standard Shewhart Chart2: Standard EWMA
Chart3: Standard CUSUM Chart4: The Proposed Method
Figure (1): The Control Charts of the Case Study
4
3
2
1
0
-1
-2
-3
-4
4
-4
20100
Subgroup Number
Upper CUSUM
Lower CUSUM
20100
3
2
1
0
-1
-2
-3
Observation Number
X=0.000
3.0SL=3.000
-3.0SL=-3.000
20100
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
Sample Number
EWMA
X=0.000
3.0SL=0.6831
-3.0SL=-0.6831
Cumulative Sum
Individual Value