1. No category

Cairo University
Faculty of Economics and Political Science
Department of Statistics - English Section
Graduation Project
Statistical Performance of
Shewhart S - Chart with Variable
Sample Size and Estimated
Parameters
Presented by:
Madonna Magdy Besada
Maria Ashraf Sobhy
Marina Maher Joseph
Maureen Hany Sadek
Under the Supervision of
Prof. Mahmoud Al-Said Mahmoud
Submission date: 24/6/2012
0
Abstract
In this report we discuss the behaviour of the variable sample size Shewhart Schart when the parameters of the underlying process are unknown and thus have to
be estimated. We focus on the effect of estimating the process standard deviation.
The in-control average run length (ARL0) of the control chart with estimated
parameters is compared with the ARL0 of the chart of known parameters case, fixing
the in-control average sample size (ASS0) for both. Additionally, we give some
recommendations on the choice of Phase I sample size and number of Phase I
samples in the context of Shewhart S-charts with variable sample size and estimated
parameters.
1
Abbreviations
SPC
Statistical Process Control
QC
Quality characteristic
CL
Centre line
UCL
Upper control limit
UWL
Upper warning limit
LCL
Lower control limit
LWL
Lower warning limit
RL
Run length
ARL
Average run length
ARL0
In-control average run length
ARL1
Out-of-control average run length
ASS
Average sample size
ASS0
In-control Average Sample Size
ASS1
Out-of-control average sample size
overall probability of false alarm
EWMA
Exponentially weighted moving average
CUSUM
Cumulative sum control chart
VSS
Variable sampling size
VSI
Variable sampling interval
n
Sample size
h
Sampling interval
k
Control limit coefficient
ns
Small sample size
nL
Large sample size
m
number of Phase I samples
n0
SAS
Phase I sample size
Statistical package for statistical analysis
system
2
Table of contents
Section 1: Introduction……………………………………………………….6
1.1 Definition of control charts………………………….…………...……..6
1.2 Types of control charts…………………………………………..……...8
1.3 Adaptive control chart ……………………………………………...…..10
1.4 The objective……………………………………………………………12
1.5 The methodology………………………………………………….........13
1.6 literature review………………………………………………...…...….13
Section 2: VSS S-Chart………………………………………………….…..14
2.1 The design of S-chart……………………………………………….….14
2.2 Measures of performance …………………….……………………….15
Section 3: The Methodology (Simulation)…………...…..…………………17
3.1 Simulation steps……………………………………………….……….17
3.2 Simulation results…………………………………………….……......18
Section 4: An illustrative example …………………………………............24
Section 5: Conclusion and Recommendations. ………………..……..…...27
5.1 Conclusion ……………………………………………………………27
5.2 Recommendation……………………………………………………...27
References ……………………………………..……………………….……29
List of Bibliography ………………………………………………………… 31
Appendices ………………………………………………..………………….33
1. Appendix (1): Simulation results tables…………………………………….………….33
2. Appendix (2): Example data………………………………………………………….………47
3. Appendix (3): Simulation results figures………………………………..69
4. Appendix (4): SAS programs………………………………………………………………...71
3
Index of Figures
1.1 Shewhart control chart for monitoring the process parameter………….8
1.2 Adaptive VSS Shewhart control chart for monitoring the process mean
parameter……………………………………………..…………………12
3.1 ARL0 vs. m at ASS0=5 and nS=4 and nL=7………………………….…19
3.2 ARL0 vs. n0 at ASS0=5 and nS=4 and nL=7……………………………20
3.3 ARL0 vs. m at ASS0=7 and nS=5 and nL=10………………………......20
3.4 ARL0 vs. n0 at ASS0=7 and nS=5 and nL=10…………………………..21
4.1 Monitoring company (A) performance (at m=50)……………………….25
4.2 Monitoring company (B) performance (at m=100)……………………...25
4
Index of Tables
3.1 Best n0 and m for each combination at ASS0 = 3…………..……………22
3.2 Best n0 and m for each combination at ASS0 = 5…………..……………22
3.3 Best n0 and m for each combination at ASS0 = 7…………..……………23
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Section 1: Introduction
Recently, controlling and improving quality have been considered one of the most
important business strategies to achieve customer satisfaction globally. Statistical Process
Control (SPC) is a collection of statistical and analytical tools that can be used to achieve process
stability and variability reduction about the process target value. The most important tool in SPC
is the control charts which was first introduced by Shewhart in the 1920's.
1.1 Definition of control charts:
Control chart is an online graphical representation used to plot sample points of certain
quality characteristic (QC) in a production process. QC is an important physical or temporal
characteristic of a product or service such as; weight of a product or the time consumer has to
wait to get a service.
In any production process we need to make sure that one or more of the process
parameters has/have to satisfy a target value(s) (which represents the optimal value of the
interested QC). Practically, we expect variability around the target value. There are two reasons
for this variability; common and special causes.
First, common causes are natural or random variation as their effect exists in all the
process output. This variation is small and cannot be removed by statistical tools for quality
control. For example, the same worker in the same production condition does not fill the same
amount of the company product in the designated package. As long as the process operates with
only common causes of variation, the process is said to be in-control.
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Second, special causes of variation are occurred due to something wrong has happened
in the process. The reasons for special causes of variations are usually classified into four
reasons; which are machine, workers, raw material, and environment. For example, the machine
might become less effective due to the destruction of one of its components. The variability
caused by special causes is substantial. It is usually assumed that when special causes occur, the
distribution of the QC is changed (shifted). A process is said to be out-of-control if it operates in
the presence of special causes. This type of variability can be detected by control charts.
In fact, the main purpose of a control chart is to separate common causes from special
causes. When a special cause of variation is detected by a control chart, the process is stopped,
and investigations are carried on to determine the reason and eliminate it (if possible). The
process is resumed when the special cause of variation is removed.
Since special causes lead to changes in the process parameters, control charts are used to
detect any changes in one or more of these parameters. The control chart is built up by taking
samples every fixed time interval, say h hours. Then plot the appropriate sample statistic for each
sample. For example, if we are interested in monitoring changes in the process mean, we plot the
sample average.
Figure (1.1) shows the design of a Shewhart control chart for monitoring a process
parameter. The horizontal axis represents the sample number or the time when the sample was
drawn. The vertical axis represents the value of the sample statistic. Shewhart chart includes
three main lines which are the centre line (CL), the lower control limit (LCL) and the upper
control limit (UCL). The process is considered to be in-control if the sample point lies between
the LCL and the UCL. However, if the sample point lies outside these limits, the process is
deemed out-of-control. As shown in Figure (1.1), all sample points are plotted between the LCL
and UCL except the 7th sample point which indicates that the illustrative process has gone out-ofcontrol at this time.
7
Figure (1.1): Shewhart control chart for monitoring a process parameter
Chart statistic
4.0
3.5
UCL
3.0
2.5
2.0
CL
1.5
1.0
0.5
LCL
0.0
1
2
3
4
5
6
7
Sample number
1.2 Types of control charts:
There are many classifications for the control charts. First, based on number of quality
characteristics under investigation. We have two control charts, the univariate control chart
which is used to monitor one quality characteristic and the multivariate control chart which is
used to monitor more than one quality characteristic.
Second, based on the type of quality characteristic, we have variable control chart on
which the QC can be measured on numerical scale. If a QC cannot be represented numerically,
we use the attribute control chart where the items are classified as conforming and nonconforming to the specifications on the QC, as the QC is said to be attribute.
Third, based on whether or not the process parameters (location and dispersion
parameters) are known. When the parameters are known we use Phase II control charts to
monitor any change in the process parameter. To measure the performance of these charts we use
8
the average run length ARL which is the average number of points plotted until the chart gives a
signal. There are two types of ARL; in-control average run length ARL0 and out-of-control
average run length ARL1. First, ARL0 is the average number of points plotted before the chart
gives a false alarm i.e. despite being in-control, the chart signals. Second, ARL1 is the average
number of points plotted until the chart detect a shift. When comparing between Phase II charts,
we fix the ARL0 for all of them and compare the performance based on the ARL1 values. Then
the chart with the least ARL1 is the best.
Practically, the process parameters are usually unknown so we estimate them using
historical data set of m in-control samples and the control charts used in this case are called
Phase I control charts. The overall probability of a signal is used to measure the performance of
Phase I control charts. Similarly, there are two types for the overall probability of a signal;
overall probability of false alarm (
) and overall power which are defined as:
(
) ,
,
where
and
are the marginal probabilty of Type I and Type II error, respectivily. To compare
the performance of Phase I charts, we fix the overall probability of false alarm and compare
according to the overall power. The chart with the largest power is the best.
Fourth, based on the type of plotted statistics we have Shewhart and Non-Shewhart
charts. Shewhart control charts take decisions based on the current chart statistic. The Shewhart
control charts for variable include ̅-chart (the sample means are plotted in order to control the
mean of the QC), R-chart (the sample ranges are plotted in order to control the variability of the
QC), S-chart (the sample standard deviations are plotted in order to control the variability of the
QC), S2-chart (the sample variances are plotted in order to control the variability of the QC). The
Shewhart control charts for attribute include C-chart (plotting the number of defectives per item),
U-chart (plotting the rate of defectives per item), np-chart (plotting the number of defective items
in a sample), P-chart (plotting the percentage of defective items in a sample).
9
On the other hand, Non-Shewhart control charts are based on the current and the previous
chart statistics. In the 1950s, the exponentially weighted moving average (EWMA) control chart
and the cumulative sum control chart (CUSUM) were introduced. It was proved in several
studies that they are effective in detecting small and moderate shifts quickly. The EWMA
statistic is defined as;
(
where
)
is the current sample statistic and
smoothing parameter, small values for
values for
,
0<
1
is the previous chart statistic.
is called the
are chosen to detect small shifts faster, while large
to detect large shifts faster. It should be noted that Shewhart chart is a special case
from the EWMA chart when
= 1.
Shewhart chart is better for Phase I analysis because the out-of-control samples can be
removed as the decision is based on the current statistic. On the other hand, EWMA and
CUSUM are not recommended for Phase I analysis as they are based on the current and the
previous statistics. As mentioned before, EWMA and CUSUM are better for Phase II in
detecting small shifts, while Shewhart charts are better for detecting large shifts. Sometimes in
Phase II analysis, a combination of Shewahrt and Non-Shewhart charts is used to assure
detecting small and large shifts taking into account the increase of the probability of false alarm.
1.3 Adaptive control charts:
Control chart usually has three design parameters: the sample size (n), the sampling
interval (h) and the control limit coefficient (k). Standard Shewhart charts are used with fixed
design parameters. There were many contributions in control charts; such as Non-Shewhart
charts (EWMA and CUSUM) or adaptive control charts, which are used to overcome the major
disadvantage of the Shewhart chart; that is the inefficiency of detecting small shifts. In this report
we focused on the later type of charts.
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Adaptive control chart has at least one variable design parameter (variable sampling size
VSS, variable sampling interval VSI, or variable control limit coefficient), where switching
among different parameters depends on the location of the current plotted sample statistic in the
chart.
It consists of three regions which are action region, warning region and central region.
Action region is the region where we get a signal (the process is out-of-control), which occurs
when the sample point falls outside the interval (
for the process parameter,
,
), where
is the largest allowable value
is the smallest allowable value for the process parameter. Warning
region is the region where sample point falls between (
) or (
), which means that the
process is in-control, but there is an evidence that a signal might occur. Central region is the
region where the sample point falls between (
,
). Here, the process is in-control and this is
the best region where there is no evidence that any signal might occur as it includes the central
line which is the optimal value for the process parameter.
In adaptive control chart, if the current plotted statistic lies in the central region then
there is no indication that the process parameters have changed so the next sample will be taken
with a smaller sample size and/or a larger sampling interval, and/or a larger control limit
coefficient. On the contrary, there is an indication that the process parameters have changed if
the current sample statistic lies in the warning region, therefore the next sample will be taken
from the process with larger sample size, smaller sampling interval, and/or smaller control limit
coefficient will be used.
It is shown in Figure (1.2), point A lies in the central region; therefore the next sample
should be taken with smaller sample size (ns), while point B lies in the warning region so the
next sample size should be larger (nL), at the 8th sample point, the process has gone out-ofcontrol.
11
Figure (1.2): Adaptive VSS Shewhart control chart for monitoring the process mean parameter
𝑋 ̅−𝑐ℎ𝑎𝑟𝑡
𝑋 ̅j
16
14
12
10
8
6
4
2
0
A
B
1
2
3
4
5
6
7
c
c
c
c
c
c
c
8c
K1
W1
CL
W2
K2
9
10
Sample number
1.4 The objective of this project:
The process performance depends on the location and dispersion parameters but the
priority is for controlling dispersion parameter first. The majority of studies in this area focused
on control charts for location parameters, such as the process mean. However, only few studies
focused on monitoring the dispersion parameters, such as the process standard deviation.If the
process standard deviation is not stable, then the accuracy of the control chart for the location
parameter is questionable.
Consequently, this report focuses on Shewhart S-Chart with Variable Sample Size to
monitor the standard deviation of an intended QC. The performance of the VSS S-chart has been
studied in the literature assuming known parameter case. Practically, in most cases the
parameters are unknown and have to be estimated from an in-control Phase I data set. The
performance of the VSS S-chart with estimated parameters has not been studied, yet. Our report
is similar to that made by Castagliola P. et al (2011) who studied VSS ̅ -chart with estimated
parameters. The VSS ̅ -chart is used to monitor the process mean. Our aim is to study the
statistical performance of the VSS S-chart with estimated parameters.
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1.5 Methodology:
In this report we used a Monte Carlo simulation technique to evaluate the performance of
the VSS S-chart with estimated parameters from an in-control Phase I data set. We will show the
effect of some important Phase I factors, such as the number of samples and the sample size, on
Phase II performance. We used the SAS software to perform the necessary calculations in our
simulation. Exactly 30,000 data sets are used to evaluate the performance in terms of the incontrol Average Sample Size (
) and in-control Average Run Length (
). Data sets are
generated from Normal distribution.
1.6 Literature Review:
The properties of the variable sampling interval (VSI) ̅ -Chart were first studied by
Renyolds et al. (1988). See also Reynolds (1990), Runger and Montgomery (1993), Amin and
Miller (1993). Moreover, the properties of ̅ -chart with variable sample size and variable
sampling interval were studied by Prabhu et al. (1994) who studied it under the assumption that
the process starts in a state of out-of-control, in 1995 Costa studied the properties of VSSI ̅ Chart when the process mean is out-of-control (assuming exponential distribution), while in
1996, he extended the VSSI control charts to the joint X-bar and R-chart. Zhang and Hua (2002)
also studied the np chart with variable sample size or variable sampling interval. The ̅ -Charts
with estimated parameters were studied by Del Castillo (1996). Jones and Champ (2002) studied
the design of EWMA charts with estimated parameters. While the dispersion-type control charts
(S2, S, and R control charts) were studied by Chen (1998) and Shahriari et al. (2009).
The rest of this report is organized as follows; we will discuss the VSS S-chart in Section
2. The simulation results are presented in Section 3. In Section 4, we will illustrate the use of the
VSS S-chart with estimated parameters using an illustrative example. Finally, conclusion and
recommendation are given in Section 5.
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Section 2: VSS S-chart
2.1 The design of the VSS S-chart:
Let
, i= 1, 2, 3... and j= 1, . . . ,
We assume that
denote certain quality characteristic of a process.
’s are independent and normally distributed Phase II samples each of size
with mean (a+ ) and standard deviation (b ). If a=0 and b=1 the process is in-control, otherwise
the process is out-of-control.
Process standard deviation can be monitored by plotting the sample standard deviation
(Si) on S-chart
Si =√∑
(
̅)
̅ i= ∑
(
/
)
.
In this report we will consider only the upper control limit to detect the increase in the
standard deviation. We will assume that the lower control limit is zero; we rely on the fact that in
S-chart the standard deviation reduction is corresponding to a desirable improvement in the
quality, so we care only about its increase. Some control charts users care about the lower limit
to detect the decrease in the standard deviation and investigate the reason of this reduction to use
it for improving the production process. Similarly, in this report we do not have a lower warning
limit.
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2.2 Measures of performance:
Control charts are used to monitor the process parameters to achieve two objectives.
First, when the process is in-control, we want the chart to signal infrequently. Statistically, if the
process is in-control, we want the probability that the computed statistic is plotted as out-ofcontrol to be as small as possible, i.e. probability of false alarm is as small as possible.
Second, when the process is out-of-control, we want the chart to signal as soon as
possible. Statistically, if the process is out-of-control, we want the probability that the computed
statistic is plotted as in-control to be as small as possible, i.e. the probability of true signal is as
large as possible. Concerning the previous two objectives, the measures of performance of VSSchart that we will use are in-control average run length ARL0 and in-control average sample size
ASS0.
RL is the number of samples until the chart signals. Consequently, RL follows geometric
distribution with probability of success p. Therefore, ARL = 1/p where p is the probability of
signal. The in-control average run length ARL0 is 1/p where p= , i.e.
= Pr (signal/ in-control
process). On the other hand, the out-of-control average run length ARL1 = 1/p where p=1- , i.e.
1-
= Pr (signal/ out-of-control process). Decreasing
will increase ARL0 which is a desired
result, but also ARL1 will increase resulting in undesired consequences; this is due to the inverse
relation between
and
. Most statisticians consider ARL0 = 370 is the desired value for
ARL0 as it achieves a balance between
and .
ASS is the average sample size until the chart signals. This measure should be taken into
account when VSS-chart is used because the sample size is not fixed. ASS0 is the in-control
average sample size and ASS1 is the out-of-control average sample size. In this report we assume
that ASS0 can take only three possible values {3, 5 and 7}. ASS is given by:
15
ASS= E ( ).
During the process monitoring,
small) or
=
will take only two possible values
("L" stands for large), where
, otherwise,
=
<
("S" stands for
. If the statistic lies inside the central region,
. nS and nL are given by:
– 1},
={
={
}.
In case of S-chart with unknown process parameters, S-pooled, the mean sample standard
deviation and the mean sample range are considered the traditional estimators. For deriving
estimates of the in-control standard deviation, we consider only the estimator based on the mean
sample standard deviation
̅= ∑
/
where m is number of Phase I samples each of size
70, 80 and 100} and
,
. In this report we will take m = {20, 50,
= {3, 5, 7 and 10}. An unbiased estimator for
is given by:
̂= ̅/ .
Some statisticians put the estimator in the limits; but in this case we will have variable
limits. Others put it in the plotted statistic to ease the comparison. We prefer using limits free
from the estimator, so we will plot Si/ in case of known parameters as following:
,
≤
.
In case of unknown parameters
̂
≤
̂
16
̂
.
Section 3: Methodology
3.1 Simulation steps:
In our analysis we performed a Monte Carlo simulation study using the statistical
package for statistical analysis system (SAS) to study the effect of estimated process parameters
on the in-control average run length and the in-control average sample size. Our concern is to
achieve
=370 and
= {3, 5 or 7}.
In the first stage of simulation, we used known process parameters to get the upper
warning limits and the upper control limits that satisfy
combination of
and
= 3 and
=370 for each
(which are determined according to the value of ASS0 as mentioned in
the previous section). First, we assumed an arbitrary value for UCL and UWL. Then, we
generated samples each of size ni from normal distribution with mean
deviation
= 4. The sample size ni is either
equal UWL, otherwise ni =
or
= 25 and standard
, where ni = nS if the statistic is less than or
.
Afterwards, we plotted the statistic (Si/ ) for each sample and compared it to the UCL
and the UWL. We continued generating until the chart signals, then RL and SS are recorded. We
used 30,000 simulation runs to calculate ARL0 and ASS0. If ARL0 ≠ 370 or/and ASS0 ≠ 3, we
tried other values for UCL and UWL until both ARL0 =370 and ASS0 =3 are satisfied. The SAS
program is in appendix (4) SAS program (1).
In the second stage, we used the results of stage one (the values of UCL and UWL
corresponding to each combination of nS and nL) to study the effect of the process parameter
estimation on the value of ARL0 and ASS0. We considered five values for number of Phase I
samples (m = 20, 50, 70, 80 and 100) and four values for Phase I sample size (n0 = 3, 5, 7 and 10)
17
for each combination of nS and nL. The following procedure was used in our simulation study:
1. Generate m samples each of size n0 (Phase I samples).
2. Calculate the standard deviation of each sample.
3. Calculate ̅ where ̅= ∑
/ m to get ̂= ̅/ .
4. Generate samples of size ni from normal distribution with mean
deviation
. The sample size ni is either
or equal UWL, otherwise ni=
or
, where ni=
and standard
if the statistic is less than
.
5. Plot the statistic (Si/ ) for each sample.
6. Repeat steps 4 and 5 until a signal is given, then record RL and SS.
We repeated the above procedure 30,000 simulation runs to calculate ASS0 and ARL0.Similarly,
we repeated the two stages for ASS0 = {5 and 7}. SAS program of this stage is in appendix (4)
SAS program (2).
3.2 Simulation results:
Our aim was to investigate the effect of using estimated process parameters on the
performance of VSS S-chart using the same design parameters (UWL and UCL) we got from the
known parameters case which satisfy ARL0=370 and ASS0={3, 5 or 7}. The results of both
simulation stages are in appendix (1) tables (1), (2) and (3).
Using the mentioned simulation procedure, we got the UWL and UCL. Unfortunately,
there was a problem with some of the UWL and UCL values. Our chart statistic was Si/̂, so the
target value was c4. This can be illustrated as follows:
Target value = E(Si/ ̂) = (c4 )/ = c4.
18
Inconveniently, the UWL of some combinations were less than their target values. This
problem can never be solved using simulation method. We recommend trying other methods as
Golden ratio or Markov to get the ASS0 and the ARL0 for these combinations. We will focus on
the correct results only in the analysis. The wrong results are colored in brown in appendix (1)
tables (1), (2) and (3).
After analyzing the results, we found that ASS0 was slightly affected by the estimation
while ARL0 was greatly affected. For illustrating the simulation results, see figures (3.1), (3.2),
(3.3) and (3.4).
Figure (3.1) ARL0 vs. m at ASS0=5 and nS=4 and nL=7
800
700
ARL0
600
500
n=3
400
n=5
n=7
300
n=10
200
100
0
20
50
70
80
19
100
m
Figure (3.2) ARL0 vs. n0 at ASS0=5 and nS=4 and nL=7
800
700
600
m=20
ARL0
500
m=50
400
m=70
300
m=80
m=100
200
100
0
3
5
7
10
n0
Figure (3.3) ARL0 vs. m at ASS0=7 and nS=5 and nL=10
600
500
ARL0
400
n=3
n=5
300
n=7
200
n=10
100
0
20
50
70
80
20
100
m
Figure (3.4) ARL0 vs. n0 at ASS0=7 and nS=5 and nL=10
600
500
ARL0
400
m=20
m=50
300
m=70
m=80
200
m=100
100
0
3
5
7
10
n0
From the previous figures we reached the following results:
1) Regardless the value of m when n0 increases the ARL0 increases.
2) Regardless the value of n0 when m increases the ARL0 decreases.
3) At ASS0=5 and ASS0=7 the values of ARL0 become close to each other starting from
m> 50.
The decrease in ARL0 value may reflect good chart performance or chart performance
deterioration. If ARL0 decreases from 500 to 400, this reflects good performance because the
value of ARL0 becomes close to 370. On the other hand, if ARL0 decreases from 300 to 200, this
reflects bad performance. Similarly, the increase in ARL0 value may reflect good or bad
performance.
The following tables show the best Phase I sample size, the best Phase I number of
samples, and Phase II design parameters (nS, nL, UWL and UCL) that should be used for VSS Schart with estimated parameters to achieve good performance.
21
Table (3.1) Best n0 and m for each combination at ASS0 = 3
nS
nL
UWL
UCL
n0
m
ARL0
2
4
0.9
2.8325
3
100
492.5283
2
5
0.975
2.8625
3
100
505.8207
2
6
1.1
2.9
3
100
513.2068
2
7
1.2025
2.9255
3
100
518.7394
2
8
1.28
2.937
3
100
520.2759
2
9
1.34
2.94775
3
100
525.4843
2
10
1.425
2.955
3
100
529.1514
Table (3.2) Best n0 and m for each combination at ASS0 = 5
nS
nL
UWL
UCL
n0
m
ARL0
2
10
0.975
2.84875
5
100
390.2085
3
8
0.975
2.31775
5
100
397.1912
3
9
1.04875
2.347725
5
100
403.1647
3
10
1.099975
2.359975
5
100
400.9127
4
7
1.05
2.105
5
100
409.787
4
8
1.125
2.119975
5
100
409.493
4
9
1.2225
2.135748
5
100
408.8869
4
10
1.25
2.139975
5
100
411.5765
22
Table (3.3) Best n0 and m for each combination at ASS0 = 7
nS
nL
UWL
UCL
n0
m
ARL0
5
10
1
1.9411
5
50
362.2793
6
9
1.05
1.8625
5
50
366.2986
6
10
1.125
1.872225
5
50
366.4989
The results of studying the chart performance for ASS0=3 were extremely bad, as shown
in table (3.1). The best performance among these results is at ARL0= 492.5283 which would be
achieved if you used Phase I control chart with n0=3 and m=100, and Phase II control chart with
nS=2, nL=4, UCL=2.8325 and UWL=0.9.
For ASS0=5, the chart performance improved as ARL0 became closer to 370. The results
are shown in table (3.2). The best performance is achieved at n0=5, m=100, nS=2 and nL=10,
UCL=2.84875 and UWL= 0.975. ARL0 in this case is 390.2085.
Finally, for ASS0=7 the chart performance became better as ARL0 is much closer to 370.
The results are shown in table (3.3). The best performance is achieved at n0=5, m=50, nS=6 and
nL=10, UCL=1.872225 and UWL= 1.125. ARL0 in this case is 366.4989. We can notice that for
higher ASS0 the ARL0 becomes closer to 370.
23
Section 4 : An Illustrative Example
In our report we are concerned about shifts in the standard deviation only. In the
following illustrative example, we will show that increasing the number of samples used for
estimation in Phase I will improve the estimator and help detect the shifts quicker.
Assume we have two companies A and B. They produced the same product.
These companies applied a program for monitoring the quality of their product. To achieve this
purpose, the quality control department in their companies designed an adaptive Shewhart chart
to monitor their product (bag of detergent). The objective was to observe the changes that happen
in the standard deviation of the weight of the detergent bag .The weight follows normal
distribution with hypothesized weight 25 kg and standard deviation 4 kg. For Phase I analysis,
company A selected one sample per hour for 50 hours, each sample was of size 5 units. The data
is shown in appendix (2) table (1). After generating the data, the estimated parameter was
calculated as follows:
̂ A= ̅/ = 4.19038 / 0.94 = 4.4578513
For Phase II analysis, company A selected one sample per hour for 100 hours. The first
50 samples were from in-control process and the last 50 samples were from out-of-control
process with a shift of size 2. Our interest was to study VSS S-chart, so we chose one of the
design parameters combinations we studied in the simulation section which are; nS=4, nL=8,
UWL= 1.125and UCL=2.119975. Phase II data is shown in appendix (2) table (2).
Similarly, company B followed exactly the same procedure in Phase I and Phase II,
except for taking one sample per hour for 100 hours in Phase I analysis instead of 50 hours. The
data of Phase I and Phase II are shown in appendix (2) table (3) and (4) respectively. The
estimated parameter was calculated using Phase I data as follows:
24
̂ B= ̅/ = 3.781457 / 0.94 = 4.02282664
The following figures are used to compare between the strategies of the two companies
(different number of Phase I samples):
Figure (4.1) Monitoring company (A) process (at m=50)
3.5
Chart statistic
3
2.5
UCL
2
1.5
UWL
1
0.5
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
0
Sample number
Figure (4.2) Monitoring company (B) process (at m=100)
Chart statistic
3.5
3
2.5
UCL
2
1.5
UWL
1
0.5
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
0
Sample number
25
Figure (4.1) and (4.2) show that company A Phase II control chart detected the shift of
size 2 at sample 63, while company B Phase II control chart detected the shift at sample 52.
Obviously, the performance of company B strategy (at m=100) is better than the performance
of company A (at m=50). Statistiaclly, S-chart with larger number of Phase I samples can
detect large shifts faster because it gives better estimators.
26
Section 5: Conclusion and Recommendations
5.2 Conclusion:
In this study, first we used simulation to obtain the design parameters (UWL and UCL)
that satisfy ARL0=370 and ASS0= {3, 5 or 7} in case of known process parameters (known mean
and standard deviation). The purpose of this research was to study the effect of estimating the
process parameters on certain measures of performance using the design parameters we got. This
research focused only on two measures of performance; in-control average run length ARL0 and
in-control average sample size ASS0. See appendix (1) tables (1), (2) and (3).
We concluded that the process parameters estimation affects the performance of VSS Schart. The value of ASS0 was slightly affected. On the other hand, ARL0 was greatly affected by
estimation. Thus, we became interested in choosing the best Phase I sample size and number of
Phase I samples needed to achieve better performance in terms of ARL0 (fixing the value of
ASS0) as shown in tables (3.1), (3.2) and (3.3).
In section 4, an example was introduced to illustrate VSS S-chart performance. It was
shown that using larger number of Phase I samples (m=100) resulted in better estimator for .
Besides, it is important to highlight the effect of using variable sample size in detecting the shifts
in the process parameters quickly regardless the number of Phase I samples. According to these
two facts, the detection of the shift was faster in case of using larger number of Phase I samples
in VSS S-chart (m=100).
5.2 Recommendations:
We recommend the best combination of Phase I sample size and number of Phase I
samples (no and m) for each ASS0 that achieve approximately ARL0=370. For ASS0=3, it is
27
better to use n0=3 and m=100. At ASS0=5, the best performance is achieved at n0=5 and m=100.
While at ASS0=7, the best combination is n0=5 and m=50. We recommend using ASS0=7
because it gives closer values to ARL0=370.
28
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32
Appendices
Appendix (1): Simulation results tables
Appendix (1): Table (1) Simulation results for ASS0=3
nS
nL
UWL
UCL
m
n0
20
50
70
80
100
830.7177
558.34383
521.57183
511.0031
492.52833
{2.8206935}
{2.8106527}
{2.8117056}
{2.8089539}
{2.8093582}
1049.3142
841.37517
811.18393
797.7199
787.98173
{2.7289849}
{2.725225}
{2.724044 }
{2.7238004}
{2.7230566}
1157.0136
979.59687
957.02847
953.59697
942.27513
{2.6976791}
{2. 6950168}
{2. 6951908}
{2.6943061}
{2.6947701}
1189.2452
1094.1973
1068.019
852.6498
846.3793
{2.678151}
{2.6761402}
{2.6742349}
{2.706988}
{2.707254}
895.2317
583.0657
535.19817
524.3308
505.82073
{3.0880327}
{3.0706531}
{3.0695544}
{3.0670902}
{3.0669358}
1093.2846
879.56533
838.40807
834.02327
817.6487
{2.9445634}
{2.9332573}
{2.9318637}
{2.9310207}
{2.9309073}
1190.9295
1033.5364
1012.5344
987.90347
985.90767
{2.8982892}
{2.891009}
{2.8874321}
{2.8869289}
{2.8870907}
1269.5777
1146.9982
1132.9841
1131.2802
1109.4508
{2.8662096}
{2.8593018}
{2.8587333}
{2.8590298}
{2.8573894}
3
nS= 2
5
nL=4
UWL=0.9
7
UCL=2.8325
10
3
nS=2
5
nL=5
UWL=0.975
7
UCL=2.8625
10
33
nS
nL
UWL
UCL
m
n0
20
50
70
80
100
978.2085
598.06157
544.2676
528.76887
513.20683
{3.1264244}
{3.0943069}
{3.0898139}
{3.0902017}
{3.0851222}
1166.8025
907.55867
883.84077
864.36943
848.07143
{2.9482809}
{2.9345723}
{2.9312109}
{2.9286012}
{2.9275043}
1276.2923
1087.0715
1053.0303
1042.159
1016.7965
{2.8900184}
{2.881696}
{2.8794345}
{2.8789216}
{2.8785394}
1362.4022
1202.5722
1168.2178
1183.3471
1163.4993
{2.8548366}
{2.847577}
{2.8476471}
{2.8457299}
{2.8459788}
1025.8117
612.10573
557.73347
540.25173
518.73943
{3.1327356}
{3.0953952}
{3.0922913}
{3.0831962}
{3.080942}
1237.2425
945.1171
917.46753
901.29363
868.91687
{2.9385205}
{2.9218051}
{2.9165122}
{2.9167236}
{2.9150016}
1377.9306
1130.0363
1099.1163
1071.1088
1069.4083
{ 2.8754751}
{2.8631577}
{2.8660938}
{2.865242}
{2.8620348}
1431.8819
1280.5475
1239.1372
1224.0034
1221.6898
{ 2.8381861}
{2.8319459}
{2.8294495}
{2.830373}
{2.8292443}
990.83343
615.1339
554.12467
542.76
520.27857
{3.1538742}
{3.1121768}
{3.1032549}
{3.0982546}
{3.0958086}
1291.7739
962.90053
917.24947
919.8364
883.6263
{2.9493464}
{ 2.9273639}
{2.9271479}
{2.9250441}
{2.9236683}
1397.3746
1144.0931
1111.1059
1088.1227
1087.4325
{2.8825168}
{2.8718922}
{2.8727323}
{2.871157}
{2.8672918}
1461.544
1289.9679
1251.2875
1255.552
1232.3188
{2.8431295}
{2.8383225}
{2.8360887}
{2.8365332}
{2.8368816}
3
5
nS=2
nL=6
7
UWL=1.1
UCL=2.9
10
3
nS=2
5
nL=7
UWL=1.203
7
UCL=2.9255
10
3
nS=2
5
nL=8
UWL=1.28
7
UCL=2.937
10
34
nS
nL
UWL
UCL
m
n0
20
50
70
80
100
1070.0346
629.74017
563.76843
546.8887
525.4843
{3.1813104}
{ 3.1444414}
{3.1332977}
{3.1347166}
{3.129763}
1302.261
985.87827
944.07773
925.72607
894.26517
{2.9736911}
{2.9514613}
{2.9473061}
{2.9499934}
{2.9496766}
1444.2028
1171.4501
1128.2567
1121.4723
1108.4966
{2.904519}
{2.8938071}
{2.893744}
{2.8922616}
{2.8911643}
1518.891
1324.7252
1287.1724
1281.3717
1257.4755
{2.8652357}
{2.8578041}
{2.8559548}
{2.8567531}
{ 2.8549075}
1112.371
622.10683
568.2804
555.865
529.15143
{3.1512392}
{3.1095274}
{3.0986938}
{3.0999411}
{3.0962339}
1320.0685
762.6036
733.2447
721.10127
709.85797
{2.9402719}
{2.9876623}
{2.9810579}
{2.9805331}
{2.978788}
1448.9734
906.2489
880.881
878.3022
867.04487
{2.8714646}
{2.9248658}
{2.921802}
{2.9239006}
{2.9190563}
1517.5265
1021.6116
1006.7165
997.35533
991.9533
{2.8299706}
{2.8831883}
{2.8814369}
{2.883563 }
{2.8821892}
3
5
nS=2
nL=9
7
UWL=1.34
UCL=2.9478
10
3
nS= 2
5
nL= 10
UWL= 1.425
7
UCL=2.955
10
35
Appendix (1): Table (2) Simulation results for ASS0=5
m
nS nL
n0
UWL UCL
20
50
70
80
100
347.49403
318.01827
314.32317
312.31057
308.66643
{5.1395433}
{5.1504492}
{5.1554126}
{5.1597924}
{5.156914}
408.426
385.9964
382.46642
381.8797
376.48887
{4.9889761}
{4.9889081}
{4.9904546}
{4.991574}
{4.9919638}
434.62997
416.32417
412.92427
411.63693
408.2074
{4.9228305}
{4.9267095}
{4.9270356}
{4.9276648}
{4.9303082}
454.45983
438.26817
434.37423
434.56527
432.07683
{4.8815954}
{4.8832787}
{4.8857062}
{4.8834536}
{4.8847832}
367.4344
306.0997
300.63967
296.13293
294.75873
{5.3346994}
{5.3375134}
{5.3330314}
{5.3416941}
{5.34244}
437.0443
389.317
384.10617
383.1026
382.8847
{5.0494457}
{5.0472527}
{5.0440191}
{5.043832}
{5.0427695}
459.1716
429.30763
427.74107
423.82433
418.8936
{4.9413883}
{4.9399988}
{4.9419183}
{4.9377656}
{4.9371193}
483.56387
460.5784
453.18713
450.4667
453.3275
{4.8668633}
{4.8634479}
{4.8646264}
{4.8660165}
{4.8662022}
377.41203
300.07447
289.8238
283.88303
281.18227
{5.4225791}
{5.4075725}
{5.405255}
{5.403886}
{5.4018602}
3
nS=2
5
nL=6
UWL= 0.65
7
UCL= 2.543
10
3
nS=2
5
nL=7
UWL= 0.78
UCL=2.6968
7
10
nS=2
3
nL=8
36
m
nS nL
n0
UWL UCL
UWL= 0.875
20
50
70
80
100
454.29507
403.62327
385.57803
388.561
381.33713
{5.0288264}
{4.9992025}
{4.9998536}
{4.9999209}
{4.9987088}
478.94707
448.27383
437.32943
434.87297
435.27337
{4.879629}
{4.8644923}
{4.8630738}
{4.8625908}
{4.8576847}
512.14097
487.47893
478.3044
478.46873
476.20237
{4.7843917}
{4.7732047}
{4.7703181}
{4.7659286}
{4.7656128}
386.86763
294.97303
286.94403
284.1685
277.3502
{5.6262437}
{5.5810917}
{5.5771233}
{5.5652738}
{5.5620208}
458.35003
406.88987
391.5146
392.20017
383.92747
{5.1295232}
{5.0970606}
{5.0884232}
{5.079443}
{5.0797967}
500.74703
451.9459
448.57053
441.64157
442.43387
{4.9500582}
{4.9253476}
{4.9195101}
{4.9220789}
{4.9130387}
531.99343
496.27007
486.18007
481.95623
479.8295
{4.8340808}
{4.8114984}
{4.816998}
{4.8147537}
{4.807591}
385.16737
292.58387
284.04933
279.09547
273.66377
{5.7976619}
{5.7283813}
{5.7160664}
{5.7039292}
{5.681832}
466.16467
410.40393
399.53577
392.6051
390.20817
{5.2100298}
{5.1551898}
{5.1496231}
{5.1440046}
{5.1380357}
515.1088
457.83343
454.61023
450.22677
445.87113
{5.0072978}
{4.975459}
{4.9625932}
{4.9666687}
{4.9564815}
5
UCL=2.7825
7
10
3
nS=2
nL=9
5
UWL= 0.93
UCL= 2.82
7
10
3
nS=2
nL=10
UWL= 0.975
5
UCL=2.84875
7
37
m
nS nL
n0
UWL UCL
20
50
70
80
100
536.02507
503.2348
496.22497
495.02293
494.45817
{4.8761425}
{4.8495088}
{4.8517777}
{4.8435877}
{4.842676}
364.1484
276.47691
264.99055
258.79829
256.84966
{5.1656737}
{5.171741}
{5.1725262}
{5.1726795}
{5.172032}
469.69652
402.999
389.67926
388.06776
383.60203
{5.0194361}
{5.0221009}
{5.0214302}
{5.0210156}
{5.0213268}
508.69358
456.78033
450.62777
446.41984
442.87183
{4.9633919}
{4.9643327}
{4.9643116}
{4.9647848}
{4.9655057}
540.43203
501.06588
498.44778
492.08499
491.65019
{4.9257772}
{4.9261083}
{4.9254264}
{4.9265106}
{4.9261763}
416.0369
288.03743
270.50223
266.6709
258.7782
{5.2848149}
{5.2818514}
{5.2754298}
{5.2748732}
{5.2783577}
507.52737
421.62563
405.22003
398.81763
392.13097
nL=7
{5.0329764}
{5.0192737}
{5.0160256}
{5.0220384}
{5.0202107}
UWL= 0.9
547.92627
480.23563
472.08847
463.79897
464.73173
{4.9394031}
{4.932532}
{4.9265882}
{4.9270194}
{4.9279168}
583.85447
532.28923
520.35737
521.57367
516.71107
{4.8726714}
{4.868189}
{4.867666}
{4.8680094}
{4.8685559}
430.5749
284.63003
267.04037
262.41707
253.79733
10
3
nS=3
5
nL=6
UWL=0.75
7
UCL=2.1975
10
3
nS= 3
UCL= 2.285
5
7
10
nS=3
3
38
m
nS nL
n0
UWL UCL
20
50
70
80
100
nL=8
{5.494836}
{5.4601935}
{5.4592558}
{5.4533514}
{5.4571185}
535.49097
424.47647
416.75567
402.10643
397.19117
{5.1406607}
{5.1235842}
{5.1144767}
{5.118604}
{5.1122293}
582.6643
491.3646
477.75107
475.397
471.32533
{5.019948}
{5.0052428}
{5.0037359}
{5.003253}
{5.0013022}
610.1146
540.25043
530.74763
529.9271
523.29947
{4.9430719}
{4.9315304}
{4.9268475}
{4.9274227}
{4.9241252}
445.72237
288.45083
265.2243
263.76593
256.00047
{5.5565134}
{5.5006196}
{5.4910124}
{5.4792278}
{5.4775641}
573.75743
440.6414
416.52
411.99003
403.16473
{5.1237507}
{5.090527}
{5.0894136}
{5.0854058}
{5.0843147}
599.44547
515.5647
493.3387
487.70533
477.68223
{4.9867104}
{4.9649905}
{4.9587873}
{4.9531697}
{4.9562042}
636.6931
568.86197
556.29447
548.22023
545.71457
{4.8933959}
{4.8768546}
{4.8733479}
{4.8756145}
{4.8736412}
471.96488
283.00456
263.30841
255.20139
247.23228
{5.6447283}
{5.5715217}
{5.5503643}
{5.5469493}
{5.5442522}
572.97492
438.24441
438.24441
408.68988
400.91268
{5.1601633}
{5.1152161}
{5.1152161}
{5.1068316}
{5.102121}
609.70106
508.551
495.35564
488.36915
481.4492
UWL= 0.975
UCL=2.31775
5
7
10
3
nS=3
5
nL=9
UWL=1.04875
7
UCL=2.347725
10
nS=3
3
nL=10
UWL=1.099975
UCL=2.359975
5
7
39
m
nS nL
n0
UWL UCL
20
50
70
80
100
{5.0028318}
{4.9747796}
{4.9683672}
{4.9641866}
{4.9632665}
641.092
571.20217
558.01957
547.2025
482.19963
{4.9018703}
{4.8823113}
{4.8780414}
{4.873932}
{4.9604452}
532.94867
275.38117
252.09403
246.5967
234.2304
{5.2959972}
{5.274752}
{5.2716517}
{5.2707349}
{5.2679313}
629.51517
451.9243
428.87103
419.68273
409.787
nL= 7
{5.0847345}
{5.0719714}
{5.0685749}
{5.0700862}
{5.0686438}
UWL= 1.05
692.4716
532.80543
520.64163
509.25207
498.68127
{5.0125605}
{5.0041615}
{5.0000829}
{4.9998039}
{5.0003173}
697.52173
603.47563
589.45953
585.68143
576.1624
{4.9636601}
{4.9573173}
{4.9562025}
{4.9573299}
{4.9552992}
560.1551
283.04043
252.78823
244.45823
234.41377
{5.4515847}
{5.4097498}
{5.4045954}
{5.401137}
{5.3979297}
638.0882
456.2129
430.74493
423.5139
409.493
{5.1732766}
{5.1485196}
{5.145325}
{5.1398804}
{5.1383623}
707.4163
548.9756
519.06167
515.3648
504.86163
{5.0770245}
{5.062215}
{5.0613403}
{5.0565738}
{5.0575881}
724.4051
617.3783
602.05947
593.33047
581.68057
{5.0157726}
{5.0065338}
{5.0041152}
{5.0039118}
{5.0041513}
10
3
nS= 4
UCL= 2.105
5
7
10
3
nS= 4
nL= 8
5
UWL= 1.125
UCL=2.119975
7
10
40
m
nS nL
n0
UWL UCL
20
50
70
80
100
586.7362
282.2523
246.2558
242.71687
232.3951
{5.4073748}
{5.3514059}
{5.3394891}
{5.3343811}
{5.3288533}
683.46703
464.59413
432.81003
422.98737
408.88693
{5.0891652}
{5.0590663}
{5.0543731}
{5.0539691}
{5.0506855}
733.2419
553.51747
524.4155
520.6517
512.75503
{4.9823707}
{4.9655909}
{4.9641534}
{4.9617462}
{4.9625093}
756.408
633.1033
612.27507
602.17177
591.6479
{4.919718}
{4.9078875}
{4.9041833}
{4.9037186}
{4.9055504}
576.06517
281.1382
252.8631
240.80653
231.76893
{5.5659229}
{5.4947432}
{5.4750111}
{5.4749264}
{5.4638414}
686.18277
468.78253
428.7995
427.89327
411.5765
{5.1912022}
{5.1607694}
{5.1511647}
{5.1435069}
{5.146164}
734.06093
557.86527
528.58793
526.5062
515.29687
UWL= 1.25
{5.0756003}
{5.0538199}
{5.0481659}
{5.0461544}
{5.0453451}
UCL=2.139975
764.1768
642.11153
615.874
607.0351
595.73237
{4.9977586}
{4.9879347}
{4.9808545}
{4.9795509}
{4.9804299}
3
nS= 4
nL= 9
5
UWL= 1.2225
UCL=2.135748
7
10
3
5
nS= 4
nL= 10
7
10
41
Appendix (1): Table (3) Simulation results for ASS0=7
m
nS nL
n0
UWL UCL
20
50
70
80
100
362.59263
357.42639
355.17551
354.85831
354.37712
{7.225142}
{7.2587639}
{7.2690957}
{7.2711784}
{7.2744306}
389.80149
379.22625
375.30888
374.00282
372.46294
{7.0366933}
{7.0557665}
{7.0609377}
{7.0640829}
{7.0638366}
396.9334
388.79165
385.90822
385.52141
385.05313
{6.9654874}
{6.9776803}
{6.9761001}
{6.9781729}
{6.9798052}
404.5355
397.50921
395.70957
394.53965
393.66061
{6.9104149}
{6.917651}
{6.9186033}
{6.9179553}
{6.919947}
355.5562
327.85717
324.36953
317.51733
314.52687
{7.4857441}
{7.5404994}
{7.5466672}
{7.5473665}
{7.5622866}
390.24913
360.0912
354.93273
354.5687
353.46838
{7.1432014}
{7.1607613}
{7.1613598}
{7.1567168}
7.1621876
399.18395
382.68627
380.55523
379.06647
377.84845
{7.0018103}
{7.0095666}
{7.0112322}
{7.0124243}
{7.009953}
410.02553
398.60596
395.39311
393.9996
390.75579
{6.9022149}
{6.9047063}
{6.9039444}
{6.9059237}
{6.9060355}
349.55723
305.71177
300.79637
297.05717
291.33437
{7.7226039}
{7.748694}
{7.761356}
{7.7711462}
{7.7714946}
388.70917
359.41973
351.09953
349.68383
347.8418
{7.2125475}
{7.2075984}
{7.2067334]
{7.2108644}
{7.2114308}
408.69443
384.6618
380.30889
381.13954
374.73767
{7.0031456}
{7.0042854}
{7.0015999}
{7.0059021}
{7.0011341}
422.2475
404.89983
401.21873
400.29023
399.09611
3
nS= 2
nL= 8
5
UWL= 0.6225
UCL= 2.35
7
10
3
nS= 2
nL= 9
UWL= 0.7375
5
7
UCL=2.585025
10
3
nS= 2
nL= 10
5
UWL= 0.8075
UCL= 2.6825
7
10
42
nS nL
m
n0
UWL UCL
20
50
70
80
100
{6.8644555}
{6.8679793}
{6.8520202}
{6.8556495}
{6.8582459}
294.39003
266.75113
262.14073
261.05641
257.95419
{7.2041182}
{7.2347613}
{7.2396615}
{7.2410482}
{7.2428473}
369.08447
347.50783
344.72973
344.39947
343.37672
{7.0181013}
{7.0280222}
{7.0322945}
{7.0295976}
{7.0317858}
398.4005
379.4391
378.2103
377.51134
378.22231
{6.9362075}
{6.9473792}
{6.9513388}
{6.948444}
{6.9520486}
421.51857
409.06767
401.6866
401.55133
402.74563
{6.8845192}
{6.8903107}
{6.8886736}
{6.8911033}
{6.8927586}
326.07817
273.19447
266.9366
268.8484
262.36153
{7.5069632}
{7.5270244}
{7.5381179}
{7.5399108}
{7.538056}
399.68177
354.46177
348.6765
348.06373
345.5687
{7.1608846 }
{7.1738226}
{7.174419}
{7.1748078}
{7.1714454}
423.3455
390.99547
385.37647
382.24753
380.16303
{7.0280799}
{7.0315637}
{7.0323672}
{7.0304848}
{7.0380519}
438.76923
420.75637
417.67907
415.74397
410.58157
{6.9407978}
{6.9361204}
{6.9376909}
{6.9324877}
{6.9375819}
340.1866
267.30707
254.59527
251.55873
248.60703
{7.6724055}
{7.6874445}
{7.6829585}
{7.6772007}
{7.6807957}
409.7832
351.9274
348.30677
341.0378
336.87567
{7.1782566}
{7.1635395}
{7.1633402}
{7.1637178}
{7.1653322}
441.14787
392.6479
388.34653
384.81723
380.24827
{6.9848968}
{6.9807683}
{6.968634}
{6.9754557}
{6.9739387}
457.59433
429.81923
423.97013
421.54027
417.2171
3
nS= 3
nL= 8
5
UWL= 0.6975
UCL= 2.0825
7
10
3
nS= 3
5
nL= 9
UWL= 0.7975
7
UCL= 2.1875
10
3
nS= 3
nL= 10
5
UWL= 0.875
UCL= 2.25125
7
10
43
nS nL
m
n0
UWL UCL
20
50
70
80
100
{6.8503359}
{6.845569}
{6.8476428}
{6.8436382}
{6.8431503}
308.40507
222.6421
214.16363
210.18683
207.62007
{7.2245402}
{7.2476698}
{7.2517048}
{7.2561145}
{7.2575918}
395.29483
341.8548
330.0463
327.99257
326.30697
{7.0402166}
{7.0504674}
{7.0555058}
{7.0530919}
{7.0576395}
435.69563
390.3271
388.439
383.97163
379.32633
{6.9706034}
{6.9790382}
{6.9778623}
{6.9778556}
{6.9771432}
465.1767
426.269
427.0679
421.47467
423.99693
{6.9200973}
{6.9243079}
{6.9236455}
{6.9235101}
{6.923056}
337.0157
238.65457
230.56863
224.0654
222.3882
{7.5318117}
{7.5544238}
{7.5538334}
{7.5597171}
{7.5580101}
415.48913
349.64373
341.0591
337.28847
334.85403
{7.214113}
{7.2138855}
{7.2198789}
{7.2147814}
{7.2161511}
452.92397
399.93947
392.46213
389.97377
386.88597
{7.0851325}
{7.0898258}
{7.0903878}
{7.0924561}
{7.0906113}
484.4473
441.60477
440.5674
435.44807
433.8797
{7.0029959}
{7.0056257}
{7.0047041}
{7.002589}
{7.0031404}
359.03417
240.86393
228.7687
225.2532
220.25463
{7.7264554}
{7.7369265}
{7.7333547}
{7.7316219}
{7.7296516}
437.07277
356.71543
340.57423
338.18693
331.50393
{7.2814067}
{7.2646018}
{7.2618015}
{7.2568266}
{7.2604293}
478.75403
411.26933
393.15133
390.95443
388.67857
{7.1073816}
{7.0952894}
{7.0901553}
{7.094398}
{7.0921729}
498.79577
450.36627
443.48467
435.77883
433.03957
3
nS= 4
nL= 8
5
UWL= 0.75
UCL= 1.95
7
10
3
nS= 4
nL= 9
5
UWL= 0.85
UCL= 2.01
7
10
3
nS= 4
nL= 10
5
UWL= 0.925
UCL= 2.05
7
10
44
nS nL
m
n0
UWL UCL
20
50
70
80
100
{6.9869815}
{6.9827565}
{6.9798593}
{6.9806218}
{6.9788286}
391.9301
211.55333
194.44433
186.8999
177.01853
{7.2160804}
{7.2314596}
{7.2327717}
{7.2324514}
{7.2354709}
479.10373
352.3669
335.66647
325.5408
320.9643
{7.0373048}
{7.0420249}
{7.0464557}
{7.0447783}
{7.0453003}
508.1343
424.24247
412.95783
404.20763
398.50507
{6.9723212}
{6.9752331}
{6.97364}
{6.9759253}
{6.9768656}
533.99187
474.1511
465.38737
455.54867
458.5659
{6.9273617}
{6.9277542}
{6.9272151}
{6.9271062}]
{6.9274438}
415.1945
227.34897
206.0848
200.00403
189.06887
{7.5135764}
{7.5078636}
{7.5159849}
{7.512105}
{7.5136001}
472.97277
347.6286
337.6987
332.65517
326.15927
{7.2156958}
{7.2079815}
{7.2031812}
{7.2062774}
{7.2042969}
518.2361
423.04703
407.66807
400.5212
397.37787
{7.1052257}
{7.0914596}
{7.0935724}
{7.09413}
{7.0908641}
541.10827
469.7394
460.9053
458.09203
453.27777
{7.0253135}
{7.0211434}
{7.0184232}
{7.0195889}
{7.0196618}
399.4851
224.01647
204.17087
200.23927
192.29573
{7.7033031}
{7.6796805}
{7.6698684}
{7.6668194}
{7.667765}
508.01107
362.2793
342.88467
335.72773
330.63277
{7.2812802}
{7.2569438}
{7.255246}
{7.2545351}
{7.2495163}
539.28183
429.19227
414.1092
406.1938
397.15763
{7.1296858}
{7.1143028}
{7.1082887}
{7.1111344}
{7.1091136}
567.42207
478.43787
467.93573
464.28607
454.04353
3
nS= 5
nL= 8
5
UWL= 0.825
UCL=1.889725
7
10
3
nS= 5
nL= 9
5
UWL= 0.96
7
UCL=1.93
10
3
nS= 5
nL= 10
5
UWL= 1
UCL= 1.9411
7
10
45
nS nL
m
n0
UWL UCL
20
50
70
80
100
{7.0321024}
{7.0162901}
{7.016494}
{7.0179632}
{7.0154901}
504.18157
212.75803
183.40623
173.37433
165.0262
{7.2674293}
{7.2712458}
{7.2724762}
{7.2718066}
{7.2734444}
573.00517
368.49743
334.5075
333.96833
318.2218
{7.1254643}
{7.1207764}
{7.1214983}
{7.1184572}
{7.1190418}
614.8968
441.80227
421.39553
414.8631
404.09447
{7.0657817}
{7.0660829}
{7.0650786}
{7.0637913}
{7.0655493}
630.356
514.69057
494.34653
492.28653
482.90253
{7.0314559}
{7.0277966}
{7.026916}
{7.0269756}
{7.0275835}
537.48097
216.69683
186.69883
181.54907
172.00737
{7.4535562}
{7.4316794}
{7.4273943}
{7.4276184}
{7.4231153}
600.1056
366.2986
341.24703
338.14453
322.86963
{7.1950446}
{7.1794038}
{7.173584}
{7.1721823}
{7.1720573}
616.16283
460.35807
421.0729
415.9359
406.6061
{7.1013629}
{7.0875107}
{7.0898654}
{7.0857867}
{7.0858237}
630.7139
511.14437
492.95247
488.52063
479.44673
{7.0378443}
{7.032147}
{7.0301044}
{7.0296995}
{7.0284753}
596.572
215.8954
187.93277
178.02823
171.14973
{7.2498921}
{7.5422155}
{7.5424056}
{7.5391132}
{7.530666}
600.7617
366.49893
341.6559
333.6262
321.24367
{7.2433143}
{7.2172922}
{7.210883}
{7.208688}
{7.2045199}
613.49167
449.18147
426.33387
416.6484
409.0556
{7.1265635}
{7.1100888}
{7.1048806}
{7.1042994}
{7.1014975}
628.12247
515.53833
490.4218
487.29733
479.3889
3
nS= 6
nL= 8
5
UWL=0.96
UCL=1.786875
7
10
3
nS= 6
nL= 9
5
UWL= 1.05
UCL= 1.8625
7
10
3
nS= 6
nL= 10
5
UWL= 1.125
UCL=1.872225
7
10
46
nS nL
m
n0
UWL UCL
20
50
70
80
100
{7.0518181}
{7.0375455}
{7.0340141}
{7.0335914}
{7.0305968}
Appendix (2): Example data
Appendix (2): Table (1) Phase I data at m=50
Sample
number
X1
X2
X3
X4
X5
Si
1
30.71794
22.2279
23.93068
21.23911
24.80067
3.702649
2
22.06866
25.58311
28.97462
23.05629
15.73338
4.899191
3
29.63639
27.80255
27.10247
22.79182
29.56188
2.791004
4
22.00582
22.61273
30.31951
24.2794
23.84555
3.318996
5
22.34031
21.87609
19.63214
28.3818
24.16253
3.277504
6
25.97584
27.10878
24.3136
30.58465
31.41043
3.029803
7
20.04454
24.28207
27.35507
21.30563
31.05447
4.492762
8
19.77189
22.16026
32.66188
28.0866
22.98927
5.186089
9
25.99876
27.90263
26.42466
23.62161
22.78051
2.102453
10
21.28601
27.93638
19.60436
31.56608
20.32734
5.310667
11
28.06931
18.54163
22.28181
22.66765
22.09858
3.41519
47
Sample
number
X1
X2
X3
X4
X5
Si
12
26.59283
22.91594
20.48925
18.93078
30.7294
4.774455
13
24.57792
25.82286
22.25023
30.52503
19.04363
4.268335
14
18.38306
25.08297
23.58846
25.16948
25.86982
3.042591
15
27.3552
23.89447
33.68988
23.74916
18.30478
5.654179
16
23.58983
21.88072
25.86806
27.84089
26.15499
2.337428
17
20.12008
29.23318
30.87898
24.15052
22.76822
4.502028
18
18.50092
21.09137
35.96619
29.37644
25.21877
6.922094
19
22.40449
25.62116
23.84317
22.38962
32.64946
4.273786
20
28.42472
25.33203
18.35393
24.62963
24.10407
3.657018
21
30.92483
20.50541
27.99147
25.76358
21.01655
4.483744
22
22.86647
28.65738
29.76481
25.91908
22.69112
3.240665
23
24.79937
25.91247
23.11333
34.15864
24.21865
4.43138
24
19.08109
26.48289
30.72051
25.67657
28.40058
4.365461
25
27.67686
18.48512
26.36193
15.19129
29.42817
6.22371
26
24.05555
24.13965
34.36899
31.72814
21.65247
5.521012
27
24.01597
20.68679
24.37194
23.98101
31.25487
3.874058
28
21.7425
30.63848
27.01278
18.6859
32.97417
5.970352
48
Sample
number
X1
X2
X3
X4
X5
Si
29
32.11368
29.11135
33.30566
28.2279
21.51724
4.60242
30
16.95546
24.32637
24.68464
16.93719
29.63938
5.494667
31
21.88428
16.47901
18.77898
25.18165
31.25489
5.789732
32
27.62869
29.33574
19.03768
23.11226
29.19583
4.478698
33
27.0172
24.71359
24.50575
21.02096
21.33518
2.522317
34
26.40537
22.92255
25.75225
24.09142
21.44804
2.02862
35
25.82082
18.4702
22.30899
31.10787
29.45022
5.165575
36
20.08523
24.89055
26.64827
23.11782
21.26086
2.655999
37
20.41609
26.32462
29.60847
20.64638
36.78875
6.828262
38
20.52267
15.71798
13.54324
20.5709
25.7332
4.754823
39
22.36264
30.93097
31.91967
28.43435
28.76617
3.7197
40
21.31362
28.31571
20.66141
20.83377
29.79451
4.483879
41
28.7613
25.26554
22.47744
30.78595
28.6822
3.299173
42
19.46183
28.01682
23.05871
23.9943
28.32635
3.697958
43
30.2293
19.28706
21.60873
26.57068
28.541
4.641758
44
23.02573
22.74216
27.67521
19.48278
28.66251
3.797309
45
20.52318
25.94091
28.54966
18.31087
28.7031
4.749152
49
Sample
number
X1
X2
X3
X4
X5
Si
46
15.83228
21.77333
23.8825
24.0321
21.85495
3.331986
47
24.28239
27.04635
24.21217
24.13293
22.48289
1.642712
48
23.09807
24.87452
29.63141
31.72215
27.61499
3.485337
49
18.64003
29.42383
20.66486
27.62872
24.38276
4.540784
50
27.90051
28.31643
27.04998
22.09423
17.32772
4.739548
Appendix (2): Table (2) Phase II data at m=50
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
1
22.9643
18.6852
25.1085
23.1700
−
−
−
−
0.60775
2
24.7771
26.331
26.3672
31.3211
−
−
−
−
0.63856
3
27.1310
22.6834
30.476
23.2006
−
−
−
−
0.81525
4
23.6044
25.1708
28.3346
23.4703
−
−
−
−
0.50745
5
25.2160
28.2574
30.8717
26.4110
−
−
−
−
0.55248
6
25.9734
21.6917
24.3888
27.4487
−
−
−
−
0.55252
7
27.2944
26.2271
24.2726
18.7537
−
−
−
−
0.85258
8
21.6124
31.4095
29.7428
26.4186
−
−
−
−
0.96901
50
9
25.4349
16.3136
29.4134
24.8596
−
−
−
−
1.23671
10
26.3651
22.1776
25.0080
22.6473
25.6698
27.3098
28.1209
28.8555
0.54302
11
24.1635
23.9453
19.132
33.3443
−
−
−
−
1.33204
12
20.7379
25.7347
25.8050
24.6179
14.0057
20.6508
21.3156
21.0896
0.86205
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
13
27.9569
23.3863
25.5035
27.0004
−
−
−
−
0.44681
14
28.3459
24.6097
30.1902
26.9913
−
−
−
−
0.52705
15
25.4624
32.2792
20.5957
26.6423
−
−
−
−
1.07656
16
24.0439
22.0911
31.0975
29.0267
−
−
−
−
0.94265
17
20.9494
22.2084
28.6726
20.9004
−
−
−
−
0.832166
18
27.6165
25.9940
29.6312
17.4044
−
−
−
−
1.20712
19
28.0731
27.3548
22.0534
30.5618
27.6481
31.2553
21.9288
21.8630
0.871304
20
20.7452
24.8837
25.5175
23.2164
−
−
−
−
0.477998
21
18.1498
22.9356
27.6187
16.8918
−
−
−
−
1.098329
22
25.7564
31.8563
21.9583
32.0896
−
−
−
−
1.107346
23
29.0763
28.0829
33.0551
19.6781
−
−
−
−
1.261428
24
23.1888
23.3290
22.2040
27.8749
25.1271
26.4044
23.4197
23.4576
0.431513
25
21.8608
24.0901
28.7693
21.5694
−
−
−
−
0.746438
51
26
27.2008
23.2136
21.8299
22.5737
−
−
−
−
0.538032
27
20.6178
26.5612
21.1369
24.1967
−
−
−
−
0.623747
28
26.6169
25.4845
24.7588
27.7743
−
−
−
−
0.296309
29
25.0803
24.2729
33.1002
20.8991
−
−
−
−
1.15953
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
30
25.1217
28.1678
24.0309
28.2407
26.0721
35.2681
29.8677
29.1557
0.781816
31
23.3642
24.1254
27.2601
25.6818
−
−
−
−
0.387832
32
29.8469
26.1329
31.1765
28.0457
−
−
−
−
0.491911
33
23.8253
25.5186
26.4131
24.3954
−
−
−
−
0.259202
34
24.7283
29.4999
30.7847
27.3334
−
−
−
−
0.59523
35
22.9726
22.9374
24.4810
19.0010
−
−
−
−
0.525922
36
30.1250
21.3971
25.5744
29.7510
−
−
−
−
0.919685
37
20.2220
26.1639
24.7067
21.9676
−
−
−
−
0.599484
38
21.2900
24.5503
17.2053
19.0504
−
−
−
−
0.70917
39
25.0008
28.3862
21.6351
31.3436
−
−
−
−
0.941979
40
26.4776
24.8795
28.808
28.5460
−
−
−
−
0.415695
41
28.4058
21.7312
24.4973
20.9703
−
−
−
−
0.75458
42
26.7329
26.2349
32.9246
25.2879
−
−
−
−
0.778806
52
43
25.6930
28.2341
27.0033
24.4347
−
−
−
−
0.368057
44
23.0199
22.2204
26.6339
18.3089
−
−
−
−
0.766147
45
20.5206
24.3486
24.2339
27.6041
−
−
−
−
0.649469
46
24.4484
19.0577
22.0315
22.1530
−
−
−
−
0.495757
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
47
31.4093
24.8765
20.7703
19.3464
−
−
−
−
1.212957
48
27.8282
26.6870
24.1461
15.2504
22.5559
22.9647
29.1820
25.9876
0.97252
49
22.0438
26.3124
39.2404
25.9888
−
−
−
−
1.67912
50
20.8186
24.0817
26.0140
23.716
24.4136
19.8765
26.4943
32.1090
0.846173
51
30.7781
30.7263
26.9660
11.7222
−
−
−
−
2.03270
52
38.3928
23.8050
25.0198
25.1798
11.0386
33.1857
35.7813
23.4272
1.95218
53
24.3057
10.9071
26.4859
17.4950
27.3064
29.1412
33.0661
17.0163
1.66020
54
28.3763
13.9222
30.1004
35.3812
12.9102
30.0502
24.9192
19.8724
1.82511
55
20.9030
18.5299
20.4178
35.6147
36.4853
24.6850
13.2855
27.381
1.83091
56
10.8408
15.6951
21.7083
28.4016
10.5393
17.1082
19.2772
21.4624
1.33644
57
25.887
32.3918
28.0550
18.6617
34.3731
16.8672
26.5368
28.8619
1.36525
58
42.2296
29.8631
25.4091
19.1940
34.7803
31.5140
17.2854
25.8280
1.83322
59
7.88652
32.6539
23.17
20.4108
25.4219
18.7969
28.0719
27.8173
1.69742
53
60
8.96938
23.7950
21.1579
19.955
32.202
28.4551
15.58
28.0865
1.69836
61
35.1199
38.4218
32.8742
27.1086
20.599
14.4711
30.671
17.1967
1.96735
62
33.1907
21.4754
15.3750
25.4283
33.5678
28.5409
29.1854
26.5005
1.35673
63
48.5793
29.5302
23.6680
21.2491
36.2045
21.1275
15.8342
22.5832
2.36593
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
64
33.147
19.1438
24.0040
18.0994
17.9358
35.8989
23.7948
16.9326
1.62929
65
24.7851
19.5971
25.1343
27.2072
23.2206
25.4617
18.0509
31.9687
0.97317
66
20.2766
38.8939
26.1658
22.5124
−
−
−
−
1.86560
67
32.4504
21.5457
11.6449
40.3730
23.9312
32.8158
16.4062
24.1045
2.10708
68
20.0888
25.8380
32.6324
35.0936
33.6265
14.6202
35.5987
42.0369
2.03190
69
36.8908
39.0299
28.1607
51.6563
18.9804
15.2092
30.5374
15.6049
2.87170
70
23.4140
36.2722
18.6243
21.5165
28.1148
12.8178
23.1666
9.60917
1.88287
71
23.5228
29.272
21.1197
16.7333
31.6397
34.8448
28.5137
28.2541
1.32336
72
25.7173
13.6907
27.2615
26.6450
28.5129
8.38440
34.7460
35.6055
2.13162
73
26.632
32.4614
32.7966
16.799
24.3571
23.0955
31.0827
29.2759
1.233
74
25.1145
22.5152
37.5899
20.9708
21.6670
27.2004
29.7281
28.7704
1.23006
75
23.7596
22.4300
34.9732
27.8542
34.3304
14.9197
31.5903
34.005
1.60372
76
28.1282
24.4901
12.5025
9.62470
20.5327
20.2136
29.9433
12.6147
1.70282
54
77
24.4993
10.9184
28.4357
33.9673
18.1559
37.4188
31.3188
32.5965
1.99203
78
22.6782
38.9365
40.4688
18.1274
15.3294
27.4059
27.6365
15.9056
2.19437
79
39.5483
14.5130
20.8055
22.6957
35.4476
11.8261
26.375
17.3266
2.19724
80
6.34836
16.2919
33.6722
27.4048
24.741
22.2113
32.9579
36.6132
2.26037
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
81
34.6485
26.4922
21.0012
16.2856
17.0035
24.0146
19.3251
26.7704
1.37074
82
36.6499
10.6753
20.7837
12.1097
22.6954
17.7765
2.34469
21.2267
2.27742
83
22.4724
18.6365
49.4350
32.3223
23.4038
17.6650
26.1545
20.3546
2.34230
84
37.5901
25.1434
24.3679
29.0489
31.4332
32.4945
22.5497
33.0661
1.15224
85
32.7723
30.4482
9.95599
25.3650
37.4695
21.2078
19.3916
24.3873
1.92993
86
18.3276
39.1223
40.8870
28.4780
19.8756
17.9914
35.595
20.5803
2.18024
87
29.9267
38.7908
14.8182
18.1471
20.8556
30.2442
23.5341
26.732
1.72463
88
30.6503
25.4527
17.8609
22.3504
11.1558
32.6601
30.1732
13.3911
1.83568
89
5.6414
27.5687
19.2036
9.32583
24.1684
22.5191
21.5990
19.8518
1.67765
90
24.7987
31.301
23.3091
20.4046
18.1258
21.3697
22.4337
32.8725
1.16590
91
25.3286
14.6384
29.355
22.532
33.126
29.9162
12.3568
17.4657
1.71715
92
18.2714
25.4236
27.7947
38.9046
23.6219
13.5099
17.8667
30.5384
1.81848
93
25.9091
37.4526
20.8885
30.8705
23.3977
21.8567
27.3659
20.83
1.2956
55
94
19.3102
15.7388
17.1784
29.3463
7.43339
33.8811
26.0552
42.5051
2.52098
95
20.0664
24.7374
25.1219
24.6986
25.3781
18.786
16.7457
37.330
1.41135
96
16.4634
20.5755
19.2052
23.1707
31.0298
14.8813
16.9279
35.3174
1.64592
97
27.244
11.101
13.7217
41.2689
35.3610
19.6597
17.5987
36.9020
2.57360
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
98
25.0117
43.0061
22.9020
15.4416
33.6511
25.3311
21.1929
11.1545
2.24397
99
23.8072
32.3020
17.6264
25.1992
28.9089
6.40234
7.58374
38.0386
2.53364
100
25.1482
11.6278
32.0818
31.9522
41.7748
23.7524
19.0255
10.1582
2.41862
Appendix (2): Table (3) Phase I data at m=100
Sample
number
X1
X2
X3
X4
X5
Si
1
22.304374
19.36908
28.22687
28.31257
33.79446
5.654234
2
28.562168
25.4293
21.71387
21.17132
22.30489
3.115425
3
29.020901
24.93586
20.02661
26.98413
25.86685
3.351177
4
26.020022
19.23675
24.58903
35.01216
16.84531
7.053242
5
16.152864
24.00462
25.06051
20.44567
25.55445
3.947389
6
19.290106
22.60486
21.77949
23.88305
33.0489
5.265046
7
22.541
26.01317
19.28298
26.18749
29.11804
3.789325
56
8
27.184418
27.12642
24.07561
17.27258
27.67871
4.372748
9
23.402783
34.73111
25.82598
17.52871
27.3526
6.251638
10
31.683333
23.63375
19.26364
22.32478
26.82231
4.732689
11
28.933092
22.03656
23.99535
32.5362
20.39391
5.039706
Sample
number
X1
X2
X3
X4
X5
Si
12
24.943243
24.94569
30.02782
24.23386
18.63997
4.041839
13
24.321603
29.19325
30.9699
27.65934
27.97226
2.443812
14
22.182375
31.52509
28.47022
25.52352
22.85116
3.914207
15
29.185105
26.92899
27.85167
23.45652
20.87085
3.41436
16
26.076001
19.73696
27.49646
31.27568
27.48063
4.20245
17
23.84655
28.69358
22.69589
29.74538
30.64965
3.609545
18
20.974867
19.91826
12.63918
14.32663
31.43007
7.380232
19
29.330053
25.89242
28.0393
29.71686
21.60094
3.325352
20
25.4515
27.66795
26.38264
19.38773
28.10733
3.52097
21
24.859245
29.73898
22.35356
26.01688
27.53814
2.779056
22
26.575601
21.53802
16.20957
22.22308
24.49072
3.896056
23
30.232492
15.90247
16.64128
22.11534
23.62411
5.82838
24
24.889611
34.62081
24.98129
23.45996
21.69447
5.038688
57
25
22.312719
20.66856
30.67214
29.57671
30.82172
4.914458
26
23.59529
25.77908
25.21253
29.48814
27.36447
2.239755
27
27.446946
26.06515
25.70542
25.10291
28.13368
1.25872
28
27.505321
30.17372
21.83937
25.06486
31.57344
3.916421
Sample
number
X1
X2
X3
X4
X5
Si
29
28.119485
23.00637
19.68801
28.8911
29.56514
4.308636
30
28.861667
23.68996
31.20086
22.10178
28.40464
3.805428
31
25.446413
26.35577
21.65936
18.66666
24.25733
3.12323
32
30.13115
22.93029
25.2503
28.46943
25.64488
2.832781
33
21.669398
25.28813
25.67626
27.03947
28.66501
2.597893
34
28.806645
24.72606
20.9037
23.94564
26.694
2.971761
35
23.886333
22.10501
24.61906
25.75339
26.74042
1.777934
36
27.406439
21.39895
24.07882
20.68948
24.09515
2.657877
37
19.740487
19.76107
26.07981
18.55736
25.88915
3.665384
38
25.047035
32.11357
16.1457
24.27478
25.65735
5.686415
39
27.038812
22.0002
28.86566
21.44534
15.86955
5.122701
40
21.001789
28.64484
26.67759
31.551
21.98385
4.454648
41
24.463199
19.42621
25.42381
27.20493
28.68973
3.535606
58
42
23.359808
16.879
22.46901
20.01257
23.33359
2.779741
43
23.359907
24.78657
30.9557
27.62344
32.58744
3.926981
44
22.315718
28.69854
22.51587
28.37652
29.8145
3.626328
45
27.296536
37.50173
20.84041
27.28183
22.5748
6.479343
Sample
number
X1
X2
X3
X4
X5
Si
46
22.907012
21.14609
22.42503
20.87213
25.60374
1.887214
47
22.953056
29.36353
31.67197
23.17847
25.95022
3.833685
48
21.459519
27.17466
26.39921
13.70525
28.74573
6.114057
49
22.063075
28.26079
25.97983
24.12799
19.26912
3.471505
50
28.817425
23.31033
23.19194
21.73771
28.9654
3.422521
51
25.932325
23.94687
26.7616
29.07437
23.51241
2.253912
52
28.425896
20.42081
28.17863
23.89547
27.4899
3.460579
53
24.916547
23.93962
24.48946
24.25174
24.09488
0.381742
54
27.592727
22.03461
26.20781
31.36982
21.78338
4.022264
55
23.590906
23.32031
26.19287
24.02634
27.95917
1.995803
56
26.650168
25.45229
29.05984
28.08739
32.69136
2.770337
57
22.632945
23.35821
22.1159
22.50379
25.14588
1.202202
58
34.230464
24.02971
32.3774
26.73925
26.39958
4.331342
59
59
24.259812
18.70853
29.78343
23.66164
29.64341
4.640603
60
29.333927
20.09022
20.16685
31.68258
26.70739
5.292518
61
17.280305
28.93964
18.31973
28.29489
23.81304
5.432612
62
24.504857
26.32157
28.88626
22.11211
25.73184
2.48309
Sample
number
X1
X2
X3
X4
X5
Si
63
30.942542
19.78413
25.89463
27.57476
23.45328
4.211218
64
30.396036
19.32826
27.05585
24.51117
30.06615
4.564365
65
18.326026
24.77777
20.14772
28.89675
30.32969
5.254855
66
27.861904
26.94275
21.41984
30.31955
24.63215
3.375876
67
28.055004
26.59603
27.88662
23.58349
23.36887
2.282981
68
25.464198
30.46726
24.07547
24.27203
26.18813
2.594465
69
24.944118
26.91398
19.62032
26.78279
21.07572
3.345581
70
27.836273
27.07543
26.80982
21.8808
25.6138
2.354694
71
26.517943
26.70016
24.14418
24.83913
18.49148
3.339298
72
25.055053
21.49557
21.33585
28.88504
27.90572
3.508062
73
28.343789
29.21314
26.00597
35.00633
29.14199
3.319123
74
19.662548
26.69753
26.75774
33.66595
25.63815
4.973644
75
32.992728
30.72418
26.23457
25.67002
33.08992
3.59248
60
76
27.330454
23.4268
21.00095
30.38375
27.54168
3.706616
77
26.812974
26.55922
27.93744
15.44943
26.35645
5.164536
78
31.29212
22.24446
27.41078
18.0067
29.07378
5.400557
79
32.392686
28.62425
17.59272
28.18812
25.73394
5.522958
Sample
number
X1
X2
X3
X4
X5
Si
80
25.028103
19.77598
29.87107
26.65344
26.72702
3.703634
81
33.577773
24.78319
24.72658
22.80859
33.40067
5.200813
82
21.498068
25.5469
22.23539
23.16909
27.20882
2.383799
83
25.399167
27.90046
28.47127
30.81191
24.32195
2.574642
84
26.884483
23.54979
16.06403
25.23865
22.44728
4.143764
85
23.690607
26.6493
27.28144
22.94103
27.07624
2.049228
86
29.6731
23.41605
27.9679
22.68568
24.9873
2.988681
87
30.544405
24.67368
22.37952
22.27087
28.42676
3.699088
88
16.394829
31.25995
24.734
23.68747
21.26428
5.414135
89
30.649218
20.65486
24.74558
31.59601
35.09335
5.772935
90
22.11959
23.07771
29.83781
24.02338
19.61485
3.786314
91
22.974043
31.02473
25.95527
25.63952
21.7149
3.586414
92
28.838015
30.24985
29.85708
28.91596
35.57119
2.796894
61
93
23.906767
27.66859
22.56321
27.28597
30.38776
3.131557
94
21.335031
32.09357
27.63337
25.19938
21.85473
4.434972
95
19.398558
30.01492
24.47972
27.51911
18.49512
5.006716
96
21.17222
29.70328
26.17772
23.4512
25.03642
3.182035
Sample
number
X1
X2
X3
X4
X5
Si
97
25.84419
27.38556
25.206
27.00926
25.4893
0.959078
98
25.177369
31.6126
29.80326
28.328
26.86227
2.501786
99
21.68748
27.72069
24.11686
18.62069
25.32541
3.486387
100
29.287117
26.89339
25.57923
24.33132
23.86723
2.18593
Appendix (2): Table (4) Phase II data at m=100
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
1
24.66622
22.39661
30.79950
31.30162
−
−
−
−
1.104619
2
25.33557
36.79787
24.33061
19.77684
−
−
−
−
1.799969
3
31.55639
16.82216
25.31923
23.96761
21.29423
30.95843
21.48973
21.58136
1.253979
4
23.01840
23.6688
24.49703
28.03792
21.61484
16.68803
22.12755
20.26909
0.818913
62
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
5
24.49865
23.60125
14.70208
24.45865
−
−
−
−
1.183268
6
26.34652
24.31916
21.10157
25.03504
19.66488
23.02091
20.72452
20.19455
0.617843
7
31.18505
22.55900
20.6966
25.65466
−
−
−
−
1.140575
8
18.20714
21.968
24.62013
31.6588
25.20090
17.24620
28.18846
26.66757
1.217996
9
25.36123
27.20385
28.47294
19.42135
21.66734
27.15169
24.51784
22.40967
3
0.780727
10
24.95719
22.84642
26.74429
24.31794
−
−
−
−
0.401491
11
21.88096
25.06356
22.8606
23.54754
−
−
−
−
0.33265
12
29.58199
24.45267
26.99834
18.01675
−
−
−
−
1.233154
13
23.4176
23.87995
25.9608
19.94572
23.53644
30.61322
27.25483
23.28928
0.793551
14
23.99645
28.65781
25.48315
24.8842
−
−
−
−
0.50438
15
29.03625
26.04321
22.6755
27.05247
−
−
−
−
0.661079
16
32.29358
26.71380
23.54200
21.30744
−
−
−
−
1.18501
17
22.83718
24.74590
27.33962
21.67170
25.27723
21.96572
24.09211
22.37533
0.484036
18
23.95328
22.49139
30.36909
25.51458
−
−
−
−
0.850587
19
23.66810
24.34328
24.04326
26.83070
−
−
−
−
0.356245
20
24.09457
26.57100
32.97248
25.57405
−
−
−
−
0.972989
21
24.4064
13.72123
26.85114
24.19808
−
−
−
−
1.451935
63
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
22
31.24089
22.45659
24.19105
30.82013
29.81208
28.23386
20.37520
22.67852
1.064276
23
34.89056
30.78866
30.10428
28.69964
−
−
−
−
0.661069
24
26.20125
21.29985
24.23991
16.96933
−
−
−
−
0.997834
25
17.00519
22.59578
24.56522
29.17686
−
−
−
−
1.253252
26
28.02556
18.68103
20.81956
25.53059
28.39101
22.27628
23.79369
16.33723
1.069909
27
22.95455
21.24368
28.42263
28.55919
−
−
−
−
0.933731
28
23.41504
21.17361
18.83911
23.89492
−
−
−
−
0.576804
29
29.6983
21.93815
19.66801
27.07526
−
−
−
−
1.143923
30
23.30469
23.22855
26.84190
23.82400
28.75089
22.26864
24.76947
29.60520
0.682307
31
21.09616
24.49523
23.4883
29.86814
−
−
−
−
0.921227
32
18.27474
25.83930
27.63955
18.29843
−
−
−
−
1.226822
33
21.57540
23.95005
20.8067
20.10784
23.20361
27.05674
23.84840
29.89968
0.817642
34
25.84190
31.53746
19.40795
28.80955
−
−
−
−
1.29485
35
22.76224
20.42745
24.50203
30.91690
25.48983
24.09544
25.14969
24.31628
0.740022
36
19.65223
22.86268
22.94912
24.06542
−
−
−
−
0.472484
37
16.04849
25.18625
32.28782
22.39289
−
−
−
−
1.6731
38
26.52830
26.50384
24.75171
23.18589
25.81757
30.25703
27.79625
16.95573
0.978127
64
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
39
19.801
24.81826
29.18668
27.8624
−
−
−
−
1.035797
40
22.35327
29.7041
21.09214
28.68729
−
−
−
−
1.085042
41
29.720
26.01688
28.50031
26.58730
−
−
−
−
0.425632
42
32.45486
28.20527
16.93908
20.92670
−
−
−
−
1.739334
43
20.11181
21.73726
28.46003
37.34668
24.24035
24.09345
24.67941
28.44575
1.33516
44
23.51603
25.87857
24.95382
31.56196
28.89184
25.99430
28.22675
25.01531
0.650583
45
23.72134
25.67021
17.46304
24.71412
−
−
−
−
0.921206
46
18.74863
30.6908
20.65064
24.63951
−
−
−
−
1.311982
47
24.07362
24.40153
23.54319
34.56742
20.95015
30.30933
23.94367
26.77416
1.089886
48
23.6464
29.81784
27.4884
20.79841
−
−
−
−
0.995605
49
21.6977
25.76026
27.5594
27.70803
−
−
−
−
0.695882
50
25.03729
22.78292
23.15343
26.49952
−
−
−
−
0.430053
51
33.55870
32.03135
18.28223
22.35405
−
−
−
−
1.844249
52
12.70308
13.65591
25.58953
6.296422
30.05421
31.91487
26.19721
28.11301
2.364665
53
6.246424
11.05909
25.91685
15.06623
30.74449
15.77735
29.21996
37.42129
2.713076
54
18.1410
29.16504
29.12976
21.01642
30.11743
25.83906
27.61923
27.91878
1.069532
55
29.46166
39.3509
19.43951
24.85759
−
−
−
−
2.098659
65
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
56
6.936589
46.38598
15.99708
24.22424
13.74819
24.97422
23.50305
40.07787
3.280808
57
28.86359
21.99264
28.0767
34.28816
34.31271
42.81261
23.21124
22.41656
1.811721
58
11.28661
25.35267
23.63851
26.42229
26.45064
29.91009
22.43610
17.55337
1.468782
59
21.03269
26.23260
32.03192
31.39885
23.31949
36.10868
27.39948
35.56056
1.375536
60
29.95620
28.46353
32.04863
36.71555
35.44495
32.65258
25.64839
34.12856
0.920465
61
23.18852
7.105312
19.65671
22.66702
−
−
−
−
1.871503
62
32.23488
20.67455
22.11262
31.14699
31.76352
39.81511
27.16564
21.05614
1.676381
63
42.30927
27.06980
39.67055
17.65056
26.26571
22.21945
30.81003
27.47724
2.063395
64
30.16723
28.16277
23.60426
24.01875
33.95434
21.41446
43.35830
21.65561
1.865346
65
35.71100
24.71331
18.50409
21.40227
23.95216
30.98483
22.43372
29.32661
1.412439
66
26.08384
28.3781
27.52989
30.67719
27.04244
30.63119
39.59569
39.3833
1.338445
67
27.51601
24.32882
37.92429
21.53997
35.94108
23.06464
31.27677
38.9833
1.72832
68
22.55313
20.01604
43.89131
25.46698
33.46524
39.42849
31.96792
22.92755
2.149575
69
30.11930
11.61281
28.81129
39.91431
27.39952
26.71528
12.54295
18.76242
2.38257
70
13.43543
22.62748
28.90654
30.96390
40.93333
18.00254
18.32634
23.59458
2.179791
71
13.56395
32.66568
19.40639
21.20063
22.66105
29.31648
18.70234
18.35139
1.545097
72
16.112
23.95318
26.34658
32.43591
27.63260
14.57046
35.68753
22.0618
1.820837
66
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
73
34.50746
26.33122
24.76450
25.59040
18.02033
19.72291
29.4746
33.92277
1.484825
74
21.96820
32.87121
25.79858
24.28054
16.1554
31.55834
21.83014
30.41300
1.424643
75
21.46630
20.40656
12.69804
23.69064
29.73329
22.4254
14.28528
23.81755
1.355125
76
28.045
14.78878
36.61870
35.92818
20.50025
23.96931
16.65058
28.98763
2.039323
77
14.56289
12.57247
36.33719
17.89834
23.62781
20.64880
31.92242
24.51723
2.039499
78
22.13263
28.43580
21.42702
31.60803
43.06482
31.56994
14.19543
19.70207
2.247779
79
29.73543
40.56517
15.37695
23.69245
22.84582
28.0218
21.27591
18.04895
1.959463
80
9.593159
34.55965
17.26497
16.4401
12.84941
9.387095
20.19895
28.29180
2.216839
81
27.01983
19.6171
26.62441
38.49627
21.3717
31.3073
34.61659
17.76717
1.832042
82
25.40182
29.93688
26.52416
22.6421
18.57315
27.61220
33.04253
20.63710
1.195389
83
25.75629
8.673225
15.16402
28.00786
17.63615
28.29012
17.22967
16.37150
1.740242
84
11.45220
28.19292
19.95668
22.72054
33.00920
25.56710
19.8442
49.01232
2.787337
85
15.06059
34.97335
27.1956
19.6833
29.95655
9.271865
22.69313
34.1576
2.275059
86
29.71646
39.46461
29.12422
25.16281
40.33574
27.27549
30.71312
15.46343
1.968472
87
19.77475
36.16291
20.13025
38.22056
16.63739
23.76724
17.28935
24.18977
2.058757
88
21.86603
32.06492
30.81641
22.28952
40.23482
22.78938
23.60894
27.23678
1.600548
89
22.33911
11.93150
30.34164
14.90256
46.30350
18.24658
26.10495
22.28680
2.672808
67
Sample
number
X1
X2
X3
X4
X5
X6
X7
X8
Chart
statistic
90
18.58781
27.66136
20.52006
18.58710
42.71909
28.18089
34.2406
22.22838
2.114109
91
20.60154
15.66455
22.636
30.19926
19.23577
23.94400
14.5793
25.03402
1.272079
92
31.08085
33.74933
11.22873
15.11705
24.06698
36.84843
30.88473
26.31802
2.238562
93
8.660859
13.06232
28.75968
30.57294
23.33313
26.42590
12.89378
18.57493
2.035824
94
27.07118
17.68504
26.46865
18.23253
20.45083
31.69701
30.25119
24.19652
1.320634
95
17.06035
37.56065
40.14299
42.74523
29.6715
24.84962
27.34659
22.23361
2.26972
96
30.77650
24.46225
20.8979
29.37241
26.16514
18.18758
14.86723
36.22546
1.747509
97
16.37510
26.79380
10.54130
41.04586
38.50598
23.35222
8.848993
11.6868
3.119089
98
16.23334
29.7897
17.37609
25.2030
25.2204
31.58806
34.43956
50.16354
2.675849
99
26.06075
31.59560
19.97057
25.44193
24.60509
23.82800
28.7189
16.78745
1.157611
100
10.47915
38.29868
19.64803
43.82425
26.97599
30.44068
15.56490
19.09879
2.858629
68
Appendix (3): Simulation results figures
Appendix (3): Figure (1) ARL0 vs. m for ASS0=3, ns=2 and nL=5
1400
ARL0
1200
1000
n=3
800
n=5
600
n=7
400
n=10
200
0
20
50
70
80
100
m
Appendix (3): Figure (2) ARL0 vs. m for ASS0=3, ns=2 and nL=7
1600
1400
ARL0
1200
n=3
1000
n=5
800
n=7
600
n=10
400
200
0
20
50
70
80
100
m
Appendix (3): Figure (3) ARL0 vs. m for ASS0=3, ns=2 and nL=9
1600
1400
ARL0
1200
n=3
1000
n=5
800
600
n=7
400
n=10
200
0
20
50
70
69
80
100
m
Appendix (3): Figure (4) ARL0 vs. n0 for ASS0=3, ns=2 and nL=5
1400
ARL0
1200
1000
m=20
800
m=50
600
m=70
m=80
400
m=100
200
0
3
5
7
10
n0
Appendix (3): Figure (5) ARL0 vs. n0 for ASS0=3, ns=2 and nL=7
1600
ARL0
1400
1200
m=20
1000
m=50
800
m=70
600
m=80
400
m=100
200
0
3
5
7
10
n0
Appendix (3): Figure (6) ARL0 vs. n0 for ASS0=3, ns=2 and nL=9
1600
ARL0
1400
1200
m=20
1000
m=50
800
m=70
600
m=80
400
m=100
200
0
3
5
7
70
10
n0
Appendix (4): SAS programs
Appendix (4): SAS program (1)
Used to get the design parameters that achieve ASS0=3 and ARL0=370
proc iml;
meu=25;
sigma=4;
c4=0.886;
nS=2;
nL=6;
UWL=1.1;
UCL=2.9;
noruns=30000;
tsn=0;
srl=0;
do i=1 to noruns;
s=sigma*c4;
sn=0;
rl=0;
do until (s>ucl);
if s<=w then n=nS;
else n=nL;
x=normal (repeat(-1,n))*sigma+meu;
xbar=sum(x)/n;
c=sum((x-xbar)##2);
a=c/(n-1);
s=sqrt(a);
rl=rl+1;
sn=sn+n;
ass=sn/rl;
end;
tsn=tsn+ass;
srl=srl+rl;
end;
asn=tsn/noruns;
arl=srl/noruns;
print asn arl;
quit;
71
Appendix (4): SAS program (2)
Used to study the effect of estimation on the value of ASS0=3 and ARL0=370
for a combination of nS=2, nL=6, n0=10, m=80, UWL=1.1 and UCL=2.9
proc iml;
meu=25;
sigma=4;
c4=0.886;
m=80;
n0=10;
nS=2;
nL=6;
UWL= 1.1;
UCL= 2.9;
Print ucl w;
noruns=30000;
tsn=0;
arl=0;
asnn=0;
do i=1 to noruns;
si=0;
do j=1 to m;
x=normal (repeat(-1,n0))*sigma+meu;
xbar=sum(x)/n0;
c=sum((x-xbar)##2);
a=c/(n0-1);
s=sqrt(a);
si=si+s;
end;
sbar=si/m;
sigmahat=sbar/c4;
ss=sigmahat*c4;
sn=0;
rl=0;
do until (ss>ucl);
if ss<=w then n=nS;
else n=nL;
x=normal (repeat(-1,n))*sigma+meu;
xbar=sum(x)/n;
c=sum((x-xbar)##2);
s= sqrt (c/(n-1));
ss=s/sigmahat;
sn=sn+n;
rl=rl+1;
end;
snn=sn/rl;
tsn=tsn+sn;
arl=arl+rl;
asnn=asnn+snn;
end;
asnn=asnn/noruns;
arl=arl/noruns;
print n0 m n1 n2 asn arl;
quit;
72