0

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

1

Abstract

In this report we discuss the behaviour of the variable sample size Shewhart S-

chart 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.

2

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 Phase I sample size

SAS Statistical package for statistical analysis

system

3

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

4

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

5

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

6

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-of-

control at this time.

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Figure (1.1): Shewhart control chart for monitoring a process parameter

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 non-

conforming 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

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1 2 3 4 5 6 7

CL

LCL

UCL

Sample number

Chart statistic

9

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).

10

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;

( ) , 0 < 1

where is the current sample statistic and is the previous chart statistic. is called the

smoothing parameter, small values for are chosen to detect small shifts faster, while large

values for 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 ( , ), where is the largest allowable value

for the process parameter, 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-of-

control.

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Figure (1.2): Adaptive VSS Shewhart control chart for monitoring the process mean parameter

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.

0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10

Sample number

K2

CL

W1

W2

K1

A

B

𝑋 −𝑐ℎ𝑎𝑟𝑡 𝑋 j

cccccccc

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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 in-

control 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.

14

Section 2: VSS S-chart

2.1 The design of the VSS S-chart:

Let , i= 1, 2, 3... and j= 1, . . . , denote certain quality characteristic of a process.

We assume that ’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.

15

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-of-

control 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 VSS-

chart 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:

16

ASS= E ( ).

During the process monitoring, will take only two possible values ("S" stands for

small) or ("L" stands for large), where < . If the statistic lies inside the central region,

= , otherwise, = . 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 . In this report we will take m = {20, 50,

70, 80 and 100} and = {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

≤

.

17

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 = 3 and =370 for each

combination of and (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 = 25 and standard

deviation = 4. The sample size ni is either or , where ni = nS if the statistic is less than or

equal UWL, otherwise ni = .

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)

18

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 and standard

deviation . The sample size ni is either or , where ni= if the statistic is less than

or equal UWL, otherwise ni= .

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.

19

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

0

100

200

300

400

500

600

700

800

20 50 70 80 100

AR

L0

m

n=3

n=5

n=7

n=10

20

Figure (3.2) ARL0 vs. n0 at ASS0=5 and nS=4 and nL=7

Figure (3.3) ARL0 vs. m at ASS0=7 and nS=5 and nL=10

0

100

200

300

400

500

600

700

800

3 5 7 10

AR

L0

n0

m=20

m=50

m=70

m=80

m=100

0

100

200

300

400

500

600

20 50 70 80 100

AR

L0

m

n=3

n=5

n=7

n=10

21

Figure (3.4) ARL0 vs. n0 at ASS0=7 and nS=5 and nL=10

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 S-

chart with estimated parameters to achieve good performance.

0

100

200

300

400

500

600

3 5 7 10

AR

L0

n0

m=20

m=50

m=70

m=80

m=100

22

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

23

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.

24

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:

25

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)

Figure (4.2) Monitoring company (B) process (at m=100)

0

0.5

1

1.5

2

2.5

3

3.5

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

10

0

Sample number

Chart statistic

0

0.5

1

1.5

2

2.5

3

3.5

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

10

0

Sample number

Chart statistic

UCL

UWL

UWL

UCL

26

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.

27

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 S-

chart. 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

28

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.

29

References

Amin, R. and Miller, R. (1993) A Robustness Study of X-bar Charts with Variable Sampling

Intervals, Vol.25, No.1. Journal of Qualify Technology. [available at:

http://www.triplescreenmethod.com/ProcessOptimization/100917_RobustnessStudyXbarCharts

withVariableSamplingIntervals.pdf ] [viewed on 15/3/2012].

Castagliola, P., Zhang, Y. , Costa, A. and Maravelakis, P. (2011) The variable sample size X-bar

chart with estimated parameters. Online in Wiley online library: John Wiley & Sons, Ltd. [available

at: http://onlinelibrary.wiley.com/doi/10.1002/qre.1261/abstract] [viewed on 6/12/2011].

Chen, G. (1998) The run length distributions of the R, S and S2 control charts when is estimated.

Vol.26, Issue.2. Online in Wiley online library: John Wiley & Sons, Ltd. [available at:

http://onlinelibrary.wiley.com/doi/10.2307/3315513/abstract] [viewed on 17/4/2012].

Costa, A. (1995) X-bar Charts with variable sample size and sampling intervals. Technical

Report #133. University of Wisconsin, Madison: Center for Quality and Productivity

Improvement. [available at: http://cqpi.engr.wisc.edu/system/files/r133.pdf ] [viewed on

6/12/2011].

Costa, A. (1996) Joint X-bar and R Charts with variable sample size and sampling intervals.

Technical Report #142. University of Wisconsin, Madison: Center for Quality and Productivity

Improvement. [available at: http://cqpi.engr.wisc.edu/system/files/r142.pdf ] [viewed on

6/12/2011].

Del Castillo, E. (1996) Evaluation of the Run Length Distribution of X-bar Charts with

Unknown Variance. Vol. 28, No. 1. Journal of Quality Technology. [available at:

http://asq.org/qic/display-item/?item=11442] [viewed on 17/4/2012].

Hua, L. and Zhang, W. (2002) Optimal np Control Charts with Variable Sample Sizes or

Variable Sampling Interval. Vol.17, No.1. Singapore: Heldermann, V. [available at:

http://www.heldermann-verlag.de/eqc/eqc17/eqc17004.pdf] [viewed on 6/12/2011].

30

Jones, L. and Champ, C. (2002) Phase I control charts for times between events. Online in

Wiley online library: John Wiley & Sons, Ltd. [available at:

http://onlinelibrary.wiley.com/doi/10.1002/qre.496/pdf] [viewed on 17/4/2012].

Prabhu, S., Montgomery, D. and Runger, G. (1994) A Combined Adaptive Sample Size and

Adaptive Sampling Interval Control Scheme. Vol. 26, No. 3. Journal of Quality Technology.

[available at: http://asq.org/qic/display-item/?item=11384] [viewed on 17/4/2012].

Reynolds, M. (1990) CUSUM Charts With Variable Sampling Intervals. Vol. 32, No.4.

Technomertrics. [available at: http://www.jstor.org/stable/10.2307/1270164] [viewed on

16/4/2012].

Reynolds, M., Amin, R., Arnold, J. and Nachlas, J. (1988) X-bar charts with variable sampling

intervals. Vol.30, No.2. Technometrics. [available at: http://www.jstor.org/stable/1270164 ]

[viewed on 6/12/2011].

Runger, G. and Montgomery, C. (1993) Adaptive Sampling Enhancements forShewhart Control

Charts. Vol. 25, Issue. 3. Online in Taylor & Francis: Taylor & Francis Group. [available at:

http://www.tandfonline.com/doi/abs/10.1080/00401706.2012.648869#preview] [viewed on

17/4/2012].

Shahriari, H., Maddahi, A. and Shokouhi, A. (2009) A Robust Dispersion control chart Based

on M-estimate. Vol. 2, No. 4. Faculty of Industrial Engineering, K. N. Toosi University of

Technology, Tehran, Iran. [available at: http://www.jise.info/issues/volume2no4/05-V2N4-

Full.pdf] [viewed on 17/4/2012].

31

List of Bibliography

Bischak, D. and Trietsch, D. (2007) The Rate of False Signals in X-bar Control Charts with

Estimated Limits. Quality and Reliability Engineering Lab.

Castagliola, P. and Maravelakis, P. (2010) A CUSUM control chart for monitoring the variance

when parameters are estimated. Vol.141, Issue. 4.

Jones, L. , Champ, C. and Rigdon, S. (2001) The Performance of Exponentially Weighted

Moving Average Charts with Estimated Parameters. Vol. 43, No. 2. Technomertrics.

Kramer, H. and Schmid, W. (2000) The influence of parameter estimation on the ARL of

Shewhart type charts for time series. Vol.41, No. 2.

Mahmoud, A. (2012) Control Charts. BSc Quality Control, Cairo University, 23rd

April 2012.

Mahmoud, A. and Maravelakis, P. (2010) The performance of multivariate CUSUM control

charts with estimated parameters. Vol. 39, No. 9. Online in Taylor & Francis: Taylor & Francis

Group.

Montgomery, D. (1996) Introduction to Statistical Quality Control, 3rd

edition, John Wiley &

Sons, New York. (1st edition, 1985, 2nd edition, 1991(.

Montgomery, D. (2005) Statistical Quality Control: A Modern Introduction, 6th

edition. Asia:

John Wiley & Sons, Inc.

Patterson, J. and Foster, S. (2004) Tools of Statistical Process Control X-bar and S Charts. AT

& T Statistical Quality Control Handbook. Charlotte, NC: Delmare.

Reynolds, M. (1996) Variable Sampling Interval Control Charts with Sampling at Fixed Times.

Vol. 28, Issue. 6. Online in Taylor & Francis: Taylor & Francis Group.

Riaz, M. (2008) A Dispersion Control Chart. Vol.37, Issue. 6. Pakistan.

32

Saleh, N., Mahmoud, A. and Abdel-Salam, G. (2012) The Performance of the Adaptive

Exponentially Weighted Moving Average Control Chart with Estimated Parameters. Online in

Wiley online library: John Wiley & Sons, Ltd.

Schoonhoven, M., Riaz, M. and Does, R. (2011) Design and Analysis of Control Charts for

Standard Deviation with Estimated Parameters. Vol. 43, No. 4. Journal of Quality Technology.

Schoonhovena, M. and Does, R. (2012) A Robust Standard Deviation ControlChart. Vol. 54,

Issue. 1. University of Amsterdam, The Netherlands: Institute for Business and

IndustrialStatistics.

Shahriari, H., Ahmadi, O. and Shokouhi, A. (2010) A Two-Phase Robust Estimation of Process

Dispersion Using M-estimator. Vol. 4, No. 1. Department of Industrial Engineering, K.N. Toosi

University of Technology, Tehran, Iran.

Zhang, Y., Castagliola, P. , Wu, Z. and Khoo, M. (2011) The variable sample interval X-bar

chart with estimated parameters. Online in Wiley online library: John Wiley & Sons, Ltd.

33

Appendices

Appendix (1): Simulation results tables

Appendix (1): Table (1) Simulation results for ASS0=3

nS nL

UWL

UCL

n0

m

20 50 70 80 100

nS= 2

nL=4

UWL=0.9

UCL=2.8325

3

830.7177 558.34383 521.57183 511.0031 492.52833

{2.8206935} {2.8106527} {2.8117056} {2.8089539} {2.8093582}

5

1049.3142 841.37517 811.18393 797.7199 787.98173

{2.7289849} {2.725225} {2.724044 } {2.7238004} {2.7230566}

7

1157.0136 979.59687 957.02847 953.59697 942.27513

{2.6976791} {2. 6950168} {2. 6951908} {2.6943061} {2.6947701}

10

1189.2452 1094.1973 1068.019 852.6498 846.3793

{2.678151} {2.6761402} {2.6742349} {2.706988} {2.707254}

nS=2

nL=5

UWL=0.975

UCL=2.8625

3

895.2317 583.0657 535.19817 524.3308 505.82073

{3.0880327} {3.0706531} {3.0695544} {3.0670902} {3.0669358}

5

1093.2846 879.56533 838.40807 834.02327 817.6487

{2.9445634} {2.9332573} {2.9318637} {2.9310207} {2.9309073}

7

1190.9295 1033.5364 1012.5344 987.90347 985.90767

{2.8982892} {2.891009} {2.8874321} {2.8869289} {2.8870907}

10

1269.5777 1146.9982 1132.9841 1131.2802 1109.4508

{2.8662096} {2.8593018} {2.8587333} {2.8590298} {2.8573894}

34

nS nL

UWL

UCL

n0

m

20 50 70 80 100

nS=2

nL=6

UWL=1.1

UCL=2.9

3

978.2085 598.06157 544.2676 528.76887 513.20683

{3.1264244} {3.0943069} {3.0898139} {3.0902017} {3.0851222}

5

1166.8025 907.55867 883.84077 864.36943 848.07143

{2.9482809} {2.9345723} {2.9312109} {2.9286012} {2.9275043}

7

1276.2923 1087.0715 1053.0303 1042.159 1016.7965

{2.8900184} {2.881696} {2.8794345} {2.8789216} {2.8785394}

10

1362.4022 1202.5722 1168.2178 1183.3471 1163.4993

{2.8548366} {2.847577} {2.8476471} {2.8457299} {2.8459788}

nS=2

nL=7

UWL=1.203

UCL=2.9255

3

1025.8117 612.10573 557.73347 540.25173 518.73943

{3.1327356} {3.0953952} {3.0922913} {3.0831962} {3.080942}

5

1237.2425 945.1171 917.46753 901.29363 868.91687

{2.9385205} {2.9218051} {2.9165122} {2.9167236} {2.9150016}

7

1377.9306 1130.0363 1099.1163 1071.1088 1069.4083

{ 2.8754751} {2.8631577} {2.8660938} {2.865242} {2.8620348}

10

1431.8819 1280.5475 1239.1372 1224.0034 1221.6898

{ 2.8381861} {2.8319459} {2.8294495} {2.830373} {2.8292443}

nS=2

nL=8

UWL=1.28

UCL=2.937

3

990.83343 615.1339 554.12467 542.76 520.27857

{3.1538742} {3.1121768} {3.1032549} {3.0982546} {3.0958086}

5

1291.7739 962.90053 917.24947 919.8364 883.6263

{2.9493464} { 2.9273639} {2.9271479} {2.9250441} {2.9236683}

7

1397.3746 1144.0931 1111.1059 1088.1227 1087.4325

{2.8825168} {2.8718922} {2.8727323} {2.871157} {2.8672918}

10

1461.544 1289.9679 1251.2875 1255.552 1232.3188

{2.8431295} {2.8383225} {2.8360887} {2.8365332} {2.8368816}

35

nS nL

UWL

UCL

n0

m

20 50 70 80 100

nS=2

nL=9

UWL=1.34

UCL=2.9478

3

1070.0346 629.74017 563.76843 546.8887 525.4843

{3.1813104} { 3.1444414} {3.1332977} {3.1347166} {3.129763}

5

1302.261 985.87827 944.07773 925.72607 894.26517

{2.9736911} {2.9514613} {2.9473061} {2.9499934} {2.9496766}

7

1444.2028 1171.4501 1128.2567 1121.4723 1108.4966

{2.904519} {2.8938071} {2.893744} {2.8922616} {2.8911643}

10

1518.891 1324.7252 1287.1724 1281.3717 1257.4755

{2.8652357} {2.8578041} {2.8559548} {2.8567531} { 2.8549075}

nS= 2

nL= 10

UWL= 1.425

UCL=2.955

3

1112.371 622.10683 568.2804 555.865 529.15143

{3.1512392} {3.1095274} {3.0986938} {3.0999411} {3.0962339}

5

1320.0685 762.6036 733.2447 721.10127 709.85797

{2.9402719} {2.9876623} {2.9810579} {2.9805331} {2.978788}

7

1448.9734 906.2489 880.881 878.3022 867.04487

{2.8714646} {2.9248658} {2.921802} {2.9239006} {2.9190563}

10 1517.5265 1021.6116 1006.7165 997.35533 991.9533

{2.8299706} {2.8831883} {2.8814369} {2.883563 } {2.8821892}

36

Appendix (1): Table (2) Simulation results for ASS0=5

nS nL

UWL UCL

n0

m

20 50 70 80 100

nS=2

nL=6

UWL= 0.65

UCL= 2.543

3

347.49403 318.01827 314.32317 312.31057 308.66643

{5.1395433} {5.1504492} {5.1554126} {5.1597924} {5.156914}

5

408.426 385.9964 382.46642 381.8797 376.48887

{4.9889761} {4.9889081} {4.9904546} {4.991574} {4.9919638}

7

434.62997 416.32417 412.92427 411.63693 408.2074

{4.9228305} {4.9267095} {4.9270356} {4.9276648} {4.9303082}

10

454.45983 438.26817 434.37423 434.56527 432.07683

{4.8815954} {4.8832787} {4.8857062} {4.8834536} {4.8847832}

nS=2

nL=7

UWL= 0.78

UCL=2.6968

3

367.4344 306.0997 300.63967 296.13293 294.75873

{5.3346994} {5.3375134} {5.3330314} {5.3416941} {5.34244}

5

437.0443 389.317 384.10617 383.1026 382.8847

{5.0494457} {5.0472527} {5.0440191} {5.043832} {5.0427695}

7

459.1716 429.30763 427.74107 423.82433 418.8936

{4.9413883} {4.9399988} {4.9419183} {4.9377656} {4.9371193}

10

483.56387 460.5784 453.18713 450.4667 453.3275

{4.8668633} {4.8634479} {4.8646264} {4.8660165} {4.8662022}

nS=2

nL=8

3

377.41203 300.07447 289.8238 283.88303 281.18227

{5.4225791} {5.4075725} {5.405255} {5.403886} {5.4018602}

37

nS nL

UWL UCL n0

m

20 50 70 80 100

UWL= 0.875

UCL=2.7825 5

454.29507 403.62327 385.57803 388.561 381.33713

{5.0288264} {4.9992025} {4.9998536} {4.9999209} {4.9987088}

7

478.94707 448.27383 437.32943 434.87297 435.27337

{4.879629} {4.8644923} {4.8630738} {4.8625908} {4.8576847}

10

512.14097 487.47893 478.3044 478.46873 476.20237

{4.7843917} {4.7732047} {4.7703181} {4.7659286} {4.7656128}

nS=2

nL=9

UWL= 0.93

UCL= 2.82

nS=2

nL=10

UWL= 0.975

UCL=2.84875

3

386.86763 294.97303 286.94403 284.1685 277.3502

{5.6262437} {5.5810917} {5.5771233} {5.5652738} {5.5620208}

5

458.35003 406.88987 391.5146 392.20017 383.92747

{5.1295232} {5.0970606} {5.0884232} {5.079443} {5.0797967}

7

500.74703 451.9459 448.57053 441.64157 442.43387

{4.9500582} {4.9253476} {4.9195101} {4.9220789} {4.9130387}

10

531.99343 496.27007 486.18007 481.95623 479.8295

{4.8340808} {4.8114984} {4.816998} {4.8147537} {4.807591}

3

385.16737 292.58387 284.04933 279.09547 273.66377

{5.7976619} {5.7283813} {5.7160664} {5.7039292} {5.681832}

5

466.16467 410.40393 399.53577 392.6051 390.20817

{5.2100298} {5.1551898} {5.1496231} {5.1440046} {5.1380357}

7

515.1088 457.83343 454.61023 450.22677 445.87113

{5.0072978} {4.975459} {4.9625932} {4.9666687} {4.9564815}

38

nS nL

UWL UCL n0

m

20 50 70 80 100

10

536.02507 503.2348 496.22497 495.02293 494.45817

{4.8761425} {4.8495088} {4.8517777} {4.8435877} {4.842676}

nS=3

nL=6

UWL=0.75

UCL=2.1975

3

364.1484 276.47691 264.99055 258.79829 256.84966

{5.1656737} {5.171741} {5.1725262} {5.1726795} {5.172032}

5

469.69652 402.999 389.67926 388.06776 383.60203

{5.0194361} {5.0221009} {5.0214302} {5.0210156} {5.0213268}

7

508.69358 456.78033 450.62777 446.41984 442.87183

{4.9633919} {4.9643327} {4.9643116} {4.9647848} {4.9655057}

10

540.43203 501.06588 498.44778 492.08499 491.65019

{4.9257772} {4.9261083} {4.9254264} {4.9265106} {4.9261763}

nS= 3

nL=7

UWL= 0.9

UCL= 2.285

3

416.0369 288.03743 270.50223 266.6709 258.7782

{5.2848149} {5.2818514} {5.2754298} {5.2748732} {5.2783577}

5

507.52737 421.62563 405.22003 398.81763 392.13097

{5.0329764} {5.0192737} {5.0160256} {5.0220384} {5.0202107}

7

547.92627 480.23563 472.08847 463.79897 464.73173

{4.9394031} {4.932532} {4.9265882} {4.9270194} {4.9279168}

10

583.85447 532.28923 520.35737 521.57367 516.71107

{4.8726714} {4.868189} {4.867666} {4.8680094} {4.8685559}

nS=3 3 430.5749 284.63003 267.04037 262.41707 253.79733

39

nS nL

UWL UCL n0

m

20 50 70 80 100

nL=8

UWL= 0.975

UCL=2.31775

{5.494836} {5.4601935} {5.4592558} {5.4533514} {5.4571185}

5

535.49097 424.47647 416.75567 402.10643 397.19117

{5.1406607} {5.1235842} {5.1144767} {5.118604} {5.1122293}

7

582.6643 491.3646 477.75107 475.397 471.32533

{5.019948} {5.0052428} {5.0037359} {5.003253} {5.0013022}

10

610.1146 540.25043 530.74763 529.9271 523.29947

{4.9430719} {4.9315304} {4.9268475} {4.9274227} {4.9241252}

nS=3

nL=9

UWL=1.04875

UCL=2.347725

3

445.72237 288.45083 265.2243 263.76593 256.00047

{5.5565134} {5.5006196} {5.4910124} {5.4792278} {5.4775641}

5

573.75743 440.6414 416.52 411.99003 403.16473

{5.1237507} {5.090527} {5.0894136} {5.0854058} {5.0843147}

7

599.44547 515.5647 493.3387 487.70533 477.68223

{4.9867104} {4.9649905} {4.9587873} {4.9531697} {4.9562042}

10

636.6931 568.86197 556.29447 548.22023 545.71457

{4.8933959} {4.8768546} {4.8733479} {4.8756145} {4.8736412}

nS=3

nL=10

UWL=1.099975

UCL=2.359975

3

471.96488 283.00456 263.30841 255.20139 247.23228

{5.6447283} {5.5715217} {5.5503643} {5.5469493} {5.5442522}

5

572.97492 438.24441 438.24441 408.68988 400.91268

{5.1601633} {5.1152161} {5.1152161} {5.1068316} {5.102121}

7 609.70106 508.551 495.35564 488.36915 481.4492

40

nS nL

UWL UCL n0

m

20 50 70 80 100

{5.0028318} {4.9747796} {4.9683672} {4.9641866} {4.9632665}

10

641.092 571.20217 558.01957 547.2025 482.19963

{4.9018703} {4.8823113} {4.8780414} {4.873932} {4.9604452}

nS= 4

nL= 7

UWL= 1.05

UCL= 2.105

3

532.94867 275.38117 252.09403 246.5967 234.2304

{5.2959972} {5.274752} {5.2716517} {5.2707349} {5.2679313}

5

629.51517 451.9243 428.87103 419.68273 409.787

{5.0847345} {5.0719714} {5.0685749} {5.0700862} {5.0686438}

7

692.4716 532.80543 520.64163 509.25207 498.68127

{5.0125605} {5.0041615} {5.0000829} {4.9998039} {5.0003173}

10

697.52173 603.47563 589.45953 585.68143 576.1624

{4.9636601} {4.9573173} {4.9562025} {4.9573299} {4.9552992}

nS= 4

nL= 8

UWL= 1.125

UCL=2.119975

3

560.1551 283.04043 252.78823 244.45823 234.41377

{5.4515847} {5.4097498} {5.4045954} {5.401137} {5.3979297}

5

638.0882 456.2129 430.74493 423.5139 409.493

{5.1732766} {5.1485196} {5.145325} {5.1398804} {5.1383623}

7

707.4163 548.9756 519.06167 515.3648 504.86163

{5.0770245} {5.062215} {5.0613403} {5.0565738} {5.0575881}

10

724.4051 617.3783 602.05947 593.33047 581.68057

{5.0157726} {5.0065338} {5.0041152} {5.0039118} {5.0041513}

41

nS nL

UWL UCL n0

m

20 50 70 80 100

nS= 4

nL= 9

UWL= 1.2225

UCL=2.135748

3

586.7362 282.2523 246.2558 242.71687 232.3951

{5.4073748} {5.3514059} {5.3394891} {5.3343811} {5.3288533}

5

683.46703 464.59413 432.81003 422.98737 408.88693

{5.0891652} {5.0590663} {5.0543731} {5.0539691} {5.0506855}

7

733.2419 553.51747 524.4155 520.6517 512.75503

{4.9823707} {4.9655909} {4.9641534} {4.9617462} {4.9625093}

10

756.408 633.1033 612.27507 602.17177 591.6479

{4.919718} {4.9078875} {4.9041833} {4.9037186} {4.9055504}

nS= 4

nL= 10

UWL= 1.25

UCL=2.139975

3

576.06517 281.1382 252.8631 240.80653 231.76893

{5.5659229} {5.4947432} {5.4750111} {5.4749264} {5.4638414}

5

686.18277 468.78253 428.7995 427.89327 411.5765

{5.1912022} {5.1607694} {5.1511647} {5.1435069} {5.146164}

7

734.06093 557.86527 528.58793 526.5062 515.29687

{5.0756003} {5.0538199} {5.0481659} {5.0461544} {5.0453451}

10

764.1768 642.11153 615.874 607.0351 595.73237

{4.9977586} {4.9879347} {4.9808545} {4.9795509} {4.9804299}

42

Appendix (1): Table (3) Simulation results for ASS0=7

nS nL

UWL UCL

n0

m

20 50 70 80 100

nS= 2

nL= 8

UWL= 0.6225

UCL= 2.35

3

362.59263 357.42639 355.17551 354.85831 354.37712

{7.225142} {7.2587639} {7.2690957} {7.2711784} {7.2744306}

5

389.80149 379.22625 375.30888 374.00282 372.46294

{7.0366933} {7.0557665} {7.0609377} {7.0640829} {7.0638366}

7

396.9334 388.79165 385.90822 385.52141 385.05313

{6.9654874} {6.9776803} {6.9761001} {6.9781729} {6.9798052}

10

404.5355 397.50921 395.70957 394.53965 393.66061

{6.9104149} {6.917651} {6.9186033} {6.9179553} {6.919947}

nS= 2

nL= 9

UWL= 0.7375

UCL=2.585025

3

355.5562 327.85717 324.36953 317.51733 314.52687

{7.4857441} {7.5404994} {7.5466672} {7.5473665} {7.5622866}

5

390.24913 360.0912 354.93273 354.5687 353.46838

{7.1432014} {7.1607613} {7.1613598} {7.1567168} 7.1621876

7

399.18395 382.68627 380.55523 379.06647 377.84845

{7.0018103} {7.0095666} {7.0112322} {7.0124243} {7.009953}

10

410.02553 398.60596 395.39311 393.9996 390.75579

{6.9022149} {6.9047063} {6.9039444} {6.9059237} {6.9060355}

nS= 2

nL= 10

UWL= 0.8075

UCL= 2.6825

3

349.55723 305.71177 300.79637 297.05717 291.33437

{7.7226039} {7.748694} {7.761356} {7.7711462} {7.7714946}

5

388.70917 359.41973 351.09953 349.68383 347.8418

{7.2125475} {7.2075984} {7.2067334] {7.2108644} {7.2114308}

7

408.69443 384.6618 380.30889 381.13954 374.73767

{7.0031456} {7.0042854} {7.0015999} {7.0059021} {7.0011341}

10 422.2475 404.89983 401.21873 400.29023 399.09611

43

nS nL

UWL UCL n0

m

20 50 70 80 100

{6.8644555} {6.8679793} {6.8520202} {6.8556495} {6.8582459}

nS= 3

nL= 8

UWL= 0.6975

UCL= 2.0825

3

294.39003 266.75113 262.14073 261.05641 257.95419

{7.2041182} {7.2347613} {7.2396615} {7.2410482} {7.2428473}

5

369.08447 347.50783 344.72973 344.39947 343.37672

{7.0181013} {7.0280222} {7.0322945} {7.0295976} {7.0317858}

7

398.4005 379.4391 378.2103 377.51134 378.22231

{6.9362075} {6.9473792} {6.9513388} {6.948444} {6.9520486}

10

421.51857 409.06767 401.6866 401.55133 402.74563

{6.8845192} {6.8903107} {6.8886736} {6.8911033} {6.8927586}

nS= 3

nL= 9

UWL= 0.7975

UCL= 2.1875

3

326.07817 273.19447 266.9366 268.8484 262.36153

{7.5069632} {7.5270244} {7.5381179} {7.5399108} {7.538056}

5

399.68177 354.46177 348.6765 348.06373 345.5687

{7.1608846 } {7.1738226} {7.174419} {7.1748078} {7.1714454}

7

423.3455 390.99547 385.37647 382.24753 380.16303

{7.0280799} {7.0315637} {7.0323672} {7.0304848} {7.0380519}

10

438.76923 420.75637 417.67907 415.74397 410.58157

{6.9407978} {6.9361204} {6.9376909} {6.9324877} {6.9375819}

nS= 3

nL= 10

UWL= 0.875

UCL= 2.25125

3

340.1866 267.30707 254.59527 251.55873 248.60703

{7.6724055} {7.6874445} {7.6829585} {7.6772007} {7.6807957}

5

409.7832 351.9274 348.30677 341.0378 336.87567

{7.1782566} {7.1635395} {7.1633402} {7.1637178} {7.1653322}

7

441.14787 392.6479 388.34653 384.81723 380.24827

{6.9848968} {6.9807683} {6.968634} {6.9754557} {6.9739387}

10 457.59433 429.81923 423.97013 421.54027 417.2171

44

nS nL

UWL UCL n0

m

20 50 70 80 100

{6.8503359} {6.845569} {6.8476428} {6.8436382} {6.8431503}

nS= 4

nL= 8

UWL= 0.75

UCL= 1.95

3

308.40507 222.6421 214.16363 210.18683 207.62007

{7.2245402} {7.2476698} {7.2517048} {7.2561145} {7.2575918}

5

395.29483 341.8548 330.0463 327.99257 326.30697

{7.0402166} {7.0504674} {7.0555058} {7.0530919} {7.0576395}

7

435.69563 390.3271 388.439 383.97163 379.32633

{6.9706034} {6.9790382} {6.9778623} {6.9778556} {6.9771432}

10

465.1767 426.269 427.0679 421.47467 423.99693

{6.9200973} {6.9243079} {6.9236455} {6.9235101} {6.923056}

nS= 4

nL= 9

UWL= 0.85

UCL= 2.01

3

337.0157 238.65457 230.56863 224.0654 222.3882

{7.5318117} {7.5544238} {7.5538334} {7.5597171} {7.5580101}

5

415.48913 349.64373 341.0591 337.28847 334.85403

{7.214113} {7.2138855} {7.2198789} {7.2147814} {7.2161511}

7

452.92397 399.93947 392.46213 389.97377 386.88597

{7.0851325} {7.0898258} {7.0903878} {7.0924561} {7.0906113}

10

484.4473 441.60477 440.5674 435.44807 433.8797

{7.0029959} {7.0056257} {7.0047041} {7.002589} {7.0031404}

nS= 4

nL= 10

UWL= 0.925

UCL= 2.05

3

359.03417 240.86393 228.7687 225.2532 220.25463

{7.7264554} {7.7369265} {7.7333547} {7.7316219} {7.7296516}

5

437.07277 356.71543 340.57423 338.18693 331.50393

{7.2814067} {7.2646018} {7.2618015} {7.2568266} {7.2604293}

7

478.75403 411.26933 393.15133 390.95443 388.67857

{7.1073816} {7.0952894} {7.0901553} {7.094398} {7.0921729}

10 498.79577 450.36627 443.48467 435.77883 433.03957

45

nS nL

UWL UCL n0

m

20 50 70 80 100

{6.9869815} {6.9827565} {6.9798593} {6.9806218} {6.9788286}

nS= 5

nL= 8

UWL= 0.825

UCL=1.889725

3

391.9301 211.55333 194.44433 186.8999 177.01853

{7.2160804} {7.2314596} {7.2327717} {7.2324514} {7.2354709}

5

479.10373 352.3669 335.66647 325.5408 320.9643

{7.0373048} {7.0420249} {7.0464557} {7.0447783} {7.0453003}

7

508.1343 424.24247 412.95783 404.20763 398.50507

{6.9723212} {6.9752331} {6.97364} {6.9759253} {6.9768656}

10

533.99187 474.1511 465.38737 455.54867 458.5659

{6.9273617} {6.9277542} {6.9272151} {6.9271062}] {6.9274438}

nS= 5

nL= 9

UWL= 0.96

UCL=1.93

3

415.1945 227.34897 206.0848 200.00403 189.06887

{7.5135764} {7.5078636} {7.5159849} {7.512105} {7.5136001}

5

472.97277 347.6286 337.6987 332.65517 326.15927

{7.2156958} {7.2079815} {7.2031812} {7.2062774} {7.2042969}

7

518.2361 423.04703 407.66807 400.5212 397.37787

{7.1052257} {7.0914596} {7.0935724} {7.09413} {7.0908641}

10

541.10827 469.7394 460.9053 458.09203 453.27777

{7.0253135} {7.0211434} {7.0184232} {7.0195889} {7.0196618}

nS= 5

nL= 10

UWL= 1

UCL= 1.9411

3

399.4851 224.01647 204.17087 200.23927 192.29573

{7.7033031} {7.6796805} {7.6698684} {7.6668194} {7.667765}

5

508.01107 362.2793 342.88467 335.72773 330.63277

{7.2812802} {7.2569438} {7.255246} {7.2545351} {7.2495163}

7

539.28183 429.19227 414.1092 406.1938 397.15763

{7.1296858} {7.1143028} {7.1082887} {7.1111344} {7.1091136}

10 567.42207 478.43787 467.93573 464.28607 454.04353

46

nS nL

UWL UCL n0

m

20 50 70 80 100

{7.0321024} {7.0162901} {7.016494} {7.0179632} {7.0154901}

nS= 6

nL= 8

UWL=0.96

UCL=1.786875

3

504.18157 212.75803 183.40623 173.37433 165.0262

{7.2674293} {7.2712458} {7.2724762} {7.2718066} {7.2734444}

5

573.00517 368.49743 334.5075 333.96833 318.2218

{7.1254643} {7.1207764} {7.1214983} {7.1184572} {7.1190418}

7

614.8968 441.80227 421.39553 414.8631 404.09447

{7.0657817} {7.0660829} {7.0650786} {7.0637913} {7.0655493}

10

630.356 514.69057 494.34653 492.28653 482.90253

{7.0314559} {7.0277966} {7.026916} {7.0269756} {7.0275835}

nS= 6

nL= 9

UWL= 1.05

UCL= 1.8625

3

537.48097 216.69683 186.69883 181.54907 172.00737

{7.4535562} {7.4316794} {7.4273943} {7.4276184} {7.4231153}

5

600.1056 366.2986 341.24703 338.14453 322.86963

{7.1950446} {7.1794038} {7.173584} {7.1721823} {7.1720573}

7

616.16283 460.35807 421.0729 415.9359 406.6061

{7.1013629} {7.0875107} {7.0898654} {7.0857867} {7.0858237}

10

630.7139 511.14437 492.95247 488.52063 479.44673

{7.0378443} {7.032147} {7.0301044} {7.0296995} {7.0284753}

nS= 6

nL= 10

UWL= 1.125

UCL=1.872225

3

596.572 215.8954 187.93277 178.02823 171.14973

{7.2498921} {7.5422155} {7.5424056} {7.5391132} {7.530666}

5

600.7617 366.49893 341.6559 333.6262 321.24367

{7.2433143} {7.2172922} {7.210883} {7.208688} {7.2045199}

7

613.49167 449.18147 426.33387 416.6484 409.0556

{7.1265635} {7.1100888} {7.1048806} {7.1042994} {7.1014975}

10 628.12247 515.53833 490.4218 487.29733 479.3889

47

nS nL

UWL UCL n0

m

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

48

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

49

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

50

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

51

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

52

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

53

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

54

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

55

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

56

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

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

57

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

58

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

59

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

60

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

61

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

62

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

63

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

64

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

65

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

66

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

67

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

68

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

69

Appendix (3): Simulation results figures

Appendix (3): Figure (1) ARL0 vs. m for ASS0=3, ns=2 and nL=5

Appendix (3): Figure (2) ARL0 vs. m for ASS0=3, ns=2 and nL=7

Appendix (3): Figure (3) ARL0 vs. m for ASS0=3, ns=2 and nL=9

0

200

400

600

800

1000

1200

1400

20 50 70 80 100

AR

L0

m

n=3

n=5

n=7

n=10

0

200

400

600

800

1000

1200

1400

1600

20 50 70 80 100

AR

L0

m

n=3

n=5

n=7

n=10

0

200

400

600

800

1000

1200

1400

1600

20 50 70 80 100

AR

L0

m

n=3

n=5

n=7

n=10

70

Appendix (3): Figure (4) ARL0 vs. n0 for ASS0=3, ns=2 and nL=5

Appendix (3): Figure (5) ARL0 vs. n0 for ASS0=3, ns=2 and nL=7

Appendix (3): Figure (6) ARL0 vs. n0 for ASS0=3, ns=2 and nL=9

0

200

400

600

800

1000

1200

1400

3 5 7 10

AR

L0

n0

m=20

m=50

m=70

m=80

m=100

0

200

400

600

800

1000

1200

1400

1600

3 5 7 10

AR

L0

n0

m=20

m=50

m=70

m=80

m=100

0

200

400

600

800

1000

1200

1400

1600

3 5 7 10

AR

L0

n0

m=20

m=50

m=70

m=80

m=100

71

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;

72

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;