AN ABSTRACT OF THE THESIS OF
Shooka Darabi for the degree of Master of Science in Industrial Engineering presented on
July 12, 2013.
Title: Modeling and Analyzing Processes with Infrequent Failures: Implications for
Process Monitoring Policies
Abstract approved: ______________________________________________________
David S. Kim
Continues improvement of industrial systems require constant measuring process
variation and eliminating extraneous variation whenever possible to push the target
measures as close to the ideal value as possible. Statistical Processes Control (SPC)
technique is a very efficient statistical technique for monitoring and controlling processes
variation. In recent years an alternate Shewhart-type statistical quality control charts
called g-chart has been developed for controlling process variation where failure is
infrequent like hospital acquired infections.
However, g-chart, EWMA chart and probability based control charts, which are the main
control charts used for monitoring infrequent failure processes fail to detect changes in
the process quickly enough. As a solution it was proposed to inspect and follow-up on
every failure that occurs in the process.
A mathematical model for the long-run cost of a quality control policy is developed. This
model is used to compare the cost of following up on every failure policy with the cost of
using statistical control chart for monitoring processes.
By comparing the total cost per hour of using statistical control chart method with the
total cost per hour of following up on every failure, it is concluded that following up on
every failure policy results in quick detection of changes in the process and it is also a
more cost efficient method.
©Copyright by Shooka Darabi
July 12, 2013
All Rights Reserved
Modeling and Analyzing Processes with Infrequent Failures: Implications for Process
Monitoring Policies
by
Shooka Darabi
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Presented July 12, 2013
Commencement June 2014
Master of Science thesis of Shooka Darabi presented on July 12, 2013
APPROVED:
Major Professor, representing Industrial Engineering
Head of the School of Mechanical, Industrial, and Manufacturing Engineering
Dean of the Graduate School
I understand that my thesis will become part of the permanent collection of Oregon State
University libraries. My signature below authorizes release of my thesis to any reader
upon request.
Shooka Darabi, Author
ACKNOWLEDGEMENTS
First and foremost, I would like to thank my professor and adviser Dr.David Kim for
his valuable guidance and advice. Dr. Kim was helpful, supportive and offered invaluable
assistance and guidance. I also want to express my deepest gratitude to the members of
the supervisory committee, Dr. Sarah Emerson and Dr. Javier Calvo-Amodio. Without
their support and assistance this thesis wouldn’t have been successful.
Finally, an honorable mention goes to my family for their love, understanding and
support.
TABLE OF CONTENTS
1Introduction ............................................................................................................................................... 1
2Literature review ....................................................................................................................................... 4
2.1Conventional Shewhart statistical control charts and infrequent failure processes ..................... 4
2.1.1Non-Shewhart statistical methods for infrequent failure process ......................................... 6
2.1.2Statistical charts for infrequent failure processes .................................................................. 7
2.2Non-parametric charts for infrequent failure processes ............................................................... 9
2.2.1Traditional non-parametric charts for infrequent failure processes ...................................... 9
2.3Summary .................................................................................................................................... 12
3Methodology ........................................................................................................................................... 13
3.1Quality control scenario and assumptions.................................................................................. 13
3.2Review of control chart methods ............................................................................................... 15
3.2.1The Shewhart g control chart .............................................................................................. 16
3.2.2g-chart with probability distribution-based control limits................................................... 18
3.2.3Exponential Weighted Moving Average control chart ....................................................... 21
3.2.4Control chart performance measurements........................................................................... 23
3.2.5Inspection and following up every failure........................................................................... 24
3.3Control charts versus follow up of every failure – what failure probability? ............................ 25
3.4Quality control cost model ......................................................................................................... 26
3.4.1Parameters ........................................................................................................................... 26
3.4.2Cost components ................................................................................................................. 27
3.4.3Fraction of time a process is in control ............................................................................... 27
3.4.4Total cost per unit time ....................................................................................................... 28
3.4.5 and using control charts ................................................................................... 28
3.4.6Approximation .................................................................................................................... 30
3.4.7 and – follow-up each failure ............................................................................ 31
3.5Scenario1- hospital acquired infections ..................................................................................... 32
3.5.1Probability based control chart for hospital acquired infections ......................................... 33
3.5.2Shewhart g-chart for hospital acquired infections............................................................... 43
3.5.3EWMA chart for hospital acquired infections .................................................................... 47
3.5.4Following up on every failure policy .................................................................................. 50
3.6Scenario2- pace maker manufacturing ....................................................................................... 51
3.6.1Probability based control chart for pace maker manufacturing .......................................... 51
3.6.2Shewhart g-chart for pacemaker manufacturing process ................................................... 55
3.6.3EWMA chart for pacemaker manufacturing ....................................................................... 57
3.6.4Following up on every failure ............................................................................................. 58
4Results .................................................................................................................................................... 59
4.1Comparing probability limit based g-chart with follow up of every failure - hospital
acquired infections ...................................................................................................................................... 59
4.2Comparing Shewhart g-chart with follow up of every failure - hospital infections................... 62
4.3Comparing EWMA with follow up of every failure - hospital acquired infections................... 64
4.4Comparing probability limit based Shewhart g-chart with follow up of every failure pacemaker manufacturing case ............................................................................................................... 67
4.5Comparing EWMA with following up of every failure pacemaker manufacturing .................. 69
5Conclusion............................................................................................................................................... 71
6References ............................................................................................................................................... 72
LIST OF FIGURES
Figure 1: Example of monitoringthe number of inspections until failure using control charts................. 15
Figure 2 :Comparing total cost per hour of using probabilit based control charts vs. following up
on every failure when d=1 and C/F = 1 ...................................................................................................... 60
Figure 3: Comparing total cost per hour of using probability based control chart vs. following up
on every failure when d = 1 and C/F = 0.4 ................................................................................................. 62
Figure 4: Comparing total cost per hour of using g-chart vs. following up on every failure when
d = 1and C/F =1 .......................................................................................................................................... 63
Figure 5: Comparing total cost per hour of using EWMA vs. following up in every failure when
d = 1 and C/F = 1 ........................................................................................................................................ 64
Figure 6: Comparing total cost per hour of using EWMA vs. following up on every failure when
d = 1 and C/F = 0.05 ................................................................................................................................... 66
Figure 7: Comparing total cost per hour of using EWMA vs. following up on every failure when
d = 1 and C/F = 20 ...................................................................................................................................... 66
Figure 8: Comparing total cost per hour of using probability based control chart vs. following up
on every failure whe d = 1 and C/F = 1 ...................................................................................................... 67
Figure 9: Comparing total cost per hour of using g-chart vs. following up on every failure when
d = 1 and C/F = 0.1 ..................................................................................................................................... 68
Figure 10: Comapring total cost per hour of using EWMA vs. following up on every failure
when d = 1 and C/F = 1............................................................................................................................... 69
Figure 11: Comparing total cost per hour of using EWMA vs. following up on every failure
when d = 1 and C/F = 0.25.......................................................................................................................... 70
Figure 12: Comparing total cost per hour of using probability based control charts vs. following
up on every failure when d = 2 and C/F = 1 ............................................................................................. 110
Figure 13: Comparing total cost per hour of using probability based control chart vs. following
up on every failure when d = 0.5 and C/F = 1 .......................................................................................... 110
Figure 14: Comparing total cost per hour of using g-charts vs. following up on every failure
when d = 2 and C/F = 1............................................................................................................................. 111
Figure 15: Comparing total cost per hour of using g-chart vs. following up on every failure when
d = 0.5 and C/F = 1 ................................................................................................................................... 111
Figure 16: Comparing total cost per hour of using EWMA vs. following up on every failure
when d = 2 and C/F = 1............................................................................................................................. 112
Figure 17: Comparing total cost per hour of using g-chart vs. following up on every failure when
d = 0.5 and C/F = 1 ................................................................................................................................... 112
Figure 18: Comparing total cost per hour of using probability based control charts vs. following
up on every failure when d = 2 and C/F = 1 ............................................................................................. 113
Figure 19: Comparing total cost per hour of using probability based control charts vs. following
up on every failure when d = 0.5 and C/F = 1 .......................................................................................... 113
Figure 20: Comparing total cost per hour of using g-chart vs. following up on every failure when
d = 2 and C/F = 1 ...................................................................................................................................... 114
Figure 21:Comparing total cost per hour of using g-chart vs. following up on every failure when
d = 0.5 and C/F =1 .................................................................................................................................... 114
Figure 22: Comparing total cost per hour of using EWMA vs. following up on every failure
when d = 2 and C/F = 1............................................................................................................................. 115
Figure 23: Comparing total cost per hour of using EWMA vs. following up on every failure
when d = 0.5 and C/F = 1.......................................................................................................................... 115
LIST OF TABLES
Table 3.1: Shewhart g-chart formulas ....................................................................................................... 18
Table 3.2: Probability distribution based control limits for the known rate of .................................... 21
Table 3.3: Exponential Weighted Moving Average control limits ........................................................... 23
Table 3.4: values of constant parameters in hospital acquired infection case ........................................... 41
Table 3.5: Estimated values of constant parameters in pacemaker manufacturing process ...................... 54
LIST OF APPENDICES
Appendix1 ................................................................................................................................................. 75
Appendix2 ................................................................................................................................................. 83
Appendix3 ................................................................................................................................................. 90
Appendix4 ................................................................................................................................................. 98
Appendix5 ............................................................................................................................................... 102
Appendix6 ............................................................................................................................................... 100
Appendix 7-Hospital graphs.................................................................................................................... 110
Appendix8 -Pacemaker manufacturing graphs ....................................................................................... 113
1
1
Introduction
The research presented in this thesis addresses the question of when more
common statistical quality control charts should be used to monitor processes that are
considered to have “infrequent failures”. Two particular applications that will be used as
examples are the monitoring of the infection rate in a hospital, and the monitoring of
component defects in pacemaker assembly processes. These examples can be used to
clarify the general features of the quality monitoring scenario considered that are
applicable to more applications. In both applications “units” are “inspected” individually
and either pass or fail the inspection. In the infection rate application, units are patients
and inspections are patient examinations. Failure of the inspection is testing positive for
an infection. In the pacemaker assembly example units are electronic components, and
inspections are actual inspections that are either passed or failed. The probability of a
failure of an individual inspection is assumed to be the same. A variety of statistical
quality control chart methods can be applied, but some methods have been developed for
the scenario described.
Relatively recently a statistical quality control chart called a g-chart has been
developed for monitoring processes similar to the infection and pacemaker assembly
applications. They have been applied to processes with “infrequent failures” where the
failure rate is usually less than 1%. g-charts have been applied in healthcare, where the
failure rate is “infrequent”. Some examples are, using g-chart for monitoring the number
of cases between hospital acquired infections, heart surgery complications, contaminated
2
needle sticks and surgical site infections (Benneyan 1999). In the g-chart, charts monitor
the numbers of cases (patients) between two successive failures (infections, heart surgery
complications …)
Medical processes are a big national concern and various techniques for making
these processes more efficient and consistent have been applied. Hospitals have been
encouraged to use statistical process control tools such as Statistical Process Control
(SPC) charts for improving and monitoring their processes. The objective of SPC is to
monitor processes and systematically identify problems and remove them.
Motivated from the use of g-charts for health care applications, the author of this
thesis tried to develop a g-chart for monitoring the failure rate in a pacemaker
manufacturing process which, because of the type of product, also has “infrequent”
failures. A pacemaker is composed of numbers of components, and pacemaker failure is
mostly from component failure, and pacemaker failure due to other reasons, like operator
error, machine error and others can be neglected. Since pace makers are very sensitive
products, before final assembly, all the components are tested to minimize pacemaker
failure due to component failures. Because of this, the failure probability of a component
in final assembly is very small. On average, 1 in every 2000 capacitors fails (failure rate
0.05%). Thus, the monitoring process of final assembly of components in pacemakers
can be placed in the infrequent failure category and quality tools designed for infrequent
failure analysis should be used for monitoring the process.
The g-chart that was developed for monitoring pacemaker manufacturing process
failed to capture changes in the final assembly process. This was concluded by testing the
3
chart with historical data. Based on this result, the author became motivated to study
why a chart that is designed specifically for monitoring infrequent failures doesn’t
perform as it is intended? In addition, if a g-chart is not helpful in monitoring infrequent
failure processes, what are other methods that can be used in this type of situations?
The first section of this thesis is the literature review, presenting relevant research
on application of statistical process control chart methods to infrequent failure processes.
The second section describes the methodology. It includes description of a cost model
for the long run cost of monitoring a process and applications of this model to two
specific scenarios. The third section summarizes sensitivity analyses and presents more
general results as well as the specific analyses performed and results obtained. Finally,
the conclusions of this research are discussed in the last section.
4
2
Literature review
In this section, a brief summary of relevant research is presented. The section
starts with a short summary of statistical process control methods. Later the use of the
statistical process control methods and their short coming in infrequent failure processes
will be addressed. The section ends with presenting some other alternative methods that
have been developed for monitoring infrequent failure processes for both parametric and
non-parametric data.
2.1 Conventional Shewhart statistical control charts and infrequent failure
processes
Many industrial processes are complicated and operate in a noisy unpredictable
environment. This means processes are subject to variation. Some of this variation
originates from natural causes that exist in every type of environment. These types of
variation usually have been referred to as “chance” causes and are present to some degree
in every process. Beside chance causes, there are other sources of variation that occur due
to a process change called “assignable” causes. Differentiating between chance and
assignable causes and eliminating assignable causes are the objective of Statistical
Process Control (SPC) Charts.
The basic methodology of control charts consist of sampling from a process over
time and monitoring a process measurement over time. Controls charts have a center line
and an upper and lower control limit established from process measurements. When a
measurement falls within the upper and lower control limit, no action is required, but
5
whenever the measurement falls outside of the control limits, a possible assignable cause
is deemed possible and further investigation to determine the root cause and correct if
necessary (Montgomery, 2007; Oakland, 2012).
Count charts are SPC charts that are used for situations when the monitored
statistic is based on the number of nonconforming items in a sample, or the number of
defects in an item. The p-chart and the c-chart are the two most well-known types of
count charts. The p-chart is used for monitoring the ratio of the number of
nonconformities to the sample size. The c-chart is used to monitor total number of
nonconformities per unit (Montgomery, 2007). These charts have a broad range of
applications. Nevertheless, in a study conducted by Ozsan et al. 2009, it is argued that
these charts underperform in an environment where the failure occurrence is low. In this
case, failure counts in many fixed time intervals will be zero leading to narrow control
limits which will cause frequent false alarms.
Another drawback of using count charts for monitoring infrequent failure
processes is the symmetrical shape of the control charts. The 3-sigma control limits are
derived from the original Shewhart control chart which assumes the monitored data is
normally distributed or is asymptotically normal. When using p-charts and c-charts for
monitoring frequent failure processes Binomial distribution and Poisson distributions are
approximately normal and therefore having symmetrical control charts with 3- sigma
control limits is well justified. However in an infrequent failure process, the normality
assumption of the p-chart and c-chart is questionable (Chan et al. 2000).
In addition, using count charts for situations where the failure rate is low results in
a negative lower control limit. Since a negative lower control limit is meaningless, it is
6
replaced with 0. Without a positive lower control limit it is impossible to identify
downward shifts of the process which is an indication of improvement (Chan et al. 2007).
A low defect rate also negatively impacts the upper control limit. A rule of thumb
(Chan et al. 2000) says, in a p-chart the upper control limit should be smaller than the
inverse value of the sample size . In infrequent failure process this means the UCL
would be set so low that every defect exceeds the upper control limit and triggers a
signal. In this case the false alarm probability is much greater than the desired value of
0.00135 (Chan et al. 2000).
In another study, Goh 1987 also confirms that even for very high yield process
with a defect rate lower than one percent there is poor low performance of the count
charts.
2.1.1
Non-Shewhart statistical methods for infrequent failure process
The inability of common Shewhart control charts for detecting out of control
processes in environments with a low failure rate has resulted in the development of other
methods and techniques for monitoring infrequent failure processes. In this section some
other control chart techniques for low failure processes are reviewed.
When the fraction of non-conforming is low and a normal approximation which is
the basis of 3-sigma control charts in invalid, some studies suggest using control limits
derived from the distribution of the statistic monitored. Goh (2007) explains the method
used to identify control limits derived from the Poisson probability distribution for the
number of defects in a sample.
7
A similar study conducted by Benneyan (2001) utilized control charts with
probability distribution based limits for monitoring low failure processes, when the
monitored statistics has a geometric distribution. In this method control limits are
derived directly from geometric distribution.
In another study, Nelson (1994) suggested counting the total number of the
defects (denoted c) in a sample with size n and develop control charts for the
transformed variable . Although, it has been demonstrated that this method works well
for infrequent event processes, a disadvantage with this method is that the interpretation
of the original data is hard and in some cases meaningless.
2.1.2
Statistical charts for infrequent failure processes
Different methods and techniques presented above, both Shewhart and nonShewhart techniques, all showed limitations when it comes to monitoring processes with
infrequent failures. To overcome this problem some specific control charts have been
developed specifically for this purpose. In this section a brief explanation of methods
designed specifically for infrequent failure processes is presented.
Since the conventional charts like p, np, u and c chart often result in subgroups
being plotted too infrequently in an infrequent failure process. An alternative chart was
designed by Benneyan (1999; 2001) mainly for the purpose of monitoring failures in an
infrequent failure environment. The charts developed are the g-chart and h-chart. These
charts monitor the number of events between occurrences of two successive failures.
These occurrences can be the number of conforming items between two consequence
failures, or days between two failures. In the g-chart, the monitored statistics follows a
8
geometric distribution. In the h-chart, the monitored statistic is the number of conforming
items until the n-th failure, which also follows a geometric distribution (Benneyan 1999;
Benneyan 2001; Benneyan 2001; Benneyan and Kaminsky 2004; Glushkovsky 1994).
For infrequent failure processes, studies show that these charts can be applied to
both the first phase of the process, to bring the process into statistical control, as well as
the second phase of the process which is to monitor the in control process (Benneyan
2001).
An alternative SPC chart for infrequent failures when the failures occur
according to Poisson process was proposed by Chan et al (2000). A cumulative quantity
control chart (CQC) monitors the occurrence of nonconformities in a process with a
constant rate of defects ( λ) per number of products in sample. Q is defined as the
number of units until the first defect is observed and it follows an exponential
distribution. Based on the information above the control limits have been defined as,
Q
α
ln 2
λ
Q Q 1
ln 2
λ
α
ln 1 2
λ
Although the CQC chart has been showed to perform well when the failure rate
of processes is low or moderate, there is another version of the CQC chart called
CQC(2), which has the virtue of being more sensitive to smaller shifts and makes it more
desirable in infrequent failure processes. CQC(2) also assumes the occurrence of
nonconformities follows a Poisson process, but the statistic that is being monitored in this
9
case is the time until the second failure occurs. This statistic follows a gamma
distribution and control limits are based on that.
2.2
Non-parametric charts for infrequent failure processes
The alternative Shewhart-like control charts that are designed specifically for
infrequent failure processes were presented. However, in the presented methods
presented there are some assumptions about the distribution of the data, and charts are not
robust to these assumption. In the following chapter some non-parametric charts that
don’t make any assumptions about the distribution of data are presented for infrequent
failure processes.
2.2.1
Traditional non-parametric charts for infrequent failure processes
In this section the traditional non-parametric charts and their specifications and
performance when applied to infrequent failure processes are described.
Non-parametric charts which are also known as distribution free charts can be
applied to a broad range of situations. Nonparametric charts are an effective alternative
when the distribution of the data is unknown or doesn’t follow a known distribution.
Studies show that these charts are very effective in detecting shifts for non-normal
distributions, especially those with heavy tails (Chakraborti et al. 2004). Furthermore,
these charts have the advantage of having the same in-control run length for all
continuous distributions as well as being more robust to outliers (Chakraborti et al. 2004).
When failures are infrequent and the data doesn’t follow a particular distribution
non-parametric charts can be applied to the situation, using the number/time of events
10
between defective units as a statistic. The two well-known non-parametric charts are the
exponential weighted moving average EWMA and the cumulative sum chart CUSUM.
These charts are very effective in detecting minor and persistent shifts (Montgomery
2007). EWMA and CUSUM control charts can be used for monitoring either attribute or
variable type data using the entire data history. EWMA charts are developed based on
monitoring the exponentially weighted moving average of prior sample means. In
CUSUM charts a sequential technique method is used which keep tracks of the
cumulative sum of prior samples. (Montgomery 2007)
Although many studies including Montgomery (2001, p.433) have pointed out
that “It (EWMA) is almost a perfectly nonparametric (distribution-free) procedure” some
other studies give evidence against that. For instance a study conducted by Graham et. al.
(2012) shows that, for EWMA charts it is usually assumed that the underlying
distribution of the process of the sub-group averages is normal or it is asymptotically
normal (Graham et. al. 2012). In addition Human et al (2011) shows that EWMA is not
robust to non-normal distributions and would incorrectly show the process is out of
control. Borror and Montgomery (1999), also show that the statement about EWMA
being a nonparametric procedure is true when the process parameters are known or
specified. When the parameters are unknown and need to be estimated from sample of
data, EWMA is showed that is non-robust to non-normal distributions. If the parameters
are estimated from the sample, this will result in many more false alarms in the Average
Run Length chart (Graham et. al 2012).
In case of CUSUM charts, Acosta-Meja (2009) argues that in many cases
parametric charts are not effective in detecting shifts due to the violation of the
11
assumptions made about the data distribution and sees the CUSUM chart as a better
choice. However, sensitivity of Average Run Length (ARL) show significant differences
between the charts from normally distributed data and those that are based on non-normal
data.
Graham et al. (2011 a,b) proposed a nonparametric EWMA chart which is based
on a sign and rank sum test when the median is known or specified. Although this
method is non-parametric, when the median is unknown and estimated from data, the
chart underperforms with an increase in the false alarm rate.
An alternative distribution-free EWMA chart was developed by Graham et al.
(2012) called EWMA_EX. EWMA-EX is a non-parametric symmetrical EWMA chart
based on exceedance test statistics.
Further analysis indicates that the EWMA-EX chart performs as good or better
than parametric CUSUM and EWMA charts for detecting small mean shifts . Even in
comparison with other recently developed nonparametric CUSUM or EWMA chart like
the EWMA based on the Wilcoxon-Mann-Whitney statistics, EWMA-EX still performs
better. These points as are proved by comparing the ARL charts of the different methods
(Graham et al. 2012).
In a study conducted by Mukherjee et al. 2012, a non-parametric CUSUM chart is
developed based on the exceedance statistic called exceedance CUSUM chart. This new
non-parametric CUSUM chart like the EWMA_EX chart is more effective method in
detecting a shift in the unknown location of the parameter of a continuous distribution.
Further analysis show that these charts are more powerful in detecting shifts in the
process than traditional CUSUM and rank-based CUSUM charts. It is showed that these
12
charts perform very well in cases when the underlying distribution is heavy-tailed and
right-skewed.
2.3 Summary
In this section a brief summary of earlier literature was presented. The main
objective was to describe methods that were developed for use in an infrequent failure
environment, both for monitoring statistics that follows a specific distribution and when
the monitored statistic does not follow any particular distribution. No research was found
that examines when some type of statistical quality control chart should be used to
monitor a process with infrequent failures. Not using statistical control charts to monitor
a process, and investigating every failure is a reasonable alternative when failures are
infrequent and the cost of running a process out of control is high. The research in this
thesis investigates this topic.
In the next section, a general mathematical model for the long run cost of
monitoring a process is developed that facilitates the choice of appropriate monitoring
method for infrequent failure processes.
13
3
Methodology
The question that is addressed in this research is: What is an infrequent failure
probability with respect to implementing statistical quality control? Alternatively the
question can be asked as: At what failure probability will statistical quality control result
in lower average costs than follow-up of every failure?
To answer this question a mathematical model for the long-run cost of a quality
control policy is developed. This model is then used to compare the policy of following
up on every failure versus using some statistical control chart method and following up
only after alarm signals are generated.
The remainder of this section starts with a presentation of the characteristics of
the specific quality control scenario being addressed. This is followed by a review of the
different control chart methods that will be analyzed. The details of the cost model for a
quality control policy will then be explained.
3.1 Quality control scenario and assumptions
In the scenario considered the rational sub-group is equal to one; meaning a data
sample of size one is collected and monitored over time. The particular data item
represents the number of “units” inspected until a “failure” occurs. In this scenario a unit
may represent a variety of entities depending on the specific application. For example a
unit may be a product, a patient, etc. It is assumed that there are individual inspections on
each unit and a unit either passes the inspection, or fails to meet the predefined
requirements of the unit. A failure may represent a variety of conditions depending on
the application. For example if the unit is a product, a failure means the product did not
14
meet specific product requirements, and if the unit is a patient then failure may mean the
patient suffered an infection. The monitored statistic is the number of units between
failures and this statistic can be converted to time between failures if the inspection time
per unit is known.
It will be assumed that each unit has the same probability of passing an
inspection, and that this probability remains fixed between changes at discrete time
points. These changes normally represent a “shift” to an out-of-control state, or the return
of an out-of-control process back to control. A smaller failure probability can be
interpreted as an increase in process quality, and higher probability can be interpreted as
a decrease in process quality. Under the scenario described the monitored statistic (units
between failures) has a geometric distribution.
If a quality control chart is being utilized to monitor the number of units between
failures the x-axis represents the failure number, and the y-axis is the number of
inspections until failure.
Figure 1 shows an example of such a control chart.
15
Figure 1: Example of monitoringthe number of inspections until failure using control charts
When using a control chart lower and upper control limits are computed and set
according to the control chart assumptions. Every point (failure) that plots above the
upper control limit or below the lower control limit generates a signal. For the situation
described, when a point plots above the upper control limit the quality of the process may
have improved, and when a point plots below the lower control limit it is an indication
that process quality may be worse. Note that in cases where the LCL is equal to zero, the
control chart loses its ability to detect increases in the failure probability.
3.2 Review of control chart methods
Infrequent failure processes occur in situations where failing to detect problems
results in potentially catastrophic (or very costly) results. By nature the products being
monitored are very sensitive and even one failure is not acceptable.
16
Some of the examples of infrequent failure processes are heart surgery
complications, catheter related infections, surgical site infections, contaminated needle
sticks, and other patient acquired outcomes (Benneyan 1999). Some examples of
infrequent failure manufacturing processes are aerospace product production, and
medical device final assembly processes.
Monitoring processes using standard charts like (p, np, u and c chart) results in
subgroups being plotted too infrequently for timely control chart feedback (Benneyan
2001). Therefore, these control chart methods are not informative and are not helpful in
detecting process changes.
In the following sections, three control charts that were developed for the scenario
described, and which have been applied detect process changes in infrequent failure
process are introduced. These charts are the Shewhart g-chart, the g-chart with
probability based control chart limits, and the Exponential Weighted Moving Average
(EWMA) control chart.
3.2.1
The Shewhart g control chart
A control chart is developed for when the number of trials between events is
monitored, and the distribution of the number of trials between events is geometrically
distributed is called a g-chart. Each trial is an independent Bernoulli trial, and the number
of trials until the first event (failure), Z, follows a geometric distribution with probability
mass function:
!" #$ %!1 %$&
'()# 1, 2, 3, …
(1)
17
The expected value and variance of Z are:
-!"$ ./)!"$ 1
%
(2)
1%
%0
(3)
p is typically estimated from data and the maximum likelihood estimator of p is”
%̂ 1
"
(4)
Where " is the sample average of a fixed number of individual observations of Z
(for example the average number of failed capacitors or average days between
infections). The g-chart was developed specifically for a process similar to the scenario
being addressed.
The parameter k represents the number of standard deviations from the centerline
used to define upper and lower control limits. Usually k is three; meaning control limits
are “3 sigma” from the center line. The lower control limit (LCL), and upper control limit
(UCL) are:
1
1%
234 56 7 8 #9 0 :
%
%
1
1%
434 56 7 #9 0 :
%
%
;
<
(5)
(6)
For these limits the actual type I error may be less than stated due to the discrete
nature of the geometric distribution. Alternative control limits where the type I error level
is greater than stated are computed as
18
1
1%
234 56 7 8 #9 0 :
%
%
1
1%
434 56 7 #9 0 :
%
%
<
(7)
;
(8)
p Known
Upper Control Limit
(UCL)
1
1%
56 7 8 #9 0 :
%
%
1
%
Center Line (CL)
Lower Control Limit
(LCL)
1
1%
56 7 #9 0 :
%
%
p Estimated
;
<
" 8 #="!" 1$;
"
" #="!" 1$<
Table 3.1: Shewhart g-chart formulas
Table 3.1 summarizes the computation of the Shewhart upper control limit
(UCL), center line (CL) and lower control limit (LCL) for a g control chart used to
monitor individual observations of Z.
3.2.2
g-chart with probability distribution-based control limits
An alternative that is helpful for situations where Shewhart control charts limits
(particularly the LCL) would be well below zero (and thus set to zero) is to derive the
control limits directly from the assumed distribution of the data. This method can be
applied to almost any control chart but it is especially helpful when the distribution of the
19
monitored statistics is not normal or symmetric as is the case when the monitored statistic
is from a geometric distribution with a small p value. Note that this method can be
applied to any other distribution with known Cumulative Distribution Function (CDF).
Assume >?@? and >A@? are the desired type I error rates (false alarm rate) for the
lower control limit and upper control limit respectively. In most cases it is assumed
>?@? >A@? BCDEFG
0
.
The Cumulative Distribution Function (CDF) of Z is:
'!#$ !" H #$ 1 !1 %I $&
'()# 1,2,3, …
(9)
where %I is the in-control probability that a unit will fail inspection. The
probability of having a random variable Z smaller or equal to the LCL when the process
is in control is obtained by replacing k with the LCL:
!" H 434$ 1 !1 %I $?@?
where K< is x rounded down to an integer.
J
(10)
By using the geometric CDF the probability that a random variable plots at or
beyond the upper control limit when the process is in control is computed as:
!" L 234$ 1 !" M 234$
(11)
1 N1 !1 %I $A@? O if the UCL is non-integer, and
J
1 1 !1 %I $A@?< if the UCL is integer.
By setting the above results equal to the desired type I error rate result in:
>?@? !" H 434$ 1 !1 %I $?@?
>A@? !" L 234$ !1 %I $A@?
(12)
(13)
20
Solving the equations for LCL and UCL it result in having :
234 434 ln!>A@? $ ;
ln!1 %I $
(14)
PQ!1 >?@? $ <
PQ!1 %I $
(15)
For these limits the actual type I error is less than stated due to the discrete nature
of the geometric distribution. Alternative control limits where the type I error level is
greater than stated are computed as:
234 434 ln!>A@? $ <
ln!1 %I $
(16)
PQ!1 >?@? $ ;
PQ!1 %I $
(17)
Similarly the center line can be computed by setting the probability of a random
variable being smaller than Center Line (CL) to be 0.5.
!" H 34$ 1 !1 %I $@? 0.5
(18)
Solving the equation above gives:
34 PQ0.5
PQ!1 8 %I $
(19)
The results above are summarized in the table below.
Control Limits
Upper Control Limit (UCL)
Formulas for known U
234 ln!>A@? $ ;
ln!1 %I $
21
34 Control Limit (CL)
434 Lower Control Limit (LCL)
PQ0.5
PQ!1 8 %I $
ln!1 >?@? $ <
ln!1 %I $
Table 3.2: Probability distribution based control limits for the known rate of 3.2.3
Exponential Weighted Moving Average control chart
An Exponential Weighted Moving Average (EWMA) control chart is an
alternative control chart method often used for in-control processes. EWMA charts are
useful for detecting small mean shifts and are applicable to situations where the rational
subgroup size is one. The other advantage with EWMA charts is that unlike other
Shewhart control charts; EWMA charts are insensitive to the normality assumption and
thus perform well when the sample size is equal to one. However recent research showed
that robustness to the normality assumption is true under the condition that distribution
parameters are known (Borror, et al. 1999). The EWMA statistic that is monitored is
defined as:
VW XKW 8 !1 X$VW<
(20)
X is defined as smoothing parameter which is constant value between zero and 1
(0 M X M 1). KW is the current observation. The starting value VI for sample size Y 1 is
defined as:
And ZI is the process mean.
VI ZI
22
The control limits for a EWMA chart are:
234 ZI 8 4[9!
X
$1 !1 X$0W 2X
X
434 ZI 4[9!
$1 !1 X$0W 2X
(21)
(22)
L determines the width of control charts. As Y increases the term 1 !1 X$0W approaches to one. Thus, when running a process for a long time the control limits
approach a steady limit. Due to discrete nature of monitoring statistics here are two
alternatives for control limits.
X
234 ZI 8 4[96
7; 2X
X
434 ZI 4[9!
$<
2X
(23)
(24)
For these limits the actual type I error is less than stated due to the discrete nature
of the geometric distribution. Alternative control limits where the type I error level is
greater than stated are computed as:
434 ZI 4[9!
X
$;
2X
234 ZI 8 4[=0<\< \
(25)
(26)
23
The results above are summarized in the table 3.3.
Control Limits
Formulas for early stages
Formulas for steady
stage
Upper Control
Limit (UCL)
Lower Control
Limit (LCL)
ZI 8 4[ 9!
X
$1 !1 X$0W ;
2X
ZI 4[ 9!
X
$1 !1 X$0W <
2X
X
ZI 8 4[9!
$;
2X
X
ZI 4[9!
$<
2X
Table 3.3: Exponential Weighted Moving Average control limits
3.2.4
Control chart performance measurements
Control chart performance can be evaluated based on how frequent false alarms
are generated when the process is in control, and how frequent an alarm is signaled when
the process is out of control. This performance is measured by Average Run Length
(ARL). ARL0 is a measure of the average time between false alarms when in control.
ARL1 is a measure of the average time between alarms when out of control.
The number of inspections until a failure is geometrically distributed so ARL are
expressed in terms of the number of inspections until an out of control signal is
generated. That is:
]^4 1
%)(_/_YP`a(bcYdQ/P
(27)
24
]^4I ]^4 1
1 !434 M " M 234|% %I $
1
1 !434 M " M 234|% f %I $
(28)
(29)
The Average Time to Signal (ATS) is the average time until an out of control
signal is generated (Montgomery, 2001). If the inspections are conducted on average
every t time units then the ATS is:
]gh ]^4 ∗ `
(30)
In more common Shewhart control charts, plotting beyond control limits generate
signals with different meanings. Signals above the UCL normally are an indication of a
decrease in the quality, and signals below the LCL normally indicate an improvement in
process quality. For g-charts this is the opposite, since the monitored statistic is the
number of units until a failure occurs. A lower statistic means the decreased quality.
For the cases when the LCL is equal to zero, the g-chart loses the ability to detect
an increase in the process failures (Benneyan, 1999).
To overcome the problem of zero LCL, and improve the power of detection of
processes changes, several design considerations have been proposed. Some of these
design considerations are narrower limits or the use of within-limit supplementary
rules(Benneyan 2001).
3.2.5
Inspection and following up every failure
Another approach that can be used when failures are infrequent is to
conduct follow-up after every failure to determine if a process shift has occurred. This
25
can be a useful approach if failures occur infrequently and the costs of increased failures
are very high. This is the situation for the production of pacemakers.
3.3 Control charts versus follow up of every failure – what failure
probability?
The different control charts reviewed have been proposed for use in the situations
described in section 3.2. Regardless of what control chart that is being used, when the
LCL is equal to zero, the charts lose their ability to detect increase in the failure
probability and may not be very useful and informative. This occurs when the failure
probability is “low” or failures are “infrequent”. Another option is to follow-up every
failure. This method is not practical and not a cost-efficient approach when the failure
rate is not “low” or failures are not “infrequent”.
If a follow up every failure policy is used when failures are infrequent, an
important question is: what failure probability defines infrequent failures, or what is the
cut-off between infrequent failures and frequent failures? When is it better to inspect and
follow-up every failure, and when should control charts be used for monitoring
processes? Which one of these methods is more beneficial from a cost perspective?
To answer these questions a cost model is developed that can be used to identify
infrequent failures for specific scenarios. This model is presented in the next section.
26
3.4 Quality control cost model
The notation used to describe the model.
3.4.1
Parameters
3
Cost per hour of running a process which is out of control.
k
Shift fraction from p0.
-!j$ Expected time a process is in control, after control is established.
'
`
%I
%I∗
Cost of signal follow up.
Average time between inspections.
In control probability of failure
The cut-off probability between infrequent and frequent failures.
The model developed generates a cost per unit of time for following a specific
quality control policy. For simplicity and clarity hours will be used as the time unit. The
overall cost per hour is comprised of two components. The first component is the cost per
hour of signal follow-up, when the process is in control (Cost IC). The second component
is the cost per hour of operating a process when the probability of failure has shifted from
%I to %I 8 k%I , plus the cost of follow-up (Cost OC). The total cost per hour is a
weighted average of the two cost components and is computed based on the percentage of
being in control (% IC), and the percentage of being out of control (% OC). The overall
cost function is:
g(`/P3lhg %n3 ∗ 3(c`n3 8 %l3 ∗ 3(c`l3
27
3.4.2
Cost components
The long-run average in control cost per hour is the cost of signal follow-up when
the probability of a failure is%I . This cost per hour is the signal follow-up cost divided by
the average time between signals when the process is in control. The average time
between signals when the process is in control is ]ghI and 3(c`n3 is computed as:
3(c`n3 '
]ghI
where]ghI ` ∗ ]^4I
(31)
(32)
and ]^4I is the average number of inspections between signals.
The out of control cost per hour is the cost of signal follow-up when the
probability of failure has shifted from %I to %I 8 k%I and process is out of control (OC).
This cost is the cost per hour of running an out-of-control process plus the signal followup cost divided by the expected time to signal after the process is out of control.
3(c`l3 3 8
'
-o
(33)
Y is the time the process stays in an out of control state. The calculation of -o
will be presented in the next section.
3.4.3
Fraction of time a process is in control
A process can be in one of the two states: in control, or out of control. Let
j = The time until the process shifts from %I once %I is established.
o = The time the process remains in an out of control state.
28
After establishing control a process remains in control for time j; then it
switches to out of control and stays out of control for time o ; then the process is brought
back to an in-control state and remains in control for time j0; then it goes out of control
and remains out of control for time o0 ; and so forth.
If the sequence of pjq r and sequence of poq r for Q L 1, are sequences of X and Y
and are independent and identically distributed, then the long-run fraction of time the
process is in control is according to Alternating Renewal Process Theorem (Ross 1996).
%n3 -j
-j 8 -o
(34)
%l3 -o
-j 8 -o
(35)
Similarly
3.4.4
Total cost per unit time
The total cost per hour for a quality control policy is,
g(`/P3(c` -j
'
-o
6
78
!3
-j 8 -o ]ghI
-j 8 -o
'
8
$
-o
(36)
The values of ]ghI and -o are determined by the quality control policy being
adopted.
3.4.5
and using control charts
29
The average time to signal when a process is in control =]ghI ` ∗ ]^4I .
]^4I !
1
1$ ∗ -"|434 M " M 234
!cYdQ/P|% %I$
8 -"|" M 434()" s 234
(37)
-"|434 M " M 234 is the expected value of " given all the values of " (the
number of inspections until a failure) are in control and is computed as following:
-"|434 M " M 234
A@?<
t Y ∗ !" Y|434 M " M 234$
Wu?@?;
!" Y|434 M " M 234$ !" Y$
!434 M " M 234$
!434 M " M 234$ '!234$ '!434$
(38)
(39)
(40)
-"|" M 434()" s 234 is the expected value of " given the values of " (the
number of inspections until a failure) are out of control and is computed as following.
-"|" M 434()" s 234
-"|" H 434
∗
!" H 234$
!" H 434$ 8 !" L 234$
∗
!" L 234$
!" H 434$ 8 !" L 234$
8 -"|" L 234
(41)
30
-o is the average time a process remains out of control, and equals the average
time until a signal is generated when the process is out of control. -o is computed by
multiplying the average time between inspections by the average number of inspections
until a signal when the probability has shifted, from %I to %I 8 k%I .
-o|% %I 8 k%I ` ∗ ]^4
(42)
]^4 is the average number of inspections until a signal is generated if a process
is out of control. When utilizing control charts,
]^4 !
1
1$ ∗ -"|434
!hYdQ/P|%I %I 8 k%I $
M"
(43)
M 234 8 -"|" M 434()" s 234
3.4.6
Approximation
For small values of pI , the value of -"|434 M " M 234can be approximated
as 0.98 of w . As an example compare -"|434 M " M 234’s exact value and their
x
approximation value for different %I .
For %I = 0.01 when UCL =658 and LCL=0 , -"|0 M " M 658 = 99.108 and the
approximation value is 98.
For pI = 0.02 when UCL =658 and LCL=0 , EZ|0 M " M 658 = 49.998 and the
approximation value is 49.
For %I = 0.05 when UCL =129 and LCL=0 , -"|0 M " M 129 = 190.42 and the
approximation value is 196.
31
The value of average run length is computed as:
]^4 !
1
1$ ∗ -"|434 M " M 234 8 -"|" M 434()" s 234
!cYdQ/P$
Although the value of -"|" M 434()" s 234 is usually greater than
wx
, the
difference made by including this value in the computation of ]^4 is very small and the
impact of on the total cost can be ignored.
For example for %I 0.01, when UCL =658 and LCL= 0,the exact value of the
in control cost using a probability based control charts is $0.054175 and the approximate
value is $0.055355. For the same example with an out-of-control probability of0.02 the
exact value of the out of control cost is $1000 and the approximate value is also $1000.
The exact total cost is $999.978 and the approximated cost is $999.977.
If %I 0.0005, and the UCL =13212 and LCL= 2,the exact value of the in
control cost using probability based control charts is $0.006626 and the approximate
value is $0.023. For the same example with an out-of-control probability of 0.001 the
exact value of out of control cost is $1000.04 and the approximate value is $1000.041.
The approximated total cost is $997.143 and the exact value is $999.978.
The tables presented in the appendices have used the approximations above.
3.4.7
and – follow-up each failure
32
In this quality control policy each failure constitutes a signal, and the average time
to signal when a process is in control =]ghI ` ∗ ]^4I and ]^4I w . ]^4I equals
x
the average number of inspections
until a failure.
When the process is out of control the failure probability has shifted from %I to
%I 8 k%I . so -o €
wx ;wx
.
If an “infrequent” failure rate is defined as the failure probability below
which control charts are more costly than follow up of each failure, the model can
be used to identify this value (which will be denotedU∗ ). By comparing the costs
of two quality control policies as a function of the failure probability, the model
can be used to determine U∗ .
In the next section the three control charts are compared to the policy of
following up every failure for two scenarios. These are the Shewhart g-chart, gchart with probability based control limits, and the EWMA chart assuming a
known failure probability. The first scenario is monitoring hospital acquired
infection rates, and the second scenario is failures in a pacemaker manufacturing
process. Three different control charts are used for monitoring the processes and
the cost model developed is applied to examine cost as a function of failure
probability.
3.5 Scenario1- hospital acquired infections
According to a research conducted in 2007, the chance of hospital acquired
infections in the United States in 2002 was estimated at 10% ( Klevens, Monina et al
33
2007). In this scenario every patient examined is considered an inspection and every
patient that acquires a hospital related infection is considered a failure. The statistic that
is being monitored is the number of examined patients until a patient acquires an
infection (the number of inspections until failure). Under the assumptions presented
earlier the number of patient examinations until a failure follows a geometric distribution,
and the sub-group size, which is the number data points used to generate the monitored
value, is equal to one.
Since the infection rate was estimated at 10%, failure rates from 1% to10% were
considered and total cost per hour was estimated for this range.
3.5.1
Probability based control chart for hospital acquired infections
In this section probability based g-chart control limits are examined for
monitoring hospital acquired infections.
Typical 3-sigma control limits are supposed to be values such that with
probability of 0.9973, the monitored data will fall between the control limits if the system
is in control. This means the equivalent cumulative probabilities are 0.00135 for the LCL
(>?@? $ and .99865 for the UCL (>A@? $. In this case the desired type I error is 0.0027.
Using (6) and (7) the values of LCL and UCL can be computed.
For example, for %I 0.01 and the desired type I error is 0.0027, LCL = 0 and
UCL = 658. For these limits the actual type I error is less than stated due to the discrete
nature of the geometric distribution. Alternative control limits where the type I error level
is greater than stated are LCL = 1 and UCL = 657.
34
The UCL in this case is chosen as 658, can be interpreted as the case when the
signal indicates possible improvement. However, to make solid conclusions further
analysis is required. Similarly, when a point plots beyond the LCL the signal indicates a
possible reduction in the quality of the process. In this specific example since the LCL
value is 0, and a negative number of inspected patients until an infected patient is not
possible, the chart loses its ability to detect a reduction in the quality.
3.5.1.1 When the process is in control
The probability of not getting a signal when the system is in control is the
probability that a point falls between the LCL and UCL when the probability of a failure
is %I . Using formulas (10) and (11) when %I is equal to 0.01 the probability a failure does
not generate a signal is:
!‚(hYdQ/P|% 0.01$ !0 M " M 658|% 0.01$
1 !1 0.01$ƒ„… 1 !1 0.01$I 0.99864
Note that in this example control limits are chosen in a way that type I error is less
than stated.
So the probability of getting a signal when the system is in control is:
!hYdQ/P|% %I $ 1 !‚(hYdQ/P|% %I $
In this example probability of getting a signal is:
!hYdQ/P|% 0.01$ 1 0.99864 0.00136
35
This value can be interpreted as the probability the number of inspections from
the prior infected patient triggers a signal with a value that plots above the UCL or below
the LCL when the system is in control.
When the system is in control, the average number of failure/events occurring
until a signal is generated is:
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c`(cYdQ/P 1
!cYdQ/P|% %I $
So in the example when %I = 0.01 the average number of infected patients until a
signal is generated is:
]†‡)/d‡Qˆ‰_‡)(bYQb‡`‡k%/`Y‡Q`cˆQ`YP/cYdQ/P 6
1
7 737.35585
0.00135
]^4I is the average number of examined patients until an infected patient trigger
a signal. ]^4I !
1
1$ ∗ -"|434 M " M 234
!cYdQ/P|% %I $
8 -"|" M 434()" s 234
Where -"|434 M " M 234 is the expected number of inspected patients
between infections given that this number is within the control limits and -"|" M
434()" s 234 is the expected value of " given the values of " (the number of
inspections until a failure) are out of control
36
-"|434 M " M 234 A@?<
t Y ∗ !" Y|434 M " M 234$
Wu?@?;
!" Y|434 M " M 234$ !" Y$
!434 M " M 234$
!434 M " M 234$ '!234$ '!434$
-"|" M 434()" s 234
-" -"|434 M " M 234 ∗ !434 M " M 234$
/ !" H 434()" L 234$
Since Z is geometrically distributed,
!LCL M " M 234$ 1 !1 pI $Ž< !1 !1 pI $Ž $ and
W<
-"|434 M " M 234=∑A@?<
/[1 !1 pI $Ž< !1 Wu?@?; Y ∗ %I ∗ !1 %I $
!1 pI $Ž and
!" H 434()" L 234$ 1 !434 M " M 234$
Back to the example when %I = 0.01 then:
!LCL M " M 234$ 1 !1 0.99$ƒ„… !1 !0.99$I $= 0.99864
ƒ„
-"|434 M " M 234 t Y ∗ 0.01 ∗ !1 0.01$W< /0.99864 99.108
Wu
!" H 434()" L 234$ 1 !434 M " M 234$ = 0.00136
-"|" M 434()" s 234 100 99.108*0.99864]/0.00136 =756.829
37
So the ]^4I value in this example is:
]^4I 737.35585 ∗ 99.108 8 756.829 73834.7
This value can be interpreted as the number of inspected patients until an infected
patient triggers a signal when the system is in control.
3.5.1.2 When the process is out of control
In the section above, it was assumed the system is in control, the probability of a
failure is %I and all the formulas were computed accordingly. In this part of the example
it is assumed there is a shift in the failure probability from %I to %I 8 k%I . In this case k
represents the percentage of the shift. In the example the failure probability shifts from
%I 0.01 to % 0.02. In this case the probability of having an infected patient has
doubled (d = 1) and the system is out of control.
When the system is in control and has reached a steady state the UCL and LCL is
set and doesn’t change afterwards. So, when the system is out of control, the UCL and
LCL remains the same as when the system was in control. So even in this case LCL is 0
and UCL is 658. Since the LCL is equal to zero, the control chart doesn’t have the power
to detect increased failure probabilities.
The probability of not getting a signal is:
!‚(hYdQ/P|% 0.02$ !0 M " M 658|% 0.02$
1 !1 0.02$ƒ„… 1 !1 0.02$I 0.99999828
So the Probability of having a signal when the system is out of control is:
!cYdQ/P|% %I 8 k%I$ 1 !‚(hYdQ/P|% %I 8 k%I $
38
In this case the probability of getting a signal is
!cYdQ/P|% 0.02$ 1 0.99999828 0.00000172
When there is a shift in the system, the average number of failure/events
occurring until a point plots beyond control limits and triggers a signal is computed as:
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c`(cYdQ/P 1
!cYdQ/P|% %I 8 k%I $
So in the example when the shifted probability is equal to 0.02 the average
number of infected patients until a signal is:
]†‡)/d‡Qˆ‰_‡)(bYQb‡`‡k%/`Y‡Q`cˆQ`YP/cYdQ/P 6
581390.9538
1
7
0.00000172
The ]^4 is the average number of examined patients until an infected patient
trigger a signal when the process is out of control.
]^4 !
1
1$ ∗ -"|434 M "
!hYdQ/P|%I %I 8 k%I$
M 234 8 -"|" M 434()" s 234
Where -"|434 M " M 234 is the expected number of patients between
infections given that this number is within the control limits. This value can be computed
using equation (38), (39) and (40).
In the example, when shifted probability is 0.02:
39
ƒ„
-"|434 M " M 234 !t Y ∗ 0.02 ∗ !1 0.02$W< $/0.9999 49.998
Wu
-"|" M 434()" s 234 707
So the ]^4 value in this example is:
]^4 583190.9538 ∗ 49.998 8 707 29069091.91
This value can be interpreted as the number of inspection until a signal is
triggered. In this example it means when the probability of having infection is doubled it
takes on average about 28488157 inspected patients until an infected patient triggers a
signal.
3.5.1.3 Cost model for hospital acquired infections using probability distribution
based control limits
Let 3 be the cost per hour of running an out of control process. Montgomery
2007, categorizes these kinds of costs as external failure costs. External failure costs are
the costs that arise when the quality of the product doesn’t meet the requirements after it
is delivered to customers. Sub-categories of external failure costs are, complaint
adjustment, returned product, warranty charges, liability costs and other indirect costs. In
the hospital acquired infection case, these costs range from additional health care cost to
the death of a patient due to infections. Some statistics indicate the national costs of
nosocomial infections cause 8.7 million additional hospital days and death rates range
from 20,000 to 80,000 per year. (Bates et al 1997)
40
In this example it is assumed the cost per hour of running the hospital with an
increased infection probability of a specified amount is $1000 per hour.
Let ' be the cost of following up on a signal. These types of costs are categorized
as internal failure costs. The important sub-categories of internal failure costs are scrap,
rework, retest, failure analysis, downtime, yield losses and downgrading (Montgomery
2007)
In the hospital example, some of these costs can be named as additional
healthcare costs for infected patients and additional costs for searching locating and
eliminating the source of infections. In this example it is assumed the total cost per follow
up on a signal is $1000.
-jis the expected time that a process is in control after control is established. In
the hospital example it is assumed on average a process goes out of control after one
month. If it is assumed every month consist of 160 working hours, -j = 160 hours.
` is the average time between inspections. In the hospital example this can be
viewed as the arrival rate of patients. In this example it is assumed patients arrive every
15 minutes.
k 1 is the shift fraction. All of these parameters are assumed deterministic
values that are specified for the system.
The summary of the values used in the example is presented below.
41
Parameter
Value
‘ Cost of running process with a shift
1000
($/hour)
’ Cost of following up on a signal
1000
($/Signal)
“ Average time between inspections
0.25
(hours/unit)
”Mean hours to U shift after control
160
established
• Shift Fraction
1
Table 3.4: values of constant parameters in hospital acquired infection case
The total cost per hour of using a quality control policy was modeled as,
g(`/P3(c` -j
'
-o
'
6
78
!3 8
$
-j 8 -o ]ghI
-j 8 -o
-o
When the system is in control the in control cost per hour is:
3(c`n3 This can be written as:
3(c`n3 '
]ghI
'
]^4I ∗ `
The]^4I was computed as 72260.88 so the in control cost per hour is:
3(c`n3 1000
0.054175
73834.7 ∗ 0.25
42
This cost is the total cost per hour of using a g-chart with probability-based
control limits for monitoring a process when the process is in control. The low cost is
driven by the large ]^4I . If a probability based quality control chart is used for
monitoring the hospital acquired infection process, when the process is in control and the
failure probability is %I =0.01, on average a signal occurs after every 72261 patients are
inspected.
When a shift occurs and the probability of having a failure shifts from %I to
%I 8 k%I the cost of using probability based control chart for monitoring the process is:
3(c`l3 3 8
'
-o
-o is the expected time until an out of control process is brought back in to
control by using process control method and the value is computed as:
-o ` ∗ ]^4
In the hospital example when using probability based control chart the cost of
monitoring system when the shifted probability is 0.02 is computed as:
-o 0.25 ∗ 29069091.91 7267273
3(c`l3 1000 8
1000
1000
7267273
This cost is primarily driven by C. The follow-up cost per hour used to bring the
process back into control is low due to a high value of ]^4 . When the probability of
having an infection has increased from 0.01 to 0.02, it takes on average about 7122039
operating hours or 28488156.73 patients to be examined until an infected patient trigger a
signal. In this case since the LCL value is 0, the control chart doesn’t have the power to
detect increased shifts.
43
According to alternating renewal process theorem the in control percentage is:
%n3 -j
-j 8 -o
And the out of control percentage is:
%l3 -o
-j 8 -o
In this example the in control percentage using probability based control charts is
computed as:
%n3 160
0.00002016
160 8 7267273
And the out of control percentage using probability based control charts is:
%l3 7267273
0.999978
160 8 7267273
The total cost per hour of using a g-chart with probability distribution-based
control limits for monitoring hospital acquired infections (probability of infection = 0.01,
and d =1 ) is
g(`/P3(c` 0.00002016 ∗ 0.054175 8 0.999978 ∗ 1000 999.978
The cost per hour with %I The cost per hour for %I ranging from 0.01 to 0.1 is
presented in appendix1.
3.5.2
Shewhart g-chart for hospital acquired infections
44
The Shewhart g-chart is the second method analyzed for monitoring hospital
acquired infections. The g-chart is a control chart designed specifically for processes
monitored with a geometrically distributed statistic.
Using equations (5) and (6) the g-chart UCL and LCL, for hospital example can
be computed as:
234 I.I 8 3=I.I I.I 1; 398
434 1
1
1
39
6
17< 198
0.01
0.01 0.01
In this case since the LCL is negative it is replaced with 0. The calculations for
the cost model follow the same steps as the prior section, but with different control limits.
3.5.2.1 When the Process is in control
When the process is in control:
!‚(hYdQ/P|% 0.01$ !0 M " M 398|% 0.01$
1 !1 0.01$–—… 1 !1 0.01$I 0.9815
!hYdQ/P|% %I $ 1 !‚(hYdQ/P|% %I $
!hYdQ/P|% 0.01$ 1 0.9815 0.0185
45
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!YQb‡`‡k%/`Y‡Q`c$`(cYdQ/P 1
6
7 54.05399
0.0185
1
!hYdQ/P|% %I$
]^4I 54.0534 ∗ 91.1483 8 501 5427.929
This value can be interpreted as the number of inspected patients until an infected
patient triggers a signal when the system is in control.
3.5.2.2 When the process is out of control
When the process is out of control: !‚(hYdQ/P|% 0.02$ !0 M " M
398|% 0.02$
1 !1 0.02$–—… 1 !1 0.02$I 0.99967
!cYdQ/P|% 0.02$ 1 0.99967 0.00033
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!YQb‡`‡k%/`Y‡Q`c$`(cYdQ/P
1
1
6
7 3042.62164
!cYdQ/P|% %I 8 k%I $
0.00033
]^4 3042.62164 ∗ 49.88 8 450 152204.4055
This value can be interpreted as the number of inspection until a signal is
triggered. In this example it means when the probability of having infection is doubled it
takes about 149089 examined patients until an infected patient trigger a signal.
3.5.2.3 Cost model for hospital acquired infections using g-chart
The in control cost can be computed as:
46
3(c`n3 1000
0.737
5427.93 ∗ 0.25
This cost is the total cost per hour of using g-chart for monitoring process when
the process is in control.
When the process is out of control the cost per hour of using Shewhart g-control
chart for monitoring the process is:
-o 0.25 ∗ 152204.5 38051.1
3(c`l3 1000 8
1000
1000.26
38051.1
Using equation (34) in this example the in control percentage using g-chart is
computed as:
%n3 160
0.0042
160 8 38051.1
And using equation (35) the out of control percentage is:
%l3 38051.1
0.996
160 8 38051.1
Using information above the total cost per hour using g-chart for monitoring
hospital acquired infections when the probability of having an infection is 0.01, and the
shifted probability is 0.02 is:
g(`/P3(c` 0.0042 ∗ 0.74 8 0.996 ∗ 1000.26 995.842
The same procedure is repeated for %I for a range of 0.01 to 0.1 and the results
are presented in appendix 2.
47
3.5.3
EWMA chart for hospital acquired infections
EWMA is the third method used for monitoring hospital infection processes.
234 1
1 0.01
0.2
9!
8 39
$; 199
0
!0.01$
0.01
2 0.2
1
1 0.01
0.2
9!
434 39
$< 1
0
!0.01$
2 0.2
0.01
In this case the UCL is 199, which means if the number of examined patients until
an infected patient is greater than 199, a signal is triggered and is an indication of
possible improvement in the quality of the process. However, to make a solid conclusion
further analysis is required. In this example the LCL is one, which is still very low.
3.5.3.1 When the Process is in control
When the process is in control:
!‚(hYdQ/P|% 0.01$ !1 M " M 199|% 0.01$
1 !1 0.01$—˜ 1 !1 0.01$ 0.85329
!hYdQ/P|% %I $ 1 !‚(hYdQ/P|% %I $
!hYdQ/P|% 0.01$ 1 0.85329 0.1467
48
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!YQb‡`‡k%/`Y‡Q`c$`(cYdQ/P 1
6.81663
0.1467
1
!hYdQ/P|% %I$
]^4I 692.53
This value can be interpreted as the number of inspected patients until an infected
patient triggers a signal when the system is in control.
3.5.3.2 When the process is out of control
When the process is out of control:
!‚(hYdQ/P|% 0.02$ !1 M " M 199$
1 !1 0.02$—˜ 1 !1 0.02$ 0.96169
!cYdQ/P|% 0.02$ 1 0.96169 0.03831
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!YQb‡`‡k%/`Y‡Q`c$`(cYdQ/P
1
1
6
7 26.1007
!cYdQ/P|% %I 8 k%I $
0.03831
]^4 1354.920267
This value can be interpreted as the number of inspection until a signal is
triggered. In this example it means when the probability of having infection is doubled, it
takes about 1279 examined patients until an infected patient trigger a signal.
49
3.5.3.3 Cost model for hospital acquired infections using EWMA
The in control cost can be computed as:
3(c`n3 1000
5.78
692.53 ∗ 0.25
This cost is the total cost per hour of using EWMA chart for monitoring in control
process.
When the process is out of control the cost per hour of using EWMA chart for
monitoring the process is:
-o 0.25 ∗ 1354.92 338.73
3(c`l3 1000 8
1000
1002.95
338.73
Using equation (34) in this example the in control percentage using g-chart is
computed as:
%n3 160
0.321
160 8 338.7301
And using equation (35) the out of control percentage is:
%l3 338.7301
0.6797
160 8 338.7301
Using information above the total cost per hour using EWMA for monitoring
hospital acquired infections when the probability of having an infection is 0.01, and the
shifted probability is 0.02 is:
g(`/P3(c` 0.321 ∗ 5.78 8 0.6797 ∗ 1002.95 683.04
50
The same procedure is repeated for %I from a range of 0.01 to 0.1 and the net
cost of using EWMA chart is estimated. The table is presented in appendix3.
3.5.4
Following up on every failure policy
3.5.4.1 When process is in control:
]ghI ` ∗ ]^4I 0.25 ∗
]^4I 1
%I
3(c`n3 š›œ ™
x
1
25
0.01
III
0„
40
3.5.4.2 When the process is out of control
I.0„
I.I0
The failure probability has shifted from %I to %I 8 k%I . so -o w
12.5
€
x ;wx
3(c`l3 3 8
'
1000
1000 8
1080
-o
12.5
g(`/P3(c` 0.92754 ∗ 40 8 0.07246 ∗ 1080 115.36232
51
3.6 Scenario2- pace maker manufacturing
A pacemaker is composed of numbers of components that are assembled in a
carefully monitored assembly process. Pacemaker failures are mostly from specific
component failures. Since pace makers are very critical products, , all the components are
tested before final assembly to minimize the chance of pacemaker failure due to
component failures. Because of this the failure probability of a component in final
assembly is very small. In order to simplify the terminology, each component will be
referred to as a “unit”.
Units are inspect individually so the number of successful units until a failure
follows a geometric distribution. The statistic being monitored is the number of good
units until a defective unit is observed.
The author of this thesis worked as an intern in a pacemaker company, and the
data being used is representative of this and similar production environments. The
average rate of unit failure in production is 1 in every 2000 units, which means the
probability of failure is about 0.0005 !%I 0.0005).
Since the failure probability was estimated at 0.0005, a range of failures from
0.0005 to 0.005 was considered and total cost was estimated for this range.
3.6.1
Probability based control chart for pace maker manufacturing
In this section a g-chart with probability based limits is developed for monitoring
unit failure in a pacemaker manufacturing company assuming the probability of having a
defective unit is 0.0005.
The LCL and UCL value is computed as 2 and 13212 respectively.
52
3.6.1.1 When the Process is in control
When the process is in control:
!‚(hYdQ/P|% 0.0005$ 0.99765
!hYdQ/P|% 0.0005$ 1 0.99765 0.00235
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!k‡b‡`Y†‡ˆQY`c$ˆQ`YPcYdQ/P 6
425.45524
1
7
0.00235
]^4I 3018322
This value can be interpreted as the number of inspected units until a defective
unit triggers a signal when the system is in control.
3.6.1.2 When the process is out of control
When the process is out of control:
!‚(hYdQ/P|% 0.001$ 0.99799
!hYdQ/P|% 0.001$ 1 0.99799 0.002
1
]†‡)/d‡Qˆ‰_‡)(b!b/YPˆ)‡c$k‡b‡`Y†‡cˆQ`YPcYdQ/P 6
7 499.79551
0.002
]^4 500311.957
This value can be interpreted as the number of inspection until a signal is
triggered. In this example it means when the probability of having a defective unit is
doubled it takes about 500312 defective units until a defective unit triggers a signal.
53
3.6.1.3 Cost model for pacemaker manufacturing using probability based control
chart
Let 3 be the cost per hour of running an out of control process. In the pacemaker
manufacturing example, the cost of running a process with a shift can be ranged from
scrapping and reworking more pacemakers, which can cost about $10,000 per pacemaker,
to delivering pacemakers that are defective or a providing a pacemaker with a life time
shorter than the specified life time and potentially causing death.
In this scenario it is assumed the total cost per hour of a running pacemaker
manufacturing process out of control is about $1000 per hour.
Let, ' be the cost of following up on a signal. This cost includes, the cost of
searching and testing for the root cause of changes plus the downtime, if the production is
stopped during this period. Additionally, this cost includes the cost of eliminating the
source of variation. In this example it is assumed the average cost of following up on a
signal is $1000.
Let,-j be the expected time that a process is in control after in control is
established. In this example it is assumed on average, a pacemaker manufacturing process
goes out of control every two months. If it is assumed every month consists of 160 hours,
then on average after 320 hours the process goes out of control.
54
Let ` be the average time between inspections. In this example, since every unit is
inspected, 1/t is the rate of assembling units. On average every 3 minutes, a unit is
assembled so the average time between inspection about 0.05 hours.
Let k be shift fraction from the actual failure probability.
The summary of values used in the example is presented in the table below.
Parameters
‘ Cost of running process with a shift ($/hour)
’ Cost of following up on a signal ($/Signal)
“ Average time between inspections (hours/unit)
”Mean hours to U shift after control
Estimated value
1000
1000
0.05
320
established
• Shift Fraction
1
Table 3.5: Estimated values of constant parameters in pacemaker manufacturing process
Using the information in table 3.5 and computed values of ]^4I the in
control cost is:
3(c`n3 0.006626
Similarly the out of control cost can be computed using equation (23).
3(c`l3 1000.04
Using equations (34) and (35) in control and out of control percentages can be
computed.
55
%n3 0.012
And the out of control percentage using probability based control charts is:
%l3 0.98
Using information above the total cost per hour, using probability based control
chart for monitoring defective units in pacemaker manufacturing process, when the
probability of having an infection is 0.0005, and the shifted probability is 0.001can be
computed as:
g(`/P3(c` 0.012 ∗ 0.006626 8 0.9838 ∗ 1000.04 987.41
The same procedure is repeated for %I from a range of 0.0005 to 0.005 and the
net cost of using probability based control chart is estimated. The table is presented in
appendix4.
3.6.2
Shewhart g-chart for pacemaker manufacturing process
In this section g-chart is used as an alternative control chart for monitoring
pacemaker manufacturing process. The UCL and LCL is 7998 and -3998 respectively. In
this case the LCL is a negative value and since negative LCL is meaningless in this
context it is replaced with zero.
3.6.2.1 When the process is in control
When the process is in control:
56
!‚(hYdQ/P|% 0.0005$ 0.98168
!hYdQ/P|% 0.0005$ 1 0.98168 0.01832
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!k‡b‡`Y†‡ˆQY`c$ˆQ`YPcYdQ/P 6
54.57086
1
7
0.01832
]^4I 363953
The ]^4I is the number of inspected units until a defective unit trigger signal.
3.6.2.2 When the process is out of control
When the process is out of control:
!‚(hYdQ/P|% 0.001$ 0.99966
!cYdQ/P|% 0.001$ 1 0.99966 0.00034
]†)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!k‡b‡`Y†‡ˆQY`c$ˆQ`YPcYdQ/P 6
2983.94392
1
7
0.00034
]^4 2984910.11
3.6.2.3 Cost model for pacemaker manufacturing process using g-chart
The in control cost can be computed as:
3(c`n3 0.055
3(c`l3 1000.007
%n3 0.0021395%l3 0.997
Using information above the total cost per hour using g-chart for monitoring
pacemaker manufacturing process is:
57
g(`/P3(c` 0.0021 ∗ 0.055 8 0.997 ∗ 1000.007 997.87
The same procedure is repeated for %I from a range of 0.0005 to 0.005 and the
net cost of using g control chart for monitoring the system is estimated. The table is
presented in appendix5.
3.6.3
EWMA chart for pacemaker manufacturing
In the pacemaker manufacturing when using EWMA for monitoring the system.
In this example the UCL and LCL values are 3999 and 1 respectively. Although the LCL
is not zero but still is very low and the chart can’t capture the increased shift in the
process.
3.6.3.1 When the process is in control
When the process is in control:
!‚(hYdQ/P|% 0.0005$ 0.86409
!hYdQ/P|% 0.0005$ 1 0.86409 0.1359
]†‡)/d‡Qˆ‰_‡)(bb/YPˆ)‡c!k‡b‡`Y†‡ˆQY`c$ˆQ`YPcYdQ/P ]^4I 37175.15
3.6.3.2 When the process is out of control
When the process is out of control:
!‚(hYdQ/P|% 0.001$ 0.98068
1
7.35819
0.1359
58
!cYdQ/P|% 0.001$ 1 0.96169 0.019316
]†‡)/d‡Qˆ‰_‡)(bk‡b‡`Y†‡ˆQY`cˆQ`YPcYdQ/P 6
]^4 52891.99
1
7 51.77154
0.019316
3.6.3.3 Cost model for pacemaker manufacturing using EWMA
The total cost per hour of using EWMA is computed as:
3(c`n3 0.538
3(c`l3 1000.38
%n3 0.108
%l3 0.892
g(`/P3(c` 0.1080816 ∗ 0.538 8 0.892 ∗ 1000.38 892.46
The same procedure is repeated for %I from a range of 0.0005 to 0.005 and the
cost of using EWMA chart is estimated. The table is presented in appendix6.
3.6.4
Following up on every failure
g(`/P3(c` 0.86486 ∗ 10 8 0.135135 ∗ 1020 146.48649
59
4
Results
In the methodology section, two infrequent failure scenarios were analyzed. Three
different control chart methods used to monitor the processes were analyzed with the cost
model developed. A step by step computation of the total cost per hour of using each of
the specific control charts was explained. In addition the total cost per hour of following
up on every failure was computed.
In this section the total cost per hour of using a specific type of control chart
(cost1), is compared to the total cost per hour of following up on every failure (cost2).
4.1 Comparing probability limit based g-chart with follow up of every
failure - hospital acquired infections
Figure 2 is a comparison of cost1 for a g-chart with probability distribution based
control limits to cost2 for %I ranging from 0.01 to 0.1 with d = 1.
60
Figure 2 :Comparing total cost per hour of using probabilit based control charts vs. following up on every
failure when d=1 and C/F = 1
The research question that was posed earlier was at what frequency of failure does
statistical quality control result in a lower average cost per hour than follow up of every
failure?
From Figure 2 it can be concluded that for the specified range and parameter
values used following up on every infected patient is more cost efficient.
When using a g-chart with probability based control limits and the process is in
control, the probability of a failure generating a signal is small, which results in large
values of ]^4I . When ]^4I is large it means the average number of inspected units until
a signal is large. This leads to less following up on infected patients which results in low
in-control cost.
However, in the out of control situation with % s %I the zero LCL value results
in a high out of control cost because the probability that a failure generates a signal goes
61
down. When the LCL is zero, the control chart loses its ability to detect an increased
failure probability which results in an increased ]^4 . This results in an increased
expected time until an out of control process is brought back in to control (-o). The
high value of -o results in a low in-control percentage and increases the out of control
percentage which leads to an increased cost per hour of using control charts when the
process is out of control.
The in control cost for following up on every failure is higher than when using
probability based control charts, because every infected patients is a signal. However, the
out of control cost is much smaller relative to the control chart since every infected
patient is a signal and the process is brought back to control much sooner. -o which is
the expected amount of time to bring an out of control process back in control is smaller.
Since -o is smaller, the in control percentage is larger.
Since the model parameters are not always known exactly, the model can be used
to conduct sensitivity analysis to examine the robustness of the prior conclusion for the
particular scenario considered, and other similar scenarios in other application contexts.
The conclusions are insensitive to the shift fraction d. When varying the shift
fraction between the 0.01 to 0.1 following up of every failure remains more efficient.
@
™
Sensitivity to the t ratio can also be analyzed. This ratio is a ratio of cost per unit
time over a cost per follow up so the exact units has no relevant interpretation, however it
is a measure of out-of-control cost, to the cost of bring a process back into control. If the
average time it takes to conduct failure follow-up is known, a dimensionless ratio can be
formed. For the specified %I range, when ™ 1, follow-up of every failure is less costly,
@
62
however as gets smaller, %I∗ also gets smaller. When = 0.4, %I∗ = 0.1 as seen in Figure
@
@
™
™
3,
Figure 3: Comparing total cost per hour of using probability based control chart vs. following up on every
failure when d = 1 and C/F = 0.4
When
@
™
= 0.05, %I∗ 0.01. These results imply that for a process with a failure
probability in the range of 0.01 to 0.1, the cost of follow up must be relatively large
before a g-chart should be used.
Some additional figures with different shift fractions and different cost ratio
values of
@
™
are presented in appendix 7.
4.2 Comparing Shewhart g-chart with follow up of every failure - hospital
infections
63
In the hospital acquired infection case, the total cost per hour was also computed
when using Shewhart g-chart for monitoring the infection process. Figures 4.3 shows a
charts that compares the total cost per hour of using a Shewhart g-chart versus following
up on every failure for %I in the range of 0.01 to 0.1.
The sensitivity analysis results are similar to those obtained for the g-chart with
probability based control limits. However, for Shewhart g-charts cost1 decreases as %I
increases, but the effect of this on the overall cost per hour is negligible.
The reason that cost1 is decreasing is because as %I is increasing the UCL limit is
decreasing and therefore more points plot above the UCL. In this case ]^4 decreases and
thus the total cost decreases.
Figure 4: Comparing total cost per hour of using g-chart vs. following up on every failure when d = 1and
C/F =1
Additional figures of this example with different shift fraction and different cost
ratio values of
@
™
are presented in appendix 7.
64
4.3 Comparing EWMA with follow up of every failure - hospital acquired
infections
In the hospital acquired infection case, the total cost per hour of using an EWMA
control for monitoring the system quality was also computed. The chart below shows the
total cost per hour of using EWMA versus following up on every failure for %I in the
range of 0.01 to 0.1.
Figure 5: Comparing total cost per hour of using EWMA vs. following up in every failure when d = 1 and
C/F = 1
With the specified parameters, when %I is between 0.001 and 0.04, it is more cost
efficient to follow up on every infected patient, but when %I is greater than 0.04 it is
more cost efficient to use an EWMA control chart for monitoring the process.
In the EWMA chart, when the process is in control, the probability of having an
out of control signal is small. This results in large values of ]^4I . When ]^4I is large it
65
means the average number of examined patients until an infected patient triggers a signal,
is large. This leads to less following up on infected patients when the process is in
control, which results in lower in control cost.
The main feature of the EWMA charts that differs from the two other control
charts is the low value of the UCL. When the process is out of control, some infected
patients generate a signal because the number of inspections since the prior failure
exceeds the UCL . The probability of getting a signal, when the process is out of control
is higher than the two other methods resulting in a smaller ]^4 . This results in smaller
-Y , and the lower value of EY increases the in control percentage of the process, and
decreases the overall cost per hour of using EWMA control charts for monitoring the
process.
Note that, when the process is out of control the control chart signals are not
necessarily because of the right reason. When the failure probability increases ideally
points plotted below the LCL will trigger a signal, but because the LCL is very small this
doesn’t happen. The primary reason signals are generated is because of the low UCL
value. These signals indicate possible quality improvements, which is not the case.
There is a negative association between shift fraction and%I∗ . As shift fraction
increases the %I∗ get smaller and vice versa. . In addition, it can be concluded that there
is a positive association between ™ and %I∗ . As ™ gets smaller, the %I∗ also gets smaller.
@
@
When the cost ratio ( ™ ) gets as small as 0.05, %I∗ occurs at 0.01. The sensitivity
@
analysis results are similar to those obtained for the g-charts.
66
Figure 6: Comparing total cost per hour of using EWMA vs. following up on every failure when d = 1 and
C/F = 0.05
Figure 7: Comparing total cost per hour of using EWMA vs. following up on every failure when d = 1 and
C/F = 20
67
In the %I range of 0.01 to 0.1, EWMA charts are a viable method that may be
more cost effective than a policy of following up every failure. A more detailed estimate
of model parameters is required.
Some Additional figures of this example with different shift fraction and different
cost ratio values of
@
™
is presented in the appedix7.
4.4 Comparing probability limit based Shewhart g-chart with follow up of
every failure -pacemaker manufacturing case
The analysis presented was conducted for g-charts with probability distributionbased control limits, but the conclusions are also applicable to the use of Shewhart gcharts, which produced very similar results. Chart 8 compares, cost1, for probability
based control chart with cost2 , for %I ranging from 0.0005 to 0.005.
Figure 8: Comparing total cost per hour of using probability based control chart vs. following up on every
failure whe d = 1 and C/F = 1
68
From the figure 8 it can be concluded that following up on every defective unit is
a more cost-efficient strategy, when the probability of having a defective unit is from
0.0005 to 0.005.
When gets as small as 0.1, %I∗ 0.0048. The cost of failure follow up must be
@
™
relatively high to justify the use of a g-chart. Results are shown in figure 9.
Figure 9: Comparing total cost per hour of using g-chart vs. following up on every failure when d = 1 and
C/F = 0.1
In most cases, the costs of running an out-of-control process such as in the
pacemaker production scenario are so high (with many difficult to quantify costs) that gcharts should rarely be used.
Some Additional graphs of this example with different shift fractions and
@
different values of cost ratio ™ is presented in the appedix8.
69
4.5 Comparing EWMA with following up of every failure pacemaker
manufacturing
The figure 10 compares the total cost1 for using EWMA chart versus cost2 for %I
ranging from 0.0005 to 0.005.
Figure 10: Comapring total cost per hour of using EWMA vs. following up on every failure when d = 1 and
C/F = 1
For %I ranging from 0.0005 to 0.005, it is more cost efficient to follow up on
every defective unit with the cost parameters used.
When
@
™
= 0.25, %I∗ 0.0048. Results are shown in figure 11.
70
Figure 11: Comparing total cost per hour of using EWMA vs. following up on every failure when d = 1 and
C/F = 0.25
Some Additional figures with different shift fraction and different cost ratio
values of
@
™
is presented in the appedix8.
71
5
Conclusion
The mathematical model developed in this research is a model for analyzing and
comparing the cost of utilizing a statistical control chart method for monitoring a
process. The model was employed to analyze the use of g-charts in a scenario (hospital
example) with relatively large failure probabilities (but still “infrequent”), and a relatively
low inspection rate of units (infection rate monitoring), and a scenario with much smaller
failure probabilities and higher inspection rate of units (pacemaker assembly). The
monitored statistic in both scenarios was geometrically distributed, for which g-charts
were developed. The scenario results as well as sensitivity analysis show that when
compared to a policy of following up every failure, g-charts should only be used when
the cost of failure follow up is very high. In most scenarios where low defect rates are
monitored carefully this will not be the case. This is a robust conclusion and the model
requires extreme scenarios to show a lower cost for g-charts.
In contrast the use of EWMA did not require extreme scenarios, except with very
low failure probabilities, to generate a lower cost. The main difference between the gcharts and the EWMA charts are the width of the control limits. This indicates that the
model developed may be useful to set specific control limits that minimize the long-run
cost of failure rate monitoring. This is a topic for future research.
72
6
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75
Appendix1
Process In-Control
p0
LCL
UCL
Prob No Signal
Prob Signal
Events to Signal
E[X|LCL<X<UCL]
ARL0
0.01
0
658
0.998644
0.00136
737.36
98
72260.88
0.011
0
598
0.998644
0.00136
737.59
89.09090909
65712.4
0.012
0
548
0.998645
0.00136
737.82
81.66666667
60255.42
0.013
0
505
0.998633
0.00137
731.40
75.38461538
55136.32
0.014
0
469
0.998637
0.00136
733.84
70
51369.09
0.015
0
438
0.998646
0.00135
738.53
65.33333333
48250.53
0.016
0
410
0.998635
0.00136
732.83
61.25
44885.99
0.017
0
386
0.998641
0.00136
736.03
57.64705882
42429.98
0.018
0
364
0.998631
0.00137
730.35
54.44444444
39763.4
0.019
0
345
0.998638
0.00136
734.28
51.57894737
37873.37
0.02
0
328
0.998648
0.00135
739.73
49
36246.77
0.021
0
312
0.998640
0.00136
735.50
46.66666667
34323.35
0.022
0
298
0.998649
0.00135
740.22
44.54545455
32973.38
0.023
0
284
0.998619
0.00138
724.17
42.60869565
30855.79
0.024
0
273
0.998650
0.00135
740.71
40.83333333
30245.75
0.025
0
261
0.998616
0.00138
722.44
39.2
28319.53
0.026
0
251
0.998620
0.00138
724.87
37.69230769
27322.06
0.027
0
242
0.998635
0.00137
732.50
36.2962963
26586.91
0.028
0
233
0.998624
0.00138
726.82
35
25438.68
0.029
0
225
0.998629
0.00137
729.28
33.79310345
24644.53
0.03
0
217
0.998611
0.00139
719.96
32.66666667
23518.53
0.031
0
210
0.998614
0.00139
721.66
31.61290323
22813.66
0.032
0
204
0.998643
0.00136
736.72
30.625
22562.05
0.033
0
197
0.998608
0.00139
718.47
29.6969697
21336.5
0.034
0
192
0.998649
0.00135
740.23
28.82352941
21336.15
0.035
0
186
0.998627
0.00137
728.53
28
20398.82
0.036
0
181
0.998639
0.00136
734.74
27.22222222
20001.27
0.037
0
176
0.998637
0.00136
733.50
26.48648649
19427.82
0.038
0
171
0.998620
0.00138
724.83
25.78947368
18693.05
0.039
0
167
0.998645
0.00136
737.76
25.12820513
18538.69
0.04
0
162
0.998601
0.00140
715.04
24.5
17518.53
0.041
0
158
0.998602
0.00140
715.28
23.90243902
17097.04
0.042
0
154
0.998591
0.00141
709.70
23.33333333
16559.75
0.043
0
151
0.998630
0.00137
729.81
22.79069767
16632.86
0.044
0
147
0.998598
0.00140
713.10
22.27272727
15882.58
76
0.045
0
144
0.998618
0.00138
723.63
21.77777778
15759.1
0.046
0
141
0.998630
0.00137
729.84
21.30434783
15548.76
0.047
0
138
0.998633
0.00137
731.60
20.85106383
15254.6
0.048
0
135
0.998628
0.00137
728.87
20.41666667
14881.06
0.049
0
132
0.998614
0.00139
721.69
20
14433.85
0.05
0
129
0.998592
0.00141
710.20
19.6
13919.85
0.051
0
127
0.998634
0.00137
731.91
19.21568627
14064.14
0.052
0
124
0.998596
0.00140
712.15
18.84615385
13421.38
0.053
0
122
0.998625
0.00138
727.20
18.49056604
13446.3
0.054
0
120
0.998648
0.00135
739.53
18.14814815
13421.07
0.055
0
117
0.998587
0.00141
707.80
17.81818182
12611.7
0.056
0
115
0.998598
0.00140
713.17
17.5
12480.46
0.057
0
113
0.998603
0.00140
715.63
17.19298246
12303.87
0.058
0
111
0.998602
0.00140
715.16
16.89655172
12083.67
0.059
0
109
0.998595
0.00141
711.74
16.61016949
11822.1
0.06
0
107
0.998582
0.00142
705.42
16.33333333
11521.85
0.061
0
105
0.998564
0.00144
696.27
16.06557377
11186
0.062
0
104
0.998629
0.00137
729.64
15.80645161
11532.99
0.063
0
102
0.998601
0.00140
714.99
15.55555556
11122.1
0.064
0
100
0.998567
0.00143
697.74
15.3125
10684.08
0.065
0
99
0.998621
0.00138
725.21
15.07692308
10933.89
0.066
0
97
0.998577
0.00142
702.59
14.84848485
10432.35
0.067
0
96
0.998624
0.00138
726.51
14.62686567
10626.62
0.068
0
94
0.998569
0.00143
698.75
14.41176471
10070.18
0.069
0
93
0.998609
0.00139
718.84
14.20289855
10209.54
0.07
0
92
0.998645
0.00136
737.99
14
10331.89
0.071
0
90
0.998576
0.00142
702.43
13.8028169
9695.477
0.072
0
89
0.998606
0.00139
717.43
13.61111111
9764.987
0.073
0
88
0.998632
0.00137
731.25
13.42465753
9816.719
0.074
0
86
0.998548
0.00145
688.76
13.24324324
9121.379
0.075
0
85
0.998568
0.00143
698.38
13.06666667
9125.554
0.076
0
84
0.998585
0.00142
706.69
12.89473684
9112.528
0.077
0
83
0.998599
0.00140
713.61
12.72727273
9082.3
0.078
0
82
0.998609
0.00139
719.11
12.56410256
9034.973
0.079
0
81
0.998617
0.00138
723.15
12.40506329
8970.746
0.08
0
80
0.998622
0.00138
725.71
12.25
8889.917
0.081
0
79
0.998624
0.00138
726.76
12.09876543
8792.88
0.082
0
78
0.998623
0.00138
726.30
11.95121951
8680.121
0.083
0
77
0.998619
0.00138
724.32
11.80722892
8552.212
0.084
0
76
0.998613
0.00139
720.84
11.66666667
8409.812
77
0.085
0
75
0.998603
0.00140
715.88
11.52941176
8253.653
0.086
0
74
0.998590
0.00141
709.46
11.39534884
8084.54
0.087
0
73
0.998575
0.00143
701.62
11.26436782
7903.343
0.088
0
72
0.998556
0.00144
692.42
11.13636364
7710.985
0.089
0
71
0.998533
0.00147
681.89
11.01123596
7508.439
0.09
0
71
0.998642
0.00136
736.38
10.88888889
8018.37
0.091
0
70
0.998617
0.00138
722.92
10.76923077
7785.325
0.092
0
69
0.998588
0.00141
708.21
10.65217391
7543.98
0.093
0
68
0.998556
0.00144
692.33
10.53763441
7295.497
0.094
0
67
0.998519
0.00148
675.37
10.42553191
7041.046
0.095
0
67
0.998623
0.00138
726.43
10.31578947
7493.699
0.096
0
66
0.998584
0.00142
706.40
10.20833333
7211.189
0.097
0
65
0.998541
0.00146
685.46
10.10309278
6925.296
0.098
0
65
0.998641
0.00136
735.84
10
7358.37
0.099
0
64
0.998595
0.00140
711.77
9.898989899
7045.78
0.1
0
63
0.998544
0.00146
687.01
9.8
6732.719
Process Out of Control
Prob
Shifted Prob
Prob Signal
0.02
Prob No
signal
0.9999983
E[X|LCL<X<UCL]
ARL1
1.72001E-06
Events to
Signal
581390.9538
0.01
49
28488156.73
0.011
0.022
0.9999983
1.70725E-06
585738.3936
44.54545455
26091982.99
0.012
0.024
0.9999983
1.69453E-06
590132.7764
40.83333333
24097088.37
0.013
0.026
0.9999983
1.71283E-06
583829.0937
37.69230769
22005865.84
0.014
0.028
0.9999983
1.68971E-06
591817.7881
35
20713622.58
0.015
0.03
0.9999983
1.65672E-06
603604.023
32.66666667
19717731.42
0.016
0.032
0.9999983
1.67117E-06
598382.4683
30.625
18325463.09
0.017
0.034
0.9999984
1.64511E-06
607863.8691
28.82352941
17520782.11
0.018
0.036
0.9999983
1.65945E-06
602610.4981
27.22222222
16404396.89
0.019
0.038
0.9999984
1.63014E-06
613444.418
25.78947368
15820408.68
0.02
0.04
0.9999984
1.59475E-06
627056.6761
24.5
15362888.57
0.021
0.042
0.9999984
1.60204E-06
624204.1857
23.33333333
14564764.33
0.022
0.044
0.9999984
1.57034E-06
636804.3098
22.27272727
14183368.72
0.023
0.046
0.9999984
1.63001E-06
613493.4901
21.30434783
13070078.7
0.024
0.048
0.9999985
1.54614E-06
646770.2564
20.41666667
13204892.74
0.025
0.05
0.9999984
1.61487E-06
619245.1658
19.6
12137205.25
0.026
0.052
0.9999984
1.59252E-06
627936.4096
18.84615385
11834186.18
0.027
0.054
0.9999985
1.54797E-06
646005.3686
18.14814815
11723801.13
78
0.028
0.056
0.9999984
1.56136E-06
640467.8335
17.5
11208187.09
0.029
0.058
0.9999985
1.53958E-06
649527.8458
16.89655172
10974780.84
0.03
0.06
0.9999984
1.56898E-06
637358.2658
16.33333333
10410185.01
0.031
0.062
0.9999984
1.55022E-06
645069.9019
15.80645161
10196266.19
0.032
0.064
0.9999985
1.4757E-06
677645.7301
15.3125
10376450.24
0.033
0.066
0.9999985
1.54166E-06
648652.7429
14.84848485
9631510.425
0.034
0.068
0.9999986
1.44025E-06
694321.7027
14.41176471
10006401.01
0.035
0.07
0.9999985
1.47688E-06
677102.6563
14
9479437.188
0.036
0.072
0.9999986
1.44091E-06
694007.6866
13.61111111
9446215.734
0.037
0.074
0.9999986
1.43523E-06
696750.5668
13.24324324
9227237.236
0.038
0.076
0.9999985
1.4596E-06
685117.2345
12.89473684
8834406.445
0.039
0.078
0.9999986
1.39744E-06
715594.6832
12.56410256
8990804.994
0.04
0.08
0.9999985
1.47856E-06
676331.7839
12.25
8285064.353
0.041
0.082
0.9999985
1.46656E-06
681866.0247
11.95121951
8149130.539
0.042
0.084
0.9999985
1.47913E-06
676071.3448
11.66666667
7887499.023
0.043
0.086
0.9999986
1.38653E-06
721224.78
11.39534884
8218607.959
0.044
0.088
0.9999986
1.44293E-06
693032.5041
11.13636364
7717861.978
0.045
0.09
0.9999986
1.38969E-06
719583.0056
10.88888889
7835459.395
0.046
0.092
0.9999986
1.35525E-06
737872.2999
10.65217391
7859944.065
0.047
0.094
0.9999987
1.33833E-06
747200.0947
10.42553191
7789958.434
0.048
0.096
0.9999987
1.33835E-06
747188.7361
10.20833333
7627551.681
0.049
0.098
0.9999986
1.35537E-06
737808.3772
10
7378083.772
0.05
0.1
0.9999986
1.39008E-06
719380.716
9.8
7049931.016
0.051
0.102
0.9999987
1.29664E-06
771225.8537
9.607843137
7409817.025
0.052
0.104
0.9999986
1.36108E-06
734708.0724
9.423076923
6923210.682
0.053
0.106
0.9999987
1.29372E-06
772965.7053
9.245283019
7146286.709
0.054
0.108
0.9999988
1.23999E-06
806459.0674
9.074074074
7317869.315
0.055
0.11
0.9999987
1.34661E-06
742607.3257
8.909090909
6615956.175
0.056
0.112
0.9999987
1.31546E-06
760189.5605
8.75
6651658.654
0.057
0.114
0.9999987
1.29592E-06
771654.954
8.596491228
6633525.043
0.058
0.116
0.9999987
1.28751E-06
776695.6066
8.448275862
6561738.745
0.059
0.118
0.9999987
1.29005E-06
775162.5911
8.305084746
6437791.011
0.06
0.12
0.9999987
1.30366E-06
767073.0983
8.166666667
6264430.303
0.061
0.122
0.9999987
1.32871E-06
752610.626
8.032786885
6045560.766
0.062
0.124
0.9999988
1.19653E-06
835751.3074
7.903225806
6605131.301
0.063
0.126
0.9999988
1.23781E-06
807877.1113
7.777777778
6283488.644
0.064
0.128
0.9999987
1.29161E-06
774226.4278
7.65625
5927671.088
0.065
0.13
0.9999988
1.18273E-06
845499.2825
7.538461538
6373763.822
0.066
0.132
0.9999987
1.25283E-06
798190.1848
7.424242424
5925957.433
0.067
0.134
0.9999988
1.15931E-06
862581.7361
7.313432836
6308433.592
79
0.068
0.136
0.9999988
1.24675E-06
802087.8717
7.205882353
5779750.84
0.069
0.138
0.9999988
1.16592E-06
857692.3651
7.101449275
6090858.824
0.07
0.14
0.9999989
1.09486E-06
913357.7892
7
6393504.525
0.071
0.142
0.9999988
1.20329E-06
831058.0899
6.901408451
5735471.324
0.072
0.144
0.9999989
1.14207E-06
875605.9588
6.805555556
5958984.997
0.073
0.146
0.9999989
1.08852E-06
918682.0907
6.712328767
6166496.225
0.074
0.148
0.9999988
1.22283E-06
817776.1582
6.621621622
5415004.291
0.075
0.15
0.9999988
1.17812E-06
848809.5164
6.533333333
5545555.507
0.076
0.152
0.9999989
1.13987E-06
877292.1892
6.447368421
5656225.957
0.077
0.154
0.9999989
1.10757E-06
902880.4114
6.363636364
5745602.618
0.078
0.156
0.9999989
1.08078E-06
925255.8152
6.282051282
5812504.48
0.079
0.158
0.9999989
1.05917E-06
944131.9182
6.202531646
5856008.1
0.08
0.16
0.9999990
1.04247E-06
959260.1454
6.125
5875468.39
0.081
0.162
0.9999990
1.03047E-06
970435.213
6.049382716
5870534.004
0.082
0.164
0.9999990
1.02302E-06
977499.71
5.975609756
5841156.803
0.083
0.166
0.9999990
1.02005E-06
980347.7389
5.903614458
5787595.085
0.084
0.168
0.9999990
1.02153E-06
978927.4985
5.833333333
5710410.408
0.085
0.17
0.9999990
1.02749E-06
973242.7208
5.764705882
5610458.038
0.086
0.172
0.9999990
1.03804E-06
963352.9125
5.697674419
5488871.245
0.087
0.174
0.9999989
1.05333E-06
949372.3805
5.632183908
5347039.844
0.088
0.176
0.9999989
1.07357E-06
931468.0642
5.568181818
5186583.539
0.089
0.178
0.9999989
1.09907E-06
909856.228
5.505617978
5009320.806
0.09
0.18
0.9999991
9.26765E-07
1079022.083
5.444444444
5874675.788
0.091
0.182
0.9999990
9.54944E-07
1047181.706
5.384615385
5638670.726
0.092
0.184
0.9999990
9.88396E-07
1011740.662
5.326086957
5388618.744
0.093
0.186
0.9999990
1.02763E-06
973116.5806
5.268817204
5127173.382
0.094
0.188
0.9999989
1.07324E-06
931755.7254
5.212765957
4857024.526
0.095
0.19
0.9999991
9.12034E-07
1096449.803
5.157894737
5655372.67
0.096
0.192
0.9999990
9.58823E-07
1042945.344
5.104166667
5323366.86
0.097
0.194
0.9999990
1.01261E-06
987545.9778
5.051546392
4988634.321
0.098
0.196
0.9999991
8.63748E-07
1157744.91
5
5788724.548
0.099
0.198
0.9999991
9.18301E-07
1088967.198
4.949494949
5389837.647
0.2
0.9999990
9.80797E-07
1019578.823
4.9
4995936.233
0.1
80
Prob
IC-Cost1
0.01
0.055355
ICCost2
40
OCCost1
1000
0.011
0.0608713
44
1000
0.012
0.0663841
48
1000
0.013
0.0725475
52
1000
0.014
0.0778678
56
1000
0.015
0.0829006
60
1000
0.016
0.0891147
64
1000
0.017
0.094273
68
1000
0.018
0.100595
72
1000
0.019
0.1056151
76
1000
0.02
0.1103547
80
1000
0.021
0.1165388
84
1000
0.022
0.12131
88
1000
0.023
0.1296353
92
1000
0.024
0.13225
96
1000
0.025
0.1412453
100
1000
0.026
0.1464018
104
1000
0.027
0.15045
108
1000
0.028
0.1572409
112
1000
0.029
0.1623078
116
1000
0.03
0.1700786
120
1000
0.031
0.1753335
124
1000
0.032
0.1772889
128
1000
0.033
0.1874722
132
1000
0.034
0.1874753
136
1000
0.035
0.1960898
140
1000
0.036
0.1999873
144
1000
0.037
0.2058903
148
1000
0.038
0.2139832
152
1000
0.039
0.2157649
156
1000
0.04
0.2283297
160
1000
0.041
0.2339586
164
1000
OC-Cost2
1080
1088
1096
1104
1112
1120
1128
1136
1144
1152
1160
1168
1176
1184
1192
1200
1208
1216
1224
1232
1240
1248
1256
1264
1272
1280
1288
1296
1304
1312
1320
1328
% In control
P/C
2.2465E-05
% In control
No P/C
0.927536232
Net Cost
1
999.9777
Net Cost 2
2.4528E-05
0.933687003
999.9756
113.2308
2.65585E-05
0.938875306
999.9736
112.0587
2.90823E-05
0.943310658
999.9711
111.6372
3.08966E-05
0.947145877
999.9693
111.814
3.2457E-05
0.95049505
999.9677
112.4752
3.49229E-05
0.953445065
999.9653
113.5345
3.65267E-05
0.956063269
999.9637
114.9244
3.90124E-05
0.958402662
999.9612
116.5923
4.04524E-05
0.960505529
999.9598
118.4961
4.16571E-05
0.962406015
999.9586
120.6015
4.39397E-05
0.964131994
999.9563
122.8809
4.51212E-05
0.965706447
999.9552
125.3114
4.89644E-05
0.967148489
999.9513
127.8739
4.84645E-05
0.968474149
999.9518
130.5523
5.27276E-05
0.96969697
999.9476
133.3333
5.40777E-05
0.970828471
999.9463
136.2054
5.45868E-05
0.971878515
999.9458
139.1586
5.70979E-05
0.972855592
999.9433
142.1846
5.83121E-05
0.973767051
999.9421
145.276
6.14745E-05
0.974619289
999.9389
148.4264
6.27641E-05
0.975417896
999.9376
151.6303
6.16743E-05
0.976167779
999.9387
154.8827
6.64441E-05
0.976873265
999.934
158.1795
6.3955E-05
0.977538185
999.9365
161.5166
6.751E-05
0.978165939
999.9329
164.8908
6.77474E-05
0.978759558
999.9327
168.2991
6.93551E-05
0.979321754
999.9311
171.7386
7.24388E-05
0.979854956
999.928
175.2071
7.11788E-05
0.980361351
999.9293
178.7023
7.72415E-05
0.980842912
999.9233
182.2222
7.85298E-05
0.981301421
999.922
185.7651
115.3623
81
0.042
0.2415495
168
1000.001
0.043
0.2404878
172
1000
0.044
0.2518482
176
1000.001
0.045
0.2538216
180
1000.001
0.046
0.2572552
184
1000.001
0.047
0.262216
188
1000.001
0.048
0.2687981
192
1000.001
0.049
0.2771263
196
1000.001
0.05
0.2873594
200
1000.001
0.051
0.2844113
204
1000.001
0.052
0.2980319
208
1000.001
0.053
0.2974796
212
1000.001
0.054
0.2980389
216
1000.001
0.055
0.3171658
220
1000.001
0.056
0.3205009
224
1000.001
0.057
0.325101
228
1000.001
0.058
0.3310253
232
1000.001
0.059
0.3383493
236
1000.001
0.06
0.3471664
240
1000.001
0.061
0.35759
244
1000.001
0.062
0.346831
248
1000.001
0.063
0.3596443
252
1000.001
0.064
0.3743889
256
1000.001
0.065
0.3658349
260
1000.001
0.066
0.3834228
264
1000.001
0.067
0.3764132
268
1000.001
0.068
0.3972123
272
1000.001
0.069
0.3917903
276
1000.001
0.07
0.3871509
280
1000.001
0.071
0.4125635
284
1000.001
0.072
0.4096268
288
1000.001
0.073
0.4074681
292
1000.001
0.074
0.4385302
296
1000.001
0.075
0.4383295
300
1000.001
0.076
0.4389561
304
1000.001
0.077
0.4404171
308
1000.001
0.078
0.4427241
312
1000.001
0.079
0.4458938
316
1000.001
0.08
0.4499479
320
1000.001
1336
1344
1352
1360
1368
1376
1384
1392
1400
1408
1416
1424
1432
1440
1448
1456
1464
1472
1480
1488
1496
1504
1512
1520
1528
1536
1544
1552
1560
1568
1576
1584
1592
1600
1608
1616
1624
1632
1640
8.11345E-05
0.981738495
999.9194
189.3294
7.7866E-05
0.982155603
999.9226
192.9136
8.29176E-05
0.982554082
999.9176
196.5164
8.16733E-05
0.982935154
999.9189
200.1365
8.14189E-05
0.983299933
999.9191
203.7729
8.21503E-05
0.983649444
999.9184
207.4245
8.38993E-05
0.983984625
999.9166
211.0903
8.67359E-05
0.98430634
999.9138
214.7696
9.07728E-05
0.984615385
999.9098
218.4615
8.63644E-05
0.984912492
999.9142
222.1654
9.24341E-05
0.985198342
999.9082
225.8804
8.9549E-05
0.985473562
999.911
229.606
8.74495E-05
0.985738734
999.9131
233.3417
9.67265E-05
0.985994398
999.9039
237.0868
9.62073E-05
0.986241057
999.9044
240.8409
9.64703E-05
0.986479178
999.9042
244.6036
9.75256E-05
0.986709197
999.9031
248.3743
9.94031E-05
0.986931521
999.9013
252.1526
0.000102154
0.98714653
999.8985
255.9383
0.000105852
0.987354578
999.8948
259.7309
9.6885E-05
0.987555998
999.9038
263.5301
0.000101844
0.987751102
999.8988
267.3356
0.000107957
0.987940183
999.8928
271.1471
0.000100402
0.988123515
999.9003
274.9644
0.000107988
0.988301357
999.8927
278.7871
0.000101441
0.988473951
999.8992
282.615
0.000110719
0.988641527
999.89
286.448
0.000105064
0.988804299
999.8956
290.2857
0.000100092
0.988962472
999.9006
294.128
0.000111574
0.989116239
999.8892
297.9747
0.000107389
0.989265779
999.8933
301.8257
0.000103776
0.989411266
999.8969
305.6806
0.000118176
0.989552863
999.8826
309.5395
0.000115394
0.989690722
999.8854
313.4021
0.000113137
0.98982499
999.8876
317.2682
0.000111377
0.989955806
999.8894
321.1378
0.000110095
0.9900833
999.8906
325.0107
0.000109278
0.990207599
999.8915
328.8868
0.000108916
0.99032882
999.8918
332.766
82
0.081
0.4549135
324
1000.001
0.082
0.4608231
328
1000.001
0.083
0.4677152
332
1000.001
0.084
0.4756349
336
1000.001
0.085
0.4846339
340
1000.001
0.086
0.4947715
344
1000.001
0.087
0.5061149
348
1000.001
0.088
0.5187404
352
1000.001
0.089
0.5327339
356
1000.001
0.09
0.4988545
360
1000.001
0.091
0.5137872
364
1000.001
0.092
0.5302241
368
1000.001
0.093
0.5482834
372
1000.001
0.094
0.5680974
376
1000.001
0.095
0.5337818
380
1000.001
0.096
0.5546936
384
1000.001
0.097
0.5775927
388
1000.001
0.098
0.5435987
392
1000.001
0.099
0.5677157
396
1000.001
0.1
0.5941136
400
1000.001
1648
1656
1664
1672
1680
1688
1696
1704
1712
1720
1728
1736
1744
1752
1760
1768
1776
1784
1792
1800
0.000109007
0.990447077
999.8917
336.6481
0.000109555
0.990562476
999.8912
340.533
0.000110569
0.990675121
999.8902
344.4207
0.000112063
0.990785109
999.8887
348.3111
0.00011406
0.990892532
999.8867
352.204
0.000116586
0.990997479
999.8842
356.0994
0.000119678
0.991100036
999.8811
359.9972
0.00012338
0.991200282
999.8775
363.8972
0.000127746
0.991298294
999.8731
367.7995
0.00010893
0.991394148
999.8918
371.704
0.000113489
0.991487913
999.8873
375.6105
0.000118755
0.991579656
999.8821
379.519
0.00012481
0.991669444
999.876
383.4295
0.000131751
0.991757336
999.8691
387.3419
0.000113154
0.991843393
999.8876
391.2561
0.00012021
0.991927672
999.8806
395.1721
0.000128275
0.992010227
999.8726
399.0898
0.000110548
0.99209111
999.8902
403.0092
0.000118728
0.992170373
999.8821
406.9302
0.000128088
0.992248062
999.8728
410.8527
83
Appendix2
Process In-Control
p0
LCL
UCL
Prob No Signal
Prob Signal
Events to Signal
E[X|LCL<X<UCL]
ARL0
0.01
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
0.02
0.021
0.022
0.023
0.024
0.025
0.026
0.027
0.028
0.029
0.03
0.031
0.032
0.033
0.034
0.035
0.036
0.037
0.038
0.039
0.04
0.041
0.042
0.043
0.044
0.045
0.046
0.047
0.048
0.049
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
398
362
332
306
284
265
248
234
221
209
198
189
180
172
165
158
152
147
141
136
132
128
123
120
116
113
110
107
104
101
98
96
94
92
89
87
85
84
82
80
0.981499978
0.981555739
0.9816114
0.981518741
0.981500204
0.981500293
0.98138816
0.981593566
0.981612304
0.981500775
0.981313115
0.981501093
0.981351023
0.981294454
0.981388983
0.981218743
0.981275834
0.981614532
0.981238088
0.981181022
0.981503175
0.981671805
0.981085555
0.981560446
0.981277254
0.981504797
0.981617834
0.981618268
0.981505933
0.981278581
0.980931804
0.981260082
0.98150764
0.981677432
0.980932466
0.980932667
0.980854643
0.981604536
0.981396579
0.981108744
0.01850002
0.01844426
0.0183886
0.01848126
0.0184998
0.01849971
0.01861184
0.01840643
0.0183877
0.01849923
0.01868688
0.01849891
0.01864898
0.01870555
0.01861102
0.01878126
0.01872417
0.01838547
0.01876191
0.01881898
0.01849683
0.01832819
0.01891444
0.01843955
0.01872275
0.0184952
0.01838217
0.01838173
0.01849407
0.01872142
0.0190682
0.01873992
0.01849236
0.01832257
0.01906753
0.01906733
0.01914536
0.01839546
0.01860342
0.01889126
54.05398848
54.21740578
54.38151991
54.10886644
54.05465149
54.05491
53.72923876
54.32882909
54.3841928
54.05631852
53.51346698
54.05724907
53.62224505
53.46007997
53.73161604
53.24457138
53.40691737
54.39078424
53.29947243
53.13784803
54.06333234
54.56074786
52.86964671
54.23124617
53.41096962
54.06807474
54.40055176
54.40183699
54.07139574
53.41475407
52.44334628
53.36202707
54.07638554
54.57750302
52.44516785
52.44571901
52.23198428
54.36122654
53.75355366
52.93454226
98
89.09090909
81.66666667
75.38461538
70
65.33333333
61.25
57.64705882
54.44444444
51.57894737
49
46.66666667
44.54545455
42.60869565
40.83333333
39.2
37.69230769
36.2962963
35
33.79310345
32.66666667
31.61290323
30.625
29.6969697
28.82352941
28
27.22222222
26.48648649
25.78947368
25.12820513
24.5
23.90243902
23.33333333
22.79069767
22.27272727
21.77777778
21.30434783
20.85106383
20.41666667
20
5297.291
4830.278
4441.157
4078.976
3783.826
3531.587
3290.916
3131.897
2960.917
2788.168
2622.16
2522.672
2388.627
2277.864
2194.041
2087.187
2013.03
1974.184
1865.482
1795.693
1766.069
1724.824
1619.133
1610.504
1539.493
1513.906
1480.904
1440.914
1394.473
1342.217
1284.862
1275.483
1261.782
1243.859
1168.097
1142.151
1112.768
1133.489
1097.468
1058.691
84
0.05
0.051
0.052
0.053
0.054
0.055
0.056
0.057
0.058
0.059
0.06
0.061
0.062
0.063
0.064
0.065
0.066
0.067
0.068
0.069
0.07
0.071
0.072
0.073
0.074
0.075
0.076
0.077
0.078
0.079
0.08
0.081
0.082
0.083
0.084
0.085
0.086
0.087
0.088
0.089
0.09
0.091
0.092
0.093
0.094
0.095
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
78
77
75
74
73
71
70
69
67
66
65
64
63
62
61
60
59
58
57
56
56
55
54
53
53
52
51
50
50
49
48
48
47
47
46
46
45
44
44
43
43
42
42
41
41
41
0.98073728
0.981283177
0.980777186
0.981226361
0.981627772
0.980935448
0.981247136
0.981516036
0.980619741
0.980798926
0.980937378
0.981035897
0.981095031
0.981115085
0.981096121
0.981037961
0.980940179
0.980802101
0.980622797
0.980401066
0.981525948
0.981256592
0.980943518
0.980584607
0.981644285
0.981240193
0.980786851
0.98028107
0.98130114
0.980748603
0.980137776
0.981127517
0.980467253
0.981422407
0.980711595
0.981636763
0.980873627
0.98003619
0.980955128
0.980057857
0.980956864
0.979995891
0.980878595
0.979849238
0.980719073
0.98155226
0.01926272
0.01871682
0.01922281
0.01877364
0.01837223
0.01906455
0.01875286
0.01848396
0.01938026
0.01920107
0.01906262
0.0189641
0.01890497
0.01888492
0.01890388
0.01896204
0.01905982
0.0191979
0.0193772
0.01959893
0.01847405
0.01874341
0.01905648
0.01941539
0.01835572
0.01875981
0.01921315
0.01971893
0.01869886
0.0192514
0.01986222
0.01887248
0.01953275
0.01857759
0.0192884
0.01836324
0.01912637
0.01996381
0.01904487
0.01994214
0.01904314
0.02000411
0.0191214
0.02015076
0.01928093
0.01844774
51.91374897
53.42786938
52.02151993
53.2661776
54.42998054
52.45337104
53.32518778
54.10095032
51.59889672
52.08041992
52.45868136
52.73120396
52.89614477
52.95231552
52.89919653
52.73694386
52.46638945
52.08903436
51.60703501
51.02318278
54.12997538
53.3520909
52.47558157
51.50552573
54.47894548
53.305452
52.04768757
50.71269022
53.47919586
51.94428235
50.3468281
52.98719928
51.19607478
53.82828698
51.84461873
54.45663064
52.28382926
50.09063831
52.50757385
50.14506137
52.51235828
49.98972963
52.29741354
49.62591389
51.8647266
54.20718213
19.6
19.21568627
18.84615385
18.49056604
18.14814815
17.81818182
17.5
17.19298246
16.89655172
16.61016949
16.33333333
16.06557377
15.80645161
15.55555556
15.3125
15.07692308
14.84848485
14.62686567
14.41176471
14.20289855
14
13.8028169
13.61111111
13.42465753
13.24324324
13.06666667
12.89473684
12.72727273
12.56410256
12.40506329
12.25
12.09876543
11.95121951
11.80722892
11.66666667
11.52941176
11.39534884
11.26436782
11.13636364
11.01123596
10.88888889
10.76923077
10.65217391
10.53763441
10.42553191
10.31578947
1017.509
1026.653
980.4056
984.9218
987.8034
934.6237
933.1908
930.1567
871.8434
865.0646
856.8251
847.157
836.1004
823.7027
810.0189
795.1108
779.0464
761.8993
743.7484
724.6771
757.8197
736.4091
714.251
691.444
721.4779
696.5246
671.1412
645.4342
671.9181
644.3721
616.7486
641.0797
611.8555
635.5629
604.8539
627.8529
595.7925
564.2394
584.7434
552.1591
571.8012
538.3509
557.0811
522.9397
540.7174
559.1899
85
0.096
0.097
0.098
0.099
0.1
0
0
0
0
0
40
40
39
39
38
0.980476
0.981300833
0.980145981
0.980965474
0.97972444
0.019524
0.01869917
0.01985402
0.01903453
0.02027556
51.21901373
53.4783178
50.36763665
52.53611365
49.32046366
10.20833333
10.10309278
10
9.898989899
9.8
522.8608
540.2964
503.6764
520.0545
483.3405
Process Out of Control
Prob
Prob No signal
Prob Signal
0.01
Shifted
Prob
0.02
E[X|LCL<X<UCL]
ARL1
0.000328664
Events to
Signal
3042.621642
0.999671336
49
149088.4605
0.011
0.022
0.999674668
0.000325332
3073.784538
44.54545455
136923.1294
0.012
0.024
0.999677975
0.000322025
3105.348543
40.83333333
126801.7322
0.013
0.026
0.999676047
0.000323953
3086.868664
37.69230769
116351.2035
0.014
0.028
0.999676739
0.000323261
3093.477958
35
108271.7285
0.015
0.03
0.999678087
0.000321913
3106.434163
32.66666667
101476.8493
0.016
0.032
0.999675501
0.000324499
3081.674359
30.625
94376.27724
0.017
0.034
0.999684012
0.000315988
3164.678126
28.82352941
91217.19303
0.018
0.036
0.999685987
0.000314013
3184.582772
27.22222222
86691.41992
0.019
0.038
0.99968347
0.00031653
3159.257434
25.78947368
81475.58646
0.02
0.04
0.999678313
0.000321687
3108.616095
24.5
76161.09432
0.021
0.042
0.999686155
0.000313845
3186.28261
23.33333333
74346.59422
0.022
0.044
0.99968234
0.00031766
3148.018548
22.27272727
70114.95858
0.023
0.046
0.999681747
0.000318253
3142.157322
21.30434783
66941.61252
0.024
0.048
0.99968634
0.00031366
3188.162616
20.41666667
65091.6534
0.025
0.05
0.999681869
0.000318131
3143.358895
19.6
61609.83434
0.026
0.052
0.999685182
0.000314818
3176.442967
18.84615385
59863.73283
0.027
0.054
0.999697931
0.000302069
3310.503759
18.14814815
60079.51266
0.028
0.056
0.999686614
0.000313386
3190.955871
17.5
55841.72775
0.029
0.058
0.999686042
0.000313958
3185.137043
16.89655172
53817.8328
0.03
0.06
0.999698179
0.000301821
3313.226898
16.33333333
54116.03933
0.031
0.062
0.999705042
0.000294958
3390.3109
15.80645161
53588.78519
0.032
0.064
0.999686921
0.000313079
3194.084887
15.3125
48909.42484
0.033
0.066
0.999704009
0.000295991
3378.482927
14.84848485
50165.35256
0.034
0.068
0.999696031
0.000303969
3289.804416
14.41176471
47411.88718
0.035
0.07
0.999704818
0.000295182
3387.737582
14
47428.32614
0.036
0.072
0.999709779
0.000290221
3445.645488
13.61111111
46899.06358
0.037
0.074
0.999711089
0.000288911
3461.26947
13.24324324
45838.43352
0.038
0.076
0.999708785
0.000291215
3433.894628
12.89473684
44279.16757
86
0.039
0.078
0.99970277
0.00029723
3364.402094
12.56410256
42270.69298
0.04
0.08
0.999692801
0.000307199
3255.223016
12.25
39876.48194
0.041
0.082
0.999704842
0.000295158
3388.014699
11.95121951
40490.90738
0.042
0.084
0.999714057
0.000285943
3497.203859
11.66666667
40800.71168
0.043
0.086
0.999720679
0.000279321
3580.108496
11.39534884
40796.58519
0.044
0.088
0.999698321
0.000301679
3314.776316
11.13636364
36914.55442
0.045
0.09
0.999699697
0.000300303
3329.969536
10.88888889
36259.66828
0.046
0.092
0.999698553
0.000301447
3317.330367
10.65217391
35336.77999
0.047
0.094
0.999723535
0.000276465
3617.092314
10.42553191
37710.11136
0.048
0.096
0.999718393
0.000281607
3551.052814
10.20833333
36250.33081
0.049
0.098
0.999710721
0.000289279
3456.873873
10
34568.73873
0.05
0.1
0.999700309
0.000299691
3336.773849
9.8
32700.38372
0.051
0.102
0.999718809
0.000281191
3556.300193
9.607843137
34168.3744
0.052
0.104
0.99970434
0.00029566
3382.262301
9.423076923
31871.31784
0.053
0.106
0.99971969
0.00028031
3567.476557
9.245283019
32982.33043
0.054
0.108
0.999733149
0.000266851
3747.403732
9.074074074
34004.21905
0.055
0.11
0.999713384
0.000286616
3488.988475
8.909090909
31083.7155
0.056
0.112
0.999724263
0.000275737
3626.647632
8.75
31733.16678
0.057
0.114
0.999733625
0.000266375
3754.10929
8.596491228
32272.16758
0.058
0.116
0.999707686
0.000292314
3420.979795
8.448275862
28901.38103
0.059
0.118
0.999714597
0.000285403
3503.821445
8.305084746
29099.53404
0.06
0.12
0.999720172
0.000279828
3573.625039
8.166666667
29184.60448
0.061
0.122
0.999724479
0.000275521
3629.489112
8.032786885
29154.91254
0.062
0.124
0.99972757
0.00027243
3670.670636
7.903225806
29010.13889
0.063
0.126
0.999729481
0.000270519
3696.601278
7.777777778
28751.34327
0.064
0.128
0.999730233
0.000269767
3706.900288
7.65625
28380.95533
0.065
0.13
0.999729831
0.000270169
3701.383593
7.538461538
27902.73785
0.066
0.132
0.999728266
0.000271734
3680.068815
7.424242424
27321.72302
0.067
0.134
0.999725514
0.000274486
3643.176023
7.313432836
26644.12315
0.068
0.136
0.999721536
0.000278464
3591.124164
7.205882353
25877.21824
0.069
0.138
0.999716274
0.000283726
3524.523249
7.101449275
25029.22307
0.07
0.14
0.999750302
0.000249698
4004.840107
7
28033.88075
0.071
0.142
0.999743957
0.000256043
3905.588256
6.901408451
26954.0598
0.072
0.144
0.9997363
0.0002637
3792.193348
6.805555556
25807.98251
0.073
0.146
0.999727222
0.000272778
3665.985342
6.712328767
24607.29887
0.074
0.148
0.999758532
0.000241468
4141.341764
6.621621622
27422.39817
0.075
0.15
0.9997486
0.0002514
3977.725309
6.533333333
25987.80535
0.076
0.152
0.999737099
0.000262901
3803.709902
6.447368421
24523.91911
0.077
0.154
0.999723848
0.000276152
3621.197493
6.363636364
23043.98405
0.078
0.156
0.999754088
0.000245912
4066.493808
6.282051282
25545.92264
87
0.079
0.158
0.999739996
0.000260004
3846.091464
6.202531646
23855.50402
0.08
0.16
0.999723862
0.000276138
3621.375467
6.125
22180.92473
0.081
0.162
0.99975313
0.00024687
4050.709741
6.049382716
24504.2935
0.082
0.164
0.99973607
0.00026393
3788.883377
5.975609756
22640.88847
0.083
0.166
0.999763605
0.000236395
4230.207498
5.903614458
24973.51414
0.084
0.168
0.999745581
0.000254419
3930.522773
5.833333333
22928.04951
0.085
0.17
0.999771696
0.000228304
4380.118328
5.764705882
25250.09389
0.086
0.172
0.999752637
0.000247363
4042.641049
5.697674419
23033.65249
0.087
0.174
0.999730758
0.000269242
3714.137394
5.632183908
20918.70486
0.088
0.176
0.999757412
0.000242588
4122.207955
5.568181818
22953.20338
0.089
0.178
0.999734162
0.000265838
3761.695723
5.505617978
20710.4596
0.09
0.18
0.999760016
0.000239984
4166.949142
5.444444444
22686.72311
0.091
0.182
0.99973522
0.00026478
3776.717925
5.384615385
20336.17344
0.092
0.184
0.999760505
0.000239495
4175.451958
5.326086957
22238.82021
0.093
0.186
0.999733942
0.000266058
3758.580052
5.268817204
19803.27124
0.094
0.188
0.999758876
0.000241124
4147.23677
5.212765957
21618.57465
0.095
0.19
0.999781525
0.000218475
4577.193208
5.157894737
23608.68076
0.096
0.192
0.999755069
0.000244931
4082.788942
5.104166667
20839.23522
0.097
0.194
0.999777635
0.000222365
4497.10854
5.051546392
22717.35242
0.098
0.196
0.999748968
0.000251032
3983.548853
5
19917.74426
0.099
0.198
0.999771637
0.000228363
4378.990522
4.949494949
21673.79147
0.1
0.2
0.999740385
0.000259615
3851.859889
4.9
18874.11345
Cost
Prob
IC-Cost1
0.01
0.755103
ICCost2
40
OCCost1
995.7555
OCCost2
115.3623
% In control
P/C
0.004274404
% In control
No P/C
0.927536232
Net Cost 1
Net Cost 2
995.75554
115.36232
0.011
0.8281097
44
995.3805
113.2308
0.004652409
0.933687003
995.38052
113.23077
0.012
0.9006661
48
995.014
112.0587
0.005021903
0.938875306
995.01401
112.05868
0.013
0.9806383
52
994.5691
111.6372
0.005470497
0.943310658
994.56906
111.63719
0.014
1.0571312
56
994.1666
111.814
0.005876318
0.947145877
994.16662
111.81395
0.015
1.1326351
60
993.7789
112.4752
0.00626733
0.95049505
993.77894
112.47525
0.016
1.2154671
64
993.3146
113.5345
0.006735688
0.953445065
993.3146
113.53445
0.017
1.2771811
68
993.0851
114.9244
0.006967337
0.956063269
993.08511
114.92443
0.018
1.3509328
72
992.7273
116.5923
0.007328405
0.958402662
992.7273
116.59235
0.019
1.4346338
76
992.266
118.4961
0.007793892
0.960505529
992.266
118.49605
0.02
1.52546
80
991.7316
120.6015
0.008333215
0.962406015
991.73158
120.6015
0.021
1.5856206
84
991.532
122.8809
0.008534859
0.964131994
991.53202
122.88092
0.022
1.674602
88
991.0264
125.3114
0.009045302
0.965706447
991.02638
125.31139
88
0.023
1.7560309
92
990.6058
127.8739
0.009470032
0.967148489
990.60579
127.87385
0.024
1.82312
96
990.342
130.5523
0.009736557
0.968474149
990.34205
130.55233
0.025
1.9164548
100
989.8028
133.3333
0.010281152
0.96969697
989.80281
133.33333
0.026
1.9870544
104
989.5093
136.2054
0.01057786
0.970828471
989.50927
136.20537
0.027
2.0261536
108
989.547
139.1586
0.010540269
0.971878515
989.54696
139.15861
0.028
2.1442185
112
988.764
142.1846
0.011331098
0.972855592
988.76402
142.18458
0.029
2.2275525
116
988.3474
145.276
0.011752212
0.973767051
988.34742
145.27597
0.03
2.2649174
120
988.4113
148.4264
0.011688208
0.974619289
988.41132
148.4264
0.031
2.3190777
124
988.2993
151.6303
0.01180185
0.975417896
988.29928
151.63029
0.032
2.4704581
128
987.1962
154.8827
0.012916396
0.976167779
987.19624
154.88275
0.033
2.483695
132
987.5129
158.1795
0.012597098
0.976873265
987.51292
158.17946
0.034
2.5982586
136
986.7989
161.5166
0.013318936
0.977538185
986.79891
161.51662
0.035
2.6421718
140
986.804
164.8908
0.013314381
0.978165939
986.80401
164.89083
0.036
2.701053
144
986.6579
168.2991
0.013462613
0.978759558
986.65789
168.29907
0.037
2.7760167
148
986.3545
171.7386
0.013769827
0.979321754
986.35446
171.73863
0.038
2.8684675
152
985.8821
175.2071
0.014247815
0.979854956
985.8821
175.20709
0.039
2.9801443
156
985.223
178.7023
0.014914697
0.980361351
985.22297
178.70228
0.04
3.1131748
160
984.3519
182.2222
0.015796041
0.980842912
984.35186
182.22222
0.041
3.1360679
164
984.586
185.7651
0.015560075
0.981301421
984.58597
185.76515
0.042
3.1701189
168
984.7017
189.3294
0.01544375
0.981738495
984.70173
189.32944
0.043
3.2157976
172
984.7009
192.9136
0.015445288
0.982155603
984.70091
192.91363
0.044
3.4243734
176
983.123
196.5164
0.017041874
0.982554082
983.123
196.5164
0.045
3.5021632
180
982.8248
200.1365
0.017344329
0.982935154
982.82482
200.13652
0.046
3.5946385
184
982.3859
203.7729
0.017789252
0.983299933
982.38588
203.77288
0.047
3.5289258
188
983.4748
207.4245
0.016688348
0.983649444
983.47485
207.42446
0.048
3.6447519
192
982.8229
211.0903
0.017348719
0.983984625
982.82294
211.09033
0.049
3.7782512
196
982.005
214.7696
0.018177305
0.98430634
982.00498
214.76962
0.05
3.9311673
200
980.9995
218.4615
0.01919594
0.984615385
980.9995
218.46154
0.051
3.8961551
204
981.8002
222.1654
0.01838638
0.984912492
981.80017
222.16536
0.052
4.0799442
208
980.5179
225.8804
0.019685452
0.985198342
980.5179
225.8804
0.053
4.0612362
212
981.1613
229.606
0.019034968
0.985473562
981.16131
229.60604
0.054
4.049389
216
981.7168
233.3417
0.018473501
0.985738734
981.71677
233.3417
0.055
4.2797973
220
980.0382
237.0868
0.020174182
0.985994398
980.03825
237.08683
0.056
4.2863689
224
980.4388
240.8409
0.019769459
0.986241057
980.43884
240.84095
0.057
4.3003507
228
980.7595
244.6036
0.019445696
0.986479178
980.75946
244.60357
0.058
4.5879798
232
978.5703
248.3743
0.021664525
0.986709197
978.57027
248.37427
0.059
4.6239321
236
978.7138
252.1526
0.021520176
0.986931521
978.71383
252.15264
0.06
4.6683972
240
978.7755
255.9383
0.021458793
0.98714653
978.7755
255.9383
0.061
4.7216747
244
978.7555
259.7309
0.021480177
0.987354578
978.7555
259.73091
0.062
4.7841147
248
978.6531
263.5301
0.021585059
0.987555998
978.65311
263.53011
89
0.063
4.8561211
252
978.4667
267.3356
0.021775119
0.987751102
978.46672
267.33562
0.064
4.9381561
256
978.1937
271.1471
0.02205303
0.987940183
978.1937
271.14713
0.065
5.0307451
260
977.8304
274.9644
0.022422516
0.988123515
977.83043
274.96437
0.066
5.1344824
264
977.3721
278.7871
0.022888432
0.988301357
977.37214
278.78708
0.067
5.2500376
268
976.8129
282.615
0.023456865
0.988473951
976.81289
282.61503
0.068
5.3781625
272
976.1454
286.448
0.024135262
0.988641527
976.14539
286.44798
0.069
5.5196998
276
975.3609
290.2857
0.024932582
0.988804299
975.36087
290.28571
0.07
5.2783007
280
977.9373
294.128
0.022319964
0.988962472
977.93735
294.12804
0.071
5.4317631
284
977.0775
297.9747
0.023193398
0.989116239
977.07754
297.97475
0.072
5.6002724
288
976.0883
301.8257
0.024198443
0.989265779
976.08832
301.82568
0.073
5.7849945
292
974.9558
305.6806
0.025349246
0.989411266
974.95583
305.68064
0.074
5.5441752
296
977.4627
309.5395
0.022806319
0.989552863
977.46266
309.53949
0.075
5.7427981
300
976.2532
313.4021
0.024035026
0.989690722
976.25322
313.40206
0.076
5.9599974
304
974.8773
317.2682
0.02543324
0.98982499
974.8773
317.26821
0.077
6.1973781
308
973.3139
321.1378
0.027022481
0.989955806
973.31388
321.13781
0.078
5.9531065
312
975.8576
325.0107
0.024440613
0.9900833
975.85764
325.01071
0.079
6.2075933
316
974.1982
328.8868
0.026127244
0.990207599
974.19824
328.8868
0.08
6.4856243
320
972.3127
332.766
0.028044438
0.99032882
972.31273
332.76596
0.081
6.2394739
324
974.8648
336.6481
0.025453091
0.990447077
974.8648
336.64807
0.082
6.537491
328
972.8612
340.533
0.02749036
0.990562476
972.86117
340.53303
0.083
6.2936335
332
975.3266
344.4207
0.02498681
0.990675121
975.32662
344.42074
0.084
6.6131674
336
973.1939
348.3111
0.027155408
0.990785109
973.1939
348.31109
0.085
6.3709189
340
975.5921
352.204
0.024719879
0.990892532
975.59211
352.20401
0.086
6.7137471
344
973.3162
356.0994
0.027034274
0.990997479
973.31619
356.09939
0.087
7.0891898
348
970.7096
359.9972
0.029686384
0.991100036
970.70961
359.99715
0.088
6.8406069
352
973.2286
363.8972
0.027126456
0.991200282
973.22865
363.89722
0.089
7.2442888
356
970.4286
367.7995
0.029975936
0.991298294
970.42857
367.79951
0.09
6.9954379
360
972.9271
371.704
0.027436344
0.991394148
972.92706
371.70396
0.091
7.4300976
364
969.9066
375.6105
0.030510808
0.991487913
969.90658
375.61049
0.092
7.1802825
368
972.4022
379.519
0.02797347
0.991579656
972.40222
379.51903
0.093
7.6490649
372
969.129
383.4295
0.031306144
0.991669444
969.12898
383.42952
0.094
7.3975801
376
971.6394
387.3419
0.028752964
0.991757336
971.63944
387.34191
0.095
7.1532053
380
973.9606
391.2561
0.026393188
0.991843393
973.96056
391.25612
0.096
7.6502202
384
970.618
395.1721
0.029796219
0.991927672
970.61795
395.1721
0.097
7.4033437
388
972.9737
399.0898
0.027400366
0.992010227
972.97374
399.08981
0.098
7.9416075
392
969.31
403.0092
0.03113182
0.99209111
969.30999
403.00917
0.099
7.6915022
396
971.7181
406.9302
0.028681813
0.992170373
971.71805
406.93016
0.1
8.2757386
400
967.6796
410.8527
0.032796776
0.992248062
967.67962
410.85271
90
Appendix3
Process In Control
p0
LCL
UCL
Prob No Signal
Prob Signal
Events to Signal
E[X|LCL<X<UCL]
ARL0
0.01
1
199
0.853299995
0.1467
6.816632353
98
668.03
0.011
1
181
0.852437176
0.14756282
6.776774639
89.09090909
603.749
0.012
1
166
0.851574405
0.1484256
6.737382452
81.66666667
550.2196
0.013
1
153
0.850161846
0.14983815
6.673867605
75.38461538
503.1069
0.014
1
142
0.849023834
0.15097617
6.623562017
70
463.6493
0.015
1
133
0.848986372
0.15101363
6.621918908
65.33333333
432.632
0.016
1
124
0.846470106
0.15352989
6.513389519
61.25
398.9451
0.017
1
117
0.846161574
0.15383843
6.500326538
57.64705882
374.7247
0.018
1
111
0.846398761
0.15360124
6.510364167
54.44444444
354.4532
0.019
1
105
0.844988144
0.15501186
6.451119448
51.57894737
332.742
0.02
1
99
0.841912166
0.15808783
6.325597447
49
309.9543
0.021
1
95
0.842989314
0.15701069
6.368993275
46.66666667
297.2197
0.022
1
90
0.839911321
0.16008868
6.246537882
44.54545455
278.2549
0.023
1
86
0.838631282
0.16136872
6.196987957
42.60869565
264.0456
0.024
1
83
0.8395778
0.1604222
6.233551219
40.83333333
254.5367
0.025
1
79
0.836209409
0.16379059
6.105356816
39.2
239.33
0.026
1
76
0.835349396
0.1646506
6.073466939
37.69230769
228.923
0.027
1
74
0.837406978
0.16259302
6.150325448
36.2962963
223.234
0.028
1
71
0.835025872
0.16497413
6.061556507
35
212.1545
0.029
1
68
0.83178424
0.16821576
5.944746186
33.79310345
200.8914
0.03
1
66
0.831909842
0.16809016
5.949188283
32.66666667
194.3402
0.031
1
64
0.831469838
0.16853016
5.933655963
31.61290323
187.5801
0.032
1
62
0.830470205
0.1695298
5.898668135
30.625
180.6467
0.033
1
60
0.828910082
0.17108992
5.844879776
29.6969697
173.5752
0.034
1
58
0.826781739
0.17321826
5.773063372
28.82352941
166.4001
0.035
1
57
0.829002931
0.17099707
5.848053468
28
163.7455
0.036
1
55
0.825910763
0.17408924
5.744180489
27.22222222
156.3694
0.037
1
54
0.827420753
0.17257925
5.794439466
26.48648649
153.4743
0.038
1
52
0.823347191
0.17665281
5.660821376
25.78947368
145.9896
0.039
1
51
0.824173763
0.17582624
5.687433324
25.12820513
142.915
0.04
1
49
0.819064894
0.18093511
5.526843425
24.5
135.4077
0.041
1
48
0.819210505
0.1807895
5.531294825
23.90243902
132.2114
0.042
1
47
0.819064439
0.18093556
5.526829535
23.33333333
128.9594
0.043
1
46
0.818631511
0.18136849
5.513636921
22.79069767
125.6596
0.044
1
45
0.81791476
0.18208524
5.491933328
22.27272727
122.3203
91
0.045
1
44
0.816915485
0.18308451
5.461958382
21.77777778
118.9493
0.046
1
43
0.815633251
0.18436675
5.423971538
21.30434783
115.5542
0.047
1
42
0.814065876
0.18593412
5.37824892
20.85106383
112.1422
0.048
1
41
0.812209411
0.18779059
5.325080475
20.41666667
108.7204
0.049
1
40
0.810058087
0.18994191
5.264767457
20
105.2953
0.05
1
39
0.807604259
0.19239574
5.197620244
19.6
101.8734
0.051
1
39
0.812190558
0.18780944
5.324545924
19.21568627
102.3148
0.052
1
38
0.809353637
0.19064636
5.245313807
18.84615385
98.85399
0.053
1
37
0.806201106
0.19379889
5.159988161
18.49056604
95.4111
0.054
1
37
0.810455809
0.18954419
5.27581454
18.14814815
95.74626
0.055
1
36
0.806925558
0.19307444
5.179349415
17.81818182
92.28659
0.056
1
35
0.803055606
0.19694439
5.077575353
17.5
88.85757
0.057
1
35
0.807044258
0.19295574
5.182535601
17.19298246
89.10324
0.058
1
34
0.802787
0.197213
5.070659645
16.89655172
85.67666
0.059
1
33
0.798154044
0.20184596
4.954273151
16.61016949
82.29132
0.06
1
33
0.801932546
0.19806745
5.048785057
16.33333333
82.46349
0.061
1
32
0.796887146
0.20311285
4.923371328
16.06557377
79.09679
0.062
1
32
0.800504657
0.19949534
5.012648332
15.80645161
79.23218
0.063
1
31
0.795032895
0.20496711
4.878831652
15.55555556
75.89294
0.064
1
31
0.798508623
0.20149138
4.962991538
15.3125
75.99581
0.065
1
30
0.792591237
0.20740876
4.82139705
15.07692308
72.69183
0.066
1
30
0.795942689
0.20405731
4.900584029
14.84848485
72.76625
0.067
1
29
0.789554798
0.2104452
4.751830847
14.62686567
69.50439
0.068
1
29
0.792797977
0.20720202
4.826207713
14.41176471
69.55417
0.069
1
28
0.785908695
0.2140913
4.670904317
14.20289855
66.34038
0.07
1
28
0.789058266
0.21094173
4.74064558
14
66.36904
0.071
1
28
0.792093434
0.20790657
4.809852899
13.8028169
66.38952
0.072
1
27
0.784699607
0.21530039
4.644673359
13.61111111
63.21917
0.073
1
27
0.787660872
0.21233913
4.709447619
13.42465753
63.22272
0.074
1
27
0.790516735
0.20948326
4.773651015
13.24324324
63.21862
0.075
1
26
0.782589071
0.21741093
4.599584787
13.06666667
60.10124
0.076
1
26
0.785388495
0.21461151
4.659582429
12.89473684
60.08409
0.077
1
25
0.776835864
0.22316414
4.481006745
12.72727273
57.03099
0.078
1
25
0.779589469
0.22041053
4.536988302
12.56410256
57.00319
0.079
1
25
0.782250596
0.2177494
4.592435083
12.40506329
56.96945
0.08
1
24
0.773066769
0.22693323
4.406582479
12.25
53.98064
0.081
1
24
0.775696512
0.22430349
4.458245424
12.09876543
53.93927
0.082
1
24
0.778240394
0.22175961
4.509387521
11.95121951
53.89268
0.083
1
24
0.780700356
0.21929964
4.55997093
11.80722892
53.84062
0.084
1
23
0.770888971
0.22911103
4.364696042
11.66666667
50.92145
92
0.085
1
23
0.77333451
0.22666549
4.4117876
11.52941176
50.86532
0.086
1
23
0.775701871
0.22429813
4.458351937
11.39534884
50.80448
0.087
1
22
0.76512787
0.23487213
4.257635849
11.26436782
47.95958
0.088
1
22
0.767492095
0.2325079
4.300929037
11.13636364
47.89671
0.089
1
22
0.769783346
0.23021665
4.343734406
11.01123596
47.82988
0.09
1
22
0.772003129
0.22799687
4.386025107
10.88888889
47.75894
0.091
1
21
0.760653377
0.23934662
4.178040982
10.76923077
44.99429
0.092
1
21
0.762883441
0.23711656
4.217335163
10.65217391
44.92379
0.093
1
21
0.765046619
0.23495338
4.256163479
10.53763441
44.84989
0.094
1
21
0.767144223
0.23285578
4.294503714
10.42553191
44.77249
0.095
1
21
0.769177542
0.23082246
4.332334084
10.31578947
44.69145
0.096
1
20
0.75703965
0.24296035
4.115897915
10.20833333
42.01646
0.097
1
20
0.759097859
0.24090214
4.151063146
10.10309278
41.93858
0.098
1
20
0.761095711
0.23890429
4.185776671
10
41.85777
0.099
1
20
0.763034332
0.23696567
4.220020595
9.898989899
41.77394
0.1
1
19
0.749905365
0.25009464
3.998486408
9.8
39.18517
Process Out of Control
Prob
Prob No signal
Prob Signal
0.01
Shifted
Prob
0.02
E[X|LCL<X<UCL]
0.038313147
Events to
Signal
26.10070099
0.961686853
0.011
0.022
0.959761301
0.012
0.024
0.013
ARL1
49
1278.934349
0.040238699
24.85169797
44.54545455
1107.030182
0.957835648
0.042164352
23.71671677
40.83333333
968.4326016
0.026
0.955762662
0.044237338
22.60533854
37.69230769
852.0473757
0.014
0.028
0.953763422
0.046236578
21.62789803
35
756.9764312
0.015
0.03
0.95205808
0.04794192
20.85857208
32.66666667
681.3800211
0.016
0.032
0.949690818
0.050309182
19.87708713
30.625
608.7357933
0.017
0.034
0.947913828
0.052086172
19.1989535
28.82352941
553.3816008
0.018
0.036
0.946279591
0.053720409
18.6148994
27.22222222
506.7389281
0.019
0.038
0.944208708
0.055791292
17.92394406
25.78947368
462.2490836
0.02
0.04
0.941694532
0.058305468
17.15105004
24.5
420.2007259
0.021
0.042
0.940284319
0.059715681
16.74602017
23.33333333
390.7404707
0.022
0.044
0.937771438
0.062228562
16.0697912
22.27272727
357.9180768
0.023
0.046
0.935735329
0.064264671
15.56064922
21.30434783
331.5094833
0.024
0.048
0.934289543
0.065710457
15.21827797
20.41666667
310.7065086
0.025
0.05
0.931700416
0.068299584
14.64137765
19.6
286.971002
93
0.026
0.052
0.929776772
0.070223228
14.24030244
18.84615385
268.3749306
0.027
0.054
0.928619873
0.071380127
14.00950146
18.14814815
254.246508
0.028
0.056
0.926297296
0.073702704
13.56802328
17.5
237.4404074
0.029
0.058
0.923743796
0.076256204
13.11368707
16.89655172
221.5760918
0.03
0.06
0.922081136
0.077918864
12.8338626
16.33333333
209.6197558
0.031
0.062
0.920267139
0.079732861
12.54188027
15.80645161
198.2426236
0.032
0.064
0.918305969
0.081694031
12.24079646
15.3125
187.4371958
0.033
0.066
0.916198127
0.083801873
11.93290752
14.84848485
177.1855965
0.034
0.068
0.913940447
0.086059553
11.61986045
14.41176471
167.4626947
0.035
0.07
0.912819131
0.087180869
11.47040646
14
160.5856905
0.036
0.072
0.910315584
0.089684416
11.15020923
13.61111111
151.7667367
0.037
0.074
0.909002608
0.090997392
10.98932592
13.24324324
145.5343162
0.038
0.076
0.90624705
0.09375295
10.66633102
12.89473684
137.5395316
0.039
0.078
0.904759651
0.095240349
10.49975153
12.56410256
131.9199551
0.04
0.08
0.901726754
0.098273246
10.17570942
12.25
124.6524404
0.041
0.082
0.900068938
0.099931062
10.00689854
11.95121951
119.5946411
0.042
0.084
0.898331821
0.101668179
9.835919268
11.66666667
114.7523915
0.043
0.086
0.896518495
0.103481505
9.663562626
11.39534884
110.1196671
0.044
0.088
0.894631077
0.105368923
9.490464293
11.13636364
105.6892614
0.045
0.09
0.892670746
0.107329254
9.317124286
10.88888889
101.4531311
0.046
0.092
0.890637765
0.109362235
9.143924291
10.65217391
97.4026718
0.047
0.094
0.88853148
0.11146852
8.971142726
10.42553191
93.52893481
0.048
0.096
0.886350304
0.113649696
8.798967696
10.20833333
89.82279523
0.049
0.098
0.884091675
0.115908325
8.627507999
10
86.27507999
0.05
0.1
0.881751996
0.118248004
8.456802392
9.8
82.87666344
0.051
0.102
0.881231246
0.118768754
8.419722919
9.607843137
80.89537706
0.052
0.104
0.878805231
0.121194769
8.251181172
9.423076923
77.75151489
0.053
0.106
0.876292768
0.123707232
8.083601776
9.245283019
74.73518623
0.054
0.108
0.875664413
0.124335587
8.042749629
9.074074074
72.98050589
0.055
0.11
0.873070262
0.126929738
7.878374423
8.909090909
70.18915395
0.056
0.112
0.870378583
0.129621417
7.714774462
8.75
67.50427654
0.057
0.114
0.869679008
0.130320992
7.673360883
8.596491228
65.96397952
0.058
0.116
0.86690281
0.13309719
7.513306613
8.448275862
63.4744869
0.059
0.118
0.864011509
0.135988491
7.353563492
8.305084746
61.07196798
0.06
0.12
0.863271943
0.136728057
7.313787816
8.166666667
59.72926717
0.061
0.122
0.860285455
0.139714545
7.157450948
8.032786885
57.49427811
0.062
0.124
0.859494553
0.140505447
7.117161798
7.903225806
56.24853679
0.063
0.126
0.856406875
0.143593125
6.964121734
7.777777778
54.16539126
0.064
0.128
0.85557541
0.14442459
6.924028638
7.65625
53.01209426
0.065
0.13
0.852377873
0.147622127
6.774052241
7.538461538
51.06593228
94
0.066
0.132
0.851515642
0.148484358
6.734716137
7.424242424
50.00016526
0.067
0.134
0.848196681
0.151803319
6.587471248
7.313432836
48.17702853
0.068
0.136
0.847312749
0.152687251
6.549335296
7.205882353
47.19373963
0.069
0.138
0.84385754
0.15614246
6.404407867
7.101449275
45.48057761
0.07
0.14
0.842960448
0.157039552
6.367822529
7
44.5747577
0.071
0.142
0.841998645
0.158001355
6.329059629
6.901408451
43.67942561
0.072
0.144
0.838448353
0.161551647
6.189970929
6.805555556
42.12619105
0.073
0.146
0.837484009
0.162515991
6.153240637
6.712328767
41.30257414
0.074
0.148
0.836460771
0.163539229
6.114740829
6.621621622
40.48950008
0.075
0.15
0.83280219
0.16719781
5.980939588
6.533333333
39.07547198
0.076
0.152
0.831785771
0.168214229
5.944800301
6.447368421
38.32831773
0.077
0.154
0.827932889
0.172067111
5.811685883
6.363636364
36.98345562
0.078
0.156
0.826930582
0.173069418
5.778028321
6.282051282
36.29787022
0.079
0.158
0.825875354
0.174124646
5.743012385
6.202531646
35.62121606
0.08
0.16
0.821868924
0.178131076
5.613843593
6.125
34.38479201
0.081
0.162
0.82083624
0.17916376
5.581485904
6.049382716
33.76454436
0.082
0.164
0.819754077
0.180245923
5.547975706
5.975609756
33.15253776
0.083
0.166
0.818624857
0.181375143
5.51343466
5.903614458
32.54919257
0.084
0.168
0.814513086
0.185486914
5.391215895
5.833333333
31.44875939
0.085
0.17
0.813414902
0.186585098
5.359484829
5.764705882
30.89585372
0.086
0.172
0.812272221
0.187727779
5.326862136
5.697674419
30.35072612
0.087
0.174
0.807945686
0.192054314
5.206860386
5.632183908
29.32599528
0.088
0.176
0.806841811
0.193158189
5.177103831
5.568181818
28.82705543
0.089
0.178
0.805695474
0.194304526
5.146560517
5.505617978
28.33499611
0.09
0.18
0.804508592
0.195491408
5.115314319
5.444444444
27.85004463
0.091
0.182
0.800008554
0.199991446
5.000213855
5.384615385
26.92422845
0.092
0.184
0.798868193
0.201131807
4.971864045
5.326086957
26.48058024
0.093
0.186
0.797688718
0.202311282
4.942878073
5.268817204
26.04312103
0.094
0.188
0.796471819
0.203528181
4.913324509
5.212765957
25.61201074
0.095
0.19
0.795219117
0.204780883
4.883268329
5.157894737
25.18738401
0.096
0.192
0.790589315
0.209410685
4.775305523
5.104166667
24.37395527
0.097
0.194
0.789390147
0.210609853
4.748115935
5.051546392
23.98532792
0.098
0.196
0.788155997
0.211844003
4.720454599
5
23.602273
0.099
0.198
0.786888311
0.213111689
4.692375187
4.949494949
23.22488729
0.1
0.2
0.781985601
0.218014399
4.58685301
4.9
22.47557975
95
Cost
Prob
IC-Cost1
0.01
5.9877553
ICCost2
40
OCCost1
1003.128
OCCost2
1080
% In control
P/C
0.333518445
% In control
No P/C
0.927536232
Net Cost 1
Net Cost 2
670.56307
115.36232
0.011
6.6252696
44
1003.613
1088
0.366335972
0.933687003
638.3807
113.23077
0.012
7.2698251
48
1004.13
1096
0.397902902
0.938875306
607.47668
112.05868
0.013
7.9505959
52
1004.695
1104
0.428940803
0.943310658
577.15041
111.63719
0.014
8.6272095
56
1005.284
1112
0.458132282
0.947145877
548.68345
111.81395
0.015
9.2457323
60
1005.87
1120
0.484342119
0.95049505
523.16312
112.47525
0.016
10.026442
64
1006.571
1128
0.512518343
0.953445065
495.82363
113.53445
0.017
10.674503
68
1007.228
1136
0.536291157
0.956063269
472.7853
114.92443
0.018
11.284989
72
1007.894
1144
0.558104364
0.958402662
451.68199
116.59235
0.019
12.021328
76
1008.653
1152
0.580631011
0.960505529
429.97789
118.49605
0.02
12.905129
80
1009.519
1160
0.603659274
0.962406015
407.9039
120.6015
0.021
13.458059
84
1010.237
1168
0.620912847
0.964131994
391.32414
122.88092
0.022
14.37531
88
1011.176
1176
0.641335211
0.965706447
371.89253
125.31139
0.023
15.1489
92
1012.066
1184
0.658768659
0.967148489
355.32827
127.87385
0.024
15.714828
96
1012.874
1192
0.673183568
0.968474149
341.60279
130.55233
0.025
16.713326
100
1013.939
1200
0.690420734
0.96969697
325.43362
133.33333
0.026
17.473125
104
1014.905
1208
0.704554891
0.970828471
312.15935
136.20537
0.027
17.918415
108
1015.733
1216
0.715686328
0.971878515
301.61068
139.15861
0.028
18.854186
112
1016.846
1224
0.729394264
0.972855592
288.91659
142.18458
0.029
19.911253
116
1018.052
1232
0.742824698
0.973767051
276.60853
145.27597
0.03
20.582468
120
1019.082
1240
0.753278153
0.974619289
266.93416
148.4264
0.031
21.324225
124
1020.177
1248
0.763502096
0.975417896
257.55088
151.63029
0.032
22.142667
128
1021.34
1256
0.773472601
0.976167779
248.48835
154.88275
0.033
23.044764
132
1022.575
1264
0.78317582
0.976873265
239.76713
158.17946
0.034
24.038453
136
1023.886
1272
0.792606277
0.977538185
231.40054
161.51662
0.035
24.428153
140
1024.909
1280
0.799414738
0.978165939
225.10983
164.89083
0.036
25.580459
144
1026.356
1288
0.808318878
0.978759558
217.41028
168.29907
0.037
26.062988
148
1027.485
1296
0.814732071
0.979321754
211.59436
171.73863
0.038
27.399211
152
1029.083
1304
0.823109275
0.979854956
204.5877
175.20709
0.039
27.988666
156
1030.321
1312
0.829101509
0.980361351
199.28582
178.70228
0.04
29.540425
160
1032.089
1320
0.83698157
0.980842912
192.97436
182.22222
0.041
30.254569
164
1033.446
1328
0.842554654
0.981301421
188.20244
185.76515
0.042
31.017525
168
1034.858
1336
0.847960215
0.981738495
183.64116
189.32944
0.043
31.832021
172
1036.324
1344
0.8531972
0.982155603
179.29427
192.91363
0.044
32.701023
176
1037.847
1352
0.858266349
0.982554082
175.164
196.5164
0.045
33.627768
180
1039.427
1360
0.863169866
0.982935154
171.25142
200.13652
0.046
34.615798
184
1041.067
1368
0.867911149
0.983299933
167.55673
203.77288
96
0.047
35.668995
188
1042.768
1376
0.872494553
0.983649444
164.07954
207.42446
0.048
36.791626
192
1044.532
1384
0.876925199
0.983984625
160.81909
211.09033
0.049
37.988383
196
1046.363
1392
0.881208811
0.98430634
157.77444
214.76962
0.05
39.264437
200
1048.264
1400
0.885351585
0.984615385
154.94469
218.46154
0.051
39.095027
204
1049.447
1408
0.887784858
0.984912492
152.47177
222.16536
0.052
40.463718
208
1051.446
1416
0.891673492
0.985198342
149.97989
225.8804
0.053
41.923842
212
1053.522
1424
0.895436537
0.985473562
147.70008
229.60604
0.054
41.777087
216
1054.809
1432
0.897640251
0.985738734
145.4708
233.3417
0.055
43.343242
220
1056.989
1440
0.901168367
0.985994398
143.52349
237.08683
0.056
45.015861
224
1059.256
1448
0.904588172
0.986241057
141.78632
240.84095
0.057
44.891744
228
1060.639
1456
0.906561834
0.986479178
139.80132
244.60357
0.058
46.687159
232
1063.017
1464
0.909770023
0.986709197
138.39062
248.37427
0.059
48.607802
236
1065.496
1472
0.912887734
0.986931521
137.19128
252.15264
0.06
48.506315
240
1066.969
1480
0.914639461
0.98714653
135.44283
255.9383
0.061
50.570955
244
1069.572
1488
0.917570251
0.987354578
134.56697
259.73091
0.062
50.484536
248
1071.113
1496
0.919211986
0.987555998
132.93908
263.53011
0.063
52.705827
252
1073.848
1504
0.921970481
0.987751102
132.38505
267.33562
0.064
52.634482
256
1075.454
1512
0.923504806
0.987940183
130.8753
271.14713
0.065
55.026815
260
1078.33
1520
0.926105557
0.988123515
130.64324
274.96437
0.066
54.970541
264
1080
1528
0.92753601
0.988301357
129.24825
278.78708
0.067
57.55032
268
1083.027
1536
0.929993263
0.988473951
129.34061
282.61503
0.068
57.509133
272
1084.757
1544
0.931323967
0.988641527
128.05644
286.44798
0.069
60.295102
276
1087.95
1552
0.933651544
0.988804299
128.47839
290.28571
0.07
60.269067
280
1089.737
1560
0.934886939
0.988962472
127.30087
294.12804
0.071
60.250474
284
1091.576
1568
0.936111248
0.989116239
126.14059
297.97475
0.072
63.271952
288
1094.953
1576
0.938242818
0.989265779
126.98565
301.82568
0.073
63.268393
292
1096.846
1584
0.939377047
0.989411266
125.92694
305.68064
0.074
63.272496
296
1098.791
1592
0.940499449
0.989552863
124.88642
309.53949
0.075
66.554366
300
1102.366
1600
0.942457836
0.989690722
126.15721
313.40206
0.076
66.573365
304
1104.361
1608
0.94349592
0.98982499
125.21263
317.26821
0.077
70.1373
308
1108.156
1616
0.945370222
0.989955806
126.84406
321.13781
0.078
70.171516
312
1110.199
1624
0.946328575
0.9900833
125.99129
325.01071
0.079
70.213073
316
1112.293
1632
0.947276351
0.990207599
125.15531
328.8868
0.08
74.100647
320
1116.33
1640
0.949013097
0.99032882
127.24072
332.76596
0.081
74.15748
324
1118.467
1648
0.94988673
0.990447077
126.49127
336.64807
0.082
74.221582
328
1120.654
1656
0.950750334
0.990562476
125.75805
340.53303
0.083
74.293349
332
1122.891
1664
0.951603254
0.990675121
125.04206
344.42074
0.084
78.552353
336
1127.191
1672
0.95316283
0.990785109
127.66762
348.31109
0.085
78.639048
340
1129.467
1680
0.953948361
0.990892532
127.03141
352.20401
0.086
78.733221
344
1131.793
1688
0.954724109
0.990997479
126.41142
356.09939
97
0.087
83.403573
348
1136.398
1696
0.956185782
0.991100036
129.53969
359.99715
0.088
83.513043
352
1138.759
1704
0.956899089
0.991200282
128.99509
363.89722
0.089
83.629723
356
1141.168
1712
0.957603603
0.991298294
128.46554
367.79951
0.09
83.753953
360
1143.626
1720
0.958298955
0.991394148
127.95174
371.70396
0.091
88.900174
364
1148.565
1728
0.959629254
0.991487913
131.67964
375.61049
0.092
89.039687
368
1151.054
1736
0.960268039
0.991579656
131.2356
379.51903
0.093
89.186385
372
1153.591
1744
0.960898746
0.991669444
130.80596
383.42952
0.094
89.340584
376
1156.177
1752
0.961521111
0.991757336
130.39125
387.34191
0.095
89.502586
380
1158.81
1760
0.962134904
0.991843393
129.992
391.25612
0.096
95.20079
384
1164.11
1768
0.963312898
0.991927672
134.41596
395.1721
0.097
95.377582
388
1166.769
1776
0.96387672
0.992010227
134.07974
399.08981
0.098
95.561716
392
1169.475
1784
0.964433104
0.99209111
133.75748
403.00917
0.099
95.753474
396
1172.229
1792
0.964981882
0.992170373
133.44962
406.93016
0.1
102.07944
400
1177.971
1800
0.966073346
0.992248062
138.58084
410.85271
98
Appendix4
Process In Control
p0
LCL
UCL
Prob No Signal
Prob Signal
Events to Signal
E[X|LCL<X<UCL]
ARL0
0.0005
2
13212
0.997649577
0.00235042
425.4552433
1960
833892.3
0.0006
2
11010
0.997449998
0.00255
392.1565217
1633.333333
640522.3
0.0007
1
9437
0.997949814
0.00205019
487.7606166
1400
682864.9
0.0008
1
8257
0.99784972
0.00215028
465.0557222
1225
569693.3
0.0009
1
7339
0.997749356
0.00225064
444.3172473
1088.888889
483812.1
0.001
1
6605
0.997649532
0.00235047
425.4472176
980
416938.3
0.0011
1
6004
0.997549033
0.00245097
408.002192
890.9090909
363492.9
0.0012
1
5504
0.997449885
0.00255012
392.1391422
816.6666667
320247
0.0013
1
5080
0.997349115
0.00265088
377.2325346
753.8461538
284375.3
0.0014
0
4717
0.998649156
0.00135084
740.2781531
700
518194.7
0.0015
0
4402
0.998648386
0.00135161
739.8563766
653.3333333
483372.8
0.0016
0
4127
0.998648969
0.00135103
740.1752039
612.5
453357.3
0.0017
0
3884
0.998648739
0.00135126
740.0496675
576.4705882
426616.9
0.0018
0
3668
0.99864851
0.00135149
739.924148
544.4444444
402847.6
0.0019
0
3475
0.998648957
0.00135104
740.1689893
515.7894737
381771.4
0.002
0
3301
0.998648593
0.00135141
739.9693846
490
362585
0.0021
0
3144
0.998649446
0.00135055
740.4366152
466.6666667
345537.1
0.0022
0
3001
0.998649487
0.00135051
740.4592972
445.4545455
329841
0.0023
0
2870
0.998648176
0.00135182
739.7410265
426.0869565
315194
0.0024
0
2750
0.998647134
0.00135287
739.1713861
408.3333333
301828.3
0.0025
0
2640
0.998647446
0.00135255
739.3420327
392
289822.1
0.0026
0
2539
0.998649653
0.00135035
740.550137
376.9230769
279130.4
0.0027
0
2444
0.998646445
0.00135356
738.7952331
362.962963
268155.3
0.0028
0
2357
0.998647841
0.00135216
739.5580632
350
258845.3
0.0029
0
2276
0.998649236
0.00135076
740.3217658
337.9310345
250177.7
0.003
0
2200
0.998649007
0.00135099
740.1962366
326.6666667
241797.4
0.0031
0
2129
0.998649184
0.00135082
740.2931238
316.1290323
234028.1
0.0032
0
2062
0.998647465
0.00135253
739.3525611
306.25
226426.7
0.0033
0
2000
0.998649943
0.00135006
740.7095369
296.969697
219968.3
0.0034
0
1941
0.998649444
0.00135056
740.4356147
288.2352941
213419.7
0.0035
0
1885
0.998647183
0.00135282
739.1985039
280
206975.6
0.0036
0
1833
0.998649256
0.00135074
740.3328841
272.2222222
201535.1
0.0037
0
1783
0.998647267
0.00135273
739.2439403
264.8648649
195799.7
0.0038
0
1736
0.998647173
0.00135283
739.1926115
257.8947368
190633.9
0.0039
0
1691
0.998644908
0.00135509
737.9573004
251.2820513
185435.4
99
0.004
0
1649
0.998646714
0.00135329
738.9418742
245
181040.8
0.0041
0
1609
0.998648246
0.00135175
739.7795938
239.0243902
176825.4
0.0042
0
1570
0.998644625
0.00135537
737.8034421
233.3333333
172154.1
0.0043
0
1534
0.998647923
0.00135208
739.6028438
227.9069767
168560.6
0.0044
0
1499
0.998647423
0.00135258
739.3292203
222.7272727
164668.8
0.0045
0
1466
0.998649632
0.00135037
740.5386117
217.7777778
161272.9
0.0046
0
1434
0.998649132
0.00135087
740.264678
213.0434783
157708.6
0.0047
0
1403
0.998646327
0.00135367
738.73095
208.5106383
154033.3
0.0048
0
1374
0.998648132
0.00135187
739.7170566
204.1666667
151025.6
0.0049
0
1346
0.998648716
0.00135128
740.0366156
200
148007.3
0.005
0
1319
0.998648486
0.00135151
739.9111589
196
145022.6
Process Out of Control
Prob
0.0005
Shifted
Prob
0.001
Prob No
signal
0.997999182
0.0006
0.0012
0.997599624
0.0007
0.0014
0.0008
Prob Signal
E[X|LCL<X<UCL]
ARL1
0.002000818
Events to
Signal
499.7955093
980
489799.5991
0.002400376
416.6013545
816.6666667
340224.4395
0.998598185
0.001401815
713.361106
700
499352.7742
0.0016
0.998398186
0.001601814
624.292352
612.5
382379.0656
0.0009
0.0018
0.998198187
0.001801813
554.9964246
544.4444444
302164.7201
0.001
0.002
0.997998188
0.002001812
499.5474762
490
244778.2633
0.0011
0.0022
0.997798188
0.002201812
454.1714084
445.4545455
202312.7183
0.0012
0.0024
0.997598192
0.002401808
416.3529466
408.3333333
170010.7865
0.0013
0.0026
0.997398191
0.002601809
384.3479298
376.9230769
144869.6043
0.0014
0.0028
0.999998192
1.80794E-06
553115.0474
350
193590266.6
0.0015
0.003
0.999998191
1.8088E-06
552851.2176
326.6666667
180598064.4
0.0016
0.0032
0.999998194
1.80605E-06
553695.5663
306.25
169569267.2
0.0017
0.0034
0.999998195
1.80546E-06
553875.2404
288.2352941
159646392.8
0.0018
0.0036
0.999998195
1.80487E-06
554055.0581
272.2222222
150826099.1
0.0019
0.0038
0.999998198
1.80248E-06
554790.5885
257.8947368
143077572.8
0.002
0.004
0.999998198
1.80226E-06
554859.7244
245
135940632.5
0.0021
0.0042
0.999998201
1.79878E-06
555930.811
233.3333333
129717189.2
0.0022
0.0044
0.999998203
1.79748E-06
556334.7532
222.7272727
123910922.3
0.0023
0.0046
0.9999982
1.79978E-06
555624.1903
213.0434783
118372110.1
0.0024
0.0048
0.999998199
1.80136E-06
555136.9868
204.1666667
113340468.1
0.0025
0.005
0.999998201
1.79933E-06
555763.3607
196
108929618.7
0.0026
0.0052
0.999998208
1.79226E-06
557954.7697
188.4615385
105153014.3
0.0027
0.0054
0.9999982
1.7996E-06
555679.901
181.4814815
100845611.7
100
0.0028
0.0056
0.999998205
1.79469E-06
557200.5896
175
97510103.17
0.0029
0.0058
0.99999821
1.78979E-06
558725.7964
168.9655172
94405393.18
0.003
0.006
0.999998211
1.7892E-06
558908.4946
163.3333333
91288387.45
0.0031
0.0062
0.999998212
1.78754E-06
559427.9395
158.0645161
88425706.57
0.0032
0.0064
0.999998209
1.7909E-06
558377.3358
153.125
85501529.54
0.0033
0.0066
0.999998217
1.78315E-06
560806.2399
148.4848485
83271229.56
0.0034
0.0068
0.999998217
1.78328E-06
560764.9851
144.1176471
80816130.21
0.0035
0.007
0.999998212
1.78807E-06
559262.6738
140
78296774.33
0.0036
0.0072
0.999998219
1.78139E-06
561358.27
136.1111111
76407097.86
0.0037
0.0074
0.999998215
1.78546E-06
560079.0463
132.4324324
74172630.46
0.0038
0.0076
0.999998215
1.78452E-06
560375.3637
128.9473684
72258928.47
0.0039
0.0078
0.999998211
1.78931E-06
558873.6538
125.6410256
70217459.06
0.004
0.008
0.999998217
1.78335E-06
560743.5541
122.5
68691085.37
0.0041
0.0082
0.999998222
1.77811E-06
562394.2092
119.5121951
67212966.46
0.0042
0.0084
0.999998214
1.78647E-06
559761.607
116.6666667
65305520.82
0.0043
0.0086
0.999998223
1.77659E-06
562877.5993
113.9534884
64141865.97
0.0044
0.0088
0.999998223
1.77671E-06
562836.8642
111.3636364
62679559.88
0.0045
0.009
0.99999823
1.76972E-06
565062.0999
108.8888889
61528984.21
0.0046
0.0092
0.99999823
1.76984E-06
565021.5205
106.5217391
60187075.01
0.0047
0.0094
0.999998224
1.77603E-06
563054.2588
104.2553191
58701401.45
0.0048
0.0096
0.99999823
1.7701E-06
564940.4985
102.0833333
57671009.22
0.0049
0.0098
0.999998233
1.76738E-06
565809.0845
100
56580908.45
0.005
0.01
0.999998233
1.7668E-06
565996.2065
98
55467628.24
Cost
Prob
IC-Cost1
0.0005
0.0239839
ICCost2
10
OCCost1
1000.041
OCCost2
1020
% In control
P/C
0.012898035
% In control
No P/C
0.864864865
Net Cost 1
Net Cost 2
987.14258
146.48649
0.0006
0.0312245
12
1000.059
1024
0.018463788
0.884792627
981.59449
128.58986
0.0007
0.0292884
14
1000.04
1028
0.012654404
0.899598394
987.38551
115.80723
0.0008
0.0351066
16
1000.052
1032
0.016461792
0.911032028
983.59023
106.39146
0.0009
0.0413384
18
1000.066
1036
0.020741192
0.920127796
979.32448
99.309904
0.001
0.0479687
20
1000.082
1040
0.025479912
0.927536232
974.60094
93.913043
0.0011
0.0550217
22
1000.099
1044
0.030664159
0.933687003
969.43335
89.771883
0.0012
0.0624518
24
1000.118
1048
0.036278961
0.938875306
963.83668
86.591687
0.0013
0.0703296
26
1000.138
1052
0.042308566
0.943310658
957.82662
84.163265
0.0014
0.0385955
28
1000
1056
3.30584E-05
0.947145877
999.96705
82.334038
0.0015
0.0413759
30
1000
1060
3.54366E-05
0.95049505
999.96468
80.990099
101
0.0016
0.0441153
32
1000
1064
3.77413E-05
0.953445065
999.96238
80.044693
0.0017
0.0468805
34
1000
1068
4.0087E-05
0.956063269
999.96004
79.43058
0.0018
0.0496466
36
1000
1072
4.24312E-05
0.958402662
999.9577
79.094842
0.0019
0.0523874
38
1000
1076
4.4729E-05
0.960505529
999.95541
78.995261
0.002
0.0551595
40
1000
1080
4.70772E-05
0.962406015
999.95307
79.097744
0.0021
0.0578809
42
1000
1084
4.93357E-05
0.964131994
999.95082
79.374462
0.0022
0.0606353
44
1000
1088
5.16473E-05
0.965706447
999.94852
79.802469
0.0023
0.063453
46
1000
1092
5.40639E-05
0.967148489
999.94611
80.362681
0.0024
0.0662628
48
1000
1096
5.64638E-05
0.968474149
999.94372
81.039092
0.0025
0.0690079
50
1000
1100
5.87501E-05
0.96969697
999.94144
81.818182
0.0026
0.0716511
52
1000
1104
6.086E-05
0.970828471
999.93933
82.688448
0.0027
0.0745836
54
1000
1108
6.34593E-05
0.971878515
999.93674
83.640045
0.0028
0.0772662
56
1000
1112
6.56299E-05
0.972855592
999.93458
84.664495
0.0029
0.0799432
58
1000
1116
6.77881E-05
0.973767051
999.93243
85.75446
0.003
0.0827139
60
1000
1120
7.01026E-05
0.974619289
999.93012
86.903553
0.0031
0.0854598
62
1000
1124
7.23719E-05
0.975417896
999.92786
88.106195
0.0032
0.0883288
64
1000
1128
7.48469E-05
0.976167779
999.92539
89.357483
0.0033
0.0909222
66
1000
1132
7.68514E-05
0.976873265
999.9234
90.653099
0.0034
0.0937121
68
1000
1136
7.91858E-05
0.977538185
999.92107
91.989218
0.0035
0.0966298
70
1000
1140
8.17336E-05
0.978165939
999.91853
93.362445
0.0036
0.0992383
72
1000
1144
8.37548E-05
0.978759558
999.91652
94.769754
0.0037
0.1021452
74
1000
1148
8.62778E-05
0.979321754
999.914
96.208437
0.0038
0.1049131
76
1000
1152
8.85625E-05
0.979854956
999.91172
97.676068
0.0039
0.1078543
78
1000
1156
9.11371E-05
0.980361351
999.90916
99.170463
0.004
0.1104724
80
1000
1160
9.31621E-05
0.980842912
999.90714
100.68966
0.0041
0.1131059
82
1000
1164
9.52107E-05
0.981301421
999.9051
102.23186
0.0042
0.116175
84
1000
1168
9.79913E-05
0.981738495
999.90233
103.79547
0.0043
0.1186517
86
1000
1172
9.97689E-05
0.982155603
999.90055
105.37901
0.0044
0.1214559
88
1000
1176
0.000102096
0.982554082
999.89824
106.98116
0.0045
0.1240134
90
1000
1180
0.000104005
0.982935154
999.89633
108.60068
0.0046
0.1268162
92
1000
1184
0.000106324
0.983299933
999.89402
110.23647
0.0047
0.1298421
94
1000
1188
0.000109014
0.983649444
999.89134
111.88751
0.0048
0.1324279
96
1000
1192
0.000110962
0.983984625
999.8894
113.55285
0.0049
0.1351284
98
1000
1196
0.0001131
0.98430634
999.88727
115.23164
0.005
0.1379096
100
1000
1200
0.000115369
0.984615385
999.88501
116.92308
102
Appendix5
Process In-Control
p0
LCL
UCL
Prob No Signal
0.0005
0
7998
0.9816752
0.0006
0
6665
0.981677033
0.0007
0
5713
0.0008
0
0.0009
Prob Signal
E[X|LCL<X<UCL]
ARL0
0.0183248
Events to
Signal
54.57085551
1960
106958.9
0.01832297
54.57631514
1633.333333
89141.31
0.981680699
0.0183193
54.58723659
1400
76422.13
4998
0.981669701
0.0183303
54.55448316
1225
66829.24
0
4443
0.981677035
0.01832296
54.57632006
1088.888889
59427.55
0.001
0
3998
0.981666033
0.01833397
54.54357008
980
53452.7
0.0011
0
3635
0.981677037
0.01832296
54.57632516
890.9090909
48622.54
0.0012
0
3332
0.981677038
0.01832296
54.57632826
816.6666667
44570.67
0.0013
0
3075
0.981662365
0.01833763
54.53265974
753.8461538
41109.24
0.0014
0
2856
0.981680707
0.01831929
54.58725956
700
38211.08
0.0015
0
2665
0.981666036
0.01833396
54.54357919
653.3333333
35635.14
0.0016
0
2498
0.981655025
0.01834498
54.51083958
612.5
33387.89
0.0017
0
2351
0.981655025
0.01834497
54.51084122
576.4705882
31423.9
0.0018
0
2221
0.981677047
0.01832295
54.57635452
544.4444444
29713.79
0.0019
0
2104
0.981675214
0.01832479
54.57089762
515.7894737
28147.09
0.002
0
1998
0.981647681
0.01835232
54.48902653
490
26699.62
0.0021
0
1903
0.981655029
0.01834497
54.51085143
466.6666667
25438.4
0.0022
0
1817
0.981677055
0.01832295
54.57637934
445.4545455
24311.3
0.0023
0
1738
0.981678892
0.01832111
54.58185065
426.0869565
23256.61
0.0024
0
1665
0.981655033
0.01834497
54.51086293
408.3333333
22258.6
0.0025
0
1598
0.981638496
0.0183615
54.4617684
392
21349.01
0.0026
0
1537
0.981662382
0.01833762
54.53270953
376.9230769
20554.64
0.0027
0
1480
0.981660548
0.01833945
54.52725555
362.962963
19791.37
0.0028
0
1427
0.98165504
0.01834496
54.51088336
350
19078.81
0.0029
0
1378
0.981667897
0.0183321
54.54911584
337.9310345
18433.84
0.003
0
1332
0.981666064
0.01833394
54.54366132
326.6666667
17817.6
0.0031
0
1289
0.981666067
0.01833393
54.54366972
316.1290323
17242.84
0.0032
0
1248
0.981625626
0.01837437
54.42362231
306.25
16667.23
0.0033
0
1211
0.981677088
0.01832291
54.57647781
296.969697
16207.56
0.0034
0
1175
0.981655054
0.01834495
54.510925
288.2352941
15711.97
0.0035
0
1141
0.98162931
0.01837069
54.43453763
280
15241.67
0.0036
0
1110
0.9816771
0.0183229
54.57651237
272.2222222
14856.94
0.0037
0
1080
0.981678939
0.01832106
54.58199268
264.8648649
14456.85
0.0038
0
1051
0.981640356
0.01835964
54.46728576
257.8947368
14046.83
0.0039
0
1024
0.981638518
0.01836148
54.46183614
251.2820513
13685.28
103
0.004
0
998
0.981610903
0.0183891
54.38004862
245
13323.11
0.0041
0
974
0.981638524
0.01836148
54.46185168
239.0243902
13017.71
0.0042
0
951
0.981655079
0.01834492
54.51100105
233.3333333
12719.23
0.0043
0
929
0.98166611
0.01833389
54.54379919
227.9069767
12430.91
0.0044
0
908
0.981677136
0.01832286
54.57662073
222.7272727
12155.7
0.0045
0
887
0.981610911
0.01838909
54.38007147
217.7777778
11842.77
0.0046
0
868
0.981636698
0.0183633
54.45643845
213.0434783
11601.59
0.0047
0
850
0.981678988
0.01832101
54.58213825
208.5106383
11380.96
0.0048
0
832
0.981655104
0.0183449
54.51107351
204.1666667
11129.34
0.0049
0
815
0.981655108
0.01834489
54.51108687
200
10902.22
0.005
0
798
0.981592477
0.01840752
54.32561427
196
10647.82
Process Out of Control
Prob
Shifted Prob
Prob No signal
Prob Signal
E[X|LCL<X<UCL]
ARL1
0.000335127
Events to
Signal
2983.94392
0.0005
0.001
0.999664873
0.0006
0.0012
0.999665074
980
2924265.042
0.000334926
2985.737076
816.6666667
2438351.945
0.0007
0.0014
0.999665343
0.000334657
2988.129813
700
2091690.869
0.0008
0.0016
0.999665075
0.000334925
2985.740267
612.5
1828765.914
0.0009
0.0018
0.999665477
0.000334523
2989.330376
544.4444444
1627524.316
0.001
0.002
0.999665209
0.000334791
2986.939832
490
1463600.518
0.0011
0.0022
0.999665745
0.000334255
2991.730315
445.4545455
1332679.867
0.0012
0.0024
0.999665879
0.000334121
2992.931608
408.3333333
1222113.74
0.0013
0.0026
0.999665478
0.000334522
2989.340773
376.9230769
1126751.522
0.0014
0.0028
0.999666281
0.000333719
2996.536902
350
1048787.916
0.0015
0.003
0.999665881
0.000334119
2992.941704
326.6666667
977694.2898
0.0016
0.0032
0.999665613
0.000334387
2990.548157
306.25
915855.3729
0.0017
0.0034
0.999665748
0.000334252
2991.750261
288.2352941
862328.0165
0.0018
0.0036
0.999666684
0.000333316
3000.157958
272.2222222
816709.6663
0.0019
0.0038
0.999666752
0.000333248
3000.7641
257.8947368
773881.2679
0.002
0.004
0.999665883
0.000334117
2992.961759
245
733275.6309
0.0021
0.0042
0.999666285
0.000333715
2996.567543
233.3333333
699199.0934
0.0022
0.0044
0.999667221
0.000332779
3004.993294
222.7272727
669293.961
0.0023
0.0046
0.999667421
0.000332579
3006.807046
213.0434783
640580.6316
0.0024
0.0048
0.999666688
0.000333312
3000.189833
204.1666667
612538.7576
0.0025
0.005
0.99966622
0.00033378
2995.987859
196
587213.6203
0.0026
0.0052
0.999667223
0.000332777
3005.01847
188.4615385
566330.4039
0.0027
0.0054
0.999667291
0.000332709
3005.627866
181.4814815
545465.7979
104
0.0028
0.0056
0.999667225
0.000332775
3005.032038
175
525880.6066
0.0029
0.0058
0.999667826
0.000332174
3010.468784
168.9655172
508665.4152
0.003
0.006
0.999667893
0.000332107
3011.080506
163.3333333
491809.8159
0.0031
0.0062
0.999668027
0.000331973
3012.296912
158.0645161
476137.2538
0.0032
0.0064
0.999666693
0.000333307
3000.241457
153.125
459411.9732
0.0033
0.0066
0.999668695
0.000331305
3018.364278
148.4848485
448181.3625
0.0034
0.0068
0.99966803
0.00033197
3012.32221
144.1176471
434128.7891
0.0035
0.007
0.999667231
0.000332769
3005.084686
140
420711.856
0.0036
0.0072
0.999669097
0.000330903
3022.029809
136.1111111
411331.8351
0.0037
0.0074
0.999669297
0.000330703
3023.860421
132.4324324
400457.1909
0.0038
0.0076
0.999668034
0.000331966
3012.35825
128.9473684
388435.6691
0.0039
0.0078
0.999668102
0.000331898
3012.972569
125.6410256
378552.9638
0.004
0.008
0.999667235
0.000332765
3005.12722
122.5
368128.0844
0.0041
0.0082
0.999668371
0.000331629
3015.413339
119.5121951
360378.6674
0.0042
0.0084
0.999669103
0.000330897
3022.092889
116.6666667
352577.5038
0.0043
0.0086
0.999669636
0.000330364
3026.964167
113.9534884
344933.126
0.0044
0.0088
0.999670168
0.000329832
3031.844445
111.3636364
337637.2223
0.0045
0.009
0.999667908
0.000332092
3011.217502
108.8888889
327888.128
0.0046
0.0092
0.999668975
0.000331025
3020.924064
106.5217391
321794.0851
0.0047
0.0094
0.999670636
0.000329364
3036.150002
104.2553191
316534.7874
0.0048
0.0096
0.999669908
0.000330092
3029.458986
102.0833333
309257.2715
0.0049
0.0098
0.999670042
0.000329958
3030.689858
100
303068.9858
0.005
0.01
0.999667914
0.000332086
3011.271444
98
295104.6015
Cost
Prob
IC-Cost1
0.0005
0.1869878
ICCost2
10
OCCost1
1000.007
OCCost2
1020
% In control
P/C
0.002183805
% In control
No P/C
0.864864865
Net Cost 1
Net Cost 2
997.82343
146.48649
0.0006
0.2243629
12
1000.008
1024
0.002617853
0.884792627
997.39092
128.58986
0.0007
0.2617043
14
1000.01
1028
0.003050392
0.899598394
996.95994
115.80723
0.0008
0.2992702
16
1000.011
1032
0.003487423
0.911032028
996.52452
106.39146
0.0009
0.3365443
18
1000.012
1036
0.00391695
0.920127796
996.09661
99.309904
0.001
0.3741626
20
1000.014
1040
0.00435374
0.927536232
995.66149
93.913043
0.0011
0.4113318
22
1000.015
1044
0.004779401
0.933687003
995.2375
89.771883
0.0012
0.4487256
24
1000.016
1048
0.005209547
0.938875306
994.80907
86.591687
0.0013
0.4865087
26
1000.018
1052
0.005647965
0.943310658
994.37243
84.163265
105
0.0014
0.5234084
28
1000.019
1056
0.00606527
0.947145877
993.95686
82.334038
0.0015
0.5612438
30
1000.02
1060
0.006503442
0.95049505
993.52053
80.990099
0.0016
0.5990196
32
1000.022
1064
0.00693951
0.953445065
993.08633
80.044693
0.0017
0.6364583
34
1000.023
1068
0.007367093
0.956063269
992.66062
79.43058
0.0018
0.6730881
36
1000.024
1072
0.007775392
0.958402662
992.25414
79.094842
0.0019
0.7105529
38
1000.026
1076
0.008202171
0.960505529
991.82929
78.995261
0.002
0.7490742
40
1000.027
1080
0.008652441
0.962406015
991.38108
79.097744
0.0021
0.7862131
42
1000.029
1084
0.009070306
0.964131994
990.96517
79.374462
0.0022
0.8226628
44
1000.03
1088
0.009471744
0.965706447
990.56565
79.802469
0.0023
0.8599704
46
1000.031
1092
0.009892104
0.967148489
990.14732
80.362681
0.0024
0.898529
48
1000.033
1096
0.01034028
0.968474149
989.70132
81.039092
0.0025
0.9368114
50
1000.034
1100
0.010781424
0.96969697
989.26237
81.818182
0.0026
0.9730165
52
1000.035
1104
0.011174542
0.970828471
988.87125
82.688448
0.0027
1.0105412
54
1000.037
1108
0.011597022
0.971878515
988.45094
83.640045
0.0028
1.0482835
56
1000.038
1112
0.012023733
0.972855592
988.02645
84.664495
0.0029
1.0849612
58
1000.039
1116
0.012425606
0.973767051
987.62671
85.75446
0.003
1.1224859
60
1000.041
1120
0.012845993
0.974619289
987.20857
86.903553
0.0031
1.1599019
62
1000.042
1124
0.013263225
0.975417896
986.79361
88.106195
0.0032
1.1999591
64
1000.044
1128
0.013739449
0.976167779
986.31997
89.357483
0.0033
1.233992
66
1000.045
1132
0.014078888
0.976873265
985.98248
90.653099
0.0034
1.2729147
68
1000.046
1136
0.014527995
0.977538185
985.5359
91.989218
0.0035
1.3121921
70
1000.048
1140
0.014984365
0.978165939
985.08212
93.362445
0.0036
1.3461723
72
1000.049
1144
0.015320834
0.978759558
984.74767
94.769754
0.0037
1.383427
74
1000.05
1148
0.015730335
0.979321754
984.34058
96.208437
0.0038
1.4238092
76
1000.051
1152
0.016209275
0.979854956
983.86446
97.676068
0.0039
1.461424
78
1000.053
1156
0.016625408
0.980361351
983.45084
99.170463
0.004
1.5011508
80
1000.054
1160
0.017088171
0.980842912
982.99088
100.68966
0.0041
1.5363684
82
1000.055
1164
0.017449215
0.981301421
982.63212
102.23186
0.0042
1.5724218
84
1000.057
1168
0.017828415
0.981738495
982.25533
103.79547
0.0043
1.6088924
86
1000.058
1172
0.01821633
0.982155603
981.8699
105.37901
0.0044
1.6453184
88
1000.059
1176
0.018602638
0.982554082
981.4861
106.98116
0.0045
1.6887939
90
1000.061
1180
0.019145161
0.982935154
980.947
108.60068
0.0046
1.7239018
92
1000.062
1184
0.019500656
0.983299933
980.5939
110.23647
0.0047
1.7573215
94
1000.063
1188
0.019818243
0.983649444
980.27852
111.88751
0.0048
1.7970511
96
1000.065
1192
0.020275155
0.983984625
979.82464
113.55285
0.0049
1.8344892
98
1000.066
1196
0.020680586
0.98430634
979.42198
115.23164
0.005
1.8783187
100
1000.068
1200
0.021226873
0.984615385
978.87933
116.92308
106
Appendix6
Process In- Control
p0
LCL
UCL
Prob No Signal
Prob Signal
Events to Signal
E[X|LCL<X<UCL]
ARL0
0.0005
1
3999
0.864097021
0.13590298
7.358190429
1960
14422.05
0.0006
1
3333
0.864037647
0.13596235
7.354977153
1633.333333
12013.13
0.0007
1
2857
0.863951189
0.13604881
7.350303114
1400
10290.42
0.0008
1
2499
0.863756376
0.13624362
7.339793033
1225
8991.246
0.0009
1
2222
0.863737653
0.13626235
7.338784505
1088.888889
7991.121
0.001
1
1999
0.863529269
0.13647073
7.32757852
980
7181.027
0.0011
1
1818
0.863537659
0.13646234
7.32802908
890.9090909
6528.608
0.0012
1
1666
0.863356373
0.13664363
7.318306896
816.6666667
5976.617
0.0013
1
1538
0.863283477
0.13671652
7.314404835
753.8461538
5513.936
0.0014
1
1428
0.863156374
0.13684363
7.307611077
700
5115.328
0.0015
1
1333
0.863097032
0.13690297
7.304443551
653.3333333
4772.236
0.0016
1
1249
0.862847891
0.13715211
7.291174805
612.5
4465.845
0.0017
1
1176
0.862856378
0.13714362
7.29162602
576.4705882
4203.408
0.0018
1
1111
0.862837696
0.1371623
7.290632846
544.4444444
3969.345
0.0019
1
1052
0.862602144
0.13739786
7.278133945
515.7894737
3753.985
0.002
1
999
0.862393594
0.13760641
7.267103558
490
3560.881
0.0021
1
952
0.862456391
0.13754361
7.270421415
466.6666667
3392.863
0.0022
1
909
0.862437727
0.13756227
7.269434965
445.4545455
3238.203
0.0023
1
869
0.862188583
0.13781142
7.256292869
426.0869565
3091.812
0.0024
1
833
0.862156405
0.13784359
7.25459897
408.3333333
2962.295
0.0025
1
799
0.861825672
0.13817433
7.237234416
392
2836.996
0.0026
1
769
0.861983538
0.13801646
7.245512516
376.9230769
2731.001
0.0027
1
740
0.861693574
0.13830643
7.230322026
362.962963
2624.339
0.0028
1
714
0.861756431
0.13824357
7.233609528
350
2531.763
0.0029
1
689
0.861507152
0.13849285
7.22058945
337.9310345
2440.061
0.003
1
666
0.861393572
0.13860643
7.214672599
326.6666667
2356.793
0.0031
1
645
0.861497145
0.13850286
7.220067753
316.1290323
2282.473
0.0032
1
624
0.861030486
0.13896951
7.195822821
306.25
2203.721
0.0033
1
606
0.861337849
0.13866215
7.211773298
296.969697
2141.678
0.0034
1
588
0.861156482
0.13884352
7.202352818
288.2352941
2075.972
0.0035
1
571
0.860961483
0.13903852
7.192251605
280
2013.83
0.0036
1
555
0.86079358
0.13920642
7.183576732
272.2222222
1955.529
0.0037
1
540
0.860693583
0.13930642
7.178420212
264.8648649
1901.311
0.0038
1
526
0.860702235
0.13929777
7.178866056
257.8947368
1851.392
0.0039
1
512
0.860330446
0.13966955
7.159756537
251.2820513
1799.118
107
0.004
1
499
0.860121565
0.13987843
7.149064848
245
1751.521
0.0041
1
487
0.860116834
0.13988317
7.148823015
239.0243902
1708.743
0.0042
1
476
0.860356577
0.13964342
7.161096297
233.3333333
1670.922
0.0043
1
465
0.860297305
0.13970269
7.158058061
227.9069767
1631.371
0.0044
1
454
0.859939243
0.14006076
7.139758655
222.7272727
1590.219
0.0045
1
444
0.859893623
0.14010638
7.137433867
217.7777778
1554.374
0.0046
1
434
0.859575985
0.14042402
7.12128904
213.0434783
1517.144
0.0047
1
425
0.859625658
0.14037434
7.123808971
208.5106383
1485.39
0.0048
1
416
0.859430428
0.14056957
7.113915095
204.1666667
1452.424
0.0049
1
408
0.859656684
0.14034332
7.125383846
200
1425.077
0.005
1
399
0.85898521
0.14101479
7.09145471
196
1389.925
Process Out of Control
Prob
0.0005
Shifted
Prob
0.001
Prob No
signal
0.980684367
0.0006
0.0012
0.0007
Prob Signal
E[X|LCL<X<UCL]
ARL1
0.019315633
Events to
Signal
51.77153715
980
50736.10641
0.980499025
0.019500975
51.27948823
816.6666667
41878.24872
0.0014
0.980306354
0.019693646
50.77779921
700
35544.45945
0.0008
0.0016
0.980084377
0.019915623
50.21183561
612.5
30754.74931
0.0009
0.0018
0.979910028
0.020089972
49.77607716
544.4444444
27100.30867
0.001
0.002
0.979684386
0.020315614
49.22322207
490
24119.37881
0.0011
0.0022
0.979517365
0.020482635
48.82184448
445.4545455
21747.91254
0.0012
0.0024
0.979299061
0.020700939
48.30698645
408.3333333
19725.3528
0.0013
0.0026
0.97911006
0.02088994
47.86993104
376.9230769
18043.2817
0.0014
0.0028
0.978906405
0.021093595
47.40775673
350
16592.71486
0.0015
0.003
0.978721066
0.021278934
46.99483477
326.6666667
15351.64602
0.0016
0.0032
0.978484424
0.021515576
46.47795588
306.25
14233.87399
0.0017
0.0034
0.978317426
0.021682574
46.11998636
288.2352941
13293.40783
0.0018
0.0036
0.978143062
0.021856938
45.7520629
272.2222222
12454.72823
0.0019
0.0038
0.977910122
0.022089878
45.2696031
257.8947368
11674.79238
0.002
0.004
0.977684459
0.022315541
44.81182008
245
10978.89592
0.0021
0.0042
0.977532128
0.022467872
44.50799753
233.3333333
10385.19942
0.0022
0.0044
0.977357757
0.022642243
44.16523494
222.7272727
9836.802328
0.0023
0.0046
0.97712117
0.02287883
43.70852813
213.0434783
9311.816862
0.0024
0.0048
0.976943159
0.023056841
43.37107592
204.1666667
8854.928
0.0025
0.005
0.976684515
0.023315485
42.8899497
196
8406.430141
0.0026
0.0052
0.976557833
0.023442167
42.65817235
188.4615385
8039.424788
0.0027
0.0054
0.976310233
0.023689767
42.21231927
181.4814815
7660.754239
0.0028
0.0056
0.976157874
0.023842126
41.9425687
175
7339.949522
108
0.0029
0.0058
0.975921268
0.024078732
41.53042675
168.9655172
7017.210037
0.003
0.006
0.975721287
0.024278713
41.18834397
163.3333333
6727.429515
0.0031
0.0062
0.975579885
0.024420115
40.94984825
158.0645161
6472.717949
0.0032
0.0064
0.975284613
0.024715387
40.46062445
153.125
6195.533118
0.0033
0.0066
0.975198207
0.024801793
40.31966541
148.4848485
5986.859409
0.0034
0.0068
0.974979962
0.025020038
39.96796423
144.1176471
5760.088963
0.0035
0.007
0.974758034
0.025241966
39.6165655
140
5546.319169
0.0036
0.0072
0.974543407
0.025456593
39.28255377
136.1111111
5346.792042
0.0037
0.0074
0.974347095
0.025652905
38.98193981
132.4324324
5162.47311
0.0038
0.0076
0.974180071
0.025819929
38.72977281
128.9473684
4994.102283
0.0039
0.0078
0.973910459
0.026089541
38.32953637
125.6410256
4815.762262
0.004
0.008
0.973684755
0.026315245
38.00078622
122.5
4655.096312
0.0041
0.0082
0.973514177
0.026485823
37.7560472
119.5121951
4512.30808
0.0042
0.0084
0.973409439
0.026590561
37.60732976
116.6666667
4387.521805
0.0043
0.0086
0.973224082
0.026775918
37.34699186
113.9534884
4255.820003
0.0044
0.0088
0.972958275
0.027041725
36.97988991
111.3636364
4118.215012
0.0045
0.009
0.97277662
0.02722338
36.73313127
108.8888889
3999.82985
0.0046
0.0092
0.972521645
0.027478355
36.39228102
106.5217391
3876.569065
0.0047
0.0094
0.972365695
0.027634305
36.18690548
104.2553191
3772.677379
0.0048
0.0096
0.972143726
0.027856274
35.89855555
102.0833333
3664.644212
0.0049
0.0098
0.972035257
0.027964743
35.75931337
100
3575.931337
0.005
0.01
0.971684978
0.028315022
35.31694209
98
3461.060325
Cost
Prob
IC-Cost1
0.0005
1.3867651
ICCost2
10
OCCost1
1000.394
OCCost2
1020
% In control
P/C
0.112013233
% In control
No P/C
0.864864865
Net Cost 1
Net Cost 2
888.49214
146.48649
0.0006
1.6648451
12
1000.478
1024
0.132564875
0.884792627
868.07009
128.58986
0.0007
1.9435544
14
1000.563
1028
0.152582727
0.899598394
848.19065
115.80723
0.0008
2.2243857
16
1000.65
1032
0.172252542
0.911032028
828.6689
106.39146
0.0009
2.5027778
18
1000.738
1036
0.191043016
0.920127796
810.03213
99.309904
0.001
2.785117
20
1000.829
1040
0.209702826
0.927536232
791.53654
93.913043
0.0011
3.0634403
22
1000.92
1044
0.227370324
0.933687003
774.03674
89.771883
0.0012
3.3463745
24
1001.014
1048
0.244972768
0.938875306
756.61254
86.591687
0.0013
3.6271731
26
1001.108
1052
0.261830636
0.943310658
739.93729
84.163265
0.0014
3.9098179
28
1001.205
1056
0.278349035
0.947145877
723.6091
82.334038
0.0015
4.1909072
30
1001.303
1060
0.294230606
0.95049505
707.92196
80.990099
0.0016
4.4784362
32
1001.405
1064
0.310169579
0.953445065
692.18878
80.044693
109
0.0017
4.758044
34
1001.505
1068
0.324981844
0.956063269
677.58
79.43058
0.0018
5.0386153
36
1001.606
1072
0.339437404
0.958402662
663.33363
79.094842
0.0019
5.327672
38
1001.713
1076
0.354084289
0.960505529
648.90867
78.995261
0.002
5.616588
40
1001.822
1080
0.36826275
0.962406015
634.95645
79.097744
0.0021
5.8947261
42
1001.926
1084
0.381288291
0.964131994
622.15083
79.374462
0.0022
6.1762653
44
1002.033
1088
0.394166282
0.965706447
609.49996
79.802469
0.0023
6.4686991
46
1002.148
1092
0.407336723
0.967148489
596.57114
80.362681
0.0024
6.751523
48
1002.259
1096
0.419536559
0.968474149
584.607
81.039092
0.0025
7.0497106
50
1002.379
1100
0.432244636
0.96969697
572.15333
81.818182
0.0026
7.3233225
52
1002.488
1104
0.443230952
0.970828471
561.40007
82.688448
0.0027
7.6209663
54
1002.611
1108
0.455167617
0.971878515
549.7236
83.640045
0.0028
7.8996325
56
1002.725
1112
0.465795015
0.972855592
539.3402
84.664495
0.0029
8.1965155
58
1002.85
1116
0.476999315
0.973767051
528.40104
85.75446
0.003
8.4861079
60
1002.973
1120
0.487528803
0.974619289
518.13195
86.903553
0.0031
8.7624255
62
1003.09
1124
0.497175501
0.975417896
508.73464
88.106195
0.0032
9.0755601
64
1003.228
1128
0.508116643
0.976167779
498.08267
89.357483
0.0033
9.3384714
66
1003.341
1132
0.516676567
0.976873265
489.76302
90.653099
0.0034
9.63404
68
1003.472
1136
0.526311939
0.977538185
480.4033
91.989218
0.0035
9.9313227
70
1003.606
1140
0.535729869
0.978165939
471.26479
93.362445
0.0036
10.22741
72
1003.741
1144
0.544829599
0.978759558
462.44519
94.769754
0.0037
10.519056
74
1003.874
1148
0.553514801
0.979321754
454.03739
96.208437
0.0038
10.802684
76
1004.005
1152
0.561694098
0.979854956
446.129
97.676068
0.0039
11.116556
78
1004.153
1156
0.570625505
0.980361351
437.50109
99.170463
0.004
11.418648
80
1004.296
1160
0.57891852
0.980842912
429.50107
100.68966
0.0041
11.70451
82
1004.432
1164
0.586493705
0.981301421
422.20371
102.23186
0.0042
11.969436
84
1004.558
1168
0.593278059
0.981738495
415.67714
103.79547
0.0043
12.259624
86
1004.699
1172
0.600610746
0.982155603
408.62942
105.37901
0.0044
12.576884
88
1004.856
1176
0.608468261
0.982554082
401.08584
106.98116
0.0045
12.866912
90
1005
1180
0.615394684
0.982935154
394.44665
108.60068
0.0046
13.182663
92
1005.159
1184
0.622775944
0.983299933
387.38008
110.23647
0.0047
13.464478
94
1005.301
1188
0.62913624
0.983649444
381.3008
111.88751
0.0048
13.770081
96
1005.458
1192
0.635889343
0.983984625
374.85406
113.55285
0.0049
14.034332
98
1005.593
1196
0.641544111
0.98430634
369.46436
115.23164
0.005
14.389264
100
1005.779
1200
0.649017427
0.984615385
362.34964
116.92308
110
Appendix 7-Hospital graphs
Figure 12: Comparing total cost per hour of using probability based control charts vs. following up on
every failure when d = 2 and C/F = 1
Figure 13: Comparing total cost per hour of using probability based control chart vs. following up on
every failure when d = 0.5 and C/F = 1
111
Figure 14: Comparing total cost per hour of using g-charts vs. following up on every failure when d = 2
and C/F = 1
Figure 15: Comparing total cost per hour of using g-chart vs. following up on every failure when d = 0.5
and C/F = 1
112
Figure 16: Comparing total cost per hour of using EWMA vs. following up on every failure when d = 2 and
C/F = 1
Figure 17: Comparing total cost per hour of using g-chart vs. following up on every failure when d = 0.5
and C/F = 1
113
Appendix8 -Pacemaker manufacturing graphs
Figure 18: Comparing total cost per hour of using probability based control charts vs. following up on
every failure when d = 2 and C/F = 1
Figure 19: Comparing total cost per hour of using probability based control charts vs. following up on
every failure when d = 0.5 and C/F = 1
114
Figure 20: Comparing total cost per hour of using g-chart vs. following up on every failure when d = 2 and
C/F = 1
Figure 21:Comparing total cost per hour of using g-chart vs. following up on every failure when d = 0.5
and C/F =1
115
Figure 22: Comparing total cost per hour of using EWMA vs. following up on every failure when d = 2 and
C/F = 1
Figure 23: Comparing total cost per hour of using EWMA vs. following up on every failure when d = 0.5
and C/F = 1