Master Thesis
Multivariate Statistical Process Control in Industrial Plants
Section: Energy & Industry (E&I)
Student: Wenchang Chen (Vincent)
Student no.: 1190229
Supervision committee: Prof. Dr. ir. M.P.C. Weijnen (E&I)
Dr. ir. Zofia Verwater-Lukszo (E&I)
Dr. Eric Molin (TLO)
Dr. Alessandro Di Bucchianico (TUE)
A.M.J. Sinon - B.Sc. (Sappi)
Department of Engineering and Policy Analysis
Faculty of Technology, Policy and Management
Delft University of Technology, August 2005
Executive summary
Operational management is a common issue; it addresses the problem that the academic
research knowledge is not well implemented into practical filed. Multivariate statistical process
control (MSPC) is one of these issues. Therefore, the research objective of this thesis research is
to make recommendations for implementing multivariate statistical process control (MSPC)
in a process-industry plant by providing clear interpretations of MSPC and suggestions to
quality management staff in the plant. Being trained in the Faculty of Technology, Policy and
Management, we will look at the technical problem with a broader view and provide promising
solutions while considering several relevant aspects, such as finance, management, organization,
etc. In this research, both academic development of MSPC and the status of its application in
practical field will be investigated. By knowing exactly what the gap is between academic and
practical fields, we will further propose practical tools and recommendations to fill up the gap –
facilitate the implementation of MSPC in the industrial plants.
Before discussing what MSPC is, some fundamental background knowledge of statistical process
control (SPC) is necessary be introduced. The aim of SPC is to achieve higher quality of final
product and lower the production loss due to the defect products. Process monitoring with control
chart is a basic tool of SPC. Control chart monitors the behavior of a production process and
signals the operator to take necessary action when abnormal event occurs. One of the most
widely adopted control charts was developed by Dr. Shewhart (Shewhart, 1931); it is also called
Shewhart control charts. Although the Shewahrt control chart is considered as simple and easy to
understand, it monitors the variables separately and the relations between variables are ignored.
Nowadays the production process has become more complex than it was in the past. Numerous
variables need to be monitored, and they are often mutually correlated, which means a certain
relationship existing between variables. Under such circumstance, assuming the variables are
independent of each other can be insufficient on detecting process variation.
Speaking of the relationship between variables, we come to the issue of multivariate
statistical process control (MSPC). MSPC can be traced back to Hotelling’s T2 method (Hotelling,
1931). This method considers the correlation between variables, and monitors more than one
variable simultaneously. By monitoring the relationship between variables, MSPC reflects the
process situation more precisely and is able to detect the out-of-control event due to
anti-correlation. Despite of these advantages, nevertheless MSPC has some drawbacks, for
example, involving complex statistics, difficult to interpret the result from MSPC, losing systematic
pattern, etc. These barriers indeed have weakened the chance of implementing this technique
into practical field.
On the other hand, what is the perspective from practical field on the issue of implementing
MSPC? Various aspects were investigated, such as MSPC seems powerful, but really complex;
are the quality people able to understand and conduct MSPC; will it be profitable to proceed the
implementation and so on. These practical concerns indeed captured our attention during this
research. Combining the perspectives from academic field and practical field provides us a
clearer overview of the gap between them, so we would be able to develop practical tools to fill up
the gap and support the industrial plants to enjoy the benefit by applying advanced technology.
During this research, we developed MSPC Implementation Guideline, which contains four
elements, MSPC Plan, MSPC Training, Team Approach and Management Involvement.
Especially in MSPC Plan, two practical tools were constructed, which are Method Model of
(M)SPC and MSPC Diagnosis. Method model of (M)SPC is a decision flow chart, which supports
the practitioners to apply the proper statistical process control chart for different circumstances. It
covers the situations of using Shewhart control charts and using MSPC control chart. MSPC
Diagnosis is designed to interpret the result of MSPC. Because the MSPC only signals the
occurrence of an out-of-control event; it does not provide further information about what the
problematic variable(s) are. We developed these tools based on the existing theoretical methods
while considering the acceptance of practitioners, for instance we tried to design the approach as
simple as possible to increase the workability while maintaining the correctness and effectiveness.
The other parts of MSPC Implementation Guideline emphasize how to practically implement this
technique into industrial plants and the concerns of financial, organizational, managerial aspects
were incorporated as well.
The research development is validated with a case study. The case is a part of the paper-making
process. We started with process investigation and tried to understand the mechanism of the
process. After that, three variables were selected to apply Method Model of (M)SPC and MSPC
Diagnosis. The analysis result validated the tools are effective and workable, even when the
variables are not highly correlated in this case. It raises our confidence and the value of MSPC
technique, because the probability that an out-of-control cannot be detected by Shewhart control
charts but can be signaled by MSPC will become larger when the variables are highly correlated.
However, by analyzing this case, we have also realized dealing with real problem is more difficult
than analyzing the simulated data from scientific paper. Because the real process system is
actually very complex and the simulated data has been often simplified to have a clear framework
for explanation and demonstration. The process system in this case, several automatic control
devices and loop control systems were included. Some variables change all the time due to
automatic controller, and these variables are not suitable to be monitored by using statistical
process control chart. Under such circumstance, we applied another technique – multiple
regression analysis to analyze the relation between variables. All in all, several ideas generated
from this case study. First, the process investigation is very important. Understanding the process
correctly and monitoring the critical variables are fundamental of applying MSPC technique.
Secondly, other SPC techniques are definitively required, for example design of experiment
(DoE), multiple regression analysis and etc. Choosing the proper technique for different situations
is crucial to achieve overall process performance improvement, and MSPC is one of these
techniques.
The entire research can be summarized as follows. We started with the literature study of MSPC.
The nature of the MSPC and especially regarding the diagnosis of responsible variable(s) from
MSPC result were discussed in detail. Then we turned to investigate the perspective from
practitioners regarding the implementation of MSPC technique in practice. Implementing a
particular technique is not simply a technical issue, in stead, financial, managerial, organizational
aspects were involved as well. With a broader view of this issue, we developed the MSPC
Implementation Guideline and applied it to a case study – a process unit from paper-making
production. Several findings and recommendations, which can be a good reference for the
process industrial plants, were generated during the entire period of research.
Although advanced techniques are often more complex than the existing one, it is still
possible to apply them in the practical field. It should be aware that the concerns of academic
researchers and of practitioners may differ, and they are not simply technical issues. By
investigating and understanding the gap with broader perception, we will be able to construct the
link between both sides and support the practical field to utilize the benefit of advanced
technology.
Acknowledgement
Although the so-called TBM thesis market was opened in the October of 2004, where the
possible thesis research topics were presented to EPA program students, the seed of my decision
already started to bud in the beginning of 2004. At that time, I took the elective courses of
Integrated Plant Management, and Operation Analysis for Quality Management, and they did
capture my attention. The manufacturing industries, the quality of production, using statistical
technique to improve the process performance, how to lower the production cost ,how to increase
the profit of company and so on motivate me to investment the last period of master study on this
subject.
This master thesis is a product from six-month process (without statistical process control) but it
contains the contributions and efforts by numerous supportive experts. I would like to thank my
supervision committee: Professor Weijnen, Dr. Verwater-Lukszo, Dr. Molin, Dr. Bucchianico and
Mr. Sinon for their knowledgeable comments and instructions. Especially to Dr. Verwater-Lukszo,
being my daily supervisor, her intensive dedication is highly appreciated. Besides, I would like to
thank Mr. Telman, Mr. Mooiweer, and Mr. Proper for their valuable experience and knowledge.
I would consider this thesis research period is a fantastic experience in my life. Not only academic
enrichment, many tacit gains, such as cultural impact, cultivation of independent thinking,
strengthening my confidence and so on, are very precious. And the most important thing is…….
I do enjoy it!!
05/Aug/2005,
Wenchang Chen (Vincent)
Table of Contents
Chapter 1. Introduction ..............................................................................................................1
1.1. Research Background ................................................................................................... 1
1.2. Research Questions and the Objective......................................................................... 2
Chapter 2. Statistical Process Control .....................................................................................5
2.1.
Shewhart Control Charts .............................................................................................. 5
2.1.1. Control Limits ...................................................................................................... 6
2.1.2. Patterns of Process Behavior.............................................................................. 7
2.1.3. Control Charts for Attributes................................................................................ 9
2.1.4. Control Charts for Variables ................................................................................ 9
2.2. Multivariate Statistical Process Control ....................................................................... 10
2.2.1. Hotelling’s T2 Statistic........................................................................................... 12
2.2.2. T2A & SPE Plot .................................................................................................. 15
2.3. Diagnostic Approaches for Hotelling’s T2 Method ....................................................... 18
2.3.1. MYT T2 Decomposition ..................................................................................... 18
2.3.2. T2 Diagnosis with Principal Component Analysis (PCA)................................... 23
2.4. Approaches Discussion............................................................................................... 26
Chapter 3. Industrial Practice..................................................................................................29
3.1. Selection of the Interviewees ...................................................................................... 29
3.2. Objectives of the Interviews ........................................................................................ 30
3.3. Insights from the Interviews......................................................................................... 30
3.4. The Gap between Academic Field and Practical Field of Statistical Issues................ 32
3.5. Conclusions of the Interviews...................................................................................... 35
Chapter 4. MSPC Implementation Guideline..........................................................................38
4.1.
MSPC Plan ................................................................................................................. 38
4.1.1. Method Model for (M)SPC ................................................................................ 38
4.1.2. MSPC Diagnosis ............................................................................................... 43
4.1.3. Process control ................................................................................................. 44
4.2.
MSPC Training ........................................................................................................... 47
4.3.
Team Approach .......................................................................................................... 47
4.4.
Management Involvement .......................................................................................... 48
Chapter 5. Case Study..............................................................................................................49
5.1. Case Briefing ............................................................................................................... 49
5.2.
MSPC Implementation ............................................................................................... 50
5.3. Result of MSPC Implementation ................................................................................. 57
5.4. Reflection..................................................................................................................... 60
5.5. Process Performance Improvement............................................................................ 61
Chapter 6. Conclusions and Recommendations...................................................................65
6.1. Conclusions ................................................................................................................. 65
6.2. Recommendations....................................................................................................... 68
6.3. Future Research Prospect .......................................................................................... 69
Reference .....................................................................................................................................70
Appendix A. Formulas of Shewhart Control Charts..............................................................72
Appendix B. Constants for Selected Control Charts ............................................................77
Appendix C. Hawkins’ Data Set and T2 Statistics of Measurements. ..................................78
Appendix D. Questionnaires for Interviews. ..........................................................................80
Appendix E. Multiple Regression Analysis of Sub-process. ...............................................85
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
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List of Figures
Figure 1-1 Structure of the thesis. ............................................................................................... 4
Figure 2-1 A generic Shewhart control chart. .............................................................................. 6
Figure 2-2 Typical systematic patterns. ....................................................................................... 8
Figure 2-3 The Western Electric run rules................................................................................... 8
Figure 2-4 An example of misleading information generated from Shewhart control chart. ......11
Figure 2-5 A generic bivariate Hotelling’s T2 control region....................................................... 12
Figure 2-6 A generic T2 control chart. ........................................................................................ 14
Figure 2-7 T2 control chart of measurements 36 to 50. ............................................................. 15
Figure 2-8 TA2 control chart based on first two principal components. ...................................... 17
Figure 2-9 SPE chart based on first two principal components................................................. 17
Figure 2-10 Normalized PCA scores. ........................................................................................ 24
Figure 2-11 Variable contribution plot of principal component 2................................................ 25
Figure 2-12 Variable contribution plot of principal component 3. .............................................. 25
Figure 2-13 Overall average contribution per variable. ............................................................. 26
Figure 2-14 A structure of multivariate statistical process control approaches. ........................ 27
Figure 4-1 MSPC Implementation guideline.............................................................................. 38
Figure 4-2 Method model for (M)SPC ....................................................................................... 42
Figure 4-3 MSPC diagnosis....................................................................................................... 44
Figure 4-4 An example of Causal-and-effect diagram............................................................... 45
Figure 4-5 The Shewhart/Deming wheel (PDCA)...................................................................... 46
Figure 5-1 Scheme of paper-making process unit..................................................................... 50
Figure 5-2 I-chart of HBtotal with one-minute measurement interval. ....................................... 51
Figure 5-3 Histogram of HBtotal measurements. ...................................................................... 52
Figure 5-4 I-chart of HBtotal with 20-minute interval. ................................................................ 52
Figure 5-5 MSPC decision path of case study........................................................................... 53
Figure 5-6 T2 control chart of in-control measurements. ........................................................... 55
Figure 5-7 T2 control chart for future observations. ................................................................... 56
Figure 5-8 Overall average contribution of every variable from observation 22 to 28. ............. 57
Figure 5-9 I-chart of Filler, HBtotal, HBash and BW. ................................................................. 59
Figure 5-10 Ellipse control chart of HBtotal and HBash. ........................................................... 60
Figure 5-11 Sub-process. .......................................................................................................... 61
Figure 5-12 Contour plot of HBtotal and Ret with respect to BW. ............................................. 63
List of Tables
Table 2-1 Unique MYT decomposition terms. (cited from Mason, Young & Tracy, 1997) ......... 20
Table 2-2 Individual Ti2 and its status......................................................................................... 22
Table 2-3 Bivariate conditional term and its status. ................................................................... 22
Table 3-1 List of interviewees .................................................................................................... 29
Table 3-2 The gap between academic and practical fields........................................................ 32
Table 5-1 Correlation of process variables ................................................................................ 54
Table 5-2 In-control process data .............................................................................................. 56
Table 5-3 The result of MSPC diagnosis. .................................................................................. 58
Table 5-4 Result of regression model of sub-process. .............................................................. 62
Table 5-5 Possible recipe of HBtotal and Ret ............................................................................ 64
Chapter 1.
Introduction
Chapter 1. Introduction
In this chapter, fundamental information of this research will be introduced, such as research
background, research motivation, research scope, research questions and objectives. At the end
of this chapter, an overview of the whole report will be provided.
1.1. Research Background
There are various definitions of quality; one is that “Quality is the totality of features and
characteristics of a product or service that bear on its ability to satisfy stated or implied needs”
defined by International Organization for Standardization (ISO). In an easier expression, quality
means to what extent the products can meet the requirements defined by the customers. High
quality of product is the vital concern for most of the companies that will survive in this highly
competitive global market. One of the most effective approaches to achieve high product quality
is Statistical Process Control (SPC).
Statistical Process Control (SPC) has become an important approach for process
industries since 1920s. The aim of SPC is to achieve higher product quality and lower the
production cost due to the minimization of the defect product. One of the greatest tools is the
statistical process control chart developed by Dr. Walter A. Shewhart (Shewhart, 1931). He also
pointed out an important fact that variation of a process is resulted from two sources. One is
termed as common causes which are inherent in the production system and it is not possible to
remove it, and the other is termed as special causes which are resulted from several particulate
reasons (e.g. problems with raw material, operator mistakes, machine failures, etc.) and special
causes may lead to serious damage to the product quality. In general, statistical process control
techniques help us to monitor the production process and to detect abnormal process behavior
due to special causes. The idea is very straightforward, once the special causes of abnormal
process behavior can be detected and further eliminated; the process can be improved, so as the
quality of product.
However, an important characteristic of Shewhart control chart is that it can only monitor
single process variable at a time. Nowadays, the modern production process has dramatically
become complex and integrated. Monitoring the process variables separately ignores the
possible correlation or interaction between them and thus Shewhart’s approach is criticized as
inadequate to reflect the process situation sufficiently. For example (Kourti & MacGregor, 1996),
in a high-pressure low-density polyethylene (LDPE) reactor, an increase of impurity in ethylene
inhibits the polymerization process; moreover, fouling of the reactor walls by sticky polymer
impedes heat transfer and cooling of the reactor. Both impurities and fouling cannot be measured
directly. When these problems occur they affect several process variables and eventually the
product quality. Their existence can therefore be detected by the effect they have on the process
and quality variables. This effect is not a simple shift of mean of one or more variables; both the
magnitudes and the relationships of the variables to each other will change. Using traditional
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
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Chapter 1.
Introduction
Shewhart control chart is not capable to reflect such complex process behavior and to detect the
problem of process.
Therefore some improvement of Shewhart control chart was developed. One of the most
often discussed methods is Hotelling’s T2 method (Hotelling, 1931). Hotelling’s T2 method
considers the correlation between process variables, and it can generate control limits to monitor
whether the process behavior is stable and detect variation resulted from special cause.
Hotelling’s T2 method can simultaneously monitor more than one process variables at a time, and
that is why it is also called multivariate statistical process control (MSPC) chart. Nevertheless,
along with the advantage of MSPC, there are also several shortcomings that need to be
mentioned. First, the result of MSPC compares the synthetic statistic value generated from more
than one process variables with the calculated control limit generated from a period of in-control
historical data set. MSCP can detect an abnormal event but does not provide a reason for it. It is
difficult for a user to identify which process variable or set of process variables is responsible for
the abnormal event and to take necessary action. Second, the application of MSPC involves too
much statistic knowledge for plant staff in the industries. Due to the complex nature of MSPC,
most of the industrial plants are still not able to adopt MSPC and really enjoy the benefit of
improving product quality.
The motivation of this research is to facilitate MSPC implementation in industrial plants while
incorporating opinions and needs from the industrial field. MSPC is theoretically proven as a
precise statistical process control technique. It can help an industrial plant to monitor the
production more properly, detect the abnormal process event more effectively and thus reduce
the production cost of a company with a lower defect product rate. However, due to the barriers of
implementing MSPC mentioned above, it seems the theoretical knowledge is not successfully
transferred into practical field.
1.2. Research Questions and the Objective
After knowing the background of the research project, the main research question is defined as
follows. “What are the difficulties of multivariate statistical process control (MSPC)
implementation and how quality management staff can be supported to facilitate MSPC in
a process industry plant?” The main research question is elaborated into five sub-questions for
better understanding.
1). What is the essence of MSPC?
2). What are the expectations from quality management staff in the process industry plant?
3). What tools can be provided to make the interpretation of MSPC results easier?
4). What advices can be provided to cope with the time axis problem for MSPC?
5). What recommendations can be provided to quality management staff in the process
industry plants?
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
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Chapter 1.
Introduction
All the sub-questions will be answered when the research work is complete. The first
sub-question basically addresses the background knowledge about MSPC and how MSPC works
in the process control. After knowing MSPC, we will investigate the opinions from practice, and
the intention is to accommodate the whole research work more workable in practice. We will
develop several approaches based on the opinions from practice and these approaches will
answer question three and four. For the last question, we will provide general recommendations
to quality staff based on the findings from both technical aspect and practical aspect.
The research objective of this thesis research is to make recommendations for implementing
multivariate statistical process control (MSPC) in a process-industry plant by providing
clear interpretations of MSPC and suggestions to quality management staff in the plant.
The main contribution of this research work will contain several targets. First, an analysis
of Shewhart control charts and various MSPC methods will be concluded. Second, in order to
obtain the opinions and needs from the practical field, several interviews with industrial plant staff,
statistical process consultants and statisticians will be conducted. Very often the academic theory
development does not receive positive feedback derived from the practical implementation
because the voice of practical field is missing. Third, by knowing the characteristics of different
SPC techniques and combining with the expectations from the practical field, MSPC
Implementation Guideline will be developed. This guideline is meant to support the industrial
plants to implement MSPC technique. In the guideline, there are two practical tools, which are
Method model for (M)SPC and MSPC Diagnosis. The first one supports the practitioners to
choose the proper SPC technique under different circumstances. MSPC Diagnosis will serve as a
generic Out-of-Control-Action-Plan (OCAP) of the plant while using MSPC. The MSPC Diagnosis
can help the plant staffs correctly react when they encounter the out-of-control measurement, and
lower the defect product rate. During this research period, these two approaches will be
accomplished at the conceptual level. Nevertheless, it is foreseeable that developing profound
software which can cover the complex statistic calculation will be a key to lower the barrier for
implementing MSPC in a plant. The approaches proposed in this research can serve as
functional specifications of software development.
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Chapter 1.
Introduction
Research Background
SPC
Research Objective
Shewhart Control Chart
Gap between Academic
and Practical Fields
Research Questions
MSPC Control Chart
Industrial Expectations
Interviews
MSPC Plan
Case Briefing
MSPC Training
MSPC Implementation
Team Approach
Result
Management
Involvement
Process Performance
Improvement
Conclusions
Recommendations
Future Research
Figure 1-1 Structure of the thesis.
The structure of this thesis is shown in Figure 1-1 and details are described as follows.
Chapter 2 will introduce some statistical process control (SPC) techniques. Shewhart control
charts will be briefly introduced and multivariate statistical process control (MSPC) will be
emphasized with several approaches. Hotelling’s T2 statistic and several successive approaches
such as T2A and SPE plot, MYT T2 decomposition and Principal Component Analysis (PCA)
application will be addressed and discussed. Chapter 3 will involve the views of practice. The
opinions and the expectations from industrial plant staff, statistical process consultants, and
statisticians will be concluded and further incorporated into our research output. MSPC
Implementation Guideline, including the scheme of Method model for (M)SPC and MSPC
Diagnosis will be illustrated in Chapter 4 with detailed explanation. In Chapter 5, the effectiveness
of these two approaches will be validated with a case application. At the end of this thesis,
conclusions and recommendations will be presented in Chapter 6.
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Chapter 2.
Statistical Process Control
Chapter 2. Statistical Process Control
The aim of statistical process control (SPC) is to achieve higher quality of final product and lower
the production loss due to defect product. Process monitoring with control chart is a basic tool of
statistical process control. It monitors the behavior of a production process and signals the
operator to take necessary action when abnormal event occurs. A stable production process is
the key element of quality improvement. In this chapter, the traditional control chart – Shewhart
control charts, which is a univariate statistical process control technique will be introduced. After
that, a multivariate statistical process control (MSPC) technique – Hotelling’s T2 method and its
advantages/drawbacks will be discussed. With the same idea of Hotelling’s T2 method, an
adjusted approach T2A and SPE plot will be introduced as well. Knowing the problem of
interpretation of the result of Hotelling’s T2 method, two diagnostic methods: (1). Application of
Principal Component Analysis (PCA) and (2). MYT T2 decomposition will be reviewed with an
example. In the end of chapter, comparison and discussion will be made for these methods.
2.1.
Shewhart Control Charts
The originator of statistical process control chart is Dr. Walter A. Shewhart. The basic idea of
Shewhart control chart requires an analyst to take samples from the process periodically and
calculate a statistic to summarize the process behavior. The measurements are plotted on the
chart against time or observation series and compared to control limits drawn on the chart
(Stephen, et al. 1999). A generic Shewhart control chart is shown in Figure 2-1. The center line
represents the expected value of the quality characteristic during in-control process. The upper
control limit (UCL) and lower control limit (LCL) are chosen based on the nature the process
behavior, which means only a certain probability that the process falls within in-control limits
(please see 2.1.1 Control limits).
The control limits are directly calculated from the process data. It should be noted that
control limits are not specification limits defined by customer. Therefore, in-control process does
not mean that the product meets the specification limits, it only means that the process behavior
is consistent and predictable. From the interview with industrial statisticians, we discovered that it
is sometimes a common mistake that the quality people take the customer-defined specification
limits as control limits while applying Shewhart control charts.
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
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Chapter 2.
Statistical Process Control
Figure 2-1 A generic Shewhart control chart.
One of Dr. Walter A. Shewhart’s (Shewhart, 1931) fundamental concepts was that, the
variation of a process results from two sources. One is called common cause, which is inherent in
the production system and it is not possible to remove the common cause from the process
unless some changes of the existing process system have been taken. The variation from
common cause is a very minor fluctuation and does not harm the final product quality. The other
is called special (assignable) cause, which is resulted from several particular reasons (e.g.
problem with raw material, operator mistakes, machine failures, etc.) and special cause may lead
to serious damage to the final product quality and cause the loss of a company. The Shewhart
control charts serve as a tool to detect the abnormal event caused by special causes and signal
the operator to analyze the problem.
2.1.1. Control Limits
A point falling within the control limits means it fails to reject the null hypothesis that the
process is statistically in-control, and a point falling outside the control limits means it rejects the
null hypothesis that the process is statistically in-control. Therefore, the statistical Type I error α
(Rejecting the null hypothesis H0 when it is true) applied in Shewhart control chart means the
process is concluded as out-of-control when it is truly in-control. Same analog, the statistical Type
II error β (Failing to reject the null hypothesis when it is false) means the process is concluded as
in-control when it is truly false. According to Engineering Statistics Handbook (NIST/SEMATECH
e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, retrieved on
01/may/2005), the UCL is pictured when the probability of a measurement falls above of UCL is
0.001. The same situation holds for LCL. Therefore, the probability will be 0.002 when a
measurement falls either outside UCL or LCL, and probability of 0.002 is practically considered
acceptable quality control. Compared with the normalized standard distribution probability, the
probability of a measurement falling outside of the limit which locates 3 sigma (standard deviation)
away from average is 0.00135. For both sides, the probability is 0.0027 of a measurement falling
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Chapter 2.
Statistical Process Control
outside of UCL/LCL. Therefore 3 sigma has become a customary distance between central line
and UCL/LCL and generally it gives good results in practice.
Average Run Length (ARL)
The performance of control charts can also be characterized by their average run length. Average
run length is the average number of points that must be plotted before a point indicates an
out-of-control condition (Montgomery, 1985). We can calculate the average run length for any
Shewhart control chart according to,
ARL =
1
p
where p is the probability that an out-of-control event occurs. Therefore, a control chart with 3
sigma control limits, the average run length will be
ARL =
1
1
=
= 370
p 0.0027
This means that if the process remains in-control, in average, there will be one false alarm every
370 samples.
2.1.2. Patterns of Process Behavior
Apart from all the measurement should fall with the control limits, the process can be viewed as
in-control when there is no systematic pattern shown in the process behavior. Systematic patterns
occurring in Shewhart control charts have often been interpreted as indicators of extraneous
sources of process variation (Mason, et al. 2003). The process will be improved if the causes of
systematic pattern in the process are diagnosed and further eliminated.
Typical patterns are shown in Figure 2-2. Cyclic pattern may be caused by systematic
environment change, such as seasonal temperature or operators shifting. Trend pattern is usually
due to wearing out of a tool/machine or catalyst deterioration. A shift in process level may be
caused by the feeding of new material, or by the operation run by a new worker.
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Chapter 2.
Statistical Process Control
Figure 2-2 Typical systematic patterns.
The Western Electric Handbook (1956) provides a set of guidelines to detect the systematic
patterns in the process. A brief summary is shown below. A process is considered as
out-of-control if any of the following conditions holds:
1). One point falls outside the 3-sigma control limits (beyond Zone A).
2). At least two out of three consecutive points fall on the same side of the center line,
and are beyond the 2-sigma control limits (in Zone A or beyond).
3). At lease four out of five consecutive points fall on the same side of the center line and
are beyond the 1-sigma limits (in Zone B or beyond).
4). At least eight successive points fall on the same side of the center line.
Figure 2-3 The Western Electric run rules.
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Chapter 2.
Statistical Process Control
2.1.3. Control Charts for Attributes
We often come to the situation when we are not able to measure the quality characteristic of a
product, for example, checking the surface scratches of a product or proportion of malfunctioning
lamps. The result of checking can be classified into whether conforming or nonconforming to the
specification on the quality characteristic. In this case, control charts for attributes should be
applied. The characteristics of each control chart and its application will be introduced as follows.
The control limits are constructed according to customary 3-sigma distance away from the center
line. Comprehensive formulas for computing the centerline and control limits of control charts of
attributes can be found in Appendix A.
p-chart
The p-chart graphs the proportions of defective items from successive subgroups. It tells us the
defect rate of the product. It should be noted that the sample size should be large enough to
contain defective products; otherwise the control chart will lose the meaning of detection if most of
the p values from the samples are zero.
np-chart
The np-chat is slightly different from p-chat. Instead of plotting the proportions of defective items,
the number of defectives np is plotted. In order to make the number of defectives comparable, it is
important that the sizes of sample have to be the same. For shop floor operators, the information
from np-chart is more straightforward than p-chart and easier to understand.
C-chart
The c-chart is applicable when the large product is inspected. The quality can be monitored in
terms of counting the number of nonconformities on each product (sample size is one).
U-chart
The u-chart is a modification of the c-chart. The number of nonconformities per unit (ui = ci / ni) is
plotted, so the sample size does not need to be one. The probability of the occurrence of
nonconformity can be increased with larger sample size n to avoid too many detecting results (u
value) are zero.
2.1.4. Control Charts for Variables
A single measurable quality characteristic, such as a dimension, weight, or volume, is called a
variable. Control charts for variables are used extensively. They usually lead to more efficient
control procedures and provide more information about process performance than attributes
control charts (Montgomery, 1985). When a variable is monitored, it is a standard practice to
control both the mean and the variability of the variable. The mean of variable is monitored with
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x-bar chart (mean chart) and the variability of the variable is monitored with S-chart (standard
deviation chart) or R-chart (range chart). The x-bar chart can tell us whether the process is stable
with respect to its level. S-cart and R-chart can tell us whether the variability of the process is
stable over time. Significant shifting of the mean and the unusual large variability are the
indications of special causes, which need to be detected and eliminated. Therefore, it is important
to monitor both simultaneously. The control limits are constructed according to customary
3-sigma distance away from center line. Traditionally, quality-control engineers have preferred the
R-chart to the S-chart because of the simplicity of calculating R from each sample. However, the
R-chart is relatively insensitive to small or moderate shifts for small sample size. Thus, in the
situation that tight control of process variability is needed, moderately large sample sizes will be
required, and the S-chart should be used (Montgomery, 1985).
I-chart
In the situation that the production rate is very slow, and difficult to accumulate more than one
sample unit before analysis, I-chart (Individual chart) should be applied. Again, the mean and the
variability of the process should be monitored simultaneously. The mean of a process is
computed directly from the mean of the entire individual sample and the variability is computed
from the moving range. Due to the sample size is one, the process variability is estimated with the
moving range MR=│Xi – Xi-1│, which is the absolute value of the difference between two adjacent
observations. Comprehensive formulas for computing the centerline and control limits of control
chart of variables can be found in Appendix A.
CUSUM Control Chart and EWMA Control Chart
Apart from various types of Shewhart control chart, there are two effective alternatives, which
should be shortly introduced. The first one is cumulative-sum (CUSUM) control chart (Page 1954)
and the other is exponentially weighted moving-range (EWMA) control chart (Roberts 1959).
CUSUM control chart accumulates the deviations of each measurement from the center line,
while the weighted average gives more weight to the more recent measurements and less to
those in the past. Since CUSUM contains all the information from the whole sequence of
measurements not only the last one, it is more effective than Shewhart control charts on detecting
small process shifts. Similar idea for exponentially weighted moving-range (EWMA) control chart.
The only difference is that for EWMA, the weights that are given to the measurements decrease
geometrically with the age of the sample mean. Additional reference can be referred to D.C.
Montgomery (Introduction to Statistical Quality Control).
2.2. Multivariate Statistical Process Control
The Shewhart control charts have been widely applied in a variety of industries because it is very
easy to implement and the information generated from the Shewhart control charts is also easy
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for plant staff to understand. However, monitoring each process variable with separate Shewhart
control chart ignores the correlation between variables and does not fully reflect the real process
situation. Nowadays, the process industry has become more complex than it was in the past and
inevitably that number of process variables need to be monitored has increased dramatically.
Very often, these variables are multivariate in nature and using Shewhart control charts becomes
insufficient. Figure 2-4 demonstrates how misleading information could be generated from
Shewhart control chart in the multivariate circumstance.
Figure 2-4 An example of misleading information generated from Shewhart control chart.
Assuming a doll production, and there are two quality variables with positive correlation
between them (for the convenience of visual illustration, product quality variable Height & Weight
are used as of process variables). The multivariate control chart on the right side is simply two
Shewhart control charts superimposed together according to vertical and horizontal axes. Due to
the positive correlation between H and W, it is expected that the measurement plots will locate
within the elliptical region. Doll 2 and 3 are considered as in-control process because they fulfill
the control limits of Shewhart control chart as well as the multivariate control chart (the elliptical
region). While doll 4 will be easily detected as an out-of-control event since it falls outside the
UCL of Shewhart control chart. The critical doll is plot 3, which falls outside of the expected region
(the ellipse) while both Shewhart control charts appear to be in-control. In multivariate
circumstance, an out-of-control signal can be caused by (1). Extraordinary value of a variable or a
set of variables, (2). Due to the relationship between two or more variables which contradicts the
pattern established by the historical data or (3). A combination of the former two causes.
Very often the process variables are not independent. They sometimes influence each
other following certain predictable patterns. For example, one variable should become larger
while the other variable becomes larger (positive relation), whereas one variable should become
smaller while the other variable becomes larger (negative relation). By using Shewhart control
charts is not able to signal the process is out-of-control when the relation between process
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variables deviates from its predicted pattern. In this case, the plant staff may lose the chance to
detect problematic process and further investigate the problem in time.
2.2.1. Hotelling’s T2 Statistic
Hotelling H. (1931) can be viewed as the originator of multivariate control charts. Hotelling
proposed a concept of generalized distance between a new observation to its sample mean. We
first illustrate how this method works with a bivariate case. Assuming these x1 and x2 are
distributed according the bivariate normal distribution. Referring to Figure 2-5, say X1 and X2
are the mean, σ1 and σ2 are the standard deviation of these two variables respectively. The
covariance σ12 is used to estimate the dependency between x1 and x2. The generalized distance
between point A and its mean can be calculated as:
X0 2 =
1
⎡ s11 ( x 2 - x 2 )2 − 2( x 2 - x 2 )( x1 - x1 ) + s22 ( x1 - x1 )2 ⎤
⎦
s11 s22 − s212 ⎣
This statistic follows the Chi-square distribution with two degrees of freedom. An ellipse can be
graphed with the x1 and x2 in this equation. Moreover, all the points lying on the ellipse will
generate the same Chi-square statistic. As a consequence, every observation can be determined
whether its generalized distance exceeds the ellipse by comparing X02 and X22,α ,where X22,α is
the upper α percentage point of the Chi-square distribution with 2 degrees of freedom. The
observation will be considered as out-of-control if X02 > X22,α.
Figure 2-5 A generic bivariate Hotelling’s T2 control region.
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With the same concept of the generalized distance, it can be extended from bivariate to a
'
multiple p variables. Let Xi = (Xi1, Xi2 ,...., Xip ) represent a p dimensional vector of measurements
made on a process time period i. The value Xij represents an observation on the jth characteristic.
Assuming that when the process is in control, the Xi are independent and follow a multivariate
normal distribution with mean vector µ and covariance matrix Σ. Normally µ and Σ are unknown,
but we can use X and S estimated from a historical data set with n observations.
Phase I and Phase II
The application of Hotelling’s T2 statistic shall be categorized into two phases. Phase I tests
whether the preliminary process was in control and phase II tests whether the future observation
remains in-control (Alt, 1985). Phase I operation refers to the construction of in-control data set.
Same idea as Shewhart control chart, control limits are estimated from a period of in-control data.
To obtain this in-control data, the raw data set needs to be purged. For instance, the outliers need
to be removed and the missing data needs to be substituted with an estimate. During phase I
operation, Hotelling’s T2 statistic is calculated for each measurement and compared to the control
limit, which will follows Chi-square distribution (according to Richard, A.J. & Dean, W.W., 2002.)
T 2 = (Xi - X)' S-1(Xi - X) ~ X2α,p (Chi - square distribution)
(eq. 2 − 1)
Also other research shows that the control limit follows Beta distribution (Mason, Young & Tracy,
1992).
T 2 = (Xi - X)' S-1(Xi - X) ~
(n-1)2
B p n −p −1
( α, ,
)
n
2
2
(eq. 2 − 2)
n: number of preliminary observations
Both control limits will be approximate when the number of observations is large. The control limit
based on Chi-square distribution is established on the assumption that X and S are true values
µ and Σ, which is just an approximate situation (Mason, Young & Tracy, 1992). Beta distribution is
more precise and is a recommendable choice. After purging the raw data with Hotelling’s T2
statistic, the in-control data set is ready for monitoring future observations which is termed as
phase II operation. The control limit for determining future observation is different from the one in
phase I. It follows an F distribution with p and (n-p) degrees of freedom.
p(n+1)(n-1)
F(p,n −p, α )
n(n − p)
n: number of preliminary observations
T 2 = (Xi − X)' S−1(Xi − X) ~
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Where sample mean is X = (X1, X2 ,...., Xp )'
and the covariance of sample
⎡ s11 s12 s13 .... s1p
⎢
s22 s23 .... s2p
⎢
S=⎢
O
⎢
⎢⎣
spp
⎤
⎥
⎥
⎥
⎥
⎥⎦
The idea of using Hotelling’s T2 statistic in phase I and phase II is the same. Each measurement
is examined whether it is out-of-control by checking if it deviates extraordinarily from its sample
mean. It should be reminded to choose the correct upper control limit on different purposes.
The Hotelling’s T2 statistic can be extended for more than two variables. Instead of a
2-dimensional ellipse control region, the result will be presented in a similar way as Shewhart
control chart. The T2 statistics calculated from all the observation will be plotted in a chart against
time or observation serious and compared to the upper control limit. Figure 2-6 is a generic T2
control chart. It should be noticed that there is no center line and the lower control limit is set to
zero, because the meaning of T2 statistic is a generalized distance between the observation and
its sample mean.
Figure 2-6 A generic T2 control chart.
Case Demonstration
We will demonstrate how Hotelling’s T2 statistic helps us to determine whether a measurement is
in-control with an example. The data set was used in Hawkins’ paper (1991). Data set can be
found in Appendix C. The data contained 50 measurements of 5 variables and measurements 1
to 35 were considered as in-control process. In this case, the data set was already purged, so we
only perform the phase II operation-monitoring the future observations. An upward shift of 25% of
a standard deviation was introduced to X5 while the marginal standard deviation of X1 was
introduced by 50 % for measurement 36 to 50.
X and S are estimated from the measurement 1 to measurement 35. As we know
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that the Hotellings’s T2 statistic will follow an F distribution with p and (n-p) degrees of freedom,
the upper control limit is calculated as,
p(n+1)(n-1)
F(p,n −p, 0.05 )
n(n − p)
5(35+1)(35-1)
=
* 2.53
35(35 − 5)
= 14.75
UCL =
Recall the idea from Figure 2-5, the value of UCL means the generalized distance between the
mean (the center point) of the measurements obtaining from a period of in-control process to the
control limit (the ellipse). So now we can compute Hotelling’s T2 statistic (eq. 2-3) for
measurement 36 to measurement 50 to determine its status. A measurement will be considered
as out-of-control if its Hotelling’s T2 statistic is larger than UCL. Figure 2-7 shows the plot for each
measurement, and measurement 48 is detected as an out-of-control situation.
Figure 2-7 T2 control chart of measurements 36 to 50.
2.2.2. T2A & SPE Plot
The Hotelling’s T2 statistic is very effective and easy to understand the result. However, using T2
with highly correlated the variables; the covariance matrix is often very ill-conditioned. When the
number of variables is large, the covariance matrix is often nearly singular and may not be
inverted (Kourti & MacGregor, 1995). Without covariance inversion matrix, the Hotelling’s T2
statistic is not possible to obtain. With this concern, another approach was proposed by Kourti
and MacGregor (1995). The details will be explained as follows. The traditional T2 equation
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T 2 = (Xi − X)' S−1(Xi − X) can be expressed (Mardia, Kent and Bibby, 1989; Kourti and MacGregor,
1994) as,
q
T2 = ∑
i =1
q
q
A
t i2
t2
t2
t2
= ∑ i2 = ∑ i2 + ∑ i2
λi
i =1 si
i =1 si
i = A +1 si
ti = Pi (X − X)
where λi and Pi are the eigenvalues and eigenvectors of principal components that generated
from the original data. So the Hotelling’s T2 statistic based on the first A principal components is,
A
TA 2 = ∑
i =1
t i2
si2
When TA2 does not utilize all the principal components, it is just an approximated value of T2;
therefore it will be equivalent to T2 if all the principal components are utilized. To construct TA2, it is
not necessary to obtain the inversion of covariance matrix anymore; besides, the dimensionality
can be reduced as well. Due to TA2 is only an approximated value of T2, it only can detect whether
there is an abnormal variance occurs in the plane constructed with A principal components. If a
totally new type of special even occurs which was not present in the reference data used to
develop the in-control PCA model, then new principal component will appear and the new
observation will move off the plane (Kourti & MacGregor, 1995). So we need another support
which is squared prediction error (SPE). The SPE is the squared perpendicular distance of an
observation xi from the projection space and it tells us how close the observation xi is to the space
constructed with A principal components. To determine the status of a new observation, both TA2
and SPE are needed.
A
t 2new,a
a =1
sa 2
TA 2 = ∑
UCL =
A ⎡ '
P (X − X) ⎤
= ∑ ⎢ a new
⎥
sa
a =1 ⎣
⎢
⎦⎥
2
(eq. 2 − 4)
p(n+1)(n-1)
A(m+1)(m-1)
F(p,n −p, α ) =
F( A,m −Α, α )
n(n − p)
m(m − A)
2
n
SPE x,new = ∑ (Xnew,j − Xnew,j )
(eq. 2 − 5)
j =1
Xnew =
a=A
∑ (t
a =1
P)
new,j a
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Case Demonstration
We will apply the same data set that we used in the previous sections for demonstration. Since
the eigenvalues of the first two principal components are 87% of the total eigenvalues, which
means the first two principal components already explained most of the variance from the original
data. Under such circumstance, the TA2 of this case will be computed with two principal
components. Applying the formula above, TA2 and SPE are computed (eq. 2-4 & eq.2-5) and
plotted for measurement 36 to measurement 50 in Figure 2-8 and Figure-2-9.
Figure 2-8 TA2 control chart based on first two principal components.
Figure 2-9 SPE chart based on first two principal components.
From the T2 chart, the measurement 48 appears to be out-of-control. SPE chart also shows
measurement 48 has relative higher value than others, which means this measurement is far
from the projection model constructed from in-control historical data. The result from this
approach is similar with the one applied Hotelling’s T2 approach, but it avoids the problem of
inversion of covariance matrix.
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Both Hotelling’s T2 statistic and T2A & SPE plot offer a great support on the deficiency of Shewhart
control chart, which cannot monitor the correlation between variables. In addition, both
approaches can reduce a large amount of individual Shewhart control chart into a synthetic
control chart. However, since the T2 statistic (either using Hotelling’s T2 statistic or T2A & SPE plot)
is the synthetic statistic value generated from more than one process variables, it is difficult for
user to determine which variable or set of variables is responsible when the abnormal event
occurs. Without knowing this information, it is difficult for plant staff to search the root causes of
the abnormal event and further eliminate them. In this case, Hotelling’s T2 statistic and T2A & SPE
plot approaches are able to signal the operator when something goes wrong in the process, yet
they are not able to tell the operator what is wrong or how exactly the process goes wrong. In the
following section, two approaches will be discussed for diagnosing responsible variable(s) once
abnormal event is detected.
2.3. Diagnostic Approaches for Hotelling’s T2 Method
In order to support Hotelling’s T2 statistic and T2A & SPE plot approaches to identify the source of
an abnormal signal, two approaches have been proposed. First, Mason, Young & Tracy proposed
a decomposition approach to breakdown T2 into orthogonal components (1995, 1997). Second,
Kourti and MacGregor (1995, 1996) provided another diagnostic approach based on Principal
Component Analysis to identify the responsible variable(s) for abnormal measurement. These
two approaches will be further explained with a case demonstration.
2.3.1. MYT T2 Decomposition
Mason, Young and Tracy (1995, 1997, 1999) proposed an approach (hereafter is referred as MYT
approach) to decompose the Hotelling’s T2 statistic into orthogonal components. Findings from
Hotelling’s T2 can thus be interpreted in a way that most people can follow. The MYT approach is
applied after the abnormal measurement is detected by either Hotelling’s T2 statistic or T2A & SPE
plot for identifying the responsible variable or set of variables. For a p-dimensional vector, one
form of the MYT approach can be expressed as,
T 2 = T12 + (Τ2 2•1 + .... + T 2p•1,2,....,p −1 )
2
(x j − x j•1,2,....,j−1 )
(x − x1 )
= 1 2
+
s1
s2 j•1,2,....,j−1
2
j = 1, 2, ... , p
The first term T12 is an unconditional Hotellings’ T2 for the first variable of the measurement. The
rest of the terms are referred as conditional terms. It should be noted that the ordering of the
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Chapter 2.
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individual conditional terms is not unique. There are p! different partitionings that can generate
the same overall T2 statistic (Mason, Young & Tracy, 1995). For example, we can start with
selecting any one of the p variables. Then we can choose any of the (p – 1) remaining variables
to condition on the first selected variable. Next we can choose any of the remaining (p – 2)
variables to condition on the first two selected variables. Iterating the same procedure will
generate all the decomposition equations which compose the same over T2 statistic. Taking a
case of three variables as an example, it can be decomposed as,
T 2 =T12 + Τ 2 2•1 + Τ 23•1,2
=T12 + Τ 23•1 + Τ 22•1,3
=T2 2 + Τ 23• 2 + Τ 21•2,3
=T2 2 + Τ 21•2 + Τ 23•1,2
=T3 2 + Τ 21•3 + Τ 22•1,3
=T3 2 + Τ 22•3 + Τ 21•2,3
It is obvious that with the increase of the number of variables, the number of terms will also
increase dramatically which makes the computation become troublesome. Nevertheless, the two
terms of greatest interest are often the unconditional term and the term containing the adjusted
contribution of one of the variables after adjusting for the other (p – 1) variables (Mason, Young
and Tracy, 1995).
Unconditional Term
The unconditional term has a similar function of a univariate Shewhart control chat. It
calculates the squared standardized variance of jth variable. A signal will occur if jth variable is too
far away from the sample mean. T2j will follow an F distribution which can be used as upper
control limit.
Tj =
2
(x j − x j )
sj
2
2
∼(
n +1
) F(1,n-1,α )
n
(eq. 2 − 6)
Conditional Term
The conditional term is a standardized observation of the jth variable adjusted by
estimates of the mean and variance from the conditional distribution associated with xj•1,2,…j-1 . The
most important function of conditional term is that it measures whether the jth variable is
consistent to the relationship pattern with other variables established from historical in-control
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data. The conditional term T2j˙1,2,…j-1 will follow an F distribution which can be used as upper
control limit.
T
2
j•1,2,....,j −1
=
(x j − x j•1,2,....,j−1 )
s
2
j•1,2,....,j −1
2
⎛ (n+1)(n-1) ⎞
∼⎜
⎟ F(1,n-k-1,α)
⎝ n(n-k-1) ⎠
κ=(j-1)
(eq. 2-7)
T2j˙1,2,…j-1 can be re-expressed as (Mason, Young and Tracy, 1997),
T 2 j•1,2,....,j−1 =
(x j − x j•1,2,....,j−1 )2
s2 j (1 − R 2 j•1,2,....,j−1 )
(eq. 2-8)
The numerator is the squared residual between the observation and the predicted point based
regressed by the variables x1, x2, … xj-1. R2j•1,2,…j-1 is the squared multiple correlation coefficient
between xi and x1, x2, … xj-1. From the equation eq.2-8, it is noticed that the conditional T2 term
will become large if xj is significantly far from what is predicted from the historical data, unless the
Rj•1,2,…j-1 is close to 1.
Reduced Computation Scheme
As stated previously the number of the unique decomposition terms will increase dramatically
when the number of variable increases. Table 2-1 provides more detailed information.
Table 2-1 Unique MYT decomposition terms. (cited from Mason, Young & Tracy, 1997)
Therefore, a reduced computation scheme was proposed also by Mason, Young & Tracy (1997).
Here we summarized this reduced computation scheme as follows.
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Step 1. Compute the individual statistic Ti2 (according to eq. 2-6) for every component of
the X vector. The variables with significant T2 statistic are out of individual control
and it is not necessary to check how they relate to other variables. Check whether
the subvector with remaining k variables produces a signal.
Sept 2. (Optional but useful for very large p). Examine the correlation structure of the
subvector. The variable with very weak correlation (0.3 or less) can be removed.
Step 3. If the subvector still produces a signal, then compute all the T2i,j terms (according to
eq. 2-8). T2i,j terms tell us something is wrong with bivariate relationship, if T2i,j is
significant. Continue to check the T2 statistic for the remaining subvector. If no
signal occurs, then it is concluded that the individual variable from step 1 and the
relationship between the bivariate are the sources of the abnormal measurement.
Step 4. If the subvector of remaining variables still produces a signal, then compute all the
T2i,j,k terms. Follow the same rule from previous steps and examine all the
conditional terms.
Step 5. Repeat the same procedures until the T2 statistic of the remaining subvector is not
significant.
Case Demonstration
We will apply the same data set that we used in the previous sections for demonstration. The
abnormal situation is detected with Hotelling’s T2 statistic approach.
1). Hotelling’s T2 statistic was computed with eq. 2-3 and plotted for measurements 36 to
measurement 50. Measurement 48 is detected as an abnormal measurement. Please refer to
Figure 2-7 T2 control chart of measurement 36 to 50.
2). Table 2-2 shows all the individual Ti2 calculated from eq. 2-6 and compared with upper control
limit.
.
n +1
) F(1,n-1,0.05)
n
35 + 1
=(
) F(1,35-1,0.05)
35
= 4.248
UCL = (
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Table 2-2 Individual Ti2 and its status.
3). Remove x1and check whether the subvector signals or not. The result shows the subvector is
still significant.
T 2 - T12 = 22.92 - 7.61 = 15.31
>
14.75
4). It is concluded that not only x1 is problematic; the relationship between variables in the
subvector is also the possible cause. So we continue to check bivariate statistic T2i,j. with eq. 2-8.
A summary table can be found in Table 2-3.
⎛ (n + 1)(n - 1) ⎞
UCL = ⎜
⎟ F(1,n-k -1,α )
⎝ n(n - k - 1) ⎠
⎛ (35 + 1)(35 - 1) ⎞
=⎜
⎟ F(1,33,0.05)
⎝ 35(35 - 1- 1) ⎠
= 4.39
Table 2-3 Bivariate conditional term and its status.
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T25˙4 is found to be significant, which means the relation between x4 and x5 is problematic. So we
may conclude that both x4 and x5 are potential causes of the abnormal observation.
5). Remove x4 , x5 and check whether the subvector signals or not. The result shows the
subvector is not significant anymore. So the computation can stop.
T 2 - T12 - T4 2 - T5 2 = 22.92 - 7.61 - 0.58 - 3.87 = 10.86
<
14.75
So far, we can conclude that x1 is individually out-of-control. Moreover the meaning of significant
value of T25˙4 is that the x5 (conditioned by x4) deviates from the variable relation pattern
established from historical data. Finally, x1 and x5 are determined as the responsible sources of
abnormal measurement 48, which is within our expectation.
2.3.2. T2 Diagnosis with Principal Component Analysis (PCA)
Another diagnostic approach is based on the idea of a well-known statistic technique-Principal
Component Analysis (PCA). The most noticeable advantage of Principal Component Analysis
(PCA) is that it can generate a new set of orthogonal principal components (principal component
is a linear combination of all original variables) based on the original data set and the information
from the original data can be explained by fewer components. The complete set of principal
components can reproduce the total variance of the original data set. However, most of the
variance can be captured by the small number k principal components and thus the
dimensionality can be reduced. Once a PCA model is constructed based on an in-control
historical data, the original variables are considered simultaneously and the relation between
variables are also captured. A new observation can be detected as out-of-control if it significantly
deviates from the PCA model.
The key elements of principal components are the eigenvectors (pi) and eigenvalues (λi)
generated from the covariance matrix S of in-control historical data. Eigenvectors (pi) serve as the
axes of principal components, while eigenvalues (λi) are the variances of the principal
components. The T2 statistic can be expressed in terms of the principal components (Mardia,
Kent, and Bibby 1979, Jackson 1991)
n
ta2
t2
= ∑ a2
a =1 λ a
a =1 sa
n
T2 = ∑
n
t a = Pa' (x - x) = ∑ pa,j (x j - x j )
(eq. 2 − 9)
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j =1
Chapter 2.
Statistical Process Control
where ta are the scores from the principal component transformation, and xj is the jth quality
characteristic. In addition, the normalized PCA scores (ta/sa) can be calculated from each
principal component and we can compare which principal component(s) contributed most to the
abnormal measurement once it is detected by Hotelling’s T2 or T2A & SPE plot approaches (see
section 2.2.). The principal component(s) with higher normalized scores (ta/sa) can be further
investigated with contribution plotting (MacGregor, et al. 1994). The contribution of each variable
to a particular principal component is,
pa,j (x j − x j )
(eq. 2 − 10)
If we plot the contribution of every variable, those with higher values are more likely to be
responsible for the abnormal measurement and need to be investigated.
Case Demonstration
We will apply the same data set that we used in the previous sections for demonstration. The
abnormal situation is detected with Hotelling’s T2 statistic approach.
1). Hotelling’s T2 statistic was computed with eq. 2-3, and plotted for measurements 36 to
measurement 50. Measurement 48 is detected as an abnormal measurement. Please refer to
Figure 2-7 T2 control chart of measurement 36 to 50.
2). Normalized PCA scores (ta/sa) are calculated with eq. 2-9, and plotted in Figure 2-10.
The normalized score of principal component 2 and 3 are relatively high. In addition, the limits for
the normalized scores are roughly used as a guide, and 2.7σ would be equivalent to
Bonferroni-type limits (I.e., replace α/2 with α/2n) 95% confidence on type I error for five
variables.
Figure 2-10 Normalized PCA scores.
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Chapter 2.
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3). Knowing normalized score 2 and 3 contributed most to this abnormal measurement, we can
construct the contribution plot (using eq. 2-10) of 5 variables for score 2 and score 3 (see Figure
2-11 and 2-12) to see which variable or set of variables contributed most.
Figure 2-11 Variable contribution plot of principal component 2.
Figure 2-12 Variable contribution plot of principal component 3.
From the contribution plot of score 2, it is clear that x1 contributed most among all variables. In
contribution plot of score3, x5 contributed most and x1 also had relatively high value. It should be
noted that score 3 was negative in the score plot. Thus, in contribution plot of score 3, we should
only look at the variables with negative value because the positive contributions only make the
score smaller. Therefore, it is suggested that variables with high contributions but with the same
sign as the score should be investigated (Kourti & MacGregor, 1996). So far, we can conclude
that x1 and x5 are the variables that need further investigation for this abnormal measurement.
It can also happen that more than one score with high value, for instance, there are two
scores with relatively high value in our case demonstration. Overall average contribution per
variable is suggested (Kourti & MacGregor, 1996). The steps of constructing overall average
contribution are summarized as below.
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Chapter 2.
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1). Select the k normalized scores with high values. In this case, score 2 and score 3 were
selected.
2). Calculate contribution of a variable xj in the normalized score.
Cont a,j =
ta
pa,j (X j − X j )
λa
Conta, j is set to zero if it is negative. (i.e., the sign of variable contribution is opposite to
the value of score)
3). Calculate the total contribution of variable xj.
k
CONTj = ∑ (cont a,j )
j =1
The overall average contribution per variable generates an overview contribution of each variable
in one plot, which is very convenient. Here again, we see that variable x1 and x5 are responsible
for the abnormal situation.
Figure 2-13 Overall average contribution per variable.
2.4. Approaches Discussion
In order to clearly demonstrate a structure of multivariate statistical process control approaches
included in this research, a tree diagram is provided in Figure 2-14.
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Chapter 2.
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Legend :
Decision point
MSPC
approaches
Decision option
Action
Stage I
Yes
Inversion of
covariance matrix
available?
T2 control
chart
No
TA2 and SPE
control chart
Stage II
PCA diagnosis
and overall
contribution
plot.
MYT T2
decomposition
PCA diagnosis
and overall
contribution
plot.
MYT T2
decomposition
Figure 2-14 A structure of multivariate statistical process control approaches.
After the discussion of the mechanisms of several approaches (Hotelling’s T2, T2A & SPE plot,
MYT T2 decomposition and PCA application,) presented above, a general discussion on their
applications will be made. With better understanding of their characteristics, we can make better
choice among them under different circumstances. The discussion will be categorized into two
stages: Stage I focusing on the detection the abnormal measurement, and stage II focusing on
the diagnosing the sources of abnormal measurement.
Stage I – Detecting the Out-of-Control Event
1). As we have discussed in Chapter 2.2., MSPC can not only monitor the status of variables
also monitor the relationship between variables, particularly when variables are highly correlated
of course. Whereas Shewhart control charts are not possible to monitor the relationship between
variables.
2). Hotelling’s T2 statistic is an effective approach for detecting abnormal measurement in the
process. Yet, when the number of variables is large, the covariance matrix is often nearly singular
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and may not be inverted (Kourti & MacGregor, 1995). T2A and SPE plot approach can cope with
the problem of covariance matrix inversion and thus would be an alternative choice to Hotelling’s
T2 method.
3). Large process systems are often comprised with several process units in it. Breaking down
the process system into logical sections, which have highly correlated variables within section but
less correlation between sections can be a recommendable idea. Analyzing smaller sections with
Hotellin’g T2 statistic separately can reduce the complexity of a large number of variables; in
addition, it would become much easier when diagnosing the sources of abnormal measurement.
Stage II – Diagnosing the Sources of an Out-of-Control Event
4).
MYT T2 decomposition which breaks down the overall Hotelling’s T2 into orthogonal
component provides useful information to identify the sources of abnormal measurement.
However, even with the reduced computation scheme, the remaining numerous computations,
especially when the number of variable is large, may still discourage practitioner to apply it.
Programming the computation and breaking down the process system are both considerable
ideas to conduct MYT T2 decomposition approach.
5). Diagnosing abnormal measurement with the normalized scores principal components and
contribution plot is clear and effective. In addition, the idea of finding high score of principal
components and further investigating the variables with high contribution to the selected principal
component is quite straightforward. The implementation can be further facilitated with simple
calculation sheet such as Excel, which is considered as an easier approach. Another advantage
is that the result can be easily understood and communicated between different levels of
practitioners with graphical presentation.
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Chapter 3. Industrial Practice
Multivariate statistical process control (MSPC) techniques, including the approaches that we
have discussed in Chapter 2, have been discussed and published in academic journals for years.
These approaches appear to be effective and may have great opportunity to improve the quality
of industry to a higher level. Yet, MSPC has not become a popular technique in industrial plants
as expected. It turns out that the link between theory and practice remains missing. Due to this
concern, apart from paying attention on theoretic development, we would like to investigate the
perspective of MSPC from practical field and why they feel difficult to adopt MSPC. We believe
that the barrier of MSPC implementation can be lowered if the opinions of practice are taken into
account.
In this chapter we will present how the interviews with people from practical fields were
conducted. The interviewees are from, for example in manufacturing plants, SPC consultant
companies, and in academic institute. Conclusions of the interviews will be made and the
information from interviews will be further incorporated in our research developments.
3.1. Selection of the Interviewees
In order to obtain objective and direct information regarding the application of MSPC, three
different target groups, namely, statistical process control (SPC) consultants, industrial
statisticians, and academic statisticians, were selected for the interview. We would like to know
from industrial statistician what current statistical process control techniques are being used in the
process plants and the perspective on MSPC technique. We also expect that the SPC
consultants have more practical knowledge on SPC and MSPC, and we may have information on
the application of MSPC on real cases. Finally academic statisticians are expected to address
theoretical perspective on MSPC.
The selection of the interviewees started with a collection of possible candidates, mainly
locating within the Netherlands. Considering the complexity of the topic, we decided to conduct a
face-to-face interview, and the interviewees must be reachable. In addition, having face-to-face
communication can lead to more insights during the discussion. Table 3-1 is the list of
interviewees. Due to privacy concern, the details of the interviewees and their companies are not
provided.
Table 3-1 List of interviewees
Position of Interviewees
Working Environment
SPC Consultant
Research institute
SPC Consultant
Research center of a consumer electronic
manufacturing company.
Industrial Statistician
Food manufacturing company
Industrial Statistician
Paper-making company
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Chapter 3.
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Chemometrician
Pharmaceutical Company
Academic Statistician
University
3.2. Objectives of the Interviews
To perform an effective interview, we must know what information that we intend to obtain from it.
Therefore, we need to formulate the objectives beforehand, and they are listed as bellows.
1). To obtain information of manufacturing industrial companies, regarding the technique of
statistical process control.
2). To understand the perspectives and expectations of practitioners regarding the application of
MSPC technique.
3). To analyze the potential workability of MSPC technique in industrial plants.
4). To understand the barriers of MSPC implementation in industrial plants.
3.3. Insights from the Interviews
Different questionnaires were designed for each target groups, the questionnaires are provided in
Appendix D. Basically the questions were designed in a style of open questions, because we are
mainly interested in the qualitative information, instead of statistical analysis. The important
comments generated from the interviews were screened and listed in the following section. Most
of the comments seem straightforward and practical. They cover various aspects, such as
technical, economical, organizational and etc. These opinions have broadened our insights on
SPC implementation as well as MSPC implementation. The barriers and the niches of MSPC
implementation also have been raised.
1. Choosing the right tool and using it correctly is the fundamental to achieve successful
SPC. Shewhart control chart is still an effective tool if it is well applied (Statistician of
a consumer electronic manufacturing factory).
2. Process system needs to be well investigated. It is in vain to monitor the variables
which are not critical to the quality of product. Using the right tool and using it in a
correct way both are necessary to achieve the successful result (Statistician of a
consumer electronic manufacturing factory).
3. The nature of production shall be considered before implementing MSPC. Certain
types of production, for example the chemical process, food production may seem
suitable to implement MSPC, because the process is highly complex. Applying
MSPC has a better chance to control the process and lower the defect rate before
the products are manufactured (Statistician of a consumer electronic manufacturing
factory).
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4. However, there are also many other niches for MSPC. It may not be efficient to use
MSPC to monitor the process variables if the process is relatively simple. Yet MSPC
also can be applied to monitor the product quality. For example if a product has many
attributes of quality, these attributes can be monitored with one MSPC control chart
instead of numerous Shewhart control charts (Statistician of a Food manufacturing
factory).
5. The SPC activities should be categorized into different levels. For instance,
shop-floor operator and a R&D center should are working on different levels of
sophisticated problem. Utilizing different specializations of the employees is more
efficient. MSPC has been applied in our R&D center and it is an effect tool for high
level SPC problem (Statistician of a Food manufacturing factory).
6. The interruption to the process due to SPC activity needs to be reduced as much as
possible. It would not be surprising that a plant rather continues the process with
foreseeable higher defect rate than stops the production process for minor
improvement. It also implies that application of MSPC needs to be simple, effective,
and efficient. Economical concerns of the entire company can never be put aside.
(Statistician of a consumer electronic manufacturing factory).
7. SPC education is necessary. It is very often to see the plant operators overreact on
the variance due to common causes. Besides, preparing plant staff the knowledge of
SPC may increase the motivation of plant staff and further improve the process
performance (Industrial SPC consultant).
8. Reacting correctly when the process shows out-of-control signals is important.
Shewhart control chart or MSPC control chart only gives operators a hint that
something is wrong in the process. Without investigating the root causes and taking
right action, the process does not fix its own problems automatically. (Statistician of a
consumer electronic manufacturing factory).
9. Complexity is a major barrier of MSPC implementation. For a long time, MSPC bears
the image of complexity, difficult to use, difficult to interpret. Computer aid can
efficiently help plant staff to perform the complex calculation and also produce the
results in a graphical way which is easier for managers, engineers and operators to
communicate in the same language. Thus, computer aid is considered a great
catalyst of MSPC implementation. (Industrial SPC consultant).
10. SPC is a necessary approach to achieve higher quality level. However, the company
should choose a proper tool depending on several criteria. For instance, the types of
the production and the current stage of quality performance. For simple process
system which contains less process variables or the variables are quite independent,
Shewhart control charst would be sufficient. Also Shewhart control can be a good
choice for the company to improve the current quality performance to a certain high
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Chapter 3.
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status. For complex process system or aiming at really high quality performance,
adopting MSPC can be a proper choice. The company should choose an economical
choice, because any implementation takes price. (Industrial SPC consultant).
11. SPC is not simply a statistical issue. To support a company to implement SPC or
even MSPC technique involves the organizational aspect as well. In many case, the
poor process quality control resulted from a lack of integration. (Industrial SPC
consultant). and economic aspects need to be considered. From the view of a
12. Organization
company owner, MSPC is valuable when it can generate extra value to the total profit.
MSPC is only part of the quality management of a plant. The level of the whole
quality management should be improved to a certain level; otherwise, MSPC
implementation is not an economic choice. (Statistician of a consumer electronic
manufacturing factory).
13. The SPC technique to be implemented is expected to be simple, easy to use and
robust. (Statistician of a paper manufacturing factory).
14. Pharmaceutical production requires highly specialized manufacturing process.
MSPC in fact is practically adopted to improve the quality of production and it works
very well. (Chemometrician from pharmaceutical company).
15. The quality of the final product is determined by the quality of the entire process. So
the process needs to be well monitored and controlled from the beginning to the end.
Any abnormal variation during the process can lead to defect product in the end and
should be avoided as much as possible. (Chemometrician from pharmaceutical
company).
3.4. The Gap between Academic Field and Practical Field of Statistical Issues
So far we have looked the issue of MSPC implementation from two aspects. Theoretical
development was discussed in Chapter 2, and the survey of practical field was performed in the
previous section as well. By looking at this issue from both sides, we will study the gap between
academic and practical fields, and try to understand how they are disconnected. The result of this
analysis can be a good guideline for us to further construct the bridge between these two fields.
Table 3-2 is the summary of the gap analysis.
Table 3-2 The gap between academic and practical fields.
Academic field
Practical field
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Chapter 3.
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‰ Scientific oriented. The theory
‰ Difficult to understand what
‰ Continuous education and
involves too much
MSPC is. Therefore, people
present the MSPC theory in a
mathematics and statistics.
hesitate to use the tool that
less scientific format can lower
they are not familiar with.
the reluctance of practitioners.
‰ Academic development is still
‰ Several different theoretical
‰ Complex and ill-defined
ongoing. Various approaches
approaches are available.
knowledge may easily
are still being explored and
Practitioners may have
overwhelm the practitioners’
experimented.
difficulty to understand them;
motivation. Screening the
and they do not know exactly
approaches and presenting a
which approach to follow.
clear structure of the theory
overview can be a good start.
Also providing a clear
guidance and instruction of
MSPC application can be a
great support.
‰ Academic research often
‰ What practical field needs is
‰ The theoretical part of MSPC
concentrates on theoretical
an effective, robust tool. It
is indeed more complex than
exploration. Although the
should be easy to understand,
the traditional control chart.
effectiveness of MSPC is
easy to operate and provide
However, it can be overcome
validated theoretically, the
prompt information.
by additional support, for
feasibility of practical
instance, software
implementation seems not
development, automatic
much emphasized.
monitoring/alarm system, etc.
The implementation of MSPC
can be boosted more easily if
the complex statistics is
supported with computer
software.
‰ Traditional control chart
‰ The investment of
(Shewhart control chart) may
Implementing MSPC can be
be still sufficient. The effort
formidable, in terms of finance,
and cost to implement a
human resource, hardware,
complex technique is too high.
etc. To maximize the utility of
this MSPC, the nature of
production should be
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Chapter 3.
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investigated in advance. For
instance, MSPC may appear
to be a proper technique for
complex process where
variables are highly correlated.
Whereas traditional control
chart is a more economical
choice for a simple process
system.
‰ In order to have a clear focus,
‰ Statistical process control
‰ A production plant is absolutely
the framework of MSPC
(SPC) is just part of the entire
not just forwarding input into
research is often simplified.
quality management work.
process and output being
The contribution of SPC may
produced. In addition to SPC,
be limited if other parts of the
many other techniques need to
process are not improved in
be involved. For example,
parallel.
design of experiment (DoE),
automatic process control
(APC) are often applied to
improve a process system.
Proper technique should be
well chosen for different
situations.
‰ The value of SPC or even
‰ How to implement the
MSPC can reveal only when
advanced technique in a
its contribution can reflect on
proper timing is an important
the overall benefit of the
concern. For example, a plant
company.
currently with poor production
quality, then traditional
statistical process control chart
could be already sufficient with
respect to the cost of
implementation.
‰ MSPC is one of the techniques
to improve process
‰ The quality staffs still often rely
on experiences and feelings.
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technique, such as MSPC,
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Chapter 3.
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performance. To implement a
Lacking of statistical process
requires tremendous
technique is different oriented.
control knowledge is a barrier
investment and very often
There are many aspects need
to implement MSPC.
encounters foreseeable
to be taken into account. For
resistance. Hierarchical
instance, how to set up the
organization of quality
plan, how to train the
management department can
employees, estimate the
be a good advice. Higher level
investment and etc.
statistician should have
sufficient MSPC knowledge,
whereas lower level operators
should receive less theoretical
but more practical knowledge.
Such organization is more
efficient in terms of human
resource expenditure and the
company is also able to
perform the internal training to
improve the quality of
employees.
3.5. Conclusions of the Interviews
In this chapter, we performed the interviews with quality people working in the practical fields, and
we studied the reasons why the academic development is not implemented into practice. With
referring to the objectives that we have set in Chapter 3.2., we would like to make some
conclusions.
1). Statistical process control (SPC) is well recognized by industries as an effective tool to achieve
higher production quality and traditional control charts, namely Shewhart control charts are often
used in practice. Nevertheless, the actual quality improvement sometimes is limited because the
quality people do not apply the Shewhart control charts in a correct way. Common mistakes
sometimes happen, for example, quality people are confused with control limits of control charts
and the specification limits (please refer to Chapter 2.1.), or the variables being monitored are not
critical to the quality of final product. This situation shows the education in terms of SPC
knowledge, of quality people still needs to be strengthened. The theory of MSPC technique is
considered even more complex than Shewhart control charts. Therefore, the education indeed is
a necessary efforts, and this is just part of the investment of MSPC implementation.
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2). Regarding the workability of MSPC in manufacturing industries, we may conclude that MSPC
appears to more effective than Shewhart control charts in the continuous process and batch-wise
industries, whereas less effective in discrete process industries. In continuous process and
batch-wise process, for instance, chemical industry or plastic injection industry, there are often
many variables need to be monitored and controlled during the processing period, therefore
MSPC has a greater chance to detect out-of-control events than Shewhart control charts. On the
other hand, for discrete process, for example, automotive assembling industry, the quality of final
product is generally determined by the quality of the assembling components, and MSPC seems
to be little help on quality improvement.
Nevertheless, the niches of MSPC application can be created. A nice example was
introduced by one of the interviewees. The MSPC is applied to check all the quality
characteristics of two types of coffee. More than 30 quality characteristics are monitored,
including the color, the taste, the bitterness, the smell, etc. Some of these quality characteristics
are correlated, and MSPC can be a very good technique to monitor them. These two types of
coffee are produced with different prices of raw materials. So the company may lower the
production cost by using the cheaper raw material, if they can successfully make the quality
characteristics of the coffee with cheaper material achieving the same level as the one produced
with expensive material. In this case, MSPC is not applied during the process but for the quality
characteristics of final product.
3). Due to the intention of this research is to facilitate the implementation of MSPC into practice;
we also would like to conclude what the expectations are from the view of practical fields. These
expectations will be always taken into consideration in the further research development.
3.1). The implementation of MSPC needs to be a gradual, gentle progress. Harsh and
sudden extra work may induce great resistance. It is important to consider the shop floor
plant people may have difficulty understanding the sophisticated statistical theory.
Minimizing the extra work derived from MSPC implementation and present the scientific
theory in an easier format can facilitate the process.
3.2). The tool needs to be simple and easy to adopt. In general, Shewhart control charts
are well-known in the industrial plants. However, it is found that still very often Shewhart
control charts are not correctly applied and lead to little help of process improvement.
Obviously MSPC is even more complex than Shewhart control charts, so there must be a
clear and simple approach which can step-by-step instruct the plant staff to adopt it
correctly during the application.
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3.3). Computer software development can be a great support for MSPC implementation.
The main resistance from the plant staff is that they need to perform complex calculations
to obtain the result, which is over their capacities. Besides, it is also not realistic to
request quality people to perform MSPC technique with manual calculation.
Most of the plant staffs are familiar with collecting data from the process and to
taking action according to out-of-control action plan (OCAP) when it is necessary. To
successfully apply MSPC technique, the critical part is something between correctly
interpreting the data from MSPC and taking right action. Thus, it is expected to construct
the computer software to support plant staff on extracting the information from a large
amount of raw data and generating clear instructions for plant staff to follow.
3.4). Graphs are more preferable than numbers and texts. In general the information is
easier to be understood with graphs than pages of texts. Besides, for communicating and
educating people from different levels, a graphical tool is a better choice and computer
software development is also possible to support largely on this requirement.
3.5). The added value from implementation. The most straightforward added value for
company is the overall profit growth from the improvement of process performance. To
reach this profit growth, the return from the process performance improvement must
climb over the investment on the implementation. Therefore, the industrial plants need a
program and tool that are able to provide foreseeable improvement of the current
situation.
3.6). A profound implementation guideline. This guideline must contain general
instructions and practical tools for a production company. Different levels of employees,
including management level, R&D specialists and shop floor operators all need to be
involved. Each role has to know how to correctly and effectively conduct the
implementation work. It is for sure not an easy task for plant to implement a new
technique, especially a complex one.
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Chapter 4.
MSPC Implementation Guideline
Chapter 4. MSPC Implementation Guideline
After understanding the theoretical development of MSPC and the perspectives from practical
field, we are going to construct an implementation guideline so as to fill up the gap and implement
MSPC into practice. There will be four elements in MSPC implementation guideline, which are
shown in Figure 4-1. The theoretical and practical concerns that we have discovered will be
incorporated in this implementation guideline and we will elaborate these four elements in this
chapter.
MSPC Implementation Guideline
1). Method Model for (M)SPC
MSPC Plan
2). MSPC Diagnosis
3). Process Control
MSPC Training
Team Approach
Management Involvement
Figure 4-1 MSPC Implementation guideline.
4.1.
MSPC Plan
We have understood the nature of MSPC is complex and it has been viewed as one of the largest
barriers for industrial practitioners. Therefore, a simple and clear instruction will be considered as
an effective tool to achieve successful MSPC implementation. We develop two practical tools
which are Method model for (M)SPC, and MSPC Diagnosis. The first one is a general instruction
for industrial practitioners to know how to choose a proper SPC technique under different
circumstances. The second one is the supplementary tool when MSPC is applied. It helps
practitioners to identify the problematic variable(s) and correctly react on it when an out-of-control
event occurs.
4.1.1. Method Model for (M)SPC
There are various types of production. Depending on the nature of the production process,
different SPC techniques should be applied in order to achieve the quality improvement
economically. Figure 4-2 is the scheme of Method model for (M)SPC, and further details are
explained as follows.
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Chapter 4.
MSPC Implementation Guideline
Step 1. Investigate the process system. Understanding the nature of the process system
is the first step toward the application of SPC. The practitioners should know what
variables need to be stable in order to achieve stable output, and then these variables are
suitable to be monitored with control chart. One way to do this is to trace backwards from
the final output and screening the possible variables that have influences to the quality of
outputs. In this stage, consulting the experienced operators and performing the site
investigation can be very helpful.
In addition, monitoring a large amount of variables is inefficient in terms of
economical concern and effectiveness concern. The critical control points should be
identified and the criterion is that the critical control point will lead to significant impact to
the quality of output when it goes out-of-control.
Step 2. Break down the process system. When the entire process system is too complex
or there are too many process steps, breaking down the process system into logical
sections, which have highly correlated variables within section but less correlation
between sections can be a good advice. Observing the actual production process and
discussing with experienced quality people can lead to a general idea of the process
system. Applying statistic software to screen the process variables can provide more
quantitative information on breaking down the process system.
In addition, with smaller monitoring system unit, it is easier to implement (M)SPC,
and easier to diagnose the responsible variable(s) when an abnormal measurement
occurs. Nevertheless, breaking down the process system should be done in a logical way;
otherwise, we may run into a risk of losing information of the process.
Step 3. When the number of monitored variables is only 1 (N=1), then it is suggested to
use Shewhart control charts. The application of Shewhart charts and relevant information
can be found in Chapter 2 and Appendix A.
When the number of monitored variables is more than 1 (N>=2), then we need to
examine whether these variables correlate with each other. The paired correlation of a
group of variables can be examined with generic statistic software (e.g. SPSS,
STAGRAPHICS). Correlation r=0.3 can be used as a criterion to decide the strength of the
correlation between variables (e.g. if statistically correlation r<0.3, it is considered little or
weak association between variables).
Step 4. Construct Hotelling’s T2 control chart for the process measurement from a period
of in-control process data, when the inversion of covariance matrix is available. If the
covariance matrix is nearly singular or not possible to calculate inversion matrix, construct
the TA2 control chart and SPE plot for the measurement. If the number of variables is large
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Chapter 4.
MSPC Implementation Guideline
and they are not correlated between each other, instead of constructing Shewhart control
charts for all of them, constructing one Hotelling’s T2 control chart to monitor all variables is
also an alternative.
There are several computer software packages which support the construction of
Shewhart control charts. SPSS, STAGRAPHICS, Conerstone are all workable. For
multivariate statistical process control, the software package for constructing the
Hotelling’s T2 control chart now is available but not many. STAGRAPHICS (version Plus
5.1) can calculate the Hotelling’s T2 statistic for each observation and also construct the
Hotelling’s T2 control chart which is very convenient.
Stept 5.
In-control data and control limit construction. It is important to purge the
preliminary data to obtain an in-control data. This in-control data is established as a norm
to monitor the future observation and to see whether it significantly deviates away from the
norm. The data purging includes identifying and removing outliers and/or substitute
missing data with an estimate.
Step 6. So far, at least one particular SPC tool (Shewhart control charts, T2 control chart
or TA2 control chart and SPE plot) shall be chosen to monitor the process. An out-of-control
situation occurs while using Shewhart control charts, then the responsible variable(s) will
reveal easily. While using MSPC control chart, the diagnosis of responsible variables(s) of
an out-of-control situation will require more analysis procedures. The detailed will be
shown in MSPC diagnosis which will be explained in next section.
Systematic Pattern in MSPC
It should be noted that in Chapter 2.1.2, we have addressed the issue of systematic pattern in
Shewhart control chart and provided Western Electric run rule as a detecting tool. Although it is
obvious that systematic pattern in the control chart indicates extraneous sources of process
variation, yet very little research addressed the detection of systematic pattern in T2 control chart.
For multivariate statistical process control chart, the information of systematic pattern is difficult to
interpret because the T2 statistic value is a synthetic information of all the variables. It is a
generalized distance between an observation to its sample mean, and it has no clear physical
meaning of any one variable in the variable set. Besides, the Western Electric run rules is not an
appropriate tool to apply, because T2 statistic has a non-normal distribution. (Mason, Young and
Tracy, 2003). We suggested that Shewhart control chart of each variable should be constructed
and examined independently when an out-of-control situation occurs in the multivariate statistical
process control chart. The procedures of examining systematic pattern of individual variable can
be referred to Chapter 2.1.2.
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Chapter 4.
MSPC Implementation Guideline
By now the MSPC has been incorporated into Method model for (M)SPC scheme. Step 1 and
step 2 do not really require statistical knowledge. People who actually operate the system are the
excellent source to consult to. Apart from that, Hotelling’s T2 control chart is already supported by
some computer software, collecting data from the process will be a familiar task for operators as
what they do in applying Shewhart control charts. From our survey, inverting covariance matrix is
rarely seen and Hotelling’s T2 control chart would be applied in most of the cases. Therefore, the
implementation of MSPC does not increase too much extra work for the plant staffs.
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Chapter 4.
MSPC Implementation Guideline
Step 1
Process
Legend :
investigation
Decision point
Decision option
Step 2
Process
breakdown
(optional)
Step 3
N=1
N: number of
variables
Action
N>=2
No
Variables
correlated?
(r > 0.3)
Yes
Step 4
Yes
No
Shewhart
control chart
N is too many to
monitor
separately.
Yes
No
Inversion of
covariance matrix
available?
T2 control
chart
TA2 and SPE
control chart
Step 5
In-control data
and control limits
construction
In-control data
and control limits
construction
In-control data
and control limits
construction
Step 6
No
Continue
process.
Monitoring future
observation.
Out-of-control
occurs?
Yes
Responsible
variable(s) are
found. Investigate
the root causes.
No
Continue
process.
Monitoring future
observation.
Out-of-control
occurs?
Yes
Apply MSPC
diagnosis.
Figure 4-2 Method model for (M)SPC
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Chapter 4.
MSPC Implementation Guideline
4.1.2. MSPC Diagnosis
MSPC Diagnosis (Figure 4-3) is applicable when an out-of-control situation occurs. Although
MSPC control chart is mainly used for monitoring two or more variables which are correlated
between each other, it is also a good tool to reduce the work of monitoring many individual
Shewhart control charts at a time. Due to the concern of plant staff’s reluctance to accept MSPC,
we tried to make the scheme as simplified as possible. The step-by-step instructions for using
MSPC diagnosis are described below (also see Chapter 2.3.2. T2 diagnosis with Principal
component analysis).
Step 1. Compute Normalized PCA Scores (ta/sa) according to the following formula. The
eigenvalue and the eigenvector of Principal Component Analysis can be obtained from a
general statistical software packages. The rest of the calculation is also possible to
perform with Excel spread sheet.
n
t a = Pa' (x - x) = ∑ pa,j (x j - x j )
j =1
sa = λ a
The principal component(s) with higher normalized scores (ta/sa) can be further
investigated with contribution plotting (MacGregor, et al. 1994).
Step2. Construct overall contribution plot.
1). Select the k normalized scores with high values. Normally the largest two or three
normalized scores will be sufficient.
2). Calculate contribution of a variable xj in the normalized score.
Cont a,j =
ta
pa,j (X j − X j )
λa
Conta, j is set to zero if it is negative.
3). Sum the total contribution of variable xj.
k
CONTj = ∑ (cont a,j )
j =1
The overall average contribution per variable generates an overview contribution of each
variable in one plot, which is very convenient. Besides, graphical information is easier to
understand. The operator can clear see which variables have with higher contribution for a
particular out-of-control event and conduct further investigation work.
The function of diagnosing the responsible variables from Hotelling’s T2 control chart
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Chapter 4.
MSPC Implementation Guideline
so far is rarely found in computer software packages. For example, STAGRAPHICS (PLUS
5.1) is one of the softwares supporting multivariate statistical process control but it only
supports to compute the Hotelling’s T2 statistic of measurement, no further diagnosis
analysis.
Legend :
MSPC
Decision point
application
Decision option
Action
Yes
No
Monitoring future
observation.
Out-of-control
occurs?
PCA diagnosis approach
Continue
process.
Compute Normalized PCA
Scores (ta/sa).
Construct overall
contribution plot of the
out-of-control observation.
Responsible variable(s) are
identified. Investigate the
root causes.
Figure 4-3 MSPC diagnosis.
4.1.3. Process control
It should be aware that the function of process control chart, either Shewhart control chart of
multivariate statistical process control chart, only monitors the behavior the variables. So we will
always have two consequences when process control chart is applied. First, let the process
continue when there is no signal from the control chart. Second, when the control chart signals,
responsible variables await to be further identified. It should be noted that identifying responsible
variables does not mean the root causes of an out-of-control situation are located. Without
identifying the root causes and taking necessary steps to bring the process back to normal status,
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Chapter 4.
MSPC Implementation Guideline
the process does not fix itself automatically. Therefore, we need additional tools that can
systematically help us find the potential root causes when out-of-control situation is encountered.
Cause-and-Effect Diagram (Figure 4-4) is a recommended tool to uncover the potential
root causes of an undesired problem. The procedures of constructing a cause-and-effect diagram
are summarized as follows (Modified from Montgomery, 1985).
1). Define the problem or the effect that needs to be analyzed.
2). Uncover potential causes through brainstorming.
3). Specify the major potential cause categories and join them as boxes connected to the
center line.
4). Identify the possible causes of each major potential cause.
5). Rank the causes to identify those that seem most likely to affect the problem.
6). Take corrective action.
The possible reasons and the remedies of an out-of-control situation shall be collected
and documented as an Out-of-Control-Action plan (OCAP). It should contain all the diagnostic
knowledge and all the operators can follow the standardized procedures to bring the process
back to in-control status.
Machines
Materials
Methods
Wrong procedure
Wrong tool
Defect supplier
Damaged handling
Insufficient warm-up
Wrong planning
Problem or
undesired effect
Wrong specification
Inexperience
High humidity
High temperature
Faulty gauge
Measurement
Poor attitude
Manpower
Environment
Figure 4-4 An example of Causal-and-effect diagram.
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Chapter 4.
MSPC Implementation Guideline
Shewhart/Deming wheel (PDCA)
The Shewhart wheel is one of the well-known strategies to achieve the process improvement. It is
known as PDCA cycle, where P stands for Plan, D stands for Do, C stands for Check and A
stands for Act. The idea of PDCA was originated also by Dr. Shewhart, and later popularized by
Dr. Deming. The book Out of the crisis (1986), by Dr. Deming is a recommendable reference for
further study.
Figure 4-5 The Shewhart/Deming wheel (PDCA).
Here we summarized the key idea of PDCA.
Plan: Identify the problem and possible causes. Cause-and-Effect Diagram can be an
effective tool to accomplish this task.
Do: Make changes that designed to correct to problem or improve the current process
situation.
Check: Study the result of these changes that have been taken.
Act: Standardize the changes if the result is successful, and document these changes
into OCAP. Whereas, search new strategy to improve the process if the changes did not
make much progress.
Combing the activities that involved in MSPC technique, such as monitoring the process,
diagnosing the responsible variables for out-of-control measurement and PDCA strategy, it forms
a continuous work flow to continuously improve the process performance.
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Chapter 4.
4.2.
MSPC Implementation Guideline
MSPC Training
Taking into account the concerns of lacking of knowledgeable employees and high
implementation cost that raised from the interview of practical filed. We recommend different
levels of training should be developed to train the employees.
First, advanced level of MSPC education should be provided to quality experts in the
R&D department. In the beginning of the implementation, the quality experts must be familiar and
execute all the steps mentioned in the MSPC Plan. MSPC Plan is a generic instruction, the
quality experts should apply and adjust (if it is necessary) the MSPC Plan to fit the real situation.
After construction the detailed procedures, they may assign the tasks down to lower level quality
people.
Second, basic level of MSPC education should be provided to all the shop floor operators,
process engineers in the plant. It is necessary for them to be familiar with concept of MSPC and
correctly conduct all the relevant tasks assigned from quality experts. The lower level employees
should be able to properly collect data, and correctly react to out-of-control event according to
Out-of-Control-Action Plan (OCAP).
Third, moderate level course for management employees should be provided. In order to
avoid the communication gap, it may not be necessary for management employees to perform
the statistical analysis, yet they should be familiar with all the procedures happening during the
process monitoring.
4.3.
Team Approach
MSPC Plan can be viewed as the tool to conduct MSPC monitoring, and MSPC Training makes
the users understand how to correctly use this tool. During the MSPC Training, we generally
define three different levels of users, advanced level (quality experts in research center), basic
level (plant people) and moderate level (management employees). The MSPC implementation
work is directly conducted by quality experts and plant people in a way of team work. Team
approach describes the work scopes of quality experts and plant people and how they should
conduct the work by using the skills that they have been trained in the previous step. The
guideline of work scope is described as follows.
Quality experts:
‹
Develop/adjust the MSPC Plan to fit the real situation.
‹
Construct OCAP.
‹
Conduct internal training, to plant people and management employees.
‹
Supervise the work of plant people.
‹
Provide expertise to plant people on complex issue.
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Chapter 4.
MSPC Implementation Guideline
Plant people:
4.4.
‹
Implement the MSPC Plan that defined by quality experts.
‹
Resolve out-of-control situation according to OCAP.
‹
Support quality experts on maintaining OCAP.
Management Involvement
Management commitment is crucial to the success of implementation work. The progress of the
entire MSPC implementation should be monitored by management employees. A regular review
system and audit system should be established. The management employees not only monitor
the process performance, but also continuously drive the plant people to improve their work. On
the other hand, except driving force, encouragement is also necessary. Motivate the plant people
with rewards, in terms of promotion, bonus, etc., and create the quality culture inside the
organization can be helpful.
In addition to monitoring the technical issues, management employees should also
coordinate with relevant department, such as finance, procurement, marketing, etc. to achieve an
overall improvement. For example, when the poor quality results from the inferior raw material,
then the task is expended to procurement and finance aspects as well.
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Chapter 5.
Case Study
Chapter 5. Case Study
In this chapter, a real case from a paper-making company will be studied. Two main analyses will
be conducted. First, we will implement multivariate statistical process control in this case, and
validate the approaches that we have developed in chapter 4 (Method model for MSPC and
MSPC diagnosis). Second, we will investigate the mechanism of the process system with multiple
regression analysis and provide practical recommendations in order to improve the process
performance.
5.1. Case Briefing
The case for further study is a process unit from a paper-making process. This process unit can
be described as follows. A flow containing fibers (named Thickstock) is mixed with another flow
containing fillers and water (named Filler). Thickstock will be called Thinstock when diluted with
water. So the Thinstock is the mixture of fibers, fillers and water. This dilution with what is called
Whitewater to control the solid content in the Thinstock. So the Whitewater increases when the
solids percentage in the Thinstock is too high, whereas it decreases when the solid percentage is
too low. In addition, a chemical (named Ret) is added before the Headbox to accelerate the
separation of solids and the water during the dewatering. Major part of the solids will form the
paper at the end, and the liquid leaving the dewatering machine is mainly recycled as Whitewater
to dilute the Thinstock again.
Finally the mixture flow (named Mixture-in) being pumped into Headbox contains fibers,
fillers, water, and chemical. There are two variables are measured in the Headbox, one is the
amount of solids of the Mixture-in (named HBtotal) and the other is the portion of the solids due
to fillers (named HBash).
Also two variables are measured in the Whitewater, one is the amount of solids (name
WWtotal) and other is the portion of solid due to fillers (named WWash). The amount of Ret is
controlled by measuring the difference between HBtotal and WWtotal.
After dewatering, the solid material will form the paper and the weight of paper is
measured (named BW). Part of the BW comes from fillers, this portion is named Paperash.
Paperash is a way of reducing the production cost, because fibers are more expensive. Within
our analysis scope, stable and correct BW value is the target. BW is also a precondition for further
processing. A conceptual scheme is provided in Figure 5-1. It should be noted that a certain level
of simplification has been embedded, because to fully reflect the real situation only when the
entire process system is considered.
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Chapter 5.
Case Study
Figure 5-1 Scheme of paper-making process unit.
All the process variables are tabulated as follows.
Table 5-1 Process variables.
Variables
Measuring unit
Description
Thickstock
[ton/hr]
Inflow of Thickstock.
Filler
[ltr/min]
Inflow of the fillers and water.
HBtotal
[g/ltr]
Amount of solid in the Thinstock.
HBash
[%]
Amount of solid in the Thinstock due to fillers.
Ret
[ltr/hr]
Chemicals
WWtotal
[g/ltr]
Amount of solid in the whitewater.
WWash
[%]
Amount of solid in the whitewater due to fillers
Paperash
[%]
Amount of paper weight due to fillers.
BW
[g/m2]
Paper average weight.
5.2.
MSPC Implementation
Before applying the MSPC to this case study, the data needs to be examined. The raw data was
provided from the plant supervisor and all the measurements of the process variables are
measured by sensors with an interval of one minute. The raw data set is a period of 1001
measurements. To obtain a rough idea what the process behavior is of the variable, we construct
the I-chart for each variable with raw material. Surprisingly, we see the control limits (with 3 sigma)
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Chapter 5.
Case Study
are extremely small and numerous of measurement fall outside of the control limits. The reason is
that, the sample size of measurement is only one (that is way we used I-chart), and the control
limits of I-chart were calculated based on the moving range of consecutive measurements. Due to
the small interval (one minute), the moving range between every two measurements is extremely
small. We recognized that the control chart with such small measuring interval is meaningless,
because it does not reflect the real process behavior. After some trials, we construct the control
chart for each variable with an interval of 20 minutes. Therefore, we have a preliminary data set
containing 51 measurements and we will continue the analysis with this data set.
Measurement Interval of I-chart
Here we will elaborate why the one-minute interval is not an appropriate choice. The idea of the
control limits in the Shewhart control charts is to represent the process behavior, which means the
control limits should correspond to the variation of the process. A graphical illustration will be
easier to understand. For upcoming explanation, we will take variable HBtotal as an example.
Figure 5-2 is the I-chart of HBtotal that constructed from 1001 measurements with one-minute
interval. The control limits are very close to centerline, and they are directly influenced by the
small moving range of consecutive measurements.
Control Chart: HBtotal
14.8
HBtotal
UCL = 14.5229
Average = 14.4954
LCL = 14.4679
14.6
14.4
14.2
14.0
989
963
937
911
885
859
833
807
781
755
729
703
677
651
625
599
573
547
521
495
469
443
417
391
365
339
313
287
261
235
209
183
157
131
105
79
53
27
1
Figure 5-2 I-chart of HBtotal with one-minute measurement interval.
With such small measurement interval (one-minute), the measurements are severely
auto-correlated. Auto-correlation means that the value of each measurement is dependent on the
previous one. Therefore, while using control charts, the assumption that the data from the
process is normally and independently distributed with its mean and standard deviation is violated.
By selecting the measurements with larger interval can lower the tensity of auto-correlation
situation.
The control limits of I-chart will become wider as we choose larger interval, because the
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Chapter 5.
Case Study
moving range between measurements also increases. Now we come to a question, to what
measurement interval is the proper choice? The percentiles of histogram plot can be a good
reference. Now, if we plot the histogram of all these measurements (Figure 5-3), we see that the
variance of the process is much wider than the control limits (14.47, 14.52) in Figure 5-2. The
control limits of 3-sigmas away from the mean in the histogram plot are approximately (14.10,
14.89).
Histogram from Dataset
240
220
200
180
160
140
Count
120
LEGEND
Normal (14.4954,0.132827)
Dataset
100
HBtotal Count
80
60
40
20
0
14
14.05
14.1
14.15
14.2
14.25
14.3
14.35
14.4
14.45
14.5
14.55
14.6
14.65
14.7
14.75
14.8
HBtotal
Figure 5-3 Histogram of HBtotal measurements.
When we adopt 20-minute interval, the control limits of I-chart has become (14.2, 14.8)
and 51 measurements remains. Figure 5-4 is the I-chart of HBtotal with 20-minute interval and it
is considered acceptable for further analysis.
Control
Chart: HBtotal
14.8
HBtotal
UCL = 14.7668
Average = 14.4848
LCL = 14.2028
14.6
14.4
14.2
14.0
51
49
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
5
7
1
3
Figure 5-4 I-chart of HBtotal with 20-minute interval.
During this case study, we intend to apply the approaches developed in Chapter 4 (Method model
for MSPC and MSPC Diagnosis) so as to validate the effectiveness of these tools. Therefore, in
the following section, we will perform the analysis step by step associated with detailed
explanation and also provide a clear overview by highlighting the decision path in Figure 5-5.
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Chapter 5.
Case Study
Step 1
Process
Legend :
investigation
Decision point
Decision option
Step 2
Process
breakdown
(optional)
Step 3
N=1
N: number of
variables
Action
N>=2
No
Variables
correlated?
(r > 0.3)
Yes
Step 4
Yes
No
Shewhart
control chart
N is too many to
monitor
separately.
Yes
No
Inversion of
covariance matrix
available?
T2 control
chart
TA2 and SPE
control chart
Step 5
In-control data
and control limits
construction
In-control data
and control limits
construction
In-control data
and control limits
construction
Step 6
No
Continue
process.
Monitoring future
observation.
Out-of-control
occurs?
Yes
Responsible
variable(s) are
found. Investigate
the root causes.
No
Continue
process.
Monitoring future
observation.
Out-of-control
occurs?
Yes
Apply MSPC
diagnosis.
Figure 5-5 MSPC decision path of case study.
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Chapter 5.
Case Study
Step 1. Process investigation.
The description of the process unit and its scheme has been provided in the section of
case briefing. As we mentioned in Chapter 4, monitoring a large amount of variables is not
efficient. Only the critical quality characteristics should be selected and monitored. Another
concern here is that the process unit contains several loop controlling systems, therefore
many variables change during processing due to automatic controller. What we need to do
is to search which variables need to be stable to achieve a stable BW.
If we look at the process scheme, we will see that the HBtotal and HBash are the
quality characteristics of the input (Mixture-in) flowing into the dewatering machine.
Therefore, when the processing inside the dewatering machine is stable; the output (BW)
should be stable if the input is stable. So HBtotal and HBash are two variables to be
monitored with statistical control chart.
As mentioned in the case briefing, the Ret is automatically controlled due to HBtotal
and WWtotal. Thickstock and the Whitewater inflow are also automatically controlled to
maintain stable quality of the Mixture-in. Only the Filler is a fixed flow and it can be
monitored with statistical control chart. Therefore, we will perform the multivariate
statistical process control for these three parameters, Filler, HBtotal and HBash.
Step 2. Process breakdown.
After conducting the preliminary investigation in step 1, we see no reason for further
decomposition. We continue the analysis according to following steps.
Step 3. Number of variables.
Filler, HBtotal and HBash are the variables to be monitored, so it is clear the number of
variables N is three. The next action is to examine the variable dependency. The
correlation between variables is a good indicator telling us how intense the variables are
related to each other. Table 5-1 is the variable correlation generated from a period of
process containing 51 measurements. Here we see that there is a moderate positive
correlation between HBtotal and HBash, so this is a good condition (but not absolutely
necessary) to adopt multivariate statistical process control.
Table 5-1 Correlation of process variables
Correlation
Filler
HBtotal
HBash
Filler
1.00
-0.17
-0.19
HBtotal
-0.17
1.00
0.58
HBash
-0.19
0.58
1.00
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Chapter 5.
Case Study
Step 4. Checking the inversion of covariance of variables and choosing T2 control chart.
Normally it will be problematic when the number of variables is really large. In this case
dealing with three variables, the inversion of covariance is not a problem (it has been
further examined with statistical software). So it is applicable to adopt T2 control chart.
Step 5.
In-control data and control limits construction. A set of data containing 51
measurements with an interval of 20 minutes was taken from a generally in-process period.
The data was further analyzed by I-chart (Individual chart) with customary plus/minus 3
sigma control limits and the problematic measurements were removed. After that, we
obtained a sample set containing 21 measurements. The T2 control chart was also
constructed (see Figure 5-6) to see whether any observation containing a problematic
relationship between parameters. In this case, no indication of out-of-control showed. The
measurements (No.1 to 21) are tabulated in Table 5-2. This data is used as a norm to
monitor future observation and to further analyze the reasons of out-of-controls, if any.
Figure 5-6 T2 control chart of in-control measurements.
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Chapter 5.
Case Study
Table 5-2 In-control process data
No.
Filler
HBtotal
HBash
No.
Filler
HBtotal
HBash
1
25.432
14.497
3.184
16
24.485
14.508
3.181
2
25.753
14.517
3.172
17
24.762
14.420
3.157
3
25.178
14.696
3.199
18
25.161
14.528
3.196
4
25.318
14.581
3.207
19
25.393
14.580
3.190
5
25.321
14.548
3.205
20
24.773
14.560
3.182
6
24.636
14.596
3.204
21
24.741
14.413
3.174
7
25.055
14.490
3.197
22
24.926
14.480
3.124
8
24.963
14.594
3.182
23
25.661
14.452
3.132
9
24.720
14.528
3.167
24
24.839
14.752
3.275
10
24.347
14.516
3.165
25
24.474
14.503
3.266
11
25.000
14.552
3.162
26
24.598
14.478
3.226
12
24.953
14.401
3.160
27
24.777
14.314
3.191
13
25.269
14.501
3.150
28
24.924
14.366
3.190
14
24.862
14.483
3.165
29
24.478
14.665
3.186
15
25.107
14.513
3.180
Step 6.
Monitoring future observation. A period of future observation with eight
measurements (No.22 to 29) will be analyzed with T2 control chart and see it if any
observation is out-of-control. The Hotelling’s T2 statistic is calculated for each new
observation based on the mean and the covariance matrix obtained from the in-control
data set. The control limit is chosen with Type I error α = 0.05. The T2 control chart (Figure
5-7) shows the observations 22, 23, 24, 25, 26, 27, and 28 are out-of-control. The control
chart signaled us that something went wrong during the observation 22 to 28, yet we do
not know which variable or set of variables is responsible for it. So we need to identify
those variables with MSPC diagnosis.
Figure 5-7 T2 control chart for future observations.
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Chapter 5.
Case Study
MSPC Diagnosis
We will apply the approach T2 diagnosis with Principal component analysis (PCA) (details please
refer to Chapter 2.3.2). The generic steps can be repeated briefly once more. Normalized PCA
scores (ta/sa) are calculated and see which one(s) has/have higher scores. Figure 5-8 shows the
chart of overall average contribution per variable is constructed based on the selected high score
Normalized PCA. The contribution of each variable to the out-of-control measurement will be
shown in this chart, and it gives us the information what problematic variables are. It should be
noted that it is suggested to perform the diagnosis approach with standardized data set. Due to
the different measuring scale of each variables, the variable with smaller measuring scale will
have relatively less weight than the one with larger measuring scale. For a particular observation,
each bar indicates the contribution of one variable. The bars represent Fillers, HBtotal and HBash
(from the left to the right).
Variable contribution
22
23
24
25
26
27
28
Figure 5-8 Overall average contribution of every variable from observation 22 to 28.
5.3. Result of MSPC Implementation
We have known the out-of-control observations detected by T2 control chart, and obtained the
overall contribution plot which tells us what problematic variables are for each observation. Now
we are going to analyze each of the out-of-control measurements and draw conclusions.
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Chapter 5.
Case Study
Table 5-3 The result of MSPC diagnosis.
Observation
Signaled
by MSPC
Potential
problematic
Status
variable(s)
Signaled
BW is
by USPC
Out-of-control?
22
Yes
HBash
Exceeding control limit.
Yes
23
Yes
HBash
Exceeding control limit.
Yes
Fillers
Within control limit
No
HBtotal
Exceeding control limit.
Yes
HBash
Exceeding control limit.
Yes
24
Yes
No
No
No
25
Yes
HBash
Exceeding control limit.
Yes
Yes
26
Yes
HBash
Exceeding control limit.
Yes
Yes
27
Yes
HBtotal
Exceeding control limit.
Yes
Yes
28
Yes
HBtotal
Within control limit
No !!
Yes
From the summary in Table 5-3, we can see that from observation 22 to 27, at least one of the
three variables exceeded control limits of the I-chart, which means these observations would also
receive an out-of-control signal by using Shewhart control chart. In observation 28, there is no
any variable falling outside of control limit, but T2 control chart signaled.
For the convenience of comparison, the I-chart of three monitoring variables (Filler,
HBtotal, HBash) and BW are constructed in Figure 5-9. If we look at these three variables
carefully in the I-chart respectively, we will see that in observation 28, the HBtotal has low value
while HBash is relatively high. This behavior contradict to the in-control process behavior that we
already found that there is a positive correlation between HBtotal and HBash.
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Chapter 5.
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Figure 5-9 I-chart of Filler, HBtotal, HBash and BW.
To provide a better picture, we plot the ellipse control chart (Figure 5-10) just for variable HBtotal
and HBash, to see how observation 28 is signaled as out-of-control. The correlation between
HBtotal and HBash has been examined. It is a moderate positive correlation, so the control region
(the ellipse) is slightly fat. In Figure 5-10, we can clear see that HBtotal and HBash are both within
individual control limits, which is traditionally statistical process control approach. But due to the
anti-correlation, it falls outside of the elliptical control region, which is signaled by multivariate
statistical process control approach.
Another point to be discussed is that, in Table 5-3; we compared the status of output (BW) and
the variables being monitored. They do not fully correspond to each other. Observations 25 to 28,
MSPC signaled, and BW appeared to be out-of-control. Yet, observation 22 to 24, MSPC
signaled but BW was actually in-control. The explanation could be that the variables that we have
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Chapter 5.
Case Study
monitored do not completely represent all the characteristics of the Mixture-in. HBtotal and
HBash only measure the amount of solid content and the portion due to fillers. Obvious the
chemical content, other possible characteristics, for instance, the viscosity, ph value, purity,
temperature, etc. of the Mixture-in are not monitored. Moreover, the processing inside the
dewatering machine has been assumed to be a constant state. Due to such circumstance, we
see the need of an integrated monitoring activity which can reflect the whole process better.
Ellipse control chart
3.28
3.25
28
HBash
3.22
3.19
3.16
3.13
3.10
14.3
14.4
14.5
14.6
14.7
14.8
HBtotal
Figure 5-10 Ellipse control chart of HBtotal and HBash.
5.4. Reflection
From the findings of this case study, several points can be concluded.
1). MSPC is a more sensitive technique than Schewhart control charts in terms of detecting
power. It monitors not only the deviation from its mean of a variable, also monitors the relation
between variables. The traditional control chart (e.g. Shewhart control chart) only signals when
the deviation of a variable is abnormal.
2). As we have mentioned in Chapter 2, the idea of SPC is to detect the variability of a process
due to special causes and improve the process performance by eliminating these causes. In this
sense, high sensitive SPC technique such as MSPC is more powerful on detecting the
occurrence of special causes than Shewhart control chart.
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Chapter 5.
Case Study
3). For the case that the variables are highly correlated with each other, the chance that the
variables are in-control of its individual control chart but fall outside of the MSPC control limit will
be much larger. Under such circumstance, using Shewhart is very likely to miss the timing to
signal when the process goes wrong. Therefore signaling out-of-control observation effectively by
MSPC helps to reduce the cost on producing defect product.
4). Modern production system is very complex. The example of this case study already has
shown that even a small unit of the entire process may contain several control loops, automatic
control devices, etc., which make the process more complicated and dynamic. Although statistical
process control chart (either Shewhart control chart or Multivariate statistical process control
chart) is just one of the process control techniques. In order to successfully deal with a complex
process system and improve the process performance, additional integrated technique is
definitely required.
5.5. Process Performance Improvement
In this section, an example will be provided to show how other technique is applied to achieve
better process performance. As already mentioned the process is quite complex, thus, for the
ease of demonstration, we will depict a smaller sub-process as an example for further analysis
(Figure 5-11). In this sub-process, BW is considered as the output that needs to achieve a certain
target value. The information about the input contains HBtotal, HBash and Ret. By investigating
and understanding the mechanism between input and output, we may discover more alternative
ways to control the process and achieve the desired quality of output
Figure 5-11 Sub-process.
A multiple regression model will be performed with a period of data set (the data set can be found
in Appendix E) and help us to understand the mechanism between inputs and output. We can
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Chapter 5.
Case Study
express the output as a function of inputs. It should be noted that non-liner relation, such as
quadratic term or interaction term can also exist between input and output. So we will experiment
the multiple regression with different types of model including linear model, linear with interaction
terms and quadratic model. Each of the regression models will be examined statistically. The
result is given in Table 5-4, detailed information about multiple regression analysis can be found
in Appendix D.
Table 5-4 Result of regression model of sub-process.
Regression model
BW = f (HBtotal, HBash, Ret)
Quadratic model
BW = 0.03 + 0.34HBtotal + 0.04HBash + 0.79Ret + 0.016HBtotal – 0.19 HBash
(Scaled data)
(Interaction term is not significant, so it is not included)
Quadratic model
BW = 378.54 – 165.06HBtotal + 583.36HBash + 0.67Ret + 5.77HBtotal – 91.41 HBash
(Original data)
(Interaction term is not significant, so it is not included)
Adjusted R
2
2
2
2
2
0.70
P r e d i c te d B W
G ra p h
3 .1 8 7 4 5
1 4. 48 48
4 3 .5 4 2 1
1 57
B W
15 5.44 8
+ /-0 .2 8 5 4 9 7
1 56
1 55
1 54
3 .1 5
3 .2
3 .2 5
H B as h
1 4 .2
1 4. 4
1 4. 6
H B t o ta l
43
Regression model
BW = f (HBtotal, HBash, Ret)
Linear with interaction term
BW = 0.22HBtotal + 0.76Ret
model
(Interaction term is not significant, so it is not included)
(Scaled data)
Linear with interaction term
BW = 110.97 – 1.13HBtotal + 0.64Ret
model
BW = 0.22HBtotal + 0.76Ret
(Original data)
44
R et
45
(Interaction term is not significant, so it is not included)
Adjusted R
2
0.66
P re dicte d BW Gra ph
1 4 .4 8 4 8
4 3 .5 4 2 1
1 57
BW
155.427 1 56
+/-0. 171292
1 55
1 4. 2
14.4
H B to tal
14.6
43
44
R et
45
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Chapter 5.
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From the analysis result in Table 5-4, we see that a quadratic model explains the relations of
inputs/output to a satisfactory level (Adj-R2=0.70). The regression model was performed with both
original and scaled data. Using scaled data, the coefficient of each term in the model is
considered as the weight of the term, so we can see which term has stronger influence to the
output. It is clear that Ret and HBtotal dominate the response variable (BW) most. In the
predicted BW graph, we can see the relation between BW and each input variable. Due to the
quadratic model term, HBtotal and HBash both have a curve regression line. Particularly in
HBtotal, we found it difficult to interpret the meaning of its relation with BW. The weight of the
paper (BW) mainly comes from the solid in the input and the graph shows either increase or
decrease of HBtotal can raise BW, which does not reflect the real situation. It should be noted that
sometimes a particular regression model may show a satisfactory statistical result, yet it does not
correspond to the reality. We should be careful on such situation and check whether there are
other types of model can lead to better explanation.
Therefore, we continued to try linear model with interaction term. The result is given in
Table 5-4, and the positive linear relations reflect our expectation better. Again we see that BW is
mainly dominated by Ret and HBtotal with positive relations, whereas HBash has no significant
relation with BW. Knowing the relation between variables, we can further control the process in a
more predictable way. Figure 5-12 gives us even clear information. To increase HBtotal and Ret
with different combination can reach a particular target BW value. An example is given in Table
5-5. The possible recipe of HBtotal and Ret can be decided due to different concerns, such, cost,
availability and etc., to achieve an efficient purpose.
Predicted BW Contour Pl ot
45
156.2
156
55.8
Ret
44
1 56.8
156.6
156.4
1 56 .2
156
156.2
1 56
155.6
155.8
55.4
155.6
1 55.8
155.2
155.4
1 55 .6
155
43
156.4
154.6
14.1
14.2
Predicted BW
155.2
155
1 54 .8
1 4. 3
LEGEND
155.4
155.2
154.8
156
155
14.4
14.5
14.6
14.7
HBtot al
Figure 5-12 Contour plot of HBtotal and Ret with respect to BW.
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Chapter 5.
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Table 5-5 Possible recipe of HBtotal and Ret
Desired BW
Estimated BW [g/m2]
HBtotal [g/ltr]
Ret [ltr/hr]
155.0
155.5
15.0
43.0
155.0
155.4
20.0
34.0
137.0
137.0
6.0
30.0
137.0
137.4
12.0
20.0
However, it has been noticed that the Adj-R2 is not really high (0.66), in the linear model, which
means the relation was not explained by the model very well. The possible reasons could be the
same as we have mentioned in Chapter 5.3. The characteristics of inputs may not be fully
measured, so the current characteristics of the input are not able to reflect the mechanism inside
the process. Another possibility is that the measurement error exists. Nevertheless, this example
demonstrated how other technique helps to achieve a better process performance. The same
exercise can be extended to other parts. For example, consider HBtotal is an output of another
sub-process, investigating what the relations are between the output and its inputs. A target value
of HBtotal can also be reached by controlling the inputs in a logical way. Once the whole process
is well investigated, the plant staff will have better understanding of the process mechanism and
be able to improve the process.
To close up this section, we can conclude that to achieve the goal of improving the production
quality of a particular industry, investigating and understanding the current process system are
very necessary and very fundamental tasks. Without such knowledge, further technique
application, such as SPC control charts or multiple regression models will have very limited
contribution. The idea is that these techniques are applied to analyze and to reflect what actually
is happening during the process and after that, we can have greater opportunity to control the
process and further improve the production quality. Nevertheless, correct and complete
understanding of the mechanism of the entire process prevents our judgment from misled by the
incorrect analytical results.
Two advices can be given. First, look at the entire production process with broader
perspective to have a thorough picture. This information is very useful when breaking down the
process into smaller units for detailed analysis and also very helpful to make the right judgment
on the results of analysis. Recall the multiple regression model (the quadratic model), the result
shows the BW would increase when HBtotal either increases or decreases, which is not true.
Understanding the process and having logic thinking help us to make the correct judgment.
Second, start with smaller part of the process while conducting detailed analysis. By doing so, it is
easier to find out what really happens inside the process. Then we may gradually expend the
scale of process unit for analysis. For instance, if we apply the multiple regression with larger
scale process unit immediately, which includes too many variables in it, we may have problem to
identify the causal relation between dependent variable and independent variable.
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Chapter 6.
Conclusions and Recommendations
Chapter 6. Conclusions and Recommendations
During this research, we began with the importance of the quality and how statistical process
control (SPC) was developed. Then we addressed the Shewhart control charts and the multivariate
statistical process control (MSPC). Due to the gap between academic development of MSPC and
its practical application, we started to investigate the situations of both sides so as to further
facilitate the implementation of MSPC and support the industrial plants to utilize the benefit of
MSPC technique. In this chapter, we will summarize the main findings of this research and provide
several recommendations. At the end, a prospect of future research will be suggested.
6.1. Conclusions
The objective of this thesis research is to make recommendations for implementing multivariate
statistical process control (MSPC) in a process-industry plant by providing clear interpretations of
MSPC and suggestions to quality management staff in the plant. We will evaluate the
achievement of this research work by reviewing the research question and its sub-questions.
The main research question is defined as, “What are the difficulties of multivariate
statistical process control (MSPC) implementation and how quality management staff can be
supported to facilitate MSPC in a process industry plant?” There are five sub-questions, which
are defined as:
1). What is the essence of MSPC?
2). What are the expectations from quality management staff in the process industry plant?
3). What tools can be provided to make the interpretation of MSPC results easier?
4). What advices can be provided to cope with the time axis problem for MSPC?
5). What recommendations can be provided to quality management staff in the process
industrial plants?
The findings and the contributions of the research which provide answers to all the sub-questions
are concluded as follows.
‰
Clear Introduction of Statistical Process Control.
Although multivariate statistical process control (MSPC) is the main topic of this research, various
types of Shewhart control charts and its application were also addressed (in Chapter 2.1) so as to
make this study more complete. In addition, with the knowledge of Shewhart control charts, it is
easier to further understand MSPC.
‰
Discussion of MSPC.
In Chapter 2, Hotelling’s T2 statistics was well elaborated. We began with simplest case, bivariate
process control chart to illustrate to idea of MSPC. After that, a generic MSPC control chart for
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Chapter 6.
Conclusions and Recommendations
multiple variables was extended. Following the explanation, a practitioner will be able to perform
the multivariate statistical process control chart to examine whether a period of process is
in-control. Apart from typical Hotelling’s T2 statistics, an adjusted type - T2A & SPE plot - was
discussed. This method can be applied in case the inversion of covariance matrix is not available
(see Chapter 2.2.2.).
To overcome the difficult interpretation of MSPC, two approaches - MYT T2 decomposition
and T2 diagnosis with Principal component analysis (PCA) were introduced with a case
demonstration. In the end of Chapter 2, we provide a clear overview of multivariate statistical
process control approaches (Figure 2-14) and comparison of these approaches.
‰
Understand the Perspective and the Need from the Practical Field.
In order to facilitate the implementation of MSPC into the practical field, we also investigated the
perspective from practical field so as to understand the gap between the academic and practical
fields. The investigation was summarized in Table 3-1, and the potential remedies were generated
as well. Several conclusions have been made and incorporated during the MSPC Implementation
Guideline in Chapter 4. For example, the tools should be simple, easy to follow, graphical format,
associated with available software is preferable. Other comments will be further elaborated as part
of the general recommendations in the next section.
‰
Development of MSPC Implementation Guideline
By knowing the situations of both academic progress and the practical field, we have developed
MSPC Implementation Guideline. This guideline contains four elements, which are MSPC Plan,
MSPC Training, Team Approach, and Management Involvement. Especially in MSPC Plan, two
practical tools were constructed, which are Method Model of (M)SPC and MSPC Diagnosis.
Method model of (M)SPC is a decision flow chart, which supports the practitioners to apply the
proper statistical process control chart for different circumstances. It covers the situations of using
Shewhart control charts and using MSPC control chart. MSPC Diagnosis is designed to interpret
the result of MSPC. Because the MSPC only signals the occurrence of an out-of-control event, it
does not provide further information about what the problematic variable(s) are.
MSPC Diagnosis basically describes the key steps of T2 diagnosis with Principal
component analysis (PCA) introduced in Chapter 2. As already mentioned, one of the drawbacks
of MSPC is that it does not tell us what the problematic variable(s) are. We have screened two
advanced approaches, and the reason why we propose T2 with PCA approach has been
addressed in Chapter 2.4 (approach discussion). So far, some statistical process control software
packages provide the function of calculating Hotelling’s T2 statistics, yet the function of MSPC
diagnosis is rarely found. MSPC diagnosis can be used as a conceptual specification for software
programming in the next stage. Getting started is important. Practitioners may be overwhelmed
with various scientific knowledge, approaches, etc., although they all can be workable. Providing a
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Chapter 6.
Conclusions and Recommendations
clear and easier guidance on selecting the approaches indeed can save plenty of time and avoid
unnecessary confusion.
The other parts of MSPC Implementation Guideline emphasize how to practically
implement this technique into industrial plants and the concerns of financial, organizational,
managerial aspects were incorporated as well. The MSPC Implementation Guideline is a
practical tool to facilitate the MSPC application in industrial plants.
‰
Systematic Pattern of MSPC.
This problem and possible solution have been addressed in Chapter 4.1. It is suggested that
Shewhart control chart of each variable should be constructed and examined independently when
an out-of-control situation occurs in the multivariate statistical process control chart. The
procedures of examining systematic pattern of individual variable can be referred to Chapter 2.1.2.
‰
Learning from the Case Study.
Although MSPC advocates often emphasize the effectiveness of this advanced technique, the
real case application and implementation is rarely found. It is understandable that to have a clear
focus and framework for a particular research, a certain simplification is inevitable. However, it is
still important to step backwards and look at the problem in a broader view. Several points are
elaborated as follows.
a. MSPC control chart is examined as an effective technique in the case study; even the
correlation of variables is not really high. It is predictable that in the case where the
number of variables is large and the correlation between them is high, then using
MSPC would become much more valuable than using Shewhart control charts.
Because the probability that an out-of-control event falls outside of MSPC control limit
but within Shewhart control chart separately will become much larger
b. The real production process is often a very complex system. It may contain various
control devices, such as, automatic controller, automatic sensors, etc. Control chart
(either Shewhart control charts or MSPC control chart) is a necessary (but not the only)
tool to conduct statistical process control and further achieve higher process
performance. Using the proper tool at right place is important; for instance, using a
control chart to monitor the fuel consumption of a reactor where the temperature needs
to be stable is meaningless. Temperature is the one needs to be monitored!
c.
To improve a complex production process, many techniques are required. For example,
Design of experiment (DoE), (Non)-Linear multiple regression analysis, and SPC
control charts are often applied. An integration of all these technique may utilize their
effectiveness to a higher level.
d. Process investigation is crucial. Academic papers often start to talk about MSPC
analysis immediately with “given” variables. In reality, without investigating the process
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Chapter 6.
Conclusions and Recommendations
system, we may run into the risk of monitoring “fuel” instead of “temperature”. Learning
from doing or communicating with experienced experts can be helpful because process
investigation involves large amount of tacit knowledge and experience, rather than
purely mathematics or statistics knowledge.
6.2. Recommendations
Apart from the findings and developments that have been concluded, it is necessary to address
several points in this section. These points are closely relevant to this research, nevertheless they
are rather conceptual or managerial oriented.
‰
Financial Evaluation.
The benefit will not fall off from the sky. The implementation of statistical process control is a huge
investment in terms of human resource, equipment, finance, etc., especial for the complicated
technique like MSPC. Therefore, it is recommended to perform cost-benefit analysis while
considering the nature the process and the current process performance. The idea was
stimulated from our practical survey. The value of SPC or even MSPC can reveal only when its
contribution can reflect on the overall benefit of the company. “Industry is not interested in the
latest technological offering, be it a smart sensor or the latest distributed control system, unless
its money-making potential is clear and demonstrable” (Anderson, 1997). Implementing MSPC in
the right timing with proper circumstance can achieve higher utilization.
‰
Management Involvement.
Statistical process control is not simply a collection of various statistics tools. Management
involvement and commitment to the quality-improvement process are necessary components of a
successful implementation. The manager must have strong motivation and knowledge to
continuously push the implementation, on the other hand the employees need to be re-educated
and encouraged. In the long term, the culture of quality oriented should be stimulated and
become part of the origination.
‰
Organization of Quality Team.
To implement complex technique like MSPC, expertise is a key element. Most of the companies
concern the lack of knowledgeable employees. Hierarchical organization of quality management
department can be a good advice. Higher level statistician should have sufficient MSPC
knowledge, whereas lower level operators should receive less theoretical but more practical
knowledge. Such organization is more efficient in terms of human resource expenditure and the
company is also able to perform the internal training to improve the quality of employees.
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Chapter 6.
Conclusions and Recommendations
6.3. Future Research Prospect
The scope and the depth of this research are limited due to the time constraint, however, we are
getting started and making the first move. Based on the reflection and the inspiration from this
research work, two potential research prospects are provided as follows.
‰
Software Development.
We have been pointed out that computer software is a big boost to accomplish the MSPC
implementation. It is understandable and realistic to apply complex technique with the aid of IT
system. The program should be developed for MSPC diagnosis and it will be even better to
integrate automatic alarm system. So the software can perform the MSPC calculation, and
provide quick signal and clear information to the operator. The operator will be informed where to
look at, and what the possible solutions are.
‰
Potential Benefit Analysis.
Although we have conducted several interviews with companies, the data was not enough for
quantitative analysis. A detailed investigation regarding the relation between the potential benefit
of applying MSPC and the type of production can be good evidence to motivate the company to
implement MSPC. The analysis also can be used as a guideline for company’s self-appraisal, so
the company can have a clear vision and more confidence on MSPC implementation.
The research is temporarily summed up due to the time limit; however, it always can be improved
with additional effort in the future. Finally, I would like to thank again all the contributors who did a
great support on this research.
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Appendix A.
Formulas of Shewhart Control Charts.
Appendix A. Formulas of Shewhart Control Charts.
p-chart
The p-chart graphs the proportions of defective items from successive subgroups. The center line
and the control limits of the control chart fro fraction nonconforming are shown as follows.
p(1 − p)
n
UCL = p + (3)
Center line = p
p(1 − p)
n
LCL = p − (3)
m
p=
m
∑ D ∑ p$
i =1
mn
i
=
i =1
i
m
D
p$ i = i
n
i = 1,2,...m
p
Is the average of the sample fractions nonconforming.
p$ i
is the sample fraction nonconforming, which is the ratio of the number nonconforming units in
the sample D to the sample size n.
Di is the number of nonconforming units of a sample.
n is the sample size.
m is the number of samplings. As a general rule, m should be 20 or 25 (Montgomery, 1985).
np-chart
The np-chat is slightly different from p-chat. Instead of plotting the proportions of defective items,
the number of defectives np is plotted.
UCL = np + (3) np(1 − p)
Center line = np
LCL = np − (3) np(1 − p)
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 72
Appendix A.
Formulas of Shewhart Control Charts.
C-chart
The c-chart is applicable when the large products are inspected. The quality can be monitored by
sampling the products and count the number of defects on each product. Ci is the number of
defects on each product (sample size is 1). The center line and the control limits of the c-chart are
shown as follows.
UCL = c + 3 c
Center line = c
c=
(c1 + c 2 + ......c m )
m
for m inspections.
LCL = c − 3 c
U-chart
The u-chart is a modification of the c-chart. The number of nonconformities per unit (ui = ci / ni) is
plotted, so the sample size does not need to be one.
UCL = u + 3 u / ni
m
Center line = u
where u =
∑u
i
i
m
for m inspections.
LCL = u − 3 u / ni
X-bar chart and R-chart
The x-bar chart plots the sample averages of sample size n over time. A set of sample data were
taken from a period of stable process, it is usually recommended that number of subgroups k is at
least 20 to 25 and the sample size n would be 4, 5 or 6. (Montgomery, 1985). Suppose a quality
characteristic is normally distributed with mean µ and standard deviation σ. The average of a
particular sample with size n can be calculated as,
x=
x1 + x 2 + ... + xn
n
It is known that x is normally distrusted with mean µ and standard deviation σ x =
the center line and the control limits of x-bar chart can be expressed as,
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
σ
n
. Thus,
Page 73
Appendix A.
Formulas of Shewhart Control Charts.
UCL x = µ + 3σ x = µ + 3
σ
n
CL x = µ
LCL x = µ − 3σ x = µ − 3
σ
n
However, the mean µ and standard deviation σ are usually unknown. The estimate process
average µ can be derived from the grand average, and this will be used as center line of x-bar
chart.
CL = x =
(x1 + x 2 + .... + xm )
m
To construct the control limits, σ also needs to be estimated since it is unknown. Suppose x1,
x2, … , xn is a sample of size n, then the range of this sample is the difference between largest
and smallest observations.
R= xmax - xmin
Let R1, R2, …., Rm be the ranges of the m samples. So the averages range can be calculated as,
R=
(R1 + R2 + .... + Rm )
m
The relation between estimated σ and the average range is computed as,
$= R
σ
d2
Therefore under the circumstance that µ and σ are unknown, the center line and the control limits
of x-bar chart can be expressed as,
UCL x = x +
3
d2 n
R = x + A2R
CL x = x
LCL x = x −
3
d2 n
R = x − A2R
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
where A 2 =
3
d2 n
Page 74
Appendix A.
Formulas of Shewhart Control Charts.
The constant A2 is tabulated in Appendix B for different sample size n.
Since the standard deviation can be estimated by sample range, we can also construct R-chart to
monitor the variability of a process. The standard deviation of R is σR=d3σ. Since σ is unknown,
$ R = d R . Thus the formulas of center line and control limits of
so σR will be estimated by σ
3
d2
R-chart are summarized as,
$ R = R + 3d R = R D
UCLR = R + 3σ
3
4
d2
where D4 = 1 + 3
d3
d2
where D3 = 1 − 3
d3
d2
CLR = R
$ R = R − 3d R = R D
LCLR = R − 3σ
3
3
d2
The constant D3 and D4 are tabulated in Appendix B for different sample size n.
X-bar chart and S-chart
The mean and the variability of a process also can be monitored by constructing x-bar chart and
S-chart. Instead of using the range R, the process standard deviation is used directly to monitor
the variability. Generally, x-bar chart and S-chart are preferable when either (1). the sample size n
is
moderately large, say n>10 or 12, (2). the sample size n is variable (Montgomery, 1985). Again,
the standard deviation of the process σ needs to be estimated from historical data. The average
of the m standard deviations is
m
S=
∑S
i =1
i
m
Where m is the number of samplings, and Si is the standard deviation of the ith sample.
The formulas of center line and control limits of S-chart are summarized as,
UCL S = S + 3σs = S + 3σ 1 − c 4 2 = S + 3
S
1 − c 42
c4
CLS = S
LCL S = S − 3σs = S − 3σ 1 − c 4 2 = S − 3
S
1 − c 42
c4
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
$= S
where σ s = σ 1 − c 4 2 and σ
c4
Page 75
Appendix A.
Formulas of Shewhart Control Charts.
The constants B3 and B4 are defined as,
B4 = 1 −
3
1 − c 42
c4
B4 = 1 +
3
1 − c 42
c4
So the formulas of S-chart can be re-write as,
UCL S = B4 S
CLS = S
LCL S = B3 S
$= S
Since the standard deviation σ is estimated as σ
c4
chart can be expressed as,
3S
UCL x = x +
c4 n
, the formulas of corresponding x-bar
= x + A3 S
CL x = x
LCL x = x −
3S
c4 n
= x − A3 S
where A 3 =
3
c4 n
The constant B3, B4 and A3 are tabulated in Appendix B for different sample size n.
I-chart
In the case that the sample size n is equal to 1, which means the sample only has one individual
unit. Then the control chart for individual measurement I-chart should be used. The process
variability is estimated with the moving range MR=│Xi – Xi-1│. The formulas of center line and
control limits of I-chart are summarized as,
UCL = X + 3
MR
d2
CL = X
LCL = X − 3
MR
d2
If a moving range of n=2 observation is used, then d2 =1.128
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 76
Appendix B.
Constants for Selected Control Charts
Appendix B. Constants for Selected Control Charts
(source: Ledolter, J. & Burrill, C.W. (1999), Statistical quality control –
Strategies and tools for continual improvement)
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 77
Appendix C.
Hawkins’ Data Set and T2 Statistics of Measurements.
Appendix C. Hawkins’ Data Set and T2 Statistics of Measurements.
No.
X 1
X 2
X 3
X 4
X 5
1
17.265
11.788
15.101
13.903
10.465
2
17.384
6.996
11.552
7.253
6.641
3
16.517
10.277
11.724
13.013
9.111
4
14.997
10.682
12.087
11.457
6.320
5
17.633
9.348
12.672
10.475
5.481
6
16.041
11.320
13.957
11.474
8.176
7
15.339
10.384
12.313
9.401
7.252
8
17.144
12.254
14.931
13.715
11.135
9
20.351
10.028
14.271
11.124
8.994
10
19.586
11.083
15.019
12.126
9.441
11
20.153
13.100
16.231
13.628
8.780
12
18.044
9.699
11.807
11.655
7.513
13
17.041
9.748
13.576
9.333
7.316
14
17.671
13.223
15.937
15.119
12.129
15
16.306
9.140
13.239
10.982
8.900
16
15.977
9.904
12.822
9.910
7.190
17
18.517
11.401
16.883
13.162
12.861
18
16.591
12.875
14.542
13.787
7.931
19
17.576
10.686
13.072
11.257
5.933
20
17.225
8.943
13.033
9.088
6.176
21
19.234
11.575
15.192
11.809
11.418
22
19.379
10.421
13.095
11.898
7.881
23
16.009
7.478
10.291
7.207
3.160
24
15.944
10.086
14.438
10.652
6.916
25
16.541
8.197
12.520
9.586
9.304
26
18.325
7.004
12.773
8.136
5.326
27
17.652
9.930
13.904
9.747
6.105
28
16.615
11.221
14.151
12.629
10.601
29
14.606
8.542
11.834
9.587
5.790
30
19.074
9.550
13.044
11.688
9.450
31
22.449
10.093
16.306
11.806
11.239
32
18.401
11.856
13.608
10.832
7.709
33
18.556
12.174
14.111
11.965
9.074
34
17.727
10.740
13.100
11.012
8.429
35
19.141
10.033
13.524
10.800
8.383
36
17.554
9.132
11.563
10.554
6.593
37
19.564
10.784
14.200
11.909
10.681
38
20.985
10.191
15.129
11.301
9.087
39
20.745
10.781
14.403
9.469
7.451
40
15.395
7.794
10.602
8.826
5.933
41
16.014
7.928
11.872
7.197
6.649
42
15.220
12.697
15.201
13.891
10.849
43
19.978
11.101
13.687
11.554
8.284
44
20.886
9.578
13.724
10.914
11.075
45
16.323
8.711
12.084
8.534
5.715
46
16.120
10.997
14.497
12.918
11.921
47
17.037
8.362
13.290
10.097
9.484
48
13.065
11.625
14.923
12.589
12.446
49
16.188
9.140
13.284
10.991
9.126
50
22.047
10.824
14.796
10.872
9.264
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 78
Appendix C.
Hawkins’ Data Set and T2 Statistics of Measurements.
T2 Statistics of measurements 36 to 50 and the T2 plot.
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 79
Appendix D.
Questionnaires for Interviews.
Appendix D. Questionnaires for Interviews.
(1). Questionnaires for Industrial Statistician
Part 1: Overview of existing production
Q1. How many grades/products are your producing?
Q2. What is the final product of the production mentioned in Q1?
Q3. How does this process operate? (Continuous, Discrete, Batch or combination.)
Q4. What are the important process parameters? How are they selected? Are they in the
same step or different steps of the production?
Q5. How often are these process produced (if batch-wise)?
Part 2: Case study
Q6. Can you please mention some examples of process monitoring in the plant?
Q7. Please describe the quality management system of the production line mentioned in Q6.
(For example what process monitoring technique is applied (if any), what is the frequency of
sampling, what is the sample size, what are the further actions after the sample analysis?)
Q8. What is the existing quality related performance of the process? (For example what is the
probability of abnormal situation? How often/How “abnormal” they are? What is the probability
of process conformity?)
Q9. What are the control limits of the monitoring system? (For example, how many sigma?)
Q10. If SPC (Statistical Process Control) is applicable, do you think these SPC techniques can
contribute to the improvement of operational performance of your plant?
Part 3: Correlation of monitored quality parameters
Q11. Do you check the correlation among the process parameters being monitored? Why do
you think they are correlated? Are these process parameters present in one step or in various
steps?
Q12. How significant is the correlation among the quality parameters being monitored?
(correlation >0.3 or 0.6 for instance.)
Q13. How do you cope with the correlation of the process parameters during the monitoring?
For example, what kind of technique is applied, please describe.
Q14. Do you know the wrong ratio of the correlated process parameters can be an indication
of process out-of-control?
Q15. Do you monitor the systematic pattern in the process behavior? How do further cope
with it?
Part 4: Out-of-Control-Action-Plan (OCAP)
Q16. How do you react when the abnormal situation occurs? How do you identify which
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 80
Appendix D.
Questionnaires for Interviews.
variable or set of variables or other reason (for example anti-correlation) to be responsible for
the abnormal measurement?
Q17. How is the response of the process after taking action on the abnormal situation?
Part 5: MSPC application
Q18. In general, what should or can be improved in you process monitoring system? Or what
are the requirements of the process monitoring techniques/tools that you would prefer? (e.g.
simple, easy, effective, accurate….)
Q19. If MSPC can be proven more effective on detection of process abnormal situation for
your plant, would you like to adopt it? If not, what are the barriers and difficulties (e.g. too
complex, difficult to interpret the result or else)?
Q20. We are conducting a research on the MSPC implementation for the industrial plants.
Would you like to cooperate with our research work? We may need your comments to verify
the research output and make it more applicable for the industrial plants.
(2). Questionnaires for SPC Consultants
Part 1: General information
Q1. How many studies in the area of SPC have been performed by your company?
Q2. Do you have a standard approach to perform such studies?
Q3. How long are average projects performed by you with SPC? (For example, 1 month, 6
months, or else)
Q4. What are the most difficult steps during performing these projects?
Q5. Why the industrial plant requested you to study or to support them on the SPC? (For
example, lacking of knowledge, techniques, or else.)
Part 2: Production monitoring system of a selected case
Q6. Can you please mention some examples of process monitoring in the plant that your
company studied before? (Applying multivariate statistical process control would be
preferable)
Q7. How does this process operate? (Continuous, Discrete, Batch or combination.)
Q8. What is the final product of the production mentioned in Q6?
Q9. Please describe the overview of the process monitoring system of the case mentioned in
Q6. (For example what process monitoring technique is applied (if any), what is the frequency
of sampling, what is the sample size, what are the further actions after the sample analysis?)
Q10. What was the old quality related performance of the process? (For example what was
the probability of abnormal situation? How often/How “abnormal” they were? What was the
probability of process conformity?)
Q11. What were the control limits of the monitoring system? (For example, how many sigma?)
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 81
Appendix D.
Questionnaires for Interviews.
Part 3: Correlation of monitored process quality parameters
Q12. How many process parameters were monitored?
Q13. How did you consider the correlation of these variables during the monitoring? For
example what kind of technique was applied, please describe.
Q14. How did you cope with the pattern behavior in the process parameters?
Q15. In practice, the relation between the process parameters might be non-linear. Based on
Hotelling’s T2 methods which only taking linear relation (positive/negative correlation) into
account, can you please comment on it?
Part 4: Out-of-Control-Action-Plan (OCAP)
Q16. How did you react when the abnormal situation occurred? How did you identify which
variable or set of variables or other reason (for example anti-correlation) to be responsible for
the abnormal situation?
Q17. How was the response of the process after taking action on the abnormal situation?
Part 5: MSPC application
Q18. Do you know these techniques? What is your impression about the applicability in the
industrial situation? What are the constraints/conditions of the applicability for these methods?
Q19. Based on your information, what is the level of applicability of MSPC in industrial plant
and why? (For example, it is commonly used, rarely, depends on the types of production or
there are certain conditions to be met.)
Q20. Being familiar with the characteristics of MSPC (especially Hotelling’s T2 method), what
is your opinion of the applicability in the industrial situation? Do you see any particular types of
manufacturing industry are suitable (or not suitable) for this technique?
Q21. We are conducting a research on the MSPC implementation for the industrial plants.
Would you like to cooperate with our research work? We may need your comments to verify
the research output and make it more applicable for the industrial plants.
(3). Questionnaires for SPC Statisticians
Part 1: MSPC from a scientific point of view
Q1. Please comment on MSPC (Multivariate statistical process control), especially on the
Hotelling’s T2 method? (For example, characteristics, advantages, shortcomings…)
Q2. Hotelling’s T2 method has not become widely popular in the industrial plants, since it was
developed. Can you please comment on it? What are possible reasons for this situation?
Q3. Hotelling’s T2 method started to be addressed since 1930s, but it has been criticized for its
complexity, clumsy for real application. Can you please introduce some latest development or
achievement of on this concern?
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 82
Appendix D.
Questionnaires for Interviews.
Q4. Continued with Q3, identifying the responsible variable or set of variables for the
abnormal situation can be one of the major difficulties of Hotelling’s T2 method. Can you
please introduce some latest development or achievement of on this concern?
Q5. Continued with Q3, the information of process pattern may not be available anymore
while using Hotelling’s T2 method. Can you please introduce some latest development or
achievement of on this concern?
Q6. Detecting process out-of-control is one thing. Process cannot be improved if no action is
taken after abnormal situation detected. Searching for the root causes of the abnormal
situation becomes crucial. What recommendations can you provide on this concern? (For
example, “Quality 7 tools”, or other possible techniques.)
Q7. Being familiar with the characteristics of Hotelling’s T2 method), what is your opinion on
the applicability in industrial plants? Do you see particular types of manufacturing industry are
suitable (or not suitable) for this technique?
It would be highly appreciated if you can mention some examples that you have
studied. Please continue part 2 to part 5!!
Part 2: Existing production monitoring system
Q1. Can you please mention some examples of process monitoring in the plant that you
studied before? (Applying multivariate statistical process control would be preferable)
Q2. How does the product operate? (Continuous, Discrete, Batch or combination.)
Q3. What is the final product of the production mentioned in Q1?
Q4. Please describe the overview of the process monitoring system in the case mentioned in
Q1. (For example what process monitoring technique is applied (if any), what is the frequency
of sampling, what is the sample size, what are the further actions after the sample analysis?)
Q5. What are the control limits of the monitoring system? (For example, how many sigma?)
Q6. Why the company requested you to study or to support them on the SPC? (For example,
lacking of knowledge, techniques, or else.)
Part 3: Correlation of monitored quality parameters
Q7. How many process parameters were selected?
Q8. How did you consider the correlation during the monitoring? For example what kind of
technique was applied, please describe.
Q9. How did you cope with the pattern behavior in the process parameters?
Part 4: Part 3: Out-of-Control-Action-Plan (OCAP)
Q10. How did you react when the abnormal situation occurs? How did you identify which
variable or set of variables or other reason (for example anti-correlation) to be responsible for
the abnormal situation?
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 83
Appendix D.
Questionnaires for Interviews.
Q11. How was the response of the process after taking action on the abnormal situation?
Part 5: MSPC application
Q12. Do you know these techniques? What is your impression about the applicability in the
industrial situation? What are the constraints/conditions of the applicability for these methods?
Q13. Based on your information, what is the level of applicability of MSPC in industrial plant
and why? (For example, it is commonly used, rarely, depends on the types of production or
there are certain conditions to be met.)
Q14. Being familiar with the characteristics of MSPC (especially Hotelling’s T2 method), what
is your opinion of the applicability in the industrial situation? Do you see particular types of
manufacturing industry are suitable (or not suitable) for this technique?
Q15. We are conducting a research on the MSPC implementation for the industrial plants.
Would you like to cooperate with our research work? We may need your comments to verify
the research output and make it more applicable for the industrial plants.
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 84
Appendix E.
Multiple Regression Analysis of Sub-process.
Appendix E. Multiple Regression Analysis of Sub-process.
Data set for sub-process analysis
ORIGINAL DATA
Hbtotal
STANDARDIZED DATA
Hbash
Ret
BW
Hbtotal
Hbash
Ret
BW
1
14.310
3.157
43.600
155.087
1
-1.22
-0.77
0.07
-0.47
2
14.244
3.142
43.551
154.835
2
-1.69
-1.15
0.01
-0.81
3
14.385
3.150
42.503
155.033
3
-0.70
-0.95
-1.21
-0.54
4
14.497
3.184
43.003
154.883
4
0.08
-0.08
-0.63
-0.75
5
14.297
3.157
42.978
154.037
5
-1.31
-0.77
-0.66
-1.91
6
14.517
3.172
42.914
154.748
6
0.22
-0.38
-0.73
-0.93
7
14.665
3.186
43.012
156.707
7
1.26
-0.03
-0.62
1.76
8
14.696
3.199
43.012
155.692
8
1.48
0.31
-0.62
0.36
9
14.581
3.207
43.005
154.901
9
0.67
0.51
-0.63
-0.72
10
14.548
3.205
43.016
155.217
10
0.44
0.46
-0.61
-0.29
11
14.596
3.204
42.934
154.853
11
0.78
0.44
-0.71
-0.79
12
14.490
3.197
43.018
155.422
12
0.03
0.26
-0.61
-0.01
13
14.594
3.182
43.012
154.882
13
0.76
-0.13
-0.62
-0.75
14
14.528
3.167
42.971
154.799
14
0.30
-0.51
-0.67
-0.86
15
14.516
3.165
42.985
154.938
15
0.22
-0.56
-0.65
-0.67
16
14.552
3.162
42.898
155.052
16
0.47
-0.64
-0.75
-0.51
17
14.401
3.160
42.961
154.800
17
-0.59
-0.69
-0.68
-0.86
18
14.501
3.150
43.052
154.793
18
0.11
-0.95
-0.57
-0.87
19
14.570
3.140
43.022
155.025
19
0.59
-1.21
-0.61
-0.55
20
14.452
3.132
43.037
154.800
20
-0.23
-1.41
-0.59
-0.86
21
14.483
3.165
42.939
155.315
21
-0.01
-0.56
-0.70
-0.15
22
14.513
3.180
43.076
155.030
22
0.20
-0.18
-0.54
-0.54
23
14.299
3.188
43.031
154.950
23
-1.30
0.03
-0.60
-0.65
24
14.399
3.197
43.013
154.618
24
-0.60
0.26
-0.62
-1.11
25
14.508
3.181
42.946
155.017
25
0.16
-0.15
-0.70
-0.56
26
14.420
3.157
43.012
154.800
26
-0.45
-0.77
-0.62
-0.86
27
14.552
3.140
43.071
155.321
27
0.47
-1.21
-0.55
-0.15
28
14.480
3.124
43.012
154.737
28
-0.03
-1.62
-0.62
-0.95
29
14.087
3.113
43.002
155.080
29
-2.78
-1.90
-0.63
-0.48
30
14.150
3.132
42.984
154.804
30
-2.34
-1.41
-0.65
-0.85
31
14.161
3.151
42.936
155.127
31
-2.27
-0.92
-0.71
-0.41
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 85
Appendix E.
Multiple Regression Analysis of Sub-process.
32
14.528
3.196
42.899
155.380
32
0.30
0.23
-0.75
-0.06
33
14.580
3.190
42.956
155.056
33
0.66
0.08
-0.68
-0.51
34
14.560
3.182
42.926
155.453
34
0.52
-0.13
-0.72
0.04
35
14.413
3.174
43.000
154.825
35
-0.50
-0.33
-0.63
-0.83
36
14.568
3.212
43.615
157.083
36
0.58
0.64
0.09
2.27
37
14.601
3.227
44.954
157.125
37
0.81
1.03
1.65
2.33
38
14.668
3.209
45.497
156.933
38
1.28
0.56
2.28
2.07
39
14.366
3.190
45.570
156.863
39
-0.83
0.08
2.37
1.97
40
14.378
3.170
45.570
156.527
40
-0.75
-0.44
2.37
1.51
41
14.314
3.191
45.041
156.387
41
-1.20
0.10
1.75
1.32
42
14.478
3.226
44.488
156.087
42
-0.05
1.00
1.11
0.91
43
14.367
3.227
44.483
156.091
43
-0.83
1.03
1.10
0.91
44
14.503
3.266
44.520
155.901
44
0.13
2.03
1.14
0.65
45
14.690
3.253
44.472
156.177
45
1.43
1.69
1.09
1.03
46
14.752
3.275
44.569
155.923
46
1.87
2.26
1.20
0.68
47
14.583
3.272
44.511
155.804
47
0.69
2.18
1.13
0.52
48
14.585
3.255
44.574
155.934
48
0.70
1.74
1.21
0.70
49
14.646
3.239
44.444
155.949
49
1.13
1.33
1.05
0.72
50
14.580
3.227
44.565
155.964
50
0.66
1.03
1.20
0.74
51
14.573
3.233
44.485
155.997
51
0.62
1.18
1.10
0.78
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 86
Appendix E.
Multiple Regression Analysis of Sub-process.
Quadratic regression model (scaled data & original data)
Linear regression model with interaction term (scaled data & original data)
Master Thesis: Multivariate Statistical Process Control in Industrial Plants.
Page 87