Multivariate Process Control Master Thesis W Chen

Department of Engineering and Policy Analysis

Faculty of Technology, Policy and Management

Delft University of Technology, August 2005

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)

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. T2

A & 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

Master Thesis: Multivariate Statistical Process Control in Industrial Plants. Page 1

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

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 TA

2 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 Ti

2 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



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



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?



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.



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.

Research Background Research Objective Research Questions

SPC Shewhart Control Chart MSPC Control Chart

Interviews Gap between Academic and Practical Fields Industrial Expectations

MSPC Plan MSPC Training Team Approach Management Involvement

Case Briefing MSPC Implementation Result Process Performance Improvement

Conclusions Recommendations Future Research

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.



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



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,

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

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.

1ARLp

=

1 1ARL 370p 0.0027

= = =



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.



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



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 (54)

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



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



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

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:

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 X2

2,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 > X2

2,α.

Figure 2-5 A generic bivariate Hotelling’s T2 control region.

1 2X and X

2 2 21 ⎡ ⎤= ( − 2( )( ) + ( )⎣ ⎦ − 2 2 1 10 11 2 2 1 22 12

11 22 12

X s x - x ) x - x x - x s x - xs s s



With the same concept of the generalized distance, it can be extended from bivariate to a

multiple p variables. Let 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 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.)

Also other research shows that the control limit follows Beta distribution (Mason, Young & Tracy,

1992).

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 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.

= 'i i1 i2 ipX (X , X ,...., X )

X and S

2 ' 1i i (p,n p, )

p(n+1)(n-1)T (X X) S (X X) ~ F (eq.2 3)n(n p)

n: number of preliminary observations

− − α= − − −

−

2 ' -1 2i i ,p T (X - X) S (X - X) ~ X (Chi - square distribution) (eq.2 1)α= −

22 ' -1

i i p n p 1( , , )2 2

(n-1)T (X - X) S (X - X) ~ B (eq.2 2)n

n: number of preliminary observations

− −α

= −

X and S



Where sample mean is and the covariance of sample

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.

are estimated from the measurement 1 to measurement 35. As we know

= '1 2 pX (X , X ,...., X )

.... ⎡ ⎤⎢ ⎥ .... ⎢ ⎥= ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦

O

11 12 13 1p

22 23 2p

pp

s s s s

s s sS

s

X and S



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,

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

− 0.05=−

=−

=

(p,n p, )p(n+1)(n-1)UCL F

n(n p)5(35+1)(35-1) * 2.53

35(35 5)14.75



can be expressed (Mardia, Kent and Bibby, 1989; Kourti and MacGregor,

1994) as,

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,

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.

2 2 2 2q q qA2 i i i i

2 2 2i 1 i 1 i 1 i A 1i i i i

i i

t t t tT

s s s

t P (X X)

= = = = +

= = = +λ

= −

∑ ∑ ∑ ∑

2 ' 1i iT (X X) S (X X)−= − −

2A2 i

A 2i 1 i

tT

s=

= ∑

22 'A Anew,a2 a new

A 2a 1 a 1 aa

(p,n p, ) (A,m , )

x

t P (X X)T (eq.2 4)

ss

p(n+1)(n-1) A(m+1)(m-1)UCL F Fn(n p) m(m A)

SPE

= =

− α −Α α

⎡ ⎤−= = − ⎢ ⎥

⎢ ⎥⎣ ⎦

= =− −

∑ ∑

2n

new,j,new new,jj 1

a A

new new,j aa 1

(X X ) (eq.2 5)

X (t P )

=

=

=

= − −

=

∑

∑



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.



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,

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

2 2 21 p 1,2,....,p 1

22j 1,2,....,j 11 j1

2 21 j 1,2,....,j 1

T T T )

(x x )(x x )j 1,2, p

s s

22•1 • −

• −

• −

= + (Τ + .... +

−− = + = ... ,



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,

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.

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

2 2 2 21 2 1 3 1,2

2 2 21 3 1 2 1,3

2 2 22 3 2 1 2,3

2 2 22 1 2 3 1,2

2 2 23 1 3 2 1,3

2 2 23 2 3 1 2,3

T =T + Τ + Τ

=T + Τ + Τ

=T + Τ + Τ

=T + Τ + Τ

=T + Τ + Τ

=T + Τ + Τ

• •

• •

• •

• •

• •

• •

2jj2

j (1,n-1, )2j

(x x ) nT F (eq.2 6)ns α

− +1 = ∼ ( ) −



data. The conditional term T2j˙1,2,…j-1 will follow an F distribution which can be used as upper

control limit.

T2j˙1,2,…j-1 can be re-expressed as (Mason, Young and Tracy, 1997),

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.

2j 1,2,...., j 1j2

j 1,2,...., j 1 (1,n-k-1,α)2j 1,2,....,j 1

(x x ) (n+1)(n-1)T F κ=(j-1) (eq. 2-7)n(n-k-1)s

• −

• −• −

− ⎛ ⎞= ∼ ⎜ ⎟

⎝ ⎠

2j 1,2,...., j 1j2

j 1,2,...., j 1 2 2j j 1,2,...., j 1

(x x )T (eq. 2-8)

s (1 R )• −

• −• −

−=

−



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 T2

i,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.

.

(1,n-1,0.05)

(1,35-1,0.05)

nUCL Fn

35 F35

4.248

+1= ( )

+1 = ( )

=



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.

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.

Table 2-3 Bivariate conditional term and its status.

2 21T - T 22.92 - 7.61 15.31 14.75 = = >

(1,n-k-1, )

(1,33,0.05)

(n 1)(n -1)UCL F n(n - k -1)

(35 1)(35 -1) F 35(35 -1-1)

4.39

α

⎛ ⎞+= ⎜ ⎟

⎝ ⎠⎛ ⎞+

= ⎜ ⎟⎝ ⎠

=



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.

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)

2 2n n2 a a

2a 1 a 1a a

n'

ja a a,j jj 1

t tT

s

t P (x - x) p (x - x ) (eq.2 9)

= =

=

= =λ

= = −

∑ ∑

∑

2 2 2 21 4 5T - T - T - T 22.92 - 7.61 - 0.58 - 3.87 10.86 14.75 = = <



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,

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.

ja,j jp (x x ) (eq.2 10)− −



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.



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.

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.

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.

aja,j a,j j

a

tCont p (X X )= −

λ

k

j a,jj 1

CONT (cont )=

= ∑



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

Inversion of covariance matrix

available?

MSPC approaches

T2 control chart

Yes No

Legend :

Decision point

Decision option

Action

TA2 and SPE

control chart

MYT T2 decomposition

PCA diagnosis and overall contribution plot.

MYT T2 decomposition

PCA diagnosis and overall contribution plot.

Stage I

Stage II



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.

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



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).



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



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). 12. Organization and economic aspects need to be considered. From the view of a

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 Potential remedies



Scientific oriented. The theory

involves too much

mathematics and statistics.

Difficult to understand what

MSPC is. Therefore, people

hesitate to use the tool that

they are not familiar with.

Continuous education and

present the MSPC theory in a

less scientific format can lower

the reluctance of practitioners.

Academic development is still

ongoing. Various approaches

are still being explored and

experimented.

Several different theoretical

approaches are available.

Practitioners may have

difficulty to understand them;

and they do not know exactly

which approach to follow.

Complex and ill-defined

knowledge may easily

overwhelm the practitioners’

motivation. Screening the

approaches and presenting a

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

concentrates on theoretical

exploration. Although the

effectiveness of MSPC is

validated theoretically, the

feasibility of practical

implementation seems not

much emphasized.

What practical field needs is

an effective, robust tool. It

should be easy to understand,

easy to operate and provide

prompt information.

Traditional control chart

(Shewhart control chart) may

be still sufficient. The effort

and cost to implement a

complex technique is too high.

The theoretical part of MSPC

is indeed more complex than

the traditional control chart.

However, it can be overcome

by additional support, for

instance, software

development, automatic

monitoring/alarm system, etc.

The implementation of MSPC

can be boosted more easily if

the complex statistics is

supported with computer

software.

The investment of

Implementing MSPC can be

formidable, in terms of finance,

human resource, hardware,

etc. To maximize the utility of

this MSPC, the nature of

production should be



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,

the framework of MSPC

research is often simplified.

Statistical process control

(SPC) is just part of the entire

quality management work.

The contribution of SPC may

be limited if other parts of the

process are not improved in

parallel.

The value of SPC or even

MSPC can reveal only when

its contribution can reflect on

the overall benefit of the

company.

A production plant is absolutely

not just forwarding input into

process and output being

produced. In addition to SPC,

many other techniques need to

be involved. For example,

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.

How to implement the

advanced technique in a

proper timing is an important

concern. For example, a plant

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.

The implementation of a new

technique, such as MSPC,



performance. To implement a

technique is different oriented.

There are many aspects need

to be taken into account. For

instance, how to set up the

plan, how to train the

employees, estimate the

investment and etc.

Lacking of statistical process

control knowledge is a barrier

to implement MSPC.

requires tremendous

investment and very often

encounters foreseeable

resistance. 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.

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.



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.



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.

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.

1). Method Model for (M)SPC

MSPC Plan 2). MSPC Diagnosis

3). Process Control

MSPC Training

Team Approach

MS

PC

Implem

entation Guideline 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.



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



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.



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.



Figure 4-2 Method model for (M)SPC

Shewhart control chart

T2 control chart

N: number of variables

N=1 N>=2

Legend :

Decision point

Decision option

Action

Variables correlated?

(r > 0.3)

No Yes

Monitoring future observation.

Out-of-control occurs?

Continue process.

Responsible variable(s) are

found. Investigate the root causes.

No Yes

Apply MSPC diagnosis.

No Yes

N is too many to monitor

separately.

Yes No

Process

investigation

Process breakdown (optional)


available?

Yes No

TA2 and SPE

control chart

Step 3

Continue process.

In-control data and control limits

construction


construction


construction



Step 1

Step 2

Step 4

Step 5

Step 6



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.

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.

Conta, j is set to zero if it is negative.

3). Sum the total contribution of variable xj.

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

n'

ja a a,j jj 1

a a

t P (x - x) p (x - x )

s=

= =

= λ

∑

aja,j a,j j

a

tCont p (X X )= −

λ

k

j a,jj 1

CONT (cont )=

= ∑



PCA diagnosis approach

MSPC application

Monitoring future

observation.


No Yes


identified. Investigate the root causes.

Legend :

Decision point

Decision option

Action

Continue

process.

Compute Normalized PCA Scores (ta/sa).

Construct overall contribution plot of the

out-of-control observation.

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.

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,



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.

Figure 4-4 An example of Causal-and-effect diagram.

Wrong tool

Insufficient warm-up

Wrong specification

Faulty gauge Poor attitude

Methods Materials Machines

Environment Manpower Measurement

Problem or undesired effect

Wrong procedure Defect supplier

Damaged handling

Wrong planning

Inexperience High humidity

High temperature



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.



4.2. 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.



Plant people:

Implement the MSPC Plan that defined by quality experts.

Resolve out-of-control situation according to OCAP.

Support quality experts on maintaining OCAP.

4.4. 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.

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.



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)



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.

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

1 27 53 79 105 131 157 183 209 235 261 287 313 339 365 391 417 443 469 495 521 547 573 599 625 651 677 703 729 755 781 807 833 859 885 911 937 963 989

14.0

14.2

14.4

14.6

14.8HBtotalUCL = 14.5229Average = 14.4954LCL = 14.4679

Control Chart: HBtotal



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).

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.

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.

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 14.8HBtotal

0 20 40 60 80

100 120 140 160 180 200 220 240

Count LEGEND

Normal (14.4954,0.132827)

DatasetHBtotal Count

Histogram from Dataset

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 5114.0

14.2

14.4

14.6

14.8HBtotalUCL = 14.7668Average = 14.4848LCL = 14.2028

Control Chart: HBtotal



Figure 5-5 MSPC decision path of case study.

Shewhart control chart

T2 control chart

N: number of variables

N=1 N>=2

Legend :

Decision point

Decision option

Action

Variables correlated?

(r > 0.3)

No Yes



Continue process.


found. Investigate the root causes.

No Yes

Apply MSPC diagnosis.

No Yes

N is too many to monitor

separately.

Yes No

Process

investigation

Process breakdown (optional)


available?

Yes No

TA2 and SPE

control chart

Step 3

Continue process.


construction


construction


construction



Step 1

Step 2

Step 4

Step 5

Step 6



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



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.



Table 5-2 In-control process data

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.

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



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).

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.

22 23 24 25 26 27 28

Variable contribution



Table 5-3 The result of MSPC diagnosis.

Observation Signaled

by MSPC

Potential

problematic

variable(s)

Status Signaled

by USPC

BW is

Out-of-control?

22 Yes HBash Exceeding control limit. Yes No

23 Yes HBash

Fillers

Exceeding control limit.

Within control limit

Yes

No No

24 Yes

HBtotal

HBash



Yes

Yes 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.



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



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.

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.

HBtotal

HB

ash

Ellipse control chart

28

14.3 14.4 14.5 14.6 14.7 14.8

3.28

3.25

3.22

3.19

3.16

3.13

3.10



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



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

(Scaled data)

BW = 0.03 + 0.34HBtotal + 0.04HBash + 0.79Ret + 0.016HBtotal2 – 0.19 HBash2

(Interaction term is not significant, so it is not included)

Quadratic model

(Original data)

BW = 378.54 – 165.06HBtotal + 583.36HBash + 0.67Ret + 5.77HBtotal2 – 91.41 HBash2


Adjusted R2 0.70

Regression model BW = f (HBtotal, HBash, Ret)

Linear with interaction term

model (Scaled data)

BW = 0.22HBtotal + 0.76Ret


Linear with interaction term

model (Original data)

BW = 110.97 – 1.13HBtotal + 0.64Ret

BW = 0.22HBtotal + 0.76Ret


Adjusted R2 0.66

1 55

1 56

1 57

B W155.427

+/-0.171292

1 4. 2 14.4 14.6H B total

43 44 45R et

14.4848 43 .5421Predicted BW Graph

1 5 4

1 5 5

1 5 6

1 5 7

B W1 5 5 . 4 4 8

+ / - 0 . 2 8 5 4 9 7

3 . 1 5 3 . 2 3 . 2 5H B a s h

1 4 . 2 1 4 . 4 1 4 . 6H B t o ta l

4 3 4 4 4 5R e t

3 .1 8 7 4 5 1 4 . 4 8 4 8 4 3 . 5 4 2 1P r e d i c t e d B W G r a p h



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.

Figure 5-12 Contour plot of HBtotal and Ret with respect to BW.

14.1 14.2 14. 3 14.4 14.5 14.6 14.7HBtotal

43

44

45

Ret

154.6 154.8

154.8

155

155

155

155.2

155.2

155.2

55.4

155.4

155.4

155.6

155.6

155.6

55.8

155.8

155.8

156

156

156

156

156.2

156.2

156.2

156.4

156.4

156.6 156.8

LEGENDPredicted BW

Predicted BW Contour Plot



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.

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



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



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



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.



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.

Reference


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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.

Is the average of the sample fractions nonconforming.

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.

−= +

=

−= −

p(1 p)UCL p (3)n

Center line p

p(1 p)LCL p (3)n

$

$=1 =1= = = =∑ ∑m m

iii i i

i

D pD

p p i 1,2,...mmn m n

p

$ip

= + −

=

= − ( −

UCL np (3) np(1 p)

Center line np

LCL np 3) np(1 p)



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.

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.

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,

It is known that is normally distrusted with mean µ and standard deviation . Thus,

the center line and the control limits of x-bar chart can be expressed as,

= +

= =

= −

1 2 m

UCL c 3 c

(c + c + ......c )Center line c c for m inspections.

m

LCL c 3 c

= +

= =

= −

∑

i

m

ii

i

UCL u 3 u /n

uCenter line u where u for m inspections.

m

LCL u 3 u/n

+ + + = 1 2 nx x ... x

xn

xσ

σ =x 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.

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,

The relation between estimated σ and the average range is computed as,

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,

+ + += = 1 2 m(x x .... x )CL x

m

σ= µ + σ = µ +

= µ

σ= µ − σ = µ −

x x

x

x x

UCL 3 3n

CL

LCL 3 3n

+ + + = 1 2 m(R R .... R )

Rm

$σ =2

Rd

= + = +

=

= − = − =

2x2

x

2 2x2 2

3UCL x R x A Rd n

CL x

3 3LCL x R x A R where Ad n d n



$σ =R 32

Rdd

==∑m

ii 1

SS

m

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,

so σR will be estimated by . Thus the formulas of center line and control limits of

R-chart are summarized as,

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

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,

$

$

= + σ = + = = +

=

= − σ = − = = −

3RR 3 4 4

2 2

R

3RR 3 3 3

2 2

dRUCL R 3 R 3d RD where D 1 3d d

CL R

dRLCL R 3 R 3d RD where D 1 3d d

$

= + σ = + σ − = + −

=

= − σ = − σ − = − − σ = σ − σ =

2 2S s 4 4

4

S

2 2 2S s 4 4 s 4

4 4

SUCL S 3 S 3 1 c S 3 1 cc

CL S

S SLCL S 3 S 3 1 c S 3 1 c where 1 c and c c



$σ =4

Sc

The constants B3 and B4 are defined as,

So the formulas of S-chart can be re-write as,

Since the standard deviation σ is estimated as , the formulas of corresponding x-bar

chart can be expressed as,

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,

= − −

= + −

24 4

4

24 4

4

3B 1 1 cc

3B 1 1 cc

=

=

=

S 4

S

S 3

UCL B S

CL S

LCL B S

= + = +

=

= − = − =

3x4

x

3 3x4 4

3SUCL x x A Sc n

CL x

3S 3LCL x x A S where Ac n c n

2

22

MRUCL X 3d

CL X

MRLCL X 3 If a moving range of n=2 observation is used, then d =1.128d

= +

=

= −

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)

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

Appendix C. Hawkins’ Data Set and T2 Statistics of Measurements.


T2 Statistics of measurements 36 to 50 and the T2 plot.

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



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.)


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?)



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


the abnormal situation?

Q17. How was the response of the process after taking action on the abnormal situation?


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?




(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?



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.)


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


the abnormal situation?



Q11. How was the response of the process after taking action on the abnormal situation?


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?




Appendix E. Multiple Regression Analysis of Sub-process.



Data set for sub-process analysis

ORIGINAL DATA STANDARDIZED DATA

Hbtotal 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



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



Quadratic regression model (scaled data & original data)

Linear regression model with interaction term (scaled data & original data)

Multivariate Process Control Master Thesis W Chen

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