Fuzzy control charts construction and applications

Therefore, this thesis aims at providing detailed procedures to extendsome traditional control charts, including x-chart, s-chart, p-chart, np-chart, u-chart, and MaxGWMA chart to become

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NATIONAL KAOHSIUNG UNIVERSITY OF

APPLIED SCIENCES

GRADUATE INSTITUTE OF MECHANICAL &

PRECISION ENGINEERING DEPT INDUSTRIAL ENGINEERING & MANAGEMENT

DOCTORAL THESIS

Fuzzy Control Charts – Construction and Applications

Ph.D Candidate: Nguyen Thanh Lam

Supervisors: Prof Ying-Fang Huang, Ph.D

Taiwan, May 2014

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A thesis submitted in fulfilment of the requirements

for the degree of Doctor of Philosophy

in the Industrial Engineering and Management Graduate Institute of Mechanical and Precision Engineering

Taiwan - May 2014

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I, Nguyen Thanh Lam, declare that this thesis titled, “Fuzzy Control Construction and Applications”, and the work presented in it are my own I con-firm that:

Charts- This work was done wholly or mainly while in candidature for a researchdegree at this University

Where any part of this thesis has previously been submitted for a degree orany other qualification at this University or any other institution, this hasbeen clearly stated

Where I have consulted the published work of others, this is always clearlyattributed

Where I have quoted from the work of others, the source is always given.With the exception of such quotations, this thesis is entirely my own work

I have acknowledged all main sources of help

Where the thesis is based on work done by myself jointly with others, I havemade clear exactly what was done by others and what I have contributedmyself

Signature: Nguyen Thanh Lam

Date: May 2, 2014

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Kahlil Gibran

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AbstractIndustrial Engineering and ManagementGraduate Institute of Mechanical and Precision Engineering

Doctor of PhilosophyFuzzy Control Charts- Construction and Applications

by Nguyen Thanh Lam

Due to the mass production of industrial products, building and implementing aneffective quality control program has become one of the most important issues forany industrial manufacturer The key quality characteristics of physical productsshould be fully monitored and controlled to reduce their variability In practice,using control charts to monitor process variability, to achieve process stability aswell as to improve production capability, is preferred However, the traditionalcontrol charts need precise data while several practical applications cannot berecorded or collected precisely as well as the human subjectivity, such as mood,optimism, intelligence and perception, plays an important role in decision-makingprocess Therefore, this thesis aims at providing detailed procedures to extendsome traditional control charts, including x-chart, s-chart, p-chart, np-chart, u-chart, and MaxGWMA chart to become fuzzy ones to adapt to fuzzy environment

To achieve the above objective, we proposed two new ranking indexes pared to some existing approaches, one of our proposed ranking index providesthe strongest discriminative power while other one provides simpler and effectiveranking index Moreover, in the development of our fuzzy control charts, therandomness and fuzziness inherent in sample data are fully taken into accountand tested, which hasn’t been seriously considered in previous papers Also, ourproposed control charts can effectively classify the process conditions into multi-intermittent states between in-control and out-of-control, noteworthy information

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Com-is of importance in avoiding unnecessary adjustment to the current process if theset-up cost is large and/or intolerable Eventually, compared with the traditionalcharts, our fuzzy control charts have shown to be more effective in many aspects

of process surveillance and classification In addition, the notion of “rather” in thefuzzy judgment is quantitatively developed in this research to provide more objec-tive and deterministic bases in making proper decisions on the process conditions.Especially, in order to judge manufacturing conditions in the fuzzy environment,

we proposed a thorough and quantitative classification approach which makes ourprocess evaluation more sufficient and justified than those of the traditional con-trol charts These conclusions were proved and drawn from four practical casestudies, including surface roughness of optical lenses, the quality of offset prints,nonconformities on dyed cloth, and the coating thickness of industrial drill bits,which served as the typical examples to illustrate the practical applicability of ourproposed fuzzy control charts

Keywords: Quality management, Fuzzy control chart, Fuzzy ranking method,Left-right areas, Centroids, Surface roughness, Offset prints, Dyed cloth, Coatingprocess, Process classification, Process evaluation

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My uttermost thanks to God, Buddha, Maitreya, and Bodhisattva for thekind arrangement of my life with lucks.

Without Taiwan Government Scholarship, I would hardly have completed mydoctoral program within three years at National Kaohsiung University of AppliedSciences, Kaohsiung, Taiwan; therefore, I would like to send my deep thanks

to Taiwan Government in general and Taipei Economic and Cultural Office inHochiminh City, Vietnam in particular for offering me the special financial aid tosupport my study and living in Taiwan- the heart of Asia, for the last three years

I am extremely grateful to my two great supervisors, Prof Ying-Fang Huang,Ph.D., and Prof Ming-Hung Shu, Ph.D., who have encouraged me with helpfulinstructions, support and patience Their professional advice and knowledge shar-ing have been invaluable on not only an academic but also a social and personallevel, which I tremendously appreciate I would like to especially express my sin-cere thanks to Prof Shu, an expert in Quality Management and Fuzzy statistics,who has taught me a lot about the quality control in fuzzy environment Myexceptional thanks to Prof Bi-Min Hsu for her knowledgable guidance and finan-cial support to my research proposals as well as overseas trips to attend severalinternational conferences

I would like to thank my beloved university, National Kaohsiung University

of Applied Sciences (KUAS), for offering me a good study & living environmentwith an excellent library, well-equipped research lab, comfortable dormitory, andvarious physical activities I would like to take this opportunity to express mygratitude to Prof Chia-Nan Wang to consider my application and give me theopportunity to have an open interview before accepting me as a Ph.D candidate

I am also thankful to the Office of International Affairs in general, Ms Wendi and

Ms Helen in particular, for their special helps in providing useful information

to clarify my questions and concerns, and dealing with paperwork since I firstcame to KUAS My deep thanks to Department of Industrial Engineering andManagement and their staffs, Ms Wendy and Ms Juice, for their kind assistanceduring my study in KUAS

I would like to acknowledge the special support from my employer, Dr DoHuu Tai- President of Lac Hong University, Dr Tran Hanh- former Rector, and

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and International Economics (BAIE) With their financial support and ment, I could totally focus on my study in Taiwan Exceptional thanks to all of

arrange-my colleagues in Lac Hong University for their loves and encouragement

With all of my sincerity and special honor, I inscribe the good deed, meritand extreme expectation of my parents, Nguyen Van Hoa and Pham Thi Ton, whohave spared no pains to bring me up and tried their best to provide me necessaryfacilities to have a smooth and continuous study environment since I was a smallchild till now Without their noble sacrifices, I don’t have this glorious day today,definitely Besides, I would also sincerely and honorably thank my parents-in-law,Hua Van Tai and Phan Thi Nhi, who gave birth to my virtuous wife, has helped

my small family when I wasn’t at home and encouraged me to complete my study

My profound thanks to all of my brothers and sisters, and my brothers-in-law fortheir love and encouragement

I dedicate this thesis to my virtuous wife, Hua Thi Thuy, who has takengood care of my small family and given birth to my dear son, Nguyen TruongLong, and my lovely daughter, Nguyen Thanh Thuy Due to her special diligence,

I have fully kept my mind in my study Our beloved children are the driving force

to make me more conscious and motivated The completion of my doctoral degree

is just a small compensation for their lack of my love in the last three years.During my time in Taiwan, I have met several friendly Taiwanese My bestregards to Mr Ding Zheng-Hsiung, Mr Wang Yi-Jin, and several venerablelecturers at Tsu-Der-Kung Temple, who have taught me a lot about the filialpiety, the sense of responsibility to the society and the family, patriotism, etc.With their ethical lessons, my morality has been greatly and positively improved

I would also thank my many Taiwanese friends and lab-mates, Jui-Chan,Jenny, Peng-Jen, Rick, Cheng-Zhe, Wen-Hung, Hsiu-Yun to name a few as well

as my several Vietnamese friends that I have met in KUAS, NCKU, and STUSTduring my time in Taiwan Their friendship makes me happy with good memoryabout the interesting 3-year period of study in Taiwan

Many thanks to all of my previous respectful teachers and advisers and those

I have met in my life but aren’t listed above They all have significant contribution

to my current success

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Declaration of Authorship i

1.1 The Importance of Quality 1

1.2 Traditional Methods in Monitoring and Controlling Quality 3

1.3 The Fuzziness in Data Collection 6

1.4 Thesis Structure 8

2 Fundamental Literature 10 2.1 Control Chart 10

2.1.1 Statistical Basis of the Control Chart 10

2.1.2 x-chart and s-chart 14

2.1.3 p-chart for Fraction Nonconforming 17

2.1.4 np-chart for Number Nonconforming 19

2.1.5 u-chart for Nonconformities 19

2.1.6 MaxGWMA Control Chart 20

2.2 Fuzzy Numbers 26

2.2.1 Fundamental Definitions 26

2.3 Chapter Summary 29

3.1 Ranking Fuzzy Numbers Based on Left and Right Dominance (LRD) 31

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3.2 Ranking Fuzzy Numbers Based on Left and Right Integral Values

(LRI) 37

3.3 Ranking Fuzzy Numbers Based on Left-Right Areas and Centroids (LRAC) 41

3.3.1 Proposed Ranking Index 41

3.3.2 Comparative Examples 43

3.3.3 Extended Ranking Rules 55

4 Fuzzy x and s-chart 59 4.1 Construction of Fuzzy x-chart and s-chart 59

4.1.1 Fuzzy x Control Chart 60

4.1.2 Fuzzy s Control Chart 62

4.1.3 Fuzzy Average ex and Fuzzy Mean Standard Deviation es 63

4.2 Classification Conditions 64

4.2.1 Based on Three Methods Presented in Chapter 3 64

4.2.2 Based on Simplified Left and Right Integral Values (SLRI) 71 4.3 Practical Application 75

4.3.1 Numerical Results Based on LRD Approach 78

4.3.2 Numerical Results Based on LRI Approach 81

4.3.3 Numerical Results Based on LRAC Approach 83

4.3.4 Numerical Results Based on SLRI Approach 87

5 Fuzzy p-chart 94 5.1 Construction of Fuzzy p-chart 94

6 Fuzzy np-chart 113 6.1 Construction of Fuzzy np-chart 113

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7 Fuzzy u-chart 131 7.1 Construction of Fuzzy u-chart 131

8 Fuzzy MaxGWMA Chart 150 8.1 Construction of Fuzzy MaxGWMA chart 150

8.3.1 MaxGWMA Chart for Crisp Data 165

8.3.2 Fuzzy MaxGWMA Chart for Fuzzy Data 167

9 Conclusion 178 9.1 Brief Summary 178

9.2 Our Contributions and Findings 179

9.3 Our Limitations 181

9.4 Recommendations for Future Works 181

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1.1 Quality and Profitability 2

1.2 Quality perspectives in the value chain 3

2.1 A typical control chart 11

2.2 Normal probability distribution 12

2.3 Process variability versus Fraction nonconforming 14

2.4 Typical x-chart and s-chart with variable sample size 16

2.5 Typical p-chart with variable sample size 19

2.6 Generalized triangle fuzzy numberea 27

2.7 Generalized trapezoidal fuzzy numberea 28

3.1 The left and right spreads of fuzzy numbereai andeaj 32

3.2 Four consequences in case Dβ0 i−j = 0 36

3.3 Four consequences in case DSβ0 1−2 = 0 40

3.4 Ae0i is the image of eAi 41

3.5 Fuzzy numbers eA1, eA2 and eA3 in Example 3.3.1 45

3.7 Fuzzy numbers eA1 and eA2 in Example 3.3.3 46

3.11 Fuzzy numbers eA1, eA2, eA3 and eA4 in Example 3.3.7 50

3.17 Four consequences in case 4LRACi−j4β = 0 57

4.1 Roughness and waviness profiles 75

4.2 (a) Schema of lights propagation at smooth and (b) rough optical surface 76

4.3 Schema of surface profile as produced by a stylus device 77

4.4 Fuzzy s-chart for roughness height of optical lens 78

4.5 Fuzzy x-chart for roughness height of optical lens 80

4.6 Compare the performance of the four investigated approaches 91

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5.1 Fuzzy p-chart for the quality of printed works 107

6.1 Fuzzy np-chart for the quality of printed works 124

7.1 Some common defects in textile dyeing industry 142

7.2 u-chart for monitoring the nonconformities in dyed cloth 143

8.1 Schema of a coating layer 164

8.2 The MaxGWMA chart for the coating thickness of drill bits 167

8.3 Fuzzy s-chart for roughness height of optical lens 170

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2.1 Relationships between K and the probability to detect process shifts 12

2.2 Symbols for out-of-control points with their indications 26

3.1 Summary of numerical examples used in this paper 44

3.2 Ranking results at different optimism levels in Example 3.3.1 45

3.4 Ranking the images of three fuzzy numbers in Example 3.3.4 48

3.6 Ranking results of the three fuzzy numbers in Example 3.3.6 50

3.7 Ranking results of the three fuzzy numbers in Example 3.3.7 51

4.1 The attained significance levels in Runs test 77

4.2 Process variability classification based on LRD approach 79

4.3 Continuation of Table 4.2 81

4.4 Process average classification based on LRD approach 82

4.6 Process variability classification based on LRI approach 84

4.8 Process average classification based on LRI approach 86

4.10 Process variability classification based on LRAC approach 88

4.12 Process average classification based on LRAC approach 90

4.14 Process variability classification based on SLRI approach 92

4.15 Process average classification based on SLRI approach 93

5.1 The fuzzy percentage of nonconforming offset prints 105

5.3 Fraction nonconforming classification based on LRD approach 108

5.4 Fraction nonconforming classification based on LRI approach 109

5.5 Fraction nonconforming classification based on LRAC approach 111

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5.6 Fraction nonconforming classification based on SLRI approach 112

6.1 Number nonconforming classification based on LRD approach 125

6.2 Number nonconforming classification based on LRI approach 127

6.3 Number nonconforming classification based on LRAC approach 128

6.4 Number nonconforming classification based on SLRI approach 129

7.1 The number of nonconformities in dyed cloth 142

7.3 u-chart classification based on LRD approach 145

7.4 u-chart classification based on LRI approach 146

7.5 u-chart classification based on LRAC approach 147

7.6 u-chart classification based on SLRI approach 149

8.1 Symbols used for out-of-control on the F-MaxGWMA chart 157

8.2 Symbols used for rather out-of-control on the F-MaxGWMA chart 158 8.3 Symbols used for rather in-control on the F-MaxGWMA chart 159

8.4 Symbols used for out-of-control on the F-MaxGWMA chart based on SLRI 161

8.5 Symbols used for rather out-of-control on the F-MaxGWMA chart based on SLRI 162

8.6 Symbols used for rather in-control on the F-MaxGWMA chart based on SLRI 163

8.7 The thickness (µm) of coating layer on drill bits 166

8.9 Fuzzy data of the coating layer thickness on drill bits 168

8.11 The lower-end fuzzy control limits for the F-MaxGWMA chart 171

8.12 The upper-end fuzzy control limits for the F-MaxGWMA chart 172

8.13 Process classification based on LRD at β = 0.5, 0.6, 0.7, 0.8 173

8.14 Process classification based on LRI at β = 0.5, 0.6, 0.7, 0.8 174

8.15 Process classification based on LRAC at β = 0.5, 0.6, 0.7, 0.8 175

8.16 Process classification based on SLRI at β = 0.5, 0.6, 0.7, 0.8 176

A.1 Constant c4i in constructing control charts with sample size ni [1] 182 B.1 Roughness height (µm) of optical lens 183

B.2 Continuation of Table B.1 184

B.3 Continuation of Table B.2 185

C.1 Process variability classification based on LRD approach 187

C.2 Continuation of Table C.1 188

C.3 Continuation of Table C.2 189

D.1 Process average classification based on LRD approach 191

D.2 Continuation of Table D.1 192

D.3 Continuation of Table D.2 193

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E.1 Process variability classification based on LRI approach 195

E.2 Continuation of Table E.1 196

E.3 Continuation of Table E.2 197

F.1 Process average classification based on LRI approach 199

F.2 Continuation of Table F.1 200

F.3 Continuation of Table F.2 201

G.1 Process variability classification based on LRAC approach 203

G.2 Continuation of Table G.1 204

G.3 Continuation of Table G.2 205

H.1 Process average classification based on LRAC approach 207

H.2 Continuation of Table H.1 208

H.3 Continuation of Table H.2 209

I.1 S1β for s-chart based on SLRI approach 210

I.7 S1β for x-chart based on SLRI approach 216

I.13 Process variability classification based on SLRI approach 222

I.14 Continuation of Table I.13 223

I.15 Process average classification based on SLRI approach 223

I.16 Continuation of Table I.15 224

J.1 SVL,β e u e ˆ pi , SVβ e u e ˆ pi and SVU,β e u e ˆ pi based on SLRI approach 226

K.1 SVL,β e u g N Ci , SVβ e u g N Ci and SVU,β e u g N Ci based on SLRI approach 228

L.1 SVL,β f up e ui, SVβ f up e ui and SVU,β f up e ui based on SLRI approach 230

M.1 SVL,β g U CL i , SVβ g U CL i and SVU,β g U CL i based on SLRI approach 231

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ARL Average Run Length

SLRI Simplified Left and Right integral value approach

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This chapter briefly introduces a term “Quality” and its traditional monitoringand controlling methods Due to the existence of fuzziness in the collected data,this chapter also presents some recent developments in extending the traditionalmethods to adapt to the fuzzy environment

Quality, an inherent entity of products/services, is a complex concept that is ally perceived differently among individuals; thus, in contemporary philosophy, itsdefinition and approaches to distinguish different kinds of quality are still con-troversial issues [2], and no universal definition is commonly accepted among dif-ferent consultants and/or business professionals [3] In spite of this, quality hasbeen widely recognized as one of the most important determinants of the final de-cision in the selection among competing products/services, making it mandatoryfor individuals and/or organizations to fully understand and improve the quality

usu-of their products/services so as to increase their competitiveness, growth and cess [3,4] Aaker & Jacobson [5], Anderson et al [6], Nelson et al [7], and Capon

suc-et al [8] pointed out that firms providing better quality have higher economicreturns because better quality leads to a significant reduction of rework, scrap,and returns of bad/defective products/services which are found as one of the keysources for the operational cost of the relevant organizations; thus, good qualitycan not only increase the organization productivity and profits but also make theircustomers satisfied with the products/services [3] Making customers satisfied is

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very important because the satisfaction is one the main drivers of customer alty [9 12] which is of great help to reduce the costs of future transactions [13],fortify future revenues [11,14,15], lower price elasticities [16], and gain significantclemency if the products/services fail to meet their requirements [9] The rela-tionship between the quality improvement and organizational profitability can bebriefly illustrated in Fig 1.1 [3].

loy-Figure 1.1: Quality and Profitability

Due to the complexity, Evans & Lindsay [3] considered the quality from ious perspectives, including judgmental perspective, product-based perspective,user-based perspective, value-based perspective, manufacturing-based perspective,integrating perspective, and customer-driven perspective as shown in Fig 1.2;whereas, Montgomery [1] believed that quality is traditionally defined as fitnessfor use which includes “quality of design” and “quality of conformance” In thetraditional definition, quality is often tendentiously understood as conformance-to-specifications” which leads to a widespread perception that quality is totallydecided in manufacturing [1] However, Evans & Lindsay [3] claimed that the con-cept of quality has been tremendously developed from the manufacturing roots

var-to modern service industries In the belief that unwanted or harmful ity (hereinafter mentioned as variability for brevity) in the key characteristics ofproducts/services is the critical factor affecting the perception of the quality ofthe products/services, Montgomery [1] proposed a modern definition where “qual-ity is inversely proportional to variability” and stated that quality improvement

variabil-is the reduction of variability in processes and products because less variabilityresults in less rework, fewer repairs and warranty claims, as well as lower costs;and in service industries, quality improvement is defined as the reduction of waste

in terms of money, time, and effort in correcting an error or a mistake To have aclear focus, this research only considers the quality of physical products; and thefollowing parts mention it as quality for brevity

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Transcendent quality and product-based quality

Design

User-based quality

Value-based quality

Manufacturing Manufacturing-

Figure 1.2: Quality perspectives in the value chain

Con-trolling Quality

According to Montgomery [1], a product is normally evaluated based on a number

of elements called quality characteristics including physical characteristics (length,weight, porosity, etc.), sensory characteristics (taste, color, appearance, smell,etc.) and time-oriented characteristics (durability, reliability, serviceability, etc.);and, variability between two products always exists in practice Therefore, inmanufacturing industries, target value of a quality characteristic and its certaintolerances, called specifications, are setup to identify the level of acceptable quality.The largest acceptable tolerance is called upper specification limit (USL), and thelowest one is called lower specification limit (LSL) If the variation is small andthe key quality characteristic is within the set specifications, the products are said

to be identical and produced under a stable, repeatable or in-control process, i.e.,their quality is well perceived A stable process usually has a very low percentage

of nonconformities and defects in its outputs, resulting in not only the decrease

in the manufacturing cost but also the increase in customer satisfaction throughusing consistent products So, monitoring and controlling the quality is critical tohave satisfied customers and keep them loyal users, sustain profitability, and gainmarket share with better competitive advantages [3]

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Nowadays, industrial products are usually produced in such a high volume.Any inefficiency in controlling their quality will, of course, lead to a high increase ofproduction cost for industrial manufacturers Therefore, building and implement-ing an effective quality control program has become one of their most importantissues, which makes quality control a powerful management tool dealing with thekey quality characteristics of visible products [17] Once the quality is in con-trol, which means that the key quality characteristic lies within a small variabilityaround its target value or nominal dimension, the products are said to be con-sistently manufactured, resulting in not only the decreased manufacturing costsdue to the low percentage of nonconformities and defects but also the increase incustomer satisfaction The variation of products are related to several causes thatare classified into chance causes and assignable causes Chance causes are referred

to as causes that occur randomly, unpredictably, unavoidably and for unknownreasons resulting in the variability of the process, i.e., chance causes inherentlyexist in the process; whereas assignable causes are identifiable factors causing thevariability A process operated without assignable causes is said to be in sta-tistical control; otherwise, it is called out-of-control [1, 17] In an out-of-controlprocess, there is an important shift in the key quality characteristic resulting innonconforming outputs that lie outside the expected specification limits Thus,reducing variability in the manufactured products is a permanent concern of everyindustrial organizations

In reducing variability, achieving process stability as well as improving duction capability, statistical methods are preferably employed to monitor andcontrol the quality because most quality characteristics can be statistically mea-sured or counted Particularly, Statistical process control (SPC) has been widelyemployed in practice due to its ease of use [1] SPC, one of the greatest tech-nological developments of the twentieth century, is a method that uses statisticaltechniques to measure, interpret and ultimately control product quality Thereare seven major tools in SPC, including (1) Histogram or Stem-and-leaf plot, (2)Check sheet, (3) Pareto chart, (4) Cause-and-effect diagram, (5) Defection con-centration diagram, (6) Scatter diagram, and (7) Control chart Among them,control chart, pioneered by Shewhart, is considered the most technically sophisti-cated [1] Thus, in monitoring and controlling manufacturing process quality, thisthesis only takes control chart into consideration

pro-Moreover, in quality engineering, data of continuous measurements such as

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weight, volume, thickness, height, length, width, etc., are called variables data,whereas discrete data often in the form of counts are called attributes data [17].This classification is important in using appropriate techniques to the monitoring,controlling and improvement processes; thus, the selection of which type of controlchart to be used obviously depends on the type of data collected Specifically, for aquality characteristic which can be numerically measured and is called a variable,x-chart is used to monitor its mean; whereas R-chart is used to monitor the range

of its variation and s-chart is for its standard deviation Montgomery [1] assertedthat the process variability should be first kept under control before the processmean is monitored because larger variability always results higher percentage ofnonconforming products although the process mean is kept unchanged at its targetvalue Therefore, a combination of x-chart with either R-chart or s-chart arecommonly observed in practice For small sample size of no more than 10, x-chartand R-chart are preferably used while x-chart and s-chart are used when samplesize is either larger than 10 or variable [1,17]

Besides the variables data, there are still many quality characteristics whichcan only be expressed linguistically, such as good/bad, conforming/nonconform-ing, etc The data of such characteristics collected in the form of counting arereferred as attributes data Hence, in dealing with this kind of data, attributescontrol charts are good substitutes For example, p-chart is designed to monitorthe fraction of nonconforming or defective products produced by a manufactur-ing process, whereas np-chart is widely used when the number of nonconformingitems need fully considered Besides, with samples of equal size, controlling thetotal number of nonconformities is usually done with c-chart while u-chart is used

to monitor the average number of nonconformities per unit Moreover, u-chart

is especially useful if the sample size is variable In attributes control charts, itshould be noted that the term nonconformity refers to a specific type of failure

of the quality characteristics, whereas defect refers to a serious nonconformity Aproduct that fails to conform one or more of its required specifications is classified

as a nonconforming product Sometimes, a nonconforming product is still usable

if its key quality characteristics are not seriously deviated from the specifications

If unusable, it is called defective product Chapter 2 further reviews the mental literature about not only these control charts but also MaxGWMA chart,

funda-a Mfunda-aximum generfunda-ally weighted moving funda-averfunda-age control chfunda-art proposed by Sheu et

al [18] due to its superior ability in detecting small shifts in process mean andvariability

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1.3 The Fuzziness in Data Collection

As a matter of fact, the above mentioned control charts are traditionally structed based on random precise data collected from a key quality characteristicand they can only classify a process into in-control and out-of-control However,data from many applications in practice cannot be recorded or collected precisely.For instance, the size of a physical item (width, length, height, thickness, etc.),the temperature in a certain location, or the transmission speed of sound in acertain environment cannot be accurately measured due to the inherent errors ofboth methods and instruments used in conducting the measurements These areknown as fuzzy data [19] In addition, the human subjectivity, such as mood,optimism, intelligence and perception, is also a critical factor in making appro-priate decisions on whether those products are conforming or nonconforming orjudging if processes are in control or not Also, some intermediate decisions in-dispensably exist in-between the binary classification [20] Therefore, to deal withthe uncertainty, including randomness and fuzziness, in the real world and thelikely increase in the variance of normal observations in the presence of fuzziness[21], the traditional control charts with binary classifications are to be extended

con-to adapt con-to these fuzzy environments [4, 22] Thus, some key points relating toFuzzy set theory are presented in Chapter 2 to provide necessary foundation forthe construction of fuzzy control charts in Chapters 4- 8

Literally, multi-grades linguistic terms such as perfect, good, medium, poorand bad, were proposed by Wang and Raz [23] to express the evaluating levels ofthe key quality characteristic In order to monitor the process mean, they proposedtwo approaches, namely the fuzzy probabilistic approach and the membershipapproach Since the underlying probability distributions of the linguistic data arenot considered in these approaches, Kanagawa et al [24] modified the construction

of control charts by estimating the probability distribution existing behind thelinguistic data Nevertheless, both of the researches have two core controversialissues: (1) The membership functions of linguistic terms are obtained arbitrarily on

a given scale regardless of the fuzziness in the judgments of experts [22,25–27]; (2)while using defuzzification methods such as z-square, fuzzy mode, fuzzy average,

or fuzzy median allow their charts to be constructed with binary classifications,their approaches risk the loss of the fuzziness information in the original data aswell as misjudgment of the manufacturing process [22,28, 29]

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In an effort to ameliorate the drawbacks in the methods, Grzegorzeski &Hryniewicz [30] proposed the necessity index of strict dominance (NISD) pre-sented by Dubois & Prade [31] However, the NISD is content-dependent becausethe ranking results may change when a new fuzzy number is added [32] Instead,G¨ulbay & Kahraman [33] employed an acceptable percentage index called a directfuzzy approach (DFA) But, Shu & Wu [22] pointed out that the DFA fails toobtain the fuzzy sample means and variances by only using the α-cuts Hence,they developed a fuzzy dominance approach (FDA) by extending fuzzy-numbersranking method proposed by Yuan [34] Under fuzzy environments, the FDA canactually categorize the manufacturing process into four different classes, includ-ing in-control, rather in-control, rather out-of-control, and out-of-control It can,though, only perform nicely at the dominance degree greater than 0.5.

Therefore, in order to overcome the aforementioned shortages, this studyconsiders both randomness and fuzziness, which has not been addressed seriously

in previous papers, in constructing the fuzzy control charts Specifically, from thetraditional control charts reviewed in Chapter 2, Fuzzy x-chart, Fuzzy s-chart,Fuzzy p-chart, Fuzzy np-chart, Fuzzy u-chart, and Fuzzy MaxGWMA chart areseparately developed in Chapters 4 - 8 The monitoring and controlling of themanufacturing process can be done by comparing observed data against their fuzzycontrol limits The comparison obviously needs certain methods of ranking fuzzynumbers because the ranking plays a vital role in providing prerequisite proceduresfor practitioners to make their proper decisions [4,35–38] Consequently, Chapter

3 presents three different ranking methods, including the left and right dominanceapproach proposed by Chen & Lu [39] due to its efficiency and the robustness

of ranking results, and the most recent ranking method based on the left andright integral values proposed by Yu & Dat [40] because of its recognized rankingconsistency, easiness and effectiveness for various types of fuzzy numbers, andone of our newly proposed methods called “Left-Right areas and centroids” whichowns a strong discriminative power These three ranking approaches are also takeninto consideration in establishing thorough classification rules for the fuzzy controlcharts

Another new ranking method proposed in this thesis is named “Simplified leftand right integral values” which is developed from the left and right integral valuesproposed by Yu & Dat [40] by adding an extra normal triangular fuzzy number e0 =(0, 0, 0) which is used as the radical number to perform the ranking of the actual

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observations versus their control limits Due to certain difference in the fuzzycontrol limits among the fuzzy control charts, this simplified approach is takeninto account in every classification conditions set for each type of fuzzy controlcharts, i.e., the simplified left and right integral values approach is repeatedlypresented in Chapters 4 - 8.

In order to illustrate the applicability of our proposed fuzzy control charts, fourpractical case studies are investigated In particular, surface roughness of opticallenses is considered in Chapter 4 with the fuzzy x-chart and s-chart whereas thequality of offset prints can be monitored with either fuzzy p-chart in Chapter5orfuzzy np-chart in Chapter 6 Chapter7 considers the application of fuzzy u-chart

in the case of defects on dyed cloth Fuzzy MaxGWMA chart presented in Chapter

8is applied to the coating process of industrial drill bits Therefore, the structure

of this thesis is arranged as the follows

• Chapter 1briefly introduces the complex term “Quality” and its traditionalmonitoring and controlling methods Due to the existence of fuzziness inthe collected data, this chapter also presents some recent developments inextending the traditional methods to adapt to the fuzzy environment, whichobviously supports the in-line trend of our research in this thesis

• Chapter 2provides not only the fundamentals about some traditional charts,including x-chart, s-chart, p-chart, np-chart, u-chart, and MaxGWMA chartbut also some key points of Fuzzy set theory so that relevant fuzzy controlcharts can be developed in Chapters4 - 8

• Chapter 3 first reviews two prominent methods of ranking fuzzy numbersproposed by Chen & Lu [39] and Yu & Dat [40] Our new ranking method

is also proposed in this chapter Then, through several numerical examples,the performance of our proposed approach is adequately compared to someother existing ranking methods to prove its validity as well as its rankingpower

• Chapter 4first presents a detailed procedure for the construction of Fuzzy chart and s-chart Then, based on the four aforementioned ranking methods,

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x-a thorough set of clx-assificx-ation conditions is suggested A cx-ase study ofmonitoring the surface roughness of optical lenses is investigated in thischapter to demonstrate the applicability of our proposed Fuzzy x-chart ands-chart in practice.

• Chapter5provides a detailed procedure for the construction of Fuzzy p-chart.Similar to Chapter4, this chapter also relies on the four aforementioned rank-ing methods to establish a thorough set of classification conditions Fuzzyp-chart is employed in monitoring the quality of offset prints to demonstrateits practical applicability

• Chapter 6offers a detailed procedure for the construction of Fuzzy np-chartwhich can be evaluated through a complete set of classification conditionsestablished from the four different ranking methods The applicability ofthis fuzzy control chart is also considered by using the case of the quality ofoffset prints discussed in Chapter 5

• Chapter 7 uses the same token as Chapters 4 - 6 by suggesting a detailedprocedure for the construction of Fuzzy u-chart It also considers the fourmentioned methods to set up entire classification conditions to judge themanufacturing process The number of nonconformities on dyed cloth isstudied in this chapter

• Chapter 8 comes up with similar approach A detailed procedure for theconstruction of Fuzzy MaxGWMA chart is first presented before thoroughclassification rules in evaluating the manufacturing process are suggested.The rules are also based on the four ranking methods This Fuzzy MaxG-WMA chart is used to monitor the thickness of the coating layer on industrialdrill bits from a practical coating process

• Chapter 9 closes this thesis with some key points obtained in this research,our main contributions and findings as well as some other remarkable con-clusions

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Fundamental Literature

This chapter introduces fundamental issues about some traditional charts, ing x-chart, s-chart, p-chart, np-chart, u-chart, and MaxGWMA chart As theyare extended to become fuzzy control charts in Chapters 4-8, some key points ofFuzzy set theory are also considered in this chapter

2.1.1 Statistical Basis of the Control Chart

Control chart is an extensively used tool for monitoring and examining turing processes Its power lies in the ability to detect process shifts and identifyabnormal conditions in the on-line manufacturing because it can provide enoughdiagnostic information as well as the value of important process parameters to pre-vent defects, reduce scrap and rework, and avoid unnecessary process adjustment,meaning that control charts can be of great help in not only improving the pro-ductivity and production capacity, but also decreasing the production cost [1, 17].Typically, a control chart consists of a center line (CL) which is the estimatedprocess target value, and two control lines- the upper control limit (U CL) andlower control limit (LCL) which are the boundaries of the normal variability used

manufac-to test if the majority of the observations are in control A typical control chart

is shown in Fig 2.1 whose control limits are determined by Eqs (2.1)

10

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Figure 2.1: A typical control chart

Based on the control chart, if the entire statistic points of collected sampledata fall within the limits and do not exhibit any systematic pattern, the variability

of the quality characteristic is in acceptable level, and the process is classified

as in statistical control; otherwise, the process is considered out-of-control, i.e.,the variability is unacceptable and the process is said to be affected by somecauses which need investigated and eliminated or solved to reduce their effects [17].However, the ability to recognize a systematic pattern and successfully discover theunderlying problems with the pattern profoundly depend on personal experienceand knowledge about the process investigated Ten practical experiences regarded

as sensitizing rules are discussed in [1] to provide fundamental remarks in quicklyrecognizing systematic patterns existing in a control chart

Control charts are constructed and operated with data collected from a cess The collected data should represent the various levels of the quality charac-teristics associated with the product Let q be a sample statistic that measures aquality characteristic of interest Its mean and standard deviation are respectivelydenoted by µq and σq Then, the control limits of Shewhart control chart aredetermined by

a smaller K make the control limits move closer to the center line resulting in morefalse alarms even though the current process is actually in control The choice of

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the control limits is similar to the choice of critical region in hypothesis testing[17].

Traditionally, K = ±3 is usually used because it scarcely generates falsealarms; particularly, an incorrect out-of-control signal will be generated about0.27% Fig.2.2 displays a normal probability distribution of a quality characteris-tic with K = ±3; and, Table 2.1 shows the relationships between the K and theprobability to detect process shifts in terms of the percentage of the quality char-acteristic inside the specification limits as well as the number of defective partsper million (ppm) [1]

Figure 2.2: Normal probability distribution

Table 2.1: Relationships between K and the probability to detect process

According to the law of large number in probability theory, it is suggested

to have larger sample sizes investigated so that small shifts in a process can beeasier detected However, for large shifts, small sample sizes are recommended foreconomical reasons Moreover, the efficiency of the control chart is significantlyaffected by the frequency of sampling It is ideal to have the investigated samplestaken as frequently as possible With high-volume manufacturing processes orhigh existence of assignable causes, rather small and more frequent samples areusually preferred in practice In constructing an effective control chart to detectprocess shifts, consecutive products are recommended to minimize the chance of

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variability within a sample due to assignable causes and maximize that betweensamples if assignable causes exist [1, 3].

Generally, using control chart to determine the current conditions of a ufacturing process is usually referred to as Phase I of control chart application

man-In phase I, Shewhart control charts are very effective in not only being easilyconstructed and interpreted but also detecting both large, sustained shifts in theprocess parameters, measurement errors, data recording and/or transmission er-rors, etc., which need settled in order to bring the process into a state of statisticalcontrol [1] Typically, 20 or 25 samples are collected from a current process that

is believed to be out-of-control to construct the trial control charts Any out ofcontrol signal or systematic pattern needs a careful investigation to find out po-tential assignable causes Once assignable causes are found, certain efforts arerequired to eliminate them from the process or reduce their effects as much aspossible before a revised control chart is constructed by excluding out-of-controlpoints or the points in systematic pattern from the data set From the revisedcontrol chart, if any further out-of-control signal or systematic pattern is found,further investigation for assignable causes is also mandatory; thus, action plan forthe assignable causes are constantly updated and expanded A new revised controlchart is then established This cycle can be repeated several times until no out-of-control signal or systematic pattern is found on the final control chart, indicatingthat the process is in statistical control after corrective actions are implemented

to deal with the assignable causes When the process is reasonably stable, it isthen of great importance to continuously monitor the process performance, which

is now referred as phase II of control chart application

Since assignable causes resulting in large shifts are systematically eliminated

in the first phase, smaller shifts must be detected in the phase II to make theproducts consistent Due to the fact that the Shewhart control charts are rela-tively insensitive to small shifts, they are obviously incompatible with the secondphase Instead, other two control charts, including the cumulative sum (CUSUM)control chart and the exponentially weighted moving average (EWMA) controlchart, are good substitutes in the phase Comparing the performance betweenthe two alternative control charts, Vargas et al [41] concluded that EWMA chartoutperforms CUSUM chart in detecting small shifts of one standard deviation orless Moreover, EWMA charts can quickly detect the small shifts at the begin-ning of the changes and do effective forecasting for the next period [41] as well as

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be set up and operated easier [1] Consequently, EWMA chart is more preferred

in practice Many scholars have paid much effort in developing this traditionalchart Among its various extensions, a superior control chart named the Maxi-mum generally weighted moving average (MaxGWMA) control chart has recentlybeen introduced by Sheu et al [18]

As mentioned in Chapter1, quality characteristics might be either measured

on numerical scales or separately counted, thus, we have variables data and tributes data, respectively [17] Some prominent control charts for each of thedata types are presented in the following sections

at-2.1.2 x-chart and s-chart

Practically, several quality characteristics, such as weight, volume, thickness, height,length, width, etc., can be numerically measured Each of such characteristics iscalled a variable; and, variables control charts are correspondingly used to mon-itor both its mean value and its variability Control of the process mean value

is normally done with the control chart for means, called x-chart Monitoringthe process variability may employ either R-chart for the range of its variation ors-chart for its standard deviation Traditionally, it is critically important to firstfully control the process variability before controlling the process mean becauselarger variability always results higher percentage of nonconforming products al-though the process mean is kept unchanged at its target value, as shown in Fig

2.3 [1] Therefore, x-chart usually goes with either R-chart or s-chart x-chartand R-chart are preferably used if sample size is small (no more than 10), whereasx-chart and s-chart are used when sample size is either larger than 10 or vari-able [17] As a consequence, in this research, the usage of x-chart and s-chart forvariable sample sizes is considered

Figure 2.3: Process variability versus Fraction nonconforming

Generally, there are several practical circumstances where collecting severalsamples of constant size encounters certain difficulties due to time limit, economic

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reasons, facility availability, etc Thus, the sample size may be varied among thecollected samples In such the cases, x-chart and s-chart are preferably used tomonitor the process mean and variability, respectively [1].

Suppose a quality characteristic X has a normal distribution with a mean µand a standard deviation σ, i.e., X ∼ N (µ, σ2) Normally, µ and σ are not known

in advance They are usually estimated from initial samples taken from a processthat is believed to be in control Conventionally, 20 or 25 samples are investigated;their grand average is used as the best estimator of µ, and their average standarddeviation can be used to obtain the estimated value of σ

Let ni be the size of the ith sample of the m samples investigated Let xij

denote the value of the quality characteristic in the sample ith at the observation

jth (i = 1, m; j = 1, ni) The average of the ith sample, denoted by xi, and thegrand average of the m samples, denoted by x, are determined by

1

Pm i=1ni− m

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where c4i is an constant determined by sample size ni as shown in Table A.1 in

AppendixA [1, 17, 42]

From the values of x and ˆσxi, the centerline (CL), upper control limit (UCL),

and lower control limit (LCL) of the x-chart are constructed by

And the control limits for s-chart are determined by

s

The control limits obtained from the Eqs (2.8) and (2.9) are usually treated

as trial control limits in the phase I application of x-chart and s-chart

Then, the control limits of the control charts are also determined by the

Eqs (2.8) and (2.9); however, each sample has its own value of c4 Therefore,

the control limits are fluctuated as shown in Fig 2.4 By plotting all of xi and si

i = 1, m against the variable control control limits, we can detect out-of-control

signal (if any)

CL UCL

-4 -6

LCL

Sample number

CL UCL

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2.1.3 p-chart for Fraction Nonconforming

In Section 2.1.2, the x-chart and s-chart are preferably used when the qualitycharacteristic can be numerically measured However, in practice, many qualitycharacteristics can only be expressed linguistically, such as good/bad, conform-ing/nonconforming, etc As mentioned in Chapter 1, the data of such characteris-tics are called attributes data Among several different control charts for attributesdata, this section discusses the first attributes control chart named p-chart which isdesigned to monitor the fraction of nonconforming or defective products produced

by a manufacturing process Especially, this section reviews p-chart for fraction ofnonconforming items with variable size because fixed sample size is a special case

of our case

In an investigated population, the ratio of the number of nonconformingproducts to its size is defined as the fraction nonconforming, i.e., if N C denotesthe number of nonconforming products in a random sample of n products, thesample fraction nonconforming ˆp is defined by

where pr(N C = x) is the probability of having x nonconforming products in the

n products; µ and σ2 are respectively the mean and variance of the binomialdistribution

Accordingly, the unbiased estimates of mean and variance of the samplefraction nonconforming ˆp, respectively denoted by µˆ and σ2

ˆ, are obtained by

µˆ = p

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Let N Ci denote the number of nonconforming products in the ith sample,and ni denote the size of the ith sample i = 1, m From Eq (2.10), the fractionnonconforming in the ith sample is calculated by

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Chapter 2 Fundamental Literature 19

CL LCL

-4 -6

LCL

Sample number

CL UCL

LCL

Sample number

p

x,s

Figure 2.5: Typical p-chart with variable sample size

Instead of monitoring the fraction nonconforming, we can directly monitor the

number of nonconforming items with an alternative control chart, named np-chart

Developed from the p-chart for fraction nonconforming, the control limits of

np-chart can be established by

rule that the upper control limit is rounded up to its next larger integer, whereas

the lower control limit is rounded down to its next smaller integer By plotting

N Ci on the np-chart, if all plotted points fall in between the two control limits, the

process is said to be in statistical control; otherwise, it is considered out-of-control

If any out-of-control signal is detected, further attention and investigation should

be implemented As such, many non-statistically trained personnel prefer to use

np-chart rather than p-chart because they can have quite direct interpretation

from the np-chart in monitoring the number of nonconforming items

As already mentioned in Section 2.1.3, nonconformity is actually a specific type of

failure that makes a product fail to meet certain specifications A nonconforming

product usually contains one or more nonconformities As a matter of fact, not

every nonconformity makes a product classified as nonconforming because the

classification heavily depends on the severity of the nonconformities Hence, in

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several practical situations such as the weld failure on a ship/ an aircraft/ a car,failures on an integrated circuit board (ICB)/ wafer, and so forth, the number ofnonconformities should be controlled rather than the fraction nonconforming orthe number of nonconforming products [1, 17].

Traditionally, in the case where the occurrence of nonconformities has aPoisson distribution, we can monitor either the total nonconformities in a samplewith c-chart if the samples are collected with equal size or the average number

of nonconformities per unit with u-chart if the sample size is either only one orvariable

In m initial samples, let xi denote the number of nonconformities in the ithsample which has niunits (i = 1, m) Then, the average number of nonconformitiesper unit, notated by ui, is calculated by

Pm i=1ni

The EWMA control chart was first introduced by Roberts [43] to detect a smallshift in the process mean Since then, it has successfully attracted the unceasingattention of many scholars as in the reviews by Xie [44], Han and Tsung [45],

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Eyvanzian et al [46], Sheu et al [47], Li and Wang [48], Zhang et al [49] andSheu et al [18].

In monitoring the process mean and variability, many different methods havebeen suggested Sweet [50] proposed the use of two EWMA charts to detect theshifts in process Some other researchers have made a lot of effort in employingonly a single chart Chan et al [51] introduced a chart which can identify theshift but it requires plotting the mean and variability separately, resulting in thedifficulty in constructing Whereas, the omnibus EWMA charts proposed by Do-mangue and Patch [52] are sensitive to shifts; but it cannot point out whetherthe shifts occur due to the process mean or variability or both [53] Gan [54]recommended a joint scheme consisting of a two-sided EWMA mean chart and

a two-sided EWMA variance chart which was found to perform well for severalout-of-control circumstances Chao and Cheng [55] came up with semicircle chartfor variable data but it also fails to track the time sequence of the plotted pointed.Gan [53] presented a method using a two-dimensional chart of elliptical in-controlregion to plot the EWMA of logarithm of sample variance (s2) against the EWMA

of sample mean (x) Chen and Cheng [56] first discussed about a new control chartcalled Max-chart by combining x-chart and s-chart under the maximum values oftheir relevant statistics The Max-chart can effectively exhibit both process meanand variability on a single chart Besides, in term of exponentially weighted mov-ing average, Max-Min EWMA chart was proposed by Amin et al [57] Amongother types of EWMA charts, such as MaxEWMA, SS-EWMA, EWMA-Max andEWMA-SC charts demonstrated by Xie [44], the MaxEWMA chart was proved

to be the most effective in terms of average run length (ARL) performance, anddiagnostic ability and have high capability of detecting small shifts in the pro-cess mean and variability as well as identifying the source and the direction of anout-of-control signal [44]

The EWMA chart was first generalized by Sheu and Lin [58] to create a newchart named the generally weighted moving average (GWMA) control chart TheGWMA chart can detect small shifts much quicker than the EWMA can However,while the EWMA chart is almost a perfectly distribution-free procedure, meaningthat EWMA chart is robust to non-normality, GWMA chart is designed under theassumption that the data set comes from a normal distribution and it is sensitive

to the departure from normality than EWMA chart when process shifts [58] Thisassumption has been used by Sheu and Yang [59], Sheu and Tai [60] and Sheu et

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