Introduction to Statistical Process Control

MUHAMMAD ASLAM, Ph.D., is a Professor in the Department of Statistics at King Abdulaziz University at Jeddah, Saudi Arabia. He was awarded the "Research Productivity Award for the year" in 2012 by Pakistan Council for Science and Technology. He is the founder of neutrosophic statistical quality control and neutrosophic inferential statistics. AAMIR SAGHIR, Ph.D., is a Professor in the Department of Mathematics at Mirpur University of Science and Technology. He received his Ph.D. in Statistics from Zhejiang University in China. LIAQUAT AHMAD, Ph.D., is an Associate Professor in the Department of Statistics and Computer Science at the University of Veterinary and Animal Sciences, Lahore, Pakistan. He's taught Statistics for over 24 years at the Ph.D. and M. Phil levels.

About the Authors xi

Preface xiii

Acknowledgments xvii

1 Introduction and Genesis 1

1.1 Introduction 1

1.2 History and Background of Control Charts 3

1.3 What is Quality and Quality Improvement? 5

Types of Quality-Related Costs 7

1.4 Basic Concepts 9

1.4.1 Descriptive Statistics 9

1.4.2 Probability Distributions 14

Continuous Probability Distributions 14

Discrete Probability Distributions 18

1.5 Types of Control Charts 19

1.5.1 Attribute Control Charts 19

1.5.2 Variable Control Charts 20

1.6 Meaning of Process Control 21

References 22

2 Shewhart Type Control Charts for Attributes 23

2.1 Proportion and Number of Nonconforming Charts 24

2.1.1 Proportion of Nonconforming Chart (p-Chart) 25

Variable Sample Size 28

Improved p-Chart 29

2.1.2 Number of Nonconforming Chart (np-Chart) 30

2.1.3 Performance Evaluation Measures 30

2.2 Number of Nonconformities and Average Nonconformity Charts 32

2.2.1 Number of Nonconformities (c-) Chart 33

2.2.2 Average Nonconformities (u-) Chart 34

2.2.3 The Performance Evaluation Measure 38

Dealing with Low Defect Levels 39

2.3 Control Charts for Over-Dispersed Data 40

2.3.1 Dispersion of Counts Data 40

2.3.2 g-Chart and h-Chart 40

2.4 Generalized and Flexible Control Charts for Dispersed Data 44

2.4.1 The gc- and the gu-Charts 45

2.4.2 Control Chart Based on Generalized Poisson Distribution 46

Process Monitoring 47

A Geometric Chart to Monitor Parameter θ 48

2.4.3 The Q- and the T-Charts 49

The OC Curve 52

2.5 Other Recent Developments 52

References 54

3 Variable Control Charts 57

3.1 Introduction 57

3.2 x̅ Control Charts 58

3.2.1 Construction of x̅ and R Charts 59

3.2.2 Phase II Control Limits 62

3.2.3 Construction of x̅ Chart for Burr Distribution Under the Repetitive Sampling Scheme 63

3.3 Range Charts 72

3.4 Construction of S-Chart 72

3.4.1 Construction of x̅ Chart 74

3.4.2 Normal and Non-normal Distributions for x̅ and S-Charts 75

3.5 Variance S2-Charts 75

3.5.1 Construction of S2-Chart 76

3.5.2 The Construction of Variance Chart for Neutrosophic Statistics 77

3.5.3 The Construction of Variance Chart for Repetitive Sampling 81

References 87

4 Control Chart for Multiple Dependent State Sampling 91

4.1 Introduction 91

4.2 Attribute Charts Using MDS Sampling 91

4.2.1 The np-Control Chart 92

4.3 Conway–Maxwell–Poisson (COM–Poisson) Distribution 98

4.4 Variable Charts 106

4.5 Control Charts for Non-normal Distributions 107

4.6 Control Charts for Exponential Distribution 109

4.7 Control Charts for Gamma Distribution 111

References 118

5 EWMA Control Charts Using Repetitive Group Sampling Scheme 121

5.1 Concept of Exponentially Weighted Moving Average (EWMA) Methodology 121

5.2 Attraction of EWMA Methodology in Manufacturing Scenario 126

5.3 Development of EWMA Control Chart for Monitoring Averages 127

5.4 Development of EWMA Control Chart for Repetitive Sampling Scheme 127

5.5 EWMA Control Chart for Repetitive Sampling Using Mean Deviation 128

5.6 EWMA Control Chart for Sign Statistic Using the Repetitive Sampling Scheme 139

5.7 Designing of a Hybrid EWMA (HEWMA) Control Chart Using Repetitive Sampling 147

References 154

6 Sampling Schemes for Developing Control Charts 161

6.1 Single Sampling Scheme 161

6.2 Double Sampling Scheme 162

6.3 Repetitive Sampling Scheme 165

6.3.1 When a Shift of μ1 = μ + kσ Occurs in the Process 169

6.4 Mixed Sampling Scheme 176

6.4.1 Mixed Control Chart Using Exponentially Weighted Moving Average (EWMA) Statistics 179

6.5 Mixed Control Chart Using Process Capability Index 180

6.5.1 Analysis Through Simulation Approach 187

References 187

7 Memory-Type Control Charts for Attributes 191

7.1 Exponentially Weighted Moving Average (EWMA) Control Charts for Attributes 191

7.1.1 Binomial EWMA Charts 192

7.1.2 Poisson EWMA (PEWMA) Chart 194

Performance Evaluation Measure 196

Calculation of ARLs Using the Markov Chain Approach 196

7.1.3 Other EWMA Charts 202

Geometric EWMA Chart 202

Conway–Maxwell–Poisson (COM–Poisson) EWMA Chart 204

7.2 CUSUM Control Charts for Attributes 209

7.2.1 Binomial CUSUM Chart 210

7.2.2 Poisson CUSUM Chart 215

7.2.3 Geometric CUSUM Chart 217

7.2.4 COM–Poisson CUSUM Chart 219

Performance Measure 219

7.3 Moving Average (MA) Control Charts for Attributes 220

7.3.1 Binomial MA Chart 221

7.3.2 Poisson MA Chart 223

7.3.3 Other MA Charts 225

References 226

8 Multivariate Control Charts for Attributes 231

8.1 Multivariate Shewhart-Type Charts 231

8.1.1 Multivariate Binomial Chart 231

Choice of Sample Size 233

8.1.2 Multivariate Poisson (MP) Chart 234

8.1.3 Multivariate Conway–Maxwell–Poisson (COM–Poisson) Chart 239

8.2 Multivariate Memory-Type Control Charts 243

8.2.1 Multivariate EWMA Charts for Binomial Process 243

Design of MEWMA Chart 244

8.2.2 Multivariate EWMA Charts for Poisson Process 245

8.3 Multivariate Cumulative Sum (CUSUM) Schemes 246

8.3.1 Multivariate CUSUM Chart for Poisson Data 247

References 248

Appendix A: Areas of the Cumulative Standard Normal Distribution 251

Appendix B: Factors for Constructing Variable Control Charts 253

Index 255