Statistical Quality Control: A Modern Introduction

Sixth Edition

Statistical Quality Control: A Modern Introduction International Student Version DOUGLAS C. MONTGOMERY Arizona State University

WILEY John Wiley & Sons, Inc.

Contents 2.7

PART 1 INTRODUCTION

MODERN QUALITY MANAGEMENT AND IMPROVEMENT Chapter Overview and Learning Objectives 1.1 The Meaning of Quality and Quality Improvement 1.1.1 Dimensions of Quality 1.1.2 Quality Engineering Terminology 1.2 A Brief History of Quality Control Shd Improvement 1.3 Statistical Methods for Quality Control and Improvement 1.4 Management Aspects of Quality Improvement 1.4.1 Quality Philosophy and Management Strategies 1.4.2 The Link Between Quality and Productivity 1.4.3 Quality Costs 1.4.4 Legal Aspects of Quality L'4.5 Implementing Quality Improvement

THE DMAIC PROBLEM SOLVING PROCESS Chapter Overview and Learning Objectives 2.1 Overview of DMAIC 2.2 The Define Step 2.3 The Measure Step 2.4 The Analyze Step 2.5 The Improve Step 2.6 The Control Step

Examples of DMAIC 2.7.1 Litigation Documents 2.7.2 Improving On-Time Delivery 2.7.3 Improving Service Quality in a Bank

PART Z STATISTICAL METHODS USEFUL IN QUALITY CONTROL AND IMPROVEMENT

54 54 56 59

61

4 4

STATISTICAL MODELS OR QUALITY CONTROL AND IMPROVEMENT 63 13 16 17 35 36 41 42

1C r *J

45 45 49 50 52 53 54

Chapter Overview and Learning Objectives 3.1 Describing Variation 3.1.1 The Stem-and-Leaf Plot 3.1.2 The Histogram 3.1.3 Numerical Summary of Data 3.1.4 The Box Plot 3.1.5 Probability Distributions 3.2 Important Discrete Distributions 3.2.1 The Hypergeometric Distribution 3.2.2 The Binomial Distribution 3.2.3 The Poisson Distribution 3.2.4 The Pascal and Related Distributions 3.3 Important Continuous Distributions 3.3.1 The Normal Distribution 3.3.2 The Lognormal Distribution 3.3.3 The Exponential Distribution 3.3.4 The Gamma Distribution 3.3.5 The Weibull Distribution 3.4 Probability Plots 3.4.1 Normal Probability Plots 3.4.2 Other Probability Plots 3.5 Some Useful Approximations 3.5.1 The Binomial Approximation to the Hypergeometric

63 64 64 66 69 71 72 76 76 77 79 80 81 81 86 88 89 91 93 93 95 96 96 IX

Contents 3.5.2 3.5.3 3.5.4

The Poisson Approximation to the Binomial The Normal Approximation to the Binomial Comments on Approximations

4.6.2 96 4.6.3 97 98

4.6.4 4.6.5

Hypothesis Testing in Multiple Regression Confidance Intervals in Multiple Regression Prediction of New Observations Regression Model Diagnostics

157 163 164 165

STATISTICAL INFERENCE IN QUALITY CONTROL AND IMPROVEMENT 103 Chapter Overview and Learning Objectives 4.1 Statistics and Sampling Distributions 4.1.1 Sampling from a Normal Distribution 4.1.2 Sampling from a Bernoulli Distribution 4.1.3 Sampling from a Poisson Distribution 4.2 Point Estimation of Process Parameters 4.3 Statistical Inference for a Single Sample 4.3.1 Inference on the Mean of a Population, Variance Known 4.3.2 The Use of P-Values for Hypothesis Testing 4.3.3 Inference on the Mean of a Normal Distribution, Variance Unknown 4.3.4 Inference on the Variance of a Normal Distribution 4.3.5 Inference on a Population Proportion 4.3.6 The Probability of Type II Error and Sample Size Decisions 4.4 Statistical Inference for Two Samples 4.4.1 Inference for a Difference in Means, Variances Known ' 4.4.2 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 4.4.3 Inference on the Variances of Two Normal Distributions 4.4.4 Inference on Two Population Proportions 4.5 What If There Are More Than Two Populations? The Analysis of Variance 4.5.1 An Example 4.5.2 The Analysis of Variance 4.5.3 Checking Assumptions: Residual Analysis 4.6 Linear Regression Models 4.6.1 Estimation of the Parameters in Linear Regression Models

104 104 105

PART D BASIC METHODS OF STATISTICAL PROCESS CONTROL AND CAPABILITY ANALYSIS 177

108 109 110 112 113 116 117 120 122 124 127 128

130 137 139 140 140 142 148 150 151

HOW SPC WORKS Chapter Overview and Learning Objectives 5.1 Introduction 5.2 Chance and Assignable Causes of Quality Variation 5.3 Statistical Basis of the Control Chart 5.3.1 Basic Principles 5.3.2 Choice of Control Limits 5.3.3 Sample Size and Sampling Frequency 5.3.4 Rational Subgroups 5.3.5 Analysis of Patterns on Control Charts 5.3.6 Discussion of Sensitizing Rules for Control Charts 5.3.7 Phase I and Phase II of Control Chart Application 5.4 The Rest of the Magnificent Seven 5.5 Implementing SPC in a Quality Improvement Program 5.6 An Application of SPC 5.7 Applications of Statistical Process Control and Quality Improvement Tools in Transactional and Service Businesses

179 179 180 181 182 182 189 191 193 195 197 198 199 205 206

213

VARIABLES CONTROL CHARTS

226

Chapter Overview and Learning Objectives 6.1 Introduction 6.2 Control Charts for x and R 6.2.1 Statistical Basis of the Charts

226 227228 228

6.2.2

6.3

6.4 6.5 6.6

Development and Use of x and R Charts 6.2.3 Charts Based on Standard Values 6.2.4 Interpretation of x and R Charts 6.2.5 The Effect of Nonnormality on x and R Charts 6.2.6 The Operating-Characteristic Function 6.2.7 The Average Run Length for the x Chart ' Control Charts for x and s 6.3.1 Construction and Operation of x and s Charts 6.3.2 The x and s Control Charts with Variable Sample Size 6.3.3 The s2 Control Chart The Shewhart Control Chart for Individual Measurements Summary of Procedures for x, R, and s Charts Applications of Variables Control Charts

ATTRIBUTES CONTROL CHARTS Chapter Overview and Learning Objectives 7.1 Introduction 7.2 The Control Chart for Fraction ''•Nonconforming '7.2.1 , Development and Operation of the Control Chart , 7.2.2 ' Variable Sample Size 7.2.3 Applications in Transactional 'and Service Businesses 7.2.4 The Operating-Characteristic Function and Average Run Length Calculations 7.3 Control Charts for Nonconformities (Defects) 7.3.1 Procedures with Constant Sample Size 7.3.2 Procedures with Variable Sample Size 7.3.3 Demerit Systems 7.3.4 The Operating-Characteristic Function

231 242 243

7.4 7.5

Contents

XI

7.3.5 Dealing with Low Defect Levels 7.3.6 Nonmanufacturing Applications Choice Between Attributes and Variables Control Charts Guidelines for Implementing Control Charts

323 326 326 330

246 246 249 251 251 255 259 259 268 268

288 288 289 289 290 301 304

306 308 309 319 321 322

DETERMINING PROCESS AND MEASUREMENT SYSTEMS CAPABILITY Chapter Overview and Learning Objectives 8.1 Introduction 8.2 Process Capability Analysis Using a Histogram or a Probability Plot 8.2.1 Using the Histogram 8.2.2 Probability Plotting 8.3 Process Capability Ratios 8.3.1 Use and Interpretation of C 8.3.2 Process Capability Ratio for an Off-Center Process 8.3.3 Normality and the Process Capability Ratio 8.3.4 More about Process Centering 8.3.5 Confidence Intervals and Tests on Process Capability Ratios 8.4 Process Capability Analysis Using a Control Chart 8.5 Process Capability Analysis Using Designed Experiments 8.6 Process Capability Analysis with Attribute Data 8.7 Gauge and Measurement System Capability Studies 8.7.1 Basic Concepts of Gauge Capability 8.7.2 The Analysis of Variance Method 8.7.3 Confidence Intervals in Gauge R & R Studies 8.7.4 False Defectives and Passed Defectives 8.7.5 Attribute Gauge Capability 8.8 Setting Specification Limits on Discrete Components 8.8.1 Linear Combinations 8.8.2 Nonlinear Combinations

344 345 345 347 347 349 351 351 354

356 357

359 364 366 367 368 368 373 376 377 381 383 384 387

XII

8.9

Contents Estimating the Natural Tolerance Limits of a Process 8.9.1 Tolerance Limits Based on the Normal Distribution 8.9.2 Nonparametric Tolerance Limits

Design of an EWMA Control Chart 9.2.3 Robustness of the EWMA to Nonnormality 9.2.4 Rational Subgroups 9.2.5 Extensions of the EWMA The Moving Average Control Chart 9.2.2

388 389 390 9.3

422 424 425 425 428

PART 4 OTHER STATISTICAL PROCESSMONITORING AND CONTROL TECHNIQUES

TIME-WEIGHTED CONTROL CHARTS Chapter Overview and Learning Objectives 9.1 The Cumulative Sum Control Chart 9.1.1 Basic Principles: The Cusum Control CJhart for Monitoring the Process Mean 9.1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean 9.1.3 Recommendations for Cusum Design 9.1.4 The Standardized Cusum .9.1.5 Improving Cusum Responsiveness for Large Shifts 9.1.6 The Fast Initial Response or Headstart Feature 9.1.7 One-Sided Cusums ' 9.1.8 A Cusums for Monitoring Process Variability 9.1.9 Rational Subgroups 9.1.10 Cusums for Other Sample Statistics 9.1.11 The V-Mask Procedure 9.1.12 The Self-Starting Cusum 9.2 The Exponentially Weighted Moving Average Control Chart 9.2.1 The Exponentially Weighted Moving Average Control Chart for Monitoring the Process Mean

397

399 400 400

400

403

408 410

410 410 413 413 414

414 415 417 419

419

ADVANCED CONTROL CHARTING TECHNIQUES Chapter Overview and Learning Objectives 10.1 Statistical Process Control for Short Production Runs 10.1.1 x and R Charts for Short Production Runs 10.1.2 Attributes Control Charts for Short Production Runs 10.1.3 Other Methods 10.2 Modified and Acceptance Control Charts 10.2.1 Modified Control Limits for the x Chart 10.2.2 Acceptance Control Charts 10.3 Control Charts for Multiple-Stream Processes 10.3.1 Multiple-Stream Processes 10.3.2 Group Control Charts 10.3.3 Other Approaches 10.4 SPC With Autocorrelated Process Data 10.4.1 Sources and Effects of Autocorrelation in Process Data 10.4.2 Model-Based Approaches 10.4.3 A Model-Free Approach 10.5 Adaptive Sampling Procedures 10.6 Economic Design of Control Charts 10.6.1 Designing a Control Chart 10.6.2 Process Characteristics 10.6.3 Cost Parameters 10.6.4 Early Work and Semieconomic Designs 10.6.5 An Economic Model of the x Control Chart 10.6.6 Other Work 10.7 Cuscore Charts 10.8 The Changepoint Model for Process Monitoring 10.9 Profile Monitoring

433 434 435 435 437 437 439 439 442 443 443 443 445 446 446 450 458 462 463 463 464 464 466 467 472 473 475 476

Contents 10.10 Control Charts in Health Care Monitoring and Public Health Surveillance 10.11 Overview of Other Procedures 10.11.1 Tool Wear 10.11.2 Control Charts Based on Other Sample Statistics 10.11.3 Fill Control Problems 10.11.4 Precontrol 10.11.5 Tolerance Interval Control Charts 10.11.6 Monitoring Processes with Censored Data 10.11.7 Nonparametric Control Charts

MULTIVARIATE SPC Chapter Overview and Learning Objectives 11.1 The Multivariate Quality-Control Problem 11.2 Description of Multivariate Data 11.2.1 The Multivariate Normal Distribution 11.2.2 The Sample Mean Vector and Covariance Matrix 11.3 The Hotelling T2 Control Chart 11.3.1 Subgrouped Data 11.3.2 Individual Observations 11.4 The Multivariate EWMA Control Chart 11.5 Regression Adjustment 11.6 Control Charts for Monitoring Variability 11.7 Latent Structure Methods ,^: 11.7.1 Principal Components > 11.7.2 Partial Least Squares

PROCESS ADJUSTMENT AND PROCESS MONITORING Chapter Overview and Learning Objectives 12.1 Process Monitoring and Process Regulation 12.2 Process Control by Feedback Adjustment 12.2.1 A Simple Adjustment Scheme: Integral Control 12.2.2 The Adjustment Chart 12.2.3 Variations of the Adjustment Chart

481 482 482 482 484 484 485

12.2.4 Other Types of Feedback Controllers 12.3 Combining SPC and EPC

XIII

539 540

PART 5 PROCESS DESIGN AND IMPROVEMENT WITH DESIGNED EXPERIMENTS

547

486 487

494 494 495 497 497 498 499 499 506 509 513 516 518 518 523

527

BASIC EXPERIMENTAL DESIGN FOR PROCESS IMPROVEMENT Chapter Overview and Learning Objectives 13.1 What is Experimental Design? 13.2 Examples of Designed Experiments In Process and Product Improvement 13.3 Guidelines for Designing Experiments 13.4 Factorial Experiments 13.4.1 An Example 13.4.2 Statistical Analysis 13.4.3 Residual Analysis 13.5 The 2k Factorial Design 13.5.1 The 2 2 Design 13.5.2 The 2k Design for k > 3 Factors 13.5.3 A Single Replicate of the 2k Design 13.5.4 Addition of Center Points to the 2k Design 13.5.5 Blocking and Confounding in the 2k Design 13.6 Fractional Replication of the 2k Design 13.6.1 The One-Half Fraction of the 2k Design 13.6.2 Smaller Fractions: The 2k~P Fractional Factorial Design

549 550 550 552 554 556 558 558 563 564 564 569 579 582 585 587 587 592

527 528 529 529 534 536

PROCESS OPTIMIZATION Chapter Overview and Learning Objectives 14.1 Response Surface Methods and Designs 14.1.1 The Method of Steepest Ascent

602 602 603 605

XIV

Contents

14.1.2 Analysis of a Second-Order Response Surface 14.2 Process Robustness Studies 14.2.1 Background 14.2.2 The Response Surface Approach to Process Robustness Studies 14.3 Evolutionary Operation

607 611 61 f

613 619

PART 6 ACCEPTANCE SAMPLING

629

BASIC ACCEPTANCE SAMPLING PROCEDURES 631 Chapter Overview and Learning Objectives 15.1 The Acceptance-Sampling Problem 15.1.1 Advantages and Disadvantages of Sampling 15.1.2 Types of Sampling Plans 15.1.3 Lot Formation 15.1.4 Random Sampling 15.1.5 Guidelines for Using Acceptance Sampling 15.2 Single-Sampling Plans for Attributes 15.2.1 Definition of a Single-Sampling Plan 15.2.2 The OC Curve " 15.2.3 Designing a Single-Sampling Plan with a Specified OC Curve f 15.2.4 Rectifying Inspection! 15.3 Double, Multiple, and Sequential Sampling 15.3.1 Double-Sampling Plans 15.3.2 Multiple-Sampling Plans 15.3.3 Sequential-Sampling Plans 15.4 Military Standard 105E (ANSI/ ASQCZ1.4, ISO 2859) 15.4.1 Description of the Standard 15.4.2 Procedure 15.4.3 Discussion 15.5 The Dodge-Romig Sampling Plans 15.5.1 AOQL Plans 15.5.2 LTPD Plans 15.5.3 Estimation of Process Average

631 632 633 634 635 635 636 637 637 637

642 643 646 647 651 652 655 655 657 661 663 664 667 667

ADDITIONAL SAMPLING PROCEDURES Chapter Overview and Learning Objectives 16.1 Acceptance Sampling by Variables 16.1.1 Advantages and Disadvantages of Variables Sampling 16.1.2 Types of Sampling Plans Available 16.1.3 Caution in the Use of Variables Sampling 16.2 Designing a Variables Sampling Plan with a Specified OC Curve 16.3 MILSTD414(ANSI/ASQCZ1.9) 16.3.1 General Description of the Standard 16.3.2 Use of the Tables 16.3.3 Discussion of MIL STD 414 and ANSI/ASQCZ1.9 16.4 Other Variables Sampling Procedures 16.4.1 Sampling by Variables to Give Assurance Regarding the Lot or Process Mean 16.4.2 Sequential Sampling by Variables 16.5 Chain Sampling 16.6 Continuous Sampling 16.6.1 CSP-1 16.6.2 Other Continuous-Sampling Plans 16.7 Skip-Lot Sampling Plans

APPENDIX

670 670 671 671 672 673 673 676 676 677 679 680

680 681 681 683 683 686

691

I.

Summary of Common Probability Distributions Often Used in Statistical Quality Control II. Cumulative Standard Normal Distribution III. Percentage Points of the %2 Distribution IV. Percentage Points of the t Distribution V. Percentage Points of the F Distribution VI. Factors for Constructing Variables Control Charts VII. Factors for Two-Sided Normal Tolerance Limits VIII. Factors for One-Sided Normal Tolerance Limits

BIBLIOGRAPHY ANSWERS TO SELECTED EXERCISES INDEX

692 693 695 696 697 702 703 704

705 721 729