W. L. Pearn Samuel Kotz
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Series on Quality, Reliability and Engineering Statistics Vol. 12 Encyclopedia and Handbook of Process Capability Indices R Comprehensive Exposition of Quality Control Neosures Encyclopedia and Handbook of
H Comprehensive Exposition of Qualitij Control Measures SERIES IN QUALITY, RELIABILITY & ENGINEERING STATISTICS
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EncLiclopedia and Handbooh of Process Capability Indices A Comprehensive Exposition of Qualify Control Neosures
W. L. Pearn National Chiao Tung University, Taiwan Samuel Kotz George Washington University, USA
^p World Scientific
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ENCYCLOPEDIA AND HANDBOOK OF PROCESS CAPABILITY INDICES A Comprehensive Exposition of Quality Control Measures Series on Quality, Reliability and Engineering Statistics, Vol. 12 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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About the Authors xi Introduction 1 1. The Cp Index 7 1.1 Process precision and the Cp index 7 1.2 Estimating and testing Cp based on a single sample 10 1.2.1 Estimation of CP 10 1.2.2 The r-th moment of Cp 11 1.2.3 Statistical properties of the estimated CP 11 1.2.4 Confidence interval for Cp 12 1.2.5 Sample size determination for estimation of CP 14 1.2.6 Hypothesis testing with CP 15 1.3 Estimating and testing C based on multiple samples 15 1.3.1 Estimation of CP and its properties 15 1.3.2 Lower confidence bound on CP 17 1.3.3 Hypothesis testing with Cp _ 18 1.4 Estimating and testing Cp based on (X , R) control chart samples 20 1.4.1 Estimation of Cp based on (X , R) samples 21 1.4.2 Hypothesis testing for Cp based on (X , R) samples 23 1.5 Estimating and testing Cp based on ( X , S) control chart samples 25 1.5.1 Estimation of Cp based on ( X , S) samples 25 1.5.2 Hypotheses testing for Cp based on (X , S) samples 27 1.6 A Bayesian approach to assessment of CP 28
2. The Ca Index 31 2.1 Process accuracy and the Ca index 31 2.2 Estimating and testing C„ based on a single sample 33 2.2.1 The first two moments of Ca 34 2.2.2 Confidence interval on C. 35 vi Encyclopedia and Handbook of Process Capability Indices
2.3 Estimating and testing Ca based on multiple samples 35 2.4 Bayesian-like estimator of Ca 38
3. The Cpk Index 40 3.1 Process capability and the Cpk index 40 3.2 Estimating and testing Cpk based on a single sample 45 3.2.1 The r-th moment of Cpk 46 3.2.2 Distributional properties of Cpk 47 3.2.3 Confidence intervals for Cpk 50 3.2.4 Hypothesis testing with Cpk 55 3.3 Estimating and testing Cpk based on multiple samples 58 3.4 The Bayesian approach to Cpk 61 3.5 The Bayesian-like estimator of Cpk 64
4. The Cpm Index 67 4.1 Process capability and the Cpm index 67 4.2 Estimating and testing Cpm based on one single sample 73 4.2.1 Estimation and distributional properties of estimators 73 4.2.2 Confidence intervals for Cpm 78 4.2.3 Sample size determination for Cpm 83 4.2.4 Hypothesis testing procedure (using Cpm) 86 4.3 Estimating and testing Cpm based on multiple samples 87 4.4 The Bayesian approach to Cpm 90
5. The Loss Indices 95 5.1 Process loss and the Le index 95 5.2 Estimation of Lpe, Lot, and Le 98 5.2.1 Estimating the process relative inconsistency loss, Lpe 98 5.2.2 Estimation process relative the off-target loss, Lot 101 5.2.3 Estimation of process expected relative loss, Le 103 5.3 Upper confidence bounds of Lpe, Lot, and Lc 105 5.3.1 An upper confidence bound on Lpe 105 5.3.2 An upper confidence bound of Lot 106 5.3.3 An upper confidence bound of Le 106 5.4 Testing process capability based on the process loss 107
6. The Cpmk Index 110 6.1 Process capability and the Cpmk index 110 6.2 Estimating and testing Cpmk based on a single sample 117 6.2.1 Estimation and the distribution of the estimated Cpmk 117 6.2.2 Confidence intervals on Cpmk 123 6.2.3 Hypothesis testing with Cpmk 124 Contents vn
6.3 Estimating and testing Cpmk based on multiple samples 127 6.4 Bayesian-like estimation of Cpmk 130
7. The Spk Index 134 7.1 Process capability and the Spk index 134 7.2 Estimating and testing Spk based on a single sample 136 7.2.1 Estimation of Spk 136 7.2.2 Confidence intervals for Sfk 142 7.3 Hypothesis testing with Spk 143
8. The CPU/CPL Index 146 8.1 Process capability and the index CPU/CPL 146 8.2 Estimating and testing CPU/CPL: Based on single sample 148 8.2.1 Estimations of CPU and CPL 148 8.2.2 r-th moment 149 8.2.3 Distribution 150 8.2.4 Testing hypothesis with CPU and CPL 151 8.2.5 Lower confidence bound for CPU / CPL 153 8.3 Estimating and Testing CPU/CPL: Based on multiple samples 154 8.3.1 Estimations of CPUand CPL based on multiple samples 154 8.3.2 Testing CPU and CPL based on multiple samples 157 8.4 Estimating and Testing CPU/CPL: Bayesian approach with single sample 158 8.4.1 Bayesian approach to the assessment of CPU and CPL based on single sample 158 8.4.2 A Bayesian approach to assessment with CPU and CPL based on multiple samples 160
9. Multi-Process Performance Analysis Chart (MPPAC) 162 9.1 Introduction 162 9.2 The modified Cpk MPPAC 165 9.3 The Cpm MPPAC 169 9.4 The Spk MPPAC 172 9.5 The Le MPPAC 176
10. PCIs with Asymmetric Specification Limits 181 10.1 Introduction 181 10.2 The Cpk index for asymmetric tolerances 182 10.3 The Cpm index for asymmetric tolerances 195 10.4 The Cpn index for asymmetric tolerances 202 10.5 The Cpmk index for asymmetric tolerances 205 10.6 The loss index for asymmetric tolerances 210 viii Encyclopedia and Handbook of Process Capability Indices
11. Supplier Selection Based on PCIs 218 11.1 Introduction 218 11.2 Tseng and Wu's MLR selection rule based on Cp 219 11.3 Chou's approximate selection rule based on Cpu and Cpl 222 11.4 Huang and Lee's approximate selection rule based on Cpm 224
12. Acceptance Sampling Plans Based on PCIs 231 12.1 Introduction 231 12.2 Acceptance sampling plans based on Cpk 237 12.3 Acceptance sampling plans based on Cpm 240 12.4 Acceptance sampling plans based on Cpmk 243 12.5 Acceptance sampling plans based on Cpuand Cpl 246
13. Process Capability Measures in Presence of Gauge Measurement Errors 249 13.1 Introduction 249 13.2 Estimating and testing Cp in presence of gauge measurement errors 252 13.3 Estimating and testing Cpk in presence of gauge measurement errors 257 13.4 Estimating and testing Cpm in presence of gauge measurement errors 263 13.5 Estimating and testing Cpmk in presence of gauge measurement errors 269 13.6 Estimating and testing Cp„and Cpl in presence of gauge measurement errors 272
14. Process Capability Assessment with Tool Wear 279 14.1 Introduction 279 14.2 A review of various approaches 281
15. Process Capability Assessment for Non-normal Processes 293 15.1 Introduction 293 15.2 A brief review of various approaches 297 15.2.1 Probability plotting approach 297 15.2.2 Clements' approach 299 15.2.3 Box-Cox power transformation approach 301 15.2.4 Johnson transformation approach 302 15.2.5 Other quantile transform approaches 303 15.2.6 Distribution-free tolerance intervals approach 304 15.2.7 Flexible index Cjkp 305 15.2.8 The Wright's Cs index 307 Contents IX
15.2.9 A superstructure capability indices CNp(u, v) 309 15.2.10 The Cpc index 313 15.2.11 The (general) Weighted Variance (WV) method 315 15.2.12 The Weighted Standard Deviation (WSD) method 320
16. Multivariate Process Capability Indices 326 16.1 Introduction 326 16.2 Multivariate PCIs 328 16.3 Concluding remarks 342
Bibliography 345 Index 375
About the Authors
W. L. Pearn is a Professor of Operations Research and Quality Assurance at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan. He received his Ph.D. degree in operations research from University of Maryland, College Park, MD, U.S.A. He was a quality research scientist at Bell Laboratories before joining National Chiao Tung University. He is currently Editor-in-Chief of the Quality Technology and Quantitative Management journal, an author and co-author of over 120 refereed journal papers. Pearn's awards include Research Assign Time Award, California State University, Performance Award, Bell Laboratories, Distinguished Research Award, National Science Council, Thomas L. Saaty Prize, and Distinguished Research Award, National Chiao Tung University. Samuel Kotz is an Emeritus Professor of Statistics at University of Maryland, College Park, MD, U.S.A. He received his Ph.D. degree from Cornell University. He is currently a senior research scholar at the School of Engineering and Applied Sciences, George Washington University, Washington D.C., U.S.A. He was the senior Editor-in-Chief of the 13-volume Encyclopedia of Statistical Sciences, (1982-2001), an author and co-author of over 200 research papers and 14 books, in particular, a Compendium on Statistical Distributions. He holds honorary Doctorates from Harbin Institute of Technology, China, University of Athens, Greece, and Bowling State University, U.S.A., and is a recipient of the Jacob Wolfowitz prize. He is a member of the Washington Academy of Sciences, Fellow of the Royal Statistical Society, Fellow of the American Statistical Association, Fellow of the Institute of Mathematical Statistics, and an elected member of the International Statistical Institute.
XI The main difficult but necessary aim is to express the desired optimal state of affairs in the national economy by a single indicator. A. N. Kolmogorov (1960)
Introduction
Since the eighties of the 20-th century, the use of capability indices to assess process acceptability has become popular and widespread. Consequently, when compiling this Encyclopedia and Handbook of Process Capability Indices, the authors have consulted a great number of publications (both theoretical and applied) in numerous fields. The first eight chapters contain an augmented presentation of the "standard" material, for the most part available in recent books and review papers, including those written by the authors. The second part (Chapters 9-16), mainly presents novel results (by the first author) available only in his published papers and technical reports. Intense international competition is now compelling corporations and firms to manufacture "defect-free" (zero defect) products. It is thus evident that quality serves as an important factor in the success of any business or engineering enterprise. The quality of a product is judged by a number of factors: performance, reliability, durability, etc. and to some extent the reputation of the manufacturer. The best way to achieve superior quality is to follow an effective quality program (where preventative activities are in focus).
Advertising billboards scream on highways and airports of quality product and logos on the sides of enormous vans promise the best quality service. Quality talk is everywhere! However, any
1 2 Encyclopedia and Handbook of Process Capability Indices overused superlative ultimately ceases to be an effective description of the object under consideration. This also applies to the word "quality". A producer of goods will always attempt to persuade potential buyers that his (her) product is the best, being the leader in the field. This stampede for quality is at present getting out of hand and beginning to cause a backlash. It creates too many standards with the noble but unrealistic aim of assuring a safer world where every facet of life is covered by standards. The most popular of all the standards is currently ISO 9000. As pointed out by the CEO of an accredited specification body: "Many companies are keen to gain ISO 9000 not because they are eager to change their working practices but because other organizations seem to have the "seal of approval" on the reception wall and they want to be part of it". Quality overkill may soon become a hindrance to the progress of industrial and business activities and could have serious cost and resource implications for business. Nevertheless, the widespread (and sometimes uncritical) use of Process Capability Indices (which are the subject matter of this Handbook) have led almost inadvertently to substantial improvements in quality, but at the same time, have been occasionally the cause of many unjust decisions, which could have been avoided by better knowledge of their properties. In particular, quantifying the "capability" of a manufacturing process is an important initial step in any quality improvement program. Capability is usually defined in dictionaries as: "the ability to carry out a task, to achieve an objective." This activity, as a rule, usually involves an element of chance, since the task may not be achievable every time but we may be able to estimate what proportion of the time it can be achieved. Statistical methodology is, therefore, essential to provide ways of measuring and modeling such situations. So far, the focus has been on examining the capability of a manufacturing process but the concept clearly lends itself to wider applications, which have not as yet been explored. In a nutshell, we are dealing with a Introduction 3 manufacturing process where a particular variable X , say, is of interest and importance. Our objective is to manufacture in such a manner that for every item produced the measured value of X will be X = T (where T is the target value). The actual reality is, however, that X will turn out to be a "random" variable. It is often assumed without sufficient justification that X ~ N(fi, a) . Clearly, the ability of the process to produce close to target will depend on 1) the magnitude of a and 2) the relation between \i and T . Ideally, we should have \i — T . For a normal distribution, the range of 6a around fi contains all but 0.27% of the population. This value, called the capability range or simply capability, can be used to give a general indication of the precision of the process1. Customers or manufacturers may wish to define a required level for the product values x . This may take the form of a specification centered about the target, T ± t or the form of lower and upper specification limits ( LSL , USL ). Any item outside these limits will be regarded as scrap or as needing reworking. The use of specification limits allows for the possibility that the midpoint m (between LSL and USL ) may not be the target. Situations may occur when only one limit is needed. [For example in the case of a continuous chemical process, USL may define the limiting value for an undesirable impurity. Over the last 20 years, a number of measures have been devised to compare the requirements of specification with the capability of the manufacturing process. These measures take the form of indices constructed to take the value 1 for a properly defined balance between process capability and specification limits. These indices are standardized and unitless and directly related to the customer's specification. In this sense, they provide a common
The above property of the normal distribution resulted in a six-sigma program of industrial performance conceptualized by Motorola as a quality goal in the mid 1980's. It was not long before many U.S. giants - XEROX, Boeing, GE, Kodak were following Motorola's lead. It is claimed that using the six-sigma program (in spite of its theoretically unsound foundations) resulted in savings for Allied Signals of some $340 millions in 1996. General Electric announced savings due to the use of this program in 1998 in the amount of $1.2 billions.•