Table Of ContentSTATISTICAL
QUALITY
CONTROL
A Loss Minimization Approach
SERIES ON APPLIED MATHEMATICS
Editor-in-Chief: Frank Hwang
Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum
Vol. 1 International Conference on Scientific Computation
eds. T. Chan and Z.-C. Shi
Vol. 2 Network Optimization Problems — Algorithms, Applications and
Complexity
eds. D.-Z. Du and P. M. Pandalos
Vol. 3 Combinatorial Group Testing and Its Applications
by D.-Z. Du and F. K. Hwang
Vol. 4 Computation of Differential Equations and Dynamical Systems
eds. K. Feng and Z.-C. Shi
Vol. 5 Numerical Mathematics
eds. Z.-C. Shi and T. Ushijima
Vol. 6 Machine Proofs in Geometry
by S.-C. Chou, X.-S. Gao and J.-Z. Zhang
Vol. 7 The Splitting Extrapolation Method
by C. B. Liem, T. Lu and T. M. Shih
Vol. 8 Quaternary Codes
by Z.-X. Wan
Vol. 9 Finite Element Methods for Integrodifferential Equations
by C. M. Chen and T. M. Shih
Vol. 10 Statistical Quality Control — A Loss Minimization Approach
by D. Trietsch
Vol. 11 The Mathematical Theory of Nonblocking Switching Networks
by F. K. Hwang
Series on
Applied Mathematics
Volume 10
STATISTICAL
QUALITY
CONTROL
A Loss Minimization Approach
Dan Trietsch
MS1S Department
University of Auckland
New Zealand
World Scientific
SiK$W!^f&\MM%ty'London 'Hong Kong
Published by
World Scientific Publishing Co. Pte Ltd.
P O Box 128, Farrer Road, Singapore 912805
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Library of Congress Cataloging-in-Publication Data
Trietsch, Dan
Statistical quality control : a loss minimization approach / by
Dan Trietsch.
p. cm. — (Series on applied mathematics; v. 10)
Includes bibliographical references and index
ISBN 9810230311
1 Quality control - Statistical methods I Title. II Series
TS156.T75 1998
658.5'62-dc21 98-10545
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Copyright © 1999 by World Scientific Publishing Co Pte Ltd.
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Printed in Singapore by Uto-Print
Dedicated to my parents
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Preface
This book originated as part of a proposed textbook in Total Quality (TQ). The main
theme was that TQ should be studied as a system. By now it's almost universally agreed that
TQ should be applied as widely as possible, i.e., to the whole system, but looking at TQ
itself as a system is less prevalent. My own premise is that TQ is a system whose aim is to
maximize quality by continuously increasing it. Therefore, once we define quality, we should
be able to judge what tools and philosophies belong to the TQ system. The test is simple:
does it support the system aim (increase quality)?
But unless we define quality broadly, maximizing it may cause suboptimization. Some
definitions of quality are not wide enough for this purpose. For example, defining quality
as conformance to specifications implies that conformance should be sought without regard
to cost. Further, it suggests that the design to which we try to conform is necessarily good.
Fitness for use, Juran's definition, implies both good design and good conformance, but
neglects the cost element. Deming — who inspired me more than any other quality teacher
-- wrote in his last book: "A product or service has quality if it helps someone and enjoys
a good and sustainable market." This includes fitness for use and economic cost. As such
it is less amenable to suboptimization. But it may still happen that third parties will suffer
as a result of a deal that is good for both seller and buyer (a classic example is pollution).
To capture the interests of third parties, known as externalities in economics, I adopted the
following definition: Total Quality is the pursuit of maximal net utility to society. Another
side of the same coin is: Total Quality is the continuous reduction and elimination of any
waste, including waste of opportunities. Thus quality is net utility to society — value, if you
will. Since intangibles also have utility, this two-sided definition takes into account the value
of leisure, culture, etc. Furthermore, it is practically impossible to measure societal utility
-- at least not without destroying it! - so we'll have to assume that societal utility is the sum
of the individual utilities of the members of society. And we'll try to increase societal utility
by methods that demonstrably increase that sum, rather than by transferring monies within
society -- a respectable endeavor, perhaps, or maybe not, but strongly outside our scope.
Having said that, we must recognize that within the sub-field of statistical quality
control (SQC) quality is measured by the degree to which design objectives are achieved.
vii
viii Statistical Quality Control
That is to say, quality is measured by conformance. But conformance no longer means
merely falling within preset tolerances. Rather, conformance is measured by the distance
between the design ideal and the actual performance. More than anyone else, Taguchi was
instrumental in making this point. The Taguchi loss function, often a quadratic function,
measures quality (of conformance) by the loss imparted to society by deviating from the ideal
design. One objective of SQC, then, is to minimize this loss. But in addition to the loss to
society, we have to take into account the cost of production, which is also part of societal
net utility. While Taguchi definitely recognized that production costs and conformance loss
are equally important, he chose to define quality by loss to society alone. The definition
adopted here is wider, and considers both elements as part of quality. Often, however, the
cost of production is not a direct function of the conformance, and in such cases decreasing
Taguchi's loss function is operationally equivalent to increasing quality.
While many texts on quality today espouse the Taguchi loss function, they rarely
draw the conclusions from this endorsement. Furthermore, many consider reducing this loss
to be a higher priority than reducing production costs. These deficiencies motivated me to
take a closer look at established SQC methods. It turns out that using the Taguchi loss
function implies some changes in the way we practice SQC, and sheds new light on questions
relating to SQC. One example is the correct relationship between statistical capability and
tolerance setting. An economical tolerance setting technique suggested by Taguchi highlights
this connection. Another example is a myth according to which Taguchi's loss function
approach calls for perfect centering. We'll discuss process adjustment issues and show that
the myth is false. While doing so we'll also extend Taguchi's tolerance setting technique to
processes with drifts that cannot be designed away economically. Deming devoted a lot of
energy to eradicating improper adjustment methods that cause tampering and increase losses
due to inferior centering. But he did not spell out how to adjust properly. I attempt to do so
here, and it's one of the issues on which I had to go against Deming's teachings, at least as
they are usually presented.
The book covers other issues that are usually neglected in the SQC literature, and
come to light when we look at SQC as a (sub)system - rather than a mere collection of
techniques. Minimizing loss, or maximizing quality, is the driver of the system, but it has
many interactions that need to be discussed explicitly, e.g., measurements, determining how
many points to sample to obtain reliable control charts, and more. We also examine
Deming's kp rule - a clear economic loss reduction application - and discuss appropriate
ways to inspect incoming lots when the rule is not applicable. Also, where conformance is
measured by a continuous loss function, the kp rule has to be changed: we outline how.
Preface ix
Taguchi's most famous contribution is the use of Design of Experiments (DOE) to
reduce the loss function. Unfortunately, Taguchi's techniques leave much to be desired, and
this created much controversy. Western statisticians, led by George Box, have shown that
there are better ways to achieve his ends. In particular, his use of external arrays and signal-
to-noise ratios have been criticized. Dorian Shainin developed yet other alternatives.
Personally, I acknowledge Taguchi as a leader in terms of where to go, but not necessarily
how to get there. DOE itself, although fascinating, is outside our scope. So are many other
engineering techniques that are indispensable today. Chief among them are poka-yoke (error-
proofing) and setup reduction techniques devised by Shigeo Shingo. All these vital methods
are concerned more with breakthrough than with control. And they can easily fill another
book of this size.
The book is intended for both practitioners and theoreticians, and it can also be used
for teaching. To gain full benefit, however, readers should have at least an introductory level
knowledge of probability and statistics. The aim to serve practitioners led me to concentrate
on basic techniques, and to organize the coverage so that some mathematical details can be
skipped without loss of continuity. The service to theoreticians, therefore, is not by
reviewing complex models, but rather by coverage of the theoretical underpinnings of the
basic ones in more detail than the norm. The service to both communities is by addressing
the issues in a systemic way, stressing connections and interactions, and covering key
subjects that are missing in practically all other SQC books.
Nine chapters and seven supplements provide the coverage. In general, supplements
are attached to chapters, and they either discuss optional material or deal with mathematical
details that are not necessary for practitioners. Some smaller mathematical expositions that
are not vital for practitioners are given in unnumbered boxes ("Technical Notes"), which
may be skipped without loss of continuity. In contrast, numbered boxes are used occasionally
to facilitate cross-reference.
The first chapter introduces classical control charts, Shewhart charts, which generally
follow Shewhart's concepts. Since this text concentrates on statistical control methods, and
not on graphic improvement tools and so on, control charts are our main focus. Chapter 2
discusses measurements. Supplement 2 elaborates on some technical details associated with
calibration and efficient measurement. Chapter 3 introduces loss functions, and covers
Taguchi's economical tolerance setting method. It also shows how to change the kp rule
where the quadratic loss function applies. Supplement 3 discusses asymmetrical loss
functions. Chapter 4 presents economical adjustment methods that avoid tampering. It also
extends Taguchi's tolerance setting technique to processes with drift. Chapter 5 is the first
x Statistical Quality Control
in a block of four that present various aspects of classic control charts. It deals with control
charts for attributes. Motivated by Deming's vehement objections to doing so, Supplement
5 compares control charts with hypothesis testing, and concludes that the only major
difference is that with control charts we should not claim we know the statistical risks
exactly. Chapter 6 is devoted to charting continuous variables. Three supplements to this
chapter deal with mathematical details: Supplement 6.1 compares the efficiency of various
dispersion statistics, and shows that a recommendation made by Shewhart should be
reversed; Supplement 6.2 shows how control chart constants are derived; Supplement 6.3
discusses optimal subgroup size when we want to detect a process deviation of a given size
by ±3cr control limits. Chapter 7 deals with pattern tests that apply to control charts. Some
results in this chapter correct prevalent errors in the literature. Supplement 7 briefly
discusses SQC extensions to processes with statistical dependence. Chapter 8 introduces a
new graphic tool, diffidence charts, that can help determine how many items we should
sample to establish a reliable control chart (it's more than many authorities claim). Another
classic SQC subject — inspection — is presented in Chapter 9.
A traditional coverage of SQC could be limited to Chapters 1, 5, 6, 7, and 9. I hope
readers will find that even these chapters are not completely conventional. Chapters 2, 3, and
4 provide necessary parts that are not covered traditionally. As for Chapter 8, I believe that
studying diffidence charts is highly conducive to understanding variation. As Deming said,
understanding variation is imperative. Thus, even if diffidence charts will not become part
of the tool-kit of every SQC practitioner, the chapter is still useful in a practical sense.
Acknowledgement
While I'm responsible for the results, I did not work in a vacuum. Many helped me
generously with their time, advice, and more. Some reviewers of the material that found its
way into this book were anonymous, but their impact is considerable. In contrast, quite a few
of those I know by name are also my relatives or good friends. As such, they'll understand
how difficult it is to draw the line after naming a finite number of people. Thank you all,
and please forgive me for taking the easy way out.
