practical steps to robust design

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KZast astern E conom y

^Tdition

T aguchi M ethods E xplained PRACTICAL STEPS TO ROBUST DESIGN

R s. 175

TAGUCHI METHODS EXPLAINED Practical Steps to Robust Design

Tapan P. Bagchi

(Ph.D., Toronto)

This w ell-organized, com pact volume introduces the reader to Taguchi Methods — a revolutionary approach innovated in Japan to engineer quality and performance into new products and manufacturing processes. It explains on-the-job application of Taguchi Methods to make products and processes perform consistently on target, hence making them insensitive to factors that are difficult to control. Designed for practising engineering managers with responsibility in process performance, quality control, and R&D, and for students of engineering and process design, the text provides all the essential tools for planning and conducting prototype development and tests which guarantee improved final field performance of products and manufacturing processes. Replete with examples, exercises, and actual case studies, the book shows how electronic circuit devicest mechanical fabrication methods, and chemical and metallurgical processes can be made robust and stable to consistently provide on-target performance, in spite of the presence of 'noise' — raw material quality variations, environmental changes, voltage fluctuations, operator's inconsistency, and so on —, all of which are external factors that cannot be economically controlled. The book also shows the reader how to plan reliable and efficient tests with physical prototypes and computer models to evaluate products and processes during development to improve them. Finally, it explains state-ofart methods to make even complex systems robust, where design variables “interact”, making conventional design optimization methods difficult to apply. (continued on back flap)

TAGUCHI METHODS EXPLAINED Practical Steps to Robust Design

. *

TAPAN P. BAGCHI Professor, Industrial and Management Engineering Indian Institute o f Technology, Kanpur

Prentice^Hall of India

u

New Delhi-110001

1993

a

OTnl

Rs. 175.00

TAGUCHI METHODS EXPLAINED : Practical Steps to Robust Design by Tapan P/Bagchi •*

*

..

PRENTICE-HALL INTERNATIONAL, INC., Englewood Cliffs. PRENTICE-HALL INTERNATIONAL (UK) LIMITED, London. PRENTICE-HALL OF AUSTRALIA PTY. LIMITED, Sydney. PRENTICE-HALL CANADA, INC., Toronto. PRENTICE-HALL HISPANOAMERICANA, S.A., Mexico. PRENTICE-HALL OF JAPAN, INC., Tokyo. SIMON & SCHUSTER ASIA PTE. LTD., Singapore. EDITORA PRENTICE-HALL DO BRASIL, LTDA., Rio de Janeiro.

© 1993 by Prentice-Hall of India Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publishers.

ISBN-0-87692-808-4

The export rights of this book are vested solely with the publisher.

Published by Prentice-Hall of India Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Bhuvnesh Seth at Rajkamal Electric Press, B-35/9, G.T. Karnal Road Industrial Area, Delhi-110033.

To

the Fond Memory of

Bhalokaku

Contents Preface 1. What Are Taguchi Methods?

ix 1 -1 7

1.1

The Road to Quality Starts at Design

1

1.2

Achieving Quality—Taguchi* s Seven Points

2

1.3

Optimized Design Reduces R&D, Production, and Lifetime Cost

3

1.4

Taguchi’s Definition of Quality

6

1.5

What Causes Performance to Vary?

9

1.6

Prevention by Quality Design

11

1.7

Steps in Designing Performance into a Product

12

1.8

Functional Design: The Traditional Focus

13

1.9

Parametric Design: The Engineering of Quality

14

1.10 Statistical Experiments Discover the Best Design Reliably and Economically E x e r c is e s

2- Handling Uncertainty

16 17

18-40

2.1

The Mystique of Probability

18

2.2

The Idea of a Random Variable

21

2.3

Some Useful Formulas

25

2.4

‘Hypothesis Testing’: A Scientific Method to Validate or Refute Speculations

27

2.5

Comparing Two Population Means Using Observed Data

30

2.6

Cause-Effect Models and Regression

32

2.7

Evaluating a Suspected “Cause” Factor

33

2.8

The F-Statistic

37

2.9

An Alternative Approach to Finding F: The Mean Sum of Squares

39

E xercises

40

3L Design of Experiments 3-1

Testing Factors One-at-a-Time is Unscientific

4 1 -6 0 41

vi

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

3.2

The One-Factor Designed Experiment

44

3.3

ANOVA Helps Compare Variabilities

49

3.4

The F-Test Tells If Factor Effects are Statistically Significant

53

3.5 3.6

Formulas for Sum of Squares and the F-Test Summary

58

E x e r c ise s

59

55

The Foundation of Taguchi Methods: The Additive Cause-Effect Model 4.1

What is Additivity?

61

4.2

Why Achieving Additivity is So Important?

62

4.3

The Verification of Additivity

65

4.4

The Response Table: A Tool That Helps Find Main Effects Quickly

65

4.5

Graphic Evaluation of Main Effects

68

4.6

Optimization of Response Level and Variability

70

4.7

Orthogonal Arrays vs. Classical Statistical Experiments

72

4.8

Summary

77

E x e r c ises

78

Optimization Using Signal-to-Noise Ratios 5.1 5.2

6.

6 1 -7 8

7 9 -8 9

Selecting Factors for Taguchi Experiments

79

To Seek Robustness One Should Measure Performance by SIN Ratios

81

5.3

SIN Ratio in Optimization— An Example

84

5.4

Not All Performance Characteristics Display Additivity

85

5.5

The OA as the Experiment Matrix

86

5.6

The Axiomatic Approach to Design

87

5.7

Summary

88

E x e r c ise s

88

Use of Orthogonal A rrays

90-106

6.1

What are Orthogonal Arrays?

90

6.2

OAs are Fractional Factorial Designs

92

6.3

Not All Factors Affect Performance the Same Way

94

6.4

Identifying Control and Noise Factors: The Ishikawa Diagram

95

At What Levels Should One Study Each Factor?

97

6.5

CONTENTS

Vii

6.7

Reaching the Optimized Design Testing for Additivity

99 100

6.8

The Optimization Strategy

100

6.9

Taguchi’s Two Steps to On-Target Performance with Minimum Variability

103

6.6

6.10 Summary E x e x c ise s

104 104

7 . C am S tv iy 1: Process Optimization — Optical Filter

1

7.1

H r ftocess far Manufacturing Optical Filters

107

12

Test Settings of Control Parameters and the OA

108

73

rVrformanre Measurements and the S/N Ratio

110

7.4

Minimizing logjo (s2), the Variability of Thickness

111

7.5

The Confirmation Experiment

111

7 j6

Adjusting Mean Crystal Thickness to Target

112

S c k d k g Orthogonal Arrays and Linear Graphs

114-122

8.1

Sizing up tbe Design Optimization Problem

114

8.2

Linear Graphs and Interactions

116

83

Modification of Standard Linear Graphs

118

8.4

Estimation of Factor Interactions Using OAs

119

8.5

Summary *

122

E x er c ise

122

9. Case Stady 2: Product Optimization — Passive Network F ile r Design

123-139

9.1

The Passive Network Filter

123

9.2

Formal Statement of the Design Problem

125

9.3

The Robust Design Formulation of the Problem

125

9.4

Data Analysis and Estimation of Effects

129

9_5

Effects of the Design Parameters

131

9.6

Discussion on Results

135

9.7

Filter Design Optimization by Advanced Methods

136

1#. A Direct Method to Achieve Robust Design

140-161

10.1 Re-Statement of the Multiple Objective Design Optimization Problem

140

10.2 Target Performance Requirements as Explicit Constraints

141

VIII

TAGUCHI METHODS EXPLAINED — PRACTICAL STEPS TO ROBUST DESIGN

10.3 Constraints Present in the Filter Design Problem

142

10.4 Seeking Pareto-Optimal Designs

143

10.5 Monte Carlo Evaluation of S/N Ratios

144

10.6 Can We Use C (or R2) as the Independent DP instead of /?3?

146

10.7 Some Necessary Mathematical Tools

147

10.8 Developing a Multiple Regression Model

150

10.9 Rationale of the Constrained Robust Design Approach

153

10.10 Application of the Constrained Approach to Real Problems

155

10.11 Discussion of the Constrained Design Optimization Approach

159

11. Loss Functions and Manufacturing Tolerances

162-171

11.1 Loss to Society is More Than Defective Goods

162

11.2 Determining Manufacturing Tolerances

164

11.3 Loss Functions for Mass-Produced Items

170

11.4 Summary

171

E x e r c ise s

171

12. Total Quality Management and Taguchi Methods

172-183

12.1 Why Total Quality Management?

172

12.2 What Really is Quality?

174

12.3 What is Control?

174

12.4 Quality Management Methods

174

12.5 The Business Impact of TQM

176

12.6 Control of Variability: Key to QA

177

12.7 How is Statistics Helpful?

178

12.8 Practical Details of Planning a Taguchi Project

179

Appendix A: Standard Normal, t, Chi-square, and F-Tables

185-190

Appendix B: Selected Orthogonal Arrays and Their Linear Graphs

191-196

Glossary

197-202

References

203-204

Index

205-209

Preface Taguchi methods are the most recent additions to the toolkit of design, process, and manufacturing engineers, and Quality Assurance (QA) experts. In contrast to Statistical Process Control (SPC), which attempts to control the factors that adversely affect the quality of production, Taguchi methods focus on design—the development of superior performance designs (of both products and manufacturing processes) to deliver quality. Taguchi methods lead to excellence in the selection and setting of product/ process design parameters and their tolerances. In the past decade, engineers have applied these methods in over 500 automotive, electronics, information technology, and process industries worldwide. These applications have reduced cracks in castings, increased the life of drill bits, produced VLSI with fewer defects, speeded up the response time of UNIX V, and even guided human resource management systems design. Taguchi methods systematically reveal the complex cause-effect relationships between design parameters and performance. These in turn lead to building quality performance into processes and products before actual production begins. Taguchi methods have rapidly attained prominence because wherever they have been applied, they have led to major reductions in product/process development lead time. They have also helped in rapidly improving the manufacturability of complex products and in the deployment of engineering expertise within an enterprise. The First objective of Taguchi methods— which are empirical — is reducing the variability in quality. A key premise of Taguchi methods is that society incurs a loss any time a product whose performance is not on target gets shipped to a customer This loss is measurable by the loss function, a quantity dependent on the deviation of the product’s performance from its target performance. Loss functions are directly usable in determining manufacturing tolerance limits. Delivering a robust design is the second objective of Taguchi methods. Often there are factors present in the environment on which the user of a product has little or no control. The robust design procedure adjusts the design features of the product such that the performance of the product remains unaffected by these factors. For a process, the robust design procedure optimizes the process parameters such that the quality of the product that the process delivers, stays on target, and is unaffected by factors beyond control. Robust design minimizes variability (and thus the lifetime cost of the product), while retaining the performance of the product on target. Statistically designed experiments using orthogonal arrays and signal-to-noise {SIN) ratios constitute the core of the robust design procedure. This text provides the practising engineer an overview of the state-of-the-art in Taguchi methods— the methods for engineering superior and lasting performance into products and processes.

X

PREFACE

Chapters 1-3 introduce the reader to the basic ideas in the engineering of quality, and the needed tools in probability and statistics. Chapter 4 presents the additive cause-effect model, the foundation of the Taguchi methodology for design optimization. Chapter 5 defines the signal-to-noise ratio—the key performance metric that measures the robustness of a design. Chapter 6 describes the use of orthogonal arrays (OAs), the experimental framework in which empirical studies to determine the dependency of performance on design and environmental factors can be efficiently done. Chapter 7 illustrates the use of these methods in reducing the sensitivity of a manufacturing process to uncontrolled environmental factors. Chapter 8 provides the guidelines for the selection of appropriate orthogonal arrays (OAs) for real-life robust design problems. A case study in Chapter 9 shows how one optimizes a product design. Chapter 10 presents a constrained optimization approach which would be of assistance when the design parameter effects interact. Chapter 11 shows how Taguchi loss functions can be used in setting tolerances for manufacturing. Chapter 12 places Taguchi methods in the general framework of Total Quality Management (TQM) in an enterprise. Throughout the text, examples and exercises have been provided for enabling the reader to have a better grasp of the ideas presented. Besides, the fairly large number of References should stimulate the student to delve deeper into the subject. I am indebted to Jim Templeton, my doctoral guide and Professor—from him I had the privilege of imbibing much of my knowledge in applied probability. I am also grateful to Birendra Sahay and Manjit Kalra, whose enormous confidence in me led to the writing of this book. I wish to thank Mita Bagchi, my wife, and Damayanti Singh, Rajesh Bhaduri and Ranjan Bhaduri whose comments and suggestions have been of considerable assistance in the preparation of the manuscript. The financial assistance provided by the Continuing Education Centre, Indian Institute of Technology, Kanpur to partially compensate for the preparation of the manuscript is gratefully acknowledged. Finally, this book could not have been completed without the professionalism and dedication demonstrated by the Publishers, Prentice-Hall of India, both during the editorial and production stages. Any comments and suggestions for improving the contents would be warmly appreciated.

Tapan P. Bagchi

%

What Are Taguchi Methods? i

1.1

THE ROAD TO QUALITY STARTS AT DESIGN

Quality implies delivering products and services that meet customers’ standards and fulfill their needs and expectations. Quality has been traditionally assured by Statistical Process Control (SPC) — a collection of powerful statistical methods facilitating the production of quality goods by intelligently controlling the factors that affect a manufacturing process. SPC attempts to achieve quality by reacting to deviations in the quality of what the manufacturing plant has recently produced. In this chapter, however, we present an overview of a some­ what different approach for assuring quality — consisting essentially of certain specially designed experimental investigations. Collectively known as the Taguchi methods, these methods focus on improving the design of manufacturing processes and products. A designer applies Taguchi methods off-line — before production begins. When applied to process design, Taguchi methods can help improve process capability. These methods also reduce the sensitivity of the process to assignable causes, substantially reducing thereby the on-line SPC effort required to keep the quality of production on target. The significance of beginning Quality Assurance (QA) with an improved process or product design is not difficult to gauge. Experience suggests that nearly 80 per cent of the lifetime cost of a product becomes fixed once its design is complete. Recent studies suggest that a superior product design ranks among the foremost attributes of a successful enterprise [1]. The application of Taguchi methods leads to superior performance designs known as robust designs. Statistical experimentation and analysis methods have been known for over the past 60 years [2, 3]. However, the Japanese appear to have been the fir§t to use these methods formally in selecting the best settings of process/product design parameters [4]. In the West, the most notable user of Taguchi methods has been AT&T, U.S.A., whose product development efforts now incorporate parametric optimization [5]. The foundation of the Taguchi methods is based on two premises: L Society incurs a loss any time the performance of a product is not on target. Taguchi has argued that any deviation from target performance results in a loss to society. He has redefined the term ‘quality’ to be the losses a product imparts to society from the time it is shipped. *

2. Product and process design requires a systematic development, progressing stepwise through system design, parametric design, and finally, tolerance design. Taguchi methods provide an efficient, experimentation-based framework to achieve this. 1

2

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

The first premise suggests that whenever the performance of a product deviates from its target performance, society suffers a loss. Such a loss has two components: The manufacturer incurs a loss when he repairs or rectifies a returned or rejected product not measuring up to its target performance. The consumer incurs a loss in the form of inconvenience, monetary loss, or a hazardous consequence of using the product. The second premise forms the foundation of quality engineering, a discipline that aims at engineering not only the function, but also quality performance into products and processes. Taguchi’s original work circulated mainly within his native country, Japan, until the late ’70s, when some translations became available in other countries. The American Society for Quality Control published a review of Taguchi’s methods, especially of “off-line quality control”, in 1985 [6 ], Since then, many engineers outside Japan have also successfully applied these methods [7], The Taguchi philosophy professes that the task of assuring quality must begin with the engineering of quality — product and process design optimization for performance, quality, and cost. To be effective, it must be a team effort involving marketing, Research and Development (R&D), production, and engineering. Quality engineering must be completed before the product reaches its production stage. One can often take countermeasures during process and product design. Such coxmtcmeasAKes can eftecVw eVy assure that the product a manufacturing process delivers will be on target and that, it will continue to perform on target measures require weW-planned, systematic, and an essentially empirical investigation during process/product desigp and development. For tivvs, ressoa, TagucYtt. c&Wed this procedure “off-line” [8]; it precedes on-line Quality Control (QC) done during manufacturing, using control charts and other reactive methods (see Section 12.4).

1.2

ACHIEVING QUALITY — TAGUCHI’S SEVEN POINTS

Achieving superior performance calls for an attitude that must continuously search for incremental improvement. The Japanese call this kaizen. This trait is different from the commonly applied method of relying only on new technologies and innovations as the route to quality improvement. The following seven points highlight the distinguishing features of Taguchi’s approach (as different from the traditional approach) which is aimed at assuring quality: 1. Taguchi defined the term ‘quality’ as the deviation from on-target performance, which appears at first to be a paradox. According to him, the quality of a manufactured product is the total loss generated by that product to society from the time it is shipped. 2. In a competitive economy, Continuous Quality Improvement (CQI) and cost reduction are necessary for staying in business. 3. A CQI programme includes continuous reduction in the variation of product performance characteristic in their target values.

WHAT ARE TAGUCHI METHODS?

3

4. Customer’s loss attributable to product performance variation is often proportional to the square of the deviation of the performance characteristic from its target value. 5. The final quality and cost (R&D, manufacturing, and operating) of a manufactured product depend primarily on the engineering design of the product and its manufacturing process. 6.

Variation in product (or process’) performance can be reduced by exploiting the nonlinear effects of the product (or process) parameters on the performance characteristics. 7. Statistically planned experiments can efficiently and reliably identify the settings o f product and process param eters that reduce perform ance variation. One achieves kaizen by formally integrating design and R&D efforts with actual production in order to get the process right and continually improve it. A large number of design, process, and environmental factors are usually involved in such a task. Consequently, there is no effective way of doing kaizen except by the pervasive use of scientific methods. Statistically designed experiments, in particular, can generate highly valuable insights about the behaviour of a process or product, normally using only a surprisingly small number of experiments. The consequence of superior performance is the superior fit of the manufacturing process or product to its users’ requirements. Subsequently this reduces the product’s lifetime cost of use.

1.3

OPTIMIZED DESIGN REDUCES R&D, PRODUCTION, AND LIFETIME COST

Cost trade-offs in quality decisions are not new. This is how industry sometimes justifies its QA programmes. Most managers believe that quality requires action when quality-related operating costs — which belong to one of the three following categories — go out of line: Failure costs result from inferior quality products in the form of scrap, rejects, repair, etc. Failure costs are also involved in the returns from customers, loss of goodwill, or a plant failure causing loss of production, property, or life at the customer’s site. Appraisal costs are incurred while inspecting, appraising, and evaluating the quality of the products one manufactures, or the materials, parts, and supplies one receives. Prevention costs are incurred when one attempts to prevent quality problems from occurring by (a) engaging process control, optimization experiments and studies; (b) training operators on correct procedures; and (c) conducting R&D to produce close-to-target products. A manufacturer often trades off one of these costs for another. Some manufacturers choose not to invest on prevention, engaging instead a team of technicians to do warranty service. When there is a monopoly, sometimes the warranty service is also cut — regardless of its effect on customers.

4

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

A large sum of money spent on appraisal can help screen out defective products, preventing them from getting to customers. This is inspection-based QA. As of today, most Very Large Scale Integration (VLSI) chips have to be produced this way. It should be clear, however, that QA based on appraisal is reactive and not preventive — it takes action after production. If resources can be directed to prevention instead, one increases the likelihood of preventing defects and quality problems from developing. With preventive action, prevention costs for an enterprise may rise, but failure costs are often greatly reduced [9]. Reduction of defective production directly cuts down in-house scraps and rejects. This also reduces returns from customers and their dissatisfaction with the product. Also, the producer projects a quality image, which often gives a marketing edge. It may be possible, of course, to go overboard with quality if we disregard real requirements. The ISO 9000 Standards document [10] as well as QFD [34] also emphasize the value of establishing the customer’s real needs first. Business economists suggest that the target for quality should be set at a level at which the profit contribution of the product is most favourable (Fig. 1.1).

*

c o

+ >

C o n trib u tio n

3 JD C

o

o

M a rke t value

CO

o

O Q> 3 O >

M a n u fa c tu rin g co st C o n trib u tio n

c

55 o 0) o t _

c

/

\ In c re a s in g

Fig. 1.1

p re c is io n —►

N

\

Contribution and precision of design

In his writings Taguchi has stated that delivering a high quality product at low cost involves engineering, economics, use of statistical methods, and an appropriate management approach emphasizing continuous improvement. To this end Taguchi has proposed a powerful preventive procedure that he calls robust design. This procedure optimizes product and process designs such that the final performance is on target and it has minimum variability about this target. One major outcome of off-target performance, be it with ill-fitting shoes, defective keyboards, or a low yielding chemical process, is the increase in the lifetime cost of the product or process (see Table 1.1). We may classify this total cost as the cost that the product/process imposes on society — the producer, the consumer, and others who may not even be its direct users — as follows:

WHAT ARE TAGUCHI METHODS?

5

O perating cost: The costs of energy, consumables, maintenance, environ­ mental control, inventory of spare parts, special skills needed to use the product, etc. constitute the product’s operating cost. Generally, with robust design this cost can be greatly reduced. M anufacturing cost: Jigs, special machinery, raw and semi-finished materials, skilled and unskilled labour, QC, scrap, rework, etc. constitute the manufacturing cost. Again, with robust design, the requirements of special skills, raw materials, special equipment, controlled environment, on-line QC effort, etc. can be substantially reduced. R&D cost: Engineering and laboratory resources, expert know-how, patents, technical collaborations, prototype development, field trials, etc. constitute the R&D cost of the product. R&D aims at producing drawings, specifications, and all other information about technology, machinery, skills, materials, etc. needed to manufacture products that meet customer requirements. The goal here is to develop, document and deliver the capability for producing a product with the optimum performance — at lowest manufacturing and operating cost. Robust design can play a key role in this effort too. T A B L E X.l I N I T I A L P R I C E vs. L I F E T I M E C O S T O F P R O D U C T S I N C O M M O N U S E *

Product

Air Conditioners Dishwasher Electric Dryer Gas Dryer Freezer, 15 cu. ft. Electric Range Gas Range Frost-Free Refrigerator B&W Television Colour Television Electric Typewriter Vacuum Cleaner Washing Machine Industrial Process Equipment

Initial Price

Lifetime Cost

($)

($)

200 245 182 207 165 175 180 230 175 540 163 89 235 75,000

665 617 670 370 793 766 330 791 505 1086 395 171 852 432,182/yr**

* F.M. Gryna (1977): Quality Costs — User vs. Manufacturer, Quality Progress, June, pp. 10-13. ** Includes repairs (part, material and labour), contract labour, defective product produced and lost production.

Generally, the producer incurs the R&D and manufacturing costs and then passes these on to the consumer. In addition, the consumer incurs the operating cost as he uses the product, especially when performance deviates from target. The knowledge emerging from Taguchi’s work affirms that high quality means lower operating cost and vice versa. Loss functions provide a means to quantify this statement.

6

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

The robust design method — the key QA procedure put forth by Taguchi — is a systematic method for keeping the producer’s costs low while delivering the highest quality to the consumer. Concerning the manufacturing process, the focus of robust design is to identify process setting regions that are most sensitive to inherent process variation. As will be shown later, this eventually helps improve the quality of what is produced — by minimizing the effect of the causes of variation — without necessarily eliminating the causes.

1.4

TAGUCHI’S DEFINITION OF QUALITY

What quality should a manufacturer aim to deliver? To resolve this fundamental dilemma as the debate intensified, Juran and others [9] defined the quality of a product to be its “fitness for use” as assessed by the customer. Taguchi has given an improved definition of this: a product has the ideal quality when it delivers on target performance each time its user uses the product, under all intended operating conditions, and throughout its intended life [4]. This ideal quality serves as a reference point even though it may not be possible to produce a product with ideal quality. A manufacturer should not think of quality except in terms of meeting customer expectations, which may be specific and many. People using pencils, for example, may desire durable points providing clear lines, and erasers that last until at least half the pencil is used. Pencil chewers would additionally want that the paint be lead-free! The ideal quality is performance at target rather than within some specification tolerance limits. This has been best shown by a study of customer preference of colour TV sets manufactured using identical designs and tolerances, but with different quality objectives. The Asahi newspaper reported this study on April 17, 1979 [5]. A Sony-U.S.A. factory aimed at producing sets within colour density tolerance m ± 5. It produced virtually no sets outside this tolerance. A SonyJapan factory produced identical sets but it aimed directly at hitting the target density m, resulting in a roughly normal distribution of densities with a standard deviation 5/3 (see Fig. 1.2). Careful customer preference studies showed that American customers who bought these TVs preferred the sets made in Japan over those made in U.S.A. Even if the fraction of sets falling outside the spec limits in U.S. production was lower than that in the Japanese production, the proportion of “Grade A” sets (those judged to do the best) from Japan was considerably higher and that of “Grade C” sets considerably lower. Thus the average grade of sets made by Sony-Japan was better than that by Sony-U.S.A. This reflected the higher quality value of sets made by Sony-Japan. At least two other major industry studies involving automobile manufacture have led to identical conclusions [ 1 , 1 1 ]. Reflecting on experiences such as above, Taguchi suggested that a product imparts a loss to society when its performance is not on target. This loss includes any inconvenience, and monetary or other loss the customer incurs when he uses the product. Taguchi proposed that manufacturers approach the ideal quality by

WHAT ARE TAGUCHI METHODS?

7

Sony -U .S .A Sony - Ja p a n

m -5

m

D

B Fig. 1.2

A

m+ 5 B

D

C o lo u r d e n s ity • Grade

Distribution of colour density in television sets. CSource: The Asahi, April 17, 1979.)

examining the total loss a product causes because of its functional variation from this ideal quality and any harmful side effect the product causes. The primary goal of robust design is to evaluate these losses and effects, and determine (a) process conditions that would assure the product made is initially on target, and (b) characteristics of a product, which would make its performance robust (insensitive) to environmental and other factors not always in control at the site of use so that performance remains on target during the product’s lifetime of use. To enforce these notions Taguchi (re)-defined the quality of a product to be the loss imparted to society from the time the product is shipped. Experts feel this loss should also include societal loss during manufacturing [6 ], The loss caused to a customer ranges from mere inconvenience to monetary loss and physical harm. If Y is the performance characteristic measured on a continuous scale when the ideal or target performance level is r, then, according to Taguchi, the loss caused L (Y ) can be effectively modelled by a quadratic function (Fig. 1.3) L(Y) = k ( Y - t ) 2 Note here that the loss function relates quality to a monetary loss, not to a ‘gut ftTliag or other mere emotional reactions. As will be shown later, the quadratic Io b ta c tk m provides the necessary information (through signal-to-noise ratios) Id achieve effective quality improvement. Loss functions also show why it is not good o r a g h for products to be within specification limits. Parts and components that most fit together to function are best made at their nominal (or the midpoint specification) dimensions than merely within their respective specification tolerances [11]. When performance varies, one determines the average loss to customers by statistically averaging the quadratic loss. The average loss is proportional to the mean squared error of Y about its target value r, found as follows: If one

8

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

L oss

P erfo rm a n c e characteristic Fig. 1.3

The relationship between quality loss and performance deviation from target.

produces n units of a product giving performances y2, ^ 3, . - . , yn respectively, then the average loss caused by these units because of their not being exactly on target r is ^ - [ L ^ ) + L (y2) + ... + L(y„)]= £ [(?! - r f + (y2 - t ) 2 + ... + {yn - x )2] /

a

n — 1

= k Ol-T) + —

2

O'

where fj, = E y-Jn and a 2 - Z (yi - fJ)2/(n - 1). Thus the average loss, caused by variability, has two components: 1. The average performance (jj)f being different from the target r, contributes the loss k(ii - r)2. 2. Loss k a 2 results from the performance {#} of the individual items being different from their own average \x. Thus the fundamental measure of variability is the mean squared error of Y (about the target t), and not the variance a 2 alone. Interestingly, it may be noted that ideal performance requires perfection in both accuracy (implying that jx be equal to r) as well as precision (implying that a 2 be zero). A high quality product performs near the target performance value consistently throughout the life span of the product.

WHAT ARE TAGUCHI METHODS?

9

Whenever available, a quantitative model that describes how the performance of a product or process design depends on the various design parameters is of great help in the optimization of designs. This dependency may become evident by invoking scientific and engineering principles, or by conducting experiments with a physical prototype. In the trial-and-error method of experimentation, intuition rather than a systematic procedure guides what levels of variable settings one should try. This approach appeals to many investigators for its apparent ‘simplicity’ [12]. In this approach, chance plays an important role to deliver the optimized design. The next popular approach is the one-variable-at-a-time experimental search to find the optimum setting. This method too is simple, but (a) the one-at-a-time approach is inefficient when the number of variables are many and (b) it can miss detection of critical interactions among design variables [12]. By sharp contrast to trial-and-error approach, statistical design o f experi­ ments is a systematic method for setting up experimental investigations. Several factors can be varied in these experiments at one time. This procedure yields the maximum amount of information about the effect of several variables and their interactions while using the minimum number of experiments. In a statistically designed experiment, one varies the levels of the independent input variables from trial to trial in a systematic fashion. A matrix of level settings defines these settings such that maximum information can be generated from a minimum number of trials. Moreover, some special statistical experiments require mere simple arithmetical calculations to yield sufficiently precise and reliable information. Classical statistical experiments, called fu ll factotial designs, require trials under all combinations of factors. Taguchi has shown that if one runs orthogonally designed experiments instead, many product and process designs can be optimized — economically and effectively, and with surprising efficiency. Taguchi’s robust design experiments for most part use only orthogonal arrays (OAs) rather than full factorial designs. Orthogonally designed parametric optimization experiments act as an efficient distillation mechanism that identifies and separates the effect each significant design or environmental factor has on performance. This in turn leads to products that (a) deliver on-target performance and (b) show minimum sensitivity to noise or uncontrolled environmental factors.

1.5

WHAT CAUSES PERFORMANCE TO VARY?

Variation of a product’s quality performance arises due to (a) environmental factors; (b) unit-to-unit variation in material, workmanship, manufacturing methods, etc.; and (c) aging or deterioration (see Table 1.2). The Taguchi approach focusses on minimizing variations in performance by determining the ~vital few” conditions of manufacture from the “trivial many”, economically and efficiently, such that when one finally manufactures the product, it is highly probable that it is, and remains on, target. Robust design aims specifically at determining product features such that performance becomes insensitive to the

10

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

environmental and other factors that the customer would perhaps not be able to or wish to control. T A B L E 1.2 F A C T O R S A F F E C T IN G P R O D U C T A N D P R O C E S S P E R F O R M A N C E

Product Perform ance

Process Perform ance Outer Noise

Consumer’s usage conditions Low temperature High temperature Solar radiation Shock Vibration Humidity Dust

Ambient temperature Humidity Dust Incoming material Operator performance Voltage and frequency Batch-to-batch variation Inner Noise

Deterioration of parts Deterioration o f material Oxidation

Machinery aging Tool wear Shift in control Between Products

Occurrence o f piece-to-piece variation when the pieces are supposed to be the same

Occurrence of process-toprocess variation when the processes are supposed to be the same Controllable Factors

All design parameters such as dimension, material, configu­ ration, packaging, etc.

All process design parameters All process setting parameters

Most real-life manufacturing processes lead to unit-to-unit variation in production and to what is produced not being always on target. Such variations may be caused by raw material differences, operator’s errors and inconsistencies, and factors such as vibration, temperature changes, humidity, etc. When one produces items in a batch, batch-to-batch process setting differences also introduce variation in product performance. In addition, manufacturing processes have a tendency to drift, causing off-target production as time passes. The first step toward robust process design is the tentative identification of all the above mentioned factors. Such a step, to be effective, requires contributions from technology experts, workers, designers, marketers, and even customers. One then includes the factors found in the statistical experiments so that their effects (individual, or interactive) may be estimated and countermeasured, if necessary. The challenges in product design are similar. The opening/closing of a refrigerator door, the amount of food kept in it, and the initial temperature of food, variation in ambient temperature and power supply voltage fluctuation are environmental factors that can effect a refrigerator’s performance. For a solar cooker, all but the last aspect might be important. One requires engineering and operational experience with the product and sound scientific judgment to ensure that all relevant factors are included in robust product design studies. Only then experiments to optimize the design may be planned.

WHAT ARE TAGUCHt METHODS?

11

An efficient tool for locating and identifying the potential factors that may affect product or process performance is the Ishikawa Cause-Effect diagram (Fig. 1.4). Operator

Machine

tired amplitude s ^ y s i c a l

^

_____

c o n d itio n

cutting

^

nervous

lm—large

delay ro ta tio n a l frequency

inadequate

return

time crock

V — Short

does not join pressure

P block i l t e r fluid 7/ / '’fruitsr Tiuia contaminant crack / / depleted removing shaving fluid grinding remains V agent V * — large hord mixing particles lathe

removal solution

Material

Fig. 1.4

\

contaminants mixed in

L

7

ratio

L •— light

^----------------*— g rin d e r

training

awareness

weight

Mony grinding cracks

grinding long

time bonding filled in pitch

Method

Cause and effect diagram for potential causes leading to cracks during contact lens grinding.

Engineers sometimes use screening experiments to review a large number of potentially important factors to separate the key factors. In such experiments the objective is to identify input factors having the largest impact on the process — the vital few from the trivial many. Taguchi experiments can be used to optimize and confirm the settings of these vital factors.

1.6

PREVENTION BY QUALITY DESIGN

Next to quality, manufacturing cost is a primary attribute of a product. However, it may appear impossible or at best difficult to reduce manufacturing cost while one is seeking on-target performance plus low variability. Somewhat surprisingly, Taguchi methods deliberately and consciously seek designs that use inexpensive components and parts and yet deliver on-target performance. The premise of this approach is that 80% of a lifetime cost (Table 1.1) of a product is fixed in its design stage. If the design calls for steel instead of plastic, then manufacturing can only aim at the remaining 20% (mostly labour) by seeking productivity during production. It is very important, therefore, that besides those affecting performance, the design engineer identifies aspects that have a significant bearing on the cost and manufacturability of the product, and then through statistical experiments >ets these also at optimal levels.

12

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

One is often unaware of the dependency of the output (performance) and the input (design and environmental factors) even if the technology is familiar and the manufacturing plant has made the product many times over. For instance, it is possible for forgings to show cracks even after a plant has made thousands of them. In practice, one does not generally know the effect of all the control factors that can be manipulated by the product/process designer. Also, one is often unaware of the noise factors that are uncontrollable but present during production or in the environment in which the product is used. However, achieving robust quality design requires that one finds out these effects systematically and countermeasures them. The Japanese discovered in 1953 that the most effective solution of quality problems is during product and process design. In that year, the Ina Tile Company used statistical experiments to reduce successfully finished tile size variability by a factor of 10 [4, 5]. Many investigations have now confirmed that it is too late to start thinking about quality control when the product is coming out of the reactors or exiting the production line. The remedy here, as proposed by Taguchi, is a three-step approach to correctly designing the product. These steps must precede production to maximize the product’s chances of delivering on-target performance with minimum variability.

1.7

STEPS IN DESIGNING PERFORMANCE INTO A PRODUCT

Designing with the objective of building quality into a product involves three steps [4]: 1. System (or concept or functional) design. This is the first step in design and it uses technical knowledge to reach the initial design of the product that delivers the basic, desired functional performance. Several different types of circuits or chemical reactions or mechanisms may be investigated, for instance, to arrive at a functional audio amplifier, a synthetic lubricant or a braking device. The technology of a special field often plays a major role in this step to reach the functional design — the initial, acceptable settings of the design parameters. 2. P aram eter design. In this step, one finds the optimum settings of the design parameters. To achieve this, one fabricates or develops a physical or mathematical prototype of the product based on the functional design (from step 1) and subjects this prototype to efficient statistical experiments. This gives the parameter values at which performance is optimum. Two types of experiments are conducted here: The first aims at identifying process parameter values or settings such that the product made by the process performs on target. The second aims at the type of experiments determining the effects of the uncontrolled, environmental, and other product design parameters to find design parameter settings such that performance suffers minimal deviation from target (i.e., it is robust) when one actually uses the product in the field. Parameter design identifies the optimum nominal values of the design parameters. 3. Tolerance design. Here, one determines the tolerances on the product design parameters, considering the loss that would be caused to society should the performance of the product deviate from the target.

WHAT ARE TAGUCHI METHODS?

13

In the functional design, one develops a prototype design (physical or mathematical) by applying scientific and engineering knowledge. From this effort one produces a basic design that broadly meets the customer’s requirements. Functional design is a highly creative step in which the designer’s experience and creativity play a key role. Good judgment used in functional design can reduce both the sensitivity of the product to environmental noise and its manufacturing cost. In parameter design, one conducts extensive empirical investigation to systematically identify the best settings of (a) process parameters that would yield a product that meets (the customer’s) performance requirement and (b) the design parameters of the product such that the product’s performance will be robust (stay near the target performance) while the product is in actual field use. Parameter design uses orthogonal arrays and statistical experiments to determine parameter settings that deliver on-target performance as also minimum variability for the product’s quality characteristics. In tolerance design, one determines manufacturing tolerances that minimize the product’s lifetime and manufacturing costs. The special device used here for expressing costs and losses is the Taguchi Loss Function, mentioned earlier. The objective in tolerance design is to achieve a judicious trade-off between (a) the quality loss attributable to performance variation and (b) any increase in the product’s manufacturing cost. The loss function philosophy acknowledges that society (consumers, manu­ facturers, and those affected indirectly by the product) incurs a loss with the product whenever the product’s performance deviates from its expected target performance. Thus, it is not enough for a product to “meet specifications”. Its performance must be as close to the target as possible. The loss function-based approach to robust design (through measures known as signal-to-noise ratios) also reduces problems in the field and is thus a preventive quality assurance step. As will be explained later, a third major advantage of aiming at on-target production (rather than only meeting specifications) is the reduction of catastrophic stack-up of deviations [1]. Loss functions help in bringing the customer requirement orientation into a plant. They also eliminate inequitable assignment Of manufacturing tolerances between departments making parts that should fit and function together. Each department then views the department following it as its customer and sets its own manufacturing tolerances using the loss function. Chapter 10 discusses these techniques. In this manner the manufacturing organization makes tolerance adjustments in whichever departments they are most economical to make, resulting in the reduction of the total manufacturing cost per unit [8].

1.8

FUNCTIONAL DESIGN: THE TRADITIONAL FOCUS

Functional design ideally creates a prototype process or product that delivers functional performance. Sometimes a product has to meet more than one Functional Requirement (FR) [13]. This requires research into concepts, technologies, and specialized fields. Many innovations occur at this stage and the core of this effort is concept design.

14

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

A functional design sometimes produces a mathematical formula; by using it, performance can be expressed as an explicit function of the (values of) design parameters. For instance, developing a mathematical representation of the functional design of passive filter type devices is a common activity in electrical engineering. By using the Kirchhoff current law, the transfer function (V0/Vs) for the circuit shown in Fig. 9.1 may be obtained as o

g

V,

3

( * 2 + Rg)(R s + u 3) + R 3RS + (R 2 + R g)R 3RsCs'

where s' is the Laplace variable. From this transfer function, the filter cutoff frequency coc and the galvanometer full-scale deflection D may be found respectively as fl,

=

( * 2 + * ,) ( * , + *3) + *3 ------------------------------------------------------------------------------------------------------------------------------------------

2n(R 2 + R )R 3RSC Vs

D= sen

g

s

G s c n [ ( R 2 ^ R g ) ( R s ^ R 3) + R s R , ]

The design parameters (DPs) that the designer is free to specify are R2, R$, and C. Another design example, from chemical engineering, illustrates a similar functional process model — also a mathematical relationship between the design parameters and performance. Many chemical processes apply mechanical agitation to promote contacting of gases with liquids to encourage reaction. Based on reaction engineering principles, the relationship between the utilization of the reacting gas and the two key controllable process variables may be given by Utilization (%) = K (mixing HP/1000 g)A (superficial velocity)5 As will be illustrated through a case study in Chapter 9, such mathematical models can be as useful as physical prototypes in achieving a robust design. Traditionally, product and process design receive maximum attention during functional design. Most engineering disciplines expound the translation of scientific concepts to their applications so that the designer is able to develop the functional design. Refinements to this initial design by trial and error may be attempted on the shop floor — combined possibly with limited field testing of the prototype. True optimization of the design, however, is rarely thus achieved or attempted [12]. The Taguchi philosophy sharply contrasts with this traditional approach to design. Taguchi has contended that, besides building function into a product, its design should engineer quality also. In his own words: “quality is a virtue of design.”

1.9

PARAMETRIC DESIGN: THE ENGINEERING OF QUALITY

A quality product, during its period of use, should have no functional variation. The losses caused by it to society by repairs, returns, fixes, adjustment, etc., and by its

WHAT ARE TAGUCHI METHODS?

15

harmful side effects are designed to be small. During its design, one takes countermeasures to assure this objective. The use of Taguchi methods makes it possible that measures may be taken at the product design stage itself to achieve (a) a manufacturing process that delivers products on target and (b) a product that has robust performance and continues to perform near its target performance. As already stated, the performance of a robust product is minimally affected by environmental conditions in the field, or by the extent of use (aging) or due to item-to-item variation during manufacturing. Besides, robust product design aims at the selection of parts, materials, components, and nominal operating conditions so that the product will be producible at minimum cost. The three steps involved in robust design are: 1. Planning the statistical experiments is the first step and includes identification of the product’s main function(s), what side effects the product may have, and factor(s) constituting failure. This planning step spells out the quality characteristic Y to be observed, the control factors {0i, fy, #3}> the observable noise factors {wi, w2, vv3}, and the levels at which they will be set during the various test runs (experiments). It also states which orthogonal design will be employed (see Fig. 1.5) to conduct the statistical experiments and how the observed data {yj. y2, >’3, . ..} will be analyzed.

D esign m a t r i x

Noise

m a trix

(Control fa c to rs )

(Noise fa c to rs )

Computed performance statistic

Observed p e rfo rm a n c e characteristic

R U

M |

d,

f

1

2 3

9

1

02 03 1 2

W1 \

1 2

1

3

3

2

t

2

2

2

3

2

3

1

3

%

3

3

2

t

3

3

2

c

W2 w 3 •

1

-

y,

y2 y3

1

2

2

2

1

2

-

2

2

1

► y4

— Z (0),

The outer orthogonal array made up of observable noise f a c t o r levels; each noise factor has two distinct levels. 1

\

1

1

2

2

2

1

2

2

2

t

y33 y34 y35 ^36

— Z (0)9

The inner orthogonal array constructed using the different design factor treatments; three treatments for each factor are available.

Fig . 1-5

A parameter design experiment plan.

16

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

2. Actual conducting of the experiments. 3. Analysis of the experimental observations to determine the optimum settings for the control factors, to predict the product’s performance at these settings, and to conduct the validation experiments for confirming the optimized design and making plans for future actions. Taguchi has recommended that one should analyze this, using a specially transformed form of the performance characteristic Y, known as the signal-to-noise ratio {Z(0)j}, see Section 4.2, rather than using the observed responses {y*} directly. One conducts all the required experiments needed, guided by the principles of design of experiments (see Chapter 3). This assures that the conclusions reached are valid, reliable, and reproducible. Briefly stated, the novel idea behind parametric design is to minimize the effect of the natural and uncontrolled variation in the noise factors, by choosing the settings of the control factors judiciously to exploit the interactions between control and noise factors, rather than by reaching for high precision and expensive parts, components and materials and plant control schemes. Such possibility was perceived by Taguchi before anyone else.

1.10 STATISTICAL EXPERIMENTS DISCOVER THE BEST DESIGN RELIABLY AND ECONOMICALLY M any engineers hesitate to use statistical form ulas or analysis in their work. Sound decisions about quality, how ever, require that one obtains the appropriate data (in the m ost efficient way) and analyzes it correctly. That is precisely w hat statistics helps us to do. In particular, when several factors influence product or process perform ance, statistically designed experim ents are able to separate reliably the vital fe w factors that have the m ost effect on perform ance from the trivial m any. This separation results in m athem atical models that m ake true product and process design optim ization possible. Also, statistical experiments produce the supporting data for verifying some hypothesis about a response (dependent) variable — usually with the smallest number of individual experiments. An example can best illustrate this point. If four types of tyre materials (Mu M2, M3, and M4) are available, four vehicle types (Vi, V2, V3, and V4) are present, and four types of terrains (7\, T2j T3, and T4) exist on which the vehicles will be used, then the total number of ways to combine these factors to study them is 43 or 64. At first, 64 may appear to be the number of tests with the different vehicles, terrains etc. that one must run. However, if prior knowledge suggests that tyre wear is unaffected by which tyre material is used on which vehicle, and on which terrain, (i.e., there are no interactions (Section 3.1)) and the objective is to identify the main effect of materials and the effects of changing vehicle type and the driving terrain, one will need to run only 16 (Latin-square designed) statistical experiments (Fig. 1.6) to grade the materials based on wear. This is a substantial saving of effort.

WHAT ARE TAGUCHI METHODS?

17

ro


_
0

1 2 3 4 5 6 7 8

1 1 2 2 1 1 2 2

1 1 2 2 2 2 1 1

1 2 1 2 1 2 1 2

1 2 1 2 2 1 2 1

1 2 2 1 1 2 2 1

1 2 2 1 2 1 1 2

Let the following terms represent the quantities as indicated:

Ai

= X observations (where one sets Factor A at “1”)

NAi

= yi + y2 + )?3 + ,)'4 - number of observations with Factor A set at “1”

Abarj = Average of observation with Factor A set at “1” = A xINai =

()’i + yi + y^ + yd/*

y\ yi ya ys >6 yi ys

56

TAGUCHI METHODS EXPLAINED— PRACTICAL STEPS TO ROBUST DESIGN

Similarly, one defines Abar2, ZJbarj, /?bar2, Cbarj, etc. One may evaluate the “main factor effects” or the main effect dependencies as follows: Average Effect^ = Abar2 - Abarx Average Effect^ = Z?bar2 - Bbaxx Average Effect^ = Cbar2 - Cbarx Average Effect^ = £>bar2 - Dbarj Average Effect^ = £bar2 - Ebari The “two-factor interactions” are calculated as follows: Let AjCi = Sum of observations with factor A set at “1” and factor C set at “ 1” = yi + ?2

A\C2 - Sum of observations with factor A set at “1” and factor C set at “2” = ^3 + y4 A2Cx = Sum of observations with factor A set at “2” and factor C set at “1” = >>5 + >>6

A2C2 = Sum of observations with factor A set at “2” and factor C set at “2” = yi + Js Then, if we define A,C) bar as A{C-}12, we have Interaction^ = [(AiC\ bar + A2C2 bar) - (AXC2 bar + A2C\bar)]/4 To carry out ANOVA of the observations, the sums of squares of certain deviations are required. One determines these sums of squares as follows: Let r = L all observations = yx + y2 + y3 + y* + ys + y