Methods for experimental design: principles and applications for physicists and chemists

R

Methods for experimental design

DATA HANDLING IN SCIENCE AND TECHNOLOGY Advisory Editors: B.G.M. Vandeginste and S.C. Rutan

Other volumes in this series: Microprocessor Programming and Applications for Scientists and Engineers by R.R. Srnardzewski Volume 2 Chemometrics: A Textbook by D.L. Massart, B.G.M. Vandeginste, S.N. Deming, Y. Michotte and L. Kaufman Volume 3 Experimental Design: A Chemometric Approach by S.N. Derning and S.L. Morgan Volume 4 Advanced Scientific Computing in BASIC with Applications in Chemistry, Biology and Pharmacology by P. Valko and S. Vajda Volume 5 PCs for Chemists, edited by J. Zupan Volume 6 Scientific Computing and Automation (Europe) 1990, Proceedings of the Scientific Computing and Automation (Europe) Conference, 12-15 June, 1990, Maastricht, The Netherlands. edited by E.J. Karjalainen Volume 7 Receptor Modeling for Air Quality Management, edited by P.K. Hopke Volume 8 Design and Optimization in Organic Synthesis by R. Carlson Volume 9 Multivariate Pattern Recognition in Chemometrics, illustrated by case studies, edited by R.G. Brereton Volume 10 Sampling of Heterogeneous and Dynamic Material Systems: theories of heterogeneity, sampling and homogenizing by P.M. Gy Volume 11 Experimental Design: A Chemornetric Approach (Second, Revised and Expanded Edition) by S.N. Derning and S.L. Morgan Volume 12 Methods for Experimental Design: principles and applications for physicists and chemists by J.L. Goupy

Volume 1

DATA HANDLING IN SCIENCE AND TECHNOLOGY -VOLUME

12

Advisory Editors: B.G.M. Vandeginste and S.C. Rutan

Methods for experimental design principles and applications for physicists and chemists

JACQUES L. GOUPY 7, Rue Mignet, 75016 Paris, France

ELSEVIER Amsterdam - London - N e w York -Tokyo

1993

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands

Translation and revised edition of: La Methode des Plans d’Experiences. Optimisation du Choix des Essais et de I’lnterpretation des Resultats 0 Bordas, 1988 0 Dunod for updatings Translated by: C.O. Parkes

ISBN

0-444-89529-9

0 1993 Elsevier Science Publishers B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521,1000 A M Amsterdam, The Netherlands. Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

To my wife Nicole

This Page Intentionally Left Blank

PREFACE This book is devoted to researchers who, because of limited time and resources, must use a minimal number of experiments to solve their problems. It was written with the aim of avoiding theoretical statistics or mathematics. It is not intended to replace the texts on analysis of variance, regression analysis or more advanced statistical treatments. It was written for experimenters by an experimenter. It is an introduction to the philosophy of scientific investigation. This book has grown out of my consulting practice and a series of short courses given to industrial researchers. These experiences taught me that a good method and solid concepts are more useful than complex theoretical knowledge. Therefore, I have attempted to preserve the balance between the practice necessary to carry out a study and the theory needed to understand it. I have tried to write a book that is usehl and clear. While mystudentsmay have learned something from me, I have certainly learned from them. As a result this new English edition contains considerable additional material not included in the original French book. The presentation makes extensive use of examples and the approach and methods are graphical rather than numerical. All the calculations can be performed on a personal computer. Conclusions are easily drawn from a well designed experiment, even when rather elementary methods of analysis are employed. Conversely even the most sophisticated statistical analysis cannot salvage a badly designed experiment. Readers are assumed to have no previous knowledge of the subject. The presentation is such that the beginner may acquire a thorough understanding of the basic concepts. There is also sufficient material to challenge the advanced student. The book is therefore suitable for an introductory or an advanced course. The many examples can also be used for self-tuition or as a reference.

ACKNOWLEDGEMENTS I am gratehl to the many researchers whose work provided the examples cited in this book and who have asked me so many questions on how to use experimental designs efficiently. I am also grateful to Owen Parkes who translated this book and was a continual source of advice. I wish to thank the staff at Dunod, particularly Maryvonne Vitry and Jean-Luc Sensi, and the staff at Elsevier, for their encouragement and support. A special thanks to my wife who sustained me with love and made this work possible.

Paris February 1993 Jacques GOUPY

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ix

CONTENTS Preface

vii

Acknowledgements

vii

Chapter 1

Research strategy : Definition and objectives

1. Introduction 2. The process of knowledge acquisition 2.1. Gradual acquisition of results, 4 2.2. Selection of the best experimental strategy, 4 2.3. Interpretation of results, 4 3. Studying a phenomenon 3.1. The classical method, 5 3.2. Experimental design methodology, 6 4. Historical background Chapter 2

Two-level complete factorial designs:2*

1. Introduction 2. Two-factor complete designs: 22 2.1. Example: The yield of a chemical reaction, 10 3. General formula of effects 4. Reduced centred variables 5 . Graphical representation of mean and effects 6. The concept of interaction 6.1. Example: The yield of a catalysed chemical reaction, 21 7. General formula for interaction Chapter 3

Two-level complete factorial designs: 2k

1. Introduction 2. Complete three factor design: 23 2.1. Example: The stability of a bitumen emulsion, 29 3. The Box notation 4. Reconstructing two 22 designs from a Z3 design 5 . The relationship between matrix and graphical representations of experimental design 6. Construction of complete factorial designs 7. Labelling of trials in complete factorial designs 8. Complete five factor designs: 25

1

1

2

4

7 9

9 10 15 16 19 21 24 29 29 29 34 35 36 37 38 38

X

8.1, Example: Penicillium chrysogenum growth medium, 38 9. Complete designs with k factors: 2k 10. The effects matrix and mathematical matrix 10.1. Matrix transposition, 45 10.2. Matrix multiplication, 45 10.3. Inverse of X, 46 10.4. Calculation of X'X, 47 10.5. Measurement units, 47

Chapter 4

Estimating error and significant effects

43 44

49

1. Introduction 49 50 2. Definition and calculation of errors 2.1. Arithmetic mean, 5 1 2.2. Dispersion, 51 53 3. Origin of the total error 56 4. Estimating the random error of an effect 4.1. The investigator knows the experimental error of the response, 56 4.2 The experimental error of the response is unknown, 59 Several measures on the same experimental point, 59 Repeat the whole experimental design, 60 4.3. The experimental error of the response is unknown, and the experimenter does not want to perform any supplementary experiments, 62 5. Presentation of results 63 5.1. Numerical results, 63 5.2. Illustration of results, 63

Chapter 5

The concept of optimal design

1. Introduction 2. Weighing and experimental design 2.1. Standard method, 70 2.2. Hotelling method, 70 2.3. Strategy for weighing four objects, 71 3. Optimality criteria 3.1. Unit matrix criterion, 76 3.2. Maximum determinant criterion, 77 3.3. Minimum trace criterion, 79 3.4. "The largest must be as small as possible" criterion, 80 4. Positioning experimental points 4.1. Positioning experimental points for one factor, 8 1 5. Measurement of an electrical resistance Example: Measuring an electrical resistance, 83 6. Positioning experimental points for two factors 7. Positioning the experimental points for k factors

67

67 69

76

80 83 85 89

xi

Chapter 6

Two-level fractional factorial designs: 2k-P The Alias theory.

1. Introduction 2. First fractional design: Z3-' 2.1 Example: Bitumen emulsion stability (continued from Chapter 3), 3. Interpretation of fractional designs 4. Calculation of contrasts 5. Algebra of columns of signs Alias generators, 99 6. Construction of fractional designs (one extra factor) 7. Notation of fractional designs 8. Construction of fractional designs (two extra factors) 9. Construction of fractional designs (p extra factors) 10. Practical rules 10.1 Going from AGS to contrasts, 110 10.2 Going from contrasts to the AGS, 110 11. Choosing the basic design 1 1.1. Total number of factors to be studied, 1 11 1 1.2. Number of trials to be performed, 112

Chapter 7

Two-level fractional factorial designs: 2k-P Examples

91

91 92 92 94 95 98 100 104 104 108

110 111

115

1. Introduction

115

2. 2*-' fractional design 2.1. Example: Minimizing the colour of a product, 116 2.2. Techniques for dealiasing main effects from interactions, 119 2.3. Construction of the complementary design, 121 2.4. Contrast calculation, 122 2.5. Interpretation, 123

116

3. 274 fractional designs 3.1. Example: Settings of a spectrofluorimeter, 127 3.2. Calculation of contrasts, 130 3.3. Interpretation of the initial design, 132 3.4. Construction of the complementary design, 134 3.5. Interpretation of the initial and complementary designs, 140 4. Studying more than seven factors 5. The concept of resolution 5.1. Definition of resolution, 142

127

5.2. An example of a 2:

design: Plastic drum fabrication, 144

142 142

xii

Chapter 8

Types of matrices

1. Introduction 2. The experimental matrix 3. The effects matrix 4. The basic design matrix for constructing fractional designs Chapter 9

Trial sequences: Randomization and anti-drift designs

151

151 151

153 154 159

1. Introduction 1.1. Drift errors, 161 1.2. Block errors, 161 2. Small uncontrollable systematic variations 3. Systematic variations: Linear drift 4. #en should trials be randomized? Example: The powder mill, 166 4.1. Powder mill: First investigator's strategy, 167 4.2. Powder mill: Second investigator's strategy, 168 4.3. Powder mill: Third investigator's strategy, 171 5 . Randomization and drift

159

Chapter 10

179

Trial sequences: Blocking

162 162 166

175

1. Introduction 179 2. Block variations 180 3. Blocking 180 Example: Preparation of a mixture, 180 185 4. Blocking on one variable Example: Penicillium chrysogenum growth medium (continued), 185 190 5 , Blocking on two variables 5.1. Example: Yates' bean experiment, 190 5.2. Interpretation of experimental results, 194 20 1 6.Blocking of a complete design

Chapter 11

Mathematical modelling of factorial 2k designs

1. Introduction 2. Mathematical modelling of factorial designs 3. Formation of the effects matrix 4. Evaluation of responses throughout the experimental domain 4.1. Example: Study of paste hardening, 210 4.2. Interpretation, 21 1 5. Test of the model adopted 6. Selection of a research direction 6.1. Mathematical model, 21 5 6.2. Isoresponse curves, 216

203

203 204 208 209 213 213

...

Xlll

6.3. Steepest ascent vector, 2 17 7. Choice of complementary trials 8. Analysis of variance and factorial designs Example: Sugar production, 220 8.1. Analysis of the problem by factorial design (one response per trial), 220 8.2 Analysis of the problem by analysis of variance (one response per trial), 223 8.3 Analysis of the problem by factorial design (two responses per trial), 226 8.4 Analysis of the problem by analysis of variance (two responses per trial), 227 9. Introduction to residual analysis 10. Error distribution

230 234

Chapter 12

239

Choosing complementary trials

220 220

1. Introduction 2. A single extra trial Example: Clouding of a solution, 240 3. Two extra trials Example: Clouding of a solution ( block effect), 244 4. Three extra trials 5. Four extra trials 5.1, Reconstruction of the experimental design, 252 5.2. Presentation of results, 252

239 240

Chapter 13

257

Beyond influencing factors

1. Introduction 1.1. IdentifLing the domain of interest, 257 1.2. Looking for an optimum, 258 1.3. Finding the minimum response sensitivity to external factors, 258 2. Identifying the domain of interest 2.1. Example: Two-layer photolithography,258 2.2. Examination of the results for response Lz, 263 2.3. Examination of the results for response L,, 265 3. Finding an Optimum 3.1. Example: Cutting oil stability, 268 3.2. Interpretation, 270 4. Finding a stable response 4.1. Example: thickness of epitaxial deposits, 271 4.2. Interpretation, 275

244 246 250

257

258

268 27 1

xiv

Chapter 14

Practical method of calculation using a quality example

1. Introduction 2. A quality improvement example Example: Study of truck suspension springs, 284 3. Interpretation, step 1 3.1. Calculation of responses, 287 3.2. Analysis of results (interpretation, step 1) , 289 4. What is a good response for dispersion? 4.1. Variance, 292 4.2. Logarithm of variance, 293 4.3. Comparison of variance and logarithm of variance, 293 4.4. The signal-to-noise ratio, 295 5. Interpretation, step 2 5.1. Calculation of responses, 295 5.2. Analysis of results (second step of interpretation), 296 6. Optimization

283 283 284 287 292

295

302

Chapter 14 (continued) Detailed calculations for the truck suspension springs example

309

1. Calculation for the first interpretation 2. Calculation for the second interpretation 3. Calculation for optimization

309 316 326

Chapter 15

Experimental designs and computer simulations

333

1. Introduction 2. Example 1, Propane remover optimizing 2.1. The problem, 335 2.2. Simulation, 337 2.3. Interpretation, 337 3. Example 2: Optimization of a hydroelastic motor suspension 3.1. The problem, 340 3.2. Calculations, 343 3.3. Interpretation, 345 3.4. Conclusion, 346

333 335

4. Example 3: Natural gas plant optimization 4.1, The gas production system and the problem to be solved, 347 4.2. Choice of responses, 349 4.3. Choice of calculation design, 351 4.4. Calculations, 352 4.5. Interpretation, 353 4.6. Optimization, 361 4.7. Conclusion, 363

347

339

xv

Chapter 16

Practical experimental designs

365

1. Introduction 2. Calculation of effects and interactions when an experimental point is misplaced 3. Calculation of effects and interactions when all the experimental points are misplaced 4. Error transmission 5. Experimental quality

365

Chapter 17

391

Overview and suggestions

1 . Introduction 2. Selection of the best experimental strategy 2.1. Defining the problem, 392 2.2. Preliminary questions, 394 2.3. Choice of design, 397 3. Running the experiment 4. Interpretation of results 4.1. Critical examination of the results, 398 4.2. Follow up, 400 5. Gradual acquisition of knowledge 6. What experimentology will not do

Appendix 1

Matrices and matrix calculations

1. Introduction 2. Definitions 2.1. General, 403 2.2. Definitions for square matrices, 405 3. Matrix operations 3.1. Operation on array, 407 3.2. Operations between arrays, 408 3.3. Calculation of an inverse matrix, 41 I 4. Matrix algebra 5 . Special matrices

Appendix 2

Statistics useful in experimental designs

1 . Normal distribution Population, 4 18 Sample, 418 Variance, 419 2. Variance Theorem One random variable, 419 Error of the mean, 420

366 372 377 388

391 392

398 398 40 1 402

403 403 403 407

413 414

417 417

419

xvi

Appendix 3 Order of trials that leaves the effects of the main factors uninfluenced by linear drift. Application to a Z3 design

421

Bibliography Author index Example index Subject index

43 1 440 443 447

CHAPTER I

RESEARCH

STRATEGY:

DEFINITION AND OBJECTIVES

1.

INTRODUCTION

Experimental scientists and technicians employed in laboratories, industry, medicine or agriculture throughout the world run experiments. The classical experimental approach is to study each experimental variable separately. This one-variable-at-a-time strategy is easy to handle and widely employed. But is it the most efficient way to approach an experimental problem? The first people to ask this question were English agronomists and statisticians working at the beginning of the century. Agronomy is somewhat different from most experimental sciences in that there are almost always a large number of variables and each experiment lasts a long time. As they could not run large numbers of trials, they worked to develop the best research strategy. They found that the classical method was not appropriate and developed a revolutionary approach which guaranteed experimenters an optimal research strategy.

2

Since then, many investigators have contributed with such topics as: optimal experimental designs, study of residuals, composite designs, Latin squares, fitting equations to data, multivariate calibration, empirical model building, response surface methodology, etc. All these techniques can be thought as components of a new discipline which, strangely, has no name. The name suggested for the field is Experimentics or Experimentology [ 11. But the scientific community has yet to decide. The factorials designs which are the subject of this book form just one part of Experimentics. This first chapter outlines the areas in which experimental designs can be applied, defines objectives and raises the general problem of how to study a phenomenon. The main points covered will be: I . The general process by which experimental knowledge is acquired. 2 . The three essential aspects of knowledge acquisition using the methodology of Experimental Designs:

Gradual acquisition of results. Selection of the best experimental strategy Interpretation of results. 3 . A comparison of classical approach and experimental design to study a phenomenon 4. A brief historical background.

2.

THE PROCESS OF KNOWLEDGE ACQUISITION

Any search for new information begins by the investigator asking a number of questions (Figure 1.1). For example, if we want to know the influence of a fertiliser on the wheat yield of a plot of land, we could ask several questions, such as : How much fertiliser is needed to increase the yield by 10% ? How does rainfall affect fertiliser efficiency ? Is the wheat quality influenced by the fertiliser ? These questions define the problem and determine the work to be carried out to solve it. It is therefore important to ask the right questions: those that can help us to resolve the problem. This is not quite as simple as it may appear. Before actually beginning any experiments, it is always wise to check that the information required does not already exist. The experimenter should first prepare an inventory of the available information, by compiling a bibliography, consulting experts, theoretical calculations, or any other method which provides himher with answers to the questions asked without actually carrying out any experiments. This preliminary survey may answer all the questions, resolving the problem. If it does not, some questions may remain to be answered, or they may be modified in the light of the information obtained. It will then be necessary to carry out experiments to obtain all the answers required. This preliminary study is a routine part of all experimental work and we shall not discuss it further. Our concern is not with this initial phase, but with those that follow. These are the steps in which the experimenter thinks about the experiments to be

3

performed, and our problem is how to select the experiments that must be done and those which need not be done. Is there a single ideal strategy? Such an ideal strategy should: give the desired results as quickly as possible. avoid carrying out unnecessary experiments. ensure that the results are as precise as possible. enable the experiments to progress without setbacks. provide a model and optimisation of the phenomena studied. There is such an ideal strategy, and it is effective because it simultaneously takes into account three essential aspects of knowledge acquisition.

0

gradual acquisition of results. selection of the best experimental strategy. interpretation of results.

SYSTEM TO STUDY

QUESTIONS Q1, Q2 ...Qn

INFORMATION INVENTORY

J OlCE OF AN EXPERlMENTAL STRATEGY I I GRADUAL ACQUISITION OF RESULTS

1 1

EXPERIMENTATION

INTERPRETATION OF THE RESULTS

5 KNOWLEDGE OF THE SYSTEM STUDIED

Figure 1.1 : The boxed steps define the areas of Experimentics.

4

The experiments should be organised to facilitate the application of the results. They should also be organised to allow the gradual acquisition of relevant results.

2.1. Gradual acquisition of results The experimenter clearly does not know the results when the study begins. It is therefore wise to work progressively and to be able to reorientate the study in the light of the early trial results. A preliminary rough outline can be done and then used to select any change in research direction that may better identi@ the most important points of the study and those avenues that should be abandoned to avoid any waste of time. This is why we recommend working progressively. An initial series of trials can provide provisional conclusions. A new series of trials can be done based on these provisional conclusions. The results of both these series should then be used to obtain a better picture of the results. Then, a third series of trials can be run if necessary. In this way the experimenter accumulates only those results that he requires, and the study stops when the original questions have been answered.

2.2. Selection of the best experimental strategy This strategy should facilitate the organisation of gradual acquisition of results. It should also minimise the number of trials, but it must not compromise the quality of the experimentation. On the contrary, it should ensure that the results are the most precise possible. Experimental designs, response surface methodology, and other approaches, such as steepest ascent and Simplex, are perfect for our requirements : 0 0 0

progressive acquisition of knowledge. only the required number of experiments the most precise results.

We will see that they provide the maximum of usefid information for the minimum number of experiments.

2.3. Interpretation of results The initial choice of experiments should facilitate interpretation of the results. Results should be readily interpreted and easily understood by both specialists in the field and those that are not. The methods recommended above can help us attain both these objectives. Microcomputers have made what used to be a long and tedious process of calculating results much more accessible. Not only are the calculations done quickly and accurately, but graphical outputs are a spectacular means of displaying results.

3.

STUDYING A PHENOMENON

The study of a phenomenon can be outlined as follows: the scientist may want to know, for example, the yield of wheat from a plot of land, the profit made on a chemical product or

5

the wear on a car motor component. This yield, price, or wear depends on many variables. The grain yield will vary with the nature of the soil, the amount and type of fertiliser, the exposure to the sun, the climate, the variety of wheat seed sown, etc. The profit from sale of a chemical may depend on the quality of the feedstock, industrial production yields, product specifications, plant conditions, etc. A similar set of variables will influence the wear of the car motor component. We can assess this as the response, y. This quantity is a finction of several independent variables, xi.which we shall call factors. It is possible to link mathematically the response y to the factors, Xi, as follows :

y =f(xl,

x2>

x37.,.>

xn..,.)

The study of a phenomenon thus requires measuring the response y for different sets of factor values. Let us, first, examine briefly the "classical" method of establishing the fhction.

3.1. The classical method The levels of all the variables except one are held constant. The response y is then measured as a hnction of several values of this unfixed variable xI.

A

B

C

D

E

X

Figure 1.2 :Only the levels of the variable x, are modified, the 8 other variables are held constant.

6

At the end ofthe experiment on this first variable, a curve is drawn of y = f (x,) (Figure 1.2). If the experimenter wishes to study all the variables, the whole experiment must be repeated for each one. Using this method, if he wanted to study just seven factors, with only five points per variable, he would have to carry out = 78,125 experiments or trials. This represents an enormous amount of work, and is clearly not feasible. The experimenter must therefore find a way of reducing the number of tnals. There are only two ways of doing this: reduce the number of experimental points per variables or reduce the number of variables. Reduce the number of experimental points If he elects to examine only three points per variable instead of five, he would have to carry out 37 = 2 187 trials. Two points per variable would require 27 = 128 trials. This is still a lot of work, and is often too much for either the budget or the time available. As there must be at least two experimental points per variable, the experimenter has no option but to:

Reduce the number of variables But even a system with four variables, testing each of them at three values, requires 34, or 81 trials. This way of working is both tedious and unsatisfactory. If some variables are ignored, people could be dubious about the results, and the investigator will be obliged to apologise for presenting incomplete conclusions. The inconvenience of this approach is particularly evident when safety or large sums of money are involved. This is precisely why we shall now proceed to examine the method of experimental design.

3.2. Experimental design methodology The essential difference between the classical one-variable-at-a-time method described above and the experimental design is that, in the latter, the values of all the factors are varied in each experiment. The way in which they are varied is programmed and rational. While this may appear somewhat disturbing at first sight, this approach of multiple simultaneous variable settings, far from causing difficulties, offers several advantages. Some of these are :

0

fewer trials. large number of factors studied. detection of interaction between factors detection of optima. best result precision. optimisation of results. model-building from the results.

Experimental designs can be used to study a great number of factors while keeping the total number of trials within reason. This is why one of its major applications is the search for influencing factors.

7

Instead of limiting the number of factors studied, the experimenter initially reduces the number of experimental points per factor. The term factor will be used rather than variable because it can include both continuous and discrete variables. The search for influencing factors consists of setting only two values for each factor, these values are called the levels. studying as many factors as possible, even those that may appear, at first sight, to have little influence. Many of the factors studied will probably have no influence, only a few will act upon the response. The results can then be used to choose new experimental points to define one or more specific aspects of the study. Thus, all the influencing factors will have been detected and studied, while keeping the number of trials to a minimum. Hence, the study can be completed without waste of either time or money.

4.

HISTORICAL BACKGROUND

Agronomists were the first scientists to confront the problem of organising their experiments to reduce the number of trials. Their studies invariably include a large number of parameters, such as soil composition, effect of fertilisers, sunlight, temperature, wind exposure, rainfall, species studied, etc., and each experiment tends to last a long time. At the beginning of the century Fisher [ 2 , 31 first proposed methods for organising trials so that a combination of factors could be studied at the same time. These were the Latin square, greco-Latin square, analysis of variance, etc. The ideas ofFisher were taken up by agronomists such as Yates and Cochran, and by statisticians such as Plackett and Burman [4], Hotelling [S], Youden [ 6 ] ,and Scheffe [7], and used to develop powerful methods. However, their studies were often highly theoretical, and involved difficult calculations. These difficulties, plus the revolutionary concepts developed by these pioneers, undoubtedly hindered the rapid spread of the new methods into the worlds of industry and universities. During the World War 11, major industrial companies realised that these techniques could greatly speed up and improve their research activities. Du Pont de Nemours adapted the techniques employed in agronomy to chemical problems, some years later ICI in England and TOTAL in France began using experimental designs in their laboratories. Other major companies, such as Union Carbide Chemicals, Proctor and Gamble, Kellogs, General Foods, have also adopted this approach. But the applications of these methods have never become generally known and for the most part have remained restricted to their original discipline of agronomy. They feature in few courses and despite the efforts of certain teachers, few students have learned them. The outstanding teachers in this field include Professors Box [8], Hunter and Draper [9], Benken in the USA, Phan Tan Luu in France, and Taguchi [ 101 in Japan. Thus, although they have been known and applied in certain areas for over half a century, the techniques are poorly understood and not generally used. The calculations are no longer a problem, thanks to the widespread availability of microcomputers. The challenge now is to overcome the reticence of users by clearly demonstrating the advantages afforded by

8

experimental design. The method described in the foilowing chapters, together with the examples which are given, will, it is hoped, make experimental designs accessible to researchers in both the industrial and academic worlds.

CHAPTER 2

TWO-LEVEL COMPLETE FACTORIAL DESIGNS: 22

1.

INTRODUCTION

Two-level factorial designs are the simplest, but are widely used because they can be applied to many situations as either complete or fractional designs. This chapter deals with complete designs. We will first examine a simplified example using only two factors. We will use it to introduce several important basic concepts which will be used in later chapters: experimental matrix, effect of a factor, iInteraction between factors, reduced centred variables, etc. This chapter also indicates how to calculate the effects of each factor and the interactions between factors. The reader will find usefkl tools to facilitate the interpretation of results and the presentation of conclusions.

10

2.

TWO-FACTOR COMPLETE DESIGNS: 22 The important concept of effect is best understood with the help of an example.

2.1. Example: The yield of a chemical reaction The Problem: The yield from this reaction depends on two factors: temperature and pressure. The chemist carrying out the study needs to know if the yield increases or decreases with increasing temperature. He also wants to $2 know the effect of pressure changes on the yield. The experirnental setup allows the reaction temperature to be varied from 60°C to 80°C,and the pressure from 1 to 2 bar The experimental domain (Figure 2.1) is I thus defined by the four points I '

~

A

1

60" C

lbar

{

80" C

60" C

lhar

12har

{

80"

c

2bar

The experimenter must adopt a specific research strategy in order to obtain the responses he requires. He could, €or example, fix the temperature at 70°C and carry out 3 experiments at 1, 1.5 and 2 bar as shown in Figure 2.2. The yield increases with pressure from 70°C to 75OC and 80°C. Pressure

2 bar

1 bar

60 OC

8o

Temperature

oc

Figure 2.1: Definition of the experimental domain

.

11

The effect of temperature can then be studied by keeping pressure constant at 1.5 bar and carrying out 3 experiments at 60"C,70°C and 80°C.The yield increases with temperature, 65%, 75% and 85%, indicating that both temperature and pressure must be increased to obtain the best yield. A final experiment at 80°C and 2 bar confirms these assumptions and the study is complete. But it has taken six experiments to obtain this result. Pressure

Figure 2.2: Example of a research strategy.

The experimenter could have selected another strategy by using other experimental points. These points could be evenly distributed throughout the experimental domain, or they could be selected randomly. But is there a best strategy? Clearly it is one that minimises the number of experiments without sacrificing precision, so that the same conclusions are reached. This best strategy exists; it consists of using the points A, B, C and D, the extremities of the experimental domain (Figure 2.3). This is the strategy adopted for two-factor experimental designs. While it provides the same results as the above experiment, it requires only four experiments. Let us now see how this approach can be used to provide a hller analysis of the results. We shall use the convention of -1 for the low level of each factor and +I for the high level. We can then place all the experimental information in a table, called the Experimental Matrix or Trial Matrix (Table 2.1). Each experiment is defined in this matrix. For example, in trial no 3, factor 1 (temperature) will be held at 60°C and factor 2 (pressure) at 2 bar. The trial is run under these conditions and the yield is measured. The other three trials shown in the matrix are carried out in a similar fashion and the results entered into a specific column in the experimental matrix and on the graph representing the experimental domain (Figure 2.4).

Pressure

D 2 bar

+I

1 bar

-1

B

A t -1

+I

Temperature

80 "C

60 "C

Figure 2.3: Location of experimental points t o obtain an optimal research strategy.

TABLE2.1 EXPERIMENTAL MA= THE YIELD OF A CHEMICAL REACTION

Pressure

-1 +1

4

1

I

Level (+)

8OoC

2 bar

13

Pressure

4

2bar

+I

1 bar

-1

-1

60 OC

+I

Temperature

80 OC

Figure 2.4: Results are entered in the experimental domain.

Results: Four trials are sufficient, and the experimenter can conclude that the greatest yield is obtained by working at 80°C and with a pressure of 2 bar.

This experiment introduces the important concept of the effect of a factor. When the temperature is increased from 60°C (level -1) to 80°C (level +1), the yield increases by 10 units (Figure 2.5), regardless of the pressure. Thus the overall effect of temperature on the yield is + 10 units. The main effect or effect of temperature is by definition, hay of this value, or +5 yield units.

14

pressure A

2 bar

+I

1 bar

-1 I Temperature

+I

-1 60 OC

80 OC

Figure 2.5: Main effect of temperature: 5 YOyield units.

When pressure is increased from 1 bar (level -1) to 2 bar (level +I), the yield increases by 20 units (Figure 2.6), regardless of the temperature. Thus the overall effect is +20 units and the main effect or effect of pressure is + I 0 yield units. Pressure

1 bar

4

' 70%

-1

I

-1 60 OC

+1 80 OC

Figure 2.6: Main effect of pressure: 10 YOyield units.

Temperature

3.

GENERAL FORMULA OF EFFECTS

The above results can be generalised by using literal values. We shall call y , the response of experiment 1, and y , the response of experiment 2, etc. The global effect of temperature is defined as the difference between the average of the responses at the high temperature and the average of the responses at the low temperature (Figure 2.7).

-1

-1

+I

Xl

Figure 2.7: y+ is the average response at the high temperature level. y - is the average response at the low temperature level. There are two responses at the high temperature level, y , and y, . The average response at the high temperature level, y+, is therefore given by:

The average response at the low temperature level, y-, is:

y-

= -[Yl 1

+Y,l

2

The effect ,?I averages,

of temperature is, by definition, half the difference between these two

16

or

Similarly, the effect Ep of pressure, is given by the expression: E =-[-yl 1 -Y, +y3 P 4

+Y~I

Inserting the numerical values of the responses ( Table 2. l), we get: 1 E - -[-60+

70-80+ 901 = 5%

‘-4

E,= -[-60-70+80+90] 1

=10 %

4 The average, I, of all the responses is given by

I = - [1 + 6 0 + 7 0 + 8 0 + 9 0 ] = 7 5 % 4

The formulae for calculating the values of the effects are easily remembered: the responses occur in the order of the trials and are preceded by the signs + or - that appear in the column of the corresponding factor in the experimental matrix. Thus the sequence of signs for the temperature column is:

-

+

-

+

This method is general and we shall use it to calculate the effects in all two-factor factorial designs, whatever the number of factors. We assume that, in the above calculations, all the phenomena studied vary linearly between the experimental points. This assumption is justified by its simplicity and by all the consequencesthat can be deduced fiom it. This is a very usefbl assumption at this stage, and we shall see that it is a first step towards more complex concepts later. So that we do not forget that there are both measured responses and calculated responses, we shall indicate measured responses as filled circles and calculated responses as open circles in all fbture diagrams or figures.

4.

REDUCED CENTRED VARIABLES

The logic behind assigning the value -1 to the low level and +I to the high level merits closer examination, as it leads to two major changes. The first is a change in the unit of measurement and the second is a change in origin.

17

. . *

80 "C

60 ' C

Normal Variables (Temperature)

1 - 1

20 "C

CRV

Centred Reduced Variables

+

-1

+1

Figure 2.8: Comparison of normal units and reduced units.

Change in units of measurement The temperature increases fkom a low level (-1) of 60°C to a high level (+1) of 80°C. There are thus 20 normal temperature units between the extremes of this experimental domain. But if we use -1 and +1 there are only two temperature units between the same two extremes. The new unit introduced by the notation -1 and +1 thus has a value of 1O"C, or ten normal temperature units. This is therefore a reduced variable, and the value of the new unit in terms of normal units is a step. In this case the temperature step is 10°C. Similarly, the pressure step in the above example is 0.5 bar. Change in origin The mid point of the [-1, +1] segment is zero, and this is the origin of the measurements in the new units. In normal units, the origin is not in the middle of the [-1, +1] segment; it lies outside the 60-80°C interval. The origin of the pressure values is similarly changed. The new variables are said to be centred.

0 "C Normal Variables 1 (Temperature)

60 "C

80 "C

1

Centred Reduced Variables

c--c--I -1

Figure 2.9: Normal origin and centred origin.

0

+1

18

The value of the new origin expressed in normal units can be obtained by taking the centre of the experimental domain, which is 70°C for the temperature and 1.5 bar for pressure. Then, if A- is the low level of a variable expressed in normal units, and A+ is the high level of a variable expressed in normal units. so that A,, is the midpoint ofthe [-1,+1] segment, or zero level of the variable expressed in normal units: A +A, A,=

..

2

For temperature, this gives A"=

60+80 ~

2

=70"C

and for pressure

A,=

~

1 +2 2

=

1.5bar

Thus, assigning the value -1 to the low level value of a factor and +1 to its high level value leads to a change of units, and a change of origin. These new variables are therefore named reduced and centred variables, or coded variables.

Normal Variables (Temperature) Centred Reduced Variables

Normal Variables (Pressure)

60°C

70%

I

I

-1

0

I

I

1 bar

1.5 bar

80°C

I +I

I 2 bar

Figure 2.10: All normal variables can be transformed into centred reduced variables.

19

The use of centred reduced variables greatly simplifies the presentation of the theories underlying two-level factorial designs. These centred reduced variables will be used in all subsequent discussions. Normal variables can be converted to centred reduced variables using the formula: x=-

A-A, step

where,

.

-

is the centred reduced variable measure in units of step, A is the variable in normal units (e. g., degrees Celsius or bar), A, is the value (in normal units) of the variable at the mid-point, i.e., the point chosen as the origin for the centred reduced variable.

x

In this case A, is 70°C for temperature and the step is 10°C Applying the formula to temperature, we get: x=-

A-70 10

Substituting A for the temperatures of 60°C , 70°C and 80°C gives the values of -1, 0, and +1 for x.

5.

GRAPHICAL REPRESENTATION OF MEAN AND EFFECTS The mean of all responses, which can be denoted as I or yo,is given by the expression: I = Y o = q1[ + Y , + Y , + Y , + Y 4 ]

or

or, using the low and high temperature means: I = y = -1[ y + f Y - ] 0

2

As the response is assumed to vary linearly, the point represented by y+ is at the centre of the segment [ y,, y , 1, i.e. at the zero pressure level (Figure 2.11). The same is true for y- , the centre of the segment [ y l ,y3 1. Using the same reasoning, the mean y o ofy+ and y - is at the centre of the segment [ y+ , y - 1. The mean of the responses is thus the value of the response at the centre of the experimental domain: level zero for temperature and level zero for pressure.

20

Y

Y

+

-1

0

+I

Xl

Figure 2.11: A 22 experimental design and t h e response surface.

If we now consider the plane passing through the zero pressure level and including the three responses y+, yo and y-, we observe that the straight line joining the responses y+ and y(Figure 2.11) represents the variation in the response on going from the low to the high temperature levels. It therefore illustrates the overall effect of temperature. The mean temperature effect, or more simply, the temperature effect, is half the overall effect. The effect of temperature is shown by the change in the response on going from the zero temperature level to the high temperature level. We will use this concept frequently in the coming pages and we will use diagrams like Figure 2.12 to represent factor effect.

21

RESPONSE

EFFECT OF FACTOR 1

I

0

-1

+1

FACTOR 1

Figure 2.12: Classical diagram to represent factor effect.

6.

THE CONCEPT OF INTERACTION The following example introduces the concept of intera d o n between factors.

6.1. Examp1e:The yield of a catalysed chemical reaction The problem:

iif The same chemist studied the same reaction under the same d conditions But this time a catalyst was added in order to improve the I yield The question now is how pressure and temperature should be f+

regulated. The experimental conditions and results are summarised in J the experimental matrix (Table 2 2). 9

In the previous example, the mean effect of a factor was defined at the zero level of the other factor. But we can also define the effect of a factor for any level of the other factor, for example the low or high level. We shall now do this to examine the concept of interaction. In Figure 2.12, the effect of pressure at the low temperature level is: E

~

P(t

1 =-[SO% 1 2

- 60%] =10 %

while the effect of pressure at the high temperature level is:

22

1

Ep(t+)= -[95% 2

- 7O%] =12.5 %

Thus, the effect of a factor is not the same at the high and low levels of the other factor. It is said that there is interaction between the two factors. The interaction between temperature and pressure is defined as half the difference between the effects of pressure at the high and low temperature levels: E

Pt

1 = -[12.5% -lo%] ~ 1 . 2% 5

2

TABLE2.2

EXPERIMENTAL MATiUX THE YIELD OF A CATALYSED CHEMICAL REACTION

Trial no

Temperature

Pressure

1 2 3 4

-1 +1 -1 +1

-1 -1 +1 +1

60% 70% 80% 95%

We can calculate the interaction for the temperature in the same way. At the low pressure level it is E

1

=-

t(P-)

2

[ 70% -6O%] =5 %

While at the high pressure level it is:

E

1

+

t(P

= -[95%

1

2

-SO%] =7.5 %

The value of the interaction is thus: E

tP

1 2

= -[7.5%

-5%] ~ 1 . 2% 5

23

which is the same as we calculated earlier. This result applies whether we refer to the pressurehemperature interaction or to the temperature/pressure interaction.

Pressure

+l

-1

60 OC

Temperature

80 OC

Figure 2.13: The effect of temperature is not the same a t the low and high pressure levels: there is interaction.

The main effect of temperature is defined and calculated as in the first example - the study of the yield of a chemical reaction, i.e., which is calculated with respect to the zero pressure level. The average response at the high temperature level is: 1 y + = --[95%+70%] =82.5 % 2

and the average response at the low temperature level is: y

-

1 2

= -[60%+80%]

=70 %

The effect of temperature, E, is thus: 1

E t = -[82.5% -7O%] 2

= 6.25%

24

andthe effect of pressure is:

E

1 2

= -[87.5%

P

-65%] = 11.25%

With these results the experimenter can conclude his study.

Results:

The four trials

indicate that the best yield is obtained with a

I temperature of 80°C and pressure of 2 bar. The catalyst has no effect at 60°C or 1 bar. Increasing the temperature alone does not reveal the effects of the catalyst. Neither does increasing pressure alone. Both @ temperature and pressure must be increased for the catalyst to 8 operate

7. GENERAL FORMULA FOR INTERACTION We can now develop the general formula for calculating interaction using the responses measured at the experimental points. The definition of effects remains the same, whether or not there is interaction. The formulae for the response mean, temperature and pressure effects are thus unchanged:

I = -[+y, 1 +Y, +Y3 + Y s ] 4

E

P

= -1[ - y 4 '

-

The interaction between temperature and pressure is indicated by Etp. At the high pressure level, the effect of temperature E (p+ ) is:

while at the low pressure level, the effect of temperatureE 1 Et@- )= TEY2 - Y1

I

t(P- )

is:

25

The interaction Etp is defined as half the difference of these two effects:

This can be simplified to:

This formula looks very like the one used to calculate the mean and effects. It can be obtained by constructing a list of the +1 and -1 having the same sequence as in the formula. This is easy as long as we note that the products of the pairs of elements in the temperature and pressure factor columns give a 12 column in which the signs are in the same order as those of the interaction (Figure 2.13). We can therefore construct an effects matrix (Table 2.3) fiom which we can obtain: the mean: using a column of four + signs. the effects of factors: using the sequence of signs in columns of the experimental design (experimental matrix). interaction between factors: each sign is calculated by applying the sign rule to the corresponding factors. e.g. in trial number 1, factor one is - and factor 2 is -; thus the interaction 12 has the sign (- ) x (- ) = (+) (Figure 2.14).

Factor 1 I

+

+

Factor 2 I

Interaction

- Multiplication

I

+ I

sign

+ +

sign

I

+

Figure 2.14: Calculating the interaction column using the sign rule.

26

The effects matrix therefore has four main columns: one for calculating each effect, one for the interaction, and one for the mean. Columns for the trial number and for the responses are normally included in the table. The divisor and the calculated results are placed beneath the mean, factors and interaction columns. The arrangement is shown in Table 2.3 TABLE2.3

EFFECT MATRIX THE YIELD OF A CATALYSED CHEMICAL REACTION

Interaction

Response 60%

70% 80% 95%

Effects

76.25

6.25

11.25

1.25

27

RECAPITULATION Our analysis of the yield of a chemical reaction has shown: The strategy used in a two-level experimental design. Using experimental points that are the extremies of the experimental domain for each factor gives the best estimate of the effect of each factor. The notion of effect and the calculation of effects. The tools used: -experimental matrix, -graphical representation of the experimental domain on which are placed the experimental results. -graphing the effects in a plane passing through the centre of the experimental domain. The definition of reduced centred variables. The example of the yield of a catalysed chemical reaction: Introduces the concept of interaction. Gives the general formula for calculating interaction Shows how to construct an effects matrix.

CHAPTER 3

TWO-LEVEL COMPLETE FACTORIAL DESIGNS: 2k

1.

INTRODUCTION

Two-level factorial designs are the simplest, but are widely used because they can be applied to many situation as either complete or fiactional designs. This chapter deals with complete designs. We will first examine a simplified example using only two factors. It will allow us to introduce several important basic concepts which will be used in later chapters. We will analyse a three-factor design and extrapolate the ideas acquired in this first example to an actual experimental design having five factors. Lastly, we will use the matrix approach to interpret two-factor complete factorial designs.

2.

COMPLETE THREE FACTOR DESIGN: 23

2.1.Example: The stability of a bitumen emulsion The Problem:

A manufacturer of bitumen emulsion wants to develop a new ak formulation. He has two bitumens, A and B. He wants to know the I

30

9

2

effects of a surfactant (fatty aad) and hydrochloric acid on the stability of the emulsion

As there are three factors, he decides to use a 23 design with the following factors and response Factors = Factor 1 high and low fatty acid concentrations.

..

Factor 2 diluted and concentrated HCI. Factor 3 bitumen A and B.

Response Emulsion stability index, measured in stability points .The scientist knows that the experimental error of the response is plus or minus two stability points. He wishes to find the most stable emulsion: the one with the lowest stability index. Domain The two levels of each factor are indicated by +1 and -1 as reduced centred (or coded ) variables. The experimental domain is a cube (Figure 3 . 1 ) and the eight experimental points chosen are at the corners ofthe cube.

8

7

6

4

Figure 3.1: Distribution of experimental points within the experimental domain.of a Z3 design.

31

The experimental matrix (Table 3.1) is constructed in the same way as for the 22 design, but contains eight and not four experiments. To simpli@ table 3.1 we have used the signs + and - without the figure 1. The factors studied are not necessarily continuous variables, and two level factorial designs may include both continuous and non-continuous or discrete variables.

Trial no

Factor 1 (fatty acid)

Factor 2 (HCl)

Factor 3 (Bitumen) -

1

-

-

2

+

-

-

3

-

-

4 5

+

+ +

-

-

6 7

+

-

8

+

+ +

+ + + +

Level (-)

low conc.

diluted

A

Level (+)

high conc.

concentrated

B

-

Response

~~

The effects of each factor and the interaction values are calculated fi-om the effects matrix (Table 3.2) as they were for the 23 design, i.e. by taking the experimental matrix signs for the main factors and using the sign rule for the interactions. The effects and interactions are obtained by a three-step calculation: The response is multiplied by the corresponding sign in the factor (or interaction) column, The products obtained are added, The sum so obtained is divided by a coefficient equal to the number of experiments For example, the effect of factor 3 is obtained fiom the formula: 0

1

E, = -[-388

37-26-24

+30+ 28+ 19+ 161

= -4

similarly, the third order interaction, 123, is obtained from: 1 E 123= -[-3 8

8+37+26-24+30-28-19+

161 = 0

32

r' +

6 7 8

+ + +

Effects 27.25

TABLE3.2

~1

EFFECTS MATRIX STABILITY OF A BITUMEN EMULSION

Inter. 23

Response

+ + +

-

+

-

+

-

+

+ +

-1

-6

+ + -4

-0.25

-0.25

0.25

0

The experimenter then analyses the results by drawing up a table of effects indicating, whenever possible, the experimental error estimated by the standard deviation (Table 3.3).

TABLE3.3 TABLE OF EFFECTS STABILITY OF A BITUMEN EMULSION

Mean

27.25

k 0.7 points k 0.7 points +_ 0.7 points 0.7 points

1

-1.00

2 3

-6.00 -4.00

12 13 23

-0.25

123

*

0.25

k 0.7points k 0.7 points k 0.7 points

0.00

k 0.7 points

-0.25

33

We can now begin to interpret these results. All the interactions are smaller than the standard deviation. These can therefore be considered to be zero and neglected. Factors 2 and 3 are much greater than the standard deviation, and thus have an influence, while factor 1 is just a little larger than one standard deviation and much smaller than two standard deviations. It is thus unlikely to have any influence STABILITY

A

33.25 27.25 21.25

-1 DILUTE

o

+ CONCENTRATED

HCI CONCENTRATION

Figure 3.2: Effect of hydrochloric acid (factor 2) on bitumen emulsion stability.

STABILITY

31.25

0

27.25 23.25

-1

+l

A

B

BITUMEN

Figure 3.3: Effect of bitumen type (factor 3) on bitumen emulsion stability.

34

Therefore, the concentration of fatty acid (factor 1 ) probably has no influence on emulsion stability over the range of concentrations tested. The plane passing through the centre of the experimental domain and parallel to factor 2 shows the effect of hydrochloric acid. The plane passing through the centre of the experimental domain and parallel to factor 3 reveals the effect of bitumen. We can now state the results of the experiment: Results:

*

The fatty acid concentration has little or no influence on the emulsion stability. The hydrochloric acid concentration has a large effect The ; I type of bitumen used is also important, the best stability (lowest Q i response) will be obtained with type B and dilute HCI There IS no xx s significant interaction. B

g

Note:

A negative effect is not necessarily an undesirable one. An effect is negative when the response falls as the factor increases from -1 to + I . Conversely, a positive effect occurs when the response increases as the corresponding factor goes from -I to + I

3. THE BOX NOTATION We could also use the Box notation [8] to indicate the effects and interactions. With this notation El is represented by a bold figure 1 (I), and E, = 2, E, = 3, etc. The mean is represented by the letter I The general formulae for the effects and interactions of a 2 3 design are’ Mean

=

1 1 = -[+Y, + Y , + y 7 +Y, + Y ~+ y 6 + y 7 +y81 8

35

4.

RECONSTRUCTINGTWO 22 DESIGNS FROM A 23 DESIGN

Examining the results in a little more detail, we see that, as factor 1 is without influence, the experimental domain is reduced to a design in which only factors 2 and 3 have any influence. This also indicates that the response does not depend on the level of factor 1, but only on the levels of factors 2 and 3 . The responses can therefore be rearranged in pairs ignoring the factor 1 level, as shown in the following table (Table 3.4). TABLE3.4 EXPERIMENTAL MATRIX REARRANGED STABILITY OF A BITUMEN EMULSION.

Trial no

7

8

Response

+

37 24 28 16

38 26 30 19

+

Average 37.5 25.0 29.0 17.5

These results can also be displayed graphically, as in Figure 3.4 29 CONCENTRATED +1

[fx

16 19]i7.5

HCI

A -1

Bitumen

B

+1

Figure 3.4: The bitumen emulsion is most stable when the hydrochloric acid is dilute and bitumen B is employed.

36

5. THE RELATIONSHIP BETWEEN MATRIX AND GRAPHICAL

REPRESENTATIONS OF EXPERIMENTAL DESIGN This relationship is easy to understand for a 22 experimental design. An experimental point A can be defined: 1 . by its coordinates in a Cartesian two dimensional space: a on the Ox, axis (horizontal) and b on the Ox, axis (vertical) as show in Figure 3.5. This is the graphical representation. The coordinates of a and b can be expressed in centred reduced (or coded) units or in classical units.

"T

/A

Figure 3.5: Geometric representation of experimental points 2. by the level of the two factors studied, trial A is defined by level a of factor x, and level b of factor x2. The coordinates of experimental points are the levels indicated in the experimental matrix X2

P

b

TRIAL NAME

a'

P

P'

7b'

-

Figure 3.6: The matrix diagram of experimental points is equivalent to the geometric representation

37

A set of experiments is defined by several points with geometrical representation and by several trials with matrix representation. Figure 3.6 illustrated these two ways of representating two experimental points and the two corresponding trials. While it is also possible to produce a graphical representation of a three factor experiment in a three dimension space it is clearly impossible to do so for four and more factors. It is therefore necessary to find a way of representing experimental points in these hyper-spaces which is both convenient and applicable to any number of dimensions. The most common solution is to use matrix representation, which works for any numbers of factors. Table 3.5 shows four trials defined by the level of seven factors. TABLE3.5

The geometrical counterpart of Table 3.5 is a set of four points defined by their seven coordinates. Hence the experimental matrix gives the location of experimental points in the experimental space, Anyone producing experimental designs must learn to think in ndimensional space without graphical representation. It is easy to pass from geometrical to matrix representation for two or three factors and experimenters must become accustomed to switching from n factor matrices to n dimensional space and vice versa.

6.

CONSTRUCTION OF COMPLETE FACTORIAL DESIGNS

All factorial designs are constructed in the same way as those shown in Tables 2.2, 2.4 and 3.7. The sequence of the signs for factor 1 is: -

+

-

+

-

+

-

+

,etc.

They alternate, commencing with a negative (-), The sequence of the signs for factor 2 is a series oftwo -, followed by two +: _ _

+ +

- -

+ + ,etc.

38

The sequence for factor 3 is four negatives (-), followed by four positives (+). Any hrther factors have 8, 16, 32, - signs followed by 8, 16, 32 + signs. There is always the same number of + and - signs in the column for each factor.

7.

LABELLING OF TRIALS IN COMPLETE FACTORIAL DESIGNS

When the + and - signs for each factor are laid out as shown above, the trials are numbered sequentially using whole numbers.(see Tables 2.2, 2.4 and 2.7). This is Standard numbering. As we will see later, the order of the trials can be changed, for randomisation, drift or blocking designs. But the number of each trial will be retained, regardless of its position in the layout. For example, trial number 23 of a complete 25 design (Table 3.7) always has the sequence of levels taken by factors 1 , 2 , 3 , 4 and 5:

-++-+ There are other ways of labelling trials, but we shall not discuss them here

8.

COMPLETE FIVE FACTOR DESIGNS: 25

8.1. Example:

Penicillium chrysogenum growth medium

The Problem: This design was used in a study to increase the yield of a penicillin production plant It was reported by Owen L Davies [Illin his book "The design and analysis of industrial experiments" Penicillium chrysogenum is grown in a complex medium, and the experimenter wanted to know the influence of five factors

8 *3. i

y.

f %-

F

-f

5

1 concentration of corn liquor 2 concentration of lactose

3 concentration of precursor 4 concentration of sodium nitrate 5 concentration of glucose

The response was the yield of penicillin, as weight (the units were not given in the original text). The experimental matrix of the 22 design summarizes the experimental data and the results of each of 32 trials.

39

TABLE 3.6

EXPERIMENTAL MATRIX PENlClLLlUM CHRYSOGENUM GROWTH MEDIUM

Trial no 1 2 3 4

Factor 1 (corn liq.)

Factor 2 (lactose)

Factor 3 :precursor)

-

-

-

-

-

-

+ +

-

-

+ -

+

5

-

-

6

+

-

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 LevelLevel +

+

+ +

+ + + +

-

-

-

+

-

-

+ +

-

-

-

+ -

-

+

-

+

+ +

+ + + +

-

-

-

+

-

-

-

+

-

+

+

-

-

+ -

+ -

+ + +

1

2% 3%

-

-

-

-

+ + + + + + + + + + + + + + + +

-

+ + + +

+ +

+

Factor 5 (glucose)

-

-

-

2% 3%

-

-

-

I

-

+ +

-

I

+ +

+ + + +

Factor 4 (sod.nit.)

1 I

0 0.05%

[

0 0.3%

I I

0

0.5%

Response 142 114 129 109 185 162 200 172 148 108 146 95 200 164 215 118 106 106 88 98 113 88 166 79 101 114 140 72 130 83 145 110

40

The effects were calculated by the standard procedure and the results are shown in the table of effects (Table 3.7). TABLE 3.7

TABLE OF EFFECTS PENlClLLlUM CHRYSOGENUMGROWTH MEDIUM

Mean

129 6

1 2 3 4 5

-17 6 06 16 1 10 20 9

12 13 14 15 23 24 25 34 35 45

-5 9 -6 1 -5 0 26 44 -1 0 30 -1 0 -10 5 22

123 124 125 134 135 145 234 235 245 345

-I 3

1234 1235 1245 1345 2345

40 26 18 42 11

12345

63

-2 9 -1 6 17 -3 2 28 -2 6 27 23 06

41

Analysis of the effects of the factors showed that two factors have no influence: Factor 2, the concentration of lactose. Factor 4, the concentration of sodium nitrate And that the effects of three factors are significant:

0

Factor I, the concentration of corn liquor Factor 3, the precursor concentration. Factor 5, the glucose concentration.

A second order interaction appears to be significant: 0

0

Interaction 35, between precursor and glucose. interaction 12345 seemed to be abnormally large. We will leave this for the time being, but come back to it later.

TABLE3.8 EXPERIMENTAL MATRIX REARRANGED PENlClLLlUM CHRYSOGENUM GROWTH MEDIUM

Trial no 1 2 5 6 17 18 21 22

3 4 7 8 19 20 23 24

9 1 1 10 12 13 15 14 16 25 27 26 28 29 31 30 32

Factor Factor 1 3 -

-

+

-

+

+ +

-

-

-

+ -

-

+ +

+ -

Results

Average -

142 129 148 114 109 108 185 200 200 162 172 164 106 88 101 106 98 114 113 166 130 88 79 83

146 95 215 118 140 72 145 110

141.25 106.50 200.00 154.00 108.75 97.50 138.50 90.00

If we look at the three factors which do influence the growth of Penicillium chrysogenum, we see that there are 32 trials, but we know that only 8 trials are required to study three factors. We can therefore group together the trials having the same levels for factors 1 , 3, and 5, regardless of the levels of 2 and 4. For example, trials 1, 3, 9, and 1 1 were carried out at the low level of factors 1, 3 and 5, so that the results of four trials should be the same, allowing for experimental error. The 32 trials are used as if four 23 designs had been performed. Table 3.9 shows the rearrangement of trials and the mean responses for each group. Thus, it appears as if a three factor design was repeated four times.

42

TABLE3.9

TABLE OF EFFECTS PENICILLIUM CHRYSOGENUM GROWTH MEDIUM

Mean

129.6i6

1

-17.6i-6 16.1 k 6 -20.9+6

3 5

-6.1 + 6 2.6*6 35 - 1 0 . 5 f 6 13 15

135

-3.2+6

The experimental domain is reduced to a cube for the three influencing factors. We can therefore introduce the mean of each response at each corner of the cube to facilitate interpretation (Figure 3.7)

138

90

/+ 108

97

GLUCOSE

200

(5) 9 . 0 (

Figure 3.7: Diagram showing the results of the trials on Penicillium chrysogenum medium.

43

A high percentage of corn liquor (factor 1) evidently reduced the yield of penicillin. At a low level of factor 1, the yield was clearly improved by the addition of precursor and the absence of glucose. The presence of glucose reduced the effectiveness of the precursor. Results:

I

Under the experimental conditions used, the best yield of penicillin is obtained

b

0

Y

with a low (2%) concentration of corn liquor

I 0 with precursor.

Q

0

without glucose, which reduces the yield and inhibits the precursor

138

108

141

200

Precursor 0%

0.05 %

Figure 3.8: Influence of precursor and glucose at a corn liquor concentration of 2%. The interaction 12345 appears to be too great; and we will look at the reason for this in the chapter on blocking (Chapter 10).

9.

COMPLETE DESIGNS WITH k FACTORS: 2k

We have seen that 22 , 23 and 25 designs can be used to study two, three or five factors. A 2k design can be used when there are more factors, with k having any desired size. The experimental matrix and the effects matrix are constructed according to the same rules as were used previously. The calculation of the k major effects and the 2k-k-1 interactions are similarly performed. There is thus no theoretical limit to the number k of

44

factors that may be studied. But in practice the number of trials needed quickly becomes very large. A total of 27 (128) trials are required to study only 7 factors. This is a considerable number, and is rarely compatible with the facilities generally available in industry or university. This brings us to a most troublesome problem. We must find a way of reducing the number of trials without reducing the number of factors studied. We will examine this problem in Chapter 6.

10. THE EFFECTS MATRIX AND MATHEMATICAL MATRIX The effects can be calculated from the experimental results using an effects matrix, which, for a 22 experiment, looks like (Table 3.10).

Trial no

Mean

Factor 1

Factor 2

1 2 3

+I

-1 +I

-1 -1

-1

+I +I

4

+I +1 +1

+1

7 Interaction

This array of numbers can be used for a calculation; it is thus a mathematical tool, a Matrix. It can be written: +I

-1

-1

+1

+I +I +I

+1 -1 -1 +1

-1 -1

+I

+1

+1

A mathematical matrix is simply a table containing elements (here they are numbers) arranged in rows and columns. When the number of rows equals the number of columns the matrix is said to be square - otherwise it is rectangular. A matrix may contain just a single row and several columns (a linear matrix) or a single column and several rows ( a column matrix, or vector matrix). We will use matrices to express experimental results. Theyi may be shown in a rather special table because its contains only one column. They response vector matrix is:

Y=

Y2 Y3 Y4

An analogous matrix can be written for the effects:

45

E=

Before we use these matrices we will examine the operations which can be performed on a single matrix, or between matrices themselves. The operations we need are transposition for a single matrix, and matrix multiplication for two or more matrices.

+1

+I

+I

+1

x t = -1

+1 -1

-1 +1

+1

+1 -1

-1

+1

-1

+l

The second operation we will need is the multiplication oftwo (or more) matrices 10.2. Matrix multiplication Any reader not familiar with matrix calculations should read Appendix 1 before continuing with this Chapter. If we multiply matrix Xt by Y we get: +I

-1 Xt Y = -1

+I

+I +1 +I -1 -1 +1

+1

-1

+1

-1

+1

+1

y1 y1

y2 y4

The first element of the matrix-product is: [+Yl

+Yz

+Y3

+Y'll

or four times the mean of the responses. Similarly the second element is: [-Y1 +Y2

-Y3

+Y,I

46

or four times El, the effect of factor 1 . The calculations for the results of the third and fourth elements of the matrix-product are similar. We can therefore write: +I -1

+1

+l

+1

+I

-1

+1

-1

-1 -1

+I

+1

~3

E,

-1

+I

y,

El,

+I

y, y 2 =4

I El

which can be condensed to X'Y = 4E or

E = -1X t Y 4

This relationship for a 22 design can be extended to all two level complete factorial designs. When n is the number of trials we have 1 n

E = -X'Y We now have, in the form of a matrix, the technique we used to calculate the effects and interaction of 2k designs. The matrix form clearly shows that the experimental responses y j have been transformed by the matrix Xt so as to be more readily interpreted. A factor increases (or reduces) the mean of responses I by a quantity equal to its effect. In the first example we examined, the yield of a chemical reaction, the four responses 6O%, 70% 80% and 90% were difficult to interpret. But when they are transformed by the matrix Xt, the effect of each factor is obtained as if it were alone. A 10°C rise in temperature increases the yield from 75% to 8O%, while a pressure rise of 0.5 bar increases the yield fiom 75% to 85%. When the responses of a 2k are examined it is impossible to distinguish the influence of each factor. But the transformation by the Xt matrix displayed the useful information in the set of responses more clearly, revealing the effect of each factor as if it were alone. As the matrix X is the mathematical translation of the location of the experimental points, it is clearly most important that these points should be optimally placed in the experimental domain. Poorly positioned experimental points obscure the information instead of highlighting it. Well positioned experimental points clarifl the information (Chapter 16). The analysis of the specific X matrices which are use in all two level factorial designs can be developed a little. These are the Hadamard matrices, and they have quite remarkable properties. Let us first calculate the reciprocal of the X matrix and then examine the product of X and its transpose, XtX. 10.3. Inverse of X

The calculation of the inverse of a matrix is complicated for the general case, requiring a computer for high order matrices. But the calculation for X matrices of factorial designs (Hadamard matrices) is greatly simplified because of the following relationship:

47

The inverse of X can be obtained by transposing X and dividing all the elements of the Xt matrix by n, the number of trials. The relationship X'Y =nE then becomes X-'Y = E

or

Y=XE This formula can be used to calculate the responses from the effects

+1

+1 +I

-1 +1 --1 -1

-1 +1 +1 +1

+1 -1

-1

+1

+1

+1 -1 -1 +1 +1 -1 +1 - 1 + I

+1 -1 -1

+I

+1

+1 +1

4 -

0

0 4

0

0

1 0 0 0

0 0 0 1 0 0 = 4 4 0 0 0 1 0

0

0

0

0 0 4

0 0 0 1

which can be condensed to XtX =41 For this design, the product of the matrix of effects by its matrix transpose is 4 times the unit matrix. The general form of the formula for all two level complete factorial designs is X'X =nI where n is the number of trials. The matrix XtX is equal to n times the matrix unit in the case of two level factorial designs. It can be demonstrated that, in this case, the precision obtained for the effects is the best than might be hoped for (see Chapter 5). Experiments in which a two level complete factorial design is used are certain to provide calculated effects with maximum precision. 10.5. Measurement units

The responses yi were measured with a unit, metre, centimetre, volt, etc., or a less usual unit, such as an index, percentage or variance. The matrix Xt does not change the unit in which yi is measured, it simply transforms the trials results into a system that is easier to interpret. As a result, the mean, the main effects and the interactions are evaluated in the unit used to measure the responses.

48

RECAPITULATION. We have used the example of bitumen emulsion stability to: Extend the concepts acquired with 22 design to a z3 design. Extend the concept of interaction. Examine the rules for calculating effects and interactions from an effects matrix. Introduce Box notation. Present the results as a table of effects. Show that both continuous variables (temperature and pressure) and discrete variables (type of bitumen) can be studied simultaneously within the same experimental design.

Using a 25 design allowed us to: Apply the principles acquired to a real case. Use the fact that some factors were without influence to construct a replicate factorial design. The mathematical matrix representation of two-level factorial designs was used to: Calculate effects, interactions and mean from the responses. Introduce transposed matrices, product matrices and vector matrices. To simplify the interpretation of results which are transformed into mean, effects and interactions. Guarantee that the effects and interactions calculated have the highest possible precision. Define the units for measuring effects and interactions.

CHAPTER 4

ESTIMATING ERROR

AND SIGNIFICANT EFFECTS

1.

INTRODUCTION

Let us now examine a problem that we touched on lightly when we discussed the bitumen emulsion stability example in Chapter 3. The problem raises two questions, which we shall attempt to answer in this chapter: When can an effect be considered significant? On what criteria can such a conclusion be based? The method generally used to answer them requires estimating the error AE in the determination of the effect E, and comparing this error with the effect itself. There are three possible situations: The effect is much larger than the error: E>> AE In this situation, there is no problem, the effect clearly has an influence.

50

The effect is smaller than the error:

E

I

12

1.5

2

3

4

f

(4

1

1 09

1.22

1.41

1.73

2

In the paste-hardening example the standard deviation of the responses was estimated as & 2 minutes. Eight trials were run. If we apply the above formula, we get:

1

v(y)= -D; n

[I+ x;

4

+ x4‘ + x:xqz]

=8 f2

(x)

1 2 f 2 (x)

-

The standard deviation of the responses calculated from the mathematical model,oyc, is the square root of this variance: 1 0% --- ff(ix ) We only need to divide all the values of f(x) by in the model for the paste hardening example.

fi to know the confidence we can have

Figure 1 1.10 shows the standard deviation curves cYcof the responses calculated for the paste hardening example. These deviation curves can be superimposed on the isoresponse curve network. f(x) = 2.0

- T

x2

0

f(x)= 1.41

b

’

* ’

f(x) = 1.22

\

Xl

/

a

Figure 11.10: The precision of calculated responses is not the same throughout the experimental domain. It is best at the central point. The further from this point, the less accurate the model.

237

RECAPITULATION 1. The mathematical model associated with factorial designs is a first degree polynomial

for each of the factors taken independently. 0

0

0

The effect of a factor is assumed to add algebraically to the effects of other factors. The mathematical model for factorial designs is established with coded variables. The polynomial coefficients are thus simply the mean, main effects and interactions. The model is valid for continuously varying variables. The yield of a chemical reaction example has provided us with formulae which can be used to go fi-om values in coded units to values in the more usual physical units, and vice versa. The sign rule is used to establish the + and - sign columns for the interactions derives from the mathematical model for factorial designs.

2 . The paste hardening example showed that, despite the fact that the trials were run at the extremities of the domain, they provide predictions for the whole of the experimental domain.

The validity of the model must be checked. Supplementary trials should be run, if possible, at the centre of the experimental domain. The calculated and measured responses at this point are then compared. We have also used this example to show how the mathematical model can be used to draw isoresponse curves within the experimental domain, or even outside it. 3 . The main effects of factors are the direction cosines of the steepest ascent vector

(coded values). 4. Isoresponse curves and the steepest ascent vector can be used to predict the regions

in which there is the greatest likelihood of success. Confirmatory experiments are required to verify the assumptions made. 5 . The sugar production example revealed the analogies between analysis of variance

and factorial designs: the same mathematical model, same results, and the same way of determining significant effects. These similarities are summarized in the formula: YtY

=

n E'E

The main inconvenience of analysis of variance is that this method uses squares: the signs are lost and comparisons made difficult. Factorial designs give the effects themselves, together with their signs, making interpretation much easier

23 8

6. Residuals analysis is used to extract information still contained in the responses after model-fitting. Interpretation is often delicate and relies on the good sense and

intelligence of the investigator. 7.

The mathematical model does not have the same precision throughout the experimental domain. It is most accurate at the central point.

CHAPTER 12

CHOOSING COMPLEMENTARY TRIALS

1.

INTRODUCTION

In the preceding chapters we have seen that a problem is first studied by testing several factors using a fractional experimental design. But this approach has the disadvantage of providing contrasts in which the effects are aliased. This results in ambigpities which must be resolved by carrying out more trials. The setting up of the spectrofluorimeter example showed how a complementary design was constructed to dealias the contrasts giving difficulty. It is sometimes possible to run just a few extra trials rather than a complete complementary design. This chapter shows how to make such a choice. Any doubts remaining after the initial fractional design can be eliminated by running one, two, three or four extra trials.

240

2.

A SINGLE EXTRA TRIAL

Example: Clouding of a solution The problem: ti -r;

The experimenter wishes to know what is causing a slight cloudiness in a solution containing several components. The factors chosen for study were:

.Factor 1: +.

3 9

F

. .

#

temperature.

Factor 2:

product A.

Factor 3:

product B.

Factor 4:

stirring speed.

The response is an index of opacity which accurately reflects the way in which the cloudy appearance of this solution varies with the factors studied.

The experimenter has carried out a fractional Z4-' design using I generator. The contrasts are therefore: 1 + 234 2 + 134 3 +124 4 +123 I , , = 12 + 34 h,,=13+24 3Llq = 14+ 23 h = I +1234

h, = h,= h,= h4=

=

I

1234 as alias

24 1

TABLE12.1 EXPElUMENTAL MATRIX CLOUDING OF A SOLUTION

Response

17.0

The contrasts are calculated from the experimental results and entered in the table of effects (Table 12.2)

TABLE12.2 TABLE OF EFFECTS CLOUDING OF A SOLUTION

Mean

11.01

1 2

3

3.86

4

12 + 34 13 + 24 14 + 23

1.96 0.34 0.34

1

242

There are two influencing factors.

.

Temperature : factor 1 .Product B : factor3

We can see that the sum of interactions 12 + 34 is large. Which is larger, 12 or 34? To find out, we must dealias 12 from 34. The design chosen (I=1234) gives the contrast: h,,

= h,, =

12 + 34

We must find a trial which gives

h',,

=h'34

= 12-34

This trial is in the complementary design I = -1234 (equivalence relationship), and was therefore not run during the first set of trials. Table 12.3 shows the eight possible trials.

TABLE12.3 EXPERIMENTAL MATRIX CLOUDING OF A SOLUTION

Trial no

remperature 1

l 2

Product A

9 10

I1 12 13

Product B 3

Stirring Spd 4 = - 123

-

+

-

-

-

-

-

+

+ + + +

14 15

16

-

+ + -

Level (-)

15°C

0 Yo

0 Yo

I00 rpm

Level (+)

30°C

I Yo

0.5 Yo

300 rpm

r Respo

If we assume that only influencing factors are 1 and 3 and the influencing interactions are 12 and 34, the mathematical model of the responses from this design is given by the formula:

243

yi

= 1+1+3f(12-34)

The difference 12 - 34 can be determined using any of the trials in Table 12.3, e.g., the easiest one to run under the particular experimental conditions, Let us assume that the experimenter has chosen trial number 10. The response y,, is obtained when factor 1 is at the high level (+) and factor 3 at the low level (-), The mathematical model gives the value of this response as:

ylo = l + l - 3 - ( 1 2 - 3 4 ) The numerical values of 1, 1 and 3 were calculated from the initial design and the value ofy,, was determined by the extra trial.

I = 11.01 1 = 4.34 3 = 3.86 ylo = 13.03 We get, 13.03 = 11.01 + 4.34 - 3.86 - (12 - 34)

or 12 - 34 = -1.54

The first set of eight trials gave: 12 + 34 = 1.96

and hence the system: 12 + 34 = +1.96 12 - 34 = -1.54

adding this two equations 12 = 0.21

and subtracting them: 34 = 1.75

Interactions 12 and 34 were dealiased and we can conclude that interaction 34 makes the greatest contribution to 12+34.

244

The extra trial was not run at the same time as the eight trials of the initial design. It could thus produce a distortion due to the fact that the non-controlled factors were set at different levels. This then results in a change in the mean of the mathematical model of factorial designs. When we set up the system of equations to calculate 12 and 34, we assumed that the mean I was the same as that of the initial trials. This may be a bit risky. If we want to avoid any risk, we need an additional equation to measure the mean I' of the mathematical model of the extra trial. Hence we must run two extra trials rather than one.

3.

TWO EXTRA TRIALS

Example: Clouding of a solution ( block effect) The experimenter suspects a shift in the mean. He has available the eight trials of the initial design, to which he adds the results of new two extra trials. These two trials were chosen from the eight trials of the complementary design I = -1234. The choice is very wide as 12 - 34 can be calculated from all the trials. We will use as an example the high level of factor 1 and the low level of factor 3 . This means we must run trials 10 and 12 (Table 12.3). The results of these two extra trials are: y,, y,,

=

13.03

=

9.73

These results are all shown in Table 12.4 as two blocks, one with the eight initial trials having a mean I, and the other with the two extra trials having a mean 1'. The second block may my considered as a Z4" design containing two trials and having the alias generator set:

From which we can calculate the value ofthe contrast h I 2

iI2 = 12- 34 + 2 - 123 - 134 + 4 - 23 +14 But we know that: 2=123=134=4=23=14

Thus:

h',2 = 12-

34

0

245

TABLE12.4

EXPERIMENTAL MATRIX CLOUDING OF A SOLUTION

Initial design + complementory trials

Trial no Temperature

Product A

Product B

Stirring Spd

1

2

3

4 = 123

1

-

-

-

-

2 3 4 5 6 7 8

+

-

-

-

-

-

+

+ +

+ + + +

+

5.2 9.5 1.1 12.8 11.8 17.0 8.6 22. I

Product A

Product B

Stirring Spd

Response

2

3

4 = -123

-

-

-

+

-

+

+

+ +

-

-

+

-

-

-

+

Trial no Temperature 1

10 12

+ +

Response

-

+ + -

13.03

We obtain a system without the means. In fact, they appear the same number of times with + signs and - signs in the expressions giving h,, (first block) and h,, (second block): 1 ~ 1 2= 12 + 34 =

1

-[ +YI 8

- Y2 - Y3

f

Y4 + Y s - Y6 - Y7 + Y s

Substituting the experimental data:

h,,

I 8

= 12+34 = -[+5.2-9.5-1.1+12.8+11.8-17-8.6+22.1]

]

246

15.7 h,, = 12+ 34 = __ = 1.96 8

1 -3.3 i, =, 12-34=-[-13.03+9.73]=-==1 2 2

65

hence the system: 12 + 34 = 1.96 12-34:-1.65

giving: 12 = 0.15 34 = 1.80

Again, interaction 34 clearly makes the greatest contribution to the sum 12+34. The small difference between the values of 12 and 34 obtained with a single extra trial and two extra trials indicates a shift in the mean of the two blocks from 1 1.01 to 1 1.38. The experimenter can thus come to the following conclusions: Conclusion: Product A has no influence on cloudiness , and can thus be used between 0% and 1% Clouding always occurs when the temperature is set at the high level. The low temperature (15°C) must therefore be used. At low temperature, there is always cloudmess when product B is present. Cloudiness can be avoided by I

-

3

1. Using product A 2 Working at low temperature 15°C

J

b -

1 3$

4. THREE EXTRA TRIALS Let us now leave the solution clouding example and return to the fabrication of plastic drums discussed in Chapter 7.

247

The problem was: &

The plastic drums must have a volume of two litres We need to find the fabrication condittons which provide a volume of at least two Iltres, 4, but not more than 2 002 litres At least two lttres to give clients full value, and not more than 2 002 lltres to avoid being too generous

A 284 design was run and there were four significant contrasts

h, = 3+127+146+158+245+268+478+567 h , =5+126+138+147+234+278+367+468 h8 =8+124+135+167+236+257+347+456

h,, = 15+26+38+47+.. We can see that the main effects are aliased with third order interactions. The interpretation hypotheses we have adopted allow us to assume that these interactions are negligible. There are thus only three significant effects, and their values can be calculated from the initial design (Chapter 7, section 5.2):

h, = 3 = +3.2 c m 3

hi = 5 = - 2 . 7 c m 3 h, = 8 = + 1 . 9 c m 3 h 1 5= 1S+26+38+47 = + 2.5 c m’

his.

We do not know what interaction is responsible for the large value of contrast We must therefore run extra trials in order to calculate each of the four interactions making up contrast This requires three extra trials. These three extra trials and the contrast A,,, obtained from the initial design, lead to a system of four equations and four unknowns. The signs of the iiiteractions are selected so that the system is a Hadamard matrix: -15-26+38+47 +I5 -26 - 38 + 47 -15+26-38+47 +15+26+38+47

Many of the 240 trials that were not run satisfy these conditions. We can write only those interactions having the assigned levels and deduce the signs of the main factors from them.

248

1 2 3 4 5 6 7 8

signs to deduce

15

26

-

-

C

-

+

+ +

38

+

-

47

+ +

- + + +

We can choose at random, or we could impose extra constraints. For example, we could choose the levels of influencing factors so that the new trials can be included in the calculation of the effects of these factors. To do so, factors 3, 5 and 8 should have the signs of a Hadamard matrix, while 3 and 8 should correspond to the signs of interaction 38. 1 2 3 4 5 6 7 8

+

+ -

-

-

+

+

-

-

+ +

The signs for columns 1, 2,4, 6 and 7 are deduced from those of columns 3, 5, 8, 15, 26, 38 and 47, again with a certain freedom, producing the following table:

TABLE12.5

ITriinOI

1 ;I ;;1 - + + + + -

+ +

-

Response

8

- + - + - - + + + + - + - +

3+8 471

We can now write the three responses y,,, y,, and y,, and the contrast mathematical model:

7.6

4.4

A,, using the

Y , =~1 - 3 + 5 - 8 - 15 - 26 + 38 + 47

y,, = I + 3 - 5 - 8 + 15 - 26 - 38 + 47 J J ,= ~

115 =

I- 3-5

+ 8 - 15 + 26- 38 + 47 + 15 + 26 + 38 + 47

These four relationships form a system of equations. The unknowns are the second order interactions. The known elements are the mean I and effects 3, 5 and 8, that were calculated from the results of the initial design. The value of contrast A,, is also obtained from the initial design.

249

h, h3 hj h, hi5

I = 3 = = 5 = = 8 = = 15 + 26 + 38 + 47 = =

=

+S.ISC~~ +3.2cm3 -2.7cm' +1.9cm3 +2.5 cm3

with these values the system of equations becomes:

~ 1 = 9

5 . 1 5 - 3 . 2 - 2 . 7 - 1.9- 15-26+38+47 5.15 + 3.2 + 2.7 - 1.9 + 15 -26 - 38 + 47 5 . 1 5 - 3 . 2 + 2 . 7 + 1.9- 15+26-38+47

h,,

2.50

~ 1 = 7

yi,

=

=

or Y , =~-2.65 yi8 =

- 15 - 26

+ 38 + 47

+9.15 + 15 - 26- 38+47

yI9 = +6.55 - 15+26- 3 8+47 2.5

=

+ 15 + 26-

38 +47

Replacing the responses by their numerical values: -15 -26 + 38 + 47 = +2.65 + 0.20 = +2.85 +15-26-38+47=-9.15 + 7 . 6 0 = -1.55 -15 +26 - 38 + 47 = -6.55 + 4.40 = -2.15 +15+26+38+47= = +2.50

Lastly.,resolving the system: 15 = +0.06 26 = -0.23 38 = +2.26 47 = +0.4 1

Thus, interaction 38 is significant. But we have assumed that the mean of the second block was the same as that of the first block. If we wish to take into account any shift in this mean we must run one more trial and carry out the calculations on twenty trials, sixteen initial trials and four extra ones.

250

5.

FOUR EXTRA TRIALS

In addition to trials 17, 18 and 19, we shall run trial 20, whose mathematical representation contains the expression: +15 +26 + 38 + 47

The four extra trials are shown in the following table (Table 12.6).

TABLE12.6 Trial no

1

17 18

-

19 20

+ +

2

3

4

5

6

8

15

26

38

47

- - + + + - + - - + - + - + + + - + - + + + + + + + + + + +

+ - + + - + + + - -

+ +

7

Response

+

10.2

We can write the four responsesyl, ,y,,, y,9 and y20using the mathematical model:

y,, = 1'-3+5-8-15-26+38+47

y,8 = I ' +3-5-8+15-26-38+47 y,!, = 1 ' - 3 - 5 + 8 - 1 5 + 2 6 - 3 8 + 4 7 y20 = 1' + 3 + 5 + 8 + 1 5 + 2 6 + 3 8 + 4 7 This system can be resolved by assuming that the effects and interactions are the same as those given by the initial design: A3

=

3

=

hj h, h,,

=

5

=

= =

8 = 15 + 26 + 38 + 47 =

+3.2cm3 -2.7 cm3 +19cm3 +2.5 cm3

Entering these values in the system ofequations, we get: yI7 = 1 ' - 3 2 - 2 7 - 1 9 - 1 5 - 2 6 + 3 8 + 4 7 .yI8 = 1 ' + 3 2 + 2 7 - 1 9 + 1 5 - 2 6 - 3 8 + 4 7 y,, = 1'-32+27+19-15+26-38+47 4'20 = 1 ' + 3 2 - 2 7 + 1 9 + 1 5 + 2 6 + 3 8 + 4 7

25 1 The fourth equation allows calculation of I' 10.2 = 1' + 4.9

I'

=

5.3

Substituting this value in the four equations gives a system analogous to that of the previous section, and from which the values of each interaction can be deduced. -15 -26 + 38 + 47 = + 2.50 + 0.20 = +15 -26 - 38 + 47 = - 9.30 + 7.60 = -15 +26 - 38 + 47 = - 6.70 + 4.40 = +I5 +26 + 38 -t 47 = - 7.70 + 10.20 =

+2.70 -1.70

-2.30 +2.50

or 15 = -0.10 26 = -0.20 38 = +2.30 47 = +0.30 Again, the result is similar, with interaction 38 being significant Comments

The results of four extra trials may be added to the sixteen results of the first set of trials to calculate the effects of 3, 5 and 8. The contrasts then contain twenty terms and there is, theoretically slightly better precision. Take, for example, contrast h, : 1 ---[-y, - 20

-)'2

- ) ' 3 -)'4

+)'5 + y 6 +?'7 +)'8 -)'9 -1110 -yll -)'I2

+)113 +yl4 +yl5 + y 1 6 -)'17

'y18

->'I9 +).20]

If the standard deviation of the response y is o whatever the value of the response, then the standard deviation of the contrasts is of&, where n is the number of responses. When the four extra responses are included in the calculation of contrasts, the standard deviation changes from o ~ f to i o1J20. As there are only 3 influencing factors we can:

reconstruct the experimental design as a duplicate 2, design as there are sixteen trials. We can thus calculate the effects and interactions, and adopt a model. present the results on a cube (experimental domain) with the response surface of interest y = 0.

252

5.1 Reconstruction of the experimental design All the trials with the same levels for factors 3, 5 and 8 (Table 12.7) are grouped together, and the effects and interactions calculated from the sixteen initial trials. The extra trials are shown in brackets, and not used to calculate the effects and interactions.

TABLE12.7

EXPERIMENTAL MATRIX REARRANGED PLASTIC DRUMS Reconstruction of the effects matrix from the results of the initial 2u design

Trial no

2 7 3 6

10 15 12 13 9 16

(17)

(19)

(20)

Effects

5.15 3.2 -2.7

1.9 -0.3 2.5

0.5 -0.2

The model is thus readily obtained by rounding off and neglecting small interactions: y

5.2 +3.2 x3 - 2.7 x5 + 1.9 x8 + 2.5 x3 XS

5.2 Presentation of results The values of the responses at each corner are calculated from the model, and the isoresponse surface drawn: The zero isoresponse surface, which is the limit that must not be crossed. The +1 cm3 isoresponse surface.

253

The best settings that can be suggested are chosen as follows: Changes in influencing factors have little effect on changes in drum volume. Changes in influencing factors around their settings that avoid crossing the zero isoresponse surface. Examination of Figure 12.1 shows that these conditions are satisfied when factor 5 is high and factor 8 low (segment AB in figure 12.1). Factor 3 can thus vary without causing excessively large changes in volume. An injection pressure ( 3 ) set to the middle of the selected segment AF3 should be suitable: point R in Figure 12.1. We can see that, for this point, changes in factors 5 or 8 never produce volumes less than 2000 cm3. However, factor 3 must vary within strictly defined limits to avoid producing drums that do not meet the specifications.

4.1

15.5

(3.05)

(15.45)

Dwell

Time

(8) 6.7

5.3

Y

Flow Rate

(5.65)

Injection Pressure (3) Figure 12.1: Isoresponse surfaces were drawn using the mathematical model. The calculated point R is only a first approximation which must be refined by supplementary trials. The setting point has been determined by the mathematical model giving the drum volume under specific conditions. But we must be careful, because if the mean experimental responses from the trials are recorded at the comers of the cube representing the experimental

254

domain (shown in parentheses in Figure 12 I), then the zero response area is slightly different from that obtained &om the mathematical model. The recommended point R could be too close to the real zero surface. A complementary study near point R is required to accurately define the best settings. We can thus extend to the first provisional conclusion fiom this study of plastic drum fabrication given in Chapter 7 as follows: Conclusionfrom the study of plastic drum fabrication: The volume of the plastic drums is generally too great, due to three

. Injection pressure (3)

. . .

I

g

Feedstock flow rate (5) Dwell time (8) There is a strong interaction between injection pressure (3) and dwell time (8)

c

The settings must be:

. . .

Feedstock flow rate (5) at high level Dwell time (8) at low level. Injection pressure, selected between levels 0 and +I in the domain defined for this factor.

These settings ensure that no drum has a volume less than 2000 om3, and should produce no drums larger than 2002 cm3 The sensitivity of volume to changes in the three influencing factors around the set point R should be studied to optimize fabrication conditions

8

255

RECAPITULATION 1. It is not always necessary to run a complementary design when there are ambiguities in interpreting the results of a fractional design. It is sometimes possible to carry out just a few extra trials. 2. Block effects are better taken into account by running two or four extra trials and using a Hadamard matrix than by running one or three trials. 3 . The extra trials for dealiasing doubtful interactions are selected on the basis of the mathematical model for the factorial design. This can resolve several problems. But it

is nevertheless wise not do this blindly. It is absolutely necessary to go back to the experimental results for the best interpretation and check them with confirmatory trials.

This Page Intentionally Left Blank

CHAPTER 13

BEYOND INFLUENCING FACTORS

1.

INTRODUCTION

All the examples we have described in the preceding chapters were used to detect influencing factors, but we have seen that a complementary study is often usefbl (e.g., in the study of plastic drum volume). This complementary study can be used to obtain three types of information: to identi@ the domain of interest. to find an optimum. to find the minimum response sensitivity to external factors

1.1. Identifying the domain of interest The choice of experimental domain is important. The desired solution may well be outside the domain selected for the initial design. This is useful for selecting influencing factors and producing an approximate model of the phenomenon. It is possible that the initial trials may not meet the experimenter's requirements. In this case, the initial results can be used by the

258

experimenter to define a new domain in which there is a good chance of finding the solution to the problem. This new part of the experimental domain that best fulfils the requirements of the experimenter is called the domain of interest.

1.2. Looking for an optimum Factorial designs are not suitable for optimization studies, because they use only two levels for each factor, so that the model contains no second degree terms. We have only covered the search for influencing factors in this book. This is thus only the initial stage in Experimentology. Once the influencing factors have been identified, it is often necessary to run an optimization design. These designs are no more complicated that those we have already studied, but the calculations are much longer and require a microcomputer. We will not study these designs, but simply indicate that they are readily obtained by adding extra experimental points to the original fractional design. All the results obtained in the first experimental phase are reused for optimization. This satisfies one of our original objectives: the progressive acquisition of knowledge by running the fewest possible trials. We will examine how best to verify that the mathematical model of the factorial designs is valid or invalid for previsions within the experimental domain.

1.3. Finding the minimum response sensitivity to external factors Application to Quality An important application of experimental design is in Quality improvement. It can be used for both product design and for manufacturing process development. The Japanese expert, Taguchi, has been responsible for applying factorial designs to the concept of quality in industrial development. In order to persue this and obtain clear results of all the tools of Experimentology must be used: influencing factors, optimisation and modelling, etc. However, it is possible to understand the fimdamental concepts of this method using a simple example. Even a search for influencing factors can resolve many of the problems of Quality. Modelling and optimization provide yet more power and efficacy.

2.

IDENTIFYING THE DOMAIN OF INTEREST

2.1. Example: Two-layer photolithography There are several critical steps in the production of the microchips bearing the thousands of transistors required for integrated circuits. High performance two-layer photolithography is one of the key steps in the fabrication. Two-layer photolithography involves several operations which must be closely controlled to obtain the submicroscopic sculpture of microchips.

259

The following example is quoted with permission from a study carried out by RTCCompelec (France). Some of the data have been changed for the sake of confidentiality. The problem: *

Photolithography is used to prepare two types of microchip: lift off and etching. Many steps are involved in this process, but just two of them can be used to understand the problem. Deposition of two layers of photosensitive resin (resin A and resin 6) onto a semiconductor substrate.

B

Photographic engraving of a groove in the resins

k

This groove must have a specific profile for each of the two applications. the lift off profile and the etching profile. This study was carried out to define the operating conditions providing these profiles.

I ki

il

The problem was defined at a meeting. The responses were defined first. The list of factors which it was considered may influence the responses defined above was then established. ARer considerable discussion, the levels of each factor were defined, and thus the experimental domain. Lastly, the experimental design was selected. Responses The responses chosen were L2 and L3 as indicated on Figure 13.1. The experimenter is looking for two different applications, the kjit ofland the etching application. L2

7 -

Resin A Resin B

LIFT OFF

L3

L2

I

s

.--. L3

Figure 13.1: Lift off and etching profiles

i

Resin A nB

ETCHING

260

The two applications, lift off and etching, have different profiles (Figure 13.1), so that the dimensions of L2 and L3 are not the same. They are indicated in the following table:

TABLE 13.1 PHOTOLITHOGRAPHY

Selection of Factors A total of eleven factors were initiallv identified, but after examination, seven were selected for hrther study.

-.

Factor 1: thickness of resin B. Factor 2: curing temperature. Factor 3: plasma 1 time. Factor 4: W dose (millijoules) Factor 5 : development method. Factor 6: development time. Factor 7: plasma 2 time.

...

Definition of the domain It is important to define the domain of each factor, i.e., the low and high levels. The combined levels define an experimental domain in a seven-dimension space. Level Factor 1: thickness of resin B Factor 2: curing temperature ("C) Factor 3 : plasma 1 time (min.) Factor 4: UV dose (millijoules) Factor 5 : development method Factor 6: development time (min) Factor 7: plasma 2 time (min)

Level +

thin 150 0.5

thick 200 3

500

1000

dip 1 1

stir 4 4

Choice of initial design As so often happens, the experimenter was not certain that the defined experimental domain contained the solution to the problem. He preferred to begin with an exploratory

26 1

design. He selected a Z74 design with only eight trials. He feared that certain factors that had been abandoned would have very slight influences, but as he did not wish to monitor them later, he decided to consider them as background noise and include them in experimental error. The trials were randomized. The planned trials were run according to the design shown in Table 13.2. Unfortunately, some responses were not measurable, so that three trials were unusable. Nevertheless, measurements could be made with the thin resin layers. The experimental domain chosen was not appropriate for these resins, but was suitable for the thick resin layers. TABLE13.2

DESIGN NO1 PHOTOLITHOGRAPHY

--

Temp.

Metd.

Time

Plm 2

2

5=12

6=23

7=13

Level-

thin

150°C 30sec

500

dip

1 min

1 min

Level +

thick

200°C

1000

stir

4 min

4 min

3 min

It was decided to choose different experimental domains for each of the two resins. One study was run on thin resins and one on thick resin layers, so that only six factors - 2, 3, 4, 5, 6 and 7 - were studied. It was decided that: The limits of factor 4 (UV dose) were changed only for the thin resin layers. 0

A completely different domain was defined for the thick resin layers. As the experimental domains for the thick and thin resin layers do not overlap, two separate designs must be used, one for each thickness.

We will confine our attention to the study of the thin resin layer. A design was run with the new domain (slightly smaller for factor 4, 350-750 millijoules rather than 500-1000). To

262

avoid any confusion in the overall analysis of results, the numbering of the factors was not changed and the trials were numbered subsequently. (Table 13.3) The new design is a 26-3 as there only six factors to be studied. We know that there are 23 terms in the AGS, and the columns of the design are aliased as in the initial 274 design: 4 = 123 5=12 6 = 23 7 = 13

We may therefore be tempted to write the independent generators as: 1= 1234 = 125 = 236 = 137

but we must take into account the fact that the first column (factor I, the resin factor) no longer represents a factor and thus we must remove it from the alias generator. We have: I

=

123.23 = 4.6

Column 1 now represents interaction 46. If we replace 1 with 46 in the alias generators we have: I = 46234 = 4625 = 236 = 4637

SimplifLing and ordering, 1=236=2456=236=3467

Two independent generators are equal, hence the four remaining independent alias generators are: 1=236=2456=3467

Multiplying them by twos and by threes gives the dependent generators, and we can write the AGS. 1=236=2456=3467=345=247=2357=567

From which the contrasts are: h, = 2 + 3 6 + 4 7 + ... h, = 3 + 26 + 45 + .. _ h, = 4 + 2 7 + 3 5 + ... h, = 5 + 34 + 67 +... h, = 6 + 23 + 57 f ...

263

h,= 7 f 24 + 56 i- ... h, = & = 46 + 25 + 37 +

The eight trials were run as indicated in design number 2 (Table 13.3). The results and the effects calculations of the two responses L2 and L3 are shown in the same table. This time all the responses can be measured, the domain is better defined but we must still be sure that the values required for L3 and L3 lie within the domain examined.

TABLE13.3

DESIGN NO2 PHOTOLITHOGRAPHY Plm 3

-22

45

150°C 30 sec

350

dip

1 min

1 rnin

Level + 200°C 3 rnin

750

stir

4 rnin

4 rnin

Level-

L,

3.75

11.25

5.5

0.75

17

14.25

2.5

65

L,

0.5

8

9.25

-2.5

13.25

33.5

-0.75

46.75

2.2. Examination of the results for response L2 The influencing factors are:

.. .

Plasma 1 (3). The development time (6). Plasma 2 (7).

264

0 0 0

Factor 4 may have a slight influence. Curing temperature (2) has no influence. The development method (5) has no influence.

Examhation of the contrasts shows that there is no significant interaction between the factors. If we adopt this hypothesis we can use the following model (neglecting factor 4): L 2 = 6 5 + l l x 3 + 17x6+14x7 where L2 is measured in hundredths of microns. This relationship can be represented by drawing the isoresponse surfaces in the experimental domain (Figure 13.2).

85

107

Plasma 2 (7)

23

45 *

c

Time (6)

Plasma 1 (3) Figure 13.2: lsoresponse curves calculated for L2 values of 0.30, 0.50 and 0.70 micron. We can see that the planned dimensions for L2 (0.3 p) is located in one comer of the domain.

265

2.3. Examination of the results for response L3 0 The influencing factors are the same as for L2 with the UV dose (4) having a larger, non-negligible influence. The curing temperature (2) and the development method (5) are both without influence. Examination of the contrasts shows that there is no ambiguity and that there are no interactions.

We will therefore adopt the following model: L3 = 4 7 + 8x3+ 9x4 + 13 x6 + 35 x7

It is difficult to plot the isoresponse surface within the experimental domain because we cannot draw them in a four-dimensional space. However, we can get a geometric representation of L3 by setting one of the factors at a given value. For example, we can compare L2 and L3 by setting x4 to 0. This is a convenient value, but another could be chosen. The model is then written: L3 = 47 + 8 x3 + 13 x6 + 35 x7

And this allows us to draw Figure 13.3

a7

103

61

Plasma 2

(7)

@ @ -9

33 ,

7

7 ,/Development Time

Figure 13.3. Isoresponse curves calculated for L3 values of 0.1 and 0.3 micron (with the UV dose set at level zero).

266

The interesting values of L3 (from 0.1 to 0.3 p) lie in the region of the domain where we found the appropriate dimensions for L2. The problem may thus be resolved We must find the region where L3 varies from 0.1 to 0.3 p and L2 remains below 0.3 p. The isoresponse surfaces indicate that this region exists, and is probably slightly outside the domain studied. We can look for this region by using mathematical models. But as we approach the domain of interest we must not neglect the influence of factor 4 on L2. The models are therefore: L2 = 65 + 11 x3 + 5 x4 + 17 x6 + 14 x7 L3 = 47 + 8 x3+ 9 x4 + 13 X6 + 35 x7

t,~~T": 1 1 -1

-2

x3

x4 x6

0

+I

Plasma 1 (sec) 750 UVdose

Development time (min) Plasma 2 (min)

x7

0

'

4

Figure 13.4: Definition of theoretical study domain.

This theoretical search will allow us to define a new domain in which there is a good chance of finding the required solution. We can illustrate the approach by setting x3 and x4 and studying the isoresponse curves in the plane x6 x7. The variable x3 is set at 30 seconds (level 1 ) and variable 4 at 150 (level-2) (Figure 13.4) L 2 = 4 4 + 17x6+ 14x7

L 3 = 2 l +13x6+35x7 Naturally, this must be confirmed experimentally. These extrapolations are only guidelines. We can draw the isoresponse curve for L2 and L3 in this domain using the above formulae (Figure 1 3.5).

267 4'

Plasma 1

2'30"

1'

1'

2'30'

Development time Figure 13.5: Region in which the experimental condition providing the lift-off (L) and etching (E) profiles will probably be found. Examination of figure 13.5 shows that the operating conditions providing the required lift off and etching profiles can be defined. The following table (Table 13.4) indicates one possible solution.

TABLE13.4 PHOTOLITHOGRAPHY

Plasma 1

w.

Development time Plasma 2

Lift off

Etching

30 sec 150 15 sec 3 min 40"

30 sec 150 78 sec 2 min 30"

But it would be unwise to consider these values as certain. We have employed several assumptions for calculating them and they may be questioned. They simply permit us to roughly define a region in which the solution to a problem may be found. It is now time to confirm that we can really obtain the lift off and etching profiles by running an extra set of trials. The new experimental domain will be reduced to four factors, as the curing temperature (2) and the development method (5) are without influence. The new domain for the remaining factors may be: Level + Level Factor 3: plasma 1 (sec) Factor 4: UV dose Factor 6 : development time (sec) Factor 7: plasma 2 (sec)

10 100 10

120

70 500 130 240

268

As the domain has been clearly defined and probably contains the required solutions, the experimenter can plan two or four measures at the central point to check the validity and quality of the model. We will not show the final results of the study. We assume that the experiments were carried out as indicated in the partial conclusions set out below.

Partial conclusion: ~

7

The thin and thick resin layers must be studied separately. For thin resin layers, there are two non-influencing factors, curing temperature and development method. We will choose:

. The lowest curing temperature (energy saving) . Dipping development method (simplest operation)

8

The trials run allowed definition of a small domain probably containing the solutions required to produce the lift off and etching profiles. The operating conditions given by the calculations must be confirmed by an experiment studying four factors: plasma 1, plasma 2, UV dose and development time. A 24 design will be run with four central points to check the quality and validity of the model.

We will now examine a case in which the methods that we have studied (factorial designs) are not powerfbl enough to provide the desired solution. This is finding an optimum.

3.

FINDING AN OPTIMUM

3.1. Example: Cutting oil stability This unpublished study was carried out in the Total laboratories. It shows how, despite their power, factorial designs are sometimes not suficient for solving a problem, and that studying the domain of interest sometimes requires even more complex methodological tools than those we have discussed so far.

The problem: Cutting oil is used to facilitate metal machining: it lubricates the machined metal and cools the cutting tool. A cutting oil is a milky f looking emulsion of water and oil. The emulsion must remain stable in 9 the machine shop, and this is ensured with a chemical additive. The investigator must determine the quantity of additive necessary to keep

1

269 '"

the emulsion stable under normal working conditions. He a stability of at least 100.

IS

looking for

*

Factors studied The investigator selected two factors:

.-

Factor 1: temperature. Factor 2: additive concentration

Response The response is the index of cutting oil stability, measured with a precision of 2 . The higher the index, the greater the stability.

Domain Low temperature level: 5°C. High temperature level: 45°C. Low additive concentration: 0.4% High additive concentration: 0.8%

Design The investigator decided to use a 22 design, but he adds two points at the centre of the experimental domain to check the validity of the model.

TABLE13.5 EXPERIMENTAL MATRIX CUTTING OIL STABILITY

Trial no Temperature

Add. conc.

I

2

1

-

-

2

+

-

3 4

-

+

+ +

5

0 0

0 0

6

Response

100

270

3.2. Interpretation The first four trials were used to calculate the effects of temperature and additive concentration. their interaction and the mean. The error on the effects was: +2

-=

A

k 1 stability point

TABLE13.6.

TABLE OF EFFECTS CUTTING OIL STABILITY

Mean

point

107.5

*

2

-3 12.5 k

point point

12

-4 k

point

1

It is wise to determine whether the calculated mean can be considered equal to the measured mean (central point) before beginning to build a model. The average of the two trials at the centre was 99 and the standard deviation was

With a 95% probability of this being true (i.e., k two standard deviations) we have: model mean: measured mean:

107.5 5 2 99 k 2.8

These two means are clearly different (Figure 13.6), making it impossible to use the factorial design model. How can we explore the domain of interest in more detail? We do not have enough information to answer this question, so we need more experimental points. But: Where should these new experimental points be placed? How can we define an optimal design? How do we do the calculations? How can we use and present the results?

271

99

107.5

Figure 13.6: The two means are clearly different.

All these questions require detailed answers, but they exceed the original objective of this book, which is to find influencing factors. Let us use this example to begin our study of modelling and optimization. These two problems fall within the range of questions treated by Experimentology. We can use matrix and statistical techniques that are as powerful as those for factorial designs to solve the problems. These techniques and their application will be the subject of a new book.

Provisional conclusion: The results show that the objective of a 100 point stability can be achieved For example, at the high additive concentration (0 8%) level the stability is over 100 at both the temperatures tested But this result is incomplete because the additive concentration is too high - it is not optimized. The economic aspect most be reconciled with demands of quality, I e , the lowest additive concentration which ensures a stability of 100 points between 5°C and 45°C The difference between the calculated and measured means makes it impossible to use the mathematical model associated with factorial designs to plot the isoresponse curves The investigator cannot make any recommendations with the information available Complementary experiments are necessary to establish a second

t degree model and plot isoresponse curves for the phenomenon

4.

FINDING A STABLE RESPONSE

4.1. Example: thickness of epitaxial deposits Taguchi [22] proposed an interesting approach for finding a robust solution to a problem. When factor levels are chosen so that the response of interest is minimally influenced

272

by factor variations, the response is said to be robust. The fundamental concept of robustness can be illustrated by a quality study carried out at AT&T in the USA, as reported by Kackar and Shoemaker [ 2 3 ] . The equipment used for preparing epitaxial deposits was set up to give thicknesses of 1415 microns. The epitaxial deposits are formed on wafers in a heated chamber. A total of 14 wafers are arranged on a support, the susceptor, that can be rotated or oscillated (Figure 13.7).

Wafer

Figure 13.7: 14 wafers placed on the susceptor.

The problem:

The deposits are not uniform, some are thinner than 14 microns, while others are thicker than 15 microns. Although the mean thickness of 14.5 microns is satisfactory, the number of rejects is high, resulting in unacceptable costs. The objective is to find new settings for the installation that give the smallest possible dispersion around the B nominal value of 14.5 microns.

8

Production set-up values The pre-study production set-up parameter values were:

..

Arsenic flux Depositing temperature

5 7%

1215°C

273

.. .

Susceptor motion Deposition time Hydrochloric acid temp. Injection nozzle position Hydrochloric acid flux

oscillation short 12OOOC 4 12%

The investigators decided to use two types of wafer in the study: type 66864 and 678D4. An eighth factor was therefore added to the seven listed above, the wafer code. They also decided to stay fairly close to the normal conditions by setting the study domain around the production set-up values. Factors The factors are the set-up parameters plus the wafer code. The domain is defined by the following table: Level +

Level Factor 1: arsenic flux Factor 2: deposition temp. Factor 3: wafer code Factor 4: susceptor movement Factor 5 : deposition time Factor 6: HCl temp. Factor 7: injection nozzle position Factor 8: HCI flux

55%

1210°C 66864 rotation long 1 180°C 2 10%

59%

1220°C 678D4 oscillation short 1215°C 6 14%

Responses The susceptor carried 14 wafers in each trial, and the thickness dispersion was measured at 5 points on each wafer.

Figure 13.8: Arrangement of points for measuring wafer thickness.

274

There are thus seventy measurements of thickness per trial. The responses chosen were the mean thickness and the dispersion of thickness. mean thickness If e, is the measured thickness and E the mean of the 70 measurements in each trial, then 1 O' e=-Eei 70 1

-

TABLE13.7 EXPERIMENTAL DESIGN

(1=2345=1346=1237=1248)

EPITAXIAL DEPOSIT

-

Trial no

1

2

3

4

5=234 5=134 7=123 P=124

1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16

Thick.

log s*

+ + + + + + + + + + + + + +

14.821 14.888 14.037 13.880 14.165 13.860 14.757 14.921 13.972 14.032 14.843 14.415 14.878 14.932 13.907 13.914

-0.4425 -1.1989 -1.4307 -0.6505 - 1.4230 -0.4969 -0.3267 -0.6270 -0.3467 -0.8563 -0.4369 -0.3131 -0.6154 -0.2292 -0.1190 -0.8625

+ +

-

55

1210 668 Rot

long

1180

2

10

Level +

59

1220 678 Osc

short

1215

6

14

Thickn -003 -005 0 0 3 -002

-041

003

007

-004

Level

I

log s2

-0005

0052

0061

0 176

-0 124 -0035

-0282

1439

-0 05 -0648

275 thickness dispersion The dispersion was defined as the log of the variance:

[

log s2 = log -X(ei-e) ;9 :7

'1

Experimental design A resolution IV fractional factorial design :2;

was selected.

4.2. interpretation Table 13.7 shows the trials run and the responses obtained. The results were interpreted by first examining the variance of the set-up factors. This approach is emphasised by Taguchi, who used it routinely to improve the quality of products or processes. The initial objective was to select the levels of factors which give the smallest possible variations in the response of interest. Once these levels have been determined for the factors influencing the variance, the nominal deposit thickness is then adjusted using the factors that influence thickness but not the variation in thickness dispersion. This provides a set-up with the correct thickness and minimal dispersion. Under these conditions the number of rejects is very small, and may even be zero if the dispersion is sufficiently small.

Thickness dispersion The factois influencing the thickness variance are shown in Table 13.8

TABLE13.8 TABLE OF EFFECTS

EPITAXIAL DEPOSIT Thickness variance Mean

-0.6484

1 2 3

-0.005

4 5

6 7 8

0.052 0.061 0176 -0.124 -0.035

-0.282 -0.050

276

The three influencing factors are, in order of importance:

..Factor 7: injector nozzle position Factor 4: susceptor rotation

.

Factor 5: deposition time.

Figu :s 13.9, 13.10 and 13.11 show that the thickness variance will be redu :d if

.

Factor 7 is set at the high level, position 6, Factor 4 is set at the low level (continuous rotation) Factor 5 remains unchanged (short deposition time).

Log s2

- 0.366 - 0.648

- 0.930 -1

(.;i /,

0

+I

i(6') ,

INJECTION NOZZLE (7)

Figure 13.9: Influence of injection nozzle position (Factor 7) on thickness variance.

277

Log

s2

t

-1 0 +I Rotation Oscillation SUSCEPTOR MOVEMENT (4)

Figure 13.10: Influence of susceptor rotation (Factor 4) on thickness variance.

s2

t

-1

Long

0

+I Short

DEPOSITION TIME (5)

Figure 13.1 1: Influence of deposition time (Factor 5) on thickness variance.

278

All that remains is to interpret the results of the experimental design for the thickness itself

Thickness Table 13.9 summarizes the effects and interactions of the factors studied

TABLE13.9 TABLE OF EFFECTS

EPITAXIAL DEPOSIT Thickness Mean

14.39 micron

1 2 3 4 5 6 7 8

-0.03 micron -0.05 micron 0.03 micron -0.02 micron -0.41 micron 0.03 micron 0.07 micron -0.04 micron

12 13 14 15 16 17 18

-0.01 micron

0.02 0.00 -0.01 0.02 0.01 -0.03

micron micron micron micron micron micron

Only one factor is influent:

.

Factor 5 : the deposition time.

There is no apparent interaction. Figure 13.12 shows the influence of deposition time on the mean thickness of the epitaxial deposit. The exact deposition time providing a thickness of 14.5 microns is readily calculated if the values of the short and long levels are known in minutes and seconds. These values are not available for obvious industrial reasons. We can however do the calculation using coded values. The mathematical model is: y

=

14.39 - 0.41 x

279

14.80

14.39 14.50

13.98

--\ -0.27 Long

Short

DEPOSITION TIME ( 5 )

Figure 13.12: Influence of deposition time (Factor 5) on the mean thickness of the epitaxial deposit.

setting y = 14.5, we get: 14.5 = 14.39 - 0 . 4 1 ~ hence: X=

-14.5+14.39 0.4 1

-

0.11 - -o,268 0.41

This value can be used to calculate the optimal deposition time in coded variable (Figure 13.12) and given as a recommendation. The information provided by the experimental design allows: Reduction of the dispersion of deposit thickness by changing the injection nozzle position and using continuous rotation instead of oscillating. The nominal thickness of 14.5 microns can be obtained by changing the deposition time. This slightly increases the thickness variance, but fortunately factor 5 is not the most important for dispersion.

280

Before giving the results and making recommendations for setting up the industrial production, the interpretation must be verified. A series of trials was therefore run with the new settings - the confirmatory trials. The standard deviation was found to be 0.24 micron, the thickness was 14.5 microns and there were almost no rejects. Conclusion:

i The dispersion of epitaxial deposit thickness can be reduced by adjusting the set-up factors as shown in the following table.

6

I

Factor Arsenic flux Deposition temperature.

1

Original setting 5 7%

I

New setting 5 7%

1215°C

1215°C

Susceptor movement

oscillation

rotation

Deposition time HCI temperature

1200°C

variable 1200°C

Injection nozzle HCl flux

12%

12%

I

The deposition time will be set to obtain the required mean thickness of 14.5 microns. The recommended settings guarantee a standard deviation of k 0.25 microns around this thickness.

This example illustrates an important concept emphasized by Taguchi: there is one setting, among all the possibilities, that minimizes the variance in the target response. In order to ensure the Quality of a product or process the stable or robust settings must be found. But Taguchi went further. Not only did he study the factors influencing fabrication, he also studied the factors that could influence the life of the product after it had left the factory. He examined all the conditions, from product design, to manufacture and subsequent client use. This approach to research and development, coupled with cost control, illustrates a particularly interesting application of experimental design to quality improvement.

28 1

RECAPITULATION 1 . The photolithography example has emphasized one of the key points in experimental

design: the search for the domain of interest. A combination of a progressive approach and detailed analysis at each stage will invariably lead to a solution whenever such a solution exists. Only experimental design can provide a rapid, reliable solution to a problem involving several factors. We must therefore again emphasize that non-influencing factors are not necessarily of no importance. No change in the response can produce savings (see also the examples on the colour of a product, bean-growing experiment and epitaxial deposits) and facilitate the production of optimal settings. 2. The cutting oil example shows that we must be extremely carefil before adopting a

model, even for a very simple case. It is vital to carry out trials at the centre of the domain to test the model. Central point trials must be run as soon as the investigator believes he is within the domain of interest and wishes to begin model-building. They are immediately usehl for estimating the standard deviation, and will remain usehl for finding a second degree model, if required. Model-building and optimization are almost always the experimenter's goal. Identifylng influencing factors and using the first degree model are often only the initial phase of the study. 3. An important application of experimental design is the use of the variance as the

response. It is possible to identifl factors that minimize the dispersion of responses so as to make them less sensitive to external factors. This is an effective way of improving the quality of a product or process, Applying this technique right fiom the design of a product, i.e. during the R&D phase, is the surest method of ensuring quality, reduced product control costs and the widest market for the product.

This Page Intentionally Left Blank

CHAPTER 14

PRACTICAL METHOD

OF CALCULATION USING A QUALITY EXAMPLE

1.

INTRODUCTION

It is much easier to interpret experimental designs and allied methods if the calculations, outlines and isoresponse curves are prepared quickly and accurately. This can only be done with a microcomputer. But this does not imply that it always requires expensive, dedicated software. All the examples in this book were prepared using a simple spreadsheet, Lotus 123. In general, the calculations involved in searching for influencing factors and evaluating variance are simple. However, those for optimization designs and identifirlng the best experimental points when preparing special optimal designs (mixture designs or designs with constraints) are not. Specific softwares are then necessary. These dedicated programs (see Nachtsheim [24] vary in complexity and ease of use, and require careful selection. The experimenter must choose carefully at all phases: design selection, aliase selection, factor

284

selection, mathematical model, residuals calculation, domain selection, etc. The software should help the experimenter make decisions and not make them for him! It is a good idea for the experimenterbeginning to use factorial designs to do the detailed calculations himself, so that he can better understand the si@cance of each result. The quality of his interpretation and conclusions depend on this. This is why we will now go step by step through an example. The presentation has been made more accessible by separating the calculations from the descriptive section. The first part of this chapter describes the problem to be studied, and references are given for each detailed calculation. The second part of the chapter covers the details of each operation; these can be reproduced by anyone with a copy of the Lotus spreadsheet or an equivalent. In this way the reader can follow the reasoning and check the calculations.

2.

A QUALITY IMPROVEMENT EXAMPLE

Example: Study of truck suspension springs This example is taken from Pignatiello and Ramberg [25].It was carried out by the firm of Eaton Yale to improve the manufacture of truck suspension springs. It is the type of design that provides quality improvement, as defined by Taguchi, by reducing response variance and then adjusting the response to the required value.

The problem Truck springs are made up of leaves having a precisely defined curvature. The curvature must be exactly eight inches, with a very small variation around this value. The leaves undergo several treatments during manufacture, including: 0

Heating to high temperature in a furnace.

0

Immersion in an annealing oil bath.

Bending in a special forming machine. The study was carried out to determine the factors influencing the curvature and the dispersion of the curvature during the three phases described above. This information was used to advise on how: 0 0

The mean curvature could be kept at 8 inches, The dispersion of curvature around 8 inches could be as small as possible.

It is important to clearly define the factors to be studied, the experimental domain and the responses, before beginning any experiments.

285

Factors The fabrication engineers believed there to be four influencing factors that may show important interactions. This was kept in mind when choosing the experimental design and the aliases. They also believed that a fifth factor (the temperature of the annealing bath) could be influencing,,but its control during fabrication would require extra equipment. They therefore preferred to consider it as a non-controlled factor contributing to the experimental error; this factor could also be said to increase the background noise. Nevertheless, steps were taken to give it two levels during the study (low and high), but these levels were defined approximately because temperature was not accurately regulated.

.

Factor 1 : furnace temperature. .Factor 2: heating time. Factor 3 : transfer time between leaving the furnace and placing in the bending machine. Factor 4: bending time. Factor 5: annealing bath temperature.

.. .beInteractions neglected.

12, 13 and 23: The experimenters believed that they could not

The level of factor 5 is difficult to keep constant because there is no regulation of the bath temperature. This factor could be considered as background noise, and the experimenters preferred ta treat it as part of the experimental error. This factor is not, therefore studied in the initial interpretation (step l), and the six responses of each set of trials will be treated as equivalent.

Domain The experimental domain for the five factors studied is shown below (Table 14.1).

TABLE14.1

TRUCK SUSPENSION SPRINGS

Factor 1 (OF) Factor 2 (sec) Factor 3 (sec) Factor 4 (sec) Factor 5 (“F)

1840 25 12 2 130-150

1880 23 10 3 150-170

286

Experimental design The engineers had defined four factors and wished to carry out only eight trials. They therefore chose a 241 design. Factor 4 is aliased with interaction 123. The reader can see that this is a resolution IV design. For each trial, three experiments are run at the low level of factor 5, and three at the high level. This provides an indication of the background noise introduced by this factor for each of the eight trials. The engineers want to know the values of interactions 12, 13 and 23, in addition to those of the four factors. The three remaining columns of the 2&' design are therefore assigned to these three interactions.

Responses The responses must be chosen so as to reflect both the value of the curvature and the dispersion ofthe curvature around the mean value. The engineers selected one response for the curvature and three responses for the dispersion: Curvature The mean curvature is selected as the sole response for curvature Dispersion of curvature The three responses are: 1 . The variance of the curvature for each set of trials. This variance will be indicated by s", with a subscript indicating the set of trials.

2. A variance fhction, Z, defined by z=10 logs2 3. The signallnoise hnction proposed by Taguchi, Z'

Z' = 10 log1 Y 2 s

Experiments Table 14.2 shows the experimental design and the six responses obtained per trial The results are interpreted in two steps: Step 1: There are four main factors, factor 5 was set at two levels but is not taken into account. It is treated as an uncontrolled factor. The six responses from each trial are therefore equivalent and are analysed together.

287

Step 2: As the influence of factor 5 could not be ignored, the experimenters use the trial results to construct a design including all five factors. As factor 5 is a controlled factor, its effect on the responses is determined.

TABLE 14.2 EXPERIMENTAL MATRIX

TRUCKSUSPENSION SPRINGS Responses ~~

~~

~

5-

6 7

7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56

+ +

3.

7.78 8.18 7.56 7.56 8.00

8.09 7.62 7.81

~

5+ 7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69

7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81

7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50

7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59

INTERPRETATION, STEP 1

3.1. Calculation of responses 3.1.1.

Mean curvature

For trial number 1, the six values are used to calculate the mean 7,

yl = -[1 6

7.78+7.78+7.81+7.50+7.25+7.12]

The same calculation is performed for the seven remaining sets of trials (see calculations, screen 14.1, p. 310).

288

3.1.2.

Dispersion of curvature

(see calculations, screen 14.2, p. 3 11)

1. Variance

For trial number 1, the variance 2 is given by the formula: (7.78-7.54) 2 +(7.78-7.54)2 +(7.81-7.54)2 +(7.50-7.54)2 +(7.25-7.54)2 +(7.12-7.54)2]

6-1 S:

= 0.0900

TABLE 14.3

EFFECT MATRIX TRUCK SUSPENSION SPRINGS

+ +

+ +

Level - 1840

25

12

2

Level+ 1880

23

10

3

289

2. Eunction Z

(see calculations, screen 14.3, p. 3 12)

For trial number 1, the Z hnction is Z, : Z, = 10 logsf = 10 log 0.09

Z,= -10.45 3. Function Z'

(see calculations, screen 14.3, p. 312)

For trial number I, the Z' hnction is Z;:

z;=

-2

10 log% = 10 log s1

~

(7.54)2 0.09

Z;= 28.00 These four responses (7,9, Z and Z') calculated from the raw experimental results can be used to calculatethe effects of each factor. The effects matrix is shown in Table 14.3, which also contains the effects and interactions. (see calculations, screens 14.4 - 14.8, p. 313, 314, 315) We can now analyses these results knowing that the influence of factor 5 remains to be examined. We will take this factor into account in step 2 of the interpretation.

3.2. Analysis of results (interpretation, step 1) 3.2.1. Mean curvature The mean standard deviation of one trial is 0.2147 (screen 14.2, p. 3 11). Each effect is calculated from the 48 experimental results, giving a standard deviation of 0.2147 OE =--

J48

0.2147 6.928

- -= 0.031

The results can be summarized in a table of effects, Table 14.4. (see calculations, screen 14.5, p. 314)

290

TABLE14.4 TABLE OF EFFECTS

TRUCKSUSPENSION SPRINGS

1

First interpretation

Mean 1

2

3 4 12 + 34 13 + 24 14 + 23

7.64 f 0.03 0.11 -0.09 -0.01 0.05

f f L f

0.03 0.03 0.03 0.03

-0.01

& 0.03

-0.01 -0.02

f 0.03 5 0.03

It appears that = Factor 1 (furnace temperature) is influent.

-

Factor 2 (heating time) is influent.

= Factor 4 (bending time) has a small influence.

Factor 3 (transfer time) and all the interactionshave no influence.

3.2.2.

Dispersion of curvature

The results can be summarized in a table of effects, Table 14.5 (see calculations, screens 14.6, 14.7 and 14.8, p. 314, 315)

29 1

TABLE14.5 TABLE OF EFFECTS TRUCKSUSPENSION SPRINGS Dispersion of spring curvature First Interpretation Effect Mean

variance

Z

Z'

0.0460

-16.02

33.67

1 2 3 4

-0.0088 -0.0300 0.0037 -0,001 1

0.29 -4.73 2.27 -1.41

-0.16 4.63 -2.28 1.47

12 + 34

0.0054 -0.0057 0.0079

1.14 -1.73 2.57

-1.15 1.72 -2.59

13+24 14 + 23

Factor 2 (heating time) had a great influence on the variance of curvature. The fbnctions Z and Z' confirm the influence of factor 2, and suggest that factor 3 and interaction 23 could be influent. 3.2.3.

Effect of factor 5 on curvature

Factor 5, which has been considered as background noise until now, remains to be examined. As this factor was studied at two levels we can calculate its effect: the average of the low level is 7.76, and the high level average is 7.50 (see calculations, screen 14.9, p. 316). 1 E5= -[7.50-7.76]= - 0.13 2

This factor has the greatest influence on spring leaf curvature! The experimenters consider it to be unreasonable to leave it unregulated during fabrication. But, before deciding to make the investment required to control it, they check that the objective could be attained. They therefore carry out a further analysis of the results.

292

Curvature 7.76

\

7.63

7.50

.

-1

+I

140°F

160°F

-0.13

TEMPERATURE (5)

Figure 14.1: Influence of annealing bath temperature (Factor 5) on curvature

4.

WHAT IS A GOOD RESPONSE FOR DISPERSION?

We will begin by studying the dispersion of curvature in order to identi@ the settings of factors that minimize it. The reader may well ask why three responses were used to define this one property. One would have been enough, but it would have to accurately define the dispersion. Unfortunately, this ideal response does not exist. In this situation we generally try to substitute quantity for quality. But as we will see, it is a vain hope, and each of the responses has a weakness. The three responses, variance, logarithm of variance and signal-tonoise ratio will be examined individually.

4.1. Variance This is, a priori, a good response as it measures the dispersion of a set of measures around the mean. But variance must always be positive, like all algebraic squares. This property may not be respected when the mathematical model of factorial design is used. There is a risk of having a negative variance. To avoid this problem, statisticians use the logarithm of variance.

293

4.2. Logarithm of variance The logarithmic function, log x, has a great advantage. It can be positive or negative, but it always gives a positive x (Figure 14.2). Using it, therefore avoids any problems of impossible variance. But, the problem with the log hnction is that it distorts the original information. It emphasises small differences in the variance when the variance is very small. As a result, the effects depend more on the difference between small variances than on the variances themselves.

Figure 14.2: Plot of the logarithmic function.

4.3. Comparison of variance and logarithm of variance Let us assume that we have to interpret the results of a 22 experimental design. We study the dispersion and have two responses, the mean variance and the log of the mean variance. The mean variance of the high level of factor 1 (indicated as s z ) is 0.30 and the mean variance

of the low level is 0.10 (indicated as s!). We can calculate the effects of the factor with the two responses .? and 10 log .?: Variance 9

EsZ

1

= -[0.30-0.10]=

2

4.10

294

z = 10 log s2-

1 2

E, =-[101og0.30-10log0.10]

E,

1

= --[-5.23-(-lo)]

2

1 = - 4.77 = 2.36 2

The mean variance for the high level of factor 2 is 0.01, and the mean variance at low level is 0.001. We can also calculate the effects of factor 2 with the two responses, variance and log of the variance. Variance 9 1 Es2 = --[0.01-0.001] = +0.0045 2

z = 10 log .9 E,

1

= 2[10

log 0.01-10 log 0.0011

1 E L = -[-20-(-30)] 2

=5

Figure 14.3 compares the two methods, showing the effects of factors on dispersion. With variance .-?,factor 1 has the greatest influence, while with 10 log ,>?- factor 2 has the greatest influence. Thus, interpretation is not easy in this case.

10 Log s2

S2

E1=0 1000 E2=0.0045

I -1

b

+I

E2= 5 E l = 2 36

-1

+I

Figure 14.3: The response selected (9or log s2) may influence the evaluation of effects.

295

4.4. The signal-to-noise ratio The signal-to-noise ratio, Z', is no better. We can write:

z' = 10 log-Y2 = 10 logy2 -10

logs 2

s2

As J 2 varies little, log p2 can be considered to be constant. We can calculate the effects E zr of a factor with the function Z': EZ,=,((1010gJ2 1 -1010gs:)-(1010gY2

E,,=

L

L

1

[(-10 logs: )

-lOlogs?)] J

-

(-10 logs-

or simply

Thus the function Z' has the same advantages and disadvantages as Z , except that the signs are inverted, providing a further risk of error in interpretation. In our study of spring leaves, we will use 2 and log 2.Let us now continue with the second stage of interpretation.

5.

INTERPRETATION, STEP 2

5.1. Calculation of reponses The results are completely reanalysed as if there are five factors and sixteen trials. A 25-1 design is constructed by adding the two levels of factor 5. The preceding 24-1design is used, first with the low level of factor 5 (trials 1-8 in Table 14.6), and then a second time using the high level of factor 5 (trials 9-16 of Table 14.6) This new design may be considered as a 24+1-1 design, which is more simply written as Z5-'.The alias generator is I = 1234, which can be used to calculate the contrasts. The four responses ( y , 9,Z and Z') are calculated: screen 14.9

296

p.316, screen 14.10 p. 317 and screen 14.11 p. 318. The results of these calculations are shown in Table 14.6.

TABLE 14.6

EXPERIMENTAL MATRIX TRUCK SUSPENSION SPRINGS

Responses

7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56 7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-

7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81 7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50

-

7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69 7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59

--

L 7.79 8.07 7.52 7.63 7.94 7.95 7.54 7.69 7.29 7.73 7.52 7.65 7.40 7.62 7.20 7.63

3 273 12 104 36 496 84 156 373 645 12 92 48 42 16 254

Z

Z'

-35.22 -15.64 -29.21 -19.81 -24.44 -13.04 -20.76 -18.06 -14.28 -11.90 -29.21 -20.34 -23.19 -23.73 -27.87 -15.94

53.06 33.77 46.73 37.43 42.43 31.04 38.30 35.77 31.54 29.67 46.73 38.01 40.57 41.37 45.02 33.60

-

The reader will note that controlling factor 5 reduces the standard deviation of one trial from 0.21 to 0.13. In general the more factors that are controlled, the smaller the experimental error (see calculations screen 14.11 p. 3 18).

5.2. Analysis of results (second step of interpretation) The experimental matrix can be used to construct the effects matrix, whose results for variance of curvature are shown in Table 14.7 and for curvature in Table 14.8. 5.2.1.

Dispersion of curvature

The results are shown in Table 14.7 (see calculations, screen 14.17 p. 324, screen 14.18 p. 325, screen 14.19 p. 325)

297

TABLE14.7

TABLE OF EFFECTS TRUCKSUSPENSION SPRINGS Dispersion of curvature Second Interpretation Effect

variance

Z

z'

0.0165

-2 1.40

39.07

1 + 234 2 + 134 3 + 124 4 + 123 5 + 12345

0.0092 -0.0074 -0.0023 0.0014 0.0020

4.10 -1.23 0.54 0.47 0.60

-3.90 1.13 -0.55 -0.41 -0.75

12 + 34 13 + 24 14 + 23 15 + 2345 25 + 1345 35 + 1245 45 + 1235

-0.0032 0.0003 0.0060 -0.00 19 -0.0017 -0.0071 0.0040

0.00 -0.92 1.45 -1.28 -1.30 -2.41 0.28

-0.01 0.91 -1.48 1.33 1.39 2.38 -0.26

125 + 345 135 + 245 235 + 145

0.0038 -0.00 18 0.0076

2.36 0.94 1.85

-2.37 -0.91 -1.88

Mean

These data show that the influent factors are different, depending on w..:ther is used for interpretation. This is not surprising.

z = 1Olon s2The influent factors and interactions are: .Factor 1 Interaction 35 Interaction 125 Interaction 235

...

4.10 2.41 2.36 1.85

2 or log 2

298

Variance .2 Factor I is again influent, while factor 2 appears to have a negative influence. Interactions 35 and 235 are influent, while interaction 125 has little influence. The high value of 125 given by Z is due to a large difference between two small values. The influent factors and interactions are thus: .Factor 1 Factor 2

.

+0.009 -0.007

.Interaction 35 Interaction 23 5

+0.007

-0.007

The influence of factor 1 on curvature variance is shown in Figure 14.4; the variance is smaller if the low level of factor 1 is selected. The influence of factor 2 is shown in Figure 14.5; here, in contrast, the high level should be chosen

S2

0.0257

0.0165

0.0073

/

i

+ 0.0092

+ 1840 F

1880 F

FACTOR (I)

Figure 14.4: Influence of furnace temperature (Factor 1) on the variance of curvature.

299

S2

0.0239

0.0165

- 0.0074 0.0091

+

-1

25 sec

23 sec

FACTOR (2)

Figure 14.5: Influence of heating time (Factor 2) on the variance of curvature.

3 00

5.2.2

Curvature

The results are shown in Table 14.8 (see calculations screen 14.16 p. 323)

TABLE14.8 TABLE OF EFFECTS TRUCKSUSPENSION SPRINGS Second interpretation Mean

7.63 k 0.02

1 + 234 2 + 134 3 + 124 4 + 123 5 + 12345

0.11 -0.09 -0.01 0.05 -0.13

f 0.02 f 0.02

12 + 34 13 + 24 14 + 23 15 + 2345 25 + 1345 35 + 1245 45 + 1235

-0.01 -0.01 -0.02 0.04 0.08 -0.03 0.01

f 0.02

125 + 345 135 + 245 235 + 145

*

0.02

k 0.02 f 0.02

*

0.02

f 0.02 f 0.02 f 0.02

k 0.02 f 0.02

*

-0.005 0.02 0.02 k 0.02 -0.02 k 0.02

It is hardly surprising that the effects are the same as those previously (first interpretation) calculated for curvature: they are derived from the same data and treated in the same way. The inclusion of factor 5 allows calculation of sixteen contrasts instead of eight and reveals new interactions. There are three influencing factors: 1, 2 and 5 , and one interaction that cannot be ignored, interaction 25.

30 I

Curvature

7.74

i

7.63

+ 0.11

7.52

-1 1840 F

+I 1880 F

FACTOR (1) Figure 14.6: Influence of furnace temperature (Factor 1) on spring curvature.

Curvature

7.72 7.63 7.54

-1

\

-0.09

+I

23 s

25 s

TIME ( 2 )

Figure 14.7: Influence of heating time (Factor 2) on spring curvature.

3 02

Curvature 7.76

7.63

1

-0.13 I

7.50

-1

+I

140°F

160°F

TEMPERATURE (5) Figure 14.8: Influence of annealing bath temperature (Factor 5) on spring curvature.

6.

OPTIMIZATION

As the average curvature is low, 7.63 inches, the levels of factors that increase it as much as possible should be selected. But we must also keep the variance of curvature to a minimum. Table 14.9 shows the elements of the discussion that we will use to choose the factor levels, listing the significant factors and interactions for the corresponding responses.

TABLE14.9

TRUCK SUSPENSIONSPRINGS Factors and interactions significantly influencing the responses

Factorsor interactions

1

2

Variance of curvature

+3

0

Curvature

.:. .:.

3

4

5

23

25

0:.

.:. .:.

35 0:.

.:.

235

*:.

303

The high level of factor 4 may be chosen to increase curvature. It is more difficult to choose the levels of the four other factors as the interactions must be taken into account. The easiest way to resolve this problem is to write mathematical models for the two responses. Curvature = 7.63 + 0.1 1 x1 - 0.09 x2 + 0.05 x4 - 0.13 xs + 0.08 x2 x5 Variance = 0.0165 + 0.009 x, - 0.007 x2 + 0.006 x2 x3 - 0.007 x3xs + 0.007 x2 x3xS The two responses, curvature and variance of curvature, are calculated (see calculations screen 14.20 p. 326, screen 14.21 p. 328, screen 14.22 p. 329, screen 14.23 p. 330) for all the possible combinations of the four factors 1, 2, 3 and 5 (Table 14.10). Factor 4 is held at the high level.

TABLE14.10 TRUCK SUSPENSION SPRINGS Calculation of curvature and dispersion of curvature for all combinations of x1 x2 x3 and xs ~~

Case no 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-1 +1 -1 +I -1 +I -1 +1 -1 +1 -1 +I -1 +1 -1 +I

-1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1

-1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1

-1 -1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 +1 +1

+I ---

Dispersion of curvature

Curvature

65 245 -55 125 225 405 65 245 345 525 -5 5 125 -55 125 65 245

7.87 8.09 7.53 7.75 7.87 8.09 7.53 7.75 7.45 7.67 7.43 7.65 7.45 7.67 7.43 7.65

The results in the table show that an eight-inch curvature is obtained in two cases:

3 04

1. Case 2: I + 2- 3- 4+ 5- produced a curvature of 8.09 inches with a variance

of 0.0245 2. Case 6: I + 2- 3+ 4+ 5- gives the same curvature (8.09 inches) but the variance of curvature is greater, at 0.0405. It would thus be best to select the low level of factor 3 to minimize the variance of curvature. This selection has no influence on the curvature itself, as curvature is independent of xj. Examination of Table 14.6 shows that Case 2 is the same as Trial number 2. The experimental values confirm the values calculated from the model. The study could therefore be stopped at this point and these setting used. But perhaps we could try to reduce the variance of curvature still further. For this purpose, we adopt a special representation of a four dimensional space. The levels of factors 1 and 2 are represented as in a 22 design (Figure 14.9). We then plot the isoresponse curves for curvature for each level of the pair of factors 1 and 2 in the plane of factors 3 and 5.

160°F

160°F

(5)

(5)

140°F

140°F

160°F

U / \

12"

(3)

10"

12l

(3)

10"

(2)

(l)

(5) 7.8

140°F

12"

140°F (3)

lo"

Figure 14.9: Isoresponse curves of curvature variance.

305

This diagram shows that a curvature of 8 inches can only be obtained in the region of comer where factor 1 = +1 and factor 2 = -1. If we assume that the smallest possible value of factor 5 is -1, we can calculate the location of points where curvature is 8 inches within the space of factors 1 and 2. For this we write that curvature is eight inches. 8 = 7.63 + 0.11xl - 0.09 x2 + 0.05 ~4 - 0.13 ~5 + 0.08 ~2 ~5

The trajectory where the curvature is 8 inches when x5 = -1 and x4 = +I is given by: 8 = 7.63 + 0.1 1 x1 - (0.09 + 0.08) ~2 + 0.05 + 0.13

or 0.17 x2 = 0.1 1 x1 - 0.19 or

11x1 17

x2 =---

19 17

this relationship is represented by a straight line (Figure 14.10). It shows all the possible settings of factors 1 and 2 to obtain a curvature of eight inches. The final choices of setting will depend on the variance, which must be as small as possible. We have:

V

0.0165 + 0.009 x1 - 0.007 x2 + 0.006 x1 x3 - 0.007 ~3 x5 + 0.007 ~2 ~3

~5

Ifwe apply the preceding hypothesis x3 = -1, x5 = -1 and x2 = 11x1/17-19/17, we get: V = 0.0162 + 0.0051 x1

This relationship can be used to place a scale on the straight line found previously. The different values of the variance of curvature are shown on Figure 14.10

306

Factor 2

A;;l*;

+I

Factor 1

v = 0.0213

-1

v= 0.0171

Figure 14.10: Optimizing the variance of curvature. The smallest variance is given when xz is at its low level and x1 equals 0.18. If we remain within the domain studied, the settings providing a curvature of eight inches with the smallest possible variance are (in coded units): x, = x2= x3= x4= x5=

0.18 -1 -1

+1 -1

In real-world units these are: .Factor I Factor 2 Factor 3 Factor 4 Factor 5

.. .

1863.6 deg F 25 seconds I2 seconds 3 seconds I30 - 150 deg F

If the manufacturing constraints allow us to move outside the experimental domain, the investigators should be able to reduce the variance of curvature still further while maintaining an average curvature of 8 inches. They could: Reduce the furnace temperature (factor 1). Increase the heating time (factor 2). More closely regulate and reduce the annealing bath temperature (factor 5).

307

Whatever the results of the theoretical study, trials must be carried out to continn that the predictions are accurate and that the objective could be attained. Conclusion

a

rli iii II

The dispersion of the curvature of truck suspension spring leaves can be reduced by setting: 0

The furnace temperature at 1863 deg F.

0

The furnace heating time to 25 sec.

0

fi 0

I! fr

41 B

The transfer time between furnace and bending machine to 12 sec. The bending time to 3 sec.

The annealing bath temperature cannot be considered to be background noise as it strongly influences the degree of curvature. If it is not regulated all the other efforts are worthless Therefore the required investment must be considered. Thus, the annealing bath temperature should be set at the lowest temperature used in this study, 130 deg F. Confirmatory trials should be run to verify the predicted settings

308

RECAPITULATION This study of the curvature of truck spring leaves has highlighted several points. 1 . The difficulty and the importance of interpretation. The experimenter must make the results speak, which requires considerable calculation to extract their useful information content. Interpretation is never automatic; it requires intelligence and imagination. The experimenter must select the most appropriate hypothesis in order to present clear, useful conclusions. 2. The use of variance as a response. This technique is often used to improve quality.

The first step is to reduce dispersion by choosing the appropriate levels that influence it; the other factors are then adjusted to reach the target value. 3. The selection of the right response is often tricky, and requires a great deal of care.

The variance example clearly illustrates this. 4. The setting of a factor influencing the response studied can reduce experimental error.

The more factors that are controlled, the more the experimental error is reduced. The next part of the chapter covers the details of calculation. It shows how to obtain the results used in the first part of the chapter. We recommend that this second section should be accompanied by the calculations run on a microcomputer.

CHAPTER 14 (CONTINUED)

DETAILED CALCULATIONS FOR

THE TRUCK SUSPENSION SPRINGS

EXAMPLE

The calculations are readily performed using a spreadsheet. This example was prepared with Lotus 123 running on a PC compatible microcomputer. The reader can transpose the calculations to hidher particular softwarehardware system.

1.

CALCULATION FOR THE FIRST INTERPRETATION

a) Calculating the mean curvature for each trial Open the first worksheet : sheet 1, and enter the 48 trial results (screen 14.1). The mean is calculated by placing the instruction :

@AVG(B9..G9)

310

in cell 19. The other means are calculated by copying this instruction to cells I10 to 116.

SCREEN 14.1 Calculation of mean curvature for each trial

Trial no Y1

Y2

Y3

Y4

Y5

Y6

7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56

7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81

7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69

7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81

7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50

7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59

Mean

8

1 2 3 4 5 6 7 8

7.540C 7.9017 7.520C 7.640C 7.670C 7.785C 7.3715 7.660C

18 19 2 0"

b) Calculating the curvature variance There are several ways of doing this calculation. This is one way. The data on screen 14.1 are used to construct the deviation squared table ( y , -Fil2. The term in cell B25 in screen 14.2 is obtained from the instruction : (B9 - $19)*(B9 - $19)

or

(B9 - $19)*2

and copying it to cells B25 to G32. The variance, .$, of a trial is obtained by adding the squares of the differences and dividing the sum by 6 - 1. For trial number 1, the instruction : @SUM(B25..G25)/5

is placed in cell 125, and copied to cells I26 to I32 (screen 14.2)

311

SCREEN 14.2 Calculation of curvature variance, SZ A

VARIANCE CALCULATION

Variance

0.0576 0.0617 0.0004 0.0025 0.0729 0.0090 0.0355 0.0100

0.0576 0.0775 0.0016 0.0064 0.1089 0.0930 0.0617 0.0225

0.0729 0.0005 0.0004 0.0121 0.0441 0.0756 0.0047 0.0009

0.0841 0.0005 0.0016 0.0121 0.0529 0.0090 0.0367 0.0256

0.0016 0.0005 0.0004 0.0001 0.1225 0.0506 0.0367 0.0225

mean

0.1764 0.2131 0.0004 0.0064 0.0529 0.0272 0.0148 0.0049

0.09004 0.07073 0.0009E 0.00792 0.09084 0.05291 0.03801 0.01728

variance

0.046088

standard deviation

0.214681

These calculations can be used to obtain the mean variance of all the trials : the variances in column I are added together and divided by 8 using the instruction @SUM(I25..I3 2)/8

placed in cell 135. The square root of the mean variance gives the standard deviation of an individual response. It is obtained by placing : @SQRT(I35) in cell I37 (screen 14.2)

c) Calculating the Z function The variance in cell I25 is used to calculate the Z hnction for trial number 1 by placing the instruction : lO*@LOG(125)

312

in cell L25 (screen 14.3). The instruction is copied to obtain the eight Z hnctions (L25 to L32).

SCREEN 14.3 Calculation of the responses s2, Z and Z'

z

Variance 0.09004 0.07073 0.00096 0.00792 0.09084 0.05291 0.03801 0.01728

-10.4556 -11.5035 -30.1772 -21.0127 -10.4172 -12.7646 -14.2002 -17.6242

Z' 28.003 29.458 47.701 38.674 28.113 30.590 31.551 35.309

d) Calculating the 2' function For trial number 1, js, (cell 19) and sf (cell 125) are used to calculate Z' with the instruction : 1 O*@LOG(I9*19/125)

in cell 025. The other values for Z are obtained by copying the formula to cells 0 2 6 to 0 3 2 (screen 14.3).

e) Calculating the effects A new worksheet is opened (sheet 2) and the matrix of effects is entered (cells B9 to 116, screen 14.4). The responses from worksheet 1 are copied to cells f9 .. M16, alongside the effects matrix (screen 14.4).

313

SCREEN 14.4 Effects matrix of the z4-' design plus the four calculated responses

EFFECT MATRIX 4 =

1

2

3

-1 1 -1 1 -1 1 -1 1

-1 -1 1 1 -1 -1 1 1

-1 -1 -1 -1

1 1 1 1

12

13

23

123

1

1 -1 1 -1 -1 1 -1 1

1 1 -1

-1 1 1 -1 1 -1 -1 1

-1 -1 1 1

-1 -1 1

-1 -1 -1 1 1

I CURV. 1 1 1 1 1

1 1

1

1.54 7.90 7.52 1.64 7.67 7.79 7.37 7.66

82

0.090 0.071 0.001 0.008 0.091 0.053 0.038 0.017

z

Z'

-10.5 -11.5 -30.2 -21.0 -10.4 -12.8 -14.2 -17.6

28.00 29.45 47.70 38.67 28.11 30.58 31.55 35.30

The data are then used to calculate the effects of each factor as follows. Using the mean curvature as an example, the instruction : +$J9*B9

in cell B26 gives the product of the mean curvature of trial 1 multiplied by - 1. This instruction is copied to the cell range B26 _ .I33 to give all the products. The columns are then added and divided by eight to give the effect (screen 14.5). The instruction in cell B35 :

gives the effect of factor 1. The effects of the other factors are obtained by copying this instruction to cells B35 to I35 (screen 14.5).

314

SCREEN 14.5 Calculation of the effects of factors on the curvature

+

B26 :

$J9*B9 CURVATLTRE 2

12

13

-7.54 -7.90 7.52 7.64 -7.67 -7.79 I.37 7.66

4=123

23

7.54 7.54 1.54 -7.90 -7.90 7.90 -7.52 7.52 -7.52 7.64 -7.64 -1.64 -7.67 -1.67 1.61 7.19 -1.79 -1.19 -1.31 -7.37 1.31 1.66 1.66 1.66 ____--__----------_ - - - - - - - -. .- - - - - - - - _ - _ 0.1106 -0.0881 -0.0143 -0.0085 -0.0098 -0.0177 -7.54 7.90 -7.52 1.64 -7.67 1.79 -7.31 7.66

I

3

-7.54 -7.50 -7.52 -1.64 7.67 7.79 1.31 7.66

1

-7.54 7.90 7.52 -1.64 7.67 -7.79. -7.31 7.66

-- - - _ _ _

I.54 7.90 7.52 I.64 7. 67 1.19 1.37 1.66

--

0.0518

7.63601

35

The effects of factors for the three other responses are obtained in the same way using an analogous series of instructions. These instructions are indicated in the top left-hand cell of each screen.

SCREEN 14.6 Calculation of effects of factors on curvature variance B44 :

i$K9*B9

VARIANCE OF THE CURVATURE

44 45 46

7

1 2 3 4 5 6

I 8

-0.0900 0.0707 -0.0010 0.0079 -0.0908 0.0529 -0.0380 0.0173

-0.0900 -0.0707 0.0010 0.0079 -0.0908 -0.0529 0.0380 0.0173

-0.0900 -0.0707 -0.0010 -0.0079 0.0908 0.0529 0.0380 0.0173

0.0900 -0.0707 -0.0010 0.0079 0.0908 -0.0529 -0.0380 0.0173

0.0500 -0.0707 0.0010 -0.0079 -0.0908 0.0529 -0.0380 0.0173

0.0900 0.0707 -0.0010 -0.0079 -0.0908 -0.0529 0.0380 0.0173

-0.0900 0.0707 0.0010 -0.0079 0.0908 -0.0529 -0.0380 0.0173

0.0900 0.0707 0.0010 0.0079 0.0908 0.0529 0.0380 0.0173

315

SCREEN 14.7 Calculation of the effects of factors on function Z

B62 :

+

$L9*B9

EQNCTION Z

1 2 3 4 5 6 8

10.46 -11.50 30.18 -21.01 10.42 -12.76 14.20 -17.62

10.46 10.46 11.50 11.50 -30.18 30.18 -21.01 21.01 10.42 -10.42 1 2 . 7 6 -12.76 -14.20 -14.20 -17.62 -17.62 _ _ _ _ _ _ - _ _ - _ - - _ -_----0 . 2 9 3 -4.734 2.268

-10.46 11.50 30.18 -21.01 -10.42 12.76 14.20 -17.62 -------

1.142

-10.46 11.50 -30.18 21.01 10.42 -12.76 14.20 -17.62

--_____

-1.736

23

4=123

I

-10.46 -11.50 30.18 21.01 10.42 12.76 -14.20 -17.62 __----2.573

10.46 -11.50 -30.18 21.01 -10.42 12.76 14.20 -17.62 -1.411

-10.4E -11.5C -30.18 -21.01 -10.42 -12.7E -14.2C -17.62 -___---16.019

4=123

I

-------

SCREEN 14.8 Calculation of the effects of factors on function Z'

B80 :

+

$M9*B9

F"CT1ON

Z'

23 1 2 3 4 5 6 7 8

-28.00 29.46 -47.70 38.67 -28.11 30.59

-31.55 35.31

______ -0.167

-28.00 -29.46 47.70 38.67 -28.11 -30.59 31.55 35.31

-28.00 -29.46 -47.70 -38.67 28.11 30.59 31.55

35.31

------ -------

4.634

-2.284

28.00 -29.46 -47.70 38.67 28.11 -30.59 -31.55 35.31

28.00 -29.46 47.70 -38.67 -28.11 30.59 -31.55 35.31

28.00 29.46 -47.70 -38.67 -28.11 -30.59 31.55 35.31

-28.00 29.46 47.70 -38.67 28.11 -30.59 -31.55 35.31

28.00 29.46 47.70 38.67 28.11 30.59 31.55 35.31 - - - - - - _ ------- - - - - - - _ - - _ _ _ _ _ ---_-_-1.150 1.726 -2.595 1.470 33.675

316

f) Calculating the mean curvature at the low level of factor 5 The data are contained in columns B, C and D of worksheet 1 (screen 14.1). They are used to calculate the mean curvature by placing the instruction : @AVG(B9..D16)

in cell El8 (screen 14.9)

g) Calculating the mean curvature at the high level of factor 5 Columns F, G and H are treated in the same way using the instruction @AVG(F9..H16)

in cell I18 (screen 14.9) SCREEN 14.9 Calculation of mean curvatures at levels 5+ and 5-

CURVATDRES AT

LEVELS 5- AND 5+ 5+

1 2

3 4 5 6

7 8

2.

1.78 8.15 1.50 1.59 7.94 7.69 7.56 7.56

7.78 5.18 1.56 1.56 8.00 8.09 7.62 7.81

1.81 7.88 1.50 7.15 7.88 8.06 7.44 1.69

1.19 8.01 1.52 7.63 7.94 1.94 7.54 7.68

1.50 7.88 1.50 7.63 7.32 7.56 7.18 7.81

1.25 1.88 1.56 7.15 7.44 7.69 7.18 7.50

1.12 7.44 1.50 7.56 1.44 1.62 1.25 7.59

1.29 7.73 1.52 7.64 1.40 7.62 7.20 7.63

Mean

1.54 7.90 1.52 7.64 1.67 1.78 7.37

7.66

CALCULATION FOR THE SECOND INTERPRETATION

Two new worksheets are opened; worksheet 3 is used to calculate the elaborated responses from the raw responses, while worksheet 4 is used to calculate the effects. The calculations themselves are analogous to the ones camed out for the Z4-'design.

317

a) Calculating the mean curvature for each trial Each trial now contains only three results. The data in worksheet 1 are copied to worksheet 3. The 48 results on 16 lines occupy the range B9..D24 (screen 14.10). The mean curvature for trial number 1 is calculated with the instruction : @AVG(B9..D9) in cell F9, and this instruction is copied to cells F10..F24.

SCREEN 14.10 Calculation of mean curvatures for the trials in the

$'-'design

Trial

no

Y1

Y2

Y3

Mean

1 2

7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56 7.50 7.88 7.50 1.63 7.32 7.56 7.18 7.81

7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81 7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50

7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69 7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59

7.79 8.07 1.52 7.63 7.94 7.94 7.54 7.68 1.29 1.73 7.52 7.64 7.40 7.62 7.20 7.63

3

4 5 6 7 8 9 10 11 12 13 14 15 16

b) Calculating s2,2 and 2' The squares of the differences are calculated in cells H9..J24 using the instruction (screen 14.11) :

copied to all the cells of this range (screen 14.11). The curvature variance of trial number 1 is obtained by placing the following instruction in cell L9 : @SUM(H9..J9)/2 This instruction is copied to cells L9 to L24 (screen 14.1I).

318

SCREEN 14.11 Calculation of variances

square deviation 1 2 3 4 5 6 7 8 9 10 11

12 13 14

15 16

7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56 7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81

7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81 7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.5

7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69 7.12 7.44 7.5 7.56 7.44 7.62 7.25 7.59

7.79 8.07 7.52 7.63 7.94 7.94 7.54 7.68 7.29 7.73 7.52 7.64 7.40 7.62 7.20 7.63

0.0001 0.0064 0.0004 0.0019 0.0000 0.0659 0.0004 0.0160 0.0441 0.0215 0.0004 0.0003 0.0064 0.0040 0.0005 0.0312

0.0001 0.0121 0.0016 0.0054 0.0036 0.0205 0.0064 0.0152 0.0016 0.0215 0.0016 0.0107 0.0016 0.0044 0.0005 0.0178

0.0004 0.0361 0.0004 0.0136 0.0036 0.0128 0.0100 0.0000 0.0289 0.0860 0.0004 0.0075 0.0016 0.0000 0.0022 0.0019

Variance 0.00030 0.02730 0.00120 0.01043 0.00360 0.04963 0.00840 0.01563 0.03730 0.06453 0.00120 0.00923 0.00480 0.00423 0.00163 0.02543

Mean variance

0.01655

Standard deviation

0.12866

The mean variance is obtained by placing : @SUM(L9..L24)/ 16

in cell L26, and the standard deviation ofeach response using the instruction BSQRT(L26)

in cell L28 (screen 14.1 1) The hnctions Z and Z' are calculated from the variance using the instructions (screen 14.12) : 10*@LOG(L9) in column P to obtain Z

3 19

I O*@LOG(F9*F9/L9) in column R to obtain Z'.

The expression 10 log y 2 can also be calculated to check that it varies little (column N in screen 14.12) SCREEN 14.12 Calculation of 9,Z and Z'

0.00030 0.02730 0.00120 0.01043 0.00360 0.04963 0.00840 0.01563 0.03730 0.06453 0.00120 0.00923 0.00480 0.00423 0.00163 0.02543

17.83075 18.13747 17.52436 17.65428 17.99641 18.00370 17.54743 17.71476 11.25455 17.76733 17.52436 17.66944 17.38463 17.64290 17.15067 17.65428

-35.22879 -15.63837 -29.2 0819 -19.81577 -24.43697 -13.04227 -20.75721 -18.05948 -14.28291 -11.90216 -29.20819 -20.34641 -23.18759 -23.73318 -27.86925 -15.94597

53.05954 33.77584 46.73254 37.4700: 42.43335 31.04597 38.30463 35.77425 31.53746 29.66945 46.73254 38.01586 40.51222 41.37607 45.01992 33.60025

320

c) Preparing the effects matrix The effects matrix of the 25-1 design is entered into worksheet 4, cells B9..Q24. The signs ofthe interaction columns are calculated according to the signs rule (screen 14.13).

SCREEN 14.13 Effects matrix of the 2'-' design

no 1 3 5

7

9 10 11 12 13 14 15 16

1

2

3

-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1

-1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1

-1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1

123

5

12

13

-1 -1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 1 -1 1 1 1

-1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1

1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1

14

15 25

1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1

1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1

1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1

35

45 125 135 235

1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1

1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1

-1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1

-1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1

-1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1

I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

32 1

The experimental results are copied to R9..T24 fiom worksheet 3, and the means of the trials entered in column U (screen 14.14).

SCREEN 14.14 Calculation of mean curvatures

Mean

8.00 8.09 7.62 7.81 7.25 7.88

7.88 8.06 7.44 7.69 7.12 7.44

7.75 7.44

7.56 7.44

7.94 7.94 1.54 7.68 7.29 7.13 7.52 1.64 7.40

3 22

The variance is shown in column AA, Z in column AC and Z in column AE (screen 14.15).

SCREEN 14.15 Calculation of variance, Z and Z'

deviation square 0.0001 0.0064 0.0004 0.0019 0.0000 0.0658 0.0004 0.0160 0.0441 0.0215 0.0004 0.0003 0.0064 0.0040 0.0005 0.0312

0.0001 0.0121 0.0016 0.0054 0.0036 0.0205 0.0064 0.0152 0.0016 0.0215 0.0016 0.0107 0.0016 0.0044 0.0005 0.0178

0.0004 0.0361 0.0004 0.0136 0.0036 0.0128 0.0100 0.0000 0.0289 0.0860 0.0004 0.0075 0.0016 0.0000 0.0022 0.0019

Variance 0.00030 0.02730 0.00120 0.01043 0.00360 0.04963 0.00840 0.01563 0.03730 0.06453 0.00120 0.00923 0.00480 0.00423 0.00163 0.02543

man variance

0.016554

atandard deviation

0.128663

Z -35.2 -15.6 -29.2 -19.8 -24.4 -13.0 -20.8 -18.1 -14.3 -11.9 -29.2 -20.3 -23.2 -23.7 -27.9 -15.9

53.1 33.8 46.1 37.5 42.4 31.0 38.3 35.8 31.5 29.7 46.1 38.0 40.6 41.4 45.0 33.6

323

d) Calculating the effect of factors on curvature The effects are calculated by multiplying the effects matrix by the mean curvatures using the instruction: +$U9*B9

copied to the range B32..Q47, then adding the columns and dividing this sum by 16 using the instruction (cell B49): @SUM(B32..B47)/16

which is copied to cells B49 - 4 4 9 (screen 14.16).

SCREEN 14.16 Calculation of the effects of factors on curvature

1

2

1-1.79-1.19 2 8.01 -8.07 3 -1.52 1.52 4 1.63 1.63 5 -7.94 -1.94 6 1.95 -1.95 1 -1.54 1.54 8 1.69 1.69 9 -1.29-7.29 10 1.13-7.73 11-1.52 7.52 12 1.65 7.65 13-1.40-1.40 14 1.62 -7.62 15-7.20 7.20 16 1.63 1.63

3

123

5

12

13

14

15

25

35

45

125 135 235

-1.19-1.79-1.19 1.19 1.79 7.19 1.79 1.19 1.19 1.19-1.79-1.19-1.79 1.1 -8.07 8.01 -8.01-8.07-8.07 8.01-8.07 8.07 8.07 -8.07 8.07 8.07 -8.07 8.0 -1.52 1.52 -7.52-7.52 1.52-7.52 7.52-1.52 1.52-7.52 7.52-7.52 1.52 1.5 -1.63-1.63-7.63 1.63-1.63-1.63-7.63-1.63 1.63 1.63-7.63 1.63 7.63 1.6 1.94 7.94 -1.94 1.94 -1.94 -1.94 1.94 1.94 -1.94 -1.94 -1.94 1.94 1.94 7.9 1.95 -1.95 -1.95 -1.95 1.95 -1.95 -7.95 1.95 -1.95 1.95 1.95 -1.95 7.95 7 . 9 1.54-7.54 -1.54-1.54-7.54 7.54 7.54 -7.54-7.54 7.54 1.54 7.54 -1.54 7 . 5 7.69 1.69-7.69 1.69 1.69 7.69-7.69-1.69-1.69-1.69-1.69 -1.69-7.69 1.6 -1.29-7.29 7.29 1.29 1.29 7.29-1.29-1.29-1.29-7.29 1.29 1.29 1.29 1.2 1.13 1.1 -7.73 1.13 1.13-1.13-1.13 1.13 1.73-1.13-7.13 1.13-7.13-1.73 -1.52 7.52 1.52-1.52 1.52-7.52-1.52 1.52-1.52 1.52-1.52 7 . 5 2 -1.52 1.5 7.65 1.65-7.65-7.65 1.65-1.65-7.65 1.6 -1.65-1.65 7.65 1.65-1.65-7.65 7.40 1.40 1.40-7.40-7.40 7.4 1.40 7.40 1.40 1.40-1.40-7.40-1.40-1.40 7.6 7.62-7.62 7.62-7.62 1.62-1.62 7.62-1.62 1.62-1.62-7.62 1.62-7.62 1.20-7.20 1.20-1.20-1.20 1.20-7.20 1.20 1.20-7.20-1.20-7.20 1.20 1.2 1.63 1.63 1.63 7 . 6 3 1.63 1.63 1.63 1.63 1.63 1.63 7.63 1.63 1.63 7.6 - _ _ _ - _ - _ _ _ _______ _ _ - - _ - _ -- _ _ _ __- _ _ _---- ---- ---- ---- - - _ _ ---- ---- ----

3 24

e) Calculating the effect of factors on curvature variance The effects of factors on curvature variance are calculated in the same way as the effect of factors on curvature itself The effects matrix is multiplied by the variance using the instruction (in cell B55): +$AA9*B9 copied to all cells in the range B55 ...Q70. The elements of each column are then added and the sum is divided by 16 (instruction in cell B72): @SUM(BSS...B70)/16

SCREEN 14.17 Calculation of the effects on curvature variance

1-0.0003 -0.0003 -0.0003 -0.0003 2 0.0273 -0.0273 -0.0273 0.0273 3-0.0012 0.0012 -0.0012 0.0012 4 0.0104 0.0104 -0.0104 -0.0104 5-0.0036-0.0036 0.0036 0.0036 6 0.0496 -0.0496 0.0496 -0.0496 1-0.0084 0.0084 0.0084 -0.0084 8 0.0156 0.0156 0.0156 0.0156 9-0.0373 -0.0313 -0.0373 -0.0313 10 0.0645 -0.0645 -0.0645 0.0645 11-0.0012 0.0012 -0.0012 0.0012 12 0.0092 0.0092 -0.0092 -0.0092 13-0.0048 -0.0048 0,0048 0.0048 14 0.0042 -0.0042 0.0042 -0.0042 15-0.0016 0.0016 0.0016 -0.0016 16 0.0254 0.0251 0.0254 0.0254 ..--. . . . .

~----

-0.0003 0.0009 0.0003 -0.0213-0.0273-0.0273

0,0003

0.0003

0,0003

0.0003

0.0003-0.0003

-0.0003 -0.0003 0,0003

0.0213-0.0213 0.0273 0.0273 -0.0273 0,0273 0.0273 -0,0273 0.0273 -0.0012-0.0012 0.0012 -0.0012 0.0012 -0.0012 0.0012 -0.0012 0,0012 -0.0012 0,0012 0.0012 -0.0101 0.0104 -0.0104 -0.0104 -0.0104 -0.0104 0.0104 0.0101 -0,0104 0.0104 0,0104 0.0104 -0.0036 0.0036-0.0036 -0.0036 0.0036 0.0036-0.0036-0.0036-0.0036 0.0036 0.0096 0.0036 -0.0496-0.0196 0.0496 -0.0496-0.0496 0.0496-0.0096 0.0496 0.0996 -0.0496 0.0496 0.0496 -0.0084 -0.0084-0.0081 0.0084 0.0084 -0.0084 -0.0084 0.0084 0.0081 0.0084 -0.0084 0.0084 -0.0156 0.0156 0.0156 0.0156-0.0156-0.0156-0.0156-0.0156-0.0156 -0.0156-0.0156 0.0156 0.0313 0.0373 0.0373 0.0373~0.0313-0.0373-0.0373-0.0373 0.0373 0.0313 0.0373 0.0173 0.0645-0.0645-0.0645 0,0645 0,0645 -0,0645-0.0645 0,0645-0.0645 -0,0645 0.0645 0.0645 0.0012-0.0012 0.0012 -0.0012-0.0012 0.0012-0.0012 0.0012-0.0012 0.0012 -0,0012 0.0012 0.0092 0.0092-0.0092 -0,0092 0.0092 0.0092-0.0092 -0,0092 0.0092 -0,0092-0.0092 0,0092 0.0048 0.0048-0.0018 -0.0048-0.0048 -0.0048 0.0018 0.0018 0.0048 -0,0048 -0.0048 0.0048 0.0042-0.0012 0.0042 -0.0012 0.0012-0.0042 0.0042 -0.0042-0.0042 0.0042 -0.0012 0.0042 0.0016-0.0016-0.0016 0.0016-0.0016 0.0016 0.0016 -0.0016-0.0016 -0.0016 0.0016 0.0016 0.0251 0 . 0 2 5 1 0.0254 0.0254 0.0254 0.0254 0.0254 0.0251 0.0254 0.0254 0.0254 0 . 0 2 5 4

. . . . .

..._. ..~. _... . . . .

. . . .

. . . .

. . . .

. . ~ . . . . ~~ ~ ...~ . .~~.~

0.00925 -0.0074 -0.0024 0.00140 0.00199-0.00320.00031 0.00601 -0.0019 0.00176-0.00710.00403 0.00386 -0.0018 0.00766 0.0165

The effects of factors on the fbnctions Z and Z are calculated in the same way, using the instructions: +$AC9*B9 and +$AE9*B9

325

SCREEN 14.18 Calculation of the effects of factors on the function Z

1 1

35.23 -15.64 29.21 -19.82 5 24.44 6 -13.04 7 20.76 8 -18.06 9 14.28 1 0 -11.90 11 2 9 . 2 1 1 2 -20.35 13 2 3 . 1 9 1 4 -23.73 1 5 21.87 1 6 -15.95

2 3 4

2

3

123

5

12

13

35.23 35.23 35.23 3 5 . 2 3 - 3 5 . 2 3 - 3 5 . 2 3 15.64 15.64 -15.64 15.64 15.64 15.64 -29.21 29.21 -29.21 29.21 29.21 -29.21 -19.82 19.82 19.82 19.82 -19.82 19.82 24.44 -24.44 -24.44 2 4 . 4 4 -24.44 24.44 1 3 . 0 4 -13.04 1 3 . 0 4 1 3 . 0 4 13.04 - 1 3 . 0 4 -20.76-20.16 20.16 20.76 20.16 20.16 -18.06 -18.06 -18.06 18.06 -18.06-18.06 14.28 14.28 14.28 -14.28 -14.28 -14.28 11.90 11.90-11.90-11.90 11.90 11.90 -29.21 2 9 . 2 1 - 2 9 . 2 1 -29.21 29.21 -29.21 -20.35 20.35 2 0 . 3 5 - 2 0 . 3 5 - 2 0 . 3 5 20.35 23.19-23.19-23.19-23.19-23.19 23.19 23.73 -23.73 23.73 -23.13 23.13 -23.13 - 2 7 . 8 1 -27.87 27.87 - 2 1 . 8 7 27.87 27.87 -15.95 -15.95 -15.95 -15.95 -15.95 -15.95

14

0 . 4 6 8 0 0 . 6 0 6 9 0.0034 - 0 . 9 2 2

25

35

45

125 135 235

-35.23 -35.23 -35.23 -15.64 15.64 -15.64 29.21 -29.21 29.21 19.82 19.82 19.82 24.44 -24.44 -24.44 1 3 . 0 4 13.04 - 1 3 . 0 4 -20.76 -20.76 20.76 -18.06 18.06 18.06 -14.28 14.28 14.28 -11.90-11.90 11.90 29.21 29.21-29.21 20.35 -20.35-20.35 23.19 23.19 23.19 23.13 -23.13 23.13 -27.81 27.87 -27.87 -15.95 -15.95 -15.95

-35.23-35.23 35.23 35.23 35.23 -35.23 15.64 -15.64 -15.64 15.64 -15.64 -29.21 29.21 -29.21 29.21 -29.21 -29.21 -19.82 -19.82 19.82-19.82-19.82 -19.82 24.44 24.44 2 4 . 4 4 - 2 4 . 4 4 - 2 4 . 4 4 -24.44 1 3 . 0 4 -13.04 - 1 3 . 0 4 13.04 -13.04 -13.04 20.16-20.16-20.76-20.76 20.16-20.76 18.06 18.06 18.06 18.06 18.06 -18.06 14.28 14.28 - 1 4 . 2 8 - 1 4 . 2 8 - 1 4 . 2 8 -14.28 11.90-11.90 11.90 11.90-11.90-11.90 29.21 -29.21 29.21-29.21 29.21 -29.21 20.35 20.35-20.35 20.35 2 0 . 3 5 - 2 0 . 3 5 -23.19-23.19-23.19 23.19 23.19-23.19 -23.73 23.13 2 3 . 7 3 - 2 3 . 1 3 23.73 -23.7 -27.87 21.87 2 7 . 8 1 2 1 . 8 7 - 2 7 . 8 7 -27.8 -15.95 -15.95-15.95-15.95-15.95 -15.9

1.4559 -1.218 -1.298

-2.412 0.2804 2 . 3 6 5 3 0 . 9 3 9 1 1 . 8 5 3 5

- _ _ _--- - - - - - - _ _ _ _ _ _ _ - ---__ - - - ____ _---

4.1059 -1.2350.5314

15

--__ ____

-15.64

_ _ _ -----

-_______

----

---

-21.41

SCREEN 14.19 Calculation of the effects of factors on the function Z'

1

2

3

123

5

12

13

14

15

25

35

45

125 135 235

1 -53.06 - 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 53.06 53.06 53.06 53.06 5 3 . 0 6 2 33.78 - 3 3 . 7 8 - 3 3 . 7 8 33.78-33.78 -33.78 -33.78 33.78-33.18 33.78 3 -46.73 46.73-46.73 46.13-46.73-46.73 46.73-46.73 46.73-46.73 4

5 6 7

8 9 10

11 12 13

14 15 16

37.47 3 7 . 4 1 - 3 1 . 4 7 - 3 7 . 4 7 - 3 7 . 4 1 37.47 -42.43-42.43 42.43 42.43-42.43 42.43 31.05-31.05 31.05-31.05-31.05-31.05 -38.30 38.30 38.30-38.30-38.30-38.30 35.77 35.77 35.71 35.77-35.77 35.77 -31.54 -31.54-31.54-31.54 31.54 3 1 . 5 4 29.67 -29.67-29.67 29.67 29.67-29.67 -46.73 46.73-46.73 46.73 46.73-46.13 38.02 3 8 . 0 2 - 3 8 . 0 2 - 3 8 . 0 2 38.02 38.02 -40.57 -40.57 40.57 40.57 40.57 40.51 41.38 -41.38 41.38-41.38 41.38-41.38 -45.02 45.02 45.02-45.02 45.02-45.02 33.60 33.60 33.60 33.60 33.60 33.60

- - _. - -- - -

----

-3.9791.1362

-0.554-0.408-0.751

-37.47 -31.11-37.47 -42.43-42.43 42.43 31.05-31.05-31.05 -38.30 38.30 38.30 35.77 3 5 . 7 7 - 3 5 . 7 1 31.54 31.54-31.54 -29.67 29.67 29.67 46.73 -46.73-46.73 -38.02 -38.02 38.02 -40.57 -40.57-40.57 41.38 - 4 1 . 3 8 41.38 -45.02 45.02-45.02 33.60 33.60 33.60

53.06 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 33.78-33.78 33.78 3 3 . 7 8 - 3 3 . 1 8 46.13-46.73 46.73-46.73 46.73 -31.47 31.41 37.47-37.47 37.47 37.47 42.43-42.43-42.43-42.43 42.43 42.43 31.05-31.05 31.05 31.05-31.05 31.05 -38.30-38.30 38.30 38.30 38.30-38.30 -35.77-35.77-35.77-35.71 -35.71-35.17 - 3 1 . 5 4 -31.54 - 3 1 . 5 4 31.54 31.54 31.54 -29.67 -29.67 2 9 . 6 7 - 2 9 . 6 7 -29.67 29.67 46.73-46.73 46.73-46.73 46.13-46.13 38.02-38.02-38.02 38.02-38.02-38.02 -40.57 40.57 40.57 40.51 -40.51-40.57 -41.38 41.38-41.38-41.38 41.38-41.38 45.02 45.02-45.02-45.02-45.02 45.02 33.60 33.60 3 3 . 6 0 33.60 33.60 33.60

53.0 33.7 46.1 37.4 42.4 31.0 38.3 35.7 31.5 29.6 46.7 38.0 40.5 41.3 45.0 33.6

- - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -0.012 0.9123 -1.4171.32891.3904

2.3808-0.263-2.372

-0,916-1.881

39.0

326

3. a)

CALCULATION FOR OPTIMIZATION Experimental design for curvature and curvature variance

Open worksheet 5 and enter the values shown in column B,C, D and E of screen 14.20. This table contains the levels of factor 1, 2, 3 and 5 in the range B9..E24, arranged in a Z4 design. Two columns are reserved for the results of the calculations of curvature (column G9..G24) and curvature variance (column F9..F24)

SCREEN 14.20 The calculated curvature and variance of curvature

-1

1

1

-1

variance

curvature

0

0

The instruction : +K53 is entered in cell F9 and copied to the rest of the cells F9..F24. The worksheet will initially show zeros in this column.

327

The instruction : +K77 is entered in cell G9 and copied to cells G9..G24. Again the column will initially contain zeros.

b) Effects matrix The values for curvature and the curvature variance are obtained from the effects matrix as follows (screen 14.21) : a. +1 is entered in cells B3 1..B46. b. The range B9..E24 is copied to C3 I..F46. c. The interaction products are calculated in cells G3 1..J46 by placing :

.

Instruction D3 1*E3 1 in cell G3 1 and copying it to G3 1..G46. Instruction D3 1*F3 1 in cell H3 1 and copying it to H3 1. .H46.

. .

Instruction E3 1*F3 1 in cell I3 1, copying it to I3 1..I46 Instruction D3 1*E3 1*F3 1 in cell J3 1, copying it to J3 1 . . J46.

328

SCREEN 14.21 Effects matrix

11 %42 12

1 1

13

1

14

1

-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1

16

1

1

1

4 5

1 1 1

6

1

I

1 1 1

3

8

1

I

I

5

23

25

35

235

1 1 1 1 -1 -1 -1 -1 -1

-1 -1 1 1

-1

1 -1 -1 1 1

-1 -1 1 1

-1 -1 -1 -1

1

1

-1 -1 1 1 -1 -1 1 1 -1 .1 1 1 -1 -1

-1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1

-1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1

1 1 -1 -1 -1 -1 1 1 1 1

-1

1 1 -1 -1 1 1 -1 -1 --1 -- 1 1 1 -1 -1

1

1

1

1

1

-1 -1 -1

1

c) Modelling the variance of curvature

The curvature variance is then calculated (screen 14.22). The values for the coefficients of the model are entered in cells B5 1 ..J51and all the coefficients are multiplied by 10,000. Cell B5 1 Cell C5 1 Cell D5 1 Cell E5 1 Cell F5 1 Cell G5 1 Cell H5 1 Cell I5 1 Cell J51

+I65 +90 -70 +O +O +60

+O -70 +70

329

The instruction +B3 1*B$51

is entered in cell B53 and copied to cells B53.. 568. The instruction @SUM(B53.. J53)

is then entered in cell K53 and copied to K53.. K68.

SCREEN 14.22 The calculated variance of curvature

DISPERSION

90

90

165

10 11 12 13 14 15 16

d)

165 165 165 165 165 165 165 165

90 -90 90

-90 90 -90 90 -90 90 -90 90

70 -70

70

-70 -70 70 70 -70 -70

70 70 -70 -70

0

0

60

0

0

-60

0 0 0

0 0

-60 60

0

0 0

0

0 0 0

0

0 0 0

0 0 0 0 0 0

60 60 60 -60 -60 -60 -60

0

-70

-70

0 0

70

-70

70

-70

70 70 -70

-70 -70 -70 -70

0 0

0 0

-70

60

Modelling the curvature

The curvature is calculated in the same way. The values of the coefficients of the model are entered in cells B75 - J75. The value in cell B75 (7.68) is obtained by adding the influence of factor 4 (0.05) to the mean (7.63).

330

Cell C75 Cell D75 Cell E75 Cell F75 Cell G75 Cell H75 Cell 175 Cell 575

+o. 1 1 -0.09 +O -0.13

+O +0.08 +O +O

The instruction +B3 1 *B$75 is entered in cell B77, and copied to cells B77..J92. The instruction @SUM(B77..J77)

is then placed in cell K77 and copied to cells K77..K92. SCREEN 14.23 The calculated curvature

I

CURVATURE 7.68

0.11

-0.09

0

-0.13

0

0.08

0

0

7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68

-0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11

0.09 0.09 -0.09 -0.09 0.09 0.09 -0.09 -0.09 0.09

0 0

0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13

0

0 0

0 0

0 0 0 0

0.08 0.08 -0.08 -0.08 0.08 0.08

0

0 0

0

-0.08

0

0

0 0 0 0 0

-0.08 -0.08 -0.08 0.08 0.08 -0.08 -0.08 0.08 0.08

0 0 0 0 0

0 0 0 0 0

0

0

0

0

0

0

0

0

curvat. 1 2 3 4

5 6 7 8

9 10 11 12 13 14 15 16

0

-0.09

0 0 0 0 0 0 0 0 0

0.09

0

0.09

0

-0.09

0 0

0.09 -0.09

-0.09

0

-0.13

0 0 0

-0.13

0

0 0 0

0 0

65 245 -55 125 225 405 65 245 345 525 -55 125 -55 125 65 245

33 1

The values for curvature variance and the curvature itself thus calculated are automatically copied to cells F9..F24and G9..G24 on screen 14.20, as shown in screen 14.24.

SCREEN 14.24 The calculated curvature and variance of curvature

Case

X1

x2

x3

x5 variance

curvature

332

RECAPITULATION 1. This calculation exercise has shown that a spreadsheet is adequate for fairly simple calculations (effects, variance or standard deviation). In general, the experimenter needs no more than this tool for his early designs. But he may eventually find it limiting. 2. This limit is reached quite soon, as it requires a lot of calculation time to test all the potential hypotheses. Graphic outputs, which are a great help, are very slow to prepare. This is why, while it is quite possible to do the calculation with a basic spreadsheet, specialized software can greatly reduce the time required for interpretation. But such software must help the experimenter to check all the hypotheses, results and conclusions. Unfortunately, this ideal program is not yet available. 3. Interpretation requires many hours of work. Therefore, anything that speeds things up is welcome. The perfect program should be transparent enough to give the experimenter the maximum freedom to use all his creativity, fast enough to prepare the calculations and graphics, and powerful enough to handle complex cases.

CHAPTER 15

EXPERIMENTAL DESIGNS AND COMPUTER SIMULATIONS

1.

INTRODUCTION

In the preceding chapters we have seen how experimental designs are used in the area of their original application: experimental science. But we will shortly see that they are equally suitable for use in computer simulation of phenomena or processes. This type of application is beginning to spread, heralding a new and promising area of application for Experimentology. The techniques can be used to organise computer manipulations so as to reduce the number of runs and provide simplified mathematical models that are much more easily used than the original simulation software. This simplified model is only valid within a limited domain, defined by the minimal and maximal values of each factor. A computer simulation is a complex program based on general scientific laws. It provides output values as a hnction of the input data. In this respect a computer simulation run is completely analogous to an experiment. The specialist who runs computer-based mathematical

334

simulations may be compared to the experimenter carrying out an experiment. As the two activities are so analogous, we can set up the following parallels (Figure 15.1):

0 0

Input variables correspond to factors. Output values correspond to the responses. A computer simulation run corresponds to an experiment. The computer calculation software is comparable to the experimental phenomena. The calculation domain within which the model is valid is analogous to the experimental domain within which the experimental design model is used. The scientific laws used to establish the mathematical simulation correspond to the natural laws governing the experimental process.

-+

+

Factors -+

-+

+ + Input -+

n Natural laws.

Definition of experimental domain.

+

R~~~~~~~~

Scientific laws software.

4

Definition of the calculation domain.

-+

-+

+ -+

-+ used in simulation

+

+ -+

output

Simplified mathematical modelling valid in experimental domain.

Simplified mathematical modelling valid in calculation domain.

Figure 15.1: Analogy between experimental designs and computer simulations. These analogies imply that the methods of experimental design are readily applied to computer-based mathematical simulations. However, there are several peculiarities in the use of experimental design methodology for mathematical simulations. These are:

0

The responses contain no random error. A computer calculation always gives the same output for the same input data. There is no response drift. There is no block effects.

The immediate consequence is that the order in which the run are performed is no longer a problem. Blocking is not required, neither are anti-drift designs, and the computer runs need not be randomised. We can thus program the whole set of runs for a design right at the start of a study. In most cases, the standard order is adopted and the computer is set up so that all the calculation are performed sequentially. The calculating power of the computer also allows calculation of the mean, effects and interactions.

335 As there is no random error analogous to experimental error it is more difficult to differentiate between the significant and non-significant effects. Discarding weak coefficients leads to differences between the initial simulation model and the simplified model. The user must judge, on the basis of his experience, whether these differences are acceptable in the context of his application. Let us now examine the application of experimental designs to three examples of simulation:

Propane remover optimizing. This study was carried out by the refining and distribution division of Total. Optimization of a hydroelastic motor suspension system. This study was camed out by the engineers at Paulstra. Optimization of industrial gas production. This was run by the explorationproduction division of Total [26].

2.

EXAMPLE 1, PROPANE REMOVER OPTIMIZING

The major component of a propane remover is a distillation tower that separates propane from butane in oil refineries. Figure 15.2 shows the main components of a propane remover. The distillation tower is fed by a propane-butane mixture. This mixture can be introduced either at plate 19 or at plate 21. The separated propane is taken off at the top of the tower while the butane is taken off at the bottom. The propane contains only traces of butane, and these are removed in an air condenser. The traces of propane in the butane taken off at the base of the tower are removed by redistillation. 2.1. The problem: The study was designed to identify the propane remover operating conditions providing the greatest possible energy economy. The engineer responsible for the study used a powerful simulation program. This software allows many of the parameters to be varied and a wide range of thermal yields to be accurately calculated. A single study could take several months if all the available options were used. But as this study was specific to a particular refinery and several parameters were fixed, the study domain may be reduced to three main factors and a single response. The engineer decided to use the methods of experimental design to organise the simulation runs.

336

C Tern

Figure 15.2: Propane remover.

Response The only element of the propane remover whose energy consumption can vary significantly is the condenser. The selected response was therefore the condenser energy consumption, which should be minimised. Factor The three factors selected are: m

Factor 1: condenser pressure. Factor 2: charge input temperature. Factor 3: the plate number at which the charge is introduced.

Domain definition

Level Factor 1 : Factor 2: Factor 3:

12.1 bar 70°C 19

Level

+

16.1 bar 80°C 21

337

The two accessible plates are chosen for charge input. It is not possible to introduce the charge at any other level with this particular installation. Thus, the choice is between plate 19 and plate 2 1. Selection of experimental design

As there are two levels for each factor a standard 23 design was selected.

2.2. Simulation The experimenter ran eight simulation runs in the standard order of the trials in a 23 design (Table 15.1) and the results are entered in the response column. TABLE 15.1 EXPERIMENTAL MATRIX PROPANE REMOVER OPTIMIZING

Run

Pressure

XI0

1

Temperature Plate numbei 2 3

Response

1 2 3 4 5

25.2 38.1 24.4

6 ‘7 8

I Level(-) I

12.1 bar

1

7OoC

I

19

I

2.3. Interpretation The results are interpreted in exactly the same way as for laboratory trials. The effects and interactions are calculated by the usual methods. The calculations for this simulation example were programmed directly into the computer using the responses in Table 15.1. The effects and interactions are shown in Table 15.2.

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TABLE15.2 TABLE OF EFFECTS PROPANE REMOVER OPTIMIZING

Mean

34.23

1 2 3

8.33 -0.40 -2.98

12

0.00

13 23

-1.87

123

0.00

0.00

Thus, two factors are influent, condenser pressure (factor 1) and plate number (factor 3), with a small interaction between them. The results are illustrated in Figure 15.3

21

Plate number

19 12.1 bar

16.1 bar

Condenser pressure

Figure 15.3: Influence of condenser pressure (1) and plate number (3) on condenser energy consumption.

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The energy consumption will be minimal if the charge is introduced at plate 21 and condenser pressure is set at 12.1 bar. 2.4 Conclusion:

The main conclusions of the interpretation are: Low condenser pressure favours energy saving. 0

The charge should be input at plate 21.

The charge temperature does not influence energy consumption. The recommendations are thus:

3.

0

Charge temperature:

70-80 "C.

0

Charge input:

Plate 21.

0

Condenser pressure: 12.1 bar.

EXAMPLE 2: OPTIMIZATION OF A HYDROELASTIC MOTOR SUSPENSION

Automobile motors are not mounted directly on the car chassis. Instead, they are mounted via a suspension system known as a hydroelmtic motor suspension. The trade name of the Paulstra system is Strafluid. This system replaces the old "silent block" system. It minimizes vibrations between the motor and the bodywork, and it is clearly of vital importance for the life of the vehicle and the comfort of the driver. The motor vibrations are filtered out by the Strafluid so that they are not transmitted to the chassis or occupants of the car. Every time a new car model is produced, this essential component must be designed, calculated and optimised for the new vehicle and its occupants. The quality of hydroelastic motor suspensions depends largely on the rubber used and its shape. Experienced engineers can design the shape of the component most likely to be suitable for a new motor and new bodywork. But the main characteristics of this preliminary approximation must be checked. Although this can be done using real prototype units, a computer simulation is cheaper and faster than component fabrication. Paulstra have therefore developed a finite elements simulation program that calculates most of the characteristics of these units. The unit features likely to be best adapted to all the requirements can then be selected. Once this unit has been defined, a prototype component must be made to ensure that it satisfies all the requirements of the car maker and will perform satisfactorily throughout the life of the vehicle. This initial prototype unit generally satisfies the majority of the requirements, but there may be one or two specifications that are not complied with. The unit must therefore be

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slightly modified to optimize performance. Here again, the computer avoids random searching, the small alterations are evaluated by the simulation software. We will examine this phase in the development of a Strafluid which the designers optimized using experimental design methodology. The example thus begins at the stage when the Paulstra engineers have prepared the first prototype designed on the basis of their experience and calculations. Laboratory tests were carried out and the prototype appeared satisfactory on all counts, except for endurance. It correctly filtered out the whole range of undesired vibrations and it was sufficiently robust. But its working life was limited to less than 400,000 cycles in stress tests, while the required working life was over 1,500,000 cycles. The engineers knew that this poor performance was due to the poor distribution of forces within the component. They also knew that this was due to the shape of the outer surface of the rubber component. Proper stress distribution depends on a small change in the shape of the component. In this way they try to retain the original properties, but with a better energy distribution throughout the new unit. The shape of the original prototype was divided into five profiles (Figure 15.4), which we will call the existing profiles. The engineers produced slightly different geometries for each of these profiles, and these will be called the proposed profiles. There are thus five existing profiles and five corresponding proposed profiles. Profile 4

n

Figure 15.4: The five profiles of the initial prototype.

3.1 The problem:

8 4 k! 8 @

3g

The new prototype may be built using a combination of existing profiles and proposed profiles. There are 32 possible combinations of the two types of profiles. The problem is to choose the best $$ combination, which produces a component with a working life of over 1,500,000 cycles. All the combinations can be calculated, but as the $8 < .