Statistical control of quality, design of experiments, Taguchi’s methodology.
Jeanne Fine, I.U.F.M. Toulouse, Laboratoire de Statistique et Probabilités, U.A. C.N.R.S. D0745, Université Toulouse III.
1. Introduction. A workshop on Design of experiments by Taguchi’s methodology, organized by the I.U.F.M. of Toulouse, has taken place on the 22nd of February of 1994. I was asked to explain, in the simplest way, the reasons why Taguchi’s methodology « works so well », and to set this approach in relation to other statistical methods used in industry for statistical control of quality. The aim of this paper is to set the contents of the lecture which tried to give answers to these questions. First, let us consider the title : statistical control of quality, design of experiments, Taguchi’s methodology. What is quality ? It is now very well known that the increasing competitiveness of enterprises needs a deep change in the management methods, sometimes summed up by the formula: zero default, zero delay, zero breakdown, zero stock, zero paper, etc. This change essentially is based on a new approach of production quality. What is Statistics ? Statistics is the set of mathematical methods which enable to analyze and to understand the most varied phenomenons, by collecting and processing data. Statistics is often confused with data themselves, for these are also called statistics (unemployment statistics, foreign trade statistics, and so on). What is a design of experiments ? Designs of experiments are one of the most known statistical methods ; they have been introduced by Fisher in 1925, and have been considerably developed thanks to their use in agronomy. Production managers are now able to use designs of experiments thanks to Taguchi who has proposed a quite simple and original presentation of the method. These designs of experiments permit to improve the product design and manufacturing processes. But, even in industry, their use may be extended to many other objectives as, for example, the optimization of complex system adjustments which we will see today. There are two parts in this talk. In the first one, we will introduce, with a simple example, the statistical and probabilistic notions which will be used, and we will present the relations between designs of experiments and other methods used in the statistical control of quality. In the second one, we will develop designs of experiments and Taguchi’s methodology.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 2
2. Statistical methods for quality control. 2.1. Example. A machine produces 5000 marbles per day, the weight of each one having to be 12.5 g with tolerance limits fixed at 11.5 g and 13.5 g. The value 12.5 g will be called the target. At the end of the day, the production is controlled. For each marble, i = 1, 2, ..., 5000, we denote by yi the weight in grammes and by ai the acceptation (yes) or the non acceptation (no) by the customer. Data table Marbles i 1 2 3 ... i ... 5000
yi
ai
12.0 12.2 13.1 ... 11.8 ... 12.9
yes yes no ... yes ... yes
In the statistical vocabulary, marbles are the statistical units, Y is a real variable and A a categorical variable.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 3
2.2. Statistics. The counts and frequency distributions of the two variables (after data have been grouped in classes of range 0.5 for the real variable Y ) are as follows :
A values
Counts
Frequencies
yes
3410
68.2%
no
1590
31.8%
Total
5000
100%
Y values
Yes 68.2%
Counts
Y 10.0 10.0 Y 10.5 10.5 Y 110 . 110 . Y 115 . 115 . Y 12.0 12.0 Y 12.5 12.5 Y 130 . 130 . Y 135 . 135 . Y 14.0 14.0 Y 14.5 14.5 Y 150 . 150 . Y
Total 20%
No 31.8%
Frequencies
29 86 221 459 749 957 953 751 460 220 84 31
0.6% 1.7% 4.4% 9.2% 15.0% 19.1% 19.1% 15.0% 9.2% 4.4% 1.7% 0.6%
5000
100%
Frequency s
15%
10%
5%
Y 0 9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
Histogram of the variable Y
Moreover, it is possible to calculate the mean and the standard deviation (the square root of the variance) of the real variable, which are equal to 12.5 and 1 respectively. So, the mean is equal to the target. Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 4
2.3. Probabilistic model. Probability distributions of random variables may be considered as idealistic frequency distributions. The very known Gaussian (or normal) distribution is only defined by its mean (also called expectation) and its standard deviation. The normal distribution with mean 12.5 and standard deviation 1, is represented as below. This may be considered as a good modelization of the real variable Y previously defined. It is possible to calculate the proportion of marbles accepted by the customer, which is only 68.2%. Indeed, the target is reached on average but the standard deviation is too large and there is a large percentage of non acceptation. Y
Probability
(12.5 ; 1)
P (11.5 Y 13.5) = 0.682
Y 0 8.5
11.5
12.5
13.5
After an adjustment of the machine, another day production is controlled and gives a normal distribution with mean 12 and standard deviation 0.5 for the real variable Y, represented below.
Probability
Y
(12 ; 0.5)
P (11.5 Y 13.5) = 0.927
Y
Though the target is no longer reached on average : E (Y ) = 12 12.5, the results are better because the standard deviation is smaller : (Y ) = 0.5. The acceptation percentage is now of 92.7%.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 5
What can we learn from this result ? In fact, if we denote by a the target, the aim is to minimize the error Y a. For this, we use the so called mean squared error, denoted by MSE, which is the expectation of the square of error. If we set : E ( Y ) = and ( Y ) = , then we obtain the following decomposition :
2 MSE E Y a a 2 2
bias
variance
In order to minimize MSE, we have to minimize both bias ( difference between the target and the mean) and variance.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 6
2.4. Receipt control by sampling. After a new adjustment, the one day production is once again controlled, but only 400 marbles, among the 5000 which have been produced, are controlled. We observe, on this sample, y 12.2 and s 0.5. 1/ What is it possible to say about the mean of the whole day production ? 2/ Is the average improvement, 12.2 instead of 12, significant ? Answers : 1/ y and s2 are observations of random variables, denoted by Y and S 2, which verify : E Y , V Y
2 and E S 2 2 n
Moreover, Y may be considered as a normal random variable. So, we have : Y P 196 . n 196 . 0.95 s We deduce that the estimation of the mean by 95% confidence interval is : s s . ; y 196 . y 196 n n
. ; 12.25 1215 In conclusion, the mean is located between 12.15 and 12.25 with an error probability of 5%. Y and S are called estimators of and respectively. 2/ The previous mean was equal to 12 and the 95% confidence interval for the production mean after the new adjustement is [12,05 ; 12,25]. Thus, the difference between 12 and 12.2 is significant ; this is not only due to sampling fluctuation, the new adjustment has improved the production quality. Statistically, conclusions on significative difference are given by using hypothese tests. Estimation and hypothese tests are parts of inference statistics. Summary We have a population of interest (here the daily production of 5000 marbles). Instead of doing the survey (here the control) on the entire population (which defines census), we do it on a subpopulation, called sample (which defines sampling). The aim is then to infer to the entire population the results obtained on the sample (inference statistics). Sampling is of course cheaper than census and the result precision (measured by small confidence intervals) is only dependent on the sample size n and not on the sampling rate n / N. Thus, it is possible to obtain a very good result precision with sampling.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 7
2.5. Statistical Process Control. Sampling may be used during the production process. Let us continue the example. At each hour of a new day, a set of 10 marbles is controlled and the mean is calculated, the so obtained values are plotted on a graph.
Control card 196 .
12.5
10
12.2
196 .
10
11.9
7
8
9
10
11
12
13
14
15
16
17
18
Hours
If the production is regular, then the distribution of Y may be considered as a normal distribution with mean 12.2 and standard deviation 0.5. We then know that the mean on 10 marbles are observations of a random variable Y , the distribution of which is normal with a mean of 12.2 and a standard deviation of 0.5 10 . The probability that an observation goes out from the proposed limits drawn on the graph is then of 5%. If such an event occurs we may conjecture that the machine is going wrong (with always an error probability of 5% !). It is also possible to construct control cards for the standard deviation or other statistical indices.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 8
2.6. Linear regression model. We look for an adjustment improvement. It has been noticed that weight Y depends on the heating temperature X (expressed in Celsius degrees). Several experiments are carried out under two different temperatures. The following results are obtained : X
Y
60
12.1
60
10.5
60
10.4
80
11.9
80
13.5
80
13.6
Y 13
13.2
12.7
12.2
11.7
Y aX b 11
11.2
10.7
X 10.2 50
60
80
Y is the variable to be explained, this is called the response. X is the explaining variable. This is here a real variable, so we use the linear regression model.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 9
The aim is to find out the line, the equation of which is Y a X b , as close as possible of the set of the 6 points (xi ,yi) i = 1, ..., 6, that is to say to look for the parameters a and b which are minimizing 6
2 1 the mean squared errors between Y and Y : y i a x i b 6 i1
The result is : a
COV X , Y V X
and b Y a X .
Here we have : X 70 , Y 12 , V X 100 and COV X , Y 10 , hence a 01 . and b 5, thus Y 01 . X 5 12 01 . X 70 . X
Y
Y 01 . X 5
60
12.1
11
1.1
60
10.5
11
-0.5
60
10.4
11
-0.6
80
11.9
13
-1.1
80
13.5
13
0.5
80
13.6
13
0.6
e Y Y
We then have the following decomposition : Y Y e 12 01 . X 70 e
The coefficient 0.1 has the following interpretation : when the variable X increases by 10 units, then the variable Y increases by 0.110 = 1 unit.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 10
2.7. Variance analysis model. Temperature may be considered as a categorical variable A (called factor) with two categories or levels : corresponding to 60 and + corresponding to 80. Model Y Yk
Vk
Error e Y Y
A
Y
12.1
10.5
10.4
0.6
+
11.9
1.1
+
13.5
+
13.6
1.1 Y1 11
Y2 13
V1 0.6
V2 0.6
0.5
0.5 0.6
We then have the decomposition of the mean and of the variance :
Y
1 Y Y 2 1 2
V Y
12
1 11 13 2
1.6
V Y
2 2 1 1 Y1 Y Y2 Y V1 V2 2 2
1
+
0.6
+
V e
V Y
We will conclude to an effect of the factor A when the difference Y1 and Y2 will be significant, that is to say when the variance of error is small in respect to the variance of Y . We use the following notation for the decomposition obtained by the analysis variance model :
Y Y e 12 1 1 A e that means Y 11 when A is at level and Y 13 when A is at level +.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 11
2.8. Designs of experiments. Weight depends on the heating temperature A but also on the calibrator diameter B, variable considered as categorical with two levels ( , +). The real variable Y is the response. The categorical variables A and B are the factors, each one with two levels. We carry out 8 experiments in accordance with the following experimental design and we obtain responses indicated in the last column. The mean and the variance of Y are 12 and 2.2875 respectively. N° essay
A
B
Y
1
–
–
11.2
2
–
–
11.8
3
–
+
10.6
4
–
+
10.4
5
+
–
11.6
6
+
–
11.4
7
+
+
14.7
8
+
+
14.3
It is possible to consider two models : one, Y1 , without interaction and one, Y2 , with interaction between A and B. Case 1. Without interaction between A and B Y 12 1 1 A 0.5 0.5 B 1
^ Response Y 1 13.5
12.5
11.5
10.5
+
A
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+
-
B
-
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 12
Case 2 With interaction between A and B 1 1 Y2 12 1 1 A 0.5 0.5 B AB 1 1 Response^2 Y 14.5 +1
13.5
12.5 -1 11.5
10.5
-1
A
+
-
+
B
+1
-
1 1 The 2 by 2 table means that we have to add 1 when A and B are both at level or level + 1 1 and to subtract 1 when A is at level and B at level + or when A is at level + and B at level .
If we define the errors : e1 Y Y1 and e2 Y Y2 then we obtain the following decompositions of the variance : 2.2875 = V(Y ) = V(Y1 ) + V(e1 ) = 1.2500 +1.0375 and 2.2875 = V(Y ) = V(Y ) + V(e ) = 2.2500 + 0.0375 2
2
For these data it is important to take into account the interaction between A and B. Summary. The statistical tool used in designs of experiments is the variance analysis. Algebraic results will permit to answer the following questions : - How many essays (or experiments) do we have to realize ? - How to combine these essays? The aim of the following part is to give the corresponding answers.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 13
3. Designs of experiments and Taguchi’s methodology. 3.1. Complete factorial designs. Factorial designs. A design with two or more factors is called factorial. The notation : 2 3 4 means that we have 3 factors with 2 levels and 1 factor with 4 levels. We then have 32 possible combinations of these factors. Complete designs. If we realize at least one essay for each combination of factors, the design is said complete. If there are k essays for each combination of factors, the design is said complete with k repetitions. If there is only one essay for each combination of factors, the design is said complete without repetition. Examples. Design 23 complete without repetition
Design 23 complete with 2 repetitions
N° Essay
A
B
N° essay
A
B
1
1
1
1
1
1
2
1
2
2
1
1
3
1
3
3
1
2
4
2
1
4
1
2
5
2
2
5
1
3
6
2
3
6
1
3
7
2
1
8
2
1
9
2
2
10
2
2
11
2
3
12
2
3
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 14
Number of parameters to estimate and number of essays. We have seen that the sum of the effects of the level of each factor is equal to zero, so when we consider a factor with p levels, we have only (p1) parameters in order to estime the effect of this factor ; in the same way, when we consider interaction between two factors A and B with respectively p and q levels, the sum of each row and the sum of each column is equal to zero, so we have only (p1) (q1) parameters in order to estimate the interaction of A and B. If, in the previous example (2 factors A and B with 2 and 3 levels respectively), we want to estimate : - the general effect (1 parameter), - the effect of each level of each factor called main effect (1 parameter for A, 2 parameters for B), - the effect of each crossed level of the two factors A and B called interaction AB (2 parameters), this needs to carry out at least 6 essays. If we choose a design without repetition (6 essays), the error is equal to zero Y Y . So, it is possible to estimate all the parameters but it is impossible to test their significance. The design with repetition enables us to test the significance of the results. Of course, it is possible to consider the effect of each cross level of 3 factors A, B, C (interaction ABC) or more than 3 factors, but it is not very used in practice. Even the interactions of two factors are not always considered.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 15
3.2. Orthogonal fractional designs. Introduction. We consider a complete design 23 without repetition, for which the levels of the 3 factors A, B, C, are denoted and + and Y the response. We carry out 8 essays. If we consider the general effect, the main effects and all the interactions, we have 8 parameters to estimate.
Jeanne Fine 1994
N° essay
A
B
C
Y
1
–
–
–
15.5
2
+
–
–
17.5
3
–
+
–
13.5
4
+
+
–
9.5
5
–
–
+
7.5
6
+
–
+
11.5
7
–
+
+
9.5
8
+
+
+
11.5
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Statistical control of quality, design of experiments, Taguchi’s methodology.
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It is easy to estimate parameters, using the following way. Columns A, B, C are given by the experimental design. First, we construct the first column I with only +, then the columns AB, AC, BC and ABC using the usual rule of the sign product (+ by + and by give +, + by and by + give ). N° essay
I
A
B
C
AB
AC
BC
ABC
Y
1
+
–
–
–
+
+
+
–
15.5
2
+
+
–
–
–
–
+
+
17.5
3
+
–
+
–
–
+
–
+
13.5
4
+
+
+
–
+
–
–
–
9.5
5
+
–
–
+
+
–
–
+
7.5
6
+
+
–
+
–
+
–
–
11.5
7
+
–
+
+
–
–
+
–
9.5
8
+
+
+
+
+
+
+
+
11.5
1 8
12
+0.5
–1
–2
–1
+1
+1.5
+0.5
For each column, we calculate the sum of responses which are given the sign denoted in the column and we divide the result by 8. We obtain the general effect 12 in the first column and the effects of the levels + of factors A, B, C, and of interactions AB, AC, BC and ABC. The effects of levels are the opposite. Here, the number of essays is equal to the number of parameters, the error is then equal to zero and it is possible to exactly reconstruct Y with the model. For example, the 6th essay response is 11.5 and verifies : 11.5 = 12 + 0.5 + 1 – 2 + 1 + 1 – 1.5 – 0.5.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 17
Factorial designs. Now we only want, with a design 23, to estimate general effect and main effects, that is 4 parameters. We propose to use a fractional design. We choose to only carry out the essays 2, 3, 5 and 8, that is, the ones for which there are + in the column of interaction ABC. We then obtain the following design (defined by columns A, B, C) : N° essay
I
A
B
C
AB
AC
BC
ABC
2
+
+
–
–
–
–
+
+
3
+
–
+
–
–
+
–
+
5
+
–
–
+
+
–
–
+
8
+
+
+
+
+
+
+
+
We observe that columns are equal 2 by 2 : I ABC (by construction) A BC B AC C AB
The main effect of A and the interaction BC are said confounded (idem for I and ABC, B and AC, C and AB). Then this design can be used in two ways : - either to estimate the general effect and the main effects of 3 factors A, B, C, each one with 2 levels, - or to estimate the general effect, the main effects of 2 factors, A and B for example, and the interaction AB which is then set in the column C. These two uses of the same design may be represented by the following graphs : vertices represent factors and edge between two vertices represent the interaction between the two factors.
A A
or B
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B AB
C
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 18
Orthogonal fractionary designs. We have built a fractionary design by choosing essays in such a way that the design so obtained verify the orthogonality property, that is : - same number of essays for each level of each factor, - for each level of each factor, same number of essays for each level of each other factor ! Let X be the matrix of the orthogonal fractionary design, with 4 rows and 4 columns corresponding to columns I, A, B and C, where + is replaced by +1 and by 1. This matrix verifies the following property : X’ X = 4 I4 , where X’ is the transposed matrix X and I4 the identity matrix of order 4. It is possible to show that, for the analysis of variance models, the variance of the parameter estimators is minimal when the matrix (X’X)1 (called Fisher’ information matrix) is diagonal. This is the case, in the context of experimentation, when we use orthogonal designs. We emphasize that the estimation of parameters depends, of course, on the response Y but that the variance of the estimators is only depending on the design and not depending on the response.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 19
3.3. Blocks and randomization. Blocks. In order to optimize the marble production, it is decided to carry out an experimental design for controlling the heating temperature A (2 levels) and the calibrator diameter B (2 levels). But the response (marble weight) is supposed to depend also on the suppliers of one of the constituent elements. This element is quite enough difficult to obtain, so, the enterprise is forced to obtain it from 8 different suppliers. Although the adjustment cannot be made on this factor, we want to control it also because its effect may hide the factors A and B effects. Such a factor is called a block factor. Factors A and B are called principal factors.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
Principal factors
page 20
A B
A B
A B
A B
– –
– +
+ –
+ +
1
3
2
3
Block factors
3
3
4
3
5
3
6
3
7
3
8
3
6
6
6
6
24
The design is then 228 ; to have one essay for each block and each cross level of principal factors needs 32 essays. The previous design is an incomplete block design (only 24 essays) but equilibrated, that is to say, same number of essays for each block (3), same number of essays for each pair of levels of principal factors (6). Randomization. We have controlled some factors : principal factors and block factors. It remains other factors which may have effect on the response and which have not been considered : for example the operators which carry out the essays. If several persons participate in the experimentation, we have to pay attention not to give one of them one type of essays and another one another type of essays. On the contrary, we have to divide up the essays into the operators according to random tables. This procedure is called randomization.
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 21
3.4. Taguchi’s methodology : the ideas. Signal and noise. We have seen that the production average has to be as close as possible as the target, in our example, the target is 12.5 g. Taguchi emphasizes that the mean of the production, y , called the signal, has to be close to the target, but, at the same time, that the variance of the production, s 2 , called the noise, has to be as small as possible. Indeed, as we have already seen, we have to minimize the MSE. n 2 1 2 2 MSE y i a y a s 2 estimation of - a 2 n i1
In order to achieve the aim, Taguchi suggests to classify controlled factors into : - those which have an effect only on the noise, denoted by x1, - those which have an effect on the signal and the noise, denoted by x2. We have then : x1 , x 2 and 2 2 x1
We search x1,0 which minimizes 2 x1 then x2,0 which minimizes x1,0 , x 2 a .
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Statistical control of quality, design of experiments, Taguchi’s methodology.
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Product design. Let us assume that factors x1 are two factors A and B with respectively 2 and 3 levels, and that factors x2 are 2 factors C and D each one with 2 levels. We propose then to carry out 24 essays (64) according to the following design, called product design, P = QR, where Q = 23, R = 22 and we consider the following models : Y I A B AB I C D S 2 I A B AB
Thus, we look for the levels of the factors A and B which minimize s 2 then the levels of factors A, B, C, and D which optimize y . R Q
C
1
1
2
2
D
1
2
1
2
A
B
1
1
y11
y12
y13
y14
y1.
s12.
1
2
y21
y22
y23
y24
y 2.
s22.
1
3
y31
y32
y33
y34
y 3.
s32.
2
1
y41
y42
y43
y44
y 4.
s 42.
2
2
y51
y52
y53
y54
y 5.
s52.
2
3
y61
y62
y63
y64
y 6.
s62.
y.1
y.2
y.3
y.4
y ..
There are other uses of the product design : the factors of design R may be exterior factors (for example, condition of use of the product) which cannot be controlled. We then have to choose the levels of controlled factors (of design P) which both optimize the signal y and minimize the noise s2 due to the uncontrolled factors (of design R).
Jeanne Fine 1994
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 23
3.5. Tables and interaction graphs. We have already seen that, among fractional designs, orthogonal designs are the most interesting for experimentation. Moreover, a fractional design may be used for several models thanks to the confounding property. Taguchi has presented the most used orthogonal fractional designs, called standard orthogonal tables, with interaction graphs describing several possible uses of each table.
The table L 8 2 7 (8 essays, 7 factors with 2 levels for each) is the following : N°
1
2
3
4
5
6
7
1
1
1
1
1
1
1
1
2
1
1
1
2
2
2
2
3
1
2
2
1
1
2
2
4
1
2
2
2
2
1
1
5
2
1
2
1
2
1
2
6
2
1
2
2
1
2
1
7
2
2
1
1
2
2
1
8
2
2
1
2
1
1
2
a
b
a
c
a
b
a
c
c
b
b
c Groups
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1
2
3
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Statistical control of quality, design of experiments, Taguchi’s methodology.
page 24
Triangle of interactions between two columns 1
2
3
4
5
6
7
(1)
3
2
5
4
7
6
(2)
1
6
7
4
5
(3)
7
6
5
4
(4)
1
2
3
(5)
3
2
(6)
1
Interaction graphs.
1 7 5
3 2
6
2
3 5
1 4
6
4 7
Comments. First, this table cannot be used if there are more than 7 factors with 2 levels ! Let us consider the interaction graphs. Vertices represent factors ; they are classified into 4 groups 1, 2, 3, 4 represented by the symbols :
,
,
,
Those of the first group are factors for which it is difficult to change levels, ..., those of the 4th group are factors for which the level change is easy. Edge between two vertices represents interaction between the corresponding factors.
Jeanne Fine 1994
http://jfstat.fr
Statistical control of quality, design of experiments, Taguchi’s methodology.
page 25
So, the first graph indicates that, if we have 4 factors A, B, C and D, each one with 2 levels, and if we consider the following model : Y = I + A + B + C + D + AB + AC + BC
then, we will have to assign the columns 1, 2 and 4 to factors A, B and C respectively, and the columns 3, 5 and 6 to interactions AB, AC and BC respectively. The second graph indicates another possible use of the table. We have 4 factors A, B, C, D, each one with 2 levels, and we consider the model : Y = I + A + B + C + D + AB + AC + AD
We assign the columns 1, 2, 4 and 7 to factors A, B, C, D respectively and the columns 3, 5, 6 to interactions AB, AC and AD respectively. Of course, it is possible to associate other graphs to this table and to consider other models, for example : Y = I + A + B + C + D + E + AB 1
represented by:
3
2
4
5
6
We assign columns 1, 2, 4, 5, 6 to factors A, B, C, D, E respectively and column 3 to interaction AB. It remains one column which is not used (column 7). The variance of error may be estimated. The interactions triangle between two columns has to be read as follows : column 3 represents interaction between columns 1 and 2, 4 and 7, 5 and 6, column 2 represents interaction between columns 1 and 3, 4 and 6, 5 and 7, column 5 represents interaction between columns 1 and 4, 2 and 7, 3 and 6, and so on.
Jeanne Fine 1994
http://jfstat.fr
Statistical control of quality, design of experiments, Taguchi’s methodology.
page 26
If we want to consider the model : Y = I + A + B + C + D + AB + CD
it is not possible to use the table L 8 2 7 because if we assign, for example, columns 1, 2, 3 to factors A and B and to interaction AB, then columns (4,5), (4,6), (5,6), (5,7), (6,7) cannot be assigned to C and D because the interaction CD would be represented respectively by columns 1, 2, 3, 3, 2, 1 which are already assigned to A and B. On the other hand, if we have three factors A, B, C with respectively 2, 2 and 4 levels, it is possible to
use the table L 8 2 7 as follows : we construct the factor C with 4 levels from columns 1 and 2 : 1
2
1
1
2
2
3
4
and we eliminate column 3 (which represented interaction between columns 1 and 2). We assign column 4 to A, column 5 to B and it is now possible to consider the model :. Y I A B C
Jeanne Fine 1994
http://jfstat.fr
Statistical control of quality, design of experiments, Taguchi’s methodology.
page 27
4. Conclusion. Experimental research methodology The presentation of the underlying mathematical tools of designs of experiments must not make one’s forget that the most important part of an experimental research is beyond the technical part. Referring to Sergent and al. (1989) experimental research methodology consists in : - formulating the problem clearly and fixing the objectives, - establishing the knowledge on the subject, - listing responses, constraints and factors which may have effect - defining the range of experimentation, - constructing the design, carrying out experiments and calculations, - interpreting results and deducing answers of the questions. In conclusion, the reasons why designs of experiments work are disconcerting : in a sampling area, the precision is all the better since a large number of experiments is carried out in the same conditions ; on the contrary, in designs of experiment area, information is collected from a small number of experiments carried out in the most different possible conditions. The quality of information does not depend on the number of experiments but on the experimental conditions (Fisher’s information matrix).
Jeanne Fine 1994
http://jfstat.fr
Statistical control of quality, design of experiments, Taguchi’s methodology.
page 28
References There are a lot of references on this subject intended to statisticians, but, for an initiation, I can only suggest the following French references, classified in order of their increasing difficulty : PILLET Maurice Introduction aux plans d’expériences par la méthode Taguchi. Les Editions d’Organisation, 1992 VIGIER Michel G. Pratique des Plans d’Expériences. Méthodologie Taguchi. Les Editions d’Organisation, 1988 Gestion de la qualité, Méthodologie Taguchi Revue de Statistique Appliquée, 1989, Vol XXXVII n°2 CERESTA, 10 rue Bertin Poirée, 75001 Paris. SERGENT Michelle, MATHIEU Didier et PHAN-TAN-LUU Roger Méthodologie de la Recherche Expérimentale. L.P.R.A.I. Marseille, 1989.
Jeanne Fine 1994
http://jfstat.fr
