Taguchi’s plans are among the most popular in the industry and particularly in design and quality services.
Introduction
Taguchi’s plans have been developed for industrial use to optimize product design and improve quality. These are actually fractional factorial designs, taking into account certain interactions deemed important and abandoning others. Contrary to fractional factorial designs, Taguchi offers tables for resolutions III, IV et V.
Le principle
Taguchi proposes to organize the experiments according to tables called L8, L16 … The Taguchi Tables are plans of Plackett and Burman in which we replaced +1 by 1 and -1 by 2 … The names of the plans were also translated. The plan 2^{3}is Table L8 and Plan 2 ^{ 4 } is the Table L16.
1 – Choose interactions
First step, the choice of interactions. We proceed by elimination:
- We delete interactions that we know are null.
- Interactions of order 3 and above are deleted.
- We keep those we know are not null
- Pour those where we have doubts, we choose according to our resources … whether we keep them or not.
2 – Choose the table
Il en existe 18 et sont identifiés par la forme L_{g}(p^{f}) avec :
- g : the number of lines, ie the number of experiments.
- p . the number of levels, the same for all factors except for tables with an x.
- f : the number of columns that corresponds to the number of factors and interactions we want to study.
- x : The x indicates that in the same table there may be factors with a number of different levels. Note that they can go up to 8 levels for a factor.
The tables are classified in 3 families :
Use | 2 level plan | 3 level plan | 2 and 3 level plan | 4 and 5 level plan | |
Study of impossible interactions | Suitable for a screening phase. | L_{12}(2^{11}) | – | L_{36(}2^{11}x3^{12}) | – |
Study of limited interactions | Also for Screening. | – | – | L_{18}(2^{1}x3^{7}) | L_{32}(2^{1}x4^{9}), L_{50}(2^{1}x5^{11}) |
Study of interactions possible | Define a precise model in a response surface study most often. | L_{4}(2^{3}), L_{8}(2^{7}), L_{16}(2^{15}), L_{32}(2^{31}), L_{64}(2^{31}) L_{64}(2^{63}) |
L_{9}(3^{4}), L_{27}(3^{13}), L_{36}(3^{13}), L_{81}(3^{40}) |
L_{36}(2^{3}x3^{13}), L_{54}(2^{1}x3^{25}) | L_{16}(4^{5}), L_{64}(4^{21}), L_{25}(5^{6}) |
2.1 – Choose the number of columns
We must look for the table that has a number of columns for our study: that is to say the one that has at least as many columns as we have of factors and interactions to study..
For example, we want to study 3 factors and only 2 interactions. We will restrict our choice of table whose interaction studies are possible and for which we have at least 5 columns.
It remains : L_{18}(2^{1}x3^{7}), L_{32}(2^{1}x4^{9}), L_{50}(2^{1}x5^{11}), L_{8}(2^{7}), L_{16}(2^{15}), L_{32}(2^{31}), L_{64}(2^{63}), L_{27}(3^{13}), L_{81}(3^{40}), L_{36}(2^{3}x3^{13}), L_{54}(2^{1}x3^{25}), L_{16}(4^{5}), L_{64}(4^{21}), L_{25}(5^{6}).
2.2 – Choose the number of levels
Among the tables we have pre-selected, we will take those that have a number of level “& nbsp; p ” which corresponds to our case. The level number must be the same for all factors. Otherwise, it will be necessary to choose a table whose name contains a « x ».
In following the example, we define that we have 2 levels by factors. So we have L_{8}(2^{7}), L_{16}(2^{15}), L_{32}(2^{31}), L_{32}(2^{31}), L_{64}(2^{63})
2.3 – Choose the number of experiments
Last step, choose the number of experiments that we are ready to do. Of course, the more experiences we choose, the more meaningful our results will be. So everything is a compromise story.
Let’s take our example. Given the tables we have left, we have the choice between a plan requiring 8, 16, 32 or 64 experiments.
Some tips
The definition of the table is not necessarily exact to our problem. Only the number of level must be exact. For example, we could very well use the table L _{ 8 } (2 ^{ 7 }), to study 4 factors with 2 levels and 2 interactions.
Par ailleurs :
- If we have 4 factors and we use table L8, it is better to use columns 1, 2, 4, 7 or 1, 3, 5, 7.
- In the same way, if we chose an L16 table to study 5 factors, it will be better to use columns 1, 2, 4, 8 and 15.
- Finally, if we chose an L16 table to study 6, 7 or 8 factors, it will be better to use odd columns or columns 1, 2, 4, 7, 8, 11, 13, 14.
3 – Build the matrix of experiences
Taguchi’s contribution is based on the construction of a graph to represent the experimental design, thus simplifying its construction..
3.1 Build the graph
The choice of the allocation of columns to factors and interactions will depend on the graphs that accompany each of the Taguchi tables. We build it in the following way:
- We indicate a number of points identical to the number of factors.
- A specific form is assigned to each point depending on the difficulty of modifying the factor (see table below)
- We will connect the points with a line to represent the interaction we want to study.
Difficulty changing levels
Symbol
Difficult, so the number of level changes must be low.
Quite difficult, so uncommon change in factor level.
Easy enough.
Easy, so frequent change of factor level.
For example, in the graph opposite we have :
- We want to study 5 factors.
- It is studied that 2 interactions, CD and AC
- Factor A is very complicated to modify, while others are rather simple.
We note that for calculations and interpretation, the points do not matter. They are just there to highlight the most complicated factors to modify to allocate them to the first columns of the table (they change positions less often).).
3.2 Comparison with a standard graph
We will compare our graph with the standards proposed by Taguchi. In our case, we will choose table L8 since we have 3 factors and 2 interactions to study and we want to make the least possible experiences. There are 2 graphs that accompany this table.
If we observe the interactions, we see that the graph on the right is the most appropriate.
3.3 Assign factors to columns
We will rank the factors by level of difficulty to modify them and assign them to the columns in an increasing way. Let’s go back to the previous example :
- We will directly position the factor A in column 1. It is indeed this factor which is the most complex to modify, it is preferable to save time to do it only once.
- We will arbitrarily put the factor C in 2 because it is quite difficult to modify.
- The factor D in 4.
3.4 Assign interactions to columns
We will now position interactions. For this, the method requires the use of triangular matrix that accompanies each of the tables. Take the example of the triangular matrix associated with the Table L_{8} :
To use it, we consider that the first factor is read vertically and the second factor is read on the horizontal axis. The reading in our case is the following :
- Factor A is associated with column 1 and Factor C with column 2. Our AC interaction is therefore in column 3.
- Similarly, factor C is associated with column 2 and factor D with column 4. The interaction CD corresponds to the column 6.
4. Perform the tests
From now on we are ready to conduct the tests. It must follow the indicated order since it took into account the fact of saving time in the settings.
5. Calculate the effects
The effects are calculated as for other types of experimental designs. The difference of the averages of the responses is made as a function of the high and low position of the factor.
For example, we have the following result table :
Factor A |
Factor C |
Interaction AB |
Factor B |
Interaction AC |
Answer |
1 |
1 |
1 |
1 |
1 |
8,3 |
1 |
1 |
1 |
2 | 2 |
7,7 |
1 |
2 |
2 |
1 | 1 |
8,1 |
1 |
2 |
2 |
2 | 2 |
8,2 |
2 |
1 |
2 |
1 | 2 |
8,65 |
2 |
1 |
2 |
2 |
1 |
8,9 |
2 |
2 |
1 |
1 |
2 |
8,5 |
2 |
2 |
1 |
2 |
1 |
9,2 |
Effect factor A = (8,65 + 8,9 + 8,5 + 9,2) / 4 – (8,3 + 7,7 + 8,1 + 8,2)/4 = 0,7375
Effect factor C = (8,1 + 8,2 + 8,5 + 9,2) / 4 – (8,3 + 7,7 + 8,65 + 8,9)/4 = 0,1125
The results for others are :
- For B : 0,1125
- For AB : 0,00375
- For AC : -0,3625
6. Validate the significance
6.1 Les hypothèses
As in everything test, the initial step is to determine the assumptions. In our case, we express the hypotheses in the form
H0 : a_{x} = 0
H1 : a_{x} ≠ 0
6.2 Calculation of differences SST
In the context of the Taguchi plans, the calculation of SSTs differs somewhat from plans built on the basis of Hadamard matrix. Mathematically, they are identical but the rearrangement of the columns and lines differs. The calculation is therefore somewhat different. The formula for each factor is as follows
SST = N * Effect^{2} / n
with :
- N : the number of tests
- n : the factor level number
- Effect : the effect of the factor in question
6.3 Calculation of total discrepancies TSS
The calculation formula is here the same as for the other plans of experiments. We simply calculate the difference between the differences between each of the answers and the average, all squared. The expression is the following :
TSS = Σ (Yi – Ycross)^{2}
6.4 Deduction of residues
Unlike the other types of plane where we calculate the residues on the interactions, with the plans of experiments of Taguchi, one can not do it. Indeed, the distribution and the treatment of the interactions being very different, one can only deduce the residues. The formula is the following :
SCR = TSS – Σ SST
6.5Calculation of mean squares
We continue the process Anova and mean squares are calculated. They are calculated as follows :
_{MSTx} = SST_{x} / dof_{x}
MSE =SCR / dof_{r}
Avec :
- _{MSSEx} : Mean Square of each of the main factors
- dof_{x} : degree of freedom of each of the factors, always equal to 1 in the context of a screening plan
- MSE : Mean Square of residues.
- dof_{r} : egree of freedom of residues, equal to the number of total dof – the sum of the dof of the factors
- dof total : number of dof in total. It is equal to the number of experiments – 1
6.6 Calculation of practical value
For each of the main factors, we will calculate a practical value. This is calculated using the following formula :
Practical value = MST_{x} / MSE
The interpretation is as follows :
- Practical value > Critical value : We reject H0, and we conclude that the factor has a strong influence within the model.
- Practical value < Critical value : We retain H0, and we conclude that the factor has no influence on the model.
6.7 Calculation of the p-Value
We can calculate the p-Value which will serve us to evaluate the influence of each factor. It follows a Fisher law. In Excel, we can calculate it using the formula :
p-Value = F.DIST.RT. (Practical value ; dof_{SSTx}, dof_{SCR})
The interpretation is as follows :
- p-Value < α : the factor is influential, it must be retain in our model.
- p-Value > α : the factor is not influential, it must be taken out of our model
Source
G. Taguchi, Y. WU (1985) – Introduction to off-line quality control
G. M. Vigier (1988) – Pratique des plans d’expériences. Méthodologie Taguchi
J. C. Chauveau, J. P. Chassaing (1994) – Introduction à la méthode des plans d’expériences par la méthode Taguchi
Alexis et Alexis (1999) – Pratique industrielle des plans d’expériences
D. Duret, M. Pillet (2005) – Qualité en production
G. Taguchi, D. Clausing (1990) – Robust quality
M. Pillet (1998) – Construire facilement des plans de résolution IV à partir des tables de Taguchi
R. N. Kacker, E. S. Lagergrenn, J. J. Filliben (1991) – Taguchi’s orthogonal arrays are classical designs of experiments