**The plans produced are original Taguchi. They cross the test matrices of the main factors with the noises to set up a plan of experiments which will make it possible to study the variability of the results according to these uncontrollable factors.**

## Introduction

Product plans test the robustness of a product to sources of “noises”. Noises are fluctuations in characteristics, variations in the working environment, the aging of the product studied …

Taguchi proposes to use these sources of noise as factors in planning experiments. The resulting ** Signal / Noise **report will report both the mean value and variability of the data around the mean.

## The ratio properties

The Signal / Noise ratio relates the average of the response to the noise (the dispersion of the response). It makes it possible to isolate the effects of noise on the average of the answers and thus to understand the behavior of our system vis-à-vis the non-controllable factors.

## 1 – Choosing Interactions and the Controllable Factors Table

Initially, we will build the test matrix based on controllable factors. We will follow for this the process “*standard;” *of the construction of a **Taguchi plan**.

## 2 – Choose the noise factor table

In the same way as for controllable factors, we will choose the most appropriate table for noise factors. In the case where the number of noise experiments is greater than the number of experiments for the factors, then the composite noise method will be used. This method consists in performing an experiment solely on the noise factors to determine one or more combinations of the factors and thus transform for example 9 noises into 3 groups of noises.

** Sample**

We study the lowering of the water temperature in the glass with the addition of an ice cube. Three noises have been identified: the location of the thermometer, the time and the amount of water. A plan of experiments is carried out according to table L4 of Taguchi. The tests show that the minimum temperature of the water is obtained with 2 combinations that are the position of the thermometer and the time at the high level and the amount of water at the low level.

We decided to set up ” *a ” *noise at 2 levels. We reduce the number of test repetitions initially to 4 to 2.

## 3 – Set up the product plan

The principle is simple, we will cross the matrix of controlled factors with the table of factors noise. This gives us a system of this type :

It is observed that we carry out repetitions of tests of the main plane, which are of “false nonspecific repetition” since they are planned with different levels of noise factors.

## 4 – Perform the experiments

The procedure is the same as for the experimental plans. The conditions must be the same, the means of measurement do not evolve between the experiments …

However, with regard to the repetition of the tests, it is necessary that the tests of the same combination of the main factors are not done one after the other.

## 5 – Calculate Signal / Noise Ratios

The calculation is done for each combination of the plan. The formula will depend on 3 cases that we detail below. It should be noted that this coefficient does not have any unity but that it is expressed in decibels.

### Case of a criterion to minimize

In the case where the smallest is best, the formula is :

** S / N = – 10 * Log (1 / n * Σ y _{ i 2 ) }**

### Case of a criterion to be maximized

In case, the bigger the better, the formula is :

**S/N = – 10 * Log (1/n * ****Σ 1/y _{i}^{2}) = -10 * Log ((1 + 3 * Variance/Y_{icross}^{2})/Y_{icross}^{2})**

with :

- Variance of the sample = Σ ((Y
_{ i – Y cross ) 2 / (n – 1))} - Y
_{ icross : Average of all responses for the i test } - n :t he number of measurements per test
- Y
_{cross average of all responses from all tests }

### Case of a criterion at nominal value

In this case, our optimal is when we are at the target. The formula is :

**S/N = 10 log (Y _{ibarre}^{2} / Variance)**

**n the number of measurements per test**- Variance of the sample = Σ ((Y
_{ i – Y cross) 2 / (n – 1))} - Y
_{cross : average of all responses from all tests }

### Case of a criterion at the target value

In this case, our optimal is when we are at the target. The formula is:

**S/N = -10 log (Variance + (Y _{icross} – Target)^{2})**

- n : the number of measurements per test
- Variance of the sample = Σ ((Yi – Ybarre)
^{2}/ (n – 1))

## 6. Validate the significance

### 6.1 Assumptions

As in all ** 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 SSSTs differs somewhat from plans built on the basis of matrices of **Hadamard**. Mathematically, they are identical but the arrangement of columns and lines is different. The formula for each factor is as follows :

**SSST = N * (Effect/n) ^{2}**

With :

**N :**the number of tests**n**: the number of factor level**Effect**: the effect of the factor in question

### 6.3 Calculation of total discrepancies TSS

The calculation formula is the same as for the other experimental plans. The sum of the difference of the differences between each of the responses is averaged and the average squared. The expression is the following :

**TSS = ****Σ (Y _{i} – Y_{cross})^{2}**

### 6.4Deduction of residues

AContrary to the other types of plan, where we calculate the residues on the interactions, with Taguchi’s plans of experiments, we 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 :

**SSE = TSS – ****Σ SST**

### 6.5 Calculation of mean squares

We continue the process of **Anova **and average squares are calculated. They are calculated as follows :

**MST**_{x} = SST_{x} / dof_{x}

**MSE = SSE / dof**_{r}

Avec :

**MST**Mean Square of each of the main factors_{x}:**dof**degree of freedom of each of the factors, always equal to 1 in the context of a screening plan_{x}:**MSE :**Mean Square of residues.**dof**degree of freedom of residues, equal to the number of total dof – the sum of the dof of the factors_{r}:**dof total :**number of dof in total. It is equal to the number of experiments – 1

### 6.6 Calculating the 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 :

**Pratical value > Critical value :**We reject H0, and we conclude that the factor has a strong influence within the model.**Pratical 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 = FDIST. (Practical value ; dof**_{SSTx}, dof_{SSE})

The interpretation is as follows :

**p-Value < α :**the factor is influential, it must be remembered in our model.**p-Value> α: :**the factor is not influential, it must be taken out of our model

## 7 – Data analysis

The analysis of the data is done on the analysis of the significance: one will not take into account the insignificant factors. The relevance of the interpretation follows the significance: the more the value of the effect of a factor is distant of 0, the more it has influence and at most the reading of the associated S / N ratio is important .

The analysis is based on a comparison of the means of the results according to the levels of the factors. Whatever the case, minimization, maximization, target or nominal, we will always look for the highest Signal / Noise ratio. Since at least there are noises that affect the average, the higher the ratio is. In other words, the greater the ratio, at least the noise has effect.

## 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

R. A. Wysk, B. W. Niebel, P. H. Cohen, T. W. Simpson (2000) – Manufacturing processes : integrated product and process design

A. Jacques (1995) – Pratique industrielle de la méthode Taguchi

G. Lasnier (2003) – Plans d’expériences industrielle

A. C. Tahhane (2009) – Statistical analysis of designed experiments

W. Y. Fowkles (2000) – l’ingénierie robust

K. Yang, Basem El-Haik (2003) – Design for Six Sigma