# Taguchi's experimental design

The Taguchi design of experiments is a statistical method used to improve the quality of products and manufacturing processes. Developed by the Dr. Genichi Taguchi The aim of this plan is to identify the factors that have the greatest influence on the variation of a product or process, while minimising the number of experiments required.

This method organises experiments systematically and efficiently, enabling the effects of several variables to be analysed simultaneously. The main objective is to make the product or process robust against external and internal variations, thus ensuring stable, high-quality performance.

## Defining factors and levels

Defining the factors involves choosing the factors to be studied. To choose the right factors, we recommend that you first understand the process or product concerned, and then carry out the following steps:

1. Organise brainstorming sessions with a multidisciplinary team to generate an exhaustive list of potential factors.
2. Assess the importance of each factor in terms of its potential impact on the final result. Prioritise the factors that are likely to have a significant effect.
3. Select the relevant factors for the experimental design based on the importance of the factor, your ability to control it and the ease of measuring it reliably.

For each factor identified, the levels studied must then be defined. It is important to choose relevant levels:

• Beach width Select levels that cover a sufficiently wide range to detect the effects of factors, but without going to unrealistic extremes that could be unnecessary or dangerous.
• Practical To ensure that the levels chosen are achievable in a real production context.

## Choice of table

Once the factors have been identified, you need to create the experimental design. To do this, we recommend that you use the Ellistat Data Analysis module which has an exclusive experiment design engine. It is capable of finding a design with the best possible strategy given a given structuring of interactions. The tables programmed in Ellistat are L4, L8, L12, L16, L20 and L32.

Here are the elements you need to take into account when creating your experimental design using Taguchi's methodology:

## Notion of interaction

A*B interaction refers to the fact that the level of A has an influence on the effect of B and vice versa.
Example:
In the previous example, the figure on the left shows :
Whatever the level of factor B, the effect of A on the Y is the same and A has an effect of 1 on the response. There is therefore no interaction between A and B.
The figure on the right shows :
When the level of factor B is at a minimum, the effect of A on the Y is 2. When the level of factor B is at a maximum, the effect of A on the Y is 1. The effect of A varies according to the level of B. There is therefore an interaction between A and B.
When there is an interaction, the effect of the interaction is modelled by a multiplicative term in the Y prediction equation, in this case A*B.
The form of the forecasting equation would be :

$Y=\alpha_0+\alpha_1*A+\alpha_2*B+\alpha_3*A*B$

The term α3 corresponds to the A*B interaction

## Notion of alias

The term alias between two factors refers to the fact that the factors have the same level for all the experiments in the experimental design.
For example, suppose we have carried out the following experimental design:
ABY
115
115
2210
2210

Factors A and B vary at the same time, so it is not possible to differentiate factor A from factor B, or to say which of the two causes the Y to vary from 5 to 10 when it goes from 1 to 2. We will say that these two factors are aliases.

Obviously, when constructing an experimental design, you choose your trials carefully so that no factor is aliased with another factor. However, it is possible for a factor to be aliased with an interaction. Let's take the following example:
In this example, we have constructed a three-factor plan using table L4. The alias table shows :
• Factor a is aliased with interaction b*c
• The b factor is aliased with the a*c interaction
• Factor c is aliased with the a*b interaction
In the same way, if two factors are aliased, it will not be possible to tell at the end of the experimental design whether the effect observed is due to factor a, interaction bc or the sum of the two. We will therefore assume that the interaction bc is zero, but this remains to be verified experimentally.
To avoid this type of problem, we generally choose experimental designs in which none of the factors are aliased with an interaction. In the previous example we could have chosen an L8 design in which none of the factors is aliased with an interaction.

## Solving an experimental design

The resolution of a design of experiment corresponds to the alias level of this design.

## Resolution III

An experimental design is of resolution III if at least one factor (order I) is aliased with a type A*B interaction (order II).
This type of design greatly reduces the number of trials, but it assumes that all interactions are zero. Care must be taken when interpreting the results, and the hypothesis of zero interactions must be validated by additional trials.
Example:
In the previous design, factor a is aliased with interaction b*c. The design is resolution III

## Resolution IV

An experimental design is of resolution IV if :
• No factor is aliased with an interaction (order II)
• At least one interaction (order II) is aliased with another interaction (order II)
This type of design limits the number of trials. It assumes that the majority of interactions are zero, except for a few that can be identified. This is generally the case.
As the factors are not aliased with other interactions, the effect of all the factors can be calculated unambiguously.
This is the type of plan most often used.
Example:
In the previous plan, none of the factors is aliased with another interaction. The interactions ab and cd are aliased, so the plan is resolution IV.

## Resolution V

An experimental design is of resolution IV if :
• No factor is aliased with an interaction (order II)
• No interaction (order II) is aliased with another interaction (order II) This type of design limits the number of trials compared with the complete design.
As the factors are not aliased with other interactions, the effect of all the factors can be calculated unambiguously.
Since interactions are not aliased with other interactions, the effect of all interactions can be calculated unambiguously.
This is the ideal type of plan, but unfortunately there is only one with fewer than 20 trials. This is a 5-factor plan using the L16 table.
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