Design of experiment

Introduction

Experimental designs consist of a series of organised experiments or assays, allowing the acquisition of new knowledge based solely on experimental results. The relationships between the input parameters (variables or factors) of the system under study and one (or several) output quantity(ies), also called response, are modelled. The model implementing a response $y$ and variables or factors $x_i$ can be presented in a generic form :

Experimental designs are used in a wide range of fields, for example :

• allow to prioritize the influence of parameters on an industrial process,
• optimize the performance of a product from prototypes,
• model a complex physical phenomenon of a material,
• or optimize the performance of a chemical mixture ...

SOSstat provides intuitive and simple functionality for trial design and analysis of experimental design results. In addition to a comprehensive library of classical designs of experiments, SOSstat offers an innovative custom design creation feature that allows user to adapt the design of experiments to the experimental constraints while minimizing the number of trials. Once the data is collected, SOSstat provides a simple and accurate analysis of the model (with many graphs), the prediction functions, then allow to find the configurations that best fit our problem.

First of all, it is important to note, that the modeling of a system cannot be done by performing tests without a prior strategy. Indeed, the organization of the trials has a decisive influence on the quality of the model, and thus of the resulting optimizations. As this property is not intuitive, it is important to follow the design of experiments methodology to guarantee the quality of the results.

Here are some of the advantages of the design of experiments methodology:

• Define the number of experiments required in advance (planning assistance)
• Propose a simple and systematic method of analysis
• Build a model to predict responses across the field of experimentation
• Maximize the accuracy of the results (for a given number of experiments)
• Reduce the number of trials (under certain conditions)

Experimental designs adapted to each problem

Over the years, the experimental plans have been enriched, with many tools to respond to a wide variety of problems. In this section, we will try to give an overview of the main experimental techniques.

Factorial designs are the oldest DOE. They allow both continuous and categorical variables to be used. The models studied are linear models with or without interaction. Factorial designs can be used in two situations:

• Doing screening, i.e. prioritizing the influence of a large number of factors
• Model a system in order to optimize it

Factorial designs

Several techniques can be used to meet these objectives:

• The full factorial designs (which exploit all combinations of factors in the design of experiments)
• Plackett and Burman Screening designs . These fractional designs which preferentially deal with two-level factors, are limited to quantifying effects and ignore interactions. The number of experiments increases quite slowly with the number of factors, since it's a multiple of 4.
• Fractional factorial designs, which may, depending on the case, do screening or modeling. These plans are generally appreciated by experimenters, as they drastically limit the number of trials.

Response surface designs

When the input variables are continuous and the model has a local optimum, it is preferable to use a second-order model. It is in this context that response surface designs develop their full power. They allow the optimization of a system by modeling its complex response surface. The best known surface planes are :

• the central composite designs, are appreciated for their sequential implementation and their ability to meet several criteria of optimality (rotational isovariance and near-orthogonality).
• the Box-Behnken designs, provide a model of reasonable and economical quality, while limiting the number of factor modalities to 3.
• It should also be noted, that response surface designs can also be algorithmically created, with D-optimal designs.This is a useful option if you want to reduce the number of experiments, or if the experimental domain has exclusion zones (impossible combinations of factors).

The case of mixtures is close to the problem of response surface designs, since we are often in a context of optimization of a non-linear system. The particularity of mixture designs is to be able to take into account the relational constraints between the different constituents of the mixture. These experimental design techniques need to be computerized, because the experiment matrices are not orthogonal, moreover the definition of the experiments requires to establish beforehand the experimentation area, which strongly depends on the constraints applied to each constituent (limitation of the experimental field).

Example of application

Using a full factorial design

The design of a DOE always starts with the identification of factors that are supposed to influence the response. Then, the experimenter determines the number of levels for each of them, as well as their levels.

When designing a design of experiments, it is also important to determine the interactions to be quantified, but in the particular case of full factorial designs, this step is not essential because all interactions can be quantified.

In the example below, 4 two-level factors are studied on a CNC lathe. The 4 factors are cutting parameters, which are assumed to have an influence on the tool life (system response).

Factors are :

• VI: Feed speed
• PP: Depth of Pass
• VC : Cutting speed
• DL: Cutting fluid flow

The implicit model is therefore (where M is the mean of the model):

Once the model is defined (list of factors with their modalities and list of interactions), SOSstat can create the matrix of experiments. In this example, it is a full factorial design. The matrix of experiments displayed in the grid describes the combination sequence of factors. It contains the following information:

• Experience number
• Random number of the experiment, if one wishes to randomize the trials
• Replication number of each experiment. Replications help improve model accuracy
• The matrix of experiments, on the following four columns
• The response column (it is possible to process several responses), where the experimenter can enter the results of each experiment

Model analysis

The model consists of a constant, effects and interactions.

Table of effects

We notice that for two-level factors the effects are opposite ( $\sum \text{Effets}=0$ )

Interaction Tables

The table below shows only some of the interactions.

For two-level factors, we observe a symmetry on the value of interactions $\sum \text{Interactions}=0$ ).

Analysis of variance of coefficients

Analysis of variance (ANOVA) is an important step in running a design of experiments, as it identifies the significant terms in the model. Only parameters with a p-Value of less than 5 percent are kept in our case. (This analysis can also be performed with a T-test table).

After exploitation of the ANAVAR, the model, which initially included 4 factors and 6 interactions, now includes only 3 factors and 2 interactions; This model can be used directly to optimize the system (SOSstat prediction tool).

DOE plots

Usually the coefficients of the model are represented graphically to facilitate their interpretation. The effects graph represents the evolution of the average response as a function of the individual fluctuations of each factor. The interaction graph represents the evolution of the averages when two factors change together.

On the effects plot, we can clearly see the major influence of the factor Cutting speed.

Only one graph of the interactions has been shown. This is the interaction Depth:Cutting speed (the most significant one). The presence of the interaction can be clearly seen because the two segments are not parallel.

Other plots are generally proposed to perform residue analysis

Optimization of results

Desirability functions

It's quite common for a design of experiments to analysis several responses. In this case, we have as many models as there are responses. In the case of two responses, for example, we have two models $y_1=f(x_1,x_2, ... , x_k)$ and $y_2=f(x_1,x_2, ... , x_k)$ , involving the same factors.

How can we optimize these two answers simultaneously, given that the models are different, and that the objectives for $y_1$ and $y_2$ are certainly different too?

To achieve a reasonable compromise, we will describe our goals using desirability functions. These functions define the attainment of the goal by a value between 0 (non-optimal) and 1 (optimal). Three families of functions are used, depending on whether we wish to minimize, maximize a response, or target a particular value. These functions must be maximized to achieve the objectives. To do this, the objectives assigned to each response are synthesized by constructing the "composite desirability" function. By maximizing the composite desirability function, the best compromise is found.

Using Taguchi's Signal to Noise Ratio

Genichi Taguchi, a Japanese engineer, developed the concept of robustness in engineering in the 1960s. The aim is to be able to optimize systems, while ensuring that their performance is not degraded by disruptive factors. With this in mind, Taguchi then had the idea of creating a summary variable that would make it possible to report on the achievement of objectives in terms of performance and robustness against disruptive factors. This variable is a signal/noise ratio that allows, depending on the case, to maximize or minimize a response, while minimizing the variance of the responses.

Bibliography

DUCLOS, E - Introduction aux plans d'expériences , LULU , 2019 , ISBN13 : 5800133354798

DROESBEKE Jean-Jacques , FINE Jeanne , SAPORTA Gilbert - Plans d'expériences, Applications à l'entreprise , Editions Technip , ISBN : 9782710807339

Goupy, J. - Introduction aux plans d'expériences , Dunod, 2017 , EAN13: 9782100778027

D.R. Cox, Nancy Reid - The Theory of the Design of Experiments , CRC Press, 6 juin 2000 - 336 pages

Douglas C. Montgomery - Design and Analysis of Experiments ,John Wiley & Sons, 2008 - 680 pages