# A Methodology to Optimize Natural By-Product Mixes for  
Rammed Earth Construction Based on the Taguchi Method  
[PERSON]  
1 Institute of Smart Cities (ISC), Department of Engineering, Public University of Navarre (UPNA), 3100 ge Pamplona, Spain; [EMAIL_ADDRESS] (E.P.) [EMAIL_ADDRESS] (A.S.)  
[PERSON]  
2 Department of Building and Public Works, Universite de Pau et des Pays de l'Adour, E2S UPPA, SIAME, 64600 Angllet, France; [EMAIL_ADDRESS]  
[PERSON]  
3 Department of Mining and Metallurgical Engineering and Materials Science, Faculty of Engineering  
Vitoria-Gasteiz, University of Basque Country UPV/EHU, 01006 Vitoria-Gasteiz, Spain; [EMAIL_ADDRESS]  
[PERSON]  
2 Department of Building and Public Works, Universite de Pau et des Pays de l'Adour, E2S UPPA, SIAME, 64600 Angllet, France; [EMAIL_ADDRESS]  
[PERSON]  
1 Institute of Smart Cities (ISC), Department of Engineering, Public University of Navarre (UPNA), 3100 ge Pamplona, Spain; [EMAIL_ADDRESS] (E.P.) [EMAIL_ADDRESS] (A.S.)  
Footnote  
==WARNING: Truncated because of repetitions==
employing an L16 OA, based on the Taguchi method illustrated in Table 4, the number of experiments decreases to 16.
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In this study, two different Taguchi studies were conducted, one to optimize the mix dry density and the other one to optimize the UCS. Nevertheless, the same samples were used to determine both properties, as dry density is a non-destructive measurement. Table 5 shows the levels of the parameters defined by the Taguchi method expressed in dry mass units and the mass percentage of each material.  
The evaluation of performance characteristics involves identifying the most significant factors and their levels. The optimization criteria are centred around computing the signal-to-noise ratio (S/N), which represents the loss function indicating the variance between the actual experimental outcomes and the desired ones. The analysis of the S/N ratio is based on three performance characteristics categorized as (i) the larger-the-better, (ii) the smaller-the-better, and (iii) the nominal-the-better [24]. In this study, the objective was to optimize the dry density and the UCS under the condition of \"larger the better\" as defined by Equation (1).  
\[S/N=-10\times log\left[\frac{1}{n}\sum_{i=1}^{n}\frac{1}{Y_{i}^{2}}\right] \tag{1}\]  
where \(S/N\) is the performance statistic, n stands for the number of repetitions for each level of a factor and, \(Y_{i}\) denotes the response value of the \(i\)th experiment. The \(S/N\) ratio values serve as a metric of robustness, contributing to the identification of control factors that mitigate variability in a process by minimizing the impact of noise factors.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experiment n\({}^{\circ}\)** & **Gravel** & **Sandy Gravel** & **Sand** & **Fines** & **MC** \\ \hline
2 & 1 & 2 & 2 & 2 & 2 \\
3 & 1 & 3 & 3 & 3 & 3 \\
5 & 2 & 1 & 2 & 3 & 4 \\
6 &  
After calculating the \(S/N\) values for each parameter, and under the condition of the \"larger-the-better\", the level of each parameter was obtained, which would increase the parameter to be optimized. The prediction values for the optimized parameter can be obtained using Equation (2).  
\[\eta_{opt}=\eta_{m}+\sum_{i=1}^{f}\left(\eta_{i}-\eta_{m}\right) \tag{2}\]  
where \(\eta_{opt}\) is the predicted value, \(\eta_{m}\) is the overall mean of the parameter to be optimized, f is the number of factors, and \(\eta_{i}\) is the mean of the parameter to be optimized at the optimal level of each factor \(i\).  
To validate the method, after calculating the predicted value, the samples must be manufactured with the combination that would optimize the parameter to be optimized, and whether the prediction made by the Taguchi method matches the actual results should be verified.