# A Methodology to Optimize Natural By-Product Mixes for
## 3 Results and Discussion
### Verification of the Results  
The optimum combinations which enhanced the dry density and maximized the UCS were manufactured. Figure 9 shows the PSD of each optimized combination. For each combination, the dry density and the UCS were obtained to verify the quality of each of the predictions made with Equation (2) by both of the Taguchi studies that were carried out. Table 12 shows the dry density and UCS measured for the samples manufactured according to the combination proposed by the Taguchi method that enhanced the samples' dry densities. The Taguchi method's predictions of dry density and UCS are also shown in this table. Table 13 shows the dry density and UCS values reached by the samples manufactured according to the combination proposed by the Taguchi method that enhanced the sample's UCS. As in Table 12, the Taguchi method's predictions of dry density and UCS are also shown. The dry density was effectively predicted in both Taguchi experiments when the dry density and UCS were the optimized parameters. In Table 12, a mean for the dry density of 2.262 g/cm\({}^{3}\) was obtained, the predicted value being 2.256 g/cm\({}^{3}\). Furthermore, this value of 2.262 g/cm\({}^{3}\) for the dry density was higher than those shown in Table 6, which proved that the Taguchi method was effective in enhancing and predicting the dry density of a natural by-product mix used for RE building. Table 13 also shows that the Taguchi method correctly predicted the dry density value when the UCS was the parameter to be optimized, with a prediction of 2.203 g/cm\({}^{3}\) versus a real dry density of 2.195 g/cm\({}^{3}\). In this case, the observed and estimated dry density values were lower than the combination which optimized the dry density shown in Table 12.  
Table 13 shows the combination proposed with the Taguchi method, which optimized the UCS. The mean UCS value for the manufactured samples reached 1.91 MPa, higher than the prediction, as it was expected to achieve 1.73 MPa. A similar trend was observed in Table 12, where the samples of the combination that enhanced dry density reached a mean UCS value of 1.40 MPa, while the predicted value was 1.26 MPa. These underestimations in the predicted value were attributed to the poor mechanical properties obtained by some of the combinations shown in Table 9. Nevertheless, the deviation in the predicted value was lower than 10%, and it was considered acceptable, as the aim of the investigation was to find a combination that optimized the UCS. Thus, the combination that was predicted to optimize the UCS was shown to be the combination that achieved the highest UCS value compared to those in Table 9, confirming that the Taguchi method was effective in enhancing the UCS of a natural by-product mix used for building. In addition, the value of 1.91 MPa achieved by the combination that optimized the UCS indicated a good performance of this natural by-product mix for use in RE construction, as the literature requirements for the UCS in RE building are between 1.3 MPa and 2 MPa [3,22]. Tables 12 and 13 demonstrate that enhancing the dry density due to a modification of the mix PSD did not directly improve the mix UCS, as the combination that achieved the highest dry density was not the same as the one that achieved the highest UCS. This was related to the materials' proportions and roles. In the case where the dry density was optimized, a higher proportion of materials with a higher bulk density was employed, as the main objective of this combination was to enhance its dry density, and the principal role of the materials was as filler. However, in the combination that optimized the UCS, a higher proportion of fines was required. Fines not only have a filler effect, but they also provided cohesion to the samples. Therefore, despite not making a prediction as accurate as for the dry density, when using the Taguchi method to optimize a mix for use in RE, the factor that should be analyzed is the mechanical strength.