## 4 Results and Discussion
### Modeling and Discussion  
A mathematical formulation is presented to numerically assess the effects of utilized materials on the concrete properties. To fulfill this purpose, the GEP method is applied. GeneXpro tools 5.0 [41] was utilized to generate the mathematical expressions between the variables and the objectives. For every objective, eight models were run with different properties such as the number of chromosomes, number of head size, linking function, and number of constants per gene (Table 5). The dataset is divided into two parts by the GEP algorithm randomly; 80% of the dataset is training data and 20% is testing data. Testing data do not have any role in the training process and building of the model. In Figures 11-13, the best results of GEP are shown as ETs, in which d\({}_{0}\), d\({}_{1}\), d\({}_{2}\), and d\({}_{3}\) represent the percentage of nano-MgO, water-to-binder ratio, curing age of concrete, and superplasticizer percentage, respectively. Figure 11: Expression trees (ETs) of Gene Expression Programming (GEP1) for compressive strength. (**a**) sub-ET1 (**b**) sub-ET2, (**c**) sub-ET3. Figure 12: Cont. Figure 12: Expression trees of GEP1 for tensile strength (**a**) sub-ET1, (**b**) sub-ET2, (**c**) sub-ET3. Figure 13: Cont. Tables 8-10 show the best mathematical formulation for the prediction of the compressive strength, tensile strength, and permeability, respectively. Every formulation consists of three genes that are linked to each other by linking functions such as addition or multiplication. For finding the proper formulation, GEP uses different operators such as powers, Sqrt, e', Ln, Log, etc. All four variables are utilized in formulations; it shows that selected variables are effective in the performance of the concrete. \begin{table}
\begin{tabular}{c c} \hline \hline
**GEP Model** & **Predictive Equation of GEP for Compressive Strength** \\ \hline \multirow{2}{*}{GEP 1} & \(\sqrt{\left(e^{(\sqrt{C_{0}\times d_{0}}-d_{0})^{3}}\right)}\) & \(+(ln2C_{0})^{3}\times\left[\left(d_{3}+C_{0}\right)^{2}+d_{3}-C_{0}\right]\) \\  & & \(+\left[\left(d_{1}\times d_{2}\right)+C_{0}^{3}\right]\times\left(C_{0}\times d _{0}\right)\times d_{3}^{1/9}\) \\ \hline \multirow{2}{*}{Gene1} & \multirow{2}{*}{\(C_{0}\)\(=7.83280702670388\)} \\  & & Gene2 \\ \multirow{2}{*}{\(C_{0}\)\(=5.57594309191839\)} \\  & & Gene3 \\ \multirow{2}{*}{\(C_{0}\)\(=-2.53441575642642\)} \\ \end{tabular}
\end{table} 
Table 8: The mathematical formulation of compressive strength of concrete samples containing nano-MgO proposed by the GEP tool. Figure 13: Expression trees of GEP6 for permeability (**a**) sub-ET1, (**b**) sub-ET2, (**c**) sub-ET3. Regression analysis is a type of statistical modeling that estimates the relation between dependent variables and independent variables. Linear regression is the most common regression. The aim of linear regression is finding the best fitting line to data. Figures 14-16 depict the regression diagrams by experimental data and the predicted data for the compressive strength, tensile strength, and permeability, respectively. The horizontal axis shows predicted values by GEP, and the vertical axis is experimental data. The error is named \"coefficient of determination\" and is shown by R\({}^{2}\). An ideal line has R\({}^{2}\) = 1, which means that the predicted values and real data are the same. The model is created by a training dataset, and it is examined by the testing dataset. The formulation of the compressive strength has an R\({}^{2}\) value of 0.98 for both the training and testing data. For the tensile strength formula, this value is 0.97 and 0.96 for the training and testing data, respectively. From Table 11, it can be concluded that GEP is an applicable and beneficial tool for predicting the concrete properties, which has been confirmed by the results.