## 4 Results
### Genetic Programming (GP)  
The genetics of programming involves applying evolutionary algorithms to generate computer programs through the use of genetic operators, such as crossover and mutation. Important parameters that affect the effectiveness of genetic programming, include population size, mutation rate, and fitness evaluation function. In this study, after a series of analyses, the best GP model was selected, based on its ability to produce high-quality programs for a given problem domain. Figure 10 shows the projected values of CBR by the best GP model, against the actual values obtained in the experiments, for both the training and testing sets. According to the findings presented in Figure 10, the best genetic programming (GP) model was capable of accurately predicting the values of the California bearing ratio (CBR) in the experiments, as indicated by the close match between the model's projected values and the experimental values. These results suggest that the GP model has a high level of accuracy in predicting CBR values. Figure 10: The results of the best GP model predicting CBR values for the (**a**) training and (**b**) testing databases. A significant distinction between genetic programming (GP) and black-box models, is that the GP produces an equation as an outcome, which can be directly utilized by the reader. In this study, Equation (13) represents the GP model outcome for predicting California bearing ratio (CBR) values. The reason for the length of this equation, is that it incorporates seven distinct inputs, namely liquid limit, plasticity index, alum sludge content, number of compaction blows, optimum moisture content of soil and mixture, and \(G_{\text{s}}\) of soil. It is worth noting, that traditional methods are incapable of producing an equation such as Equation (13), with the consideration of all seven influential inputs. Consequently, this study represents the first successful attempt to formulate an equation for predicting CBR numbers based on these seven important inputs. It is important to note, that the parameters in Equation (6) are normalized values, and values should be based on Table 1 before being used in Equation (13). \[ \begin{array}{l}\text{CBR}=((((0.234+(((X_{2}-0.339)-0.185)^{2}))-((((X_{2} \gamma^{2})-X_{3})-((0.339-X_{5})^{3})))\times X_{6}))\times((X_{4}+\\ (((X_{5}{}^{2})\times(0.339+X_{2}))\times((X_{3}\times X_{5})\times(X_{1}+0.339) )))-((((X_{4}+0.185)-X_{7})\times((X_{6}\times X_{5})-0.234))\\ -((X_{5}{}^{3})-((0.234^{2})\times(0.234+X_{1}))))))\\ \end{array}  \tag{13}\]  
where \(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{4}\), \(X_{5}\), \(X_{6}\), and \(X_{7}\) are LL, PI, AS content, number of compaction blows, OMC of soil, OMC of mixture, and G\({}_{\text{s}}\) of soil, respectively. Table 8 presents the results of the GP model in predicting the California bearing ratio (CBR) for the mixtures of alum sludge and soil. The GP model's performance was evaluated using various performance metrics on both the training and testing datasets. The mean absolute error (MAE) of the model on the training database was 0.459, indicating that, on average, the model's predictions were off by 0.459 CBR units. The MAE on the testing database was slightly higher, at 0.475, indicating that the GP model's generalization ability was slightly worse than its performance on the training database. The mean squared error (MSE) on the training database was 0.358, indicating that the GP model's predictions had a higher variance than its performance on the testing database (0.304). The root mean squared error (RMSE) on the training database was 0.598, indicating that the GP model's predictions had a higher magnitude of error than its performance on the testing database (0.552). The mean squared logarithmic error (MSLE) on the training set was 0.010, and the corresponding value on the testing set was slightly lower, at 0.009, indicating that the GP model's predictions were more accurate on the testing set.