## 5 Determination of the Effect of System Parameters on the Required AGV Number
Surrogate models are used when the outcomes of the problem cannot be easily computed, especially for complex optimization problems. They provide an efficient way to define models, which can be solved at a reduced computational cost instead of by complex simulations. The structure of the ANN used in this study is shown in Figure 5. The ANN is composed of a three-layer network. The inputs were the velocity, charging time, distance, and hauled part count and the output was the number of required AGVs. Pearson correlation coefficient (R) and mean absolute percentage error (MAPE) values were examined to find the best ANN structure [53]. The hidden layer included five neurons. The training algorithms of Levenberg-Marquardt and Bayesian regularization were used to train the neural network model. Figure 5: The structure of the ANN model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Symbol** & **Meaning** & **Min.** & **Avg.** & **Max.** \\ \hline \(Y\) & Required AGV Number & & & \\ \(A\) & Velocity (m/min) & 10 & 20 & 30 \\ \(B\) & Charging Time (min) & 4 & 6 & 8 \\ \(C\) & Distance (ratio) & 0.5 & 1 & 1.5 \\ \(D\) & Trolley Parts Capacity (ratio) & 0.5 & 1 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: System input and output parameters. Cross-validation is a statistical technique to evaluate networks by partitioning the data into subsets of specified ratios. In this research, a hold-out method for cross-validation was used by partitioning the data into subsets, which are the data used for testing and validation and the data used for neural network model training. This technique is preferred as the dataset is small. The ANN model and predictions were saved when the Pearson correlation coefficient (R) of the test was higher than 0.998, as well as when finding better mean square error (MSE) values. The reason for choosing the MSE over the R-value is to prevent overfitting and increase accuracy. For the dataset, 80% of the data are used as training, 10% for testing, and 10% for validation. The neural network model was run in Matlab [46]. The best performance for the training was realized with Bayesian regularization, and the performance was evaluated in terms of mean squared error (MSE) and Pearson correlation coefficient (R) values. The R-values for the neural network model were recorded for training and testing. All the R-values were computed as shown in Figure 6. Bayesian regularization backpropagation (BRB) and the Levenberg-Marquardt (LM) algorithm were used to train the neural network model. Bayesian regularization allows for better values for R (0.998) and the MSE (\(2.088\times 10^{-5}\)). The R-value must be close as possible to 1. The ANN estimates successfully compared to RSM and MLR in R2 and mean absolute percentage error (MAPE), which were used to evaluate the prediction accuracy. They are performance evaluation results between the predicted and true value. The effect of system parameters on the required AGV number was investigated for the velocity, charging time, distance, and trolley part capacity. These parameters and their minimum, maximum, and average values are given in Table 8.  
A design of experiment (DOE) was carried out for the four determined system variables. Box-Behnken designs are experimental designs for response surface methodology. They do not contain an embedded factorial or fractional factorial design. Box-Behnken designs are created by combining factorial design with incomplete block designs, resulting in efficient solutions in terms of the number of runs required. This method allows us to produce effective solutions with a smaller number of trials than incomplete block design [52]. The Box-Behnken method was used, and 40 trials were carried out, including four factors and one repetition. The dataset is given in Table 9. The required AGV number was obtained by running the GA. RSM, ANN, and MLR were employed to examine the AGV number and the results were compared with the GA as shown in Table 9.  
Figure 6: ANN accuracy rates with R-values. RSM was run in Minitab [50].