## 5 Results and Discussion
### Finding the Optimal Structure of the Intelligent Models  
As is known, determining an optimal structure of the networks is a substantial step in the field of artificial intelligence utilization. In this study, we executed an extensive trial and error process to achieve the appropriate architecture of ANN, GA-ANN, and ICA-ANN models. To do this, mean square error (MSE) was defined as the objective function to measure the accuracy of various structures of the networks. MSE is expressed as follows:  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}\left(Y_{i_{observed}}-Y_{i_{predicted}}\right)^{2}, \tag{4}\]  
where similar to the aforementioned criteria, \(N\), \(Y_{i}\)_observed_, and \(Y_{i}\)_predicted_ respectively symbolize the number of samples, actual and predicted values of HL (or CL).  
For ANN, a feed-forward back-propagation MLP containing one hidden layer was tested based on different numbers of hidden neurons. Also, the Levenberg-Marquardt (LM) training algorithm is applied, due to the better performance compared to conventional gradient descent (GD) technique [46; 47]. Besides, the tangent-sigmoid activation function (simply called Tansig) was used in this study, which is formulated as follows:  
\[Tansig\ (x)=\frac{2}{1+e^{-2x}}-1, \tag{5}\]  
Figure 7 depicts the results of the sensitivity analysis carried out to select the best ANN structure, as is shown. However, the MLP with 3, 4, and 5 nodes in its hidden layer yields a satisfying approximation, and the least MSE is obtained by the network with four hidden neurons. So, the used ANN model forms as \(8\times 4\times 2\) representing eight input neurons, four hidden neurons, and two output neurons, respectively. This is noteworthy that this structure was used in the following and for the GA-ANN and ICA-ANN predictive models.  
As explained supra, to enhance the capability of the typical ANN, we combined it with genetic algorithm and imperialist competition algorithm (Figure 8). Similar to ANN, the best parameters were  
Figure 7: A sensitivity analysis for artificial neural network (ANN), based on the number of hidden neurons.  
selected via a trial and error process. There are several parameters that affect the learning process of evolutionary algorithms. In the case of this study, different population sizes for GA, and a different number of decades for ICA were tested within 1000 iterations. Moreover, similar to [PERSON] et al. (2019), the most appropriate values for other parameters have been determined by a trial and error process. Table 2 denotes the results of the mentioned process.  
Next, both parameters of \"number of countries\" and \"population size\" were considered to change between 50 to 500 with 50 intervals. The objective function was calculated for each iteration, and the results of this process are shown in Figures 9 and 10. Based on these figures, no significant change was observed after the 600 th iteration for either GA and ICA algorithms. Accordingly, the lowest MSE obtained for the GA and ICA models with the number of countries and population size equal to 450 and 400, respectively.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**ICA-ANFIS** & **GA-ANN** \\
**(Num. of Countries = 200, and Num. of** **Decades = 1000)** & **(Population Size = 200, and Num. of** **Generation = 1000)** \\ \hline Revolution rate = 0.3 & \\ Num. of initial imperialists = 5 & Crossover rate = 0.50 \\ Damp ratio = 0.99 & Mutation rate = 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The optimal parameters of GA and ICA algorithms.  
Figure 8: The incorporation of MLP with the proposed imperialist competition algorithm (ICA) and genetic algorithm (GA) algorithms.