## 4 Model Development for \(\mathbf{F_{ult}}\) Estimation
### Hybrid PSO-ANN and ICA-ANN Models in Predicting \(F_{ult}\)  
Two-hybrid models of ICA-ANN and also PSO-ANN are employed to choose the most suitable predictive model among them. Therefore, to discover optimum factors in both models, many parametric studies are done. The optimal state of ANN model needs to be indicated, prior to running a parametric investigation for hybrid model factors. Parametric investigation of ANN model was performed considering a series of trial and error process (discussed in Section 4.1), and it was found that an ANN model with eight hidden neurons (or architecture of \(12\times 8\times 1\)) received better performances. Hence, the obtained architecture was confirmed and used to both hybrid intelligent systems (e.g., ICA-ANN and PSO-ANN). As previously mentioned, in order to determine the most appropriate structure of the hybrid model of PSO-ANN, there is a need for a parametric investigation according to the trial and error process. Numerous parameters such as coefficients of velocity relation (\(C_{1}\) and \(C_{2}\)), inertia weight and the number of iterations, and the number of existing particles can considerably affect network efficiency of the PSO approach. Many scholars suggested that, for providing an acceptable network performance, \(C_{1}\) and \(C_{2}\) equal to 2 as well as inertia weight (\(i_{w}\)) of 0.25 is proper. Different models of PSO are generated by distinct amounts of swarm size (i.e., 25, 50, 75, 100, 150, 200, 250, 300, 350, 400, 450, and 500) to indicate the appropriate particle size/swarm population (with \(I_{w}\) = 0.25 as well as \(C_{1}\) and \(C_{2}\) equal to 2). When 400 is the number of swarm size, the best efficiency is obtained. For the training and testing datasets, the \(R^{2}\) and \(RMSE\) are (0.9607 and 0.076) and (0.978, and 0.075), respectively. Figure 10a shows the performance result variation; for example, MSE utilized here, of PSO-ANN models that have different various population sizes. As observed, the model that has swarm sizes equal to 400 indicates the least RMSE, which demonstrates its advantage in comparison to other approaches. For selecting appropriate values for \(C_{1}\) and \(C_{2}\), a similar trial and error approach is used. In this regard, by distinct \(C_{1}\) and \(C_{2}\), around 12 models are designed, and then their efficiencies are according to total ranking and CER approaches. The most appropriate efficiencies with the total value of 69 obtained when \(C_{1}\) and \(C_{2}\) were (1.33 and 0.67) or (1.5 and 1.5), respectively. For the testing datasets, the \(R^{2}\), and \(RMSE\) are 0.9607, and 0.076, respectively. Finally, five approaches are provided by \(I_{w}\) (0.2, 0.4, 0.6, 0.8, and 1.0) to choose an appropriate amount for the \(I_{w}\). For the IW, achieved total rank of 0.2, 0.4, 0.6, 0.8, and 1.0 are 28, 12, 18, 6, and 26, respectively. It demonstrates the advantage of the PSO-ANN approach with \(I_{w}\) of 0.2. In order to take the most appropriate predictive result from the ICA-ANN hybrid approach, optimizing its influential factors is an important concern. In order to employ the best approach, two distinct ranking methods (CER and total ranking [49]) are used. Similar to the first parametric study process performed, e.g., in selecting proper ANN architecture and PSO-ANN model, this has also been obtained considering a series of trial and error progress. As stated earlier, in ICA, number of the country (\(N_{c}\)), number of the decade (\(N_{d}\)), and number of imperialists (\(N_{i}\)) are recommended as the most influential parameters on ICA performance. In order to determine the proper \(N_{c}\), several designs are created by distinct amounts of \(Nc\) (25, 50, 75, 100, 150, 200, 250, 300, 350, 400, 450, and 500). For these models, \(N_{i}\) = 5 and \(N_{d}\) = 200 were utilized. As a result, with the total rank of 68, \(N_{c}\) = 300 can provide a higher efficiency system capacity.