## 2 Materials and Methods
### Computational Methodology
#### 2.2.3 Optimization Approach by Inverse ANN  
In order to identify the fan voltages that guarantee optimal forced convection conditions to improve the operation of the solar system and achieve the maximum drying velocity, a computational strategy known as ANNi [39; 40] was implemented, which is based on the algebraic reduction of Equations (3) and (4). This approach uses the coefficients of the weights and bias of the ANN obtained after their training (\(W_{o}\), \(W_{i}\), \(b_{1}\), and \(b_{2}\)) and inverts the ANN to form the objective function defined by Equation (5).  
\[y=v_{f}-b_{2}+\sum_{j=1}^{s}W_{o}(1,s), \tag{5}\]  
where \(y\) is the function of the experimental variable to be optimized (\(y=f(In(v_{d})\)). Equations (6) and (7) present the mathematical expressions for the case of ANN architectures with hyperbolic tangent and logistic sigmoid activation functions in the hidden layer neurons, respectively.  
\[y=\frac{2\sum_{s=1}^{S}W_{o}(1,s)}{1+\exp\left(2\sum_{k=1}^{K}(X(s)+In(v_{d}) \times W_{i}(j,k))\right)}, \tag{6}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Statistical Parameter** & **Mathematical Expression** \\ \hline Root-mean-square error & \(RMSE=\sqrt{\frac{\sum_{s=1}^{s}\left(x_{out,exp}(s)-x_{out,sim}\right)^{2}}{n}}\) \\ Mean absolute percentage error & \(MAPE=\frac{\sum_{s=1}^{s}\left|\frac{x_{out,exp}(s)-x_{out,sim}}{x_{out,sim}} \right|}{x_{out,sim}(s)}\times 100(\%)\) \\ Determination coefficient & \(R^{2}=1-\frac{\sum_{s=1}^{s}\left(x_{out,sim}(s)-x_{out,sim}\right)^{2}}{\sum_{ s=1}^{s}\left(x_{out,exp}(s)-x_{out,sim}\right)^{2}}\) \\ Linear regression coefficient & \(R=\sqrt{1-\frac{\sum_{s=1}^{s}\left(x_{out,sim}(s)-x_{out,sim}\right)^{2}}{\sum_{ s=1}^{s}\left(x_{out,exp}(s)-x_{out,sim}\right)^{2}}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistical parameters for the evaluation of the model based on ANN.  
\[y=\frac{\sum_{s=1}^{S}W_{o}(1,s)}{1+\exp\left(\sum_{k=1}^{K}(X(k)+In(v_{d})\times W _{i}(j,k))\right)}, \tag{7}\]  
where \(X(s)\) is determined using Equation (8), from which \(K\) acquires all \(In\) input values except for the one corresponding to \(v_{f}\) (\(In\Big{(}v_{f}\Big{)}=In(7)\)).  
\[X(s)=-\Bigg{(}\sum_{k=1}^{K-1}W_{i}(j,k)\times In(k)+b_{1}(j)\Bigg{)}. \tag{8}\]  
Substituting Equations (4) and (5) into Equation (6), the objective functions for both network architectures can be obtained (hyperbolic tangent, Equation (9) and logistic sigmoid, Equation (10)).  
\[y=\frac{2\sum_{s=1}^{S}W_{o}(1,s)}{1+\exp\left(2\sum_{k=1}^{K}(X(s)+In(v_{d}) \times W_{i}(j,k))\right)}-b_{2}+\sum_{j=1}^{s}W_{o}(1,s). \tag{9}\]  
\[y=\frac{\sum_{s=1}^{S}W_{o}(1,s)}{1+\exp\left(\sum_{k=1}^{K}(X(k)+In(v_{d}) \times W_{i}(j,k))\right)}-b_{2}+\sum_{j=1}^{s}W_{o}(1,s). \tag{10}\]  
For the solution of both objective functions, the evaluation of three metaheuristic optimization algorithms is proposed to determine the most efficient alternative dryer operation with the least computation time. The metaheuristic algorithms evaluated were genetic algorithm (GA), particle swarm optimization (PSO), and ant colony optimization (ACO). A detailed description of their operating principle is provided in Appendix A.