## 3 Results and Discussion
### Influence of the Variable Completeness on the BP-ANN Prediction Performance  
The combinations of input variables for each response are listed in Table 3. Each combination was fed as input to BP-ANN models with the node numbers in the hidden layer proposed in Table 4. For instance, the optimal node number in the hidden layer was 4, 5, and 6 for the maximum compressive stress. That is to say, three BP-ANN modeling were conducted for each input combination. The prediction performance was evaluated by the mean values of \(RE\_MAX\), \(MRE\), and \(B\) of three BP-ANN models, shown in Figure 5. In general, for all three responses, lower \(RE\_MAX\) and \(MRE\) and larger B can be achieved with increasing variable completeness. The minimum number of variables for each response was determined by an arbitrary chosen error tolerance, i.e., 15% of \(RE\_MAX\), 3% of \(MRE\), and 0.90 of \(B\) and listed in Table 7 with contribution summations of these variables to the response. The combinations consisted of the minimum number of variables that are capable of predicting the relation between input variables and corresponding outputs. It also indicates that the minimum number of variables changes with respect to the complexity of the problem. A larger number of variables shall be considered if the problem shows significant complexity, which shall contribute a certain value to the response. For instance, this value could be 90% for the tensile stress. This information further confirms the top priority of the significant factors of lining concepts analyzed with the Taguchi method.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Literature Information**} & \multicolumn{4}{c}{**Proposed Guidelines**} \\ \hline \hline \multirow{2}{*}{**Research Topics**} & **Optimal** & \(N_{h}\)**Range** & \(N_{h}\)**Range** & \(N_{h}\)**Range** & **Total Number** \\  & \(N_{h}\) & **(\(PF\) = 10)** & **(\(PF\) = 70)** & **(\(PF\) = 10–70)** & **of Trials** \\ \hline Power output [15] & 7 & [2; 4] & [5; 7] * & [2; 7] & 6 \\ Lithology identification for shale oil & 10 & [7; 10] * & [13; 17] & [7; 17] & 11 \\ \begin{tabular}{c} Moisture content prediction in paddy \\ drying process [20] \\ \end{tabular} & 2 & [1, 3] * & [3; 5] & [1; 5] & 5 \\ \begin{tabular}{c} Mechanical behavior of mortar [22] \\ \end{tabular} & 2 & [2, 4] * & [6; 8] & [2; 8] & 7 \\ \
\begin{tabular}{c} Extraction of phenolic compounds [23] \\ \end{tabular} & 2 & [1, 3] * & [3; 5] & [1; 5] & 5 \\
\begin{tabular}{c} Damage pattern of structural systems [24] \\ \end{tabular} & 17 & [7; 9] & [13; 15] & [7; 15] & 9 \\ \hline \hline \end{tabular}  
* The range includes the optimal node number in the literature.  
\end{table}
Table 6: Comparison of optimal node numbers in literature and the predicted ranges from the proposed guidelines.  
Figure 4: The relation between \(PF\) and \(A\) (a function of \(PF\)).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Response** & **Minimum Number of Input Variables** & **Contribution to Response (\%)** \\ \hline Compressive stress & 3 & 98 \\ End temperature & 5 & 94 \\ Tensile stress & 6 & 85 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Selection of the minimum number of input variables satisfying the predefined criteria.  
Figure 5: Performance assessment (**a**) _RE_MAX_, (**b**) _MRE_, and (**c**) \(B\) of BP-ANN models with different variable completeness (the ratio of input variables to the total number of variables).