## 6 Implementation of the Proposed Method and Solution Space Exploration  
In this section, we detail the implementation of our proposed model and method. This work does not cover step 1 (performance-property), and we started with step 2, as shown in Figure 6. After running our model for each space, the solution space was explored to identify the solution set points that satisfy the requirements for each module. For solution space exploration, the parameters involved in each module were normalized between 0 and 1. The 0 value corresponds to the minimum value of a parameter, while the 1 value  
Figure 5: Inverse Flow of Information in Hot Stamping. corresponds to the maximum value. Each parameter value was normalized using the maximum value of that parameter in the output results using the following expression. \[r_{ij}=\frac{p_{ij}}{max_{1}^{j}P_{ij}} \tag{6}\]  
where \(r\) has values in the range of 0-1. \(p_{ij}\) the value of parameter \(i\) in the \(j-th\) row, and \(max_{1}^{j}P_{ij}\) shows the maximum value of the parameter \(i\) in the \(j-th\) row. _Step 2 Property-Microstructure Space:_ Step 2 of the information flow involves identification of the target goals for material properties, hardness, and tensile strength. Density was not included because no material removal or addition was involved; therefore, only two material properties were considered. The dataset for property-microstructure space was not available, so we took advantage of the fact that each individual phase fraction has its own hardness value. For each microstructure combination in the dataset obtained from [48], we obtained values for hardness and tensile strength using established empirical formulae [50]. For each individual phase fraction, we multiplied its hardness by its percentage, and then the mean was calculated for the summation of all the phase fractions, as shown in Table 5.  
For a microstructure combination of 24% martensite, 22% bainite, and 54% ferrite, the corresponding values of hardness and tensile strength are 278.4 HV and 980 MPa, respectively. We generated 50 data points using the data from the dataset of [48]. The  
Figure 6: Inverse Flow of Information in Hot Stamping (step 1 skipped). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Phase Fraction \%**} & \multicolumn{2}{c}{**Unit Hardness HV**} & \multicolumn{1}{c}{**Overall Hardness**} \\ \hline
**Martensite \%** & 24 & 435 & 435 \(\ast\) 24 = 10,440 \\ \hline
**Bainite \%** & 22 & 300 & 6600 \\ \hline
**Ferrite \%** & 54 & 200 & 10,800 \\ \hline
**Mean Hardness** & & (10,440 + 6600 + 10,800)/100 = 278.40 HV \\ \hline
**Tensile Strength** & & 278.40 \(\ast\) 3.45 = 980 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Illustration of the Method followed with the Generate Dataset for Step 2. For solution space exploration, the requirements for mechanical properties were specified as shown in Table 7. The requirements are provided in the form of ranges that satisfy a given engineering design problem. Upon mapping these requirements with the results for the ANN model, a feasible region was identified as shown by the shaded area in Figure 9. The x-axis of Figure 9 represents the number of data points in the output results. The y-axis represents the combinations of the microstructure at each data point. The red line represents the corresponding tensile strength of each microstructure combination. All microstructure combinations were plotted against the ascending order of the tensile strength so that the feasible region (green rectangular) could be identified. The ascending order allowed us to easily identify the lower and upper limits of the tensile strength requirements. The data points in this feasible region were the microstructure combinations that satisfied the tensile strength requirements. Tensile strength was selected because the target for the hot stamping process is to obtain the best tensile strength.