## 4 Optimization Analysis
Probability of Directional Cross-Over = 0.5
2. Probability of Selection = 0.05
3. Probability of Mutation = 0.2
4. DNA String Mutation Ratio = 0.05
5. Elitism = Enabled
6. Treat Constraints = Penalizing Objectives  
In the adopted workflow there are no constraints on the input variables. Such constraints, related to the congruence of the numerical model, are already implicitly implemented within the APDL-macro that generated the parametric model. The output variables are the bearing and axial forces (in a local reference system) for each pin (hole's center). In particular, since the bearing loads are the most critical values for such a problem, the last ones have been defined as objective functions while the axial loads have been set as constraint functions. As reported in the optimization workflow, the objective functions are eight and four for the deployed configuration and four for the folded configuration. Table 2 reports the limit value set for the constrain functions in terms of axial load on bolts (reaction load on the reference point of the MPC element). Further, Figure 12 reports the scheme adopted for identifying the output variables. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Optimization Analysis Constraint Functions**} \\ \hline \multirow{2}{*}{**Variable**} & **Hinge Location** & **Wing Configuration** & **Constrained Point** & **Force Type** & **Less than** \\ \hline H1C1-AX\_DP & Forward & Deployed & C1 & Axial & 7500 \\ \hline H1C2-AX\_DP & Forward & Deployed & C2 & Axial & 7500 \\ \hline H2C4-AX\_DP & Afterward & Deployed & C4 & Axial & 10,000 \\ \hline H2C5-AX\_DP & Afterward & Deployed & C5 & Axial & 10,000 \\ \hline H1C1-AX\_FD & Forward & Folded & C1 & Axial & 7500 \\ \hline H1C3-AX\_FD & Forward & Folded & C3 & Axial & 2000 \\ \hline H2C4-AX\_FD & Afterward & Folded & C4 & Axial & 10,000 \\ \hline H2C6-AX\_FD & Afterward & Folded & C6 & Axial & 2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constraint functions. Figure 12: Output variables definition. The optimization process evaluated 300 design sets and only a very reduced part (7.3%) of them were determined as unfeasible. A design is considered unfeasible if the constraints reported in Table 3 are violated. Figure 13 reports the time history of the objective functions H1 UP_BR_DP, H1 LW_BR_DP, H1 LW_BR_DP, H2 UP_BR_DP, and H2 LW_BR_DP, which are the bearing loads in the deployed configuration at the four constrained points. The results marked with a gray square are feasible, while those marked with an orange rhombus are unfeasible. Figure 14 reports the time history of the objective functions H1 UP_BR_FD, H1 LW_BR_FD, H2 UP_BR_FD, and H2 LW_BR_FD, which are the bearing loads in the folded configuration at the four constrained points. Figure 14: _Cont_. Figure 13: Objective function time history in the deployed configuration. (**A**) forward hinge in upper location; (**B**) forward hinge in lower location; (**C**) afterward hinge in upper location; (**D**) afterward hinge at a lower location. The previous figures show that it is only possible to see a clear trend in the folded configuration and, thus, the algorithm is able to provide design sets that are able to minimize the reaction forces for all hinges simultaneously. Conversely, in the deployed configuration, a sample is able to minimize only one or two objective functions and thus the best one has to be defined with a compromise. The probability density function, related only to feasible designs, for the two design variables (H1-X and H2-X) is shown in Figure 15. In an evolutive optimization process, the probability of selecting a particular sample depends on the fitness function. The latter is better for samples that satisfy the objective functions and the constraint functions with a bigger margin. Therefore, the better an individual is, the greater the likelihood that it will be re-selected and, therefore, that such a sample will survive into the next generations.