# Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage
## 6 Numerical Experimental Results  
The MORBDO problem was solved for 30 runs with \(\alpha\)-cut = 1.0, 0.5, and 0.001, and the obtained statistical results are shown in Table 2. The average computational times for each optimization run and \(\alpha\)-cut were slightly different in terms of running time, as shown in Table 2. The new technique spends only a few second in solving the problem. The plot of the best Pareto frontier is shown in Figure 7. Some selected solutions were obtained from clustering [36] and shown in Figure 8. The minimum steering error was0.03043, with the maximum worst-case scenario being 0.5867 at the parameter bound of \(\alpha\)-cut = 1.0. The maximum steering error obtained was0.2232, with the maximum worst-case scenario (WCSV) being \(-\)3.673 at the parameter bound of the same \(\alpha\)-cut. The steering error was0.05463, which was close to zero at \(\alpha\)-cut = 1.0, representing the deterministic optimum solution, which provided the WCSV as \(-\)0.05887. According to the mean value of hyper volume (HV) in Table 2, the highest value represented the best front. It can be concluded that the best front to worst were\(\alpha\)-cut = 1.0, then 0.5, and finally 0.001. From the results, the lower HV meant more reliable solutions. The Pareto frontier represents various solutions obtained from using different \(\alpha\)-cuts. The fronts depended on the WCSVs and the level of \(\alpha\)-cuts. The best and worst of the steering errors were solutions with plots of ideal (\(\theta_{OA}\)) versus actual (\(\theta_{O}\)) angles at \(\alpha\)-cut = 0.001 (solution 2 and 9 of the most reliable front in Figure 9). From the figure, the result shows the worst steering error, but it was also the most reliable. To re-examine the results of the proposed technique depending on \(\alpha\)-cuts, we selected the plot as the ideal (\(\theta_{OA}\)) versus actual (\(\theta_{O}\)) angles of solution 2 with very high value of \(\alpha\)-cut. Figure 10a shows that the steering error increased as the level of \(\alpha\)-cut =1 decreased. A similar took takes place for solution 9, as shown in Figure 10b. Figures 11-13 show the boxplots of distribution design parameters where the central mark indicates the median and the bottom and top edges of the box show the 25% and 75% percentiles, respectively. The whiskers extended to the most extreme data points do not consider outliers, and the outliers are plotted individually using the \"+\" symbol. Figure 8 shows some selected solutions while Figure 14 shows a corresponding steering linkage kinematic diagram of solution number 1. The various linkages were indifferent because of the design constraints imposed in problem (18) caused by the narrow feasible solution as mentioned in [17]. The mechanisms may have similar shapes but they have different dimensions, as shown in Figures 11-13. Such differences affect the results of the worse-case scenario value. A designer can select the conservative solution, which provides a compromise between the steering error and worse-case scenario value due to uncertainties. The motion of a steering linkage of solution number 1 in Figure 8 is illustrated in Figure 14. It can promote the designed steering linkage to work well. Figure 7: The best Pareto frontier of the test problem at each \(\alpha\)-cut = 1.0, 0.5, and 0.001. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{\(\alpha\)-Cut} \\ \hline
**Run No.** & **1.0** & **0.5** & **0.001** \\ \hline  
x** & 5.079538 & 5.06729 & 5.072948 \\
**STD** & 0.394746 & 0.41472 & 0.397889 \\
**Comp. Time a (s)** & 25.586207 & 23.103448 & 20.689655 \\ \hline \hline \end{tabular}  
* Computational time from 30 runs.  
\end{table} 
Table 2: Statistical results of the hyper volume (HV) of Pareto front explored 30 times of each \(\alpha\)-cut. Figure 8: Selected solutions of \(\alpha\)-cut = 0.001. Figure 11: Link dimension distribution in Figure 7 at \(\alpha\)-cut = 1.0.