## 5 Case Studies
### Power-Shift Steering Transmission Control System  
In heavy vehicles, the power-shift steering transmission control system (PSSTCS) plays a crucial role, enabling steering control, speed adjustment, fan operation, and lubrication. It comprises several subsystems: hydraulic oil supply, integrated pump-motor, fan control, electronic control, and hydraulic control. The functional components and structural principles of the PSSTCS are depicted in Figures 7 and 8, respectively. The multi-objective optimization problem for the PSSTCS to optimize reliability, cost, and robustness can be stated as follows:  
\[ \begin{split}\max_{R^{\text{low}},R^{\text{up}}}& \quad\ln\big{(}v(R^{\text{low}},R^{\text{up}})\big{)},\\ \max_{R^{\text{low}},R^{\text{up}}}&\quad f_{1}^{*}(R ^{\text{low}},R^{\text{up}}),\\ \max_{R^{\text{low}},R^{\text{up}}}&\quad f_{2}^{*}(R ^{\text{low}},R^{\text{up}}),\\ \text{subject to}&\quad f_{1}^{*}(R^{\text{low}},R^{ \text{up}})\geq f_{1}^{l},\\ &\quad f_{2}^{*}(R^{\text{low}},R^{\text{up}})\geq f_{2}^{l},\\ &\quad R^{l}\leq R^{\text{low}}\leq R^{\text{up}}\leq R^{\text{up }}\leq R^{\text{u}},\end{split}  \tag{5}\]  
where \(R^{\text{low}}=(R_{1}^{\text{low}},\ldots,R_{86}^{\text{low}})\) and \(R^{\text{up}}=(R_{1}^{\text{up}},\ldots,R_{86}^{\text{up}})\) are the lower and upper boundaries of the box-shaped solution space, respectively; \(\ln\big{(}v(R^{\text{low}},R^{\text{up}})\big{)}=\ln\big{(}\prod_{i=1}^{86}(R _{i}^{\text{up}}-R_{i}^{\text{low}})\big{)}\), that is, the logarithmic transformation of the volume; \(f_{1}(R)\) is the reliability of the PSSTCS, but it is a black box and therefore is given by a numerical method--the GO reliability assessment method in [42]; \(f_{1}^{*}(R^{\text{low}},R^{\text{up}})=\min_{R\in[R^{\text{low}},R^{\text{up }}]}f_{1}(R)\); \(f_{2}^{*}(R^{\text{low}},R^{\text{up}})=\min_{R\in[R^{\text{low}},R^{\text{up }}]}-f_{2}(R)\); \(f_{2}(R)=-\sum_{i=1}^{86}\frac{\ln(1-R_{i})}{\alpha_{i}}\) is the cost of the PSSTCS, where \(\alpha_{i}\) is a constant number above zero; and \(f_{1}^{l}=0.9900\) and \(f_{2}^{l}=-5\times 10^{5}\) are threshold values of the reliability and cost, respectively. Figure 6: The Pareto-optimal front is obtained using the DIRECT-NSGA-II and MOEA-IA methods. The MOEA-IA method relies on analytically known functions, rendering it unsuitable for this complex system with a black-box reliability function. Instead, we employ the developed DIRECT-NSGA-II algorithm to tackle the multi-objective optimization problem (5) using a population size of 400. The algorithm runs five times, and the best result is selected as the final result. Figure 9 illustrates the Pareto-optimal solution spaces obtained in the last generations. The upper-left plot in Figure 9 shows the non-dominated solution spaces. The remaining three plots display the pairwise relationships between the three objectives. We can observe a negative correlation between the reliability and the volume (robustness) of the box-shaped solution space, while the cost shows a positive association with the volume (robustness). Additionally, a negative correlation exists between the reliability and the cost. To make the final decision, seven compromise solutions are selected, and the details, including the boundaries of the box-shaped solution space and objective function values, are depicted in Tables 2 and A1, respectively. The seven solution spaces are marked in the Pareto-optimal fronts in Figure 9 and are numbered from No. 1 to No. 7. Among  
Figure 8: Drawing of structural principles of PSSTCS. Figure 7: Functional constituents of PSSTCS. them, solutions No. 1-3 are the solutions with the strongest robustness (the largest box-shaped solution space), the highest reliability, and the lowest cost (the highest negative cost), respectively. Solution No. 4 is the'min-max' Pareto-optimal solution with respect to the three objectives of the volume, reliability, and cost. Solutions No.