## 6 Conclusions and Discussion  
This paper focused on the box-shaped solution space, where performance requirements can be met at any design point within it. The volume of the box-shaped solution space indicates its robustness to parameter variation. Moreover, the minimum value of the performance function on the box-shaped solution space measures its performance.  
The performance and robustness of the box-shaped solution space were simultaneously considered in this paper. Since the optimization of performance and an increase  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Solution** & **No. 1** & **No. 2** & **No. 3** & **No. 4** & **No. 5** & **No. 6** & **No. 7** \\ \hline log-volume & \(-707.4685\) & \(-983.9283\) & \(-1160.7678\) & \(-833.3206\) & \(-820.7878\) & \(-826.4275\) & \(-833.3206\) \\ reliability & \(0.9900\) & \(0.9977\) & \(0.9900\) & \(0.9943\) & \(0.9958\) & \(0.9907\) & \(0.9943\) \\ cost (\(10^{5}\)s) & \(4.9999\) & \(4.9999\) & \(4.4265\) & \(4.7104\) & \(4.8929\) & \(4.5644\) & \(4.7104\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The values of the objective functions of the selected non-dominated solution spaces for the PSSTCS.  
Figure 9: The approximated Pareto-optimal front.  
in robustness are two conflicting objectives, we proposed a multi-objective optimization problem to identify the trade-off between performance and robustness, represented by the Pareto front. Additionally, we proposed a novel DIRECT-NSGA-II approach to address this multi-objective optimization problem, applicable to both analytically known and black-box performance functions. Once a set of Pareto-optimal solution spaces is identified by the DIRECT-NSGA-II approach, engineering designers can use higher-level qualitative considerations to make a choice. For example, if performance is the primary concern, then engineering designers prioritize the box-shaped solution space with high performance but poor robustness. In contrast, if robustness is vital, then priority is given to the box-shaped solution space with high robustness but low performance.  
Finally, we discuss interesting topics for future study. Firstly, we should explore the application of the proposed DIRECT-NSGA-II method to more complex nonlinear systems or even to real-world applications within a stochastic framework. Second, our study focused on the box-shaped solution space. However, a significant limitation of using box-shaped solution spaces is that they often fail to include many designs that meet performance requirements. In order to reduce this loss, further studies should focus on more complex solution spaces, such as the component solution spaces introduced in [43]. In addition, in our methodology, we first identify a representative set of Pareto-optimal solutions and then allow engineers to choose one based on their subjective preferences. How to incorporate engineers' subjective preferences into the iteration optimization process is a challenging but also interesting topic for future study.  
Conceptualization, Y.C.; methodology, Y.C.; supervision, J.S.; writing--review and editing, Y.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Youth Academic Innovation Team Construction project of Capital University of Economics and Business grant number No. QNTD202303.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.  
Details of selected non-dominated solution spaces for PSSTCS.  
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