## 2 Motivation and General Problem Statement  
### A Simple Example Problem  
A simple parallel system that consists of two components is considered. Its system reliability is given by \(R_{s}(\mathbf{R})=R_{1}+R_{2}-R_{1}R_{2}\), and the total cost is \(C_{s}(\mathbf{R})=100R_{1}+150R_{2}\). The requirements for the system are  
\[R_{s}(\mathbf{R}) \geq 0.99,\] \[C_{s}(\mathbf{R}) \leq 245.\]  
The gray region in Figures 1 and 2 is the complete solution space. For system designers, achieving higher reliability clearly requires a greater investment in design. In traditional multi-objective optimization, an optimal point is identified to enhance system reliability while minimizing overall cost. Since these two objectives are conflicting, there exists a number of Pareto-optimal points. Some Pareto-optimal points, including the reliability extreme point, the cost extreme point, and the compromise point, are shown in Figure 1, where the reliability and cost extreme points correspond to the optimal reliability and cost, respectively, and the compromise point is obtained by the'min-max' method [23]. Unfortunately, the Pareto-optimal points in Figure 1 sit at or near the boundary of the complete solution space and therefore may be non-robust. Therefore, in this paper, both optimal performance metrics (i.e., highest system reliability and lowest total cost) and robustness are considered. The box-shaped solution space in which the volume is maximum, system reliability is maximum, and the maximum total cost is minimum is sought. That is, the multi-objective optimization problem can be defined as follows:  
\[\max_{\mathbf{R}^{tot\omega},\mathbf{R}^{tot\varphi}} \prod_{i=1}^{2}(R_{i}^{up}-R_{i}^{low}),\] \[\max_{\mathbf{R}^{tot\omega},\mathbf{R}^{tot\varphi}} \min_{\mathbf{R}^{\in}[\mathbf{R}^{low},\mathbf{R}^{tot\varphi}]}R_{s}(\mathbf{R}),\] \[\min_{\mathbf{R}^{tot\omega},\mathbf{R}^{tot\varphi}} \max_{\mathbf{R}^{tot\varphi},\mathbf{R}^{tot\varphi}}C_{s}(\mathbf{R}),\] \[\text{subject to} \begin{cases}\min_{\mathbf{R}\in[\mathbf{R}^{low},\mathbf{R}^{opt}]}R_{s}(\bm {R})\geq 0.99,\\ \max_{\mathbf{R}\in[\mathbf{R}^{low},\mathbf{R}^{opt}]}C_{s}(\mathbf{R})\leq 245,\\ \mathbf{0.85}\leq\mathbf{R}^{low}\leq\mathbf{R}^{up}\leq\mathbf{1},\end{cases} \]  
Figure 1: The Pareto-optimal points. where \([\mathbf{R}^{low},\mathbf{R}^{up}]=\{\mathbf{R}=(R_{1},R_{2}):R_{i}^{low}\leq R_{i}\leq R_{i}^{up},i=1,2\}\) is a box-shaped solution space; \(\min_{\mathbf{R}\in[\mathbf{R}^{low},\mathbf{R}^{up}]}R_{s}(\mathbf{R})\), that is, the global minimum of \(R_{s}(\mathbf{R})\) for \(\mathbf{R}\in[\mathbf{R}^{low},\mathbf{R}^{up}]\), is the system reliability of the box-shaped solution space; and \(\max_{\mathbf{R}\in[\mathbf{R}^{low},\mathbf{R}^{up}]}C_{s}(\mathbf{R})\), that is, the global maximum of \(C_{s}(\mathbf{R})\) for \(\mathbf{R}\in[\mathbf{R}^{low},\mathbf{R}^{up}]\), is the system cost of the box-shaped solution space. Figure 2 shows the Pareto-optimal box-shaped solution spaces. The green, red, magenta, and cyan boxes are the volume extreme solution, reliability extreme solution, cost extreme solution, and compromise solutions obtained by the'min-max' method considering volume, reliability, and cost. For comparison purposes, the compromise solution considering only reliability and cost is also shown by the yellow box in Figure 2. Figure 3 displays the approximated Pareto-optimal front, along with its 2-D projections. The extreme boxes for reliability and cost in Figure 2 simplify to points, nearly matching the extreme points for reliability and cost in Figure 1. This indicates that our proposed optimization can achieve the optimal solutions for each performance metric. The reliability and cost of the compromise point in Figure 1 are 0.9950 and 224.1668, while those of the compromise box1 in Figure 2 are 0.9973 and 225.7504. There is no significant difference between these two compromise solutions. This implies that our results are comparable to those of traditional multi-objective optimization.