## 2 The RRAP Models  
The RRAP models, while aim to maximize the system reliability under the non-linear constraints of cost, weight, and volume, for the four benchmarks including series system, the network with series and parallel elements, a complex (bridge) system, and the overspeed protection of a gas turbine system as shown in Figures 1-4 are discussed in this section. The RRAP aims to maximize the system reliability, which has to simultaneously decide reliability and redundancy of components under the non-linear constraints of cost, weight, and volume, is modelled as following Equations (1) and (2). \[\text{Maximize }R_{\text{s}}(\textbf{R},\textbf{N}) \tag{1}\]  
\[\text{Subject to }Cst_{j}(\textbf{R},\textbf{N})\leq U_{j} \tag{2}\]  
Based on the RRAP formulations in Equations (1) and (2), model the four benchmarks of RRAP including series system, the network with series and parallel elements, a complex (bridge) system, and the overspeed protection of a gas turbine system as following Equations (3)-(18). Benchmark 1. Series system in Figure 1[24; 34; 37]. \[\text{Maximize }R_{\text{s}}(\textbf{R},\textbf{ }\textbf{N})=\prod_{i=1}^{N_{\text{c}}}R_{i} \tag{3}\]  
\[\text{Subject to }Cst_{1}(\textbf{R},\textbf{ }\textbf{N})=\sum_{i=1}^{N_{\text{c}}}\alpha_{i}(-1000/ \ln r_{i})^{\beta_{i}}(n_{i}+\exp(n_{i}/4))\leq U_{c} \tag{4}\]  
\[Cst_{2}(\textbf{R},\textbf{ }\textbf{N})=\sum_{i=1}^{N_{\text{c}}}w_{i}n_{i} \exp(n_{i}/4)\leq U_{w} \tag{5}\]  
\[Cst_{3}(\textbf{R},\textbf{ }\textbf{N})=\sum_{i=1}^{N_{\text{c}}}w_{i}w_{i}^{2}n_{i} ^{2}\leq U_{v} \tag{6}\]  
\[0\leq r_{i}\leq 1\]  
The objective function to maximize the system reliability for the series system in Figure 1 is shown in above Equation (3). The constraints of cost, weight and volume are shown in above Equations (4)-(6), respectively. Benchmark 2. Network with series and parallel elements in Figure 2[24; 34; 37]. \[\text{Maximize }R_{\text{s}}(\textbf{R},\textbf{ }\textbf{N})=1-(1-R_{1}R_{2})\{1-[1-(1-R_{3})(1-R_{4})]R_{5}\} \tag{7}\]  
\[\text{Subject to }Cst_{1}(\textbf{R},\textbf{ }\textbf{N})=\sum_{i=1}^{N_{\text{c}}}\alpha_{i}(-1000/\ln r_{i})^{\beta_{i} }(n_{i}+\exp(n_{i}/4))\leq U_{c} \tag{8}\]  
\[Cst_{2}(\textbf{R},\textbf{ }\textbf{N})=\sum_{i=1}^{N_{\text{c}}}w_{i}n_{i} \exp(n_{i}/4)\leq U_{w} \tag{9}\]  
\[\text{Subject to }Cst_{j}(\textbf{R},\textbf{N})\leq U_{j} \tag{2}\]0  
\[\text{Subject to }Cst_{j}(\textbf{R},\textbf{N})\leq U_{j} \tag{2}\]1The objective function to maximize the system reliability for the network with series and parallel elements in Figure 2 is shown in above Equation (7). The constraints of cost, weight and volume are shown in above Equations (8)-(10), respectively. Benchmark 3. Complex (bridge) system in Figure 3[24; 34; 37]. \[\text{Maximize }R_{\text{s}}(\textbf{{R}},\textbf{{N}})= \begin{array}{l}R_{1}R_{2}+R_{3} R_{4}+R_{1}R_{4}R_{5}+R_{2}R_{3}R_{5}-\\ R_{1}R_{2}R_{3}R_{4}-R_{1}R_{2}R_{3}R_{5}-R_{1}R_{2}R_{4}R_{5}-\\ R_{1}R_{3}R_{4}R_{5}-R_{2}R_{3}R_{4}R_{5}+2R_{1}R_{2}R_{3}R_{4}R_{5}\end{array}  \tag{11}\]  
\[\text{Subject to }Cst_{1}(\textbf{{R}},\textbf{{N}})=\sum_{i=1}^{N_{v}}\alpha_{i }(-1000/\ln r_{i})^{\beta_{i}}(n_{i}+\exp(n_{i}/4))\leq U_{c} \tag{12}\]  
\[Cst_{2}(\textbf{{R}},\textbf{{N}})=\sum_{i=1}^{N_{v}}w_{i}n_{i}\exp(n_{i}/4) \leq U_{w} \tag{13}\]  
\[Cst_{3}(\textbf{{R}},\textbf{{N}})=\sum_{i=1}^{N_{v}}w_{i}v_{i}^{2}n_{i}^{2} \leq U_{v} \tag{14}\]  
\[0\leq r_{i}\leq 1\]  
The objective function to maximize the system reliability for the complex (bridge) system in Figure 3 is shown in above Equation (11). The constraints of cost, weight and volume are shown in above Equations (12)-(14), respectively. Benchmark 4. The overspeed protection of a gas turbine system in Figure 4[24; 34; 37].