## 3 Fuzzy Constraints Models of RRAP
### Fuzzy Constraints Models of RRAP  
[PERSON] in 1965 first developed the fuzzy number [28], which has been shown to be effective for solving the uncertainty condition. The triangular fuzzy number, which has been defined as the following Definition 1, is one of the famous fuzzy number that has been adopted in this study. **Definition 1**. : _The triangular fuzzy number \(\underline{A}=(a_{1},\,a_{2},\,a_{3})\), where \(a_{1}<a_{2}<a_{3}\) and \(a_{i}\) belongs to the real numbers for \(i=1,\,2,\,3\). The relationship of the membership function \(u_{\underline{A}}(a)\) and the triangular fuzzy number \(\underline{A}=(a_{1},\,a_{2},\,a_{3})\) has been shown in Equation (19)._  
\[u_{\underline{A}}(a)=\left\{ \begin{array}{ll}0,&if\,\,a<a_{1}\\ (a-a_{1})/(a_{2}-a_{1}),&if\,\,a_{1}\leq a<a_{2}\\ 1,&if\,\,a=a_{2}\\ (a_{3}-a)/(a_{3}-a_{2}),&if\,\,a_{2}<a<a_{3}\\ 0,&if\,\,a_{3}<a\end{array} \right. \tag{19}\]  
Accordingly, this work adopts the triangular fuzzy number for the uncertainties of systems components in the constraints of RRAP including cost, weight, and volume, which are defined as following the triangular fuzzy number for the cost, weight, and volume:  
1. The triangular fuzzy number for the cost \(\underline{c}=(c_{1},\,c_{2},\,c_{3})\), where \(c_{1}<c_{2}<c_{3}\) and \(c_{i}\) belongs to the real numbers for \(i=1,\,2,\,3\). 2. The triangular fuzzy number for the weight \(\underline{w}=(w_{1},\,w_{2},\,w_{3})\), where \(w_{1}<w_{2}<w_{3}\) and \(w_{i}\) belongs to the real numbers for \(i=1,\,2,\,3\). 3. The triangular fuzzy number for the volume \(\underline{v}=(v_{1},\,v_{2},\,v_{3})\), where \(v_{1}<v_{2}<v_{3}\) and \(v_{i}\) belongs to the real numbers for \(i=1,\,2,\,3\). Therefore, the constraints of 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 shown in Equations (5) and (6), Equations (9) and (10), Equations (13) and (14), and Equations (17) and (18) are accordingly modelled to the triangular fuzzy number for the constraints of RRAP as following Equations (20)-(27). However, the components cost is not used within the cost constraints so that there is no need to convert Equations (4), (8), (12), and (16) to the fuzzy constraints. Benchmark 1. Series system in Figure 1.  
\[\text{Subject to }Cst_{2}(\textbf{R},\,\textbf{N})=\sum_{i=1}^{N_{\text{c}}} \underline{w}_{i}n_{i}\exp(n_{i}/4)\leq U_{w} \tag{20}\]  
\[Cst_{3}(\textbf{R},\,\textbf{N})=\sum_{i=1}^{N_{\text{c}}}\underline{w}_{i} \underline{v}^{2}n_{i}^{2}\leq U_{v} \tag{21}\]  
The triangular model of RRAP fuzzy weight and volume constraints for the series system in Figure 1 are shown in above Equations (20) and (21), where \(\underline{w}=(w_{1},\,w_{2},\,w_{3})\) and \(\underline{v}=(v_{1},\,v_{2},\,v_{3})\) are the triangular fuzzy number for weight and volume, respectively. Benchmark 2. Network with series and parallel elements in Figure 2.  
\[\text{Subject to }Cst_{2}(\textbf{R},\,\textbf{N})=\sum_{i=1}^{N_{\text{c}}} \underline{w}_{i}n_{i}\exp(n_{i}/4)\leq U_{w} \tag{22}\]  
\[Cst_{3}(\textbf{R},\,\textbf{N})=\sum_{i=1}^{N_{\text{c}}}\underline{w}_{i} \underline{v}^{2}n_{i}^{2}\leq U_{v} \tag{23}\]  
The triangular model of RRAP fuzzy weight and volume constraints for the network with series and parallel elements in Figure 2 are shown in above Equations (22) and (23), where \(\underline{w}=(w_{1},\,w_{2},\,w_{3})\) and \(\underline{v}=(v_{1},\,v_{2},\,v_{3})\) are the triangular fuzzy number for weight and volume, respectively. Benchmark 3.