## 4 SD-EAR in Detail
### Optimum Route Selection
#### 4.2.3 Example of Optimum Path Selection
Hence, opt-performance = VH  
Figure 10: Topology of zones \(Z_{A}\) and \(Z_{B}\) along with neighbors. Performance of R\({}_{2}\)  
\[\text{reft}(\text{R}_{2})=[\{1-1/(25+2)\}\,\{1-3/(3+25+20+2)\}]^{0.5}=0.9509\]  
\[\text{trim}(\text{R}_{2})=[\{1-(1+1)/(3\times 4)\}/\max\{8^{0.5},\,0.5^{0.5},1 \}]^{0.5}=0.5426\]  
\[\text{neff}(\text{R}_{2})=[\{1-0/(3)\}\,(1-1/(0+2))]^{0.5}=0.7071\]  
After fuzzification, fuzzy premise variables corresponding t reft(R\({}_{2}\)), trim(R\({}_{2}\)), andneff(R\({}_{2}\)) become VH, H, and H. Their combination, according to Tables 1 and 2, is VH. Hence, opt-performance = H  
Performance of R\({}_{3}\)  
\[\text{reft}(\text{R}_{3})=[\{1-1/(20+2)\}\,\{1-4/(3+20+25+20+2)\}]^{0.5}\]  
\[\text{i.e., reft(R}_{3})=0.94845\]  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]0  
n\({}_{\text{f}}\) is sleeping. We need to search for alternatives of n\({}_{\text{f}}\). Only n\({}_{\text{c}}\) has the same set of neighbors as n\({}_{\text{f}}\). However, n\({}_{\text{c}}\) belongs to this route R\({}_{3}\) itself. So, n\({}_{\text{c}}\) cannot be an alternative to n\({}_{\text{f}}\). So, n\({}_{\text{f}}\) has no alternative:  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]1  
After fuzzification, the fuzzy premise variables corresponding t reft(R\({}_{3}\)), trim(R\({}_{3}\)), and neff(R\({}_{3}\)) become VH, VH, and H. Their combination, according to Tables 1 and 2, is VH. Hence, opt-performance = VH  
Performance of R\({}_{4}\)  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]2  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]3  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]4  
n\({}_{\text{f}}\) is sleeping. We need to search for alternatives of n\({}_{\text{f}}\). Only n\({}_{\text{c}}\) has same set of neighbors as n\({}_{\text{f}}\). However, n\({}_{\text{c}}\) belongs to the route R\({}_{3}\) itself. So, n\({}_{\text{c}}\) cannot be an alternative to n\({}_{\text{f}}\). So, n\({}_{\text{f}}\) has no alternative. \[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]5  
After fuzzification, the fuzzy premise variables corresponding t reft(R\({}_{4}\)), trim(R\({}_{4}\)), and neff(R\({}_{4}\)) become VH, H, and H. Their combination, according to Tables 1 and 2, is H.  
Hence, opt-performance = H  
Performance of R\({}_{5}\)  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]6  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]7  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]8  
\[\text{reft}(R_{1})=[\{1-1/(20+2)\}\,\{1-3/(3+20+20+2)\}]^{0.5}=0.9438\]9In this path, only \(\rm n_{g}\) is sleeping. We need to search for alternatives of \(\rm n_{g}\). However, no node in zone \(\rm Z_{A}\) has the same set of neighbors as \(\rm n_{g}\). So, \(\rm n_{g}\) has no alternative. \[\text{trim}(R_{1})=[\{1-(2+1)/(3\times 4)\}/\max\{2^{0.5},\,0.5^{0.5}\}]^{0.5}=0.7 282\]0  
After fuzzification, the fuzzy premise variables corresponding t reft(R\({}_{S}\)), trim(R\({}_{S}\)), and eff(R\({}_{S}\)) become VH, L, and H, respectively. Their combination, according to Tables 1 and 2, is L.  
Hence, opt-performance = L  
\(\rm R_{1}\) and \(\rm R_{4}\) both have performance VH. Therefore, we compute the weight of these paths. \[\text{trim}(R_{1})=[\{1-(2+1)/(3\times 4)\}/\max\{2^{0.5},\,0.5^{0.5}\}]^{0.5}=0.7 282\]1  
\[\text{trim}(R_{1})=[\{1-(2+1)/(3\times 4)\}/\max\{2^{0.5},\,0.5^{0.5}\}]^{0.5}=0.7 282\]2  
Therefore, \(\rm R_{1}\) is chosen as optimal.