## 6 Discussion and Results
### Evolution of Proposed HMFO with Practical Instances
The percentage of ND solutions provided by a certain algorithm is expressed in terms of dominance ratio \(\Upsilon\Upsilon\) in Equation (15). If larger, the dominance ratio indicates the superiority of the given algorithm. Table 9.9  
where \(|B(\underset{i}{\cup}M_{i})\backslash B(\underset{i\
eq k}{\cup}M_{i})|\) indicates the non-dominated solutions found by the algorithm \(M\) that are unable to identify by the other standard algorithms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Indicator**} & \multirow{2}{*}{**Algorithm**} & \multicolumn{6}{c}{**Instance**} \\ \cline{3-8}  & & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{2}{*}{\(\alpha\)} & HMFEO & **8.5** & **10.8** & **9.7** & **10.6** & **8.8** \\  & NSGA II & 8.0 & 10.5 & 9.4 & 10.6 & 8.6 \\ \hline \multirow{2}{*}{\(\beta\)} & HMFEO & **8.1** & **10.7** & **9.5** & **10.6** & **8.6** \\  & NSGA II & 7.0 & 9.5 & 9.3 & 9.3 & 8.3 \\ \hline \multirow{2}{*}{\(\beta^{\text{/a}}\)} & HMFEO & **0.9529** & **0.9900** & **0.9700** & **1.0000** & **0.9700** \\  & NSGA II & 0.8750 & 0.9047 & 0.9893 & 0.8773 & 0.9651 \\ \hline \multirow{2}{*}{\(\Upsilon\)} & HMFEO & **0.5264** & **0.5079** & **0.6012** & **0.5000** & **0.5264** \\  & NSGA II & 0.4736 & 0.4921 & 0.3988 & 0.5000 & 0.4736 \\ \hline \multirow{2}{*}{K} & HMFEO & **0.0800** & **0.0100** & **0.0300** & **0.0000** & **0.0300** \\  & NSGA II & 0.1250 & 0.0953 & 0.0107 & 0.1227 & 0.0349 \\ \hline \multirow{2}{*}{\(\lambda\)} & HMFEO & **0.3693** & **0.0037** & **0.6289** & **0.0000** & **0.0042** \\  & NSGA II & 12.123 & 9.236 & 8.265 & 8.6321 & 9.5632 \\ \hline \multirow{2}{*}{\(\pi\)} & HMFEO & 0.4236 & **0.4856** & **0.4982** & **0.4932** & **0.4495** \\  & NSGA II & **0.4125** & 0.5563 & 0.6029 & 0.6765 & 0.5988 \\ \hline \multirow{2}{*}{HR} & HMFEO & **0.8526** & **0.7445** & 0.9538 & **0.6984** & **0.4495** \\  & NSGA II & 1.2536 & 1.1456 & **0.9469** & 0.9548 & 0.9854 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of performance indicators for comparison of HMFEO and NSGA II instance 1–5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Indicator**} & \multirow{2}{*}{**Algorithm**} & \multicolumn{6}{c}{**Instance**} \\ \cline{3-8}  & & **6** & **7** & **8** & **9** & **10** \\ \hline \multirow{2}{*}{\(\alpha\)} & HMFEO & **12.5** & **13.8** & **9.7** & **16.6** & **9.8** \\  & NSGA II & 12.0 & 10.5 & 9.4 & 16.6 & 8.8 \\ \hline \multirow{2}{*}{\(\beta\)} & HMFEO & **12.1** & **12.7** & **9.5** & **16.6** & **9.6** \\  & NSGA II & 11.0 & 9.2 & 9.3 & 15.0 & 8.3 \\ \hline \multirow{2}{*}{\(\beta^{\text{/a}}\)} & HMFEO & **0.9682** & **0.9202** & **0.9700** & **1.0000** & **0.9795** \\  & NSGA II & 0.9166 & 0.8761 & 0.9893 & 0.9036 & 0.9431 \\ \hline \multirow{2}{*}{\(\Omega\)} & HMFEO & **0.5151** & **0.5070** & **0.5078** & **0.5000** & **0.5057** \\  & NSGA II & 0.4848 & 0.4929 & 0.4922 & 0.5000 & 0.4942 \\ \hline \multirow{2}{*}{K} & HMFEO & **0.0800** & **0.0104** & **0.0310** & **0.0000** & **0.0400** \\  & NSGA II & 0.1250 & 0.0835 & 0.0307 & 0.1027 & 0.0349 \\ \hline \multirow{2}{*}{\(\lambda\)} & HMFEO & **0.4693** & **0.0089** & **0.5259** & **0.0000** & **0.0063** \\  & NSGA II & 12.123 & 9.236 & 8.265 & 8.6321 & 10.5632 \\ \hline \multirow{2}{*}{\(\pi\)} & HMFEO & 0.5136 & 0.4856 & **0.4982** & **0.4932** & **0.4445** \\  & NSGA II & **0.4125** & **0.4563** & 0.5129 & 0.5365 & 0.5488 \\ \hline \multirow{2}{*}{HR} & HMFEO & **0.7415** & **0.8556** & **0.8629** & **0.7875** & **0.7589** \\  & NSGA II & 0.9438 & 1.2465 & 1.1803 & 1.9888 & 0.8765 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Results of performance indicators for comparison of HMFEO and NSGA II instance 6–10. \(K(a,b)\) in Equation (16) is useful for comparison of Pareto fronts, particularly to identify the weak solutions of algorithm b with an algorithm \((a>b)\), which ultimately gives the correctness of the particular algorithm.