## 5 Experiment
### Results and Discussion
Here, the performance metrics of the three methods are compared under different initial population sizes with the same number of iterations, as shown in Tables 12-14, where IP represents the size of the initial population and the bold contents are the better values. In Table 12, the Cov value of the rule method \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **Iteration** & & **F1** & **F2** & **RE** \\ \hline \multirow{4}{*}{rules} & \multirow{2}{*}{100} & best & **852** & \(\mathbf{-4693}\) & 8.496.58 \\  & & average & 924.3 & \(\mathbf{-4384.3}\) & / \\ \cline{2-5}  & & \multirow{2}{*}{300} & best & **848** & \(\mathbf{-4731}\) & **5.834.34** \\  & & average & **897.4** & \(\mathbf{-4525.9}\) & / \\ \hline \multirow{4}{*}{mixed} & \multirow{2}{*}{100} & best & 855 & \(\mathbf{-4593}\) & **5.38**\(\mathbf{5.11}\) \\  & & average & **901** & \(\mathbf{-4358.2}\) & / \\ \cline{2-5}  & \multirow{2}{*}{300} & best & 848 & \(\mathbf{-4810}\) & 7.635.59 \\  & & average & 912.7 & \(\mathbf{-4540.9}\) & / \\ \hline \multirow{4}{*}{random} & \multirow{2}{*}{100} & best & 853 & \(\mathbf{-4529}\) & 6.584.22 \\  & & average & 909.1 & \(\mathbf{-4337.8}\) & / \\ \cline{2-5}  & \multirow{2}{*}{300} & best & 848 & \(\mathbf{-4824}\) & 7.285.94 \\  & & average & 909.7 & \(\mathbf{-4537.4}\) & / \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Values of makespan, total tardiness, and RE of 25 initial population. Figure 7: Distribution of makespan and total tardiness of 25 initial population sizes under different iteration numbers: (**a**) makespan under 100 iterations; (**b**) total tardiness under 100 iterations; (**c**) makespan under 300 iterations; and (**d**) total tardiness under 300 iterations. is generally higher than that of the traditional method. The larger is the Cov value, the higher is the dominant ability. That is, in terms of dominance, the rule method is superior to the traditional random method in comparing the size of different initial populations. At the same time, the Spacing values of the new method is generally smaller than that of the traditional method. From Equation (9), it can be seen that the distribution of the rule method is also superior to the random method in considering the initial population size as a variable. Figure 8 shows the box diagram of the Cov of the rule method vs. the random method. From the box diagram, we can find more intuitively that the Pareto solution of the rule method is superior to that of the traditional random method. Similarly, from the results in Tables 13 and 14, we can conclude that the performance of rule method in Cov is also \"rules \(>\) mixed \(>\) random\". For Spacing performance, the mixed method is better than the rule method, which shows that the mixed method can more easily obtain a uniformly distributed solution set with a limited number of iterations. Overall, the Spacing metrics of both the rule and mixed methods are superior to the random method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multirow{2}{*}{**IP**} & \multicolumn{3}{c}{**IT100**} & \multicolumn{3}{c}{**IT300**} \\ \cline{3-6}  & & & **Cov** & **Spacing** & **Cov** & **Spacing** \\ \hline \multirow{4}{*}{mixed} & \multirow{2}{*}{25} & best & **1** & **19** & **1** & **48** \\  & & average & **0.63** & **24** & 0.86 & **37** \\ \cline{2-6}  & & average & **0.45** & **16** & 0.92 & **0.45** & 20 \\ \hline \multirow{4}{*}{random} & \multirow{2}{*}{25} & best & 0.27 & \multirow{2}{*}{76} & 1 & \multirow{2}{*}{50.5} \\  & & average & 0.11 & & **0.43** & & \\ \cline{1-1} \cline{4-6}  & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Values of Cov and Spacing (mixed vs. random).