## 5 Experiment
### Results and Discussion  
In this section, we first compare the new rule method, mixed method, and traditional random method under different iteration times with the same initial population size. Then, under the same number of iterations and different initial population sizes, the performance of the three methods are compared. Finally, compared with [PERSON]'s rule-based initial population [30], the effectiveness of the ADSM is proved. To analyze the values of the two objective functions and the average distribution from the optimal solution, the result of different iterations of 100 initial population is shown in Table 9, where F1 and F2 represent the values of makespan and total tardiness, respectively, and the bold contents are better values. The rules method represents the new approach for population knowledge mining applied to the hybrid NSGA-II to solve MOJSSP. The traditional random method is the NSGA-II combined with SA for MOJSSP. The mixed method is a combination of the two methods. Figure 5 shows the box distribution of its function values of which the values in the rule method and the mixed method are overall smaller than the traditional method. To some extent, the smaller are the values, the better is the performance. Similarly, the results of different iterations of 50 and 25 initial population are shown in Tables 10 and 11. Figures 6 and 7 are the box diagrams of the respective function values. From the results, we can find that the population mining method has more excellent results than the traditional random method because the rule population contains more excellent information than the traditional random population. Although the new method has no absolute advantage in the test of 25 initial population, it should be explained that, when the initial population size is too small, the performance of the method will inevitably be reduced. In general, there is no obvious advantage or disadvantage between rule method and mixed method in terms of function values, but, in term of the index RE, the rule method is superior to the mixed method, and both are superior to the random method. That is, the more rule-based individuals there are in the population, the more excellent information they contain, and thus it is easier to obtain excellent solutions under a limited number of iterations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **Iteration** & & **F1** & **F2** & **RE** \\ \hline \multirow{3}{*}{rules} & \multirow{3}{*}{100} & best & 852 & \(-\)4568 & **5.94\(|\)3.27** \\  & & average & 902.6 & \(-\)4418.4 & / \\ \cline{2-5}  & \multirow{3}{*}{300} & best & **848** & \(-\)4835 & 8.54\(|\)4.87 \\  & & average & 920.4 & \(-\)**4599.3** & / \\ \hline \multirow{3}{*}{mixed} & \multirow{3}{*}{100} & best & **848** & \(-\)4618 & 6.03\(|\)3.52 \\  & & average & **899.1** & \(-\)**4455.6** & / \\ \cline{2-5}  & \multirow{3}{*}{300} & best & 848 & \(-\)**4868** & 10.74\(|\)6.04 \\  & & average & 939.1 & \(-\)4573.8 & / \\ \hline \multirow{3}{*}{random} & \multirow{3}{*}{100} & best & 854 & \(-\)**4630** & 6.77\(|\)4.97 \\  & & average & 911.8 & \(-\)4400 & / \\ \cline{1-1} \cline{2-5}  & \multirow{3}{*}{300} & best & 848 & \(-\)4815 & **7.88\(|\)5.12** \\  & & average & **914.8** & \(-\)4568.5 & / \\ \hline \end{tabular}
\end{table} 
Table 10: Values of makespan, total tardiness, and RE of 50 initial population. Figure 6: Distribution of makespan and total tardiness of 50 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. Since this paper optimizes two objectives at the same time, the values of makespan or total tardiness cannot be separately compared. Thus, the efficiency of the method was determined by the performance metrics of the acquired Pareto solutions, especially the dominance metric (Cov).