## 4 The DABC Algorithm for an EAHFSP
### Scout Bee Phase
Tables 6 and 7 provide the computational results of five algorithms for metric \(IGD\) and \(c\), in which \"A\", \"B\", \"C\", \"D\", and \"E\" represent \"DABC\", \"ABC\", \"TICA\", \"HPSO\", and \"CCA\". Figure 6 shows the distribution curves for the optimal solutions of all algorithms for six instances, including \(20\times 8\), \(30\times 2\), \(50\times 4\), \(60\times 6\), \(70\times 2\), and \(120\times 2\). Box plots are provided to further analyze the stability of the DABC algorithm. Figure 7 shows the box plots for five algorithms corresponding to \(IGD\) and \(c\). As stated in Tables 6 and 7, \(IGD\) of the DABC algorithm is less than that of the ABC algorithm in 44 instances, and \(c(A,B)\) exceeds \(c(B,A)\) in 44 instances. Moreover, \(c(A,B)\) is equal to 1 in 37 instances, that is, all non-dominated solutions of the ABC are dominated by those of the DABC in 37 instances. Thus, the DABC algorithm provides better results than the ABC in most instances. Similarly, the DABC generates a smaller \(IGD\) than CCA in all instances, and it also gains a smaller \(c(C,A)\) than \(c(A,C)\) in all instances. Moreover, \(c(A,C)=1\) is gained in 27 instances, that is, all solutions of CCA are dominated by the non-dominated ones of the DABC. On the other hand, the DABC obtains a smaller \(IGD\) than TICA and HPSO in 37 and 35 instances; \(c(A,D)\) is larger than \(c(D,A)\) in 34 instances; and \(c(A,E)\) is bigger than \(c(E,A)\) in 31 instances. Therefore, it is concluded that the DABC algorithm performs notably better than the ABC algorithm, TICA, HPSO, and CCA in over 70% of the instances corresponding to two metrics. This conclusion can also be drawn from the distribution curves for the five algorithms in Figure 6.

According to the box plots in Figure 7, the lower quartile (Q\({}_{1}\)), the median (Q\({}_{2}\)), the upper quartile (Q\({}_{3}\)), and the quartile difference (IQR) between Q\({}_{1}\) and Q\({}_{3}\) of the DABC for metric \(IGD\) are smaller than those of the four compared algorithms. Similarly, the DABC algorithm produces a greater value for Q\({}_{1}\), Q\({}_{2}\), Q\({}_{3}\), and IQR for metric \(c\). These comparisons and analyses demonstrate that the stability of the DABC algorithm is superior to the ABC algorithm, TICA, HPSO, and CCA. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Instance** & **DABC** & **ABC** & **TICA** & **HPSO** & **CCA** & **Instance** & **DABC** & **ABC** & **TICA** & **HPSO** & **CCA** \\ \hline \(20\times 2\) & **0.058** & 0.131 & 0.059 & 0.132 & 0.194 & \(70\times 6\) & 0.163 & 0.387 & 0.176 & **0.135** & 0.376 \\ \(20\times 4\) & **0.033** & 0.486 & 0.280 & 0.522 & 0.594 & \(70\times 8\) & **0.102** & 0.5





To analyze the statistical difference in the experimental results, a Wilcoxon test is conducted. The significant level is \(\alpha=0.05\), that is, if the test value is \(p<0.05\), then a significant difference exists between the comparative algorithms. The Wilcoxon test

Figure 6: Distribution curves for five algorithms and six instances: (**a**) \(20\times 8\); (**b**) \(30\times 2\); (**c**) \(50\times 4\); (**d**) \(60\times 6\); (**e**) \(70\times 2\); and (**f**) \(120\times 2\). Figure 7: Box plots for the five algorithms and two metrics: (**a**) box plots for \(IGD\); (**b**) box plots for \(c\). results are shown in Table 8, where \"R+\" indicates that the DABC algorithm is better than the comparative algorithm, \"R-\" indicates that the DABC is worse than the comparative algorithm, and \"Yes\" means that there is a significant difference between the DABC and the other algorithms. From Table 8, it is concluded that the DABC algorithm has more significant advantages than other algorithms. To further verify the significant difference among all algorithms, [PERSON]'s test is applied. The mean ranks generated using [PERSON]'s test are shown in Table 9. It is found that the mean rank of the DABC algorithm is the smallest among all experimental algorithms.