## 4 Experimental Study
### Results Analysis and Discussion
#### 4.4.1 Analysis of the Comparative Study (1) for FDA and dMOP Problems
Table S1 (see the Supplementary File) presents the mean and standard deviation values through the MIGD metric; the efficiency of the new DB-CSA-II system shows the best mean and standard deviation values for all test suites with different environmental changes compared to the six transfer learning-based approaches and the standard CSA algorithm. Based on the statistical results obtained using the Wilcoxon signed rank test on the following Table 8, we can determine the importance of the new DB-CSA-II with a \(p\)-value of less than 0.05, defining a significant difference in the mean values compared to MMTL-MOEA/D, KF-MOEA/D, PPS-MOEA/D, SVR-MOEA/D, Tr-MOEA/D, and RI-MOEA/D approaches. The one-way ANOVA box-plots in Figure 5 determines the importance of the DB-CSA-II algorithm compared to six transfer-learning-based methods. Compared with the DB-CSA-II algorithm, which aims to control the evolution and deterioration of optimal solutions, most TL-based approaches were designed to predict the new population after each dynamic change, but tend to be inefficient when managing dynamic change in both POS  
Second, the five standard MOEAs (DNSGA-II, dCOEA, PPS, MOEA/D, and SGEA) were compared to the new DB-CSA-II algorithm. The average and the standard deviation values for both FDA and dMOP test suites over the IGD and HVD metrics, respectively, can be seen in Tables S2 and S3 of the Supplementary File. Based on the IGD metric on Table S2, we can argue the superiority of DB-CSA-II method compared to five standard MOEAs designed for dynamic multi-objective optimization. The results based on Wilcoxon signed rank test are presented in Table 9, indicating that DB-CSA-II is the best method when compared to IGD at a statistically significant level of 0.05 compared to other MOEAs. The same conclusion is confirmed using the box plot over a one-way ANOVA test in Figure 6.  
Figure 5: One-way ANOVA Box-plots of 7 MOEAs over MIGD of FDA, dMOP for (**a**) severe with (\(\tau_{t}=5\), \(n_{t}=10\)), (**b**) moderate (\(\tau_{t}=10\), \(n_{t}=10\)), and (**c**) slight with (\(\tau_{t}=20\), \(n_{t}=10\)) environmental changes respectively. Figure 6: One-way ANOVA Box-plots of 6 MOEAs over IGD of FDA, dMOP for (**a**) severe with (\(\tau_{t}\) = 5, \(n_{t}\) = 10), (**b**) moderate (\(\tau_{t}\) = 10, \(n_{t}\) = 10), and (**c**) slight (\(\tau_{t}\) = 20, \(n_{t}\) = 10) environmental changes, respectively. Figure 7: One-way ANOVA Box-plots of 6 MOEAs over HVD of FDA, dMOP for (**a**) severe with (\(\tau_{t}=5\), \(n_{t}=10\)), (**b**) moderate (\(\tau_{t}=10\), \(n_{t}=10\)), and (**c**) slight (\(\tau_{t}=20\), \(n_{t}=10\)) environmental changes, respectively. Table S3 reports the quantitative results using an HVD quality indicator. We can conclude that the proposed DB-CSA is the winner when solving different types of DMOPs, including FDA1 in type I, with dynamic POS and static POP, FDA3, FDA5, and dMOP2 in type II, with time-varying POS and POP and dMOP1, and in type III, with unchangeable POS and dynamic POP, regarding all categories of environmental changes. Meanwhile the DB-CSA-II obtained similar results to the SGEA system when solving FDA2 function in type II, characterized by a dynamic density of the solutions and a cyclic change of the POP from convex to concave as well as the FDA4 problem in type I characterized by a time-varying spread of solutions in a severe dynamic change. In addition, the dCOEA algorithm has a closed mean value for solving the dMOP3 function, characterized by the static curvature of the estimated OP and dynamic spread of the solution set compared to the proposed DB-CSA-II. Table 9 shown the negative and positive Wilcoxon ranks, we can conclude that the DB-CSA-II algorithm is the best method compared to other comparable approaches based on the HVD quality indicator. This importance does not determine statistically significance with a \(p\)-value greater than 0.05.