## 5 Conclusions  
This paper has proposed a novel DILA to solve the problem of single-machine scheduling with sequence-dependent family setup times while minimizing the total tardiness of jobs. The DILA first generates two initial populations: the mainstream market population and the emerging market population. Then, three phases within a generation are used to evolve the quality of solutions in the two populations. Finally, the DILA allows the population to evolve for several generations until the maximum execution time is met. The MAF and MRF for the two-stage type (stage 1 = (0.50, 0.20)) and stage 2 = (0.80, 0.20)) and the phase termination time of 2 s were incorporated in the DILA to help achieve better performance. Computational experiments on 1440 benchmark instances were used to compare the results produced by the proposed DILA against those of several metaheuristics. The results show that the proposed DILA is superior to the HGA, ILS_BASIC, ILS_DP, and ILS_DP + PR heuristics. Furthermore, the DILA found the best solution in 1435 out of 1440 instances. We also analyzed the performance of the DILA across a wide variety of values for the following parameters: number of jobs (\(n\)), number of families (\(F\)), due-date range (\(r\)), and setup-time range (\(\eta\)). The DILA performs better than the other heuristics regardless of every level of the four parameters. The DILA can find all best solutions when the number of jobs is 60, the numbers of families are 2 and 5, and the setup-time range is [11; 20]. Future research may involve applying the proposed heuristic to other machine scheduling problems, for example, problems with parallel machines, flow shops, and hybrid flow shops. Furthermore, a self-adaptive MAF and MRF strategy may be the future direction for research on the DILA. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**(n,F,\(\pi\))** & **HGA** & **ILS_BASIC** & **ILS_DP** & **ILS_DP + PR** & **DILA** \\ \hline (*,0.5,*) & 48.29 & 7.71 & 4.95 & 2.38 & 0.56 \\ (*,1.5,*) & 23.82 & 1.98 & 1.99 & 1.17 & 0.28 \\ (*,2.5,*) & 21.62 & 3.30 & 3.12 & 1.96 & 0.28 \\ (*,3.5,*) & 1.94 & 2.35 & 1.39 & 1.68 & 0.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Average RDIs for instances grouped by due date ranges (\(r\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**(n,F,\(\pi\))** & **HGA** & **ILS_BASIC** & **ILS_DP** & **ILS_DP + PR** & **DILA** \\ \hline (*,*,[11; 20]) & 13.09 & 4.72 & 3.37 & 1.71 & 0.00 \\ (*,*,[51; 100]) & 18.08 & 2.23 & 1.78 & 0.94 & 0.63 \\ (*,*,[101; 200]) & 22.72 & 3.07 & 1.94 & 1.86 & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Average RDIs for instances grouped by setup time ranges (\(\eta\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**(n,F,\(\pi\))** & **HGA** & **ILS_BASIC** & **ILS_DP** & **ILS_DP + PR** & **DILA** \\ \hline (*,*,[11; 20]) & 13.09 & 4.72 & 3.37 & 1.71 & 0.00 \\ (*,*,[51; 100]) & 18.08 & 2.23 & 1.78 & 0.94 & 0.63 \\ (*,*,[101; 200]) & 22.72 & 3.07 & 1.94 & 1.86 & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Average RDIs for instances grouped by setup time ranges (\(\eta\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**(n,F,\(\pi\))** & **HGA** & **ILS_BASIC** & **ILS_DP** & **ILS_DP + PR** & **DILA** \\ \hline (*,*,[11; 20]) & 13.09 & 4.72 & 3.37 & 1.71 & 0.00 \\ (*,*,[51; 100]) & 18.08 & 2.23 & 1.78 & 0.94 & 0.63 \\ (*,*,[101; 200]) & 22.72 & 3.07 & 1.94 & 1.86 & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Average RDIs for instances grouped by setup time ranges (\(\eta\)). **Funding:** This work was supported by the Ministry of Science and Technology, Taiwan, ROC, under the contract MOST 106-2221-E-147-001. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available on request from the corresponding author.