## 4 Computational Experiments
### The Second Set of Experiments  
The second set of experiments investigated the performance of the DILA using the best parameter values determined in the first set of experiments. We compared the performance of the proposed DILA with four heuristics [9]: HGA, ILS_BASIC, ILS_DP, and ILS_DP + PR. The results of the four heuristics are provided by [PERSON] and [PERSON] [9]. For a fair comparison of these algorithms, 30 runs of the DILA were intentionally performed with a time limit of \(n^{\star}0.5\) s for 1440 test instances, which is in the same stop condition as in [PERSON] and [PERSON] [9]. Table 5 lists the RDI and NBS for different job and family sizes obtained by the DILA and the four heuristics. There are 120 instances per \(n\times F\) group. The results in Table 5 show that the DILA was able to find the best solutions for 1435 out of 1440 instances (99.7%). The average RDIs show that the DILA dominated HGA by 98.3% ((17.96 \(-\) 0.3)/17.96), dominated ILS_BASIC by 91.0% ((3.34 \(-\) 0.3)/3.34), dominated ILS_DP by 87.3% ((2.37 \(-\) 0.3)/2.37), and dominated ILS_DP + PR by 80.0% ((1.50 \(-\) 0.3)/1.50). Table 6 presents the results of the paired \(t\)-tests for HGA, ILS_BASIC, ILS_DP, ILS_DP + PR, and the DILA. The results show that the DILA dominates HGA, ILS_BASIC, ILS_DP, and ILS_DP + PR at a significance level of 0.05. A comparison with the reference heuristics is summarized in Table 7, where the number of solutions better than, equal to, and worse than those obtained by the DILA are presented. We observe that the DILA outperformed HGA, ILS_BASIC, ILS_DP, and ILS_DP + PR. Compared with HGA, ILS_BASIC, ILS_DP, and ILS_DP + PR, the proposed DILA produced 258, 113, 105, and 68 better solutions, respectively, for the 1440 instances, whereas the number of worse solutions did not exceed 4. \begin{table}
\begin{tabular}{c c c c} \hline
**Controlled Probability \((\boldsymbol{p_{a^{\prime}}}\), \(\boldsymbol{p_{r}})\)** & **Results (Groups)** & **Average RDI** \\ \hline stage 1 = (0.50, 0.20), stage 2 = (0.80, 0.20) & A & 0.2098 \\ (0.20, 0.80) & A & 0.2179 \\ (0.20, 0.60) & A & 0.2183 \\ stage 1 = (0.50, 0.20), stage 2 = (0.20, 0.20) & A & 0.2190 \\ (0.50, 0.20) & A & 0.2213 \\ stage 1 = (0.20, 0.20), stage 2 = (0.50, 0.20) & A & 0.2232 \\ stage 1 = (0.80, 0.20), stage 2 = (0.20, 0.20) & A & 0.2259 \\ stage 1 = (0.80, 0.20), stage 2 = (0.50, 0.20) & A & 0.2450 \\ (1.00, 0.00) & A & 0.2454 \\ (0.00, 1.00) & AB & 0.2492 \\ stage 1 = (0.20, 0.20), stage 2 = (0.80, 0.20) & AB & 0.2556 \\ (0.50, 0.50) & AB & 0.2582 \\ (0.80, 0.20) & AB & 0.2618 \\ (0.20, 0.20) & AB & 0.2759 \\ (0.50, 0.50) & AB & 0.2935 \\ (0.00, 0.00) & AB & 0.2958 \\ (0.20, 0.40) & C & 0.3309 \\ \hline \end{tabular}
\end{table} 
Table 4: Results of [PERSON]’s multiple range test for the controlled probability (\(\alpha\) = 0.05). \begin{table}
\begin{tabular}{c c c c c} \hline
**Source** & \
\begin{tabular}{c} **Type III Sum of** \\ **Squares** \\ \end{tabular} & **df** &
\begin{tabular}{c} **Mean** \\ **Square** \\ \end{tabular} & **F** & **Sig.** \\ \hline Phase termination time & 0.006 & 2 & 0.003 & 0.238 & 0.788 \\ Error & 3.361 & 282 & 0.012 & & \\ Total & 43.392 & 306 & & & \\ Corrected total & 24.297 & 305 & & & \\ \hline \end{tabular}  
* \({}^{\mathrm{a}}\): R Squared = 0.862 (Adjusted R Squared = 0.850).  
\end{table} 
Table 3: _Cont._We further 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 results are summarized in Tables 8-11. In Table 8, we observe that the DILA performs better than the other heuristics regardless of the number of jobs. For the case of 60 jobs, the DILA can find all the best solutions, and the average RDI (0.02) produced by ILS_DP + PR is very close to the average RDI value (0.00) generated by DILA. In Table 9, we can clearly see that the DILA performs better than the other heuristics regardless of the number of families.