## 5 Computational Experiment
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multicolumn{2}{c}{**Combinations**} & \multicolumn{3}{c}{**AveTWT**} & \multicolumn{3}{c}{**Nopt**} \\ \cline{2-10}  & **(L, TE, R)** & **MIP** & **Basic GA** & **GARI** & **TISGA** & **MIP** & **Basic** & **GARI** & **TISAB** \\ \hline
5 & (1,1,1) & 75.700 & 75.700 & 75.700 & 75.700 & 10 & 10 & 10 & 10 \\ \hline  & (1,1,2) & 108.900 & 108.900 & 108.900 & 108.900 & 10 & 10 & 10 & 10 \\ \hline  & (1,2,1) & 217.000 & 217.000 & 217.000 & 217.000 & 10 & 10  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{**Combinations**} & \multicolumn{4}{c}{**AveTWT**} & \multicolumn{4}{c}{**Nopt**} \\ \cline{2-10}  & & **(L, TF, R)** & **MIP** & **Basic GA** & **GARI** & **TISGA** & **MIP** & **Basic** & **GARI** & **TISAB** \\ \hline  & (1,2,1) & 229.800 & 229.800 & 229.800 & 229.800 & 10 & 10 & 10 & 10 \\ \hline  & (1,2  
Recall that the MIP model cannot find all optimal solutions within 7200 s, and the basic GA becomes slightly worse on solution quality when \(n=15\). Thus, we further investigate the performances of different GAs for large-sized problems in the second experiment. Please note that the computational time limits of the GAs are the same, i.e., \((n\times 0.01)\) seconds, for each \(n\). For comparison, we apply the relative percentage deviation (RPD) of each instance computed as follows. \(\text{RPD}=[(TWT(A)-Min)/Min]\times 100\%\), where \(Min\) is the lowest \(TWT\) value for a given instance obtained by any of the GA algorithms, and \(TWT(A)\) is the \(TWT\) value obtained for a given algorithm and instance. AveRPD refers to Average RPD. Table 10 shows the AveRPD and the total number of best solutions (Nbest) obtained by each GA. As depicted in the table, GARI finds slightly better solutions than the basic GA; that is, 1.937% is obtained by GARI, while the basic GA gives a mean AveRPD value of 2.341%. Overall, TISGA significantly outperforms GARI and basic GA because the total number of best solutions obtained by the proposed TISGA is 1269, 136 for basic GA, and 156 for GARI among 1280 instances. The results support our inference that GAs maintain the diversity level of the population through immigration strategy to achieve a better quality of solutions. Furthermore, the proposed trajectory-based immigration strategy enhances the effectiveness of the GA method more than the random immigration strategy. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{ \begin{tabular}{c} **Combinations** \\ **(L, TE, R)** \\ \end{tabular}  } & \multicolumn{4}{c}{**AveTWT**} & \multicolumn{4}{c}{**Nopt**} \\ \cline{2-9}  & & **MIP** & **Basic GA** & **GARI** & **TISGA** & **MIP** & **Basic** & **GARI** & **TISAB** \\ \hline
15 & (1,1,1) & 1158.800 & 1158.800 & 1158.800 & 1158.800 & 8 & 8 & 8 \\ \hline  & (1,1,2) & 1284.500 & **1293.300** & 1284.500 & 1284.500 & 8 & 7 & 8  
We also examined the performance of GAs under different combinations of (L, TF, R) for each problem. Figures 8-23 illustrate the comparison results under eight combinations of (L, TF, R) for large-sized problems. From these figures, in some cases, GARI is worse than the basic GA; that is, the performances of GARI and the basic GA are influenced by the values of (L, TF, R). However, as the number of jobs increases, GARI becomes gradually better than the basic GA for any combination of (L, TF, R) because adding randomly generated immigrants helps increase the diversity of solutions for the GA method, especially for larger job sizes. The proposed TISGA overcomes the influence of combinations of (L, TF, R) on solutions and is robust in obtaining better solutions. Better convergence is another important topic for designing a good GA method. Thus, we compared the convergence of the basic GA, GARI and TISGA using instances with \(n=50\) and \(n=200\), respectively, as shown in Figures 21 and 22.