## 5 Computational Framework and Result
While DE, CU, SDR, TTM, and MF were different from the exact method by 4.52%, 29.05%, 14.43%, 15.01%, and 13.44%, respectively, VaNAS was 0.00% different, which means that it could find an optimal solution for small problems 100% of the time. The maximum error is the difference between the best solution and the worst solution of each proposed algorithm. As shown in Table 9, the maximum error of VaNAS was the lowest among others, indicating that VaNAS had the highest solution stability. The second experiment's medium-sized random dataset was composed of 12 test instances. The number of jobs was randomly selected to be from 40 to 52 jobs, and the number of machines was set to be from 3 to 5. In this experiment, Lingo was executed for 48 h, and the computation time of all proposed methods was 30 min as the termination condition. After the simulation had been executed five times, the best solutions and the maximum error of medium-sized test instances were recorded in Table 11 and the statistical results are shown in Table 12. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**No**} & \multirow{3}{*}{**Job**} & \multirow{3}{*}{**\(m\)**} & \multirow{3}{*}{**NT**} & \multirow{3}{*}{**Lingo**} & \multirow{3}{*}{**Best**} & \multirow{3}{*}{**Max.**} & \multirow{3}{*}{**Best**} & \multirow{3}{*}{**Max.**} & \multirow{3}{*}{**Best**} & \multirow{3}{*}{**Max.**} & \multirow{3}{*}{**Best**} & \multirow{3}{*}{**Max.**} & \multirow{3}{*}{**Best**} & \multirow{3}{*}{**Max.**} & \multirow{3}{*}{**Best**} & \multirow{3}{*}{**Max.**} \\  & & & & & **Sol.** & & & & & & & & & & & \\ \hline L-1 & 80 & 5 & 6 & 47,968 & 45,871 & 1376 & 52,297 & 5229 & 39,917 & 1197 & 46,617 & 1864 & 46,521 & 1960 & 45,598 & 1823 \\ \hline L-2 & 80 & 5 & 5 & 49,351 & 41,278 & 825 & 45,580 & 4102 & 40,015 & 1200 & 42,893 & 1715 & 42,886 & 1293 & 42,361 & 1270 \\ \hline L-3 & 80 & 4 & 5 & 37,599 & 33,192 & 998 & 38,817 & 2329 & 31,108 & 311 & 34,918 & 1396 & 34,118 & 1847 & 34,287 & 1028 \\ \hline L-4 & 80 & 3 & 5 & 41,625 & 38,185 & 381 & 41,491 & 2904 & 36,614 & 366 & 38,811 & 1164 & 38,927 & 1824 & 38,891 & 1555 \\ \hline L-5 & 100 & 4 & 6 & 52,604 & 47,467 & 474 & 53,158 & 3721 & 44,581 & 891 & 48,819 & 2440 & 48,984 & 2275 & 48,816 & 1464 \\ \hline L-6 & 100 & 5 & 6 & 56,497 & 49,981 & 1499 & 57,619 & 5185 & 48,819 & 1464 & 50,176 & 1505 & 51,197 & 484 & 52,348 & 1570 \\ \hline L-7 & 100 & 6 & 6 & 55,823 & 50,187 & 1003 & 58,281 & 4079 & 47,713 & 954 & 51,810 & 1554 & 52,184 & 2216 & 52,183 & 1565 \\ \hline L-8 & 100 & 7 & 6 & 54,891 & 49,571 & 1487 & 55,819 & 3349 & 47,289 & 1418 & 50,972 & 2548 & 50,013 & 2488 & 51,197 & 2047 \\ \hline L-9 & 120 & 6 & 7 & 67,905 & 62,995 & 1889 & 68,913 & 6891 & 59,982 & 599 & 63,395 & 3169 & 64,512 & 1418 & 53,891 & 2155 \\ \hline L-10 & 120 & 7 & 7 & 67,067 & 63,417 & 1268 & 68,919 & 4135 & 58,813 & 1764 & 64,992 & 1949 & 63,857 & 4384 & 64,387 & 3219 \\ \hline L-11 & 120 & 8 & 6 & 65,596 & 60,874 & 608 & 65,817 & 5265 & 58,716 & 1174 & 59,816 & 1794 & 60,128 & 2678 & 60,016 & 3000 \\ \hline L-12 & 134 & 8 & 7 & 72,168 & 67,617 & 676 & 73,215 & 5125 & 64,480 & 644 & 67,764 & 2710 & 67,366 & 3108 & 67,952 & 2718 \\ \hline C-1 & 201 & 11 & 7 & 128,704 & 118,917 & 2378 & 129,814 & 11,683 & 110,172 & 2203 & 119,859 & 5992 & 119,347 & 4107 & 119,846 & 5992 \\ \hline \multicolumn{10}{c}{\% Improved from best objective} \\ \multicolumn{10}{c}{found by Lingo} & \multicolumn{1}{c}{9.88} & \multicolumn{1}{c}{1.42} & \multicolumn{1}{c}{16.35} & \multicolumn{1}{c}{7.89} & \multicolumn{1}{c}{8.01} & \multicolumn{1}{c}{9.33} \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Computation results of large test instances.