## 4 Experimental Results and Analysis
The formula is  
\[\text{AE}=\frac{1}{K}\cdot\sum_{i=1}^{k}\frac{z_{opt}-z_{i}}{z_{opt}}\cdot 100 \tag{14}\]  
where \(z_{opt}\) is the known upper bound value for each instance, \(z_{i}\) is the optimal makespan of each algorithm, and \(K\) is the number of instances. The average error comparison results of several algorithms are shown in Figure 7. As can be seen from the figure, the lowest AE value of HGSA in this paper was 5.58, which was lower than the lowest value of 20.516 for the DSOMA algorithm. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Problem** & **Instance** &
\begin{tabular}{c} **Upper** \\ **Bound** \\ \end{tabular} & **MA** & **IG-RIS** & **HGA** & **IIGA** & **DSOMA** & **HGSA** \\ \hline \multirow{8}{*}{\(200\times 20\)} & Ta101 & 11,195 & 14,912 & 14,912 & 20,148 & 19,681 & 15,027 & **11,642** \\  & Ta102 & 11,203 & 14,876 & 15,002 & 20,539 & 20,096 & 15,211 & **11,683** \\  & Ta103 & 11,281 & 15,057 & 15,186 & 20,511 & 19,913 & 15,247 & **11,930** \\  & Ta104 & 11,275 & 14,975 & 15,082 & 20,461 & 19,928 & 15,174 & **11,791** \\  & Ta105 & 11,259 & 14,733 & 14,970 & 20,339 & 19,843 & 15,047 & **11,728** \\  & Ta106 & 11,176 & 14,861 & 15,101 & 20,501 & 19,942 & 15,212 & **11,690** \\  & Ta107 & 11,360 & 14,988 & 15,099 & 20,680 & 20,112 & 15,168 & **11,958** \\  & Ta108 & 11,334 & 14,926 & 15,141 & 20,614 & 20,056 & 15,247 & **11,730** \\  & Ta109 & 11,192 & 14,885 & 15,034 & 20,300 & 19,918 & 15,136 & **12,138** \\  & Ta110 & 11,288 & 14,921 & 15,122 & 20,437 & 19,935 & 15,243 & **12,084** \\ \hline \multirow{8}{*}{\(500\times 20\)} & Ta111 & 26,059 & 35,677 & 35,372 & 49,095 & 46,689 & 37,064 & **26,859** \\  & Ta112 & 26,520 & 35,953 & 35,743 & 49,461 & 47,275 & 37,419 & **27,220** \\  & Ta113 & 26,371 & 35,732 & 35,452 & 48,777 & 46,544 & 37,059 & **27,511** \\  & Ta114 & 26,456 & 36,084 & 35,687 & 49,283 & 46,899 & 37,014 & **26,912** \\  & Ta115 & 26,334 & 35,774 & 35,417 & 48,950 & 46,741 & 36,894 & **26,930** \\  & Ta116 & 26,477 & 35,948 & 35,747 & 49,533 & 46,941 & 37,372 & **27,354** \\  & Ta117 & 26,389 & 35,631 & 35,395 & 48,943 & 46,509 & 36,698 & **26,888** \\  & Ta118 & 26,560 & 35,943 & 35,568 & 49,277 & 46,873 & 36,944 & **27,229** \\  & Ta119 & 26,005 & 35,658 & 35,304 & 49,207 & 46,743 & 36,862 & **28,103** \\  & Ta120 & 26,457 & 36,016 & 35,643 & 49,092 & 46,847 & 37,098 & **27,290** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 6: Gantt chart of the Ta001. (2) IP represents the improvement percentage on makespan of the HGSA algorithm compared with the other algorithms. The formula is  
\[IP=\frac{C_{HGSA}-C_{X}}{C_{HGSA}}\cdot 100\% \tag{15}\]  
where \(C_{HGSA}\) is the minimum makespan obtained by the HGSA algorithm, and \(C_{X}\) is the minimum makespan obtained by the other algorithms. Figure 8 shows the percentage improvement on makespan of the HGSA algorithm compared to the MA, IG-RIS, IIGA, and DSOMA algorithms. It can be seen from the figure that the superiority of the HGSA algorithm is not obvious compared with other algorithms when the scale of the benchmark is small. In the 20 \(\times\) 20 instances, the HGSA algorithm performed slightly weaker than the DSOMA algorithm. However, with the increase of instances scale, that is, the increasing of workpieces and machine tools number, the advantages of HGSA became increasingly obvious. In the 500 \(\times\) 20 instances, HGSA had an improvement of 31.7%, 30.5%, 80.6%, 36.1%, and 71.9% compared to MA, G-RIS, HGA, DSOMA, and IIGA, respectively. As the scale of instances increased, the improvement effect was more obvious, indicating that the algorithm performed better when the schedule was larger. (3) ARPD represents the average relative percentage deviation between the result obtained by the algorithm and the optimal value.