# Textile Flexible Job-Shop Scheduling Based on a Modified Ant Colony Optimization Algorithm
## 4 Results and Analysis
### Verification of Algorithms under Different Conditions
Figure 15 provides a visual comparison of the results obtained from 20 runs of each algorithm across different groups, while Figure 16 illustrates the optimization process of the best-performing runs among the 20 runs for each algorithm. Based on Figures 15 and 16, it is evident that the PACO algorithm can effectively address workshop scheduling problems with varying numbers of operations and product types. Figure 15 clearly illustrates that the IPACO algorithm consistently produces significantly lower overall results across 20 runs in all five experimental groups compared to the PACO algorithm. While the MPACO 2 algorithm exhibits a trend similar to that of the IPACO algorithm, the IPACO algorithm consistently achieves the lowest optimization results among the 20 algorithm runs. Furthermore, comparing with Figure 16, which depicts the iterative optimization process, reveals that the early convergence and overall iterative search of the MPACO 2 algorithm are inferior to those of the IPACO algorithm. In all five experimental groups, the optimization curve of the IPACO algorithm continuously fluctuates from the initial stage to the end of the algorithm, resulting in superior outcomes, even if the changes are sometimes slight. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  &
\begin{tabular}{c} **Quantity of** \\ **Products** \\ \end{tabular} & \multirow{2}{*}{**Product**} & \multirow{2}{*}{**Oper. 1**} & \multirow{2}{*}{**Oper. 2**} & \multirow{2}{*}{**Oper. 3**} & \multirow{2}{*}{**Oper. 4**} & \multirow{2}{*}{**Oper. 5**} & \multirow{2}{*}{**Oper. 6**} & \multirow{2}{*}{**Oper. 7**} & \multirow{2}{*}{**Oper. 8**} \\ \cline{5-10} \cline{6-10} \multirow{-2}{*}{Group 1} & 5 & Product 1 & & & & & & & & \\ \cline{5-10}  & 5 & Product 2 & 1.5 & 8 & 10 & 5 & 25 & 3 & 0 & 0 \\  & 5 & Product 3 & 3 & 12 & 15 & 11 & 50 & 10 & 0 & 0 \\ \hline \multirow{2}{*}{Group 2} & 4 & Product 1 & 5 & 3 & 8 & 13 & 0 & 0 & 0 & 0 \\  & 6 & Product 2 & 7 & 2 & 12 & 11 & 0 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{Group 3} & 3 & Product 1 & 9 & 10 & 6 & 32 & 3 & 0 & 0 & 0 \\  & 2 & Product 2 & 12 & 12 & 7 & 24 & 3 & 0 & 0 & 0 \\  & 3 & Product 3 & 10 & 14 & 10 & 35 & 5 & 0 & 0 & 0 \\  & 2 & Product 4 & 7 & 13 & 9 & 28 & 4 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{Group 4} & 2 & Product 1 & 15 & 20 & 5 & 16 & 13 & 2 & 10 & 0 \\  & 4 & Product 2 & 13 & 28 & 7 & 15 & 15 & 3 & 13 & 0 \\ \cline{1-1}  & 3 & Product 3 & 12.5 & 23 & 5 & 11 & 20 & 2 & 12 & 0 \\ \cline{1-1}  & 2 & Product 4 & 14 & 26 & 6 & 14 & 20 & 3 & 10 & 0 \\ \cline{1-1}  & 3 & Product 5 & 13 & 19 & 7 & 17 & 18 & 3 & 14 & 0 \\ \hline \multirow{2}{*}{Group 5} & 8 & Product 1 & 25 & 14 & 21 & 30 & 5 & 8 & 19 & 21 \\ \cline{1-1}  & 10 & Product 2 & 20 & 16 & 19 & 36 & 7 & 6 & 19 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The operating time (hours) and the quantity of products corresponding to each experimental group of operations. Figure 15: The visual comparison chart of the results from 20 runs of each algorithm across different experimental groups. To compare the superiority of the IPACO algorithm over the PACO algorithm, MACO 1 algorithm, and MACO 2 algorithm, we recorded the average value of the objective function \(T^{*}\) for 20 runs of each algorithm in five experimental groups, as well as the minimum \(T^{*}\) obtained for each algorithm in the 20 runs. We then compared the average value of \(T^{*}\) for each algorithm with the average value of \(T^{*}\) obtained by the IPACO algorithm, and the optimal value of \(T^{*}\) for each algorithm with the optimal value of \(T^{*}\) obtained by the IPACO algorithm. Figure 16: The comparative visualization of the iteration process for the best optimal results achieved by each algorithm across different experimental groups. algorithm. We refer to this comparison as the relative percentage error (RPE) [32]. The RPE is defined by Equation (26). \[\left\{ \begin{array}{l} RPE\_A=\frac{A\text{r}\text{p}-A\text{p}^{*}}{A\text{p} ^{*}}\times 100\%\\ RPE\_B=\frac{Best-Best^{*}}{Best^{*}}\times 100\%\end{array}