## 2 Materials and Methods
### Adaptive Selection of Algorithm Parameters
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Problem** & \(L_{m1}\) & \(L_{a1}\) & \(L_{g}\) & \(L_{ga}\) & \(L_{m2}\) & \(L_{a2}\) & \(L_{aco}\) & \(C_{ant}\) \\ \hline abz6 & 980 & 1005.3 & 945 & 977.33 & 948 & 985.26 & 1000 & 100 \\ ft06 & 55 & 55.16 & 55 & 55 & 55 & 55.16 & 30 & 10 \\ ft10 & 1017 & 1038.8 & 950 & 995.87 & 975 & 1013.84 & 1000 & 100 \\ la01 & 666 & 673.08 & 666 & 669.4 & 666 & 673.08 & 30 & 10 \\ la10 & 958 & 958 & 958 & 958 & 958 & 958 & 1000 & 100 \\ la15 & 1211 & 1220.6 & 1207 & 1215.45 & 1207 & 1220.62 & 1000 & 100 \\ la17 & 796 & 809.32 & 784 & 798.25 & 787 & 809.32 & 1000 & 100 \\ la21 & 1121 & 1168.1 & 1107 & 1150.9 & 1118 & 1154.06 & 1000 & 100 \\
3\_Plates & 657.55 & 664.782 & 657.55 & 662.08 & 657.55 & 664.782 & 30 & 10 \\
6\_Plates & 109 & 113.22 & 107 & 111.45 & 108 & 112.12 & 30 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Examples of the best sets of coefficients obtained using the genetic algorithm. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Problem** & \(L_{m1}\) & \(L_{a1}\) & \(L_{gm}\) & \(L_{ga}\) & \(L_{aco}\) & \(C_{ant}\) & **Best known** \\ \hline abz6 & 980 & 1005.3 & 945 & 977.33 & 1000 & 100 & 943 \\ ft06 & 55 & 55.16 & 55 & 55 & 30 & 10 & 55 \\ ft10 & 1017 & 1038.8 & 950 & 995.87 & 1000 & 100 & 930 \\ la01 & 666 & 673.08 & 666 & 669.4 & 30 & 10 & 666 \\ la10 & 958 & 958 & 958 & 958 & 1000 & 100 & 958 \\ la15 & 1211 & 1220.6 & 1207 & 1215.45 & 1000 & 100 & 1207 \\ la17 & 796 & 809.32 & 784 & 798.25 & 1000 & 100 & 784 \\ la21 & 1121 & 1168.1 & 1107 & 1150.9 & 1000 & 100 & 1048 \\
3\_Plates & 657.55 & 664.782 & 657.55 & 662.08 & 30 & 10 & 657.55 \\
6\_Plates & 109 & 113.22 & 107 & 111.45 & 30 & 10 & 107 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of results. Figure 4 shows experimental data on the increase in the speed of parallel calculations compared to sequential operation, depending on the number of processors (cores) used. It can be seen from the figure that the proposed approach can significantly reduce the computation time. The greater the effect of parallelization, the greater the number of calculations in solving the problem by the ACO algorithm, i.e., the greater the dimension of the problem, the number of ants, and the number of iterations. Indeed, according to [PERSON]'s law, parallel computing is more effective the greater the proportion of calculations performed in parallel. In the test examples under consideration, the proportion of such calculations was 3-5%. The greater the gain from the above technique of using two graphs, the greater the number of ants and the size of the graph. In test tasks of scheduling with 10 requirements in five stages, the calculation time was reduced by about four times; in tasks with 10 requirements in 10 stages--five times; in tasks with 50 requirements in 10 stages--seven times. ## 4 Conclusions

This study considers ways to improve the speed, accuracy, and flexibility of the ant colony optimization algorithm for solving scheduling problems. A new way of representing the problem as a problem of finding the shortest path on a graph is proposed, which is distinguished by a high level of universality and flexibility. At the same time, its use allows for obtaining acceptable job-shop scheduling problem solutions. It is shown that the use of the Genetic Algorithm as a meta-optimizer for tuning the parameters of the ACO algorithm simplifies the study, makes the algorithm adaptive to the problem being solved, and improves the resulting plans. It is determined that the sets of the best parameters of the algorithm differ from task to task. For the next steps, we plan to conduct a study on a larger basis of scheduling instances, while covering not only job-shop scheduling problems but also open-shop scheduling (OSS) problems and flexible JSS and OSS problems [6; 41; 53; 54] and considering the advantages of the proposed approach for planning problems with dynamically changing conditions [54; 55; 56].