## 6 Conclusions  
A method to solve the Collision-Free Multiple Traveling Salesman Problem (CFMTSP) applied to multiple agents was proposed. In this problem, each agent must visit all nodes in a provided graph, following the provided edges. The graph is modified into an augmented graph such that the information of nodes' position and speed options can be accommodated, and in turn, the arrival time to each node can be determined. According to the optimization of collision-free for each vehicle, an ACO involving multiple ant species is utilized on the augmented graph. The pheromone trails are not left on the edges (such as in typical graph model), but the augmented edges. As a consequence, the probability of selection is assigned to those augmented edges, as well. As a result, the solution is not the sequence of routes but also the trajectory (routes and selected speeds). The algorithm has to guarantee that the resulted trajectories are collision-free. Simulations were established with no violation of minimum allowable arrival time difference. In addition, the simulations have shown three behaviors of the resulted solutions. First, the increase of minimum allowable arrival time difference leads to the decrease of the number of successful trials, almost-strongly increases of the average, the standard deviation, and the ever-found minimum traveling times. Second, it can be concluded that there exists a threshold of the ratio \(\phi\) such that the solution for all vehicles is unable to find. It was demonstrated that the threshold of \(\phi\) is lower as the number of nodes increases. Moreover, for the same number of nodes, the increase of the average of nodes' degree leads to the number of vehicles that are able to find solutions successfully. Third, it is concluded that that the increase in the average of minimum traveling time tends to decrease the algorithm's accuracy. Third, the evaporate rate in the algorithm has a weak influence on the average, the standard deviation, and the ever-found minimum traveling times. Future works will focus on some issues, such as the continuation of speed options and the consequences of modifying the augmented graph. Another issue is in the performance of the ACO algorithm, such as the convergence of the optimal trajectories, the decrease of the standard deviation of the slowest agent's traveling time in high minimum allowable arrival time difference. A.K.P. contributed to writing the original draft, formal analysis, conceptualization, methodology, and visualization; [PERSON] contributed to validation, review and editing of the manuscript, and supervision; conceptualization, methodology, and other aspects were agreed on by all authors. All authors have read and agreed to the published version of the manuscript. This research received internal funding from Universitats Atma Jaya Yogyakarta, Indonesia. This research was supported by Universitats Atma Jaya Yogyakarta. We thank Universitats Atma Jaya for providing the funding and tools required for this research. The authors declare no conflict of interest. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**For \(t_{\text{occ}}\textgreater{}10\) s**} & \multicolumn{3}{c}{**For\(t_{\text{occ}}\textgreater{}150\) s**} \\ \hline \multirow{2}{*}{**Nodes**} & **Vehicles** & **Vehicles** & **Vehicles** & **Vehicles** & **Vehicles** & **Vehicles** \\  & **1 and 2** & **1 and 3** & **2 and 3** & **1 and 2** & **1 and 3** & **2 and 3** \\ \hline
1 & 767.8 & 364.8 & 403 & 1723.7 & 1186.2 & 537.5 \\
2 & 497.4 & 509.1 & 11.7 & 434.7 & 269.3 & 165.4 \\
3 & 175.8 & 998.3 & 1174.1 & 233.8 & 1106.6 & 872.8 \\
4 & 565.9 & 1292.6 & 726.7 & 1476.2 & 832.9 & 643.3 \\
5 & 583.7 & 471.4 & 112.3 & 337.1 & 679 & 341.9 \\
6 & 197.9 & 347.7 & 545.6 & 894 & 190.1 & 703.9 \\
7 & 179.7 & 202.7 & 382.4 & 238.1 & 686.4 & 448.3 \\
8 & 97.1 & 1010.6 & 1107.7 & 858.1 & 1106.6 & 248.5 \\
9 & 192.2 & 51.7 & 243.9 & 346.8 & 1132.1 & 785.3 \\
10 & 451.6 & 725.9 & 274.3 & 671.9 & 323.5 & 348.4 \\ \hline \hline \end{tabular}
\end{table}