## 6 Conclusions and Future Work  
In this study, we developed a deep neural network to solve the job assignment problem between multiple agents and multiple jobs for a multi-agent system within a constant time.  
To create a training dataset for the deep neural network, the PDDL domain describes the environment defined by various parameters for agents and jobs; the PDDL problem is described by varying the values of the parameters; and finally, the PDDL solver generates the assignment problem result for each case.  
Our test results showed that the job assignment based on the inference of the deep neural network increased by up to 13% compared with the results of the PDDL solver in the case of a bottleneck. The proposed DNN method had a constant execution time (approximately 20 ms), whereas that of the algorithm used for the PDDL solver or the metaheuristic method in other studies increased rapidly according to the number of agents and jobs, and other parameters.  
In future work, it will be necessary to reduce the size of the network and improve the deep neural network to solve the assignment problem. This includes the use of extended parameters to better represent real-world situations. We also intend to implement a deep neural network that predicts the order of execution of multiple jobs assigned to each agent in order to help agents perform their tasks.  
Conceptualization, J.L. and Y.C.; methodology, J.L.; software, J.L.; validation, J.L., Y.C. and J.S.; formal analysis, J.L.; investigation, J.L.; resources, J.L.; data curation, J.L.; writing--original draft preparation, J.L.; writing--review and editing, Y.C. and J.S.; visualization, J.L.; supervision, J.S.; project administration, J.S.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript.  
This research was a part of the project titled 'Development of the core technology and establishment of the operation center for underwater gliders (20200482)', funded by the Ministry of Oceans and Fisheries, Korea.  
Not applicable.  
The authors would like to thank all researchers of the smart mobility research center in the Korea Institute of Robotics and Technology Convergence, and all reviewers for very helpful comments and suggestions.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Number of** \\ **Agents (j) and** \\ **Jobs (k)** \\ \end{tabular} } & \multicolumn{3}{c}{**Time (Algorithm Running Time [s])**} \\ \cline{2-5}  & **PDDL Solver** & **DeConNet** & **Runtime** \\  & **(OPTIC)** & **(b)** & **Reduction Ratio** \\  & & **(a)** & **(c)** \\ \hline \(j\) & \(k\) & \(T_{G}\) & \(T_{P}\) & \(T_{P}/T_{G}\) \\ \hline
3 & 10 & \(7.48\times 10^{0}\) & \(2.59\times 10^{-2}\) & \(3.46\times 10^{-3}\) \\
5 & 10 & \(1.23\times 10^{2}\) & \(2.50\times 10^{-2}\) & \(2.03\times 10^{-4}\) \\
8 & 5 & \(3.49\times 10^{0}\) & \(1.41\times 10^{-2}\) & \(4.04\times 10^{-3}\) \\
8 & 8 & \(1.01\times 10^{2}\) & \(1.52\times 10^{-2}\) & \(1.50\times 10^{-4}\) \\
8 & 10 & \(4.36\times 10^{2}\) & \(2.08\times 10^{-2}\) & \(4.77\times 10^{-5}\) \\
9 & 5 & \(8.56\times 10^{0}\) & \(2.10\times 10^{-2}\) & \(2.45\times 10^{-5}\) \\
9 & 10 & \(5.62\times 10^{2}\) & \(2.48\times 10^{-2}\) & \(2.45\times 10^{-3}\) \\
10 & 5 & \(1.54\times 10^{1}\) & \(1.40\times 10^{-2}\) & \(9.09\times 10^{-4}\) \\
10 & 10 & \(6.21\times 10^{2}\) & \(2.49\times 10^{-2}\) & \(4.01\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of the execution time [s] of the ground cost to the prediction time of the DeConNet.