## 6 Conclusions and Perspectives  
In this paper, a new control policy was proposed to design an optimal PN controller by avoiding generating the reachability graph. Indeed, based on structural properties of PNs and mathematical concepts, the controller was designed. Thus, the computing time of the supervisor was reduced,  
Figure 8: Ghaffari example 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Control** & **No. of PN** & \begin{tabular}{c} **No. of** \\ **Controllers** \\ \end{tabular} & \begin{tabular}{c} **Maxlims in** \\ **the RG** \\ \end{tabular} & \begin{tabular}{c} **Maxlimally** \\ **Permissive** \\ \end{tabular} & \
\begin{tabular}{c} **No. of** \\ **MTSI** \\ \end{tabular} &
\begin{tabular}{c} **Computation** \\ **Time (ms)** \\ \end{tabular} & **Approach** \\ \hline \(R_{1}\) & 4 & 120 & Yes & 12 & 14,450 & Minimal Cuts \\ \hline \(G\) & 3 & 215 & Yes & 7 & 29,150 & RG based \\ \hline \(R_{3}\) & 3 & 0 & Yes & 7 & 9960 & Without RG \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison results of example 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Additional Control Places** & **Initial Marking \(M_{0}\)(\(P_{cd}\))** & **Incidence Vector \(C\)(\(P_{cd}\))** \\ \hline \multirow{4}{*}{\(R_{1}\)} & \(P_{c1}\) & 3 & \((0,-2,0,0,0,0,0,0,0,0,0)\) \\  & \(P_{c2}\) & 3 & \((0,0,0,0,-2,0,0,0,0,0,0)\) \\  & \(P_{c3}\) & 3 & \((0,0,0,0,0,0,0,0,0,-2)\) \\  & \(P_{c4}\) & 2 & \((0,0,0,0,0,0,0,0,-2,0)\) \\ \hline \multirow{4}{*}{\(G\)} & \(P_{c1}\) & 2 & \((0,-1,0,0,1,0,0,0,1,-1)\) \\  & \(P_{c2}\) & 2 & \((0,0,-1,0,1,0,0,1,-1,0)\) \\  & \(P_{c3}\) & 3 & \((0,0,0,0,-1,1,0,0,0,1,-1)\) \\ \hline \multirow{4}{*}{\(R_{3}\)} & \(P_{c1}\) & 3 & \((0,-2,0,0,2,0,0,0,0,2,-2)\) \\  & \(P_{c2}\) & 3 & \((0,0,-2,0,2,0,0,0,2,-2,0)\) \\ \cline{1-1}  & \(P_{c3}\) & 4 & \((0,0,0,0,-2,2,0,0,0,2,-2)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Petri net controllers of example 3.  
and then one can control the system constraints in a timely manner. Consequently, the production and the security will be directly affected. The proposed methodology was implemented in our flexible manufacturing system. The results analysis shows that our algebraic method using Petri nets is more effective than previous works based on the theory of regions. Indeed, for the same number of controllers, one can supervise the system under less computational burden. The steps of generation and analysis of the reachability graph are skipped in the control process, which allowed us to reduce the computation time using CPLEX software. Moreover, the proposed approach always generates the minimal number of MTSIs, since the application of the theory of regions is confined on the calculated paths and not on the reachability graph. Therefore, it reflects a great reduction of the resolution complexity of the TR and prevents a possible combinatorial explosion of states when the complexity of the system increases. Based on experimental results, it is self-evident to remark the considerable reduction of the computation cost of the TR. Thus, our new method can obtain an optimal maximally permissive supervisor for bounded Petri nets. Hence, in the near future, how to obtain blocking states without calculating the reachability graph is to become an important issue in the control synthesis field. Our team is working on this project from the perspective of this model. The authors of this paper are specialists in the industrial engineering field and more precisely in the supervision of discrete event systems and preventive/corrective maintenance for systems. Conceptualization, S.R., S.T., and N.R. ; Methodology, S.T. ; Software, S.R. ; Validation, S.R., S.T., and N.R. ; Formal analysis, S.T. ; Investigation, S.R. ; Resources, N.R. ; Data curation, N.R. ; Writing--original draft preparation, S.R. ; Writing--review and editing, S.R. ; Visualization, S.T. ; Supervision, N.R. ; Project administration, N.R. ; Funding acquisition, N.R. This research received no external funding.