## Appendix A
### Proof of Proposition 4  
Based on the model shown in Figure 7, the system can be scheduled as follows. At marking \(M_{0}\) (time \(\tau_{0}\)), \(M_{0}(p_{1s})=M_{0}(p_{2s})=M_{0}(p_{3c})=M_{0}(p_{4s})=1\), \(V(M_{0}(p_{1s},\)\(\phi_{1}))=\zeta_{1}\geq\sigma\), \(V(M_{0}(p_{2s},\)\(\phi_{2}))=\zeta_{2}\geq\sigma\), \(V(M_{0}(p_{3c},\)\(\phi_{3}))=\mu\times\alpha_{3}\), and \(V(M_{0}(p_{4s},\)\(\phi_{3}))=\mu\times\alpha_{3}\) hold. We fire \(t_{14}\) with color \(\phi_{1}\), \(t_{24}\) with color \(\phi_{2}\), and \(t_{33}\) with color \(\phi_{3}\) to feed \(p_{1}\), \(p_{2}\) and \(p_{3}\), respectively, and fire \(y\) with color \(\phi_{1}\) and volume \(\alpha\) to charge \(p_{1s}\). At time \(\tau_{1}=\tau_{0}+\Psi\), \(CTK_{3}\) is applied and the oil in \(CTK_{4}\) is ready for feeding under the normal mode. At this time, the system is transferred to \(M_{1}\) such that \(M_{1}(p_{1s})=M_{1}(p_{2s})=M_{1}(p_{4c})=1\), \(V(M_{1}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+\alpha_{2}+\alpha_{3}\leq\zeta_{1}\), \(V(M_{1}(p_{2s},\)\(\phi_{2}))=\zeta_{2}-\alpha_{2}\), and \(V(M_{1}(p_{4c},\)\(\phi_{3}))=\mu\times\alpha_{3}\). At this marking, we continue firing \(t_{14}\) with color \(\phi_{1}\), \(t_{24}\) with color \(\phi_{2}\), and \(t_{43}\) with color \(\phi_{3}\) to feed \(p_{1}\), \(p_{2}\), and \(p_{3}\), respectively. As \(CTK_{3}\) is emptied, we fire \(y\) with color \(\phi_{2}\) and volume \(\omega\times\alpha_{2}\) to charge \(p_{3s}\) instead of \(p_{1s}\) to increase the usage of \(CTK_{3}\). At time \(\tau_{2}=\tau_{1}+\omega\times\alpha_{2}/f_{p\max}\), \(M_{2}\) is reached such that \(M_{2}(p_{1s})=M_{2}(p_{2s})=M_{2}(p_{3s})=M_{2}(p_{4c})=1\), \(V(M_{2}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+\alpha_{2}+\alpha_{3}-(\omega\times\alpha_{2} \times f_{d81})/f_{p\max}\geq\sigma\), \(V(M_{2}(p_{2s},\)\(\phi_{2}))=\zeta_{2}-\alpha_{2}-(\omega\times\alpha_{2} \times f_{d82})/f_{p\max}\), \(V(M_{2}(p_{3s},\)\(\phi_{2}))=\omega\times\alpha_{2}\), and \(V(M_{2}(p_{4c},\)\(\phi_{3}))=\mu\times\alpha_{3}-(\omega\times\alpha_{2}\times f_{d 83})/f_{p\max}\). Then, \(t_{14}\), \(t_{24}\), and \(t_{43}\) continue feeding \(p_{1}\), \(p_{2}\), and \(p_{3}\), respectively, but transition \(y\) switches to charge \(p_{1s}\) with color \(\phi_{1}\) and volume \(\mu\times\alpha_{1}-\alpha\geq 0\), as \(\mu\geq 2f_{p\max}/(f_{d81}+f_{d82})\) and \(f_{d81}\geq f_{d82}\) hold, during \(\tau_{2}\), \(\tau_{2}+(\mu\times\alpha_{1}-\alpha)/f_{p\max}\) and charge \(p_{2s}\) with color \(\phi_{2}\) and volume \((\mu-\omega)\times\alpha_{2}\) during \([\tau_{2}+(\mu\times\alpha_{1}-\alpha)/f_{p\max}\), \(\tau_{2}+(\mu\times\alpha_{1}+\mu\times\alpha_{2}-\omega\times\alpha_{2}- \alpha)/f_{p\max}]\). Notice that, as the crude oil in \(CTK_{3}\) is ready for feeding at time \(\tau_{2}+\Psi\) under the normal mode, we fire \(t_{33}\) instead of \(t_{24}\) with color \(\phi_{2}\) to feed \(p_{2}\) during \([\tau_{2}+\Psi,\)\(\tau_{2}+\Psi+\sigma\times\Psi]\). At time \(\tau_{3}=\tau_{2}+(\mu\times\alpha_{1}+\mu\times\alpha_{2}-\omega\times\alpha_{2}- \alpha)/f_{p\max}=\tau_{2}+\Psi+\sigma\times\Psi=\tau_{0}+(\mu\times\alpha_{1}+ \mu\times\alpha_{2})/f_{p\max}\), the system is transferred to \(M_{3}\) such that \(M_{3}(p_{1s})=M_{3}(p_{2s})=M_{3}(p_{4c})=1\), \(V(M_{3}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+\mu\times\alpha_{1}-f_{d81}\times(\mu \times\alpha_{1}+\mu\times\alpha_{2})/f_{p\max}\), \(V(M_{3}(p_{2s},\)\(\phi_{2}))=\zeta_{2}+\mu\times\alpha_{2}-f_{d82}\times(\mu \times\alpha_{1}+\mu\times\alpha_{2})/f_{p\max}\leq\zeta_{2}\), and \(V(M_{3}(p_{4c},\)\(\phi_{3}))=(\mu+1)\times\alpha_{3}-f_{d83}\times(\mu\times \alpha_{1}+\mu\times\alpha_{2})/f_{p\max}\). At this marking, \(CTK_{3}\) is emptied, and \(CTK_{2}\) is used to feed \(DS_{2}\) under the SCF mode. Therefore, we fire \(y\) with color \(\phi_{3}\) and volume \(\mu\times\alpha_{3}\) to charge \(p_{3s}\), and fire \(t_{14}\) with color \(\phi_{1}\), \(t_{24}\) with color \(\phi_{2}\), and \(t_{43}\) with color \(\phi_{3}\) to feed \(p_{1}\), \(p_{2}\), and \(p_{3}\), respectively.