## Appendix A
### Proof of Proposition 2  
With the PN model shown in Figure 6, at marking \(M_{0}\) (time \(\tau_{0}\)), we have \(M_{0}(p_{1s})=M_{0}(p_{2c})=M_{0}(p_{3s})=1\), \(V(M_{0}(p_{1s},\)\(\phi_{1}))=\zeta_{1}\), \(V(M_{0}(p_{2c},\)\(\phi_{2}))=\alpha_{2}\), and \(V(M_{0}(p_{3s},\)\(\phi_{2}))=\mu\times\alpha_{2}\). At this marking, due to \(\zeta_{1}\geq\sigma\), the SCF mode is adopted for \(CTK_{1}\). Hence, we fire \(t_{14}\) with color \(\phi_{1}\) to feed \(p_{1}\), at the same time, fire \(y\) with color \(\phi_{1}\) and volume \(\alpha\) to charge \(p_{1s}\), and fire \(t_{23}\) with color \(\phi_{2}\) to feed \(p_{2}\). At time \(\tau_{1}=\tau_{0}+\Psi\), \(M_{1}\) is reached such that \(M_{1}(p_{1s})=M_{1}(p_{3c})=1\), \(V(M_{1}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+\alpha_{2}\leq\zeta_{1}\) and \(V(M_{1}(p_{3c},\)\(\phi_{2}))\)\(=\mu\times\alpha_{2}\). At this marking, we fire \(t_{14}\) with color \(\phi_{1}\) and \(t_{33}\) with color \(\phi_{2}\) to feed \(p_{1}\) and \(p_{2}\) respectively. Since \(CTK_{2}\) is empty, we fire \(y\) with color \(\phi_{1}\) and volume \(\omega\times\alpha_{1}\) to charge \(p_{2s}\) instead of \(p_{1s}\). At time \(\tau_{2}\)\(=\tau_{1}+(\omega\times\alpha_{1})/f_{p\rm max}\), \(M_{2}\) is reached such that \(M_{2}(p_{1s})=M_{2}(p_{2s})=M_{2}(p_{3c})=1\), \(V(M_{2}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+\alpha_{2}-(\omega\times\alpha_{1}\times f_{ds1})/f_{p \rm max}\geq\sigma\), \(V(M_{2}(p_{2s},\)\(\phi_{1}))=\sigma\times\alpha_{1}\leq\zeta_{2}\), and \(V(M_{2}(p_{3c},\)\(\phi_{2}))=\mu\times\alpha_{2}-(\omega\times\alpha_{1}\times f_{ds2})/f_{p \rm max}\). Then, at this marking, \(t_{14}\) and \(t_{33}\) continue their firing with color \(\phi_{1}\) and \(\phi_{2}\) to feed \(p_{1}\) and \(p_{2}\), respectively, and \(y\) fires with color \(\phi_{1}\) and volume \((\mu-\omega)\times\alpha_{1}-\alpha\geq 0\) to charge \(p_{1s}\), as \(\mu\geq(\omega\times f_{ds1}+f_{p\rm max})/f_{ds1}\) holds. At time \(\tau_{3}=\tau_{2}+\Psi\), \(M_{3}\) is reached such that the crude oil in \(CTK_{2}\) is ready for feeding under the normal mode and \(V(M_{3}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+2\alpha_{2}-(\omega\times\alpha_{1}\times f_{ds1})/f_{p \rm max}\), \(V(M_{3}(p_{2c},\)\(\phi_{1}))=\omega\times\alpha_{1}\), and \(V(M_{3}(p_{3c},\)\(\phi_{2}))=(\mu-1)\times\alpha_{2}-(\omega\times\alpha_{1}\times f_{ds2})/f_{p \rm max}\). Then, we fire \(t_{23}\) instead of \(t_{14}\) with color \(\phi_{1}\) to feed \(p_{1}\) under the normal mode. Transition \(t_{33}\) continues its firing with color \(\phi_{2}\) to feed \(p_{2}\) and transition \(y\) continues its firing with color \(\phi_{1}\) to charge \(p_{1s}\). At \(\tau_{4}=\tau_{3}+\omega\times\Psi=\tau_{0}+(\mu\times\alpha_{1})/f_{p\rm max}\), \(M_{4}\) is reached such that \(M_{4}(p_{1s})=M_{4}(p_{3c})=1\), \(V(M_{4}(p_{1s},\)\(\phi_{1}))=\zeta_{1}+(\mu-\omega)\times\alpha_{2}-(\mu\times \alpha_{1}\times f_{ds1})/f_{p\rm max}=\zeta_{1}+(\mu\times\alpha_{2}\times f_{ds 1})/f_{p\rm max}\leq\zeta_{1}\), and \(V(M_{4}(p_{2s},\)\(\phi_{2}))=(\mu+1)\times\alpha_{2}-(\mu\times\alpha_{1}\times f_{ds 2})/f_{p\rm max}\). Afterwards, the SCF mode is started again for \(CTK_{1}\) by firing transition \(t_{14}\) with color \(\phi_{1}\) to feed \(p_{1}\). Besides, \(t_{33}\) continues its firing with color \(\phi_{2}\) to feed \(p_{2}\), and \(y\) starts its firing with color \(\phi_{2}\) and volume \(\mu\times\alpha_{2}\) to charge \(p_{2s}\). At time \(\tau_{5}=\tau_{4}+(\mu\times\alpha_{2})/f_{p\rm max}=\tau_{0}+\mu\times\Psi\), \(M_{5}\) is reached such that \(M_{5}(p_{1s})=M_{5}(p_{2s})=M_{5}(p_{3c})=1\), \(V(M_{5}(p_{1s},\)\(\phi_{1}))=\zeta_{1}\), \(V(M_{5}(p_{2s},\)\(\phi  
\(\phi_{2}))=\zeta_{2}+\mu\times\alpha_{2}\times(f_{p\max}-f_{d81}-f_{d52})/f_{p \max}\leq\zeta_{2}\), and \(V(M_{1}(p_{4c},\phi_{3}))=(\mu+1)\times\alpha_{3}-\mu\times\alpha_{3}\times(f_{d81 }+f_{d82})/f_{p\max}\).