## 6 Industrial Case Study
As the conditions in Proposition 8 are satisfied, we set \(Q\) = 1 and use the idle time of \(CTK_{4}\) and \(CTK_{5}\) to feed \(DS_{4}\) under the normal mode for some time. As the initial state is not exactly the same as required by Proposition 8, we can adjust the crude oil in the charging tanks by properly scheduling the oil delivering via the pipeline. Let \(\tau_{s}\) = 0 h. According to Proposition 8, we schedule the pipeline as follows. When \(DS_{4}\) starts its operation at time \(\tau_{1}\) = 96 h, one can use \(CTK_{4}\) to feed \(DS_{4}\) under the normal mode for some time. Thus, \(CTK_{4}\) must be emptied at time \(\tau_{2}\) = \(\tau_{1}\) + \(\Psi\) = 102 h. Hence, we charge it with volume \(v_{4}\) = (\(\tau_{2}\) - \(\tau_{0}\))\(\times f_{ds2}\) - 21,000 = 9804 tons of oil #4 only for feeding \(DS_{2}\) when it is charged the first time. For this case, \(DS_{1}\) has to process oil type #2 after 38,000 tons of oil type #1. We charge \(CTK_{1}\) with 13,000 tons of oil type #1 and charge \(CTK_{7}\) with 2504 tons of oil type #2 to realize the oil type switches. To ensure the safety stock, we start charging \(CTK_{6}\) with volume 1000 tons of oil #7 at time \(\tau_{3}\) = 95.4 h. By the above pipeline transportation schedule, at \(\tau_{1}\), the initial crude oil in charging tanks satisfies Condition (10). By following the scheduling method given by Proposition 8, the volumes of crude oil that should be transported from storage tanks to charging tanks for feeding distillers \(DS_{1}\), \(DS_{2}\), \(DS_{3}\), and \(DS_{4}\) are \(V_{1}\) = \(\mu\times\)\(\alpha_{1}\) = 15,504 tons, \(V_{2}\) = \(\mu\times\)\(\alpha_{2}\) = 14,496 tons, \(V_{3}\) = \(\mu\times\)\(\alpha_{3}\) = 30,000 tons and \(V_{4}\) = \(\mu\times\)\(\alpha_{4}\) = 19,200 tons, respectively. When oil type #7 is transported, it should first be charged into \(CTK_{6}\) with volume \(v_{1}\) = \(\Psi\times f_{p\max}\) = 9900 tons, then into \(CTK_{4}\) or \(CTK_{5}\) with volume \(v_{2}\) = \(\alpha_{1}\times f_{ds1}\) = 8762 tons, and \(CTK_{6}\) again with volume \(v_{3}\) = \(V_{4}\) - \(v_{1}\) - \(v_{2}\) = 538 tons. As \(DS_{4}\) has to switch for processing different oil types, we charge \(CTK_{6}\) with \(v_{4}\) = 25,000 - \(V_{4}\) = 4800 tons of oil type #7 and charge \(CTK_{7}\) with \(v_{5}\) = \(v_{1}\) - \(v_{4}\) = 5100 tons of oil type #8. When the oil #7 in \(CTK_{6}\) for feeding \(DS_{4}\) is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tank** & **Capacity (Ton)** & **Type of Oil Filled** & **Volume (Ton)** \\ \hline \(CTK_{1}\) & 16,000 & Oil \#1 & 9000 \\ \(CTK_{2}\) & 16,000 & Oil \#1 & 16,000 \\ \(CTK_{3}\) & 16,000 & Oil \#3 & 5000 \\ \(CTK_{5}\) & 16,000 & Oil \#3 & 16,000 \\ \(CTK_{6}\) & 20,000 & & \\ \(CTK_{7}\) & 20,000 & & \\ \(CTK_{8}\) & 30,000 & Oil \#5 & 30,000 \\ \(CTK_{9}\) & 30,000 & & \\ \(CTK_{10}\) & 30,000 & Oil \#9 & 30,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial state for the charging tanks. Figure 10: PN model for the case study. used up at time \(\tau_{5}\) = 158.5 h and \(CTK_{6}\) is emptied, \(CTK_{7}\) starts feeding \(DS_{4}\) with oil type #7 under the SCF mode. The sequence of charging the charging tanks and the time point for each event are shown in Figure 11.  
Therefore, a schedule that realizes the refining schedule is obtained and the Gantt charts in Figures 11 and 12 present the oil transportation from storage tanks to charging tanks and the detailed schedules of distiller feeding. Furthermore, the detailed schedule of charging tanks \(CTK_{4}\), \(CTK_{5}\), \(CTK_{6}\), and \(CTK_{7}\) for being charged and feeding distillers is presented in Figure 13, which clearly shows how the method given by Proposition 8 schedules the system to reduce the time during which \(DS_{4}\) is fed by the charging tanks under the SCF mode.