## 6 Industrial Case Study  
This section presents a practical scenario of a refinery in China to show the application of the proposed method. The refinery has four distillers \(DS_{1}\)-\(DS_{4}\) with 10 charging tanks. A short-term schedule with 10-day scheduling horizon should be created three times a month to realize a target refining schedule. In this case, the refining schedule is shown in Figure 9, where \(\#i\) denotes crude oil type \(\#i\), \(i\in\{1,2,\ldots,9\}\). The oil feeding rates for the distillers and maximal flow rate of the pipeline are \(f_{ds1}=323\) tons/h, \(f_{ds2}=302\) tons/h, \(f_{ds3}=625\) tons/h, \(f_{ds4}=400\) tons/h, and \(f_{p\max}=1650\) tons/h, respectively, which indicates \(f_{p\max}=f_{ds1}+f_{ds2}+f_{ds3}+f_{ds4}\). Note that distillers \(DS_{1}\)-\(DS_{3}\) operate normally, while \(DS_{4}\) is under maintenance at the beginning and will be put into operation at the beginning of the fifth day. The oil residency time for charging tanks is \(\Psi=6\) h, and the safety stock is \(\sigma=1000\) tons. The initial state for the 10 charging tanks is shown in Table 1. Initially, \(CTK_{1}\), \(CTK_{4}\), and \(CTK_{8}\) are feeding \(DS_{1}\), \(DS_{2}\), and \(DS_{3}\), respectively. Note that, according to the refining schedule, the crude oil in \(CTK_{10}\) is not to be processed during the scheduling horizon, and \(CTK_{2}\) is under maintenance and cannot be used. Hence, there are three charging tanks ready for charging at the initial time. By checking the crude oil in the charging tanks and the refining schedule, we find that: (1) the pipeline needs to deliver 22,000 tons of oil type \(\#1\) from the storage tanks to the charging tanks for feeding \(DS_{1}\) after the oil initially in \(CTK_{1}\) and \(CTK_{3}\) is used up; (2) \(DS_{2}\) first processes the crude oil in \(CTK_{4}\) and \(CTK_{5}\), then these two charging tanks are ready for charging; (3) \(CTK_{8}\) will be ready for charging after the oil initially in it is used up by feeding \(DS_{3}\); and (4) \(CTK_{9}\) is empty and free to be charged. Therefore, when \(DS_{4}\) starts its operation at the beginning of the fifth day, the system has four distillers operating concurrently but only eight charging tanks are usable, i.e., there are not sufficient charging tanks in the refinery. Hence, the SCF mode has to be adopted for some time, and the system is scheduled with \(CTK_{1}\) and \(CTK_{3}\) being assigned to \(DS_{1}\), \(CTK_{4}\), and \(CTK_{5}\) to \(DS_{2}\), \(CTK_{8}\), and \(CTK_{9}\) to \(DS_{3}\), \(CTK_{6}\), to \(DS_{4}\), and \(CTK_{7}\) is used for the oil type switch if necessary. The PN model for this system is shown in Figure 10. Note that we do not model the charging tanks that are not available because they have no effect on finding a schedule. Also, in the model, \(CTK_{7}\), which is used for oil type switches, is ignored for concision. In the model, \(\{y,\,t_{i2},\,t_{i3},\,t_{i4},\,t_{i5},\,p_{is},p_{ic},p_{j3}\}\), \(i\in\{1,2,\ldots,10\}\), model \(CTK_{i}\), while \(p_{k}\) and \(t_{k}\), \(k\in\{1,2,3,4\}\), model \(DS_{k}\) and its distillation, respectively. Based on the PN model shown in Figure 10, we code the scheduling method given by Proposition 8 and the evolution rules of the model to verify the proposed method. For this case study, we have  
Figure 9: Refining schedule for the case study. \(\mu\leq\min\{\xi_{1}/\alpha_{1}\), \(\xi_{3}/\alpha_{1}\), \(\xi_{4}/\alpha_{2}\), \(\xi_{5}/\alpha_{2}\), \(\xi_{8}/\alpha_{3}\), \(\xi_{9}/\alpha_{3}\), \(\xi_{6}/\alpha_{4}\), \(\xi_{5}/\alpha_{1}\)} = 8\). To fully utilize the charging tanks, we set \(\mu\) = 8. As \(f_{ds2}<f_{ds1}<f_{ds3}\) holds, we have \(\alpha_{1}\) = (\(\mu\times f_{ds1}+\mu\times f_{ds3}+\mu\times f_{ds4}-2f_{\max}\))/(\(f_{ds4}\)\(+\)\(f_{p\max}\)) = 3.65. It can be verified that \(\mu\geq\) (\(\alpha_{1}\times f_{ds4}+f_{p\max}\))/\(f_{ds4}\) = 7.78 and \(\mu\geq 2f_{p\max}\)/(\(f_{ds1}+f_{ds3}+f_{ds4}\)) = 2.45 hold.