## 5 Scheduling for General Systems  
In the last section, we discussed the scheduling problem for different cases. In this section, we extend these results to a general system with \(K\) distillers. For a system with \(K\) distillers and \(K\) charging tanks, it is obvious that a schedule with one tank feeding one distiller under the SCF mode can be found such that \(\delta=1\). Then, we discuss the case with \(2K-N\) charging tanks, where \(N<K\). Case I: \(2k-N\) Charging Tanks  
For a system with \(K\) distillers and \(2K-N\) charging tanks, where \(N<K\), only one charging tank is assigned to each of distiller \(DS_{1-N}\), and the other distillers have two charging tanks. Let charging tank \(CTK_{i}\) be represented by \(\{p_{is},\)\(p_{ic},\)\(t_{i2},\)\(t_{i3},\)\(t_{i4},\)\(t_{i5}\}\) and distiller \(DS_{i}\) be represented by \(p_{i}\) in a PN model. Then, we present the following results. **Proposition 7**. : _Assume that: (1) there are \(K\geq 3\) distillers \(DS_{1-K}\) with feeding rates \(f_{\text{d81}}\)-\(f_{\text{d8K}}\), and \(\alpha=\Psi\)\(\times\)\(f_{\text{pmax}}\), \(\alpha_{1}=\Psi\)\(\times\)\(f_{\text{d81}}\), \(\alpha_{2}=\Psi\)\(\times\)\(f_{\text{d82}}\),..., \(\alpha_{K}=\Psi\)\(\times\)\(f_{\text{d8K}}\); (2) there are \(2K-N\) (\(N<K\)) charging tanks \(CTK_{1-(2K-N)}\) with capacities \(\xi_{1-(2K-N)}\); (3)\(f_{\text{pmax}}=f_{\text{d81}}+f_{\text{d82}}+\ldots+f_{\text{d8K}}\); and (4) initially, the volume of oil type 1 (i; j; j) in \(CTK_{1}\) (\(CTK_{i}\); \(CTK_{(2j-N-1)}\); \(CTK_{(2j-N)}\)) is \(\zeta_{1}\) (\(\zeta_{ij}\); \(\zeta_{(2j-N-1)}\)); \(\zeta_{(2j-N)}\)), \(i\in\{2,\)\(3,\)\(\ldots,\)\(N\}\) and \(j\in\{N+1,\)\(N+2,\)\(\ldots,\)\(K\}\). Then, if Condition (9) holds, a schedule can be found with \(\delta=N/K\)._  
\[\left\{ \begin{array}{l}\sigma\leq\zeta_{1}\leq\xi_{1}-\mu\times(\alpha-\alpha _{1})\times f_{\text{d81}}/f_{\text{pmax}}\\ \ldots\\ \sigma+\mu\times\sum_{k=1}^{i-1}\alpha_{k}\times f_{\text{d81}}/f_{\text{pmax}} \leq\zeta_{i}\leq\xi_{i}-\mu\times\left(\alpha-\sum_{k=1}^{i}\alpha_{k}\right) \times f_{\text{d8i}}/f_{\text{pmax}}\\ \ldots\\ \sigma+\mu\times\sum_{k=1}^{N-1}\alpha_{k}\times f_{\text{d8N}}/f_{\text{pmax}} \leq\zeta_{N}\leq\xi_{N}-\mu\times\left(\alpha-\sum_{k=1}^{N}\alpha_{k}\right) \times f_{\text{d8N}}/f_{\text{pmax}}\\ \zeta_{(N+1)}=\alpha_{(N+1)}\\ \zeta_{(N+2)}=\mu\times\alpha_{(N+1)}\\ \ldots\\ \zeta_{(2j-N-1)}=\alpha_{j}\\ \zeta_{(2j-N)}=\mu\times\alpha_{j}\\ \ldots\\ \zeta_{(2K-N-1)}=\alpha_{K}\\ \zeta_{(2K-N)}=\mu\times\alpha_{K}\\ \end{array} \right., \tag{9}\]_where \(\mu\in[f_{pmax}/(f_{ds1}+f_{ds2}+\ldots+f_{dsN})\), \(min\{\zeta_{(N+1)}/\alpha_{(N+1)}\}\), \(\xi_{(N+2)}/\alpha_{(N+1)}\),..., \(\xi_{(2j-N-1)}/\alpha_{j}\), \(\xi_{(2j-N)}/\alpha_{j}\),..., \(\xi_{(2K-N-1)}/\alpha_{K}\), \(\xi_{(2K-N)}/\alpha_{K}\)}]\)._  
Proof. : See Appendix A.7. Proposition 7 extends Propositions 1, 3, and 5 and presents a scheduling method for a general system with \(K\) distillers and \(2K-N\) charging tanks. By the method given in Proposition 7, every time a charging tank for feeding distiller \(DS_{i}\), \(i\in\{1,3,\,\ldots,\,K\}\), is charged, \(\mu\times\alpha_{i}\) of oil is charged no matter what the operation mode of the charging tank is. Also, if the normal mode is applied, a charging tank is emptied after feeding a distiller. Thus, a schedule is easily found. Note that charging tanks \(CTK_{2j-N}\) are empty during \([\tau_{0}+\Psi,\,\tau_{0}+\mu\times(\alpha_{1}+\alpha_{2}+\ldots+\alpha_{j-1})/ f_{pmax}]\), \(j\in\{N+1,N+2,\,\ldots,\,K\}\), and can be used to reduce \(\delta\). Hence, among these charging tanks, \(Q\) tanks with \(1\leq Q\leq\min[N,\,K-N]\) can be used to feed \(DS_{(N-Q+1)-N}\) under the normal mode.