## 5 Scheduling for General Systems
Let \(\omega_{1}=(\mu\times f_{ds1}+\mu\times f_{ds2}+\ldots+\mu\times f_{ds(K-Q)}-2 \times f_{pmax})/(f_{ds(N-Q+1)}+f_{pmax})\), and \(\omega_{q}=(\mu\times f_{ds1}+\mu\times f_{ds2}+\ldots+\mu\times f_{ds(q+K-Q-1 )}-2\times f_{pmax}-\omega_{1}\times f_{ds(N-Q+1)}-\omega_{2}\times f_{ds(N-Q+2 )}-\ldots-\alpha_{(q-1)}\times f_{ds(q+N-Q-1)}/(f_{ds(q+N-Q)}\)\(+f_{pmax}),q\in[2,3,\,\ldots,\,Q]\), \(\omega=\omega_{1}+\omega_{2}+\ldots+\omega_{Q}\) and \(\Omega_{q}=(\omega_{1}\times\alpha_{(N-Q+1)}+\omega_{2}\times\alpha_{(N-Q+2)} +\ldots+\omega_{q}\)\(\times\alpha_{(q+N-Q)})/f_{pmax}\). Further, let \(\varepsilon_{i}=\mu\times\alpha_{i}/f_{pmax}\) if \(1\leq i\leq N-\,Q+1\) and \(\varepsilon_{i}=(\mu-\omega_{(i-N+Q)})\times\alpha_{i}/f_{pmax}\) if \(N-\,Q+1<i\leq N\). Then, there exists an integer \(L\) such that \(\varepsilon_{1}+\varepsilon_{2}+\ldots+\varepsilon_{(L-1)}\leq\Psi\leq \varepsilon_{1}+\varepsilon_{2}+\ldots+\varepsilon_{L}\) and \(1\leq L\leq N\) hold. We present the following results. **Proposition 8**. : _Assume that: (1) there are \(K\geq 3\) distillers \(DS_{1-K}\) with feeding rates \(f_{ds1}\)-\(f_{dsK}\), and \(\alpha=\Psi\times f_{pmax}\), \(\alpha_{1}=\Psi\times f_{ds1}\), \(\alpha_{2}=\Psi\times f_{ds2}\),..., \(\alpha_{K}=\Psi\times f_{dsK}\), \(f_{ds1}\geq f_{ds2}\geq\ldots\geq f_{dsN}\) and \(f_{ds(N+1)}\geq f_{ds(N+2)}\geq\ldots\geq f_{dsK^{\prime}}\) (2) there are \(2K-N\) (N < \(K\)) charging tanks \(CTK_{1-(2K-N)}\) with capacities \(\xi_{1-(2K-N)}\); (3) \(f_{pmax}=f_{ds1}+f_{ds2}+\ldots+f_{dsK^{\prime}}\) and (4) initially, the volume of oil type 1 (\(L\); \(L\); \(l\); \(j\)) in \(CTK_{1}\) (\(CTK_{i}\); \(CTK_{L}\); \(CTK_{l}\); \(CTK_{(2)-N-1)}\); \(CTK_{(2)-N-1)}\) is \(\xi_{1}\) (\(\xi_{j}\); \(\xi_{L}\); \(\xi_{(2)}\); \(\xi_{(2)-N-1)}\); \(\xi_{(2)-N)}\), where \(i\in\{2,\,3,\,\ldots,\,L-1\}\), \(l\in\{L+1,\,L+2,\,\ldots,\,N\}\), and \(j\in\{N+1,N+2,\,\ldots,\,K\}\). Then, if Condition (10) holds, a schedule can be found with \(\delta\) = (\(N\times\mu\)\(-\omega\))/(\(K\times\mu\))/(\(K\times\mu\))._  
\[\left\{\begin{array}{l}\sigma\leq\zeta_{1}\leq\xi_{1}-\varepsilon_{1}\times f _{pmax}+\varepsilon_{1}\times f_{ds1}\\ \ldots\\ \sigma+\sum_{k=1}^{i-1}\varepsilon_{k}\times f_{dsi}\leq\zeta_{i}\leq\zeta_{i}- \varepsilon_{i}\times f_{pmax}+\sum_{k=1}^{i}\varepsilon_{k}\times f_{dsi}\\ \ldots\\ \sigma+\sum_{k=1}^{L-1}\varepsilon_{k}\times f_{dsL}\leq\zeta_{L}\leq\xi_{L}- \alpha+\sum_{k=1}^{L-1}\varepsilon_{k}\times f_{pmax}+\alpha_{L}\\ \sigma-\alpha+\sum_{k=1}^{L-1  
**Proof. ** See Appendix A.8. Proposition 8 is the extension of Propositions 2, 4, and 6 and presents a method to find a schedule for a general situation with \(2K-N\) charging tanks. The method given by Proposition 8 improves the one given by Proposition 7 by using the idle time of charging tanks to feed distillers under the normal mode. Thus, one needs to examine the idle time of such charging tanks and schedule the idle time by the method given in Proposition 8. The key is that, during the idle time of charging tank \(CT_{K(2q+2K-2Q}\)\({}_{N-1})\) or \(CT_{(2q+2K-2Q-N)}\), \(q\in\{1,\,2,\,\ldots,\,Q\}\), for feeding \(DS_{(q+K-Q)}\), it is charged with volume \(\varpi_{q}\times\alpha_{(q+N-Q)}\) of oil type \((q+N-Q)\), and feed \(DS_{(q+N-Q)}\) under the normal mode for some time. However, the total volume of oil type \(k\), \(k\in\{1,2,\,\ldots,\,K\}\) that is charged into charging tanks remains \(V_{k}=\mu\times\alpha_{k}\), which is the same as that in Proposition 7. However, for oil of type \(i\), \(i\in\{1,2,\,\ldots,\,N\}\), it may be divided into several parts and charged into the charging tanks sequentially according to the proposed method. Note that, in order to reduce the number of switches in feeding distillers, we usually take the maximal value of \(\mu\). While adopting Proposition 8, the maximal \(\mu\) is calculated first. Then, this value is used to calculate the corresponding \(\varpi\).