## 6 Example 4 of General Scheduling Problem with Interruptible Operations
It is clear that this strict order \((v_{k_{|V(k)|}},v_{k_{|V(k)|-1}},\ldots,v_{k_{2}},v_{k_{1}})\) is opposite to the strict order \((v_{k_{1}},v_{k_{2}},\ldots,v_{k_{|V(k)|}})\) determined for the corresponding problem \(G_{c}MPT[[r_{i}],[t_{ij}],pmtn,[C_{\max}\) for processing integer-time operations \(V^{(k)}\). If the precedence constraints \(v_{i}\Rightarrow v_{j}\) and the weak precedence constraints \(v_{p}\to v_{q}\) are defined on the set of operations belonging to different jobs in the corresponding problem \(G_{c}MPT[[r_{i}],[t_{ij}],pmtn,[D_{i}]|L_{\max}\), then the inverse precedence constraint \(v_{j}\Rightarrow v_{i}\) and the inverse weak precedence constraints \(v_{q}\to v_{p}\) have to be given for the problem \(G_{c}MPT[[t_{ij}],pmtn,[D_{i}]|L_{\max}\). It follows from Theorem 8 that the optimal semiactive schedule for the roblem \(G_{c}MPT[[r_{i}],[t_{ij}],pmtn,[C_{\max}\) determined by the mixed graph \(G=(V,A,E)\) can be defined by the optimal colouring \(c(G)\). This colouring \(c(G)\) also defines an optimal semiactive schedule for the problem \(G_{c}MPT[[t_{ij}],pmtn,[D_{i}]|L_{\max}\). It is clear that the optimal maximal lateness may be determined as follows: \(L_{\max}=\chi(G)-\max\{r_{i}:J_{i}\in J\}\). Thus, we proved the following claim. _Every solvable scheduling problem \(G_{c}MPT[[r_{i}],[t_{ij}],pmtn\)\(|C_{\max}\) with given integer release dates \(r_{i}\geq 0\), \(J_{i}\in J\), and integer durations of interruptible operations \(Q_{ij}\) is equivalent to the scheduling problem \(G_{c}MPT[[t_{ij}],pmtn,[D_{i}]|L_{\max}\) with integer due dates \(D_{i}=\max\{r_{j}:J_{j}\in J\}-r_{i}\), \(J_{i}\in J\), and integer durations of the interruptible operations \(Q_{ij}\), \(J_{i}\in J\)._  
Obviously, the above arguments are reversible. This allows us to prove the following claim in a similar way. _Every solvable scheduling problem \(G_{c}MPT[[t_{ij}],pmtn,[D_{i}]|L_{\max}\) with given integer due dates \(D_{i}\geq 0\), \(J_{i}\in J\), and integer durations of the interruptible operations \(Q_{ij}\), is equivalent to the scheduling problem \(G_{c}MPT[[r_{i}],[t_{ij}],pmtn,[D_{i}]|L_{\max}\) with given integer release dates \(r_{i}=\max\{D_{j}:J_{j}\in J\}-D_{i}\) and integer durations of the interruptible operations \(Q_{ij}\), \(J_{i}\in J\)._  
Theorem 9 is proved. \(\Box\)  
The following corollary follows directly from Theorem 8 and Theorem 9. Corollary 3: _If the job due dates and durations of operations \(Q_{ij},J_{i}\in J\), are integers for the solvable scheduling problem \(G_{c}MPT[[t_{ij}],pmtn,[D_{i}]|L_{\max}\), then an semiactive optimal schedule for this problem is presentable as an optimal colouring of vertices of the mixed graph \(G\) that determines the input data of this scheduling problem._  
To illustrate Theorem 9 and Corollary 3, we solve Example 5 of the problem \(G_{c}MPT[[t_{ij}],pmtn,[D_{i}]|L_{\max}\) with four jobs \(J=\left\{J_{1},J_{2},J_{3},J_{4}\right\}\), seven dedicated processors \(M=\left\{M_{1},M_{2},\ldots,M_{7}\right\}\), integer due dates \(D_{i}\geq 0\), integer processing times \(t_{ij}>0\) of the multi-processor operations, precedence constraints between operations of different and the same jobs, and allowed interruptions of the integer-time operations. **Example 5**. : _In Example 5 of the scheduling problem \(G_{c}MPT\big{|}[t_{ij}],pmtn,[D_{i}]\big{|}L_{\max}\) on the mixed graph \(G=(V,A,E)\) depicted in Figure 5, the set of jobs \(J_{1},\ J_{2},\ J_{3}\) and \(J_{4}\) are ready for processing on set \(M=\{M_{1},M_{2},\ldots,M_{9}\}\) of the dedicated processors from time zero \(t=0\). Each processor \(M_{k}\in M\) must process the same subset of the jobs as in Example 4.