## 6 Example 4 of General Scheduling Problem with Interruptible Operations  
To illustrate Theorem 8, we describe Example 4 of solvable scheduling problem \(G_{c}MPT[|r_{i}],[t_{ij}],pmtn\big{|}C_{\max}\) with a set of jobs \(J=\{J_{1},J_{2},J_{3},J_{4}\}\), seven dedicated processors \(M=\{M_{1},M_{2},\ldots,M_{7}\}\), integer release dates \(r_{i}\geq 0\), integer processing times \(t_{ij}>0\) of the multi-processor tasks, two types of the precedence constraints between operations, and allowed interruptions of the integer-time operations. **Example 4**. : _The main part of the input data for Example 4 is presented in Table 1. Other input data for Example 4 will be determined after constructing a mixed graph with a set \(W\) of the vertices corresponding to unit-time operations generated by partitioning the integer-time operations \(Q_{ij}\in Q\) based on Remark 2._  
Using the constructive proof of the first part of Theorem 8, we next show how to construct the mixed graph determining the input data of Example 4. Based on Lemma 2, we first partition the integer-time operations \(Q_{1,j}\in Q^{(1)}\) into operations \(W_{1}=\{v_{1},v_{2},v_{3}\}\) with unit durations. Similarly, we partition the integer-time operations \(Q_{2,j}\in Q^{(2)}\) into operations \(W_{2}=\{v_{4},v_{5},v_{6},v_{7},v_{8},v_{9}\}\) with unit durations. Then, we partition the integer-time operations \(Q_{3,j}\in Q^{(3)}\) into operations \(W_{3}=\{v_{10},v_{11},v_{12},v_{13},v_{14},v_{15}\}\) with unit durations. Finally, we partition the operations \(Q_{4,j}\in Q^{(4)}\) of the last job into the operations \(W_{4}=\{v_{16},v_{17},v_{18},v_{19}\}\) with unit durations. As a result, we construct the following set of operations: \(W=\{v_{1},v_{2},\ldots,v_{19}\}=\cup_{i=1}^{4}W_{i}\) with unit durations generated from the integer-time operations of four given jobs \(J=\{J_{1},J_{2},J_{3},J_{4}\}\). Due to Remark 2 and Definition 1, the mixed graph \(G=(V,A,E)\) contains a set of arcs \(A_{i}\) and a set of edges \(E_{i}\) that determine the linear order of the operations of the set \(W_{i}\) in the semiactive schedule that exists for Example 4. The set of arcs \(A_{i}\) and the set of edges \(E_{i}\) must be constructed for each job \(J_{i}\in\{J_{1},J_{2},J_{3},J_{4}\}\). Thus, we have constructed a subgraph \((W,\cup_{i=1}^{|J|}A_{i},\cup_{i=1}^{|J|}E_{i})\) of mixed graph \(G=(V,A,E)\) with inclusion \(W\subset V\); see Figure 4, where the desired mixed graph \(G\) is depicted. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Operations \(Q_{1,j}\) of the job \(J_{1}\) & \(Q_{1,1}\) & \(Q_{1,2}\) & - & - \\ \hline Processors \(M_{\mu(1,j)}\) & \(M_{1},M_{2}\) & \(M_{7}\) & - & - \\ Duration \(t_{1,j}\) of operation \(Q_{1,j}\) & 2 & 1 & - & - \\ \hline Operations \(Q_{2,j}\) of the job \(J_{2}\) & \(Q_{2,1}\) & \(Q_{2,2}\) & \(Q_{2,3}\) & - \\ \hline Processors \(M_{\mu(2,j)}\) & \(M_{3},M_{7}\) & \(M_{1}\) & \(M_{2}\),\(M_{6}\) & - \\ Duration \(t_{2,j}\) of operation \(Q_{2,j}\) & 2 & 1 & 3 & - \\ \hline Operations \(Q_{3,j}\) of the job \(J_{3}\) & \(Q_{3,1}\) & \(Q_{3,2}\) & \(Q_{3,3}\) & \(Q_{3,4}\) \\ \hline Processors \(M_{\mu(2,j)}\) & \(M_{3}\) & \(M_{1}\) & \(M_{4}\) & \(M_{5}\),\(M_{6}\) \\ Duration \(t_{3,j}\) of operation \(Q_{3,j}\) & 1 & 1 & 2 & 2 \\ \hline Operations \(Q_{4,j}\) of the job \(J_{4}\) & \(Q_{4,1}\) & \(Q_{4,2}\) & - & - \\ \hline Processors \(M_{\mu(4,j)}\) & \(M_{1},M_{3}\) & \(M_{4}\),\(M_{5}\) & - & - \\ Duration \(t_{4,j}\) of operation \(Q_{4,j}\) & 1 & 3 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A main part of the input data for Example 4. Figure 4: The mixed graph defining input data for Example 4 of the problem \(G_{c}MPT[t_{i}],\{t_{ij}\},pmtnn|C_{\max}\). For each processor \(M_{k}\in M\), the operation set \(Q_{k}\) defines the set \(V_{k}\) of operations with unit durations to be executed on that processor \(M_{k}\).