## 6 Example 4 of General Scheduling Problem with Interruptible Operations
The cardinality \(|V_{k}|\) of set \(V_{k}\) is equal to the total processing time of the integer-time operations for fulfilling a set of the jobs \(J^{(k)}=\left\{J_{k_{1}},J_{k_{2}},\ldots,J_{|J^{(k)}|}\right\}\), \(k\in\{1,2,\ldots,7\}\). Based on the Formula (8), we calculate these cardinalities as follows:  
\[ \begin{array}{c}|V_{1}|=\sum_{I_{1}\in J^{(1)}}t_{i,1}=2+1+1+1=5,\,|V_{2}|= \sum_{I_{1}\in J^{(2)}}t_{i,2}=2+3=5,\\ |V_{3}|=\sum_{I_{1}\in J^{(3)}}t_{i,3}=2+1+1=4,\,|V_{4}|=\sum_{I_{1}\in J^{(4)} }t_{i,4}=2+3=5,\\ |V_{5}|=\sum_{I_{1}\in J^{(5)}}t_{i,5}=2+3=5,\,|V_{6}|=\sum_{I_{1}\in J^{(6)} }t_{i,6}=3+2=5,\\ |V_{7}|=\sum_{I_{1}\in J^{(7)}}t_{i,7}=1+2=3.\end{array} \]  
The prohibition of simultaneously executing each pair of unit-time operations of set \(V_{k}\) on the processor \(M_{k}\) in any semiactive schedule is determined by the complete \(\left|J^{(k)}\right|\)-partite subgraph \((V_{k},\varnothing,E_{k})\) with vertex set \(\left\{V_{k}^{1},V_{k}^{2},\ldots,V_{k}^{|J^{(k)}|}\right\}\) as \(\left|J^{(k)}\right|\) parts in the \(\left|J^{(k)}\right|\)-partite subgraph \((V_{k},\varnothing,E_{k})\) and edge set \(E_{k},k\in\{1,2,\ldots,7\}\). Based on the Formula (9), we calculate the cardinalities \(|E_{k}|,k\in\{1,2,3,4\}\), as follows:  
\[ \begin{array}{c}|E_{1}|=\sum\limits_{i=1}^{|J^{(1)}|-1}\left\{\sum_{j=i+1}^{ |J^{(1)}|}\left\{|V_{1}^{i}|\times\left|V_{1}^{i}\right|\right\}\right\}=\{ \{2\times 1\}+\{2\times 1\}+\{2\times 1\}\}+\{\{1\times 1\}+\{1\times 1\} \}+\{\{1\times 1\}\}=9,\\ |E_{2}|=\sum\limits_{i=1}^{|J^{(2)}|-1}\left\{\sum_{j=i+1}^{|J^{(2)}|}\left\{| V_{1}^{i}|\times\left|V_{1}^{i}\right|\right\}\right\}=2\times 3=6,\,|E_{3}|=\sum \limits_{i=1}^{|J^{(3)}|-1}\left\{\sum_{j=i+1}^{|J^{(3)}|}\left\{|V_{1}^{i}| \times\left|V_{1}^{i}\right|\right\}\right\}=\{\{2\times 1\}+\{2\times 1\} \}+\{1\times 1\}=5,\\ |E_{4}|=|E_{5}|=|E_{6}|=|E_{2}|=2\times 3=6,\,|E_{7}|=1\times 2=2.\end{array} \]  
It is clear that, due to the 4-partite graph \((V_{1},\varnothing,E_{1})\), it is forbidden to simultaneously fulfill a pair of the operations of set \(V_{1}\) on the processor \(M_{1}\). Due to the tripartite graph \((V_{3},\varnothing,E_{3})\), it is forbidden to simultaneously fulfill a pair of operations from the \(V_{3}\) on the processor \(M_{3}\). Due to the bipartite graph \((V_{2},\varnothing,E_{2})\), it is forbidden to simultaneously fulfill a pair of the operations of set \(V_{2}\) on the processor \(M_{2}\). Due to the bipartite graph \((V_{4},\varnothing,E_{4})\), it is forbidden to simultaneously fulfill a pair of the operations of set \(V_{5}\) on the processor \(M_{5}\). Due to the bipartite graph \((V_{6},\varnothing,E_{6})\), it is forbidden to simultaneously fulfill a pair of the operations of set \(V_{6}\) on the processor \(M_{6}\). Due to the bipartite graph \((V_{7},\varnothing,E_{7})\), it is forbidden to simultaneously fulfill a pair of the operations of set \(V_{7}\) on the processor \(M_{7}\). _Let precedence constraints determined on a set of operations belonging to different jobs in Example 4 be given as follows: \(R_{\omega}=\{v_{4}\Rightarrow v_{1},\ v_{0}\Rightarrow v_{3},v_{13} \Rightarrow v_{16_{6}}\}\). Let the weak precedence constraints in Example 4 be given as follows: \(R_{\rightarrow}=\{v_{1}\to v_{13},\ v_{13}\to v_{5},v_{5} \to v_{1}\}\)._  
To include the precedence constraints \(R_{\rightarrow}\) in the input data of Example 4, we add the set of arcs \(A_{|I|+1}=\{(v_{4},v_{1}),(v_{9},v_{3}),(v_{13},v_{16})\}\) and the edge set \(E_{|I|+1}=\{(v_{1},v_{4}),\{v_{3},v_{9}\},[v_{13},v_{16}]\}\) to the mixed graph \(G=(V,A,E)\). To include weak precedence constraints \(R_{\rightarrow}\) in the input data of Example 4, we add the arc set \(A_{|I|+2}=\{(v_{1},v_{13}),(v_{13},v_{5}),(v_{5},v_{1})\}\) to the mixed graph \(G=(V,A,E)\). We see that set of arcs \(A_{|I|+2}\) determines circuit \((v_{1},v_{13},v_{5},v_{1})\) in the directed graph \((V,A,\varnothing)\). Therefore, unit-time operations of set \(\{v_{1},v_{13},v_{5}\}\) have to be simultaneously fulfilled.