# Minimum-Energy Scheduling of Flexible Job-Shop Through Optimization and Comprehensive Heuristic
## 5 Evaluation of the Proposed Scheduling Heuristic
### Illustration of Resulting Scheduling in a Flexible Job Shop  
The number of jobs \(n\) and the number of machines \(m\) are equal to 7, and the job shop is organized into three rows of machines to process raw material (job 1 and job 2), semi-raw material (job 3, job 4, and job 5), and semi-finished products (job 6 and job 7). The machines of the job shop and the possible connections between them are represented in Figure 9. There, jobs 1 and 2 have three operations; jobs 3, 4, and 5 have two operations; and jobs 6 and 7 have one operation. To these operations, a dummy initial and a dummy final operation are added for each job. For job 1, operation \(O_{12}\) can be performed by machines 1 and 2, operation \(O_{13}\) can be performed by machines 3 or 4, and operation \(O_{14}\) can be performed by machine 6. For job 2, operation \(O_{22}\) can be performed by machines 1 and 2, operation \(O_{23}\) can be performed by machines 3 or 4, and operation \(O_{24}\) can be performed by machine 7. For job 3, operation \(O_{32}\) can be performed by machine 4, and operation \(O_{33}\) can be performed by machines 6 or 7. For job 4, operation \(O_{42}\) can be performed by machine 3, and operation \(O_{43}\) can be performed by machine 5, and operation \(O_{53}\) can be performed by machine 6 or 7. For job 5, operation \(O_{52}\) can be performed by machine 5, and operation \(O_{53}\) can be performed by machine 6 or 7. For job 6, operation \(O_{62}\) can be performed by machine 6. For job 7, operation \(O_{72}\) can be performed by machine 7. This shows that there is flexibility in the assignment of machines to some operations of each job. Figure 8: GANTT diagrams of the different heuristics. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Heuristic** & **TTE** & **ETT** & **HET** \\ \hline Make-span & 38 & 43 & 42 \\ \hline Energy & 48 & 35 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of performances between heuristic and priority rules. The following tables provide data on the operations' processing times and energy per machine (Table 6), the transportation delays and energy between machines (Table 7), and finally, the release and due times (Table 8). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Machines** & \(\mathbf{m_{1}}\) & \(\mathbf{m_{2}}\) & \(\mathbf{m_{3}}\) & \(\mathbf{m_{4}}\) & \(\mathbf{m_{5}}\) & \(\mathbf{m_{6}}\) & \(\mathbf{m_{7}}\) \\ \hline Job 1 & 8/50 & 12/40 & 15/70 & 15/60 & — & 10/50 & — \\ \hline Job 2 & 10/55 & 7/40 & — & 15/70 & 15/55 & — & 8/60 \\ \hline Job 3 & — & — & — & 8/30 & — & 15/30 & 15/25 \\ \hline Job 4 & — & — & 15/50 & — & — & 10/40 & 8/45 \\ \hline Job 5 & — & — & — & — & 10/60 & 6/60 & 10/50 \\ \hline Job 6 & — & — & — & — & — & 12/50 & — \\ \hline Job 7 & — & — & — & — & — & 15/55 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Processing delays/energy per jobs and machines. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Delay/Energy** & \(\mathbf{m_{1}}\) & \(\mathbf{m_{2}}\) & \(\mathbf{m_{3}}\) & \(\mathbf{m_{4}}\) & \(\mathbf{m_{5}}\) & \(\mathbf{m_{6}}\) & \(\mathbf{m_{7}}\) \\ \hline \(\mathbf{m_{1}}\) & — & — & 5/5 & 5/5 & 8/10 & — & — \\ \hline \(\mathbf{m_{2}}\) & — & — & 8/10 & 5/5 & 5/5 & — & — \\ \hline \(\mathbf{m_{3}}\) & — & — & — & — & — & 5/15 & 8/10 \\ \hline \(\mathbf{m_{4}}\) & — & — & — & — & — & 5/5 & 5/5 \\ \hline \(\mathbf{m_{5}}\) & — & — & — & — & — & 8/10 & 5/5 \\ \hline \(\mathbf{m_{6}},\mathbf{m_{7}}\) & — & — & — & — & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Transportation delays and energy. Figure 9: The considered job shop. Here, only the delays and energy consumption resulting from the transfer of products between machines are considered; they are also supposed to be independent of the type of product that is transported. The due times were chosen so that the jobs requiring only final machining (jobs 6 and 7) are cleared first to free up space in the job shop for the remaining activities of the other jobs.