# Energy-Efficient Hybrid Flow-Shop Scheduling under Time-of-Use and Ladder Electricity Tariffs
## 4 Case Study
### Random Instances
For small-scale and medium-scale cases, the two target values of the best solutions obtained by the I-NSGA-II are both smaller than or equal to the corresponding target value of the traditional NSGA-II. It can be seen from Table 7 that the average solution of the I-NSGA-II is better than the average solution of the traditional algorithms. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Factors** & **Values** & **Number of Levels** \\ \hline Number of jobs & \(10,20,50\) & 3 \\ Number of stages & \(3,5,8\) & 3 \\ Number of parallel machines & 2, 4 & 2 \\ Processing time (min) & Discrete uniform [10; 50] & 1 \\ Power of machine (kW) & Discrete uniform [5; 10] & 1 \\ Standby power of machine (kW) & 1 (fixed) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of test data. Figures 9-11 show the Pareto diagram of each instance. Through the graph, we can see the comparison results directly. Figure 9 represent the approximate Pareto fronts found by I-NSGA-II and NSGA-II for 10 jobs. It can be seen from the figure that, on average, the Pareto solutions obtained by I-NSGA-II are better than those obtained by NSGA-II. For small-scale calculation examples where there is right-shift operation space, TEC can reach a significant decline, such as 10-3-4, 10-5-4, and 10-8-4. Figure 10 shows the Pareto fronts found by the methods for 20-job instances. The results are similar with the small instances. For medium-scale calculation examples, the degree of improvement is reduced. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n\) & **Instance** & \multicolumn{3}{c}{**NSGA-II**} & \multicolumn{3}{c}{**I-NSGA-II**} \\ \hline  & **Factors** & **Makespan (h)** & **TEC (kJ)** & **Makespan (h)** & **TEC (kJ)** & **dev (TEC)** \\ \hline
10 & 10-3-2 & 3.02 & 113.949 & 2.90 & 97.144 & 14.75\% \\  & 10-5-2 & 5.01 & 249.721 & 4.87 & 215.836 & 13.57\% \\  & 10-8-2 & 6.28 & 385.581 & 6.28 & 385.581 & 10.98\% \\  & 10-3-4 & 2.12 & 104.948 & 2.05 & 66.321 & 36.81\% \\  & 10-5-4 & 3.51 & 234.965 & 3.51 & 149.124 & 36.53\% \\  & 10-8-4 & 4.98 & 381.096 & 4.98 & 260.402 & 31.67\% \\ \hline
20 & 20-3-2 & 6.16 & 285.215 & 6.1 & 263.268 & 7.69\% \\  & 20-5-2 & 7.08 & 458.979 & 7.08 & 434.537 & 5.33\% \\  & 20-8-2 & 7.13 & 469.815 & 7.13 & 441.721 & 5.98\% \\  & 20-3-4 & 3.56 & 290.724 & 3.56 & 229.164 & 21.01\% \\  & 20-5-4 & 4.70 & 496.805 & 4.55 & 415.557 & 16.19\% \\  & 20-8-4 & 5.93 & 800.921 & 5.93 & 683.832 & 15.46\% \\ \hline
50 & 50-3-2 & 13.57 & 403.096 & 13.57 & 389.828 & 3.29\% \\  & 50-5-2 & 14.51 & 674.565 & 14.50 & 634.58 & 5.93\% \\  & 50-8-2 & 15.73 & 1007.18 & 15.73 & 974.164 & 3.28\% \\  & 50-3-4 & 7.30 & 646.214 & 7.22 & 621.997 & 3.75\% \\  & 50-5-4 & 8.18 & 976.182 & 8.18 & 1115.527 & 3.05\% \\  & 50-8-4 & 9.58 & 1391.56 & 5.58 & 1351.92 & 2.85\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparisons of algorithmic BEST performance. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & **Instance** & \multicolumn{3}{c}{**NSGA-II**} & \multicolumn{3}{c}{**I-NSGA-II**} \\ \hline  & **Factors** & **Makespan (h)** & **TEC (kJ)** & **Makespan (h)** & **TEC (kJ)** & **dev (TEC)** \\ \hline
10 & 10-3-2 & 3.02 & 120.340 & 3.04 & 109.641 & 8.89\% \\  & 10-5-2 & 5.01 & 258.162 & 5.01 & 230.943 & 10.54\% \\  & 10-8-2 & 6.62 & 400.963 & 6.62 & 362.007 & 9.72\% \\  & 10-3-4 & 2.12 & 104.948 &  
Figure 9: Pareto fronts found by each algorithm for 10-job instances. Figure 11 presents the Pareto fronts found by I-NSGA-II and NSGA-II for large instances with 50 jobs. In general, it may be concluded from the figures that the improved algorithm does not achieve ideal results, which shows that there are certain limitations. As the number of workpieces increases while machines used for processing have not increased, the space for right movement operations is reduced. Nevertheless, energy consumption can still be improved. Figure 10: Pareto fronts found by each algorithm for 20-job instances.