# Near-Optimal Heuristics for Just-In-Time Jobs Maximization in Flow Shop Scheduling
## 5 Computational Experiments and Results
### Experiment 2: Quality of the Heuristic Solutions in Relation to the Optimal Solution  
In the second part of the computational experimentation, the quality of each heuristic solution was measured by the RPD in relation to the optimal solution provided by solving the instances of mathematical model by CPLEX. The RPD is calculated by Expression (14), where \(n_{JIT}^{best}\) is the optimal solution given by CPLEX. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(n\) & **H1** & **H2** & **H3** & **H4** & **H5** & **H6** & **H7** & **H8** & **H9** & **H10** \\ \hline
15 & 3.9 & 1.5 & 10.9 & 6.4 & 0.7 & 0.4 & 59.6 & 41.3 & 26.2 & 40.1 \\
20 & 3.6 & 1.1 & 9.4 & 6.2 & 0.5 & 0.4 & 69.4 & 55.7 & 28.3 & 47.5 \\
30 & 3.3 & 0.7 & 8.9 & 6.7 & 0.3 & 0.2 & 77.7 & 69.0 & 30.5 & 57.5 \\
50 & 2.1 & 0.3 & 8.5 & 6.7 & 0.3 & 0.1 & 84.6 & 79.6 & 33.4 & 69.3 \\
80 & 1.0 & 0.1 & 8.8 & 7.3 & 0.1 & 0.1 & 89.4 & 86.3 & 34.8 & 76.3 \\
100 & 0.6 & 0.0 & 25.1 & 8.7 & 0.2 & 0.1 & 78.9 & 76.6 & 35.1 & 77.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of performances (RPD) of heuristics by number of jobs for Group 2. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(m\) & **H1** & **H2** & **H3** & **H4** & **H5** & **H6** & **H7** & **H8** & **H9** & **H10** \\ \hline
2 & 0.3 & 0.5 & 7.1 & 3.7 & 0.2 & 0.1 & 42.2 & 16.5 & 7.5 & 20.8 \\
3 & 0.9 & 0.8 & 7.4 & 3.7 & 0.3 & 0.1 & 42.3 & 16.7 & 9.6 & 21.3 \\
5 & 2.4 & 1.9 & 6.7 & 3.5 & 0.7 & 0.5 & 40.7 & 15.5 & 11.8 & 21.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of performances (RPD) of heuristics by number of machines for Group 1. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(m\) & **H1** & **H2** & **H3** & **H4** & **H5** & **H6** & **H7** & **H8** & **H9** & **H10** \\ \hline
5 & 0.5 & 0.3 & 12.2 & 7.2 & 0.2 & 0.1 & 78.0 & 69.8 & 28.6 & 61.8 \\
10 & 2.1 & 0.7 & 13.0 & 7.1 & 0.4 & 0.2 & 75.9 & 68.0 & 30.7 & 61.3 \\
15 & 3.1 & 0.6 & 11.0 & 7.1 & 0.3 & 0.2 & 76.9 & 68.2 & 32.5 & 61.3 \\
20 & 3.9 & 0.8 & 11.6 & 6.6 & 0.5 & 0.3 & 75.5 & 66.4 & 33.7 & 61.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of performances (RPD) of heuristics by number of machines for Group 2. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & **Scenario** & **H1** & **H2** & **H3** & **H4** & **H5** & **H6** & **H7** & **H8** & **H9** & **H10** \\ \hline \multirow{4}{*}{Group 1} & 1 & 0.9 & 0.8 & 8.7 & 4.6 & 0.4 & 0.3 & 42.2 & 16.5 & 10.0 & 20.8 \\  & 2 & 1.0 & 0.7 & 4.7 & 2.2 & 0.2 & 0.2 & 45.7 & 19.2 & 4.3 & 20.0 \\  & 3 & 1.9 & 1.7 & 9.3 & 5.1 & 0.8 & 0.5 & 39.3 & 14.7 & 15.0 & 22.1 \\  & 4 & 1.0 & 1.0 & 5.5 & 2.7 & 0.1 & 0.1 & 39.8 & 14.5 & 9.2 & 21.5 \\ \hline \multirow{4}{*}{Group 2} & 1 & 1.7 & 0.7 & 15.0 & 8.9 & 0.4 & 0.2 & 76.3 & 68.1 & 39.4 & 61.6 \\  & 2 & 1.6 & 0.5 & 7.8 & 4.5 & 0.1 & 0.1 & 78.6 & 71.3 & 13.8 & 62.0 \\ \cline{1-1}  & 3 & 3.7 & 1.0 & 15.2 & 9.4 & 0.7 & 0.4 & 75.1 & 65.4 & 43.0 & 60.9 \\ \cline{1-1}  & 4 & 2.5 & 0.3 & 9.7 & 5.3 & 0.1 & 0.1 & 76.3 & 67.6 & 29.3 & 61.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Performances (RPD) of heuristics by group and scenario in relation to EM for Group 1 and to the best found solution for Group 2. The optimality of the CPLEX solution was proven for the 6000 instances of Group 1. The analysis of quality the heuristics' results in relation to those optimal solutions, using 95% confidence intervals of the average RPDs, is depicted in Figure 3.  
It is remarkable that the values of graphs in Figure 2a (Group 1) and Figure 3 are very similar. Table 10 presents rankings of the solution quality of the heuristics, i.e., the values of the RPD of the optimal solutions. Table 10 shows the same ranking and values very close to the ones presented in Table 4 for relative comparisons of Group 1, which validated and reinforced the results from the previous analyses.