## 5 The Computational Results and Analysis
### Performance Analysis of the QL-Driven ABC Algorithm with Learning Effect  
The performance of the first algorithm, namely the IQABC-LE, is proven through three scenarios and 1100 executions per scenario on the designed benchmarks completed with learning indexes (\(\alpha\)). First, we assume a small common learning rate among maintenance operators, so we generate small learning indexes (SF) from the uniform distribution \(\mathcal{U}[0;0.20]\). Then, assuming that the operators learn very quickly at the same rate, large learning indices (LF) are generated from the uniform distribution \(\mathcal{U}[0.80;1]\). Finally, we assume that the learning phenomenon is random and depends on the machine characteristics, so the learning indexes are generated from the uniform distribution \(\mathcal{U}[0;1]\) per machine (FPM). The execution results are compared with the ABC learning algorithm (ABC-LE) reported in [54]. Indeed, the IQABC-LE algorithm implements a QL mechanism to guide the neighbor structure selection in the employed bees phase, instead of random selection as in ABC-LE. In addition, the IQABC-LE uses the INEH heuristic to generate one complete solution, while the ABC-LE algorithm applies the NEH heuristic [43] to generate first, the production sequence, then, the maintenance heuristic[44] to complete the solution with maintenance emplacements. The comparison between the two algorithms is made according to the ARPD values, which are calculated on the basis of the \(C_{max}\) values given in Formula (4). The results presented in Tables 3-5 clearly show the superiority of the IQABC-LE algorithm over the ABC-LE algorithm. An important note comparing the results of the IQABC algorithm when considering a learning effect and no effect, is the significant  
Figure 7: Average convergence of the IQABC for the 50x20_M1 instance. difference in ARPD values, which increasingly reinforce the need to take into account real constraints when studying scheduling problems. From Figure 8, it can be seen that the IQABC-LE algorithm with large learning indexes (LF) optimizes the \(C_{max}\) very well compared to the small and random learning indexes. Consequently, it is of great importance for a workshop to improve the ability of its employees to learn. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**M1**} & \multicolumn{2}{c}{**M2**} \\ \cline{2-5}  & **ABC-LE(LF)** & **IQABC-LE(LF)** & **ABC-LE(LF)** & **IQABC-LE(LF)** \\ \hline
20x5 & 4.73 & 3.06 & 3.29 & 3.51 \\
20x10 & 13.34 & 11.47 & 14.22 & 12.52 \\
20x20 & 25.1 & 23.23 & 26.95 & 24.88 \\
50x5 & 0.21 & \(-\)1.06 & \(-\)0.43 & \(-\)2.53 \\
50x10 & 6.85 & 3.70 & 8.15 & 2.97 \\
50x20 & 16 & 12.35 & 18.69 & 12.39 \\
100x5 & \(-\)2.82 & \(-\)3.15 & \(-\)4.3 & \(-\)4.94 \\
100x10 & \(-\)0.4 & \(-\)1.45 & \(-\)1.62 & \(-\)3.60 \\
100x20 & 7.71 & 5.83 & 6.52 & 3.75 \\
200x10 & \(-\)3.87 & \(-\)4.13 & -6.36 & \(-\)7.05 \\
200x20 & 0.7 & \(-\)0.22 & \(-\)1.36 & \(-\)2.52 \\
500x20 & \(-\)4.39 & \(-\)4.79 & \(-\)8.17 & \(-\)8.52 \\ \hline average & 5.26 & 3.73 & 4.63 & 2.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance offset from [PERSON]’s upper bounds of IQABC-LE and ABC-LE considering large learning indexes. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**M1**} & \multicolumn{2}{c}{**M2**} \\ \cline{2-5}  & **ABC-LE(SF)** & **IQABC-LE(SF)** & **ABC-LE(SF)** & **IQABC-LE(SF)** \\ \hline
20x5 & 4.47 & 2.79 & 3.35 & 3.33 \\
20x10 & 12.81 & 11.36 & 14.44 & 12.27 \\
20x20 & 24.76 & 23.24 & 27.20 & 24.88 \\
50x5 & 1.87 & 0.27 & 2.24 & \(-\)0.25 \\
50x10 & 8.98 & 5.43 & 12.26 & 6.27 \\
50x20 & 19.43 & 14.65 & 23.37 & 16.26 \\
100x5 & 0.39 & 0.25 & 0.38 & \(-\)0.14 \\
100x10 & 3.7 & 2.37 & 4.93 & 2.50 \\
100x20 & 12.39 & 10.46 & 14.46 & 11.28 \\
200x10 & 0.43 & 0.76 & 0.74 & \(-\)0.13 \\
200x20 & 6.31 & 5.01 & 7.16 & 5.66 \\
500x20 & 1.26 & 1.43 & 0.23 & 0.42 \\ \hline average & 8.07 & 6.50 & 9.23 & 6.86 \\ \hline \hline \end{tabular}
\end{table}