## 6 Conclusions  
We addressed the makespan PFSP with flexible maintenance under learning and deteriorating effects. The maintenance operations are inserted to prevent machine failures with an intelligent PHM-based policy. In addition, the learning effect on the maintenance duration is considered and modeled by a decreasing position-dependent function. On the other hand, the deteriorating effect is assumed to extend the job processing times over time. For the resolution, the well-known ABC algorithm, which has been successfully applied to solve the PFSP without learning and deterioration effects, is adapted to the problem with flexible maintenance, learning, and deteriorating effects in order to find a faster and near-optimal or optimal solution to the problem. To improve the local search in the ABC-employed bees phase, a QL algorithm, considered a common potential tool of the RL mechanism, is carefully integrated into the ABC metaheuristic. Furthermore, a newly designed heuristic, the INEH, is applied to enrich the search space with a high-quality complete solution. The adapted ABC algorithm with the new strategies is used to first solve the integrated PFSP with a learning effect (IQABC-LE) and then also with a deteriorating effect (IQABC-LDE). We conducted a comprehensive experimental study to evaluate the IQABC algorithm without considering the learning and deteriorating effects, the IQABC-LE and IQABC-LDE algorithms. The objective of evaluating these three algorithms is, on the one hand, to justify the significance of our study when we have considered the learning and deteriorating effects, and on the other hand, to prove the effectiveness and robustness of our newly proposed algorithms. For the latter purpose, the experimental results are compared with those of two existing and competitive metaheuristics in the literature: ABC and VNS. The computational results and comparisons demonstrate that the new strategies of the IQABC approach really improve its search performance and IQABC is a competitive approach for the considered problem. Moreover, the need to consider learning and deterioration effects for a more realistic and accurate study is also demonstrated and emphasized. For future work, we propose to integrate more realistic constraints such as the forgetting effect and human resource constraints in the study of the PFSP. We also expect to apply other recently introduced bio-inspired metaheuristics, such as BBO and GWO, to solve the studied scheduling problem, and compare the results with the ABC algorithm. Conceptualization, N.T. ; data curation, N.T. ; formal analysis, N.T., F.B.-S.T., and A.L. ; methodology, F.B.-S.T. and A.L. ; project administration, F.B.-S.T. ; software, N.T. ; supervision, F.B.-S.T. and A.L. ; validation, F.B.-S.T. and A.L. ; writing--original draft, N.T. ; writing--review and editing, F.B.-S.T. All authors have read and agreed to the published version of the manuscript. This research received no external funding. \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-LDE** & **IQABC-LDE** & **ABC-LDE** & **IQABC-LDE** \\ \hline
20x5 & 13.04 & 9.64 & 12.45 & 9.24 \\
20x10 & 23.66 & 19.31 & 23.02 & 19.05 \\
20x20 & 37.3 & 30.82 & 36.17 & 32.10 \\
50x5 & 11.30 & 7.96 & 12.09 & 8.59 \\
50x10 & 18.49 & 14.09 & 19.32 & 13.85 \\
50x20 & 28.54 & 23.09 & 30.06 & 23.17 \\
100x5 & 10.84 & 7.85 & 11.45 & 5.62 \\
100x10 & 13.29 & 10.41 & 13.32 & 8.87 \\
100x20 & 21.41 & 17.43 & 20.32 & 15.55 \\
200x10 & 12.86 & 9.56 & 8.91 & 7.18 \\
200x20 & 17.69 & 12.92 & 15.01 & 12.08 \\
500x20 & 13.04 & 9.84 & 12.45 & 8.71 \\ \hline average & 18.45 & 14.40 & 17.88 & 13.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance offset from [PERSON]’s upper bounds of IQABC-LDE and ABC-LDE considering random learning and deteriorating indexes per machine. Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article.