## 6 Conclusions  
In this paper, we have proposed the Late Acceptance Hill-climbing Matheuristic (LAHCM) as a general framework for solving the optimization problem. It has been successfully applied to the general lot sizing and scheduling problem with defective items and lifetime constraints (GLSP-RP). Moreover, we have discussed the differences between the original GLSP and the GLSP-RP and gave an extensive overview of solution approaches and conducted research on several extensions of the GLSP. We exemplified the proceeding of the LAHCM on an example case and later applied the algorithm on several data classes. The overall results show that LAHCM performs very well on the GLSP-RP and is able to find optimal solutions or at least better solutions in much less time than a general purpose exact solver like CPLEX. While we were still not able to solve all instances to optimality, we were able to find 48 better or equal solutions for data classes A and B and for 40 instances for class C out of 50 instances. The results of the method also showed that increasing the single-point parameter \(l\) led to better solution values while the trade-off between better solution values and computational times has to be closely monitored. Moreover, the performance of LAHCM advises its use in practical environments where, due to rework and lifetime constraints, fast and reliable solutions are required in reasonable computational times.  
Further research will be conducted towards studying the performance of LAHCM on other optimization problems, especially those where exact approaches are computationally limited because of the instance size. In that regard, a further research direction is to investigate how different solvers could be exploited using LAHCM for the same given problem. Moreover, steering mechanisms inside the LAHCM such as defining upper-level rules for determining the application of the fixing strategies will be investigated. In addition, research will be conducted in the area of different model formulations for the GLSP-RP, for example, in the form of a simple plant location model formulation.  
Conceptualization, A.G. and E.L.-R; Data curation, A.G.; Formal analysis, A.G. and E.L.-R.; Investigation, A.G. and E.L.-R.; Methodology, A.G. and E.L.-R.; Resources, S.V.; Software, A.G.; Supervision, E.L.-R. and S.V.; Validation, A.G.; Writing--original draft, A.G.; Writing--review & editing, E.L.-R. and S.V. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**CPLEX**} & \multicolumn{3}{c}{**LAHCM (\(I=100\))**} \\ \hline
**Instance** & **Objective** & **Time** & **Objective** & **Time** & **Iterations** \\ \hline
**Cla s A** & & & & & \\
21 & 5355.57 & 1800 s & 5355.57 & 843 s & 141 \\
37 & 6487.71 & 1800 s & 6387.71 & 1017 s & 140 \\
**Cla s s B** & & & & & \\
30 & 2552.00 & 144 s & 2552.00 & 209 s & 113 \\
40 & 2297.83 & 39 s & 2297.83 & 189 s & 202 \\
**Cla s s C** & & & & & \\
13 & 57,119.40 & 1800 s & 57,119.40 & 313 s & 253 \\
14 & 36,329.90 & 1800 s & 36,329.90 & 233 s & 191 \\
21 & 22,345.40 & 720 s & 22,345.40 & 240 s & 169 \\
23 & 39,030.30 & 1800 s & **39,180.24** & 225 s & 190 \\
34 & 38,211.63 & 1800 s & 38,211.63 & 220 s & 120 \\
40 & 5876.00 & 186 s & 5876.00 & 142 s & 132 \\
41 & 23,764.38 & 1800 s & **23,871.00** & 241 s & 256 \\
42 & 38,450.13 & 1800 s & 38,450.13 & 233 s & 121 \\
44 & 31,453.00 & 1800 s & 31,453.00 & 253 s & 173 \\
49 & 52,298.00 & 1800 s & 52,298.00 & 260 s & 224 \\ \hline \hline \end{tabular}  
* With: \(I=\) list length, Objective = Objective function value, Time = CPU time in seconds, Iterations = Number of iterations.  
\end{table}
Table 13: Results for LAHCM with \(l=100\) for selected instances where no better solution could be provided in previous experiments.