## 3 Resolution Approaches
### Approximate Approaches
#### 3.2.3 Simulated Annealing

Table 13: Gap from best solution in percentage and execution time (best results depicted in bold). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**n**} & \multirow{2}{*}{**m**} & \multirow{2}{*}{**k**} & \multicolumn{2}{c}{**MILP**} & \multicolumn{2}{c}{**Greedy**} & \multicolumn{2}{c}{**SA (10\({}^{4}\) iter.)**} & \multicolumn{2}{c}{**SA (10\({}^{5}\) iter.)**} \\ \cline{4-10}  & & & **avg** & **std** & **avg** & **std** & **avg** & **std** & **avg** & **std** \\ \hline
10 & 10 & 5 & 3606 & 9 & **2** & 0 & 8 & 4 & 71 & 18 \\
20 & 10 & 5 & 3612 & 9 & **2** & 0 & 11 & 4 & 106 & 19 \\
30 & 10 & 5 & 3623 & 9 & **2** & 0 & 14 & 6 & 139 & 30 \\
40 & 10 & 5 & 3638 & 10 & **2** & 0 & 18 & 6 & 162 & 26 \\
50 & 10 & 5 & 3659 & 15 & **2** & 0 & 21 & 6 & 171 & 34 \\ \hline
10 & 20 & 5 & 3612 & 9 & **2** & 0 & 8 & 4 & 72 & 17 \\
20 & 20 & 5 & 3643 & 9 & **2** & 0 & 11 & 4 & 104 & 21 \\
30 & 20 & 5 & 3720 & 26 & **2** & 0 & 15 & 4 & 136 & 33 \\
40 & 20 & 5 & 3846 & 81 & **2** & 0 & 17 & 5 & 150 & 39 \\
50 & 20 & 5 & 4130 & 238 & **2** & 0 & 22 & 6 & 176 & 35 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Execution time in seconds (best results depicted in bold). We observe that the simulated annealing approach again performs the best, finding the best solution for each problem instance, although not for all of its 50 executions, as can be noticed from the average gap not being equal to 0. We can also conclude that the execution times for this approach remain reasonable for larger problem instances, which could be attributed to the increase in resource types (\(k=10\)), making it potentially easier to assign the needed resources when constructing a solution. ## 5 Conclusions and Perspectives

In this paper, a novel scheduling problem consisting of job and machine acceptance, the assignment of jobs to unrelated parallel machines, the scheduling of jobs on selected machines, and multiple-resource allocation was considered. The machines and renewable resources (such as manpower, industrial machines or tools, and computing devices) required for the processing of jobs were rented during their operating time. The objective was to maximize the overall profit. One practical application of the proposed problem consisted of establishing a production facility without (or with minimum) investment (in machinery and equipment) or location. Modeling existing infrastructure and manpower, in this case, can be easily achieved by reducing the renting costs and setup times to 0 for a given set of machines or resources. Another application corresponds to the illustrated event-planning problem. In order to tackle this problem, we developed an exact approach based on mixed-integer linear programming as well as several heuristic approaches. Due to the complexity of the problem and following our experimental results, the exact approach may only be useful for smaller-sized problem instances. When larger, more realistic problem instances are considered, a self-adapting simulated annealing approach should be favored. Additional work may consist of adapting the proposed approaches for other machine configurations, such as flowshop, or considering other real-world constraints, such as:

* Machine unavailability;
* Machine or resource renting discount based on the duration of their use;
* Resource transport duration dependent on machine and resource location;
* Ready times for jobs or resources;
* etc. **Author Contributions:** Conceptualization, M.Z. ; methodology, A.-L.O., M.S. and M.Z. ; software, A.-L.O. ; validation, A.-L.O., M.S. and M.Z. ; investigation, A.-L.O. ; data curation, A.-L.O. ; writing--original draft preparation, M.Z. ; writing--review and editing, A.-L.O., M.S. and M.Z. ; visualization, A.-L.O. ; supervision, M.S. ; project administration, A.-L.O., M.S. and M.Z. ; All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.