# Solution Merging in Matheuristics for Resource Constrained Job Scheduling
## 5 Experiments and Results
### Study Concerning the Number of Cores Used  
Remember that the number of allowed cores influences two algorithmic components: (1) the ACO component, where the number of cores corresponds to the number of colonies and (2) the solution of the restricted MIPs (which is generally more efficient when more cores are available). However, with a growing number of allowed cores, the restricted MIPs become more and more complex, due to being based on more and more solutions. In any case, the number of allowed cores should make a significant difference to the performance of both MS-ACO and CMSA-ACO.  
The results are presented in Figures 6 and 7. The figures show the average over all instances with the same number of machines of 25 runs for the gap to the best solution found by any of the methods. The results for MS-ACO show that using 15 or 20 cores is preferable compared to using only 10 cores. The difference between 15 and 20 cores is small with a slight advantage using 20 cores.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Inst.** & **MS-MIP** & **MS + ACO** & **CMSA-MIP** & **CMSA + ACO** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 4: The contribution of the merge step of MS-ACO and CMSA-ACO, relative to the total solution improvement. The MIP results sum up the percentage improvement in the objective for every MIP solve for an instance. The MS/CMSA+ACO result is the percentage difference of the first best solution found to the best solution at the end of the run. Small instances where the best solution is the first one found have been omitted.  
As mentioned already above, the diversity of the solutions generated when using 20 cores leads to large restricted MIPs (many more variables) to be solved, which can be very time consuming and even very inefficient. Hence, on occasion, 15 cores are preferable to 20 cores. Compared to using 10 cores, using 15 or 20 cores cores provides sufficient diversity leading to good areas of the search space for the restricted MIPs within the 120 s time limit.  
The results for CMSA-ACO show that overall the use of 15 or 20 cores is preferable over only using 10 cores. In CMSA-ACO, several instances are solved more efficiently with 20 cores. Though, for the large instances (10 or more machines), the use of 15 cores is most effective. For CMSA the effect of increased number of solutions on increasing the size and complexity of the restricted MIP used in  
Figure 6: The performance of MS-ACO with 10, 15 and 20 cores. The results are averaged over instances with the same number of machines and show the gap to the best solution found by MS-ACO or CMSA-ACO.  
Figure 7: The performance of CMSA-ACO with 10, 15 and 20 cores. The results are averaged over instances with the same number of machines and show the gap to the best solution found by MS-ACO or CMSA-ACO.  
the merge step is even more pronounced than in MS. Hence, overall the use of 15 cores proves to be most effective.  
The conclusion of this comparison is that, while solution merging benefits from having a number of different solutions available, a too large pool of solutions can also have a detrimental effect. For RCJS, using 15 solutions is generally the best option irrespective of the type of merge step used (MS vs. CMSA). To better understand the effect of the alternative solution merging approaches, we next test these using exactly the same pools of solutions.