# Enhanced Hyper-Cube Framework Ant Colony Optimization for Combinatorial Optimization Problems
## 3 Enhanced Hyper-Cube ACO Algorithms
Best solution probability was \(P_{best}=0.05\) for MMACO and BWACO algorithms and lastly the coefficient of worst solution pheromone trail was \(\lambda=0.6\) for BWACO algorithm only. ### Termination Criteria

All algorithms stop as soon as one of the following conditions are satisfied:

* If there is no improvement in the solution after 10 or 56 iterations. * If the number of iterations exceeds 150 and the best solution is equal to the worst solution (means all artificial ants following the same path). * If a maximum number of iterations has been generated. ## 6 Results and Discussion

The case study described in the previous section was optimized using nine different algorithms: standard ACOA, EACO, RBACO, MMACO, BWACO, HCF/EHCF for both ACO and MMACO algorithms. Analysis of the algorithm's performance will be divided into two parts. First, the performance of ACO algorithms with Hyper-Cube Framework will be assessed. The second part is dedicated to the comparison of EHCFACO algorithm with the rest of ACO algorithms. ### Hyper-Cube Framework ACO Algorithms Results Analysis

Referring to Section 2.5, the Hyper-Cube Framework (HCF) can be applied for both versions of the standard ACOA and MMACO. Hence, this part of the analysis is devoted to determining which version of both ACO algorithms, with HCF and EHCF, could exhibit a better performance. The performance measures for the original ACOA, MMACO, HCFACO, HCFAMACO, EHCFACO, and EHCFMMACO are listed in Table 3. The performance values listed in Table 3 reveal that applying HCF to the ACOA positively affected the overall performance of ACO. The average practical reliability increased by 22-56% and the normalized price declined from 51.17 to 36.91 for the fast convergence rate and from 181.12 to 94.24 for the slow one. The performance rate doubled at the slow rate while \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{E_{1}(GPa)}\) & \(\mathbf{E_{2}(GPa)}\) & \(\mathbf{G_{12}(GPa)}\) & \(\mathbf{v_{12}}\) \\ \hline
127.59 & 13.03 & 6.41 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Graphite-epoxy lamina’s properties. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{N_{L}}\) & \(\mathbf{t(mm)}\) & \(\mathbf{a(mm)}\) & \(\mathbf{b(mm)}\) & \(\mathbf{N_{x}(N/m)}\) & \(\mathbf{N_{x}/N_{y}}\) \\ \hline
64 & 0.127 & 508 & 254 & 175 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Graphite-epoxy lamina’s geometrical and loading data. remaining the same for the fast. The FDC correlation coefficient \(r\) decreased slightly for both levels of convergence. Further improvement of HCFACO performance was acquired when the proposed local search movements were imposed. The average practical reliability became more than twice of the standard ACO, and the normalized price decreased more to hold at 28.92 instead of 51.17 and 81.49 instead of 181.2 for both convergence rates. The performance rate was slightly increased, and the FDC correlation coefficient r was partially improved. On the contrary, HCFMMACO performed poorly compared to the original MMACO. However, the performance of MMACO improved when the Enhanced HCF was applied, but the computational effort became more costly. Based on these results, we conclude that EHCFACO delivers an inexpensive solution with significant performance. Eventually, the solution convergence of the algorithms mentioned above was plotted, Figure 3: solution convergence of ACO-MMACO and their Hyper-Cube Framework variants, for the same selected experiment (seeds = 75). ### Other ACO Algorithms Results Analysis

All ACO algorithms successfully found the best-known value of the maximum critical buckling load factor, \(\lambda_{cb}\).