# Enhanced Hyper-Cube Framework Ant Colony Optimization for Combinatorial Optimization Problems
## 3 Enhanced Hyper-Cube ACO Algorithms
A different sample of optimal SSD obtained by different ACO algorithms is listed in Table 4. The solution convergence of EACO, RBACO, and BWACO for a selected experiment (seeds = 75) is graphically illustrated in Figure 4.

It is observed from Table 4 that the optimal stacking sequence design followed the same pattern of switching between two groups of 90\({}_{2}\) and \(\pm\)45 fiber orientations which confirm the results of previous studies [1; 23]. On the other hand, the solution convergence plot in Figure 4 illustrates that both RBACO and BWACO algorithms develop gradual search trends on their way to the optima whereas EACO and EHCFACO algorithms smoothly converge to the global optima. Further, the numerical experiments confirm the fluctuation of ACO algorithms in finding the global optimal solution due to their stochastic nature, as illustrated in Figure 5.

According to the introduced performance assessment criteria in Section 5, the average values of different performance measures of reliability, performance rate, solution quality, normalized price, and searching effort coefficient were determined for EACO, RBACO, BWACO at fast and slow convergence rates. These results, alongside the EHCFACO results, are plotted in Figures 6-12 to provide a sensible comparison of the performance evaluation of the proposed algorithm with other ACO algorithms. The average practical reliability of the algorithms is introduced in Figure 6. Both EACO and RBACO algorithms show low practical reliability values, with 10% and 8%, respectively, while BWACO presented \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Performance Measure** & **Convergence Rate** & **ACOA** & **HCFACO** & **EHCFACO** & **MMACO** & **HCFMMACO** & **EHCFMMACO** \\ \hline Elapsed time, & Slow & 0.87 & 0.01 & 2.16 & 1.12 & 0.96 & 1.34 \\ \(t_{s}\), min & Fast & 3 & 4.15 & 6.86 & 4.62 & 6.68 & 4.39 \\ \hline Reliability & Slow & 35.71 & 77.1 & 89.6 & 16 & 13.37 & 54.36 \\ \% & Fast & 36.45 & 93.15 & 98.95 & 87.7 & 91.05 & 98.25 \\ \hline Normalized & Slow & 51.17 & 36.91 & 28.92 & 152.23 & 169.26 & 52.2 \\ Price, \(nP_{s}\) & Fast & 181.12 & 94.24 & 81.49 & 118.53 & 117.3 & 98.25 \\ \hline Performance & Slow & 0.0196 & 0.0272 & 0.0347 & 0.0068 & 0.0063 & 0.0206 \\ Rate, \(P_{rate}\) & Fast & 0.0056 & 0.0106 & 0.0123 & 0.0088 & 0.0090 & 0.0106 \\ \hline Quality & Slow & 99.69 & 99.93 & 99.96 & 98.28 & 98.57 & 99.81 \\ \% & Fast & 99.69 & 99.97 & 99.98 & 99.96 & 98.57 & 99.98 \\ \hline Fitness-Distance & Slow & \(-0.80\) & \(-0.67\) & \(-0.79\) & \(-1\) & \(-1\) & \(-0.82\) \\ Correlation & Fast & \(-0.84\) & \(-0.71\) & \(-0.73\) & \(-0.81\) & \(-0.77\) & \(-0.75\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The performance measures of Hyper-Cube Framework ACO algorithms. better value at the slow rate of convergence, but it poorly performed at the fast rate. The EHCFACO algorithm exhibited significant reliability values of 89.6-98.95%. Furthermore, it demonstrated the highest performance rate measure (0.012-0.035), see Figure 9. The solution quality results of the algorithms are summarized and depicted in Figure 10, which reveals that all ACO algorithms produce an excellent solution quality for this particular case study. Figure 3: Solution convergence of ACO-MMACO and their Hyper-Cube Framework variants.