# Enhanced Hyper-Cube Framework Ant Colony Optimization for Combinatorial Optimization Problems
## 3 Enhanced Hyper-Cube ACO Algorithms
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Optimal Stacking Sequence Design** & \begin{tabular}{c} **Critical Buckling** \\ **Load Factor**, \(\lambda_{cb}\) \\ \end{tabular} \\ \hline EACO & \begin{tabular}{c} \([2333332333323333]s\) \\ \([\pm 45^{\circ}/90^{\circ}\!\!_{10}/\pm 45^{\circ}/90^{\circ}\!\!_{8}/\pm 45^{\circ}/90^{ \circ}\!\!_{8}]_{s}\) \\ \end{tabular} & \\ \hline RBACO & \begin{tabular}{c} \([2333333333332]_{s}\) \\ \([\pm 45^{\circ}/90^{\circ}\!\!_{8}/\pm 45^{\circ}/90^{\circ}\!\!_{8}/45^{\circ}]_{s}\) \\ \end{tabular} & \\ \hline BWCACO & \
\begin{tabular}{c} \([33333233222222]_{s}\) \\ \([90^{\circ}\!\!_{8}/\pm 45^{\circ}/90^{\circ}/\pm 45^{\circ}/90^{\circ}\!\!_{7}/\pm 45^{ \circ}/90^{\circ}\pm 452]_{s}\) \\ \end{tabular} & 3973.01 \\ EHCFACO &
\begin{tabular}{c} \([3333332322222]_{s}\) \\ \([90^{\circ}\!\!_{10}/\pm 45^{\circ}/90^{\circ}/\pm 45^{\circ}/90^{\circ}/\pm 45^{ \circ}\!\!_{8}]_{S}\) \\ \end{tabular} & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The optimal stacking sequence for 64 ply laminates subjected to biaxial loading without contiguity constraint \(\left(N_{y}=N_{x}=1\ and\ a/b=2\right)\). Figure 4: Solution convergence of EACO, RBACO, and BWACO. Figure 5: Critical buckling load factor vs. number of experiments. Figure 8: Correlation coefficient of EACO, RBACO, BWACO, and EHCFACO algorithms. Figure 6: Reliability of EACO, RBACO, BWACO, and EHCFACO algorithms. Figure 7: Normalized price of EACO, RBACO, BWACO, and EHCFACO algorithms solutions. Figure 11: Elapsed time of EACO, RBACO, BWACO, and EHCFACO algorithms to find the optimal solution. Figure 10: Solution quality of EACO, RBACO, BWACO, and EHCFACO algorithms. Figure 9: Performance rate of EACO, RBACO, BWACO, and EHCFACO algorithms. In terms of solution cost, the normalized price results plotted on the scatter chart are illustrated in Figure 7. As mentioned before, the normalized price measure reflects the balance between the solution cost and the reliability. Thus, it is quite clear that EHCFACO outperformed other ACO algorithms. BWACO comes second at a slow convergence rate, whereas RBACO and EACO deliver a costly solution. The numerical experiments results show that ACO variants with high reliability need more computational time to find the global optimal solution while the number of iterations required to find it is low. The Fitness-Distance Correlation values demonstrated the complexity of such a NP-hard optimization to be solved where ACOs with high reliabilities have low \(r\) values. Eventually, applying the Hyper-Cube Framework (HCF) to standard ACO has a significant influence on the overall performance of ACO. In addition, imposing local search movements, as an enhancement of exploitation effort, helped HCFACO to deliver a cost-effective solution. ## 7 Conclusions

A comprehensive MHs performance assessment criterion was developed in the current study. In addition to the computation time and statistical matrices, several new measures such as the practical reliability, price (computational cost), normalized price, performance rate, solution quality, and fitness landscape analysis were included. Additionally, two different convergence rates were imposed on examining the MHs at both the slow and the fast rates. The reproducibility of the numerical experiments' results was also considered

Figure 4:0 Fitness vs. distance to global optimal solution of ACO algorithms. within the procedure of MH assessment. Thereafter, the proposed criterion was employed to compare five different variants of Ant Colony Optimization (ACO). The proposed measures demonstrated a comprehensive assessment of the compared ACOs performance. The initial results of the comparison study revealed that the Hyper Cube Framework (HCF) ACO variant outperforms the others. Consequently, an investigation of further improvement led to introduce an enhanced version of HCFACO (or EHCFACO). The new variant has advanced intensification features that use both the insertion and bit-flip movements to enhance the local search effort.