## 3 Hybridization Method
### Krill Herd (KH)
To statistically evaluate the minimum value returned by the algorithms across 12 optimization problems, we also performed the [PERSON] test and the post hoc Nemenyi test (at an \(\alpha\) level of 0.05). The results (mean ranks) are visualized as a critical distance (CD) diagram in Figure 10. These results show the statistically significant superiority of the hybridized algorithm. Figure 8: Mean ranks from the Friedman test during optimization. Figure 4:0 Average ranks for 12 optimization functions in the final iteration. A smaller rank is better. Finally, we analyzed the time performance for all 12 optimization functions. The results are presented in Figure 4:1 Since the hybrid algorithm uses the iterations from both KH and BA algorithms, its performance is better than the algorithm with the worst performance (KH) but much worse than the performance of the algorithm with the best performance (BA). ### Three-Bar Truss Design Problem

The three-bar truss design is a well-known optimization problem in the realm of civil engineering (see Figure 12). The problem is very popular and has been used to benchmark optimization methods in numerous papers, most recently in [65; 66; 67; 68]. The main goal of this problem is to decrease the weight of a three-bar truss by taking two structural characteristics into account. The three major constraints are deflection, stress, and buckling. This problem is defined as follows. Figure 4:2 Time performance comparison for 12 optimization functions. Figure 4:3 Critical difference (CD) diagram comparing baseline algorithms and hybridized algorithm using an alternative stopping criterion. High-to-low rankings are presented from left to right. A higher rank means that the algorithm performed better. The three truss nodes are fully secured. The fourth node is free and has 2 degrees of freedom of movement \(u_{1},u_{2}\), and a force of magnitude \(P\) is applied to it at angle \(\theta\) to the horizon. The angle is counted clockwise and can be changed in the range from 0 to \(360^{\circ}\); \(L\) is the farm size; \(x_{1},x_{2}\), and \(x_{3}\) are cross-sectional areas of rods; \(\rho\), \(E\), and \(d\) are density, the modulus of longitudinal elasticity, and the permissible tensile and compressive stresses of the truss material, respectively. The purpose of the design is to select the cross-sectional areas of the rods \(x_{1},x_{2}\), and \(x_{3}\) (controlled parameters, design variables) such that the truss has the minimum mass and that stress and area constraints are met for the cross-sections of rods. Consider:

Figure 4:4

Minimize:

Figure 4:5

Subject to:

Figure 4:6

Figure 4:7

Figure 4:8

Variable range:

Figure 4:9

where \(l=100\) cm, \(P=2\) KN/cm\({}^{2}\), and \(\sigma=2\) KN/cm\({}^{2}\). The results are presented in Table 2, which also compares the proposed approach with other methods. Table 2 was adopted from [69] and updated with the proposed hybrid algorithm results. The algorithms used for comparison are as follows: dingo optimization algorithm (DOA), mine blast algorithm (MBA), salp swarm algorithm (SSA), PSO with differential evolution (PSO-DE), and DE with dynamic stochastic selection (DEDS). ## 6 Discussion and Conclusions

We investigated the proposed model and discovered its suitability for developing meta-heuristics to address mathematical problems of varying dimensions and features. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(x_{1}\) & \(x_{2}\) & **Optimum Weight** \\ \hline DOA & 0.788675095 & 0.40824840 & 263.8958434 \\ MBA & 0.78856500 & 0.40855970 & 263.8958522 \\ SSA & 0.7886641 & 0.40827578 & 263.8958434 \\ PSO-DE & 0.78867510 & 0.40824820 & 263.8958433 \\ DEDS & 0.78867513 & 0.40824828 & 263.8958434 \\ \hline Proposed & 0.78853476 & 0.40866456 & 263.8958434 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of performance on the three-bar truss problem. Figure 12: Schematic representation of the three-bar truss problem.