## 3 Utilized Meta-Heuristic Algorithms
### Standard Genetic Algorithm
On the other hand, it can be observed that RA, SGA, PSO, and GWO give more successful results and continue to search for the best value throughout the iterations. Besides, BIRA shows superior success in all of the test functions shown in the Figure 4 and managed to reach the global value in its initial iterations. Based on the performance of other algorithms, it is obvious that PSO offers the closest performance to BIRA. While DE and CS generally continue to get stuck locally, SGA, RA, and GWO are successful in searching the space but they are not successful in finding the global value. Figure 3: Convergence curve of the algorithms in the CEC2014. Figure 2: Convergence curve of the algorithms in benchmarks. ### Statistical Results

IBM SPSS Statistics 22 was performed to acquire the test results of the Friedman while the _signrank_ function of Matlab 2019a was utilized to obtain the test results of the Wilcoxon Signed Rank. The Friedman Test commences its procedure by assigning ranks to each row based on the values within the respective columns. Subsequently, it calculates the total rank values for each column. To evaluate this test, the \(X^{2}\) statistic (also known as Chi-square) value, and the \(\mathrm{k}-1\) degrees of freedom (df) value must be known, where \(k\) means the number of compared algorithms. The appropriate Chi-square values corresponding to the degrees of freedom can be found in [45]. For instance, in our case, with df = 6 and \(\alpha=0.05\), the expected \(X^{2}\) value is 12.592. If the computed \(X^{2}\) value exceeds the expected one, it leads to the rejection of the hypothesis that states \"There is no difference between compared algorithms\". It becomes evident which algorithm excels by referencing Table 9. An algorithm with the lowest rank value indicates superior performance in comparison to the others. Table 9 clearly shows that BIRA offers a better performance when comparing it to the other tested algorithms. Moreover, not only do the probability values fall below \(\alpha=0.05\), but the computed \(X^{2}\) values in Table 10 also surpass the expected value of 12.592. Consequently, the hypothesis must be rejected, signifying that \"There is a difference between compared algorithms\". Nevertheless, this test does not specifically identify which algorithms differ

Figure 4: Convergence curve of the algorithms in the CEC2018. from the others. Therefore, the Wilcoxon Signed Rank Test is employed to elucidate these differences. The Wilcoxon Signed Rank Test is performed in pairs: one algorithm is systematically selected from the first column of the table and then compared with all subsequent algorithms. Additionally, the \(p\) values indicate the degree of similarity between the compared algorithms. A value of '0' implies no similarity between the outcomes of the compared algorithms, while '1' indicates that they are identical. In Table 11, it can be seen that the \(p\) values resulting from the Wilcoxon Signed Rank Test are generally very close to 0. This infers that no algorithm is similar to another in terms of their performance on all test functions. However, it is obvious to say that the performance of BIRA and PSO are closer than the others since their pairwise value is 0.269221. ### Real-World Optimization Problems

To analyze the performance of the algorithms, three well-known real-world engineering problems were utilized: pressure vessel design, gear train design, and tension/compression spring design. The parameter settings given in Table 4 were applied while testing the algorithm performance in the real-life scenarios. While the algorithms used in this study were tested in real-life scenarios, each algorithm was run 30 times using 30 different random seeds and the best results obtained from these 30 repetitions were shared. The pressure vessel design and tension/pressure spring design problems used in this section contain constraints. The algorithms must produce results by taking these constraints into account. However, as can be expected, some results in the tests performed