# Multi-Strategy Hybrid Whale Optimization Algorithm Improvement
## 4 Simulation Experiments and Comparative Analysis
### An Introduction to Test Functions
The convergence curve can show the convergence trajectory of the algorithm and visually observe its accuracy. Figure 4 shows the convergence trajectory comparison of MHWOA and other standard swarm intelligence algorithms and visually compares their accuracy. The error of the Y-axis represents the difference between the optimal solution calculated by different algorithms and the actual optimal solution of the function. These values are the average iteration curve derived from the average of the iterative processes in all 30 runs. Among them, MHWOA achieves the best convergence accuracy in the test functions and its convergence speed is also among the top algorithms. Particularly, MHWOA demonstrates outstanding performance during the phase of escaping local optima. It can be seen from the comparison figure that the final position reached by MHWOA is always at the lowest point of multiple algorithms, which means that MHWOA has the smallest calculation error and the highest calculation accuracy in all tests. Meanwhile, when solving low-complexity problems such as F1 and F2, the convergence speed of RSO and SO is faster than that of MHWOA, which is caused by the algorithm structure, and MHWOA is more suitable for solving relatively complex problem types. Starting from F6, it can be found that MHWOA has a faster convergence speed than other algorithms, and the optimal solution can be found at an earlier iteration number in general. In some problems, such as F7, F14, F17, and F20, MHWOA can avoid the algorithm falling into the local optimum and missing the optimal solution due to premature convergence. The test results of MHWOA and the improved WOA are presented in Table 4. The numbers specifically labeled in bold represent the optimal terms in each set of test data. From Table 4, it can be observed that in the tests on single-peak and multi-peak functions, MHWOA and the improved swarm intelligence algorithm both achieve the optimal values in F1, F2, F4, and F5, with their standard deviations ranking first. In the remaining test functions, MHWOA demonstrates the best performance in terms of both mean and standard deviation. In the tests on hybrid functions and composition functions, MHWOA still outperforms other algorithms significantly. MHWOA and most of the compared algorithms can achieve the optimal value in terms of both numerical value and standard deviation when dealing with unimodal functions F1 to F5, so they are not comparable. In addition, when dealing with multimodal functions, the calculation accuracy of MHWOA is improved by no less than 3.26% and the calculation error is reduced by no less than 12.85% compared with other algorithms in extreme cases. When dealing with hybrid functions, in extreme cases, the calculation accuracy of MHWOA is improved by no less than 42.66% compared with other algorithms, and the calculation error is reduced by no less than 101%. When dealing with composition functions, the calculation accuracy of MHWOA is improved by no less than 28.6% compared with other algorithms in extreme cases and the calculation error is reduced by no less than 381%. In the Friedman test, MHWOA has an average ranking of 1.2166 and a final ranking of being first, followed by eWOA. In terms of computational cost, when dealing with single-peak functions, MHWOA improves the computational speed by 1.66 times compared with MSWOA, 3.66 times compared with MWOA, and 1.31 times compared with eWOA. When handling multi-peakfunctions, MHWOA achieves a computational speed improvement of 85.47% compared with MSWOA, 34.64% compared with MWOA, and 24.24% compared with eWOA. For hybrid functions, MHWOA achieves a computational speed improvement of 78.79% compared with MSWOA, 29.54% compared with MWOA, and 17.44% compared with eWOA. When dealing with composition functions, MHWOA achieves a computational speed improvement of 65.62% compared with MSWOA, 16.39% compared with MWOA, and 9.18% compared with eWOA. Figure 4: _Cont._