## 6 Evaluated Algorithms
### Experiments of Hybrid FVIM with DE Components  
In this section, we present a series of experiments designed to evaluate the impact of different components of the Differential Evolution (DE) algorithm when combined with the hybrid FVIM (Four-Variable Inertia Method). The DE algorithm is composed of three main operators: mutation, crossover, and selection, each contributing uniquely to the exploration and exploitation capabilities of the algorithm. By selectively removing or retaining these components in the hybrid approach, we aim to investigate their effects on optimization performance. These experiments provide valuable insights into how each DE component influences the solution quality in the hybrid FVIM framework. Table 16 shows the results of the experiments comparing different DE components such as mutation only (FVIMMU), crossover only (FVIMCR), selection only (FVIMSE), and a combination of mutation and crossover (FVIMMUCR) across multiple benchmark functions (F1 to F12). The results reveal that the full hybrid FVIMDE consistently outperforms the other configurations, securing the best rank across most functions. Configurations that exclude one or more DE components generally exhibit poorer performance, with FVIMMUCR (mutation and crossover) often ranking among the lowest, especially in more complex functions. The results highlight the importance of integrating all DE components (mutation, crossover, and selection) to achieve better optimization outcomes, as evidenced by FVIMDE's superior results in terms of mean performance and ranking across the majority of the benchmark functions. The experimental results demonstrate that including each DE component in isolation (mutation, crossover, or selection) generally led to worse optimization performance compared to the full integration of all components in the hybrid FVIMDE. This outcome can be attributed to the individual weaknesses of each component when applied alone. Mutation only (FVIMMU): Mutation introduces diversity by perturbing existing solutions, which helps explore the search space. However, without crossover, mutation alone lacks the ability to effectively combine good traits from different solutions. This limitation can lead to slower convergence and an inability to escape local optima, as the algorithm relies solely on random variations rather than guided search. Crossover only (FVIMCR): Crossover is designed to recombine existing solutions to explore new areas of the search space. However, when used in isolation, it tends to exploit the current population without introducing new diversity. Without mutation, crossover alone may quickly converge to suboptimal solutions, as it lacks the ability to explore beyond the recombined solutions' neighborhood. Selection only (FVIMSE): Selection ensures that only the best solutions survive to the next generation, promoting convergence. However, without mutation or crossover, the algorithm has no mechanism to generate new solutions or explore the search space. Selection alone essentially preserves the best solutions from the initial population without improving them, leading to stagnation and poor optimization results. Mutation and Crossover (FVIMMUCR): While combining mutation and crossover provides both exploration and recombination, the absence of selection weakens this configuration. Without selection, the algorithm lacks the necessary pressure to favor better solutions, leading to the retention of poor candidates and reducing the overall effectiveness of the search process. The crossover plays a role in balancing between exploration, utilizing the diversity introduced by mutation, and exploitation; in addition, crossover does not directly introduce diversity but rather controls the level of diversity generated through mutation, striking a balance between exploration and intensification. The results emphasize that each DE component plays a crucial role in balancing exploration and exploitation.