Fig. 1

Phases of the Harris Hawks Optimization process73. Where r: is a random number between 0 and 1, which is used to introduce randomness in the position updates of the hawks. It helps in diversifying the search process and is crucial for both exploration and exploitation phases. q: is another random number between 0 and 1, which is used to determine the type of behavior (strategy) the hawks will adopt. It helps in deciding whether the hawks will perform a hard or soft besiege strategy. ∣E∣: is considered as the absolute value of the escape energy of the prey. This parameter decreases over iterations, starting from a positive value and gradually becoming negative. The value of ∣E∣ determines whether the algorithm is in the exploration phase or the exploitation phase. Hence, for ∣E∣ ≥ 0.5: The hawks are in the exploration phase, meaning they are searching broadly across the solution space. While for ∣E∣ < 0.5: The hawks are in the exploitation phase, meaning they are focusing on fine-tuning the solutions around the best-found solutions.