Algorithm 2 Proposed eHHO pseudo-code

Inputs: The population size N and maximum FFEs number ($maxFFE$) Initialize population $X_i$, ($i=1,2,3,\dots N$) according to Algorithm 1 Initialize $FFEs=0$ while$FFEs\le maxFFEs$do  Calculate the fitness values of all individuals  Set $X_{best}$ as the location of current best solution  for each solution $X_i$ do   Update the initial energy $E_0$ and jump strength J   Update E   if $\left|E\right|\ge 1$ then    Exploration phase    Update the location vector   end if   if $\left|E\right|<1$ then    Exploitation phase    if $r\ge 0.5$ and $\left|E\right|\ge 0.5$ then     Soft besiege     Update the location vector by soft besiege    else if $r\ge 0.5$ and $\left|E\right|<0.5$ then     Hard besiege     Update the location vector by using hard besiege    else if $r<0.5$ and $\left|E\right|\ge 0.5$ then     Soft besiege with progressive rapid dives     Update the location vector by using soft besiege with rapid dives    else if $r<0.5$ and $\left|E\right|<0.5$ then     Hard besiege with progressive rapid dives     Update the location vector using hard besiege with progressive rapid dives    end if   end if  end for  if $\psi <FFEs$ then   Generate $X_{best}^{qr}$   Perform greedy selection between $X_{worst}$ and $X_{best}^{qr}$  else if $\psi \ge t$ and $k<bt$ then   Generate $X_{best}^{qr}$   Perform greedy selection between $X_{best}$ and $X_{best}^{qr}$  else   Generate $X_{best}^c$   Perform greedy selection between $X_{best}$ and $X_{best}^c$  end if end while Return $X_{best}$ Post-process results and visualization