Algorithm 1 Improved Kingfisher Optimization (IPKO) Algorithm

1:Step 1: Sine chaotic opposition-based initialization (Equations (5)–(10))   2:Initialize population X, evaluate fitness, find $G_{\mathrm{best}}$, $f_{\mathrm{best}}$   3:Set $stuck=0$   4:Step 2: Main loop   5:for $t=1$ to $MaxIter$ do   6:   for $i=1$ to N do   7:     Step 3: Adaptive directional search   8:     Compute $PE$   9:     if rand() $\le (1-PE)\left(Equation\right(11\left)\right)$ then 10:        $r_1=\mathrm{randi}\left([1,N]\right)$, $r=rand\left(\right)$ 11:        $X_2=(G_{\mathrm{best}}-{\mathbf{X}}_i^{(t-1)})\times r$ 12:        $B=2\times cos({\displaystyle \frac{\pi }{2}}\times (1-{\displaystyle \frac{t}{MaxIter}}))$ 13:        $\theta =\lceil 360\times rand\left(\right)\rceil$ 14:        $V=\mathrm{randi}\left(\right[2,5\left]\right)$ 15:        ${\mathbf{X}}_{\mathrm{new}}={\mathbf{X}}_2+({\mathbf{X}}_{r_1}^{(t-1)}-{\mathbf{X}}_i^{(t-1)})\times cos\theta \times B\times V\times rand\left(\right)$ 16:        $+{\mathbf{X}}_{r_1}^{(t-1)}\times sin\theta \times B\times rand\left(\right)$ 17:        ${\mathbf{X}}_i^{\left(t\right)}=\mathrm{CentroidBoundary}\left({\mathbf{X}}_{\mathrm{new}}\right)$ (Equations (20)–(23)) 18:     else 19:        ${\mathbf{X}}_i^{\left(t\right)}=\mathrm{OriginalPKODiving}\left({\mathbf{X}}_i^{(t-1)}\right)$ 20:        ${\mathbf{X}}_i^{\left(t\right)}=\mathrm{CentroidBoundary}\left({\mathbf{X}}_i^{\left(t\right)}\right)$ 21:     end if 22:     Step 4: Stochastic binary perturbation fusion 23:     if $f\left({\mathbf{X}}_i^{\left(t\right)}\right)<f_{\mathrm{best}}$ then 24:        $stuck=0$, ${\mathbf{G}}_{\mathrm{best}}={\mathbf{X}}_i^{\left(t\right)}$, $f_{\mathrm{best}}=f\left({\mathbf{X}}_i^{\left(t\right)}\right)$ 25:     else 26:        $stuck=stuck+1$ 27:     end if 28:     if $stuck\ge 5$ then 29:        Choose $r_1,r_2,r_3$ distinct and $\ne i$ 30:        $\mathbf{y}=({\mathbf{X}}_i^{\left(t\right)}+{\mathbf{X}}_{r_3}^{\left(t\right)})/2$ 31:        ${\mathbf{U}}_1=(rand(1,Dim)>0.5)$ 32:        ${\mathbf{X}}_{\mathrm{new}}={\mathbf{U}}_1·{\mathbf{X}}_i^{\left(t\right)}+(1-{\mathbf{U}}_1)·(\mathbf{y}+rand\left(\right)·({\mathbf{X}}_{r_1}^{\left(t\right)}-{\mathbf{X}}_{r_2}^{\left(t\right)}))$ (Equation (19)) 33:        ${\mathbf{X}}_i^{\left(t\right)}=\mathrm{CentroidBoundary}\left({\mathbf{X}}_{\mathrm{new}}\right)$ 34:        $stuck=0$ 35:     end if 36:   end for 37:end for 38:return ${\mathbf{G}}_{\mathrm{best}}$, $f_{\mathrm{best}}$