## 4 Description of the SAVOA
### Adaptive Weighting Factors and Enhanced Levy Flight  
In the second exploitation phase, a random tiny value is added when \(rand_{P_{3}}\leq P_{3}\):  
\[\epsilon=10^{-6}+10^{-5}\cdot rand_{e} \tag{62}\]  
Meanwhile, the denominators of the two guiding objectives \(A_{1}\) and \(A_{2}\) change to:  
\[ \begin{cases}den_{1}=BestVulture_{1}(i)-P(i)^{2}+\epsilon\cdot\left(|P(i)|+0. 1\cdot(ub-lb)\right)\\ den_{2}=BestVulture_{2}(i)-P(i)^{2}+\epsilon\cdot\left(|P(i)|+0.1\cdot(ub-lb) \right)\end{cases}  \tag{63}\]  
\(rand_{e}\) is a random number between \([0,\ 1]\); \(\epsilon\) is used to carry out the stabilization of the values to avoid too small a denominator which would lead to too large a gradient and thus destroy convergence; \(den_{1}\) and \(den_{2}\) utilize the nonlinear coupling between the current individual and the superior individual to construct an offset bootstrap mechanism that differentiates more by diversity than linearity. Keeping the weighting coefficient \(w_{t}\) between \(0.25\) and \(0.85\) allows for adaptive regulation of equilibrium:  
\[w_{t}=clip(0.65+0.25\cdot tanh(3F),\ 0.25,\ 0.85) \tag{64}\]  
The use of the hyperbolic tangent function \(tanh(3F)\) allows for a nonlinear smooth change in the weight curve, ensuring a natural transition in the middle; instead of direct averaging, the adaptive weighting coefficient \(w_{t}\) allows for greater flexibility by allowing the current individual to be biased in the direction of a more optimal bootstrap but without being overly dependent on one superior individual. On the basis of maintaining excellent individual guidance, Gaussian perturbation noise \(\sigma\) is added to maintain the diversity of local search:  
\[\sigma=0.05\cdot(ub-lb)\cdot\mathcal{N}(0,1)\cdot\delta \tag{65}\]  
where \(\mathcal{N}(0,1)\) is the Gaussian distribution; \(ub\) and \(lb\) are the upper and lower boundaries; and \(\delta\) is the dynamic decay factor. Reinforcement of Levy flight when \(rand_{P_{3}}>P_{3}\):  
\[\left\{ \begin{array}{l}scale=1.2-0.5\cdot\left(\frac{iter}{iter_{Max}}\right) \\ \phi=1+0.15\cdot sign(Bestulture_{1}(i)-P(i))\end{array} \right. \tag{66}\]  
_scale_ denotes the perturbation-scaling factor, whose value decreases with the number of iterations. It is used to control the overall scaling of the _Levy_ step, with a larger perturbation in the early stage to favor exploration and a smaller perturbation in the later stage to favor fine search; \(\phi\) is a scaling factor used to enhance positive jumps or fine tune negative jumps to form an asymmetric search mechanism, and \(sign\) is a directional indicator. In summary, the positional formula for the second exploitation phase of the SAVOA is updated as:  
\[\textit{Pr}(i+1)= \begin{cases}\textit{Equation}(70)&P_{3}\geq rand_{P_{3}}\\ \textit{Equation}(71)&P_{3}<rand_{P_{3}}\end{cases}  \tag{67}\]  
\[A_{1}^{\prime}=\textit{Best}ulture_{1}(i)-\frac{\textit{Best}ulture_{1}(i) \times P(i)}{den_{1}}\times F \tag{68}\]  
\[A_{2}^{\prime}=\textit{Best}ulture_{2}(i)-\frac{\textit{Best}ulture_{2}(i) \times P(i)}{den_{2}}\times F \tag{69}\]  
\[\textit{Pr}(i+1)=w_{t}\cdot A_{1}^{\prime}+(1-w_{t})\cdot A_{2}^{\prime}\cdot\sigma \tag{70}\]  
\[\textit{Pr}^{\prime}(i+1)=\textit{R}(i)-\big{|}d(t)\big{|}\cdot\textit{F} \cdot\textit{scale}\cdot\textit{Levy}(d)\cdot\phi \tag{71}\]  
The pseudo-code and flowchart of SAVOA are Algorithm 1 and Figure 5.  
```
1.pop: size of the population;
2.pd: the number of the producers;
3.pv: the number of cultures who perceive the danger;
4. Generating first-generation populations using Sobol sequences
5.whileiter\(<\textit{iter}_{Max}\)do
6. Find the best and the second-best individuals;
7.for\(i=1:pd\)
8. Updating the location of cultures using Equation (50);
9. Calculate the fitness of cultures;
10.endfor
11.for\(i=pd+1:pop\)
12. Calculate the starvation rate \(F\) using Equations (30) and (31);
13.if\(|F|\geq 0.5\)then
14. Updating the position using Equation (55);
15.else
16. Updating the position using Equation (67);
17.endif
18.