# Chemotherapy Regimen Optimization Using a Two-Archive Multi-Objective Squirrel Search Algorithm
## 3 Proposed Model and Method
### Two-Archive Multi-Objective Suirrel Search Algorithm
#### 3.2.1 Squirrel Search Algorithm (SSA)
** _Individuals on acorn nut trees (\(FS_{at}\)) head for hickory nut tree (\(FS_{ht}\)), new locations at iteration \(t+1\) as represented in Equation (22):_  
\[FS_{at}^{t+1}=\left\{ \begin{array}{ll}FS_{at}^{t}+d_{g}\ \times G_{c}\times \left(FS_{ht}^{t}-FS_{at}^{t}\right)&R_{1}\geq P_{dp}\\ Random\ \text{location}&\text{otherwise}\end{array} \right. \tag{22}\]  
_where \(d_{g}\) denote a random gliding distance, \(G_{c}\) is a sliding constant to balance exploration and exploitation, \(R_{1}\) is random within [0,1], \(FS_{ht}^{t}\) represents the hickory nut tree position at iteration \(t\)._  
**Case 2. ** _A small part of individuals on normal trees (\(FS_{nt}\)) proceed in the direction of acorn nut trees (\(FS_{at}\)), and their new locations at iteration \(t+1\) can be represented as Equation (23):_  
\[FS_{nt}^{t+1}=\left\{ \begin{array}{ll}FS_{nt}^{t}+d_{g}\ \times G_{c}\times \left(FS_{at}^{t}-FS_{nt}^{t}\right)&R_{2}\geq P_{dp}\\ Random\ \text{location}&\text{otherwise}\end{array} \right. \tag{23}\]  
_where \(R_{2}\) is random within [0,1]._  
**Case 3. ** _The rest individuals on normal trees (\(FS_{nt}\)) head for the hickory nut tree (\(FS_{ht}\)), and their new locations at iteration \(t+1\) can be represented as Equation (24):_  
\[FS_{nt}^{t+1}=\left\{ \begin{array}{ll}FS_{nt}^{t}+d_{g}\times G_{c}\times \left(FS_{ht}^{t}-FS_{nt}^{t}\right)&R_{3}\geq P_{dp}\\ Random\ \text{location}&\text{otherwise}\end{array} \right. \tag{24}\]  
_where \(R_{3}\) is random within [0,1]._  
* Seasonal monitoring conditions  
The foraging strategies employed by flying squirrels change seasonally, and they tend to move less in winter. Therefore, monitoring parameters are introduced for easing local optima stagnation. The seasonal constant \(S_{c}\) and the minimum value of seasonal constant \(S_{min}\) are calculated using Equation (25) and Equation (26), respectively. Subsequently, these two parameters are used to determine the season. \[S_{c}^{t}=\sqrt{\sum\limits_{k=1}^{d}\left(FS_{nt,k}^{t}-FS_{ht,k}\right)^{2}} \tag{25}\]  
where t = 1, 2, 3. \[S_{min}=\frac{10E^{-6}}{\left(365\right)^{1/(ln/25)}} \tag{26}\]  
where \(t\) and \(t_{m}\) denote current iteration numbers and maximum iteration numbers, respectively. The seasonal monitoring condition is met when \(S_{c}^{t}<S_{min}\) (i.e., winter has ended and summer comes). Flying squirrels on normal trees would be randomly relocated using the following Equation (27):  
\[\text{FS}_{nt}^{\text{new}}=\text{FS}_{L}+\text{Levy}(\text{n})\times\left( \text{FS}_{U}-\text{FS}_{L}\right) \tag{27}\]where Levy Flight is applied for more efficient exploration, it can be calculated by Equations (28) and (29) below:  
\[Levy(x)=(0.01\times r_{a}\times\delta)/|r_{b}|^{\frac{1}{7}} \tag{28}\]  
\[\delta=\Gamma(1+\gamma)\times sin\big{(}\frac{\pi\gamma}{2}\big{)}\big{/}\Gamma \big{(}\frac{1+\gamma}{2}\big{)}\times\gamma\times 2^{(\frac{\gamma-1}{2})} \tag{29}\]  
where \(r_{a}\) and \(r_{b}\) denote numbers within \([0,1]\), \(\gamma\) is a constant number equaling to 0.5 and \(\Gamma(x)\)\(=(x-1)!\). In contrast, \(S_{c}^{t}\geq S_{min}\) means summer has not come yet, flying squirrels would not enforce any extra moving behavior. The pseudocode of the SSA is shown in Algorithm 1. ```
1. Initialize location for \(n\) individuals and calculate their fitness values;
2. Rank and assign individuals by fitness value;
3. Randomly select individuals which are on normal trees to head for acorn nut trees;
4. **While** (the maximum iteration is not reached):
5. **For**\(\mathbf{m}=1\) to \(m_{1}\) (individuals on acorn trees head for hickory nut tree)
6. Update \(F_{Sat}\) as Equation (22);
7. **End For**\(\mathbf{m}=1\) to \(m_{2}\) (individuals on normal trees proceeding in the direction of acorn nut tree)
9. Update \(F_{Sat}\) as Equation (23);
10. **End For**\(\mathbf{m}=1\) to \(m_{3}\) (all individuals on normal trees head for hickory nut tree)
12. Update \(F_{Sat}\) as Equation (24);
13. **End For**
14. Calculate \(S_{c}\) and \(S_{min}\) as Equations (25) and (26);
15.