# Chemotherapy Regimen Optimization Using a Two-Archive Multi-Objective Squirrel Search Algorithm
## 3 Proposed Model and Method
### Two-Archive Multi-Objective Suirrel Search Algorithm
#### 3.2.3 Two-Archive Multi-Objective Squirrel Search Algorithm (TA-MOSSA)  
The primary issue in extending the Squirrel Search Algorithm (SSA) to the domain of multi-objective optimization lies in how to select the optimal solution. Due to the difficulty in obtaining a single optimal solution that dominates all other individuals in terms of all objective functions, it is also challenging to determine the fitness values of the hickory nut tree (\(FS_{ht}\)) and acorn nut tree (\(FS_{at}\)) simply by comparing the objective function values. Common improvement methods in existing research on the multi-objective Squirrel Search Algorithm often involve introducing a single external archive mechanism from the Non-dominated Sorting Genetic Algorithm II (NSGA-II), determining \(FS_{ht}\) and \(FS_{at}\) based on non-dominated ranks [35; 36] or grid density [37]. However, considering the reason the SSA selects the optimal and suboptimal solutions as \(FS_{ht}\) and \(FS_{at}\), respectively, it is important to guide the convergence and diversity of the solution population separately during the search process. The optimal solution guides most of the population individuals to converge towards the optimal direction, while multiple suboptimal solutions ensure that the search space is thoroughly explored and the local optima are alleviated. A single external archive cannot effectively balance convergence and diversity. The functionality assigned to \(FS_{ht}\) and \(FS_{at}\) in the SSA process essentially matches the functionality of the convergence archive (CA) and diversity archive (DA) in the Two_Arch2. Therefore, the proposed TA-MOSSA introduces the improved two-archive mechanism. In each iteration, new solutions are generated and used to update the two external archives, CA and DA, with the updated CA and DA serving as the candidates for \(FS_{ht}\) and \(FS_{at}\), respectively, in the next iteration. The dynamic foraging mechanism of the SSA was also modified accordingly. Each of the \(FS_{ht}\) individuals moving towards \(FS_{ht}\) randomly select solutions as the target from CA, ensuring a broad balance between objectives and introducing some diversity into the convergence process while converging towards the ideal Pareto front, expanding the coverage range of the Pareto solution set. The \(FS_{ht}\) individuals moving towards \(FS_{at}\) accordingly randomly select solutions from the DA. For the new solutions generated from the dynamic foraging mechanism, each \(FS_{ht}\) is updated based on the greedy principle and dominance relationship. Let \(FS_{ht}^{*}\) be the new solution generated after \(FS_{ht}\)'s gliding; if \(FS_{ht}^{*}\) dominates \(FS_{ht}\), \(FS_{ht}^{*}\) replaces \(FS_{ht}\); if they have no dominance relationship, and \(FS_{ht}^{*}\) replaces \(FS_{ht}\) with a certain probability, ensuring that the progresses towards the ideal Pareto frontier in each iteration of the search process while further introducing variability into the population. The above updating rule can be expressed as Equation (34):  
\[FS_{ht,i}= \begin{cases}FS_{ht,i}^{*},&if\;FS_{ht}^{*}\;dominates\;FS_{ht}\\ randomchoiceof\;[FS_{ht,i}^{*},FS_{ht,i-1}],otherwise\end{cases}  \tag{34}\]  
where \(i-1\) and \(i\), respectively, denote the previous and current iterations within the search process. The pseudocode for the dynamic foraging mechanism in the TA-MOSSA, improved based on the two-archive mechanism, is presented in Algorithm 2. Figure 4: Flowchart of Two_Arch2. In the original Squirrel Search Algorithm (SSA), the seasonal monitoring condition serves to evaluate overall convergence and determine whether the search process should prioritize global exploration or local exploitation. During \"summer\", the SSA infers that the solution population is nearing convergence. It then randomly relocates all \(FS_{nt}\) individuals to promote global exploration and prevent premature convergence to suboptimal solutions.