## 4 Design of the POM-PSASIDEA Algorithm
### Population Size Adaptive Strategy for the Improved Differential Evolution Algorithm
Similarly, the diversity matrix is obtained, and the diversity threshold is \(Q_{\xi}\). It is worth noting that individual similarity within a population is used to assess the degree of similarity of individuals within a population, focusing on the differences between individuals. Population diversity is used to assess the overall level of variation in the population. The calculation of individual similarity and population diversity enables a comprehensive evaluation of both dimensions of the algorithm. In the later stages of algorithm evolution, the population evolution status monitoring mechanism determines whether the algorithm is facing evolutionary stagnation through both the fitness value and the population evolution rate. Here, the fitness value is consistent with the objective function of each layer of the MSIMPLTS-MLOO model, and the population evolution rate refers to the degree of improvement in the optimal solution between two adjacent generations, which can reflect the convergence speed of the population to a certain extent. The mathematical expression for the population evolution rate is shown in Equation (18):  
\[R_{evolution}=\frac{1}{0.5G}\sum_{g=0.5G}^{G}\frac{f_{i}(optimal_{g})-f_{i}(optimal _{g-1})}{\left|f_{i}(optimal_{g-1})\right|+\mu},i=(1,2,3) \tag{18}\]  
where the population evolution rate is calculated at the late stage of evolution, so the number of iterations after 50% is \([0.5G,G]\), \(G\) denotes the number of iterations, \(g\) denotes the current number of iterations, and \(f_{i}(optimal_{g})\) denotes the value of optimal fitness among the populations in the \(g\)-th generation of the \(i\)-th objective function of the MSIMPLTS-MLOO model. If the population evolution rate is not significantly improved in generation \(m\), it can be assumed that there is a stagnation problem in the population evolution. Secondly, according to the output results of the population evolution state monitoring mechanism, the population size adaptive strategy is used to adjust the population size. For the pre-evolutionary stage of the algorithm, there is a high similarity and low diversity of individual populations, and there is an adaptive increase in population size to improve the diversity of the population, which is conducive to promoting the exploration of the algorithm to help it jump out of the local optimum; for the late stage of the algorithm, there is a stagnation of the value of the fitness of the population, as well as the rate of evolution of the population for many consecutive generations; adaptive reduction in the population size is conducive to the depth of the development to accelerate the rate of convergence of the population. The construction method of the population size adaptive strategy is shown in Equation (18):  
\[\left\{ \begin{array}{l}N_{g}=N_{initial}+(N_{max}-N_{initial})\cdot\frac{f_{i} (x)}{\max_{f_{i}}(x)},g\in[0,0.5G]\\ N_{g}=N_{0.5G}-(N_{0.5G}-N_{min})\cdot\frac{f_{i}(x)}{\max_{f_{i}}(x)},g\in[0.5 G,G]\end{array} \right. \tag{19}\]  
where \(N_{g}\) denotes the population size of the \(g\)-th iteration, \(N_{max}\) and \(N_{min}\) denote the upper and lower limits of the population size, respectively, and \(f_{i}(x)/\max f_{i}(x)\) denotes the ratio of the fitness value of the objective function to the maximum fitness value of the previous generations, which is used as a scaling factor for the adjustment of the population size. Since too large a population size may lead to excessive consumption of computational resources, while too small a population size may lead to premature convergence of the algorithm to a local optimum, the upper and lower values of the population size are set to enable the algorithm to achieve a balance between stability and robustness. The PSASIDEA algorithm is specifically described in Algorithm 5. Firstly, a random population is generated based on the initial population size, and in each iteration of the generation, a mutation operation is performed to generate new mutated individuals.