## 3 Optimization Methods
### Linearly Ensemble of Parameters and Mutation Strategies in Differential Evolution  
The EPSDE method was proposed by [PERSON] et al. in 2011 [95], and it involves the competition of a pool of mutation strategies with a pool of values corresponding to each associated parameter to produce a successful offspring population. This method selects  
Figure 6: Flow chart of DE algorithm. three mutation strategies in the DE algorithm to form a pool and enables each individual in an initial population to be randomly assigned a mutation strategy and associated parameter values (mutation and crossover factors) obtained from the respective pools. The mutation strategy and associated parameter values producing a better trial vector (offspring) is retained, with the trial vector emerging as the target vector (parent) in the next generation, whereas those that fail to produce better trial vectors are randomly reinitialized with a new mutation strategy and associated parameter values. The three mutation strategies selected in the EPSDE method to form a pool with diverse characteristics for most optimization problems are as follows:  
1. DE/rand/1 Exhibits stronger exploration capabilities, and it is faster, robust and is one of the most widely used mutation strategies in the DE literature. 2. DE/best/2 Utilizes two difference vectors, and it is more robust than a strategy that utilizes one difference vector
3. DE/current-to-rand/1  
Takes advantage of its rotation-invariant characteristics, and does not require a crossover operation. In the EPSDE method, each vector is randomly selected using the aforementioned three mutation methods, thereby utilizing the advantages of the three mutation methods. [PERSON] et al. [95] suggested that the population size should be kept constant throughout the evolution process, and the value of mutation factor should be taken in the range of [0.4, 0.9] at a step size of 0.1, and the value of the crossover factor should be taken in the range of [0.1, 0.9] at a step size of 0.1. The flow chart of the EPSDE method is shown in Figure 7.  
The characteristic feature of the EPSDE method is that the designer does not need to set too many algorithm parameters to adapt to the problem; however, this method exhibits an incomplete convergence and slower speed when dealing with higher dimension problems, resulting in the need to increase the number of groups or iterative number to reach convergence. [PERSON] et al. [98] proposed a linear-success-history-based adaptive DE (L-SHADE) method to significantly improve the performance of SHADE using a linear population size reduction (LPSR) method, a simple deterministic population resizing method which continuously reduces the population size according to a linear function, and which only requires the initial population size and population reduction frequency as the user-defined parameters. In addition, it has been experimentally confirmed to exhibit the best overall performance on 30 benchmark problems compared to other DE variants, particularly for higher dimension (D \(\in\)[30; 50]) of the optimal design problem. The LPSR formula for linear population reduction is as follows:  
\[N_{t+1}=round\left(N_{init}-\frac{NFE_{t}}{NFE_{\max}}\cdot(N_{init}-N_{\min})\right) \tag{27}\]  
where \(N_{t+1}\) is the population size in the next generation, \(N_{\min}\) is the minimum number of population size (at the end of the run), \(N_{init}\) is the initial number of population size (at \(t\) = 1), \(NFE_{t}\) is the current number of fitness evaluations, and \(NFE_{\max}\) is the maximum number of fitness evaluations. In the EPSDE method, the population size (\(Np\)) is kept constant throughout the evolution process; thus, in this study, based on the benefit of LPSR in L-SHADE, the characteristic of LPSR was incorporated to improve the EPSDE method to linearly reduce the population size during the evolution process to increase the convergence rate to the optimal solution.