## 4 Conclusions  
This study conducts an in-depth investigation into how the optimization performance of the Kriging-based parallel EGO algorithm with the PEI multi-point sampling criterion is affected by different numbers of points filled in a single optimization cycle. Meanwhile, to obtain comprehensive numerical simulation results and conduct an in-depth comparative analysis, nine benchmark test functions with different optimization problem characteristics were selected as optimization targets. The point-filling levels within a single optimization cycle were set to five values: 2, 4, 6, 8, and 10. Different fidelity surrogate models were then constructed using both OK and GEK. Finally, the optimization tasks were performed using the parallel EGO algorithm with the PEI multi-point sampling criterion. Finally, a systematic comparative analysis of the results was conducted from the perspectives of the optimization performance, efficiency, and diversity of the parallel EGO algorithm, leading to the following conclusions:  
1. The impact of different point-filling quantities on the optimization performance of the parallel EGO algorithm based on the OK model is significant. Filling two points per optimization cycle ensures the most stable optimization performance of the parallel EGO algorithm. However, as the number of filled points increases, the stability of the optimization performance gradually declines. In contrast, for the GEK model, the influence of different point-filling quantities on the optimization performance of the  
Figure 15: Exploration and exploitation curves of the GEK-based parallel EGO algorithm on all tested problems. parallel EGO algorithm is only noticeable in the early stages of optimization. As the optimization process progresses, the performance differences gradually diminish. 2. For the parallel EGO algorithm based on the OK model, optimization efficiency improves as the point-filling quantity increases. However, the rate of improvement gradually slows with the increasing number of filled points. In the case of GEK, this slowdown in optimization efficiency becomes even more pronounced. 3. The point-filling quantity has a significant impact on the optimization diversity of the parallel EGO algorithm based on the OK model. In most cases, filling two points per optimization cycle ensures a reasonable level of optimization diversity and maintains a balance between local exploitation and global exploration. In contrast, for the GEK model, the influence of different point-filling quantities on the diversity of the parallel EGO algorithm is less pronounced. 4. For practical implementation, it is recommended to employ a smaller number of filling points per optimization cycle during the initial stages when design space information remains relatively limited. However, as iterations progress, gradually increasing the filling quantity may enhance algorithmic convergence speed and improve overall optimization efficiency. Although the introduction of multi-point sampling criteria can significantly improve the efficiency of the EGO algorithm in solving practical optimization problems, its fundamental principle still follows a sequential point selection strategy within a single optimization cycle. Therefore, achieving true parallelization of the EGO algorithm requires decoupling the selection process of potential improvement points and reducing their mutual dependence. In the future, further research will focus on developing more efficient and stable point-filling strategies to better meet the needs of practical engineering optimization problems. Conceptualization, H.F. and M.T. ; methodology, H.F. and Q.W. ; software, H.F. and Q.W. ; validation, H.F. and T.N. ; formal analysis, H.F., R.B. and M.T. ; writing--original draft preparation, H.F.; writing--review and editing, Q.W., T.N., R.B. and M.T. ; supervision, M.T. ; funding acquisition, M.T. and H.F. All authors have read and agreed to the published version of the manuscript.