# Task-Driven Virtual Machine Optimization Placement Model and Algorithm
## 5 Experiment Results and Analysis
### Evaluation of the CE-VMP Method
#### 5.4.2 Efficiency
Figure 9 presents the results of the GD metrics. It can be seen that the GD metrics for the CE-NSGAIII proposed in this study are superior to those of the other compared algorithms. This suggests that the solution set generated by the CE-NSGAIII is closer to the optimal Pareto front, indicating better convergence performance. This improvement is attributed to the introduction of the PBI association strategy, which guides the search process by incorporating a penalty term. This scheme helps the algorithm approximate the Pareto frontier more accurately while avoiding the generation of infeasible solutions. Furthermore, as shown in Figure 9a,b, all algorithms exhibit significant fluctuations in GD values. These fluctuations can be explained by the limited solution space when the number of tasks is small, making the algorithms more susceptible to falling inside local optima. As a result, during the iterative process, the algorithms are prone to being trapped in various local optima, leading to considerable variations in the generated Pareto solution sets. The experiment results of the HV values are presented in Table 5. The compared algorithm obtaining the best performance is marked with dark gray shading. It is observed that when the number of tasks is high, the HV values for the CE-NSGAIII outperform those of the other compared algorithms across eight instances (\(T4\), \(T5\), \(T7\), \(...,T12\)). This performance is attributed to the incorporation of the chaotic mapping mechanism and the PBI association strategy, which collectively enhance both the convergence and diversity of the obtained Pareto solution set. However, the CE-NSGAIII shows generally in the four instances of \(T1\), \(T2\), \(T3\), and \(T6\). This is likely due to the fact that, for small-scale problems, the random initialization strategies used by the other compared algorithms are simple and effective, allowing them to cover the solution space more quickly. In contrast, when the population size is limited, the elite guidance strategy in CE-NSGAIII may prematurely  
Figure 8: Comparison of Pareto front (T7–T12). fixate on a specific set of solutions, potentially overlooking other beneficial search directions and thus reducing diversity during the initial exploration phase. In instances \(T2\), \(T3\), and \(T6\), the NSGA-III achieves the best performance, likely because its original reference point association strategy more accurately guides the population evolution compared to the PBI association strategy used in CE-NSGAIII. Figure 9: Comparison of GD. The results of the IGD values, as shown in Table 6, demonstrate that the CE-NSGAIII algorithm outperforms the other compared algorithms, with the best-performing algorithm highlighted in dark gray. The results imply that the CE-NSGAIII generates a solution set closer to the true Pareto front with a more uniform distribution. This performance advantage is primarily due to the chaotic mapping mechanism employed by the CE-NSGAIII, which produces initial solutions with high randomness and comprehensive exploration capabilities, enabling more efficient exploration of multiple regions of the problem space. Additionally, the elite guidance strategy facilitates a rapid return to promising regions, effectively enhancing the convergence of the algorithm. As a result, the CE-NSGAIII achieves better IGD metrics when solving the TLVMP problem. The experiment results of the IGD+ values are presented in Table 7. The compared algorithm obtaining the best performance is marked with dark gray shading. It can be observed that the IGD+ value of CE-NSGAIII outperforms the other comparative algorithms in 10 instances (\(T1\), \(T4\), \(T5\),..., \(T12\)), as the number of tasks increases. This improvement is attributed to the PBI association strategy, which effectively guides population evolution, and the chaotic mapping mechanism, which enhances the randomness of the generated population and broadens the search space coverage.