## 4 Results
### Comparison with Existing Algorithms  
After conducting the calibration, we compared MOWPS with three existing algorithms, using the instances from Section 4.1 for the comparison. In Figure 11, the trade-offs produced by various algorithms for different instance sizes are depicted in 2D and 3D plots, with each point on the plot representing a potential task assignment. The results indicate that MOWPS generates superior trade-off fronts compared to other algorithms, with a more evenly distributed set of solutions. This is attributed to the proposed Petri-net-based constraint amending method, which does not constrain the solution space, and the SDRS, which effectively evaluates solutions. Therefore, the task assignments produced by MOWPS offer a better trade-off between energy, cost, and makespan. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Factor Level**} & \(N_{p}\) & \(\Xi\) & \(\varepsilon\) & \(P_{reserve}\) & \(P_{bias}\) \\ \hline \multirow{8}{*}{Small} & 1 & 0.5769 & 0.4749 & 0.5584 & 0.6146 & 0.5910 \\  & 2 & 0.6050 & 0.6235 & 0.5400 & 0.6329 & 0.6565 \\  & 3 & 0.6809 & 0.7011 & 0.7154 & **0.6732** & **0.6603** \\  & 4 & **0.6859** & **0.7491** & **0.7348** & 0.6280 & 0.6409 \\  & Delta & 0.1090 & 0.2742 & 0.1949 & 0.0586 & 0.0693 \\  & Rank & 3 & 1 & 2 & 5 & 4 \\  & SPV & 50 & 0.4 & 0.4 & 0.4 & 0.2 \\ \hline \multirow{8}{*}{Medium} & 1 & 0.3869 & 0.2199 & 0.3554 & 0.4961 & 0.4611 \\  & 2 & 0.4363 & 0.5096 & 0.3742 & 0.4093 & 0.4444 \\  & 3 & 0.5455 & 0.5252 & **0.6312** & **0.5466** & 0.5207 \\  & 4 & **0.6105** & **0.7244** & 0.6184 & 0.5270 & **0.5530** \\  & Delta & 0.2236 & 0.5044 & 0.2758 & 0.1373 & 0.1086 \\  & Rank & 3 & 1 & 2 & 4 & 5 \\  & SPV & 50 & 0.4 & 0.3 & 0.4 & 0.3 \\ \hline \multirow{8}{*}{Large} & 1 & 0.2493 & 0.1459 & 0.2564 & 0.4236 & 0.3803 \\  & 2 & 0.3543 & 0.3701 & 0.2596 & 0.3429 & 0.3862 \\ \cline{1-1}  & 3 & 0.4809 & 0.4110 & **0.5422** & **0.4254** & **0.4137** \\ \cline{1-1}  & 4 & **0.5023** & **0.6598** & 0.5286 & 0.3949 & 0.4066 \\ \cline{1-1}  & Delta & 0.2529 & 0.5140 & 0.2858 & 0.0825 & 0.0334 \\ \cline{1-1}  & Rank & 3 & 1 & 2 & 4 & 5 \\ \cline{1-1}  & SPV & 50 & 0.4 & 0.3 & 0.4 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistical analyses and suggested parameter values. Figure 11: Trade-offs for all four algorithms on different instances. (**a**) A \(\times\) 1 (**b**) B \(\times\) 1 (**c**) C \(\times\) 2 (**d**) D \(\times\) 2 (**e**) E \(\times\) 5 (**f**) F \(\times\) 5. Table 9 displays the statistics of the comparison results for all algorithms. The performance of each algorithm varies significantly, with the optimal value for each instance highlighted in bold black. MOWPS outperforms all other algorithms in terms of \(aNS\), \(aHV\), and \(aDR\) in almost all cases, except for a few instances where NSGA-II has the highest \(aNS\). Figure 12 illustrates the performance of each algorithm on different instances. In Figure 12a, MOWPS and NSGA-II produce a significantly higher number of non-dominated solutions than MCHO and IMPOSQ for all instances, as MCHO and IMPOSQ's constraint processing method restricts the solution space to produce a specific task sequence. In contrast, our Petri-net-based method efficiently amends any randomly generated task sequence to satisfy the constraints, as demonstrated in Section 4.2. Figure 12b indicates that MOWPS outperforms other algorithms in terms of \(aHV\), considering the balance between convergence and diversity of the non-dominated solutions, while others only consider diversity as a submetric (e.g., IMPOSQ and NSGA-II), which enables MOWPS to obtain more diverse and high-quality solutions. Figure 12c demonstrates that MOWPS consistently generates solutions with high \(aDR\) values, validating the effectiveness of the algorithmic search mechanism. These numerical experiments confirm the exceptional performance of MOWPS for offloading tasks with dependencies in SEC.