## 4 Experimental Study
### Performance Comparison of Different Optimization Algorithms
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance** & **Algorithm** & _Avg_ & **RPE\_A (\%)** & _Best_ & **RPE\_B (\%)** \\ \hline \multirow{4}{*}{Case 1} & SA & [149, 198] & 0 & [149, 198] & 0 \\  & GA & [149, 198] & 0 & [149, 198] & 0 \\  & PSO & [149, 198] & 0 & [149, 198] & 0 \\  & PSO & [149, 198] & 0 & [149, 198] & 0 \\  & SA & [216, 279] & 0 & [216, 279] & 0 \\ Case 2 & GA & [216, 279] & 0 & [216, 279] & 0 \\ (33 \(\times\) 4) & PSO & [216, 279] & 0 & [216, 279] & 0 \\  & DPSO & [216, 279] & 0 & [216, 279] & 0 \\  & SA & [501, 553, 539] & \(-\)4.7 & [146, 251] & \(-\)1.2 \\ Case 3 & GA & [4903, 549.5] & \(-\)3.9 & [149, 509] & 0 \\ (86 \(\times\) 5) & PSO & [479, 5, 541.2] & \(-\)2.3 & [149, 509] & 0 \\  & DPSO & [471, 6, 528.8] & 0 & [149, 509] & 0 \\  & SA & [539, 549, 69.6] & \(-\)5.3 & [149, 563] & \(-\)2.3 \\ Case 4 & GA & [518, 7, 583.5] & \(-\)2.5 & [149, 555] & \(-\)0.9 \\ (92 \(\times\) 5) & PSO & [512, 45, 734.4] & \(-\)0.7 & [149, 550] & 0 \\  & DPSO & [506, 369.4] & 0 & [149, 550] & 0 \\  & SA & [795, 874.2] & \(-\)4.0 & [149, 557, 827] & \(-\)1.6 \\ Case 5 & GA & [775, 485.1] & \(-\)1.3 & [149, 514] & 0 \\ (125 \(\times\) 6) & PSO & [772, 38, 547.5] & \(-\)0.9 & [149, 514] & 0 \\  & DPSO & [762, 284.0] & 0 & [149, 514] & 0 \\  & SA & [721, 678.8] & \(-\)8.3 & [149, 514] & 0 \\ Case 6 & GA & [692, 6, 764.3] & \(-\)4.9 & [149, 514] & \(-\)5.6 \\ (148 \(\times\) 8) & PSO & [673, 72, 794.4] & \(-\)2.9 & [149, 514] & \(-\)1.7 \\  & DPSO & [658, 472.6] & 0 & [149, 514] & 0 \\ Case 7 & GA & [807, 38, 893.] & \(-\)6.4 & [149, 514] & \(-\)4.3 \\ (164 \(\times\) 8) & PSO & [786, 369.8] & \(-\)4.1 & [149, 514] & \(-\)1.6 \\  & DPSO & [756, 835.7] & 0 & [149, 514] & 0 \\  & SA & [891, 9, 984.8] & \(-\)14.1 & [149, 514] & \(-\)7.8 \\ Case 8 & GA & [802, 49, 909.4] & \(-\)5.3 & [149, 514] & \(-\)3.6 \\ (170 \(\times\) 9) & PSO & [1811, 596.2] & \(-\)3.8 & [149, 514] & \(-\)3.9 \\  & DPSO & [779, 863.4] & 0 & [149, 514] & 0 \\  & DPSO & [779, 863.4] & 0 & [149, 514] & 0 \\  & SA & [871, 597.0] & \(-\)9.3 & [149, 514] & 0 \\ Case 9 & GA & [814, 912.4] & \(-\)2.8 & [149, 514] & 0 \\ (176 \(\times\) 9) & PSO & [812, 690.7] & \(-\)2.3 & [149, 514] & \(-\)6.4 \\  & DPSO & [794, 887.2] & 0 & [149, 514] & 0 \\  & SA & [1069, 2117.9] & \(-\)11.4 & [149, 514] & \(-\)6.4 \\ Case 10 & GA & [998, 2116.3] & \(-\)5.9 & [149, 514] & \(-\)4.7 \\ (185 \(\times\) 10) & PSO & [988, 1509.1] & \(-\)3.2 & [149, 514] & \(-\)3.9 \\  & DPSO & [953, 7, 1057.4] & 0 & [149, 514, 514] & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparisons of SA-MAE and the four existing algorithms for the A)SSP on different scenarios. and SA, respectively. In terms of algorithm stability, the objectives of DPSO are on average 5.1% and 10.3% better than GA and SA, respectively. Considering the similar structure of the population-based optimization algorithms, the computational results of GA and DPSO show that the proposed encoding and initialization method based on the adjacency matrix are effective. The discretized particle update method is also more advantageous than conventional genetic operations. In addition, we compared the performance of the original PSO algorithm and the improved DPSO algorithm. For Cases 1-5, although the PSO also obtained the same best values of optimal solutions, the algorithm is not stable enough in 10 independent runs. For Cases 6-10, DPSO is obviously better than PSO in all respects. Specifically, the best values and average value of optimal solutions are increased by 2.6% and 3.3%, respectively. This result also proves the effectiveness of the proposed particle update method. In summary, the proposed DPSO algorithm is competitive in performance and stability, especially on larger-scale problems.