# SSTNDS: A SET-Based Non-Dominated Sorting Algorithm for Multi-Objective Optimization Problems
## 6 Conclusions  
A novel efficient non-dominated sorting algorithm--SETNDS for MOEA--is proposed in this paper. Unlike other non-dominated sorting algorithms, this paper adopts the positional information of solutions and combines them with set theory to find the dominant set that dominates the solution that needs to be sorted first. In SETNDS, the rank of solutions can be determined by, at most, one comparison, therefore the number of comparisons of the algorithm is far less than the existing algorithms. The experimental results indicate that SETNDS is superior to its competitors--namely the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Algorithm**} & \multicolumn{5}{c}{**DTZ11**} \\ \cline{3-6}  & & **2** & **5** & **10** & **15** \\ \hline \multirow{2}{*}{**FNDS**} & **Com.** & \(1.25\times 10^{7}\) & \(1.44\times 10^{7}\) & \(1.5\times 10^{7}\) & \(1.49\times 10^{7}\) \\  & **Time** & 139.23 & 173.11 & 176.71 & 189.61 \\  & **Com.** & \(7.07\times 10^{6}\) & \(7.78\times 10^{6}\) & \(8.07\times 10^{6}\) & \(7.89\times 10^{6}\) \\  & **Time** & 44.07 & 49.96 & 50.90 & 53.16 \\  & **Com.** & \(1.17\times 10^{7}\) & \(1.29\times 10^{7}\) & \(1.32\times 10^{7}\) & \(1.30\times 10^{7}\) \\  & **Time** & 74.64 & 85.83 & 89.00 & 97.16 \\  & **Com.** & \(1.20\times 10^{7}\) & \(1.41\times 10^{7}\) & \(1.47\times 10^{7}\) & \(1.46\times 10^{7}\) \\  & **Time** & 52.10 & 65.70 & 68.95 & 72.80 \\  & **Com.** & \(9.82\times 10^{6}\) & \(1.28\times 10^{7}\) & \(1.39\times 10^{7}\) & \(1.37\times 10^{7}\) \\  & **Time** & 58.63 & 80.52 & 83.49 & 89.87 \\  & **Com.** & \(6.39\times 10^{6}\) & \(5.06\times 10^{6}\) & \(3.74\times 10^{6}\) & \(2.97\times 10^{6}\) \\  & **Time** & 34.64 & 34.11 & 31.91 & 33.80 \\  & **Com.** & \(2.37\times 10^{4}\) & \(1.53\times 10^{4}\) & \(1.28\times 10^{4}\) & \(1.32\times 10^{4}\) \\  & **Time** & 9.16 & 11.07 & 14.28 & 16.54 \\ \hline \multicolumn{2}{c}{**Algorithm**} & \multicolumn{5}{c}{**DTZ12**} \\ \cline{3-6}  & **2** & **5** & **10** & 15 \\ \hline \multirow{2}{*}{**FNDS**} & **Com.** & \(1.46\times 10^{7}\) & \(1.49\times 10^{7}\) & \(1.55\times 10^{7}\) & \(1.57\times 10^{7}\) \\  & **Time** & 167.92 & 191.26 & 187.76 & 197.01 \\  & **Com.** & \(9.08\times 10^{6}\) & \(9.22\times 10^{6}\) & \(1.06\times 10^{7}\) & \(1.15\times 10^{7}\) \\  & **Time** & 53.16 & 59.66 & 67.05 & 80.92 \\  & **Com.** & \(1.32\times 10^{7}\) & \(1.31\times 10^{7}\) & \(1.39\times 10^{7}\) & \(1.41\times 10^{7}\) \\  & **Time** & 87.15 & 95.19 & 94.72 & 101.09 \\  & **Com.** & \(1.45\times 10^{7}\) & \(1.49\times 10^{7}\) & \(1.52\times 10^{7}\) & \(1.56\times 10^{7}\) \\  & **Time** & 64.24 & 69.10 & 77.03 & 84.11 \\  & **Com.** & \(1.32\times 10^{7}\) & \(1.39\times 10^{7}\) & \(1.50\times 10^{7}\) & \(1.53\times 10^{7}\) \\  & **Time** & 79.85 & 87.74 & 98.69 & 103.47 \\  & **Com.** & \(7.25\times 10^{6}\) & \(5.37\times 10^{6}\) & \(4.13\times 10^{6}\) & \(3.32\times 10^{6}\) \\  & **Time** & 41.93 & 42.17 & 36.42 & 39.94 \\  & **Com.** & \(1.50\times 10^{4}\) & \(1.22\times 10^{4}\) & \(0.99\times 10^{4}\) & \(0.91\times 10^{4}\) \\  & **Time** & 9.40 & 11.76 & 14.59 & 16.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of comparisons and running time (in seconds) for DTLZ1 and DTLZ2. fast non-dominated sort, the arena's principle sort, the deductive sort, the corner sort, the efficient non-dominated sort and the best order sort, especially with vast populations. The shortcoming of the proposed algorithm is that it will take more time on pre-ordering with the growth of the objectives. There are two directions of our future work: one is to parallelize the algorithm in order to increase its speed. The other one is to combine the proposed algorithm and MOEAs (e.g., NSGA II) to solve practical problems. [PERSON] and [PERSON] conceived the method, [PERSON] wrote the manuscript draft; [PERSON] and [PERSON] helped to modify it; [PERSON] and [PERSON] provided experimental data. All authors have read and agreed to the published version of the manuscript.