# Crossover Rate Sorting in Adaptive Differential Evolution
## 3 Results
### Numerical Results  
The parameter settings for the mentioned algorithms were taken from the original papers. For example, for L-SHADE-RSP, the initial population size was set to \(N_{max}=75*D^{\frac{2}{3}}\) and memory size \(H=5\); for NL-SHADE-RSP, the initial population size \(N_{max}=30*D\), and memory size \(H=20*D\); and for NL-SHADE-LBC, the population size \(N_{max}=23*D\), and \(H=20*D\). For the L-NTADE algorithm, the following values were used: \(N_{max}=20*D\), selective pressure parameter \(kp=3\), biased parameter adaptation coefficient \(pm=4\), and mutation strategy r-new-to-ptop/n/t. These values were set, as they enable us to achieve good results on both CEC 2017 and CEC 2022 benchmarks. The algorithms with crossover rate sorting were marked with _s, e.g., L-SHADE-RSP_s. It should be mentioned that the NL-SHADE-RSP and NL-SHADE-LBC had the sorting procedure in their standard setting, so here, the standard setting is denoted as NL-SHADE-RSP_s and NL-SHADE-LBC_s. The efficiency comparison of the tested algorithms was performed with the Mann-Whitney statistical test with normal approximation and tie-breaking to compare variants with and without crossover rates sorting. Using normal approximation is possible due to the sufficient sample sizes (30 or 51 independent runs). It also enables us to calculate the standard score (Z-score), which simplifies the reasoning and estimating the difference between algorithm variants. If the decision about the significance of such a difference is to be made, the significance level equal to 0.01 is used, which corresponds to the threshold Z score of \(|2.58|\) (two-tailed test). In addition to the Mann-Whitney test, the Friedman ranking procedure was used to analyze the performance of the algorithms. The ranks were calculated for every function and then summed, with lower ranks being better. In the first set of experiments, the algorithms are compared on the CEC 2017 benchmark. The results are shown in Tables 1 and 2, the latter of which contains the Friedman ranking of the results. The values in Table 1 shown in every cell are the number of wins/ties/losses of the algorithm without crossover rate sorting, and with it, the sum of these three equals the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & \(\mathbf{10*D}\) & \(\mathbf{30*D}\) & \(\mathbf{50*D}\) & \(\mathbf{100*D}\) \\ \hline L-SHADE-RSP vs. & \(1/27/2\) & \(6/24/0\) & \(4/25/1\) & \(7/22/1\) \\ L-SHADE-RSP\_s & (2.07) & (29.45) & (28.24) & (21.21) \\ \hline NL-SHADE-RSP vs. & \(3/27/0\) & \(1/28/1\) & \(1/29/0\) & \(0/28/2\) \\ NL-SHADE-RSP\_s & (12.67) & (3.23) & (8.59) & (\(-8.92\)) \\ \hline NL-SHADE-LBC vs. & \(1/27/2\) & \(3/26/1\) & \(13/15/2\) & \(11/17/2\) \\ NL-SHADE-LBC\_s & (\(-9.51\)) & (26.43) & (51.53) & (50.23) \\ \hline L-NTADE vs. & \(0/28/2\) & \(1/22/7\) & \(2/19/9\) & \(4/15/11\) \\ L-NTADE\_s & (\(-10.58\)) & (\(-27.31\)) & (\(-14.32\)) & (\(-32.69\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Algorithms with and without crossover rate sorting, CEC 2017, Mann–Whitney tests and total standard score. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & \(\mathbf{10*D}\) & \(\mathbf{30*D}\) & \(\mathbf{50*D}\) & \(\mathbf{100*D}\) & **Total** \\ \hline LSHADE-RSP & 109.00 & 99.45 & 96.85 & 96.72 & 402.02 \\ LSHADE-RSP\_s & 99.21 & 92.85 & 95.21 & 92.23 & 379.49 \\ NL-SHADE-RSP & 127.16 & 120.21 & 114.37 & 110.51 & 472.25 \\ NL-SHADE-RSP\_s & 121.92 & 119.51 & 114.37 & 110.51 & 466.31 \\ NL-SHADE-LBC & 88.43 & 104.06 & 110.52 & 108.70 & 411.71 \\ NL-SHADE-LBC\_s & 83.47 & 101.79 & 109.83 & 108.28 & 403.38 \\ L-NTADE & 109.83 & 104.54 & 98.81 & 106.25 & 419.43 \\ L-NTADE\_s & 100.98 & 97.59 & 100.03 & 106.81 & 405.41 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Algorithms with and without crossover rate sorting, CEC 2017, and Friedman ranks. number of test functions. The number in brackets is the total standard score, i.e., sum of all standard scores of the Mann-Whitney test over all test functions.