# Binary Cockroach Swarm Optimization for Combinatorial Optimization Problem
## 4 Simulation Studies
The Mann-Whitney U-test null hypothesis is that two groups of data are not different; the U statistics computed from the data determine whether the null hypothesis is accepted or rejected. The calculated \(p\)-value is the probability of obtaining either the observed difference or a more extreme value of the difference between the two groups. It is used as a basis for either accepting or rejecting the null hypothesis. If the \(p\)-value is less than a threshold value of 0.05, we reject the null hypothesis, and the result is considered significant. Otherwise, the null hypothesis is accepted. Tables 4 and 5 illustrate the U statistics conducted on the mean optimal performance of BCSO and BPSO [25]. The computed \(P\)-values asymptotic significance 0.0 is less than the threshold value 0.05, and the null hypothesis is rejected, which indicates that the performance of the BCSO algorithm is significantly different from the BPSO algorithm for solving the benchmark functions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ranks** & & & & \\ \hline  & **Algorithm** & **N** & **Mean Rank** & **Sum of Ranks** \\ \hline  & 1 & 12 & 6.88 & 82.50 \\ Fitness & 2 & 12 & 18.13 & 217.50 \\  & Total & 24 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mann–Whitney U-test statistics on the performance of BCSO and BPSO. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Benchmarks** & **Dimensions** & **BCSO** & **BPSO**[27] \\ \hline  & 3 & **0.00** & 6.82 \(\times\) 10\({}^{-9}\) \\ Sphere & 5 & **0.00** & 1.92 \(\times\) 10\({}^{-6}\) \\  & 10 & **0.00** & 0.11 \\ \hline  & 3 & **1.3 \(\times\) 10\({}^{-6}\)** & 0.09 \\ Rosenbrock & 5 & **4.4 \(\times\) 10\({}^{-5}\)** & 2.25 \\  & 10 & **0.00** & 32.83 \\ \hline  & 3 & **0.00** & 2.09 \(\times\) 10\({}^{9}\) \\ Griewangk & 5 & **0.00** & 7.4 \(\times\) 10\({}^{3}\) \\  & 10 & **0.00** & 0.06 \\ \hline  & 3 & **0.00** & 1.35 \(\times\) 10\({}^{-6}\) \\ Rastrigin & 5 & **0.00** & 3.40 \(\times\) 10\({}^{-3}\) \\  & 10 & **0.00** & 10.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the mean global best of BCSO and BPSO. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ranks** & & & \\ \hline  & **Algorithm** & **N** & **Mean Rank** & **Sum of Ranks** \\ \hline  & 1 & 12 & 6.88 & 82.50 \\ Fitness & 2 & 12 & 18.13 & 217.50 \\  & Total & 24 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mann–Whitney U-test statistics. We computed the effect size \(r\) of the significant difference of the test, \(z=-4.046\), and \(N=24\) is the total number of samples. \[ \begin{array}{ll}r&=\frac{z}{\sqrt{N}}\\ &=-\frac{4\,046}{\sqrt{24}}\\ &=-0.8\end{array} \]  
\(Z\) is negative when the observed data are below the mean and positive when above. Using [PERSON]'s guideline on effect size \(r\) (small: 0.1, medium: 0.3, large: 0.5, very-large: 1.0) [30; 31]. The statistic 0.8 is of a very large magnitude effect size, which indicates that the BCSO algorithm performance is significantly different from the BPSO algorithm. Additionally, in this work, the performances of the original CSO [5] and ICSO [10] were evaluated on the same set of benchmark problems and the same experimental settings as shown in this section and then was compared to the proposed binary version BCSO. Table 6 shows the simulation results of the original CSO, and Table 7 depicts the result of ICSO. The average performance of both the original CSO and its improved version, ICSO, were compared to the proposed BCSO in Table 8. The comparison results clearly show that the proposed binary version BCSO algorithm outperforms the existing continuous versions, CSO and ICSO. BCSO is able to solve optimally all of the tested benchmark problems, including Rosenbrock, which has been shown in the literature to be difficult to solve optimally.