## 6 Results and Discussion
### Investigation on Three Estimation Methods  
The proposed MUCPSO algorithm is examined and compared with four other algorithms, as follows: SPSO, CPSO, GA, and UCPSO to search the optimum solutions to the UCP, COCOMO, and Agile estimation method as formulated in Equation (10) to \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Parameter Settings** & **Ref.** \\ \hline \multirow{2}{*}{SPSO} & \(\omega:0.9\sim 0.4\), \(c_{1}=c_{2}=2\), \(r_{1}=r_{2}=\text{rand}\;[0,1]\), & \\  & \(p_{\text{agile}}=100\), \(p_{\text{ocomo}}=60\), \(p_{\text{UCP}}=100\) & [64] \\ \hline \multirow{2}{*}{CPSO} & \(p=40\), \(\omega:0.9\sim 0.4\), \(c_{1}=c_{2}=1.5\), \(r_{1}=\text{Singer}\), & \\  & \(r_{2}=\text{Sine}\), \(p_{\text{agile}}=100\), \(p_{\text{ocomo}}=100\), \(p_{\text{UCP}}=100\) & [24] \\ \hline \multirow{2}{*}{GA} & crossover rate : 0.8, mutation rate: 1/popSize, elitism selection, & \\  & \(p_{\text{agile}}=100\), \(p_{\text{ocomo}}=100\), \(p_{\text{UCP}}=90\) & [25] \\ \hline \multirow{2}{*}{UCPSO} & \(\omega:0.9\sim 0.4\), \(c_{1}=c_{2}=2\), \(r_{1}=r_{2}=\text{rand}[0,1]\), & \\  & \(p_{\text{agile}}=10\), \(p_{\text{ocomo}}=70\), \(p_{\text{UCP}}=100\), cosine map & [23] \\ \hline \multirow{2}{*}{MUCPSO} & \(\omega:0.9\sim 0.4\), \(c_{1}=c_{2}=2\), \(r_{1}=r_{2}=\text{chaotic}\), & \\  & \(p_{\text{agile}}=10,p_{\text{ocomo}}=100\), \(p_{\text{UCP}}=10\) & Presented \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter settings. Figure 7: Friedman mean rank calculated using 10 different population sizes \(p\) for each optimization algorithms in Agile estimation model. Equation (25). For each estimation method, the maximum particles size is set to 2500, with 30 runs to reduce the coincidence. The random seeds of the 30 initial populations (for each estimation method) are the same to obtain fairness. The evolution is illustrated using the step sizes of 40 to obtain fairness. Thus, all the algorithms show the same generations from 20 to 980. Table 5 illustrates the examination results based on the following four metrics: best solution, worst solution, mean solution, and standard deviation (STD) as explained further in the following passages. The best solution for UCP estimation is yielded by MUCPSO, whereas the highest worst and mean solution metric is obtained using UCPSO, and MUCPSO, respectively. If we compare the two lower standard deviation values between MUCPSO (0.32) and UCPSO (0.18), we can observe that the deviation value is slightly different. Hence, we can conclude that UCPSO and MUCPSO has the smallest and smaller amount of variation, respectively. In other words, the mean for MUCPSO is less reliable than the mean for MUCPSO. The lowest best and mean solution for COCOMO estimation is yielded by SPSO, MUCPSO, respectively, whereas the highest worst solution is yielded by MUCPSO. Although SPSO has the lowest best solution, however, due to the lowest mean and standard deviation is yielded by MUCPSO compared with four other algorithms, we can conclude that MUCPSO has the smallest amount of variation and most reliable for the mean value. The lowest solution value for Agile estimation is yielded by MUCPSO in terms of the best and mean solution. For the highest worst solution is also yielded by MUCPSO. This result is supported by its lowest standard deviation value (0.007), indicating that MUCPSO has the smallest amount of variation which means that the data points are most concentrated around the reliable mean solution. Furthermore, based on the Friedman mean rank (FMR), we can observe that MUCPSO yielded the first rank, followed by SPSO, CPSO, UCPSO, and GA. The Wilcoxon rank-sum test illustrated in Table 6 confirms that MUCPSO is significantly better than all the competitor algorithms for the three estimation benchmark methods. All the \(p\)-values are lower than the significance level of 0.05, except the UCP estimation where MUCPSO is worse than UCPSO with a \(p\)-value of greater than 0.05.