## 6 Results and Discussion
### Investigation on Three Estimation Methods
However, since the proposed method has 11 better results (91.67%) out of 12 significant results, we can conclude that the proposed method is better than most of the benchmark algorithms in almost all the estimation methods. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **Metric** & **SPSO** & **CPSO** & **GA** & **UCPSO** & **MUCPSO** \\ \hline UCP & Best & 1009.26 & 1009.31 & 1071.29 & 1009.46 & 1009.14 \\  & Worst & 1011.25 & 1011.51 & 1162.35 & 1010.16 & 1010.28 \\  & Mean & 1010.21 & 1010.34 & 1123.26 & 1009.9 & 1009.81 \\  & STD & 0.47 & 0.56 & 27.64 & 0.18 & 0.32 \\ \hline COCOMO & Best & 21.06 & 24.39 & 61.94 & 28.56 & 25.58 \\  & Worst & 271.68 & 301.14 & 642.35 & 49.4 & 40.32 \\  & Mean & 63.97 & 68.92 & 183.24 & 40.89 & 34.43 \\  & STD & 53.7 & 60.27 & 144.22 & 5.32 & 3.18 \\ \hline Agile & Best & 13.466 & 13.466 & 16.597 & 13.463 & 13.461 \\  & Worst & 13.587 & 13.570 & 24.883 & 13.537 & 13.493 \\  & Mean & 13.498 & 13.49 & 21.643 & 13.488 & 13.475 \\  & STD & 0.029 & 0.021 & 1.974 & 0.014 & 0.007 \\  & **FMR** & **2.87** & **3.13** & **4.07** & **3.44** & **1.67** \\  & **Rank** & **2** & **3** & **5** & **4** & **1** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of SPSO, CPSO, GA, UCPSO, and MUCPSO for 3 software estimation methods. Furthermore, we discussed the detailed investigations of the convergence analysis of MUCPSO and its competitors. For each estimation method benchmark, the maximum number of particles is set to 2500 with 30 runs. Figure 8a illustrates the evolutionary processes of four algorithms (SPSO, CPSO, UCPSO, and MUCPSO) until they converge with the optimum solution for the UCP estimation method. Figure 8b illustrates the evolutionary process of all of the algorithms until convergence with the optimum solution. In this benchmark, MUCPSO converges to a much better solution than the other algorithms. Impressively, MUCPSO evolves at the greatest speed in the initial generations and finally provides the best mean solution of 1009.81. Next, the convergence analysis is provided for the COCOMO estimation method benchmark. Figure 9 illustrates the evolutionary processes of all algorithms until convergence with the optimum solution. In this benchmark, MUCPSO converges to a much better solution than the SPSO, CPSO, GA, and UCPSO algorithm. MUCPSO demonstrates and impressive convergence from the beginning to the end of generation. Finally, Figure 10 illustrates the evolutionary processes of all algorithms until convergence with the optimum solution for the Agile estimation method. In this benchmark, MUCPSO converges stably to the better solution than the other algorithm. From the beginning of generation, MUCPSO is able to compete with four other algorithms and finally provides the best mean solution of 13.475.  
benchmark algorithms, including GA. We are able to observe that the difference between GA and the aforementioned algorithms is quite large. Figure 10: Convergence analysis for Agile. Figure 9: Convergence analysis for COCOMO. Next, we analyze the diversity of COCOMO estimation. Figure 12 shows that the median of SPSO, CPSO, UCPSO, and MUCPSO is quite similar. However, based on the interquartile box, we can observe that MUCPSO has the smallest shape compared with other algorithms. This shape indicates that MUCPSO has the lowest spread. Finally, Figure 13 describe the diversity analysis of Agile estimation. Similar to Figure 11, we split the results into two parts, presented as Figure 13a,b. Due to the large difference observed when including GA in the group of boxplots, we provide clearer results by excluding GA in Figure 13a. Based on the boxplots in Figure 13a we observe that  
Figure 11: Best solution diversity analysis for UCP (**a**) of four algorithms (SPSO, CPSO, UCPSO, MUCPSO); (**b**) of all algorithms including GA.  
Figure 12: Best solution diversity analysis for COCOMO. MUCPSO has the lowest median, whereas the rest (SPSO, CPSO, and UCPSO) are quite similar.