## 4 Experimental Results
### Quantile Comparisons Using Peak, Average, and Bad-Case Performances  
Although the mean and standard deviation values presented in the previous section are useful in performing statistical comparisons between ACO algorithms, they are not very informative [17]. For example, they do not provide any indication of the expected solution quality value in case ACO is executed once, which is the case in real-world applications. On the other hand, the quantiles of the distribution values (from multiple independent runs) address this issue as follows: if the quantile \(Q_{p}\) of the solution quality value of an ACO algorithm is \(C^{bs}\), then the probability of achieving a solution quality value better or equal to \(C^{bs}\) is greater than or equal to \(p\). Tables 3 and 4 show the quantiles \(Q_{0.10}\), \(Q_{0.50}\), and \(Q_{0.90}\) of \(\mathcal{MM}\)AS and P-ACO for the DTSP with node and weight changes, respectively. Quantile \(Q_{0.10}\) indicates the peak performance, \(Q_{0.50}\) indicates the average performance, and \(Q_{0.90}\) indicates the bad-case performance of algorithms. In addition, Figures 5-7 show the box plots of \(\mathcal{MM}\)AS and P-ACO drawn between the first and third quartile of the distribution on kroA100, rat575, and pr1002, respectively, for the different DTSP test cases. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Algorithm** & \multicolumn{3}{c}{**kroA100**} & \multicolumn{3}{c}{**rat575**} & \multicolumn{3}{c}{**pr1002**} \\ \(\boldsymbol{m\Rightarrow}\) & **0.1** & **0.25** & **0.5** & **0.75** & **0.1** & **0.25** & **0.5** & **0.75** & **0.1** & **0.25** & **0.5** & **0.75** \\ \hline \(\mathcal{MM}\)AS & 22,127 & 22,390 & 22,440 & 22,385 & 6573 & 6459 & **6375** & 6381 & **302,494** & **311,643** & **311,175** & **312,291** \\ P-ACO & **22,084** & **22,266** & **22,356** & **22,268** & **6546** & **6418** & 6397 & **6367** & 307,513 & 315,147 & 313,007 & 314,951 \\ \(\mathcal{MM}\)AS & 22,186 & 22,467 & 22,520 & 22,486 & **6570** & 6483 & **6395** & 6402 & **306,249** & 315,903 & 314,514 & 316,560 \\ P-ACO & **22,102** & **22,341** & **22,416** & **22,319** & 6583 & **6445** & 6432 & **6390** & 308,586 & **315,880** & **313,964** & **315,455** \\ \(\mathcal{MM}\)AS & 22,400 & 22,670 & 22,676 & 22,556 & 6613 & 6518 & **6431** & 6420 & 309,279 & 318,927 & 317,478 & 318,266 \\ P-ACO & **27,212** & **22,432** & **22,506** & **22,386** & **6610** & **6467** & 6481 & **6419** & **309,059** & **316,302** & **314,222** & **316,100** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results of ACO algorithms for the DTSP with node changes. Figure 5: Box plots of \(\mathcal{M}\)MAS and P-ACO drawn between the first (\(Q_{0.25}\)) and third quartile (\(Q_{0.75}\)) of the distribution on kroA100 with node (**left**) and weight (**right**) changes. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Algorithm** & \multicolumn{3}{c}{**kroA100**} & \multicolumn{3}{c}{**rat575**} & \multicolumn{3}{c}{**pr1002**} \\ \(m\Rightarrow\) & **0.1** & **0.25** & **0.5** & **0.75** & **0.1** & **0.25** & **0.5** & **0.75** & **0.1** & **0.25** & **0.5** & **0.75** \\ \hline \multirow{3}{*}{\(\mathcal{M}\)MAS} & **20,415** & **20,531** & **20,147** & **19,804** & **6594** & **6493** & **6380** & **6383** & **260,009** & **258,675** & **266,618** & **267,049** \\ P-ACO & 20,424 & 20,590 & **20,147** & 19,854 & 6658 & 6643 & 6624 & 6603 & 28,2195 & 276,460 & 275,443 & 275,157 \\  & 20,483 & **20,590** & 20,223 & **19,873** & **6618** & **6465** & **6402** & **6399** & **267,186** & **265,953** & **268,035** & **268,321** \\ P-ACO & **20,453** & 20,654 & **20,196** & 19,895 & 6767 & 6680 & 6644 & 6620 & 282,958 & 276,900 & 275,998 & 275,900 \\  & 20,881 & **20,689** & 20,308 & **19,944** & **6658** & **6499** & **6439** & **6417** & **270,615** & **269,053** & **269,799** & **270,295** \\ P-ACO & **20,519** & 20,753 & **20,245** & 19,956 & 6833 & 6706 & 6662 & 6634 & 283,574 & 277,404 & 276,372 & 276,496 \\ \hline \hline \end{tabular}
\end{table}