# The Pheromone-Based Harmony Search Algorithm for the Asymmetric Traveling Salesman Problem
## 7 Results
### Tasks from TSPLIB
The breakdown of the running time of particular methods was presented in Table 4. According to the achieved results, the total time of running the method increased by 50% in relation to the technique devoid of the long-term memory. What deserves particular attention is the increase in the time of realizing \(IT\) iterations by the PBHS for \(\rho=0.25\), in relation to the variants with different values of the indicated parameter. The shortening of the time of running the tasks was due to optimization conducted by the compiler, as a result of which the multiplication operation (of the value \((1-\rho)\)) was substituted by shifting (for \(\rho=0.5\) and \(\rho=0.75\) was \(2^{-1}\) and \(2^{-2}\) achieved). The insignificant differences which appeared when the PBHS was running with \(\rho=0.5\) and \(\rho=0.75\) were probably caused by an unequal processor overload. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test Name**} & \multicolumn{6}{c}{**Average Error**} \\ \cline{2-7}  & **NNA** & **GLS** & **HC** & **AMCPA** & **GA** & **PBHS** \\ \hline br17 & 135.9\% & 7.69\% & 7.69\% & 0.26\% & 1.54\% & 0\% \\ ftv33 & 30.87\% & 23.64\% & 23.64\% & 7.77\% & 7.79\% & 1.93\% \\ ftv35 & 21.59\% & 21.38\% & 21.38\% & 6.52\% & 6.2\% & 1.43\% \\ ftv38 & 16.21\% & 10\% & 10\% & 5.33\% & 6.92\% & 1.3\% \\ p43 & 2.63\% & 0.53\% & 0.96\% & 0.15\% & 0.27\% & 0.05\% \\ ftv44 & 24.86\% & 24.18\% & 24.18\% & 10.49\% & 8.38\% & 3.02\% \\ ftv47 & 33.67\% & 28.89\% & 28.89\% & 7.21\% & 4.86\% & 2.03\% \\ ry48p & 16.19\% & 15.21\% & 13.81\% & 2.79\% & 4.67\% & 0.77\% \\ ft53 & 37.78\% & 30.93\% & 30.14\% & 12.64\% & 11.98\% & 7.97\% \\ ftv55 & 25.12\% & 23.2\% & 23.2\% & 11.41\% & 14.25\% & 2.9\% \\ ftv64 & 43.50\% & 36.54\% & 36.22\% & 13.18\% & 14.86\% & 2.16\% \\ ft70 & 11.67\% & 8.15\% & 8.98\% & 4.4\% & 5.45\% & 3.99\% \\ ftv70 & 31.85\% & 23.85\% & 23.28\% & 13.06\% & 9.97\% & 4.62\% \\ kro124p & 31.12\% & 26.82\% & 24.77\% & 7.67\% & 10.58\% & 8.08\% \\ ftv170 & 42.4\% & 38.73\% & 36.88\% & 46.02\% & 43.28\% & 15.24\% \\ rbg323 & 30.77\% & 12.37\% & 12.67\% & 43.4\% & 60.11\% & 36.89\% \\ rbg358 & 55.8\% & 17.02\% & 20.55\% & 64.05\% & 74.89\% & 52.09\% \\ rbg403 & 43.41\% & 6.98\% & 4.87\% & 17.52\% & 20.83\% & 23.66\% \\ rbg443 & 44.19\% & 7.21\% & 7.57\% & 25.56\% & 24.29\% & 24.73\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the average error obtained by PBHS (\(\rho=0.25\)) and selected methods from the literature on the subject. Figure 2: Comparison of the average error obtained by PBHS (\(\rho=0.25\)) and selected methods from the literature on the subject. The number of iterations after which different PBHS variants have reached convergence is shown in Figure 3 (using the popular box plot, described e.g., in the paper [38]). On their basis, there were no significant differences between the individual PBHS settings to achieve convergence. In addition, the scatter of iterations indicates the choice of the not too low value of \(IT\) for the tested set (only for the smallest task br17 it could be reduced without any loss of efficiency). The average error--defined by the particular methods--was subjected to the Wilcoxon Signed-Rank Test, with the application of the R package and the wilcox.test function (applying the following values of parameters: paired = TRUE, alternative = \"less\", exact = F, correct = F). The following pairs of methods were used in the process (marked as \(O1\) and \(O2\)), and the assumed test significance level value was 0.05 (achieved \(p\)-values, which were characterized by a lower result, were distinguished by bold and they cause to accept the alternative hypothesis, according to which \(O1\) achieved lower results than \(O2\)). The results of the test were presented in Table 5. On their basis, it was established that taking into consideration only the value of the objective function of the solution, it is recommended to use the PBHS with \(\rho=0.25\).