# The Pheromone-Based Harmony Search Algorithm for the Asymmetric Traveling Salesman Problem
## 7 Results
### Tasks from TSPLIB  
The breakdown of the average error achieved by the particular methods was presented in Table 2 (the best results between the listed methods are in bold). Based on it, it was established that the algorithm devoid of a pheromone defined more favorable results than the PBHS only for two benchmark tests described by the means of a relatively small number of vertices (36 and 45). The effectiveness of the suggested modification was especially emphasized for instances described  
Figure 1: The map for case study with the points to visit. by minimum 171 nodes (for task rbg358 limitation of average error by almost 25% was noticed). Regardless of the applied value of parameter \(\rho\), usage of the pheromone substantially increased the effectiveness for the tested method (average error was limited from 13.42 to 10.15, 10.92 and 11.68% for \(\rho=0.25\), \(\rho=0.5\) and \(\rho=0.75\) respectively). Taking into consideration the total average error and number of tasks, for which the lowest result was observed - among the results defined by the analyzed methods - the most favorable PBHS variant is the technique the parameter value of which is \(\rho=0.25\). The smallest effectiveness is characteristic of the PBHS equipped in a fast pheromone evaporation mechanism (\(\rho=0.75\)), which constructed the best route only for the p43 test (difference of the average error between the created solution and results achieved by the remaining methods equals merely 0.01%). The breakdown of the average error achieved by PBHS and selected methods from the literature on the subject was presented in Table 3 (the best results between the listed methods are in bold) and Figure 2. On their basis, the effectiveness of the proposed approach was found to be significant in relation to the selected techniques described in the literature on the subject. Only for the largest tasks the method obtained a less favorable average error, indicating the need to increase the exploitation process that dominates in GLS and HC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Name**} & \multicolumn{4}{c}{**Average Error**} \\ \cline{2-5}  & **HS** & **PBHS (\(\rho=0.25\))** & **PBHS (\(\rho=0.5\))** & **PBHS (\(\rho=0.75\))** \\ \hline br17 & \(0\pm 0\) & \(0\pm 0\) & \(0\pm 0\) & \(0\pm 0\) \\ ftv33 & \(3.63\pm 2.28\) & \(1.93\pm 2.09\) & \(2.34\pm 2.08\) & \(3.41\pm 1.97\) \\ ftv35 & \(1.35\pm 0.46\) & \(1.43\pm 0.41\) & \(1.45\pm 0.39\) & \(1.47\pm 0.38\) \\ ftv38 & \(1.44\pm 0.71\) & \(1.3\pm 0.74\) & \(1.35\pm 0.98\) & \(1.34\pm 0.57\) \\ p43 & \(0.05\pm 0.03\) & \(0.05\pm 0.03\) & \(0.05\pm 0.02\) & \(0.04\pm 0.01\) \\ ftv44 & \(1.76\pm 1.73\) & \(3.02\pm 1.69\) & \(2.12\pm 1.38\) & \(2.23\pm 1.48\) \\ ftv47 & \(1.95\pm 0.99\) & \(2.03\pm 1.09\) & \(1.79\pm 0.94\) & \(2.08\pm 1.15\) \\ ry48p & \(0.89\pm 0.43\) & \(0.77\pm 0.4\) & \(1.17\pm 0.91\) & \(1.24\pm 0.76\) \\ ft53 & \(10.06\pm 3.22\) & \(7.97\pm 2.78\) & \(10.44\pm 2.99\) & \(9.89\pm 3.52\) \\ ftv55 & \(2.92\pm 1.72\) & \(2.9\pm 1.84\) & \(2.8\pm 2.01\) & \(2.83\pm 1.26\) \\ ftv64 & \(3.09\pm 1.82\) & \(2.16\pm 1.41\) & \(2.11\pm 1.4\) & \(2.42\pm 1.47\) \\ ft70 & \(4.35\pm 0.5\) & \(3.99\pm 0.43\) & \(4.28\pm 0.52\) & \(4.06\pm 0.61\) \\ ftv70 & \(4.89\pm 1.73\) & \(4.62\pm 1.33\) & \(4.92\pm 1.41\) & \(5.11\pm 1.32\) \\ kro124p & \(8.88\pm 1.62\) & \(8.08\pm 1.49\) & \(7.6\pm 1.42\) & \(8.17\pm 1.64\) \\ ftv170 & \(19.15\pm 4.23\) & \(15.24\pm 3.35\) & \(16.62\pm 4.59\) & \(17.58\pm 4.11\) \\ rbg323 & \(53.84\pm 2.31\) & \(36.89\pm 1.3\) & \(40.51\pm 1.9\) & \(44.82\pm 2.55\) \\ rbg358 & \(77.01\pm 3.86\) & \(52.09\pm 2.06\) & \(57.41\pm 3.11\) & \(62.53\pm 2.56\) \\ rbg403 & \(29.27\pm 1.47\) & \(23.66\pm 0.88\) & \(24.51\pm 0.75\) & \(25.66\pm 1.14\) \\ rbg443 & \(30.45\pm 1.28\) & \(24.73\pm 1.12\) & \(26\pm 0.87\) & \(27.09\pm 1.12\) \\ Average & \(13.42\pm 1.6\) & \(10.15\pm 1.29\) & \(10.92\pm 1.46\) & \(11.68\pm 1.45\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the average error.