# Ant-Balanced Multiple Traveling Salesmen: ACO-BmTSP
## 3 Experimental Results and Analysis
### Computational Results
On the other hand, for \(m=\{5,10\}\), an average and standard deviation of 104.75 and 76.58 were achieved, respectively. Considering the problem with 100 cities, the average costs obtained were \(\{8151.69,\,5461.89,\,4382.47\}\) for 3, 5, and 10 salesmen, and the standard deviations obtained were \(\{7.10\%,5.69\%,15.7\%\}\). Finally, for the mTSP150 problem, the average costs and standard deviations were \(\{13157.03,\,8864.47,\,5829.85\}\) and \(\{76.97,\,146.69,\,508.42\}\) for 3, 5, and 10 salesmen, respectively. Considering the small-sized group of instances (11a, 11b, 12a, 12b, and 16, which were all for three salesmen), the average costs obtained were \(\{74.73,\,88.36,\,75.51,\,1024.36,\,90.60\}\), and the standard deviations obtained were \(\{5.12,10.80,3.77,108.56,\,4.38\}\), respectively. For the problem with 128 cities and ten salesmen, the average costs obtained were \(\{4725.61,\,5026.29\}\), and the standard deviations obtained were \(\{1268.96,\,1187.69\}\). For the instance kroD with 100 cities, the average costs were \(\{8669.68,\,6561.67,\,4725.61\}\), and the standard deviations were \(\{12.4\%,14.1\%,26.9\%,15.7\%\}\) for 3, 5, and 10 salesmen, respectively. Finally, for the rat783 instance, which considered 783 cities and 20 salesmen, the average costs were \(1254.55\), and the standard deviation was \(136.22\). The results obtained enabled to conclude that salesmen's paths were generally well balanced, and the difference in the distances was minor. The algorithm found balanced routes for all salesmen. Additionally, as the number of salesmen increased, the cost of individual tours decreased. To perform the best interpretation of the results, Figures 5-7 show the real paths traveled by each salesman for 51 cities. Each color identifies the cities visited by a salesman. Salesman 1 (TSP1) follows the path whose cities are represented by blue, TSP2 by red, and TSP3 by green. In problems with five and ten salesmen, TSP4 travels through pink cities and TSP5 through gray cities. Finally, in problems with ten salesmen, TSP6 travels by cities represented by yellow, TSP7 by maroon, TSP8 by dark gray, TSP9 by cyan, and TSP10 by magenta. It is possible to see the positions of all cities in the space and the path for each salesman. Figures 8-10 show the real paths traveled by each salesman for 100 cities. Figure 5: Tour for 3 TSP51. Figure 6: Tour for 5 TSP51. Figure 7: Tour for 10 TSP51. To compare the developed algorithm with others found in the literature, its performance was compared with that of algorithms that used the minimax function as an objective, since they also somehow promoted the tours' balancing. To make this comparison possible, the longest tour was extracted from the obtained solution and compared with those in the other works that used the minimax function. The entirety of the tours of a solution was  
Figure 8: Tour for 3 TSP100. Figure 10: Tours for 10 TSP100. Figure 9: Tours for 5 TSP100. not compared because the other works only provided the minimax values. This paper's proposed technique not only considered the minimization of the longest tour, but also promoted achieving balanced tours. Additionally, the proposed algorithm did not use local heuristics, since these techniques enhance the solution quality. Table 3 shows the results of the proposed algorithm, ACO-BmTSP, and those of other algorithms found in the literature. The ACO-BmTSP algorithm used (10), while the other algorithms used a minimax objective. It can be observed that the ACO-BmTSP algorithm obtained good results in some of the problems.