## 4 Experimental Results and Discussions
### Experiment Comparison of BBO, OBBO, ROBBO and COOBBO
When compared with BBO, OBBO and ROBBO, the mean of the best solution by COOBBO algorithm is lessened, on average, almost by 35.78%, 51.27% and 51.60%, respectively. It is important to note that the overall performance of OBBO is getting worse than BBO, which may seem contradictory with the conclusion in [28]. In fact, they are not contradictory. A rational reason explained this phenomenon is the termination criterion is pretty different in the two papers. As described above, the two algorithms are not iterated until the fixed iteration is satisfied in [28], which means that the OBBO algorithm will explore more candidate solutions compared with BBO algorithm. However the same algorithms are stopped when the number of evaluation of cost function is reached the maximum number in this paper. In this case, it is not surprising then that OBBO's performance is worse than BBO. From our point of view, the performance difference of two modified algorithm, OBBO and COOBBO, is attributed to the disparate definition of opposite path. In [28], the opposite of each city is furthermost in the circle. Therefore, the difference of candidate path and its corresponding opposite path is merely style. However, the definition of opposite path proposed in this paper is differs appreciably from that method. The sequence of candidate paths and the distances between adjacent nodes in the tour are considered simultaneously. As another direct result of the differences in definition of opposite path in the two papers, the utilization rate of opposite paths in this paper is significantly higher than it in [28]. As seen from Table 2, the new method is over 13 times more effective than the traditional one (12.14 _vs._ 0.91 in average). Different from Figure 7 above, the utilization rate of opposite paths in this paper decreases from 50% to 5% over 20 times as seen in Figure 8. Although the decline trend is inevitable, the utilization rate of opposite paths are expected to decrease more slowly over time. Furthermore, as for the great difference between Figures 7 and 8, the utilization rate is no longer decline, after reaching minimal value, such as 4.93% in Figure 8. What some readers will find more surprising is that it wanders and increases slowly with the number of opposition-based generation jumping. That is to say, more and more opposite paths can be survived and further being improved by our proposed algorithm. Using the definition of opposite path from [28], such an event would have been almost inconceivable. Except for the berlin52 problem, BBO has the lowest edge entropy, which indicates clearly its population diversity is the highest among all tested algorithms. Now, there is an interesting and unexpected phenomenon to us. When compared with COOBBO algorithm, BBO has both higher population diversity and worse solution. In accordance with a common view in the field of population-based search algorithms, the higher population diversity should lead to better solutions of many combinatorial problems. In fact, in order to be successful, a search algorithm needs to establish a good ratio between exploration, which is the process of visiting entirely new regions, and exploitation, which is the process of visiting those regions within the neighborhood of previously visited points [36]. If we think that BBO, to a certain degree, has already a good balance between exploration and exploitation, the combination of opposition-based learning and BBO will break the inherent stability inevitably. Several opposite paths are created based on the original paths and the optimal path in the current population, and then their search space are limited within the neighborhood of previously visited path. Therefore, in my opinion, the advantage of opposition-based learning is the in-depth exploitation, not extensive exploration. Furthermore, the termination criterion in this paper is that the evaluation number is reached the maximum number.