## 4 Experimental Results and Discussions
### Experiment Comparison of Different COOBBO  
As stated above, three methods to define the opposite paths are proposed in this paper, which are called opposite path (Backward Ellipse), opposite path (Forward Ellipse) and opposite path (Circle), respectively. In Section 4.2, opposite path (Backward Ellipse) is employed into COOBBO, and then it is compared with BBO, OBBO and ROBBO. As shown in Table 2, it is demonstrated that the proposed COOBBO algorithm can achieve near optimal performance in terms of best solution found and utilization rate of opposite paths. In order to design better opposite path, three different definitions proposed in this paper are compared in details. Experiment results of applying different version of COOBBO to solve those benchmark functions above are given in Table 4, in which the better results for each case are also highlighted in **boldface**. From Table 4, we can see that each algorithm has own unique advantages in different comparison criteria over other methods. For instance, COOBBO (Backward Ellipse) is the best algorithm on best solution found and utilization rate of opposite paths, COOBBO (Forward Ellipse) is the best one on population diversity, and COOBBO (Circle) is the best one on computation time. We believe that, the most likely reason for this is that different definition of opposite path is embedded into classical BBO algorithm. Here let me take opposite node O\({}_{2}\) of node B\({}_{2}\) for example. As seen from Figures 3, 5 and 6, the search process for the opposite node O\({}_{2}\) is limited different search space, such as the ellipse determined by nodes A\({}_{1}\), A\({}_{2}\), and B\({}_{2}\) for Backward Ellipse method, the ellipse determined by nodes A\({}_{2}\), A\({}_{3}\), and B\({}_{2}\) for Forward Ellipse method, and the circle determined by nodes A\({}_{2}\) and B\({}_{2}\) for Circle method, respectively. Hereby we can infer that, if a backward line (such as A\({}_{1}\)B\({}_{2}\)) is considered to create the opposite path, the exploitation ability can be improved greatly, and the \begin{table}
\begin{tabular}{l c c c c} \hline
**Benchmark** & **BS** & **CT** & **UR** & **PD** \\ \hline \multirow{2}{*}{berlin52} & 0.743 & 0.000 & 0.000 & 0.000 \\  & (8802.2257 vs. 8816.3923) & (12.8949 vs. 12.7935) & (1.39 vs. 0.98298) & (32.0102 vs. 31.8362) \\ \multirow{2}{*}{bier127} & 0.960 & 0.000 & 0.001 & 0.228 \\  & (247079.3802 _vs._ 247017.627) & (28.0967 vs. 27.9017) & (0.79193 _vs._ 0._ 73474) & (40.0844 vs. 39.9542) \\ \multirow{2}{*}{ch130} & 0.520 & 0.000 & 0.000 & 0.137 \\  & (16163.1156 vs. 16110.5935) & (28.6327 _vs._ 28.4332) & (0.79987 _vs._ 0._ 72923) & (39.7573 _vs._ 39.6187) \\ \multirow{2}{*}{kroA150} & 0.140 & 0.022 & 0.000 & 0.002 \\  & (86542.4877 vs. 87258.7988) & (32.634 _vs._ 32.4983) & (0.84709 _vs._ 0._ 71816) & (41.6914 vs. 41.3216) \\ \multirow{2}{*}{kroA200} & 0.918 & 0.002 & 0.000 & 0.034 \\  & (136496.701 _vs._ 136436.9172) & (42.9461 _vs._ 42.702) & (0.77342 _vs._ 0._ 68909) & (45.0263 _vs._ 44.635) \\ \multirow{2}{*}{kroC100} & 0.174 & 0.009 & 0.000 & 0.000 \\  & (43909.6763 _vs._ 44319.1631) & (22.5343 _vs._ 22.4417) & (0.98634 _vs._ 0._ 75694) & (38.3319 _vs._ 37._ 8325) \\ \multirow{2}{*}{lin105} & 0.930 & 0.000 & 0.000 & 0.000 \\  & (32433.6636 vs. 32414.3953) & (23.576 _vs._ 23.4203) & (0.94954 _vs._ 0._ 75923) & (38.8883 vs. 38.4409) \\ \multirow{2}{*}{lin318} & 0.460 & 0.018 & 0.000 & 0.066 \\  & (324536.2956 vs. 325160.5952) & (67.0055 vs. 66.7332) & (0.72979 _vs._ 0._ 66951) & (52.315 _vs._ 51.8151) \\ \hline \end{tabular}
\end{table} 
Table 3: Performance comparison of COOBBO and ROBBO. better utilization rate of opposite paths and overall performance can be achieved. From another perspective, if forward line (such as A\({}_{3}\)B\({}_{2}\)) is considered in that process, the exploration ability and population diversity can be improved greatly. It is obvious that the method of finding opposite path (Circle) has the characteristics of simple process flow, less calculation and high velocity.