## 4 Experimental Results and Discussions  
### Experimental Setup  
Here we introduce a random version of OBBO (called ROBBO in this paper) firstly. On the basis of original OBBO, all parts of the proposed algorithm are kept untouched. In addition, instead of using opposite paths for the population initialization and the generation jumping, the random paths, which are generated uniformly in entire search space, are employed in ROBBO algorithm. According to past experience, we admit that this version must be not the best one, yet it is a basic and well-known method, which is firstly used by [PERSON] in [12]. **Benchmark Functions. ** A comprehensive set of 8 different well-known traveling salesman problems has been employed for performance verification of the proposed approach. All the benchmarks are selected from TSPLIB [33]. The benchmarks sizes vary from small problem to extra large problem to cover a wide range of problem complexity based on category criteria in [32]. **Parameter Settings. ** Note that the main point in this paper is to perform relative comparison among BBO, OBBO, ROBBO and COOBBO. In order to compare the performance without affecting other factors, all conducted experiments use the parameters as follows. * Population size: 100
* Maximum number of cost function calls: 100,000
* Number of elites: 3
* Opposition jumping rate: 0.3  
The termination criterion for all tested algorithms in this paper is that the number of evaluation of cost function is reached the maximum number. It is obvious that, in each iteration of the loop, the number of function evaluation is different for different algorithms due to opposite numbers explored more space. Therefore, the termination criterion in this paper seems more fair and advisable, especially when compared with that used in [28]. **Comparison Criteria. ** We compare the algorithm performance of BBO, OBBO, ROBBO and COOBBO with the help of four well-used criteria in this paper: best solution (BS), computation time (CT), utilization rate of opposite paths (UR), and population diversity (PD). From the definition and design goal, there is no direct relationship among these parameters. Our investigation listed later in this section confirms this conclusion clearly. Thus, we must check all experimental data carefully and discuss them depending on the situation. First of all, we compare the algorithm performance by measuring the best solution found at the end of the computation, which is the most commonly used metric in literatures. In order to minimize the effect of the stochastic nature of the heuristic algorithms on the measured metric, the reported results for each function is the average over 100 independent Monte-Carlo simulations. It is worth noting that, the data in the following tables are not the minimum value (best result), but rather the mean value of all 100 experiments. Computation time, also called process time, is the amount of CPU time required for processing instructions of a computer program. The CPU time is usually measured in clock ticks or seconds. It is well known that it depends on a lot of factors such as data structure utilized by the computer program, programming style of programmer or author, and even performance of computer running the algorithm program. However, on the other hand, it also reflects the convergence speed of tested algorithm in a sense. Utilization rate of opposite paths is a typical comparison criterion for opposition-based soft computing. Refer to early definition proposed in [34], utilization rate of opposite paths is defined as  
\[UR=\frac{OP_{res}}{N_{p}} \tag{4}\]  
where \(OP_{res}\) is the number of opposite paths reserved as offspring in the next generation, and \(N_{p}\) is the size of the population including all candidate paths and their corresponding opposite paths. A larger UR means more effects of opposite paths during the process of evolution.