## 4 Numerical Experiments, Results and Discussions  
### Experiments Settings  
In order to verify its effectiveness, we test the proposed algorithm on 12 standard benchmark instances (listed in the Table 1) from the TSPLIB [38]. The first 10 instances are small-scale problems, with sizes ranging from 51 to 150 cities, and the last two are large-scale problems, whose size is 1323 and 1400 respectively. To avoid the effects caused by the randomness of the algorithm, the experiments for the former eight instances are repeated 20 times independently, the experiments for KroA150 and KroB150 are repeated 10 times independently, and the experiments for the last two instances are repeated 5 times independently, considering the consumption of time. As the calculation method of the TSPLIB, the distance between two cities is computed using Euclidean distance equation and rounded to an integer. The proposed algorithm is coded in MATLAB R2010b. All the experiments are finished on a PC with Core 2 Duo at 2.2 GHZ, 2 GB RAM and Windows Vista Home Basic Operating system. In our tests, the revolution rate is set to 0.3 and the \(\check{\zeta}\) is set to 0.1. We specify a maximum number of iterations for each test. The algorithm stopped after getting to the iterations number. The number of initial countries, initial empires and iterations set for every instance are shown in Table 1, represented by \"Num.C\", \"Num.E\" and \"Num.Ite\" respectively. Larger scale TSP means higher solving difficulty, so we set more population size, more empire numbers and more iterations numbers for the larger scale TSP. Note that the parameters we listed here may be not the best. Actually, in our previous experiments, we found that the proposed algorithm is not very sensitive to the initial parameters. In order to examine the role of 2-opt algorithm in the proposed algorithm (abbreviated as DICA1 in the following), another discrete ICA (abbreviated as DICA2 in the following) is also tested on the mentioned instances for making a comparison. In the DICA2, the assimilation process is the same as that in the DICA1, but the revolution process is achieved by replacing the randomly selected colonies with an equal number of new randomly generated countries. For fairness, the parameters set in the DICA2 are same as those in the DICA1. Table 2 shows the experimental results. The column \"opt\" represents the length of the known optimal solution of every instance. The columns \"Best\", \"Worst\", \"Ave\" and \"StD\" represent the best, the worst found result, the average and the stand deviation of the results for every instance, respectively. The column \"N\({}_{1\%}\)\" denotes the number of the found results that are within 1% deviation \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Instance** & **Num.C** & **Num.E** & **Num.Ite** & **Instance** & **Num.C** & **Num.E** & **Num.Ite** \\ \hline eil51 & 100 & 6 & 200 & berlin52 & 100 & 6 & 200 \\ st70 & 100 & 6 & 200 & eil76 & 100 & 6 & 200 \\ pr76 & 100 & 6 & 200 & kroA100 & 100 & 6 & 200 \\ kroB100 & 100 & 6 & 200 & eil101 & 100 & 6 & 300 \\ kroA150 & 150 & 6 & 300 & kroB150 & 150 & 8 & 350 \\ rl1323 & 200 & 10 & 400 & fl1400 & 200 & 10 & 400 \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters set for every instance. of the optimality over the experiments for every instance. The last column \"Ave.time\" represents the average running time for every instance. The bold data in the table are better. Figures 5 and 6 show the trend lines of the DICA1 and the DICA2 on rl1323 and f1400. They are utilized to compare the convergence process of the two algorithms. Limited by the length of the article,