## 4 Simulation Results
### Clustering Problem Compared KHE with Seven Other Methods  
As stated before, a clustering problem is essentially an optimization problem, so it can be solved by the KHE method. Here, KHE is compared with pure FCM and the other five metaheuristic methods including the basic KH method. The dataset used in this paper is the same with the data in [67]. The data have four-hundred data samples, and its characteristic dimension is two. Now, we will divide these data samples into four categories. Therefore, each krill contains eight elements. Population size and maximum generation number are set to 16 and 25, respectively. For other algorithms, their parameter settings are the same as Section 4.1. Figure 2 is the clustering results by pure FCM clustering algorithm when its final objective function value is 3.620176. Figure 3 shows the optimization process of KHE method for clustering problem. From Figure 3, we can see, KHE has a fast convergent speed for clustering problem. Figure 4 is the clustering results by KHE algorithm when its final objective function value is 3.303485. From Figures 2 and 4, we can see that, KHE method can obtain more accurate clustering results than pure FCM. More results can be recorded in Table 3. From Table 3, on average, the KHE method has the most accurate clustering results, and both SGA and KHE have the optimal performance for the best clustering results. For the worst performance, all the methods except FCM have the similar clustering results that are significantly better than pure FCM. For standard deviation (STD), KHE has the second performance that is only inferior to HS. From Table 3 and Figures 2-4, it can be see that, the KHE method can solve the clustering problem better than other comparative methods in most cases.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Function** & **ACO** & **GA** & **HS** & **KH** & **KHE** & **PSO** & **SGA** \\ \hline \multirow{6}{*}{MEAN} & F01 & 3.36E5 & 1.25E5 & 8.26E5 & 1.55E5 & **18.90** & 2.77E5 & 8.64E3 \\  & F02 & 32.55 & 106.40 & 403.60 & 67.46 & **1.13** & 172.70 & 29.50 \\  & F03 & 8.72E4 & 3.51E4 & 2.07E5 & 3.74E4 & **1.86** & 8.36E4 & 2.29E3 \\  & F04 & 5.92E3 & 1.88E3 & 6.06E3 & 3.70E3 & **36.01** & 2.47E3 & 182.80 \\  & F05 & 17.75 & 7.92 & 54.68 & 10.14 & **4.44E\(\mathbf{-4}\)** & 13.77 & 0.66 \\  & F06 & 5.47E3 & 1.86E3 & 3.99E3 & 1.22E3 & **31.60** & 1.38E3 & 313.10 \\  & F07 & 85.19 & 21.57 & 119.50 & 20.03 & **0.04** & 50.05 & 11.02 \\ \hline \multirow{6}{*}{BEST} & F01 & 1.14E5 & 1.71E4 & 3.13E5 & 5.31E4 & **3.37** & 2.44E4 & 1.41E3 \\  & F02 & 14.90 & 33.08 & 266.10 & 35.84 & **1.02** & 91.07 & 9.64 \\  & F03 & 2.36E4 & 6.30E3 & 9.39E4 & 1.76E4 & **0.03** & 1.16E4 & 300.60 \\  & F04 & 2.43E3 & 388.10 & 2.26E3 & 1.00E3 & **2.41** & 1.01E3 & 52.56 \\  & F05 & 6.12 & 1.28 & 25.52 & 5.10 & **5.13E\(\mathbf{-6}\)** & 4.22 & 0.08 \\  & F06 & 3.73E3 & 513.10 & 2.20E3 & 697.30 & **28.19** & 508.00 & 137.50 \\  & F07 & 55.53 & 5.80 & 70.20 & 10.44 & **3.84E\(\mathbf{-3}\)** & 29.45 & 4.16 \\ \hline \multirow{6}{*}{WORST} & F01 & 8.46E5 & 3.63E5 & 1.26E6 & 2.85E5 & **167.60** & 2.39E6 & 4.90E4 \\  & F02 & 69.45 & 235.90 & 498.70 & 101.70 & **1.63** & 568.30 & 68.65 \\ \cline{1-1}  & F03 & 1.76E5 & 1.35E5 & 3.29E5 & 6.26E4 & **20.86** & 6.00E5 & 8.98E3 \\ \cline{1-1}  & F04 & 8.89E3 & 4.42E3 & 1.07E4 & 6.35E3 & **218.40** & 4.72E3 & 558.50 \\ \cline{1-1}  & F05 & 37.48 & 28.87 & 81.76 & 17.74 & **7.82E\(\mathbf{-3}\)** & 32.10 & 5.08 \\ \cline{1-1}  & F06 & 8.08E3 & 4.14E3 & 5.70E3 & 1.90E3 & **46.15** & 2.88E3 & 688.00 \\ \cline{1-1}  & F07 & 126.40 & 42.72 & 143.40 & 33.13 & **0.31** & 65.62 & 21.11 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mean, best and worst function values obtained by different methods.