# Kernel Clustering with a Differential Harmony Search Algorithm for Scheme Classification
## 4 Experiments
### Case Study
The problem of water diversion scheduling is to design the inflow of Zengjiaxiang, Qinshangang, Luojiaxiang every day, which improves the water quality as much as possible with minimum cost. Since the water quality simulation is time-consuming and MOEA cannot be adapted to this problem, after some pretreatment, we have identified a number of feasible schemes. These feasible schemes are shown in Table 12. Then we applied these schemes to the water quality model. The water quality simulation results of the schemes are shown in Table 13. Since the water quality model may have a certain amount of error, we decided to determine a set of good schemes instead of one scheme. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**1-5 Days**} & \multicolumn{3}{c}{**6–10 Days**} & \multicolumn{3}{c}{**11-15 Days**} & \multicolumn{3}{c}{**16–20 Days**} & \multicolumn{3}{c}{**21–25 Days**} & \multicolumn{3}{c}{**26–30 Days**} \\ \cline{2-19}  & \(\mathbf{q}_{z}\) & \(\mathbf{q}_{q}\) & \(\mathbf{q}_{l}\) & \(\mathbf{q}_{z}\) & \(\mathbf{q}_{q}\) & \(\mathbf{q}_{l}\) & \(\mathbf{q}_{z}\) & \(\mathbf{q}_{q}\) & \(\mathbf{q}_{l}\) & \(\mathbf{q}_{z}\) & \(\mathbf{q}_{q}\) & \(\mathbf{q}_{l}\) & \(\mathbf{q}_{z}\) & \(\mathbf{q}_{q}\) & \(\mathbf{q}_{l}\) & \(\mathbf{q}_{z}\) & \(\mathbf{q}_{q}\) & \(\mathbf{q}_{l}\) \\ \hline  
& 5 & 22.5 & 13.8 & 5 & 22.5 & 13.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Feasible schemes. In this paper, the goal is to use the DHS-KFC method to determine the good schemes. According to the requirements, we need to divide the schemes into five categories, including excellent, good, average, fair, and poor. Using the kernel cluster method explained in Section 3, we input the scheme results shown in Table 13 as the samples for the clustering, including three water quality parameters (COD, TP, TN) of five lakes, water diversion quantity, and economic cost. The weight vector of water quality, water diversion quantity, and economic cost is defined in Equation (23) according to the advice given by experts. In this case study, the parameter \(m=1.7\), and the minimum objective function value is 0.225269. The cluster results are shown in Table 14. The cluster center \(\mathbf{v}\), membership matrix \(\mathbf{u}\), and weight vector \(\mathbf{w}\) are shown as:  
\[\mathbf{w}=[0.040.040.040.040.040.040.040.040.04} \tag{19}\] \[0.040.040.040.040.040.20.2]\] \[\mathbf{v}=\left[ \begin{array}{cccccccc}0.077&0.102&0.087&1&0.084& 0.001&0.087&0.120&0.030\\ 0.459&0.461&0.417&1&0&0&0.457&0.459&0.388\\ 0.601&0.600&0.660&1&0.979&1&0.784&0.741&0.764\\ 0.761&0.554&0.769&1&0.915&0.916&0.775&0.563&0.696\\ 0.223&0.243&0.230&1&0.032&0.067&0.750&0.669&0.467\\ 0.364&0.198&0.118&0.981&0.846&0.995&1&0.946\\ 0.533&0.307&0.322&0.772&0.883&0.878&0.648&0.645\\ 0.626&0.352&0.391&0.752&0.808&0.953&0.185&0.191\\ 0.937&0.914&1&0&0.107&0&0.313&0.286\\ 0.645&0.098&0&0.88&0.773&1&0.48&0.479\\ \end{array} \right]\] (20) \[\mathbf{u}=\left[ \begin{array}{cccccccc}0.935&0.001&0.973&0.858&0.06 4&0.010&0.021&0.034&0.028\\ 0.035&0.003&0.015&0.102&0.869&0.842&0.044&0.377&0.125\\ 0.005&0.982&0.002&0.005&0.006&0.006&0.077&0.044&0.018\\ 0.004&0.006&0.001&0.004&0.004&0.003&0.813&0.018&0.008\\ 0.019&0.006&0.007&0.028&0.054&0.137&0.042&0.525&0.819\\ 0.010&0.004&0.127&0.029&0.015&0.013&0.021&0.005&0.059\\ 0.026&0.014&0.697&0.815&0.042&0.044&0.923&0.012&0.525\\ 0.873&0.915&0.019&0.015&0.824&0.780&0.005&0.035&0.106\\ 0.048&0.045&0.017&0.012&0.040&0.115&0.003&0.935&0.084\\ 0.040&0.019&0.137&0.127&0.077&0.047&0.046&0.011&0.223\\ \end{array} \right] \tag{21}\]  
Using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) model [22; 23], the similarity values of the cluster centers are shown as:  
\[\mathbf{CC}=\left[ \begin{array}{cccc}&0.950&0.592&0.13&0.052&0.816\\ \end{array} \right] \tag{22}\]  
The schemes in cluster I are considered better than the others.