# Shelved-Retrieved Method for Weakly Balanced Constrained Clustering Problems
## 4 Results
### Farmland Consolidation  
Farmland consolidation is a classical scenario in the WBCC. A large number of small-sized lots cultivated by farmers are scattered over an agricultural area. In farmland consoli \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Cost Kernel** & **Shelved–Retrieved Method** & **D-CPD Method** & **Reduction** \\ \hline \multirow{3}{*}{Case 1} & \(\hat{f}_{1}\) & **2614.56** & 2615.82 & 0.0482\% \\  & \(\hat{f}_{2}\) & 3916.60 & / & / \\  & \(\hat{f}_{3}\) & 3040.00 & / & / \\ \hline \multirow{3}{*}{Case 2} & \(\hat{f}_{1}\) & **2615.81** & 3319.67 & 21.20\% \\  & \(\hat{f}_{2}\) & 2635.42 & / & / \\  & \(\hat{f}_{3}\) & 2345.04 & / & / \\ \hline \multirow{3}{*}{Case 3} & \(\hat{f}_{1}\) & **1881.07** & 3506.45 & 46.35\% \\  & \(\hat{f}_{2}\) & 2254.69 & / & / \\ \cline{1-1}  & \(\hat{f}_{3}\) & 2069.65 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transportation costs of two methods on synthetic cases. Owing to the limitation of the D-CPD method to metric spaces, the transportation costs can only be calculated using the cost kernel \(\hat{f}_{1}\) in all the cases. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Cost Kernel** & **Shelved–Retrieved Method** & **D-CPD Method** & **Reduction** \\ \hline \multirow{3}{*}{Case 1} & \(\hat{f}_{1}\) & **0.809** & 0.813 & 0.492\% \\  & \(\hat{f}_{2}\) & 0.99 & / & / \\  & \(\hat{f}_{3}\) & 0.87 & / & / \\ \hline \multirow{3}{*}{Case 2} & \(\hat{f}_{1}\) & **0.74** & 0.91 & 18.68\% \\  & \(\hat{f}_{2}\) & 0.81 & / & / \\  & \(\hat{f}_{3}\) & 0.77 & / & / \\ \hline \multirow{3}{*}{Case 3} & \(\hat{f}_{1}\) & **0.69** & 0.94 & 26.60\% \\  & \(\hat{f}_{2}\) & 0.75 & / & / \\ \cline{1-1}  & \(\hat{f}_{3}\) & 0.72 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSSTD on synthetic cases. Since the D-CPD method can only be used in metric spaces, RMSSTD can be calculated with the objective function \(\hat{f}_{1}\) in all the cases. dation, these lots are restructured into several large connected fields. The adjacent lots in a large connected field are assigned to the same farmer, and the area of lots belonging to the farmer should not change too much. These requirements in farmland consolidation can be formulated by the WBCC problem. To verify the rationality of our method, we conduct experiments on farmland consolidation. We obtain the relative data of the agricultural area in Germany, such as the position of each lot, the barriers in the area, and the boundary of the lots. Due to the privacy of the datasets, we present the schematic map of the agricultural area in Figure 4. As Figure 4 shows, the lots are distributed over a large area, and the barriers are located in the center of the farmland. A total of 399 lots in Figure 4 are cultivated by seven farmers, and each farmer requires that the area should not change too much after reassignment. Each farmer, respectively, provides the lower and upper thresholds \(\epsilon^{-},\epsilon^{+}\), i.e., the maximal value of area deviation. The original farm area of farmer \(i\) is denoted as \(\kappa_{i}\); then, the restructured farmland area is within the interval \([\kappa_{i}-\epsilon^{-},\kappa_{i}+\epsilon^{+}]\). To harmonize expressions of formulation, we denote \(\kappa_{i}^{-}=\kappa_{i}-\epsilon^{-}\) and \(\kappa_{i}^{+}=\kappa_{i}-\epsilon^{+}\). According to the requirements in farmland consolidation, information regarding the belonging of each lot to respective farmers should be obtained. In the experiments, we set \(m=7\) and \(n=399\). Similar to the experiments on synthetic datasets, the D-CPD method is applied to handle the farmland consolidation with cost kernel \(\hat{f}_{1}\). As presented in Figure 5, the D-CPD method produces a disconnected result in the cyan cluster. Hence, the D-CPD method is unsuitable for solving the WBCC problem in farmland consolidation. Figure 4: The schematic map of the agricultural area in German.