# Shelved-Retrieved Method for Weakly Balanced Constrained Clustering Problems
## 5 Conclusions  
The WBCC problem necessitates the division of a point set into connected clusters, each with weights falling within specified intervals. To handle this problem, our study introduced the shelved-retrieved method, which incorporates adjacent relationships into power diagram construction, enabling points to be assigned to clusters based on their adjacent points. Leveraging parameters from power diagrams, this method effectively partitions the point set into connected clusters. Furthermore, it employs a specially designed loop structure to guarantee the generation of clusters that adhere to both weight and geometrical connectivity constraints.  
Our experiments, which included three synthetic cases using three cost kernels, as well as their application in farmland consolidation, consistently demonstrated the effectiveness of the shelved-retrieved method. Our results consistently met the constraints of the WBCC problem, resulting in an average reduction of transportation costs by 22.53% and 40.17% for synthetic cases and farmland consolidation, respectively.  
Our findings highlight the fact that the shelved-retrieved method not only addresses the WBCC problem effectively, but also ensures cluster connectivity, surpassing other techniques in terms of transportation costs and RMSSTD. This method's flexibility in quantifying different costs through cost kernels allows for substantial cost reductions in various scenarios. However, it is important to note that the shelved-retrieved method may face challenges when dealing with high-dimensional weights. Future research endeavors should aim to address the WBCC problem in such high-dimensional weight scenarios to further enhance the method's applicability and effectiveness.  
Conceptualization, X.H.; methodology, X.H.; software, X.H. and A.Q.; validation, X.H.; formal analysis, X.H.; investigation, X.H., L.Y., A.Q.; data curation, X.H. and L.Y.; writing--original draft preparation, X.H.; writing--review and editing, X.H., A.Q. and L.Y.; project administration, Z.Y. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Data Availability StatementData available on request due to restrictions, e.g., privacy or ethical. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy policy.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cost Kernel** & **Shelved–Retrieved Method** & **D-CPD Method** \\ \hline \(f_{1}\) & **8239.77** & 13,772.06 \\ \(f_{2}\) & 9711.75 & / \\ \(f_{3}\) & 10,538.33 & / \\ \hline \hline \end{tabular}
\end{table}
Table 4: Transportation costs of two methods on farmland consolidation. Owing to the limitation of the D-CPD method to metric spaces, the transportation costs can only be calculated using the objective function \(f_{1}\) in all the cases.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cost Kernel** & **Shelved–Retrieved Method** & **D-CPD Method** \\ \hline \(f_{1}\) & **3.24** & 4.19 \\ \(f_{2}\) & 3.46 & / \\ \(f_{3}\) & 3.67 & / \\ \hline \hline \end{tabular}
\end{table}
Table 5: RMSSTD of two methods on farmland consolidation. Owing to the limitation of the D-CPD method to metric spaces, the transportation costs can only be calculated using the objective function \(f_{1}\) in all the cases.