# Shelved-Retrieved Method for Weakly Balanced Constrained Clustering Problems
## 4 Results
### Synthetic Datasets  
Experiments were conducted on three synthetic cases with three objective functions to evaluate the validity and rationality of the shelved-retrieved method. The discrete form of the centroidal power diagram (D-CPD) method, which is specified in Appendix A, was used as a comparison method in the experiments. For all synthetic cases, 2000 points were generated in a funnel-shaped region, and chosen to effectively represent concave conditions. Three separate cases were constructed to evaluate the effectiveness of the proposed method on datasets containing different numbers of clusters. In Case 1, three subarea capacity intervals were set as [1969.49, 2089.49], [5819.85, 5939.85], and [2182.77, 2302.77]. In Case 2, we increased the number of subareas and the capacity intervals were set as [1969.49, 2089.49], [3819.85, 3939.85], [2182.77, 2302.77], and [1819.85, 2039.85]. In Case 3, we set five subareas with capacity intervals [1969.49, 2089.49], [2819.85, 2939.85], [2182.77, 2302.77], [1819.85, 2039.85], [819.85, 1045.85]. Transportation costs in different scenarios served as objective functions. Three distinct cost kernels were constructed to quantify different transportation costs and validate the effectiveness of our method across a range of scenarios. First, the following Euclidean function was used as the cost kernel in the experiments as expressed below:  
\[\hat{f}_{1}(\mathbf{x}_{j},\mathbf{s}_{i})=\|\mathbf{x}_{j}-\mathbf{s}_{i}\|_{2}.\]  
In addition to the Euclidean function, two additional cost kernels, \(\hat{f}_{2}\) and \(\hat{f}_{3}\), were used in the experiments as expressed below:  
\[\hat{f}_{2}(\mathbf{x},\mathbf{y})=(\mathbf{x}-\mathbf{y})M_{1}(\mathbf{x}-\mathbf{y})^{T},\]  
\[\hat{f}_{3}(\mathbf{x},\mathbf{y})=(\mathbf{x}-\mathbf{y})M_{2}(\mathbf{x}-\mathbf{y})^{T}.\]  
Here,  
\[M_{1}= \begin{pmatrix}1&0\\ 0&4\end{pmatrix} \]  
is a positive-definite matrix, but  
\[M_{2}= \begin{pmatrix}1&1\\ 2&4\end{pmatrix} \]  
is not. Because the D-CPD method is applicable exclusively in metric spaces, we applied it to address the WBCC problem with the cost kernel \(\hat{f}_{1}\). The results are shown in Figure 2. These results show that the D-CPD method yields disconnected results when applied to the synthetic datasets. Consequently, it can be inferred that the D-CPD method is unsuitable for solving the WBCC problem in all scenarios. Further, the shelved-retrieved method was applied to address the WBCC problem. The results corresponding to the three distinct cost kernels are presented in Figure 3. The visual representations in Figure 3 show that our method consistently satisfies the connectivity requirements of the clusters. Hence, the shelved-retrieved method can produce connected results when applied in diverse scenarios. Figure 3: **(a–i)** Results of three synthetic cases with three cost kernels. In each subfigure, the color of a point represents the cluster to which it belongs. The results in the same row represent variations of a single case under different cost kernel functions, while those in the same column correspond to a common cost kernel. The cost kernels involved in the three columns are \(\hat{f}_{1}(\mathbf{x},\mathbf{y})=\|\mathbf{x}-\mathbf{y}\|_{2}\), \(\hat{f}_{2}(\mathbf{x},\mathbf{y})=(\mathbf{x}-\mathbf{y})M_{1}(\mathbf{x}-\mathbf{y})^{T}\), and \(\hat{f}_{3}(\mathbf{x},\mathbf{y})=(\mathbf{x}-\mathbf{y})M_{2}(\mathbf{x}-\mathbf{y})^{T}\), respectively. Figure 2: Results on the three synthetic cases. In each subfigure, the color of a point represents the cluster to which it belongs. (**a**) The result in Case 1; (**b**) The result in Case 2; (**c**) The result in Case 3. To compare the shelved-retrieved method to the D-CPD method, we evaluated all the results using two metrics. For each of the different cost kernels \(\hat{f}_{k},k=1,2,3\), the transportation cost can be calculated using formula \(\sum_{i=1}^{\eta}\sum_{\mathbf{x}_{j}\in\mathbb{C}_{i}}\hat{f}_{k}\) for each \(k\).