4.1. 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:
In addition to the Euclidean function, two additional cost kernels,
and
, were used in the experiments as expressed below:
Here,
is a positive-definite matrix, but
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
. 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.
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
, the transportation cost can be calculated using formula
for each
k. The results are presented in
Table 2. The transportation costs obtained using the shelved–retrieved method were considerably lower than those obtained using the D-CPD method. Compared to the D-CPD method, the transportation costs of the shelved–retrieved method were reduced by an average of 22.53% for the three cases. Consequently, the shelved–retrieved method effectively reduced transportation costs.
Additionally, we utilized RMSSTD as a metric to measure the similarity between the clusters.
All the results were evaluated using RMSSTD, and the values are presented in
Table 3.
Table 3 shows that the RMSSTD of the shelved–retrieved method is considerably lower than that of the D-CPD method in each case. Consequently, the clustering results obtained using the shelved–retrieved method outperform those obtained using the D-CPD method.
In synthetic cases, compared with the D-CPD method, our method can produce connected and more compact clusters, and the corresponding transportation costs are much lower. The shelved–retrieved method is more suitable for solving the WBCC problems compared to the D-CPD method.
4.2. 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 consolidation, 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
, i.e., the maximal value of area deviation. The original farm area of farmer
i is denoted as
; then, the restructured farmland area is within the interval
. To harmonize expressions of formulation, we denote
and
.
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
and
. Similar to the experiments on synthetic datasets, the D-CPD method is applied to handle the farmland consolidation with cost kernel
. 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.
Furthermore, the shelved–retrieved method is applied to solve the farmland consolidation with
in the experiments. Three cost kernels
are used to measure different transportation costs in farmland consolidation. The results corresponding to the three distinct objective functions are shown in
Figure 6. Compared with the result of the D-CPD method in
Figure 5, the shelved–retrieved method produces connected results but the D-CPD method does not. Thus, the shelved–retrieved method can reasonably solve the WBCC problem in farmland consolidation.
We also evaluate all results by the transportation costs and RMSSTD to compare the two methods. Transportation costs are presented in
Table 4. In the experiments, the transportation cost generated by the results of the shelved–retrieved method is reduced by 40.17% compared to the D-CPD method. Therefore, the shelved–retrieved method can reduce the cost of equipment transportation between lots.
The RMSSTD value of each result is presented in
Table 5.
Table 5 shows that the RMSSTD of the shelved–retrieved method is lower than that of the D-CPD method with a 22.67% reduction. Thus, the shelved–retrieved method outperforms other methods in farmland consolidation.