# Globally Optimal Facility Locations for Continuous-Space Facility Location Problems
## 4 Numerical Examples
### Numerical Results  
First of all, the model \(\mathcal{M}1\) is applied and the facility locations are presented in Figures 3-6 for Instances 24-27. As shown in Figures 3-6, different numbers of the facilities (i.e., 3, 4, 5, and 6) are tested for each of the instances, where the black hollow triangle stands for the demand point and the solid square is represented as the facility. It is obvious that the facility locations are well scattered in the planning area. Since the difference between the models \(\mathcal{M}1\) and \(\mathcal{M}2\) is subject to whether the fixed cost is considered  
Similarly, the globally optimal facility locations can also be obtained when the facility capacities are considered in model \(\mathcal{M}3\). As shown in Figure 7, the facility locations are determined for Instance 24 when different numbers of facilities are considered, which verifies the effectiveness of the proposed approach in obtaining the globally optimal facility locations, even though the facility capacities are involved. Note that similar results can be obtained when Instances 25-27 are considered. It is also obvious that the globally optimal facility locations obtained from model \(\mathcal{M}3\) are different from those in models \(\mathcal{M}1\) and \(\mathcal{M}2\), which indicates that the facility capacities have a great influence on the multi-facility location determination problem. Next, the influence of different numbers of facilities and fixed costs on the objective function values are tested for models \(\mathcal{M}1\), \(\mathcal{M}2\), and \(\mathcal{M}3\). Regarding model \(\mathcal{M}1\), the objective function value is only affected by the number of facilities. In this sense, the relationship between the number of facilities and the objective function value is presented in Figure 8 given different shipping costs. It is obvious that the increase in the number of facilities always decreases the total cost, no matter how the shipping cost changes. Figure 4: Globally optimal facility locations in the planning area for Instance 25. (**a**) \(m=3\); (**b**) \(m=4\); (**c**) \(m=5\); (**d**) \(m=6\). When the fixed cost is considered at facilities, in addition to the number of facilities, the fixed cost also has a great influence on the objective function value. To further provide a visual illustration of the influence of the parameters \(m\) and F on the total cost, a cross-validation experiment is carried out when the number of facilities \(m\) goes from 2 to 18 and fixed cost F goes from 60 to 360. Then, the results under different shipping costs are presented in Figures 9-13. Specifically, as shown in Figure 9, the growth in the fixed cost from 60 to 360 increases the total cost. However, when the number of facilities \(m\) goes from 2 to 18, the total cost decreases first, since more facilities lead to lower transportation costs, and then the total cost increases because too many facilities contribute to marginally lower transportation costs, but result in higher fixed costs. Similar results are obtained in Figures 10-13. As a consequence, an appropriate number of facilities needs to be determined so that the globally minimum total cost can be obtained. To investigate the influence of the capacitated facilities on the optimal facility locations and objective function values, the proposed three models \(\mathcal{M}1\), \(\mathcal{M}2\), and \(\mathcal{M}3\) are compared in terms of Instance 24 and the corresponding comparison results are provided in Table 5. Regarding the models \(\mathcal{M}1\) and \(\mathcal{M}2\), the same facility locations are obtained because the fixed cost does not influence the determination of the facility locations. When the facility capacities are considered, the globally optimal facility locations in model \(\mathcal{M}3\) can still be obtained, which is different from that in model \(\mathcal{M}2\).