# A Dynamic Framework for Community-Facility Siting with Inter-Community Competition
## 5 Results
### Results of Different Models  
To compare the performance differences among the BOMSDM, Models I, II, and III were tested across three scenarios set in Section 3.1: (i) In Scenario I, with 72 facilities, the strategy of Model III aligned with that of BOMSDM, while Model I shared its strategies with Model II. (ii) In Scenario II, we set the total number of facilities to 79, which is equal to the number of communities requiring facilities, and subsequently computed all four models. In this scenario, BOMSDM and Model III inherited the median solutions from their Pareto sets obtained in Scenario I, specifically [SDGC = 0.009, SSR = 0.412] and [SDGC = 0.025, SSR = 0.392], respectively. (iii) In Scenario III, we increased the total number of facilities by 40, making the total number of facilities in this scenario 1.5 times the number of communities that needed to be allocated. Similarly, we computed all four models in this scenario, with BOMSDM and Model III inheriting the median solutions from the Pareto sets in Scenario II, namely [SDGC = 0.049, SSR = 0.418] and [SDGC = 0.007, SSR = 0.440]. Based on the emergence of multiple Pareto optimal solution sets in each computation (Table S2), we calculated the average and median values of SDGC and SSR for each set, along with the maximum SSR and minimum SDGC obtained. Additionally, we computed the community allocation rate (CAR) for each solution, representing the proportion of communities that were allocated facilities relative to the total number of communities to illustrate the degree of balance in the distribution of facilities among communities (Table 3). The results indicate that, compared to Model I, the BOMSDM exhibits higher average SDGC values in Scenarios I and II, surpassing Model I by 0.019 and 0.027, respectively,  
Figure 4: The comparison between the BOMSDM and the realistic solution. while still maintaining an optimal level of fairness. Additionally, the average SSR for the BOMSDM is lower than that of Model I by 0.022 and 0.01. Thus, it is evident that the BOMSDM incurs a loss in efficiency and fairness compared to Model I, which is unconstrained by facility-community allocation and incremental strategy limits. When the number of allocated facilities is further increased to a redundant level, in Scenario III, the average SDGC of the BOMSDM is 0.009, which is still higher than 0.006 in Model I. However, in terms of SSR, the average value of the BOMSDM surpasses that of Model I by 0.029. This trend is similarly reflected in the median and optimal values available in each scenario. This indicates that in Scenario III, the fairness of the solutions obtained by the BOMSDM remains lower than that of Model I, whereas, in terms of efficiency, the BOMSDM now matches or exceeds Model I. However, BOMSDM achieves a significant increase in CAR, with enhancements of 30.2%, 27.8%, and 17.8% across the three scenarios. The situation for Model II mirrors that of Model I: in Scenario II, the BOMSDM sacrifices 0.039 in average SDGC and 0.02 in average SSR for a 36.5% increase in CAR. In Scenario III, all performance metrics of the BOMSDM surpass those of Model II. When compared to Model III, BOMSDM's advantages are particularly pronounced in Scenario III, where the use of a random allocation strategy leads to enhanced fairness and efficiency. Notably, the BOMSDM's average SSR exceeds that of Model III by 17.5%. Figure 5 presents the site selection decisions for selected examples from the four models, focusing on the median solutions from all Pareto sets. The figure illustrates the distribution of facilities in each scenario, along with the newly constructed facilities at each stage. The results indicate that the BOMSDM and Model II are more cost-effective compared to the other two modes. Through an incremental strategy, they reduce the number of new facilities in the last two scenarios: in Scenario II, the BOMSDM and Model II require the construction of only seven new facilities, whereas Model I and Model III require nearly nine times that number on average.