# A Dynamic Framework for Community-Facility Siting with Inter-Community Competition
## 3 Model Development  
### Problem Description  
BOMSDM is a multi-period incremental model that imposes no limitations on the number of periods. Based on the cumulative total of facilities at a given period, the proposed model establishes three distinct scenarios: (i) Scenario I: a facility-insufficient scenario, where the total number of available facilities accumulated over all previous periods falls short of meeting the needs of all communities; (ii) Scenario II: facility-sufficient scenario, where the cumulative facility count precisely matches the number of communities; (iii) Scenario III: facility-redundant scenario, where the cumulative number of facilities exceeds the community demand. The BOMSDM includes two facility-community allocation strategies: (i) the average allocation strategy, in which facility sitting is constrained by community boundaries, ensures that each community has an equal number of facilities (when the number of facilities is a multiple of the number of communities) or a difference of no more than one (when the number of facilities is not a multiple of the number of communities); and (ii) the random allocation strategy, which permits facility sitting without boundary constraints, selecting randomly from all candidate locations. Corresponding facility-community allocation strategies are set for the three scenarios as illustrated in Figure 1: (i) in Scenario I, the average allocation strategy is employed to ensure that community facilities are allocated to as many communities as possible; (ii) in Scenario II, the average allocation strategy ensures a one-to-one correspondence between community facilities and communities; and (iii) in Scenario II, facilities exceeding the number of communities are allocated using a random strategy, where each community initially receives one facility, with the remaining community facilities allocated randomly. In addition, we set two fundamental objectives for BOMSDM: (i) the fair distribution of community facilities among different demographic groups to minimize service distribution inequality. We assessed service distribution equity using the service distribution Gini coefficient (SDGC). (ii) The second objective aimed to maximize the service supply of community facilities to ensure efficient utilization, and the objective was reflected by the service supply rate (SSR). Constraints were imposed on facility-community allocation to ensure community facilities were allocated to different communities and on the number of community facilities to reflect a varying facility number constraint. In real-world scenarios, it is hard to simultaneously achieve these two objectives. This study introduces Pareto non-dominated solutions to enable decision-makers to choose the final solution based on their preferences, providing an ideal optimization decision-making process of seeking the optimal solution first and making the decision later [63; 64]. To simplify the computational analysis, we made the following assumptions: (i) Both present and future development needs are adequately considered when setting community facilities, so capacity limitation can be ignored. (ii) Residents access community facilities on foot, unaffected by road conditions or vehicular congestion. Figure 1: Illustrations of dynamic strategies in BOMSDM. (**a**) An illustration of a solution in Scenario I with 11 communities, 34 candidate points, and 9 community facilities. (**b**) An illustration of a solution in Scenario II with 11 communities, 34 candidate points, and 11 community facilities. (**c**) An illustration of a solution in Scenario III with 11 communities, 34 candidate points, and 15 community facilities. The sets, parameters, and decision variables in the BOMSDM are defined as follows:  
**Sets**  
_I_: Set of demand points, indexed by \(i\);  
_J_: Set of candidate points of community facilities, indexed by \(j\);  
_C_: Set of communities, indexed by \(c\);  
_T_: Set of periods indexed by \(t\).