# A Dynamic Framework for Community-Facility Siting with Inter-Community Competition
## 1 Introduction  
Facility location is a well-known classic problem referring to locating at least one new facility among several candidate locations to optimize at least one objective function like cost, profit, revenue, travel distance, etc. [1]. At the same time, the facility location is a critical element in strategic planning for a wide range of private and public firms, as it provides the foundation for the development of industry, services, politics, commerce, and the economy [2]. Currently, the emergence of homeworking and an aging population has led to an increasing reliance on communities, calling for more comprehensive public services at the community level, which has raised concerns about community facilities. Community facilities encompass diverse services, albeit with variations in scope between different countries or regions. Generally, community facilities refer to facilities such as education, healthcare, sanitation, and cultural amenities that are provided at the community level, typically funded by the government or publicly supported [3,4]. The decentralization of public service facilities at the community level helps enhance the multi-level public facilities system, which serves to alleviate pressure on public resources during times of public crisis, which is a consensus that has been solidified in the post-pandemic era [5]. Additionally, through the construction of community facilities, the social assets within communities, particularly in deteriorated communities, can be preserved and enhanced [6,7]. While the construction of community facilities offers numerous benefits, the locating of community facilities presents significant challenges for policymakers. Unlike city or regional-level facilities, community facilities are smaller in scale but greater in number, requiring substantial financial and time investments from local governments or firms. In China, the government has adopted an efficient way to address these challenges: local governments fund their development using the jurisdiction of the community committee (the primary-level administrative organization in China) as the physical boundary of the community, with one facility allocated per community. The type and number of facilities serve as key performance indicators for evaluating the community committee. This strategy has enabled the rapid advancement of community-facility construction in China in recent years. However, this strategy faces some practical challenges. First, the number of facilities allocated is uncertain, generally depending on the government's budget, which is limited and variable, often disbursed in phases. Even with sufficient budget to cover all communities, certain ones will be selected for pilots to assess the effectiveness of community facilities when a new community-facility project is carried out. Second, performance evaluations of community facilities have sparked inter-community competition, with the intensity of competition typically tied to the number of facilities allocated. When budgets are tight, every community ties to have facilities allocated within their jurisdiction. Conversely, when there is a surplus, this competition may diminish. Based on the aforementioned scenarios, we pose a central question: how can we optimize the location of community facilities to maximize the efficiency and equity of service while accommodating dynamic changes in the number of facilities and inter-community competition? The siting of community facilities is often a long-term, enduring decision that evolves through sustained governmental budget allocations or policy updates. To enhance flexibility in decision-making, dynamic strategies for facility location are frequently employed [8, 9, 10]. Dynamic facility location strategies typically account for variables such as the number of facilities, their capacity, and resident demand over the planning horizon and involve periodic revisions to facility location decisions to adapt to changing infrastructure investments and service needs [11].