# A Dynamic Framework for Community-Facility Siting with Inter-Community Competition
## 2 Literature Review
### Dynamic Facility Location Problems  
Facility location problems under uncertainty are a key area of focus within the broader study of facility location problems, encompassing uncertainties in the number of facilities [47], facility capacity [48,49], and demand [50,51]. Facility location decisions must be dynamically adjusted to address such uncertainties based on varying real-world needs. Generally, dynamic facility location problems involve two main considerations: first, the inheritance and updating of facilities across different scenarios, and second, the corresponding location decisions tailored to each scenario.  
In previous studies, early in 1973, [PERSON] was among the first to expand the static facility location problem by allowing positional adjustments for facilities at different stages [52]. Currently, [PERSON] and [PERSON] are developing a multi-criteria model, testing dynamic location strategies under varying facility quantity scenarios [53]. [PERSON] and [PERSON] innovatively designed and applied mixed-integer programming optimization models to address dynamic, multi-stage, multi-commodity location allocation challenges [25]. Research by [PERSON] and [PERSON] [47], [PERSON], [PERSON], [PERSON] and [PERSON] [54], [PERSON], [PERSON], [PERSON], and [PERSON] [55] similarly proposed multi-stage facility update models that tackle uncertainty in decision variables through the dynamic relocation and augmentation of facilities. Moreover, scholars have developed multi-period incremental service facility location models for non-essential service facilities such as libraries, senior care facilities, and recycling stations, considering the high costs of facility relocation. In 2009,Albareda-Sambola first introduced the multi-period incremental service facility location problem, aiming to establish new facilities within a limited timeframe to dynamically meet the demands of a specified customer group [56]. [PERSON] and [PERSON] provided an incremental framework establishing a sequence of open facilities and a sequence of service customers constrained by fixed correspondences in 2015 [57]. Building on [PERSON]'s work, in 2020, [PERSON], [PERSON], [PERSON], and [PERSON] incorporated user preferences into a multi-period bi-objective integer programming model [58]. The richness of findings in this area is well-documented, as evidenced by studies such as those by [PERSON] [23], [PERSON] and [PERSON] [59], [PERSON], [PERSON], and [PERSON] [60].  
Considerations have been made to the heritage and updating of facilities across different periods. Building upon this foundation, a limited number of scholars have introduced constraints on facility inheritance and updating methods. [PERSON], [PERSON], [PERSON], and [PERSON] presented a computational study on two variants of the dynamic multi-period two-level incapatitated facility location problem, focusing on dynamic decision-making regarding allocation relationships between primary factories and secondary facilities in 2021 [61]. Later, [PERSON], [PERSON], and [PERSON] introduced three interrelated models, conducting analyses under two scenarios of facility redundancy to offer optimal decision support tailored to various redundancy contexts [62].  
To the best of our knowledge, previous studies on dynamic facility location problems have placed limited emphasis on dynamic decision-making for facility siting and have not considered the spatial allocation relationships between facilities and their managing entities. As mentioned in Chapter One, competition for community facilities exists among communities, with the intensity of this competition fluctuating following facility number constraints. Thus, it is essential to develop allocation strategies for facilities that are tailored to varying facility number scenarios.