## 1 Introduction  
The issue of facility location planning, especially in the domain of public health facilities, is significant [1]. The term \"facility location\" refers to the process of determining the optimal locations for facilities where services are provided to a certain population. Facility location involves considerations such as accessibility, service coverage, cost-effectiveness, and the socio-economic impact of the facility location on the surrounding community. In the context of healthcare facility location, it entails determining the optimal locations for the establishment of medical facilities to provide timely and effective healthcare services to the population [2]. The problem of facility location modeling has gained significant attention in recent years due to its critical role in optimizing infrastructure and resource allocation. The field of facility location modeling has a rich history, with foundational studies providing the basis for modern optimization approaches. Early works, such as those by [PERSON] (1963) [3] and [PERSON] (1965) [4], established the principles of optimal location for facilities, focusing on minimizing distances and maximizing service coverage. These studies introduced the concept of balancing cost and service quality, which remains a cornerstone of location optimization today. Over time, these foundational principles have been extended to multi-objective optimization, addressing increasingly complex problems in various domains, including healthcare, logistics, and telecommunication networks. Early contributions to facility location modeling laid the groundwork for current advancements in the field. For example, [PERSON]'s seminal work (1963) on the p-median problem and [PERSON]'s foundational study (1965) on location theory introduced optimization techniques that balanced cost-efficiency and service delivery. These studies emphasized the importance of mathematical rigor in solving facility location problems, which has influenced subsequent research in multi-objective optimization [3,4]. In recent decades, the focus has shifted toward addressing dynamic and multi-objective problems, as highlighted in the works of [PERSON] (1995) and [PERSON] (2002). These studies incorporated probabilistic and environmental factors, paving the way for more comprehensive models. Our research builds on these advancements by incorporating sustainability and environmental disruption factors into the hub location problem, a dimension that has not been extensively explored in prior studies [5,6]. Several studies and much research have been conducted in the field of hub location so far [7, 8, 9, 10], including both emergency and non-emergency facilities. However, a crucial aspect that has received less attention is the comprehensive evaluation of facility locations' efficiency [11,12]. This entails assessing whether the decisions made over time regarding the location of facilities, considering various needs and constraints, are collectively optimal. Therefore, it is essential to comprehensively examine whether the chosen hub locations are optimal across different time periods and in relation to each other [13]. Therefore, the topic of optimal hub location is raised in a multidimensional manner, where the aim is to comprehensively and systematically examine the optimality of hub locations across the entire target region, and if necessary, to carry out hub location research [14]. This hub location consideration may involve establishing new hubs, removing existing hubs, or increasing or decreasing the capacity of a hub. If the location is not considered for the target area, we will face a number of local optimal solutions, each obtained at a specific time, which not only may not provide us with a globally optimal solution but also may not consider the changes in demand over time in the covered areas after the location is determined [15]. One of the challenges in the hub location problem is considering the hubs at different points equally and with similar services and capacities.