# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 2 Related Work
[PERSON] [16] originally discussed the complexity of the \(k\)-balance problem with two different objectives: (_i_) minimizing the maximum number of allocated clients to any center, and (_ii_) minimizing the difference between the maximum and the minimum number of clients allocated to the center. He showed NP-hardness of the \(k\)-balance problem for both objectives in the plane under both Manhattan and Euclidean metrics. FL in the context of healthcare is a multifaceted challenge that involves optimizing accessibility, resource allocation, and patient outcomes. The related work in this field encompasses various modeling techniques, GIS applications, and patient-centric approaches to address the complex task of facility placement [17; 18]. [PERSON] et al. [19] focused on healthcare FL in low and middle-income countries, particularly the Philippines. They introduced a novel cooperative covering maximal model to optimize primary care facility placement using open-source data, considering equity and efficiency parameters. The approach holds promise for evidence-based healthcare facility decisions in resource-limited settings and can be adapted to other sectors. [PERSON] et al. [20] aimed to explore the principles and factors impacting the choice of locations for emergency medical facilities during public health crises. They delved into the process of identifying optimal facilities and introduced a logistic regression model to establish a site selection framework tailored for emergency medical facilities in megacities during public health emergencies. [PERSON] et al. [21] addressed equity in stochastic healthcare FL models, examining how uncertainty affects disparities. They focused on modeling uncertainty, equity, and FL, encompassing aspects and outcomes like tractability, fairness, and access metrics. [PERSON] et al. [22] studied the FL problem in China's evolving healthcare landscape, particularly,location-allocation challenges in growing cities. They introduced a hierarchical model balancing social, economic, and environmental factors, using a bi-level multi-objective particle swarm optimization algorithm for complex decisions. [PERSON] et al. [23] addressed the global challenge of an aging population, whereby healthcare decision-makers face the complexities of optimizing home healthcare for the elderly and ensuring its sustainability. They introduced a robust multi-objective optimization model for home healthcare, considering factors like caregiver scheduling, care continuity, patient availability, service times, and quality standards. Finally, they presented a metaheuristic to tackle the problem. [PERSON] et al. [24] studied a multi-period vaccination planning problem, optimizing vaccination recipients' travel distance and operational costs. The problem involves deciding when to open vaccination sites, how many stations to launch, recipient assignments, and site replenishment. Initially framed as a bi-objective mixed integer linear program, they introduced a weighted-sum, \(e\)-constraint and used genetic algorithms to solve the problem. [PERSON] et al. [25] discussed the COVID-19 vaccination center location problem with the objectives of enhancing accessibility and minimizing costs. They employed maximal coverage models with a focus on minimizing transportation time and travel distance. [PERSON] et al. [26] studied multi-period FL decisions in scenarios emphasizing pandemics with volatile demand, and including opening, relocating, closing, and utilizing mobile facilities. They employed NSGA-II to balance economic costs and CO\({}_{2}\) emissions. [PERSON] et al. [27] presented a vaccination station location model, incorporating multi-period planning for medical professionals, vaccine procurement, and inventory decisions amidst demand uncertainties. Formulated as a complex two-stage stochastic problem, they utilized a Benders decomposition-based heuristic for effective resolution.