# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 6 Conclusions  
In this paper, we have tackled a critical concern related to the establishment of diagnostic test centers for infectious diseases, drawing inspiration from the testing capacity limitations repeatedly exposed during the COVID-19 pandemic. Our primary objectives were to reduce workload disparities among centers while concurrently minimizing the average travel distance for individuals seeking testing. This posed a multifaceted challenge with significant real-world implications. To address this complex problem, we introduced an integer linear programming model. Additionally, we proposed two distinct approaches for its solution. The first is a local search algorithm, named TCLA, which leverages Voronoi diagrams to efficiently uncover a set of non-dominated solutions in a single execution. The second approach employs the \(\epsilon\)-constraint method, solved using the Gurobi solver. We conducted comprehensive testing across a range of problem instances, rigorously assessing the performance of these approaches in terms of computational time and the quality of the resultant non-dominated solutions. Here, quality is gauged by the proximity of the obtained solutions to the Pareto-optimal solutions. In light of the trade-off between computational time and solution quality, our comparative analysis demonstrates that TCLA emerges as an efficient algorithm for identifying Pareto-optimal solutions within a reasonable timeframe. This efficiency is particularly evident in the context of larger problem instances, where TCLA outperforms Gurobi. This suggests its practical utility in real-world scenarios where time constraints are critical.  
The models and approaches presented in this paper hold practical significance across a spectrum of real-world applications extending beyond healthcare systems. These principles can be applied to a wider range of facility location challenges where achieving workload equilibrium among centers is of utmost importance. Furthermore, by integrating elements such as uncertainties related to demand fluctuations or variations in travel times, as well as leveraging geographic information system data and spatial analysis, it is possible to create more realistic models that better align with real-world scenarios. Furthermore, in certain scenarios, the feasible facility center locations can be continuous, allowing for the possibility of opening centers in various positions throughout the city. For instance, during the COVID-19 pandemic, small kiosks offered antigen tests, illustrating this flexibility. In such cases, the model presented in this study may not be applicable, necessitating the development of a new formulation.  
Conceptualization, M.D.; methodology, M.D.; software, M.D.; validation, M.D. and J.M.C.; formal analysis, M.D.; investigation, M.D. and J.M.C.; resources, M.D. and J.M.C.; data curation, M.D.; writing--original draft preparation, M.D.; writing--review and editing, M.D. and J.M.C.; visualization, M.D. and J.M.C.; supervision, M.D. and J.M.C.; project administration, J.M.C.; funding acquisition, J.M.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The authors declare no conflicts of interest.  
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**Nomenclature** & Number of demand points \\ \(n\) & Set of weighted demand points \\ \(\ \