Medical centers location and specialists’ allocation: a healthcare planning case study

Horizonte, MG, Brasil *joao.flavio@dep.ufmg.br Abstract Paper aims: To set the locations of new medical centers to meet the population’s secondary care needs, the additional number of specialists, equipment, and an installation sequence at municipalities. Originality: We developed descriptive cost functions models and adopted aggregate data from official sources to set parameters of an integrated MILP model. Research method: A case study at the Brazilian state of Minas Gerais. Main findings: For every scenario, the recommended locations set centers dispersed over the state area, in cities with the minimum required infrastructure. We also propose a scenario of secondary care network re-design and demonstrate the reduced cost of such a strategy. Implications for theory and practice: To automate the decision process, we developed a web-based system, providing flexibility and scientific-based results. Finally, we propose a sequence for installing 43 new medical centers and improving the capacity of 27 existing infrastructure The authors and the IT lab team developed a system to provide flexibility and automate the analysis of the secondary care planning process. The system provides the results of optimizations run, with the location of medical centers, the number of medical specialties, the assignment of patients’ demand to medical centers on municipalities. Simultaneously, the analyst can evaluate the assignment of patients’ demand for exams to medical centers and the number of the equipment for procurement within a maximum distance. The system, presented in Figure 12, enables scenario investigations by changing the quantity of new equipment and a maximum distance of coverage.

have been a topic of both political and methodological relevance, pressuring leaders to strive on setting priorities in allocating resources.
In this study, we evaluate and analyze the secondary care of Minas Gerais (MG), a Brazilian state, where the 2000 and 2010 census demonstrate the fast aging diagnostic (see Figure 1). We acquire health needs on demographic projections (Instituto Brasileiro de Geografia e Estatística, 2019) and the Ministerial Decree 1.631, and adopt mathematical programming models to solve an optimization problem of location and resource allocation (Brasil, 2015). Although SUS has improved access to primary and tertiary care over the past years, the provision of secondary is still problematic. The three service levels are interdependent (Macedo & Martin, 2014). The primary care consists of low-cost prevention events, reaching in 2010 nearly 98 million people in 85% of Brazilian municipalities. On the extreme side, public teaching hospitals and contracted private sectors providers execute tertiary care, paid by the SUS at about market value. With intermediate technological density between primary and tertiary care, secondary care is responsible for performing medium complexity procedures (Minas Gerais, 2019) and providing specialized services to outpatient at the hospital level.
The provision of secondary care is problematic. Medium complexity procedures are often restricted to patients with private health plans. Since there is little regulation at the second level, SUS is highly reliant on the private sector. In Brazil, only 24.1% of Computed Tomography (CT) scanners and 13.4% of Magnetic Resonance Imaging (MRI) scanners are public . Furthermore, the medical centers, qualified for healthcare on the second level, are not geographically dispersed, which motivates this study of medical centers' location and equipment allocation for health care.
Healthcare location problems have been an active research area (Rais & Viana, 2011). The models are an extension of the classic p-median, p-centers (Hakimi, 1964), set-covering (Toregas et al., 1971), and the max-covering model (Church & ReVelle, 1974), and the location-allocation model (Schilling et al., 1979). The p-centers and the location-allocation models are of particular importance to this work because, collectively, they address issues of equity, minimizing the distance between remote patients, and the equipment allocation to facilities.
From a historical perspective, we refer the interested readers to a comprehensive review (Hale & Moberg, 2003), successful case studies (Brandeau, 2016) and a recent survey (Ahmadi-Javid et al., 2017). Applications include modeling a 3-level location system of perinatal facilities (Galvão et al., 2002); the use of a genetic algorithm for a maximal covering location for health care (Shariff et al., 2012); and a facility location model for primary care system approaching physicians' preferences (Güneş et al., 2014). Following, Khodaparasti et al. (2016) address a hierarchical location model for community-based organizations with health care providers by applying multi-objective programming. Additional applications consists on the development of a system to optimize the equipment allocation (Treurnicht & Van Dyk, 2014); the tactical workforce planning and capacity allocation minimizing salary costs considering vacation and subcontracting opportunities ( Van der Veen et al., 2015); and a performance evaluation of 87 private and public primary care health units (Novignon & Nonvignon, 2017). On health care location-allocation problem, examples include the location of preventive health care units (Gu et al., 2010), two-phase procedures to set the location and size of medical departments (Stummer et al., 2004), and for improving spatial accessibility (Luo et al., 2017), and an integrated approach aiming inequality reduction (Sathler et al., 2019).
Despite the rich literature, most studies address primary care on the municipality level. Little attention, however, has been paid to location-allocation problems for health care on the secondary level, so there are potential gaps and open issues yet to investigate, besides, the literature presents few studies that address realistic problems (Güneş et al., 2014;Ahmadi-Javid et al., 2017). Our model: (i) adopts descriptive models of cost functions and dada aggregation to elaborate parameters; (ii) evaluates the existing equipment idle capacity on a hierarchical health care system; (iii) considers both the population requirements for local multiple medical services and physicians' preferences on living on municipalities with minimum infrastructure; and (iv) the government financial condition, as stated in objective function.
The authors developed the method following flexibility principles (De Neufville & Scholtes, 2011) for dealing with the problem faced in a project. Since we recognized that the context was of a generic style, representing an unstudied component of real health care location-allocation problems (Almeida et al., 2018), this work can directly contribute to general health care planning by supporting policymaking and providing scenario analyses.
In the following sections, we present a Mixed Integer Linear Programming (MILP) location-allocation model, a descriptive model based on cost functions for setting health care costs, and a strategy for installing health care units based on equality principles. Next, we provide a case study applying the proposed method to solve the problem in MG. We evaluate scenarios by distance, by budget availability are discuss a sequence for health units' installation. We finish the paper by providing conclusions and suggestions for future research.

Method
This paper presents a MILP model for healthcare planning at the state level, particularly, secondary care. Cost functions descriptive models and aggregate data are the base for the MILP model's parameters. The mathematical model aids the planner to select secondary care facilities locations and designate medical specialists, and equipment to them. The planner, here represented by the government health care management team, must combine patients' and physicians' inclinations, adequately size capacity, and guarantee service quality to justify the public infrastructure financing. As medical procedures often depend on equipment, we include these recourses and its' demands into the model considering official standards' amount of equipment per inhabitants and the actual available capacity.
The aim is minimizing the annual overall cost of a strategy that includes (i) the maintenance costs of medical centers; (ii) the cost of procurement of new equipment for existing infrastructure and new medical centers, and balance with (iii) a social opportunity cost of society, estimated by cost functions. We also evaluate available specialists and equipment in a hierarchical system. Secondary care specialists working in primary or tertiary care includes the available capacity. Their production is decreased from the estimated demand, set on Ministerial Decree 1.631. Datasus Outpatient and Hospital Information System and Applications (Departamento de Informática do SUS, 2018a, b) provides a history of outpatient and hospital production.
We denote a state with I municipalities, offering E medical specialties services and Q equipment. Patients move within their municipality or via available paths ( ) , i j K I I ∈ ⊂ × between pairs of municipalities to meet E medical specialties (see Table 1).
The number of inhabitants i B per municipality i is sourced from Instituto Brasileiro de Geografia e Estatística (2019) projection to 2019. The observed physicians' preference for municipalities with a population higher than a minimum amount min B and better infrastructure is considered. A basic measure for selecting acceptable distances is defining a lower bound parameter of the maximal allowed patients' displacement as for critical medical specialties services, like cardiology. Open Route online Services provides the distance between municipalities. The data helps setting candidates based on the maximum displacement eij DM of patients, which results from the combination of medical specialty and distances (or time) to be established according to government parameters for service levels and budget availability. Therefore, the candidate municipalities are a set of locations that meet the following requirements , and | | i N must not be empty for any municipality. Services often don't meet nominal capacities based on projections, therefore, the global efficiency of services j E provided on municipality j can be adjusted based on observations combined with data envelopment analysis (DEA), moreover, the decision-maker may have the option to consider j P municipalities that are provider of specialties services with available capacity on healthcare services.
The demand of patients for medical specialties services ie HE and exams on equipment iq HQ is available on official government publications; however, the decision-maker can estimate demand based on benchmarking indicators of OECD countries (Organisation for Economic Co-operation and Development, 2017), for instance.
The demand does not remain constant through every geographical region but decays proportionally to the distance. We propose a simplified decay function based on distance; however, this function can be improved by a floating catchment area (FCA) method based on real data (Bauer & Groneberg, 2016).
The (i) nominal capacity of the equipment q K is different from the (ii) available capacity of the equipment iq KQ . The first consists in the Original Equipment Manufacturer (OEM) information of annual production capacity of each equipment, individually, while the second is related to the municipality capacity on providing the service of exams on equipment, therefore, the information is obtained by the number of available equipment on healthcare units. The availability of physicians ie KE of each specialty is based on historical data of similar periods in the past and consists of registries of the historical production of physicians on each municipality. A matrix eq M assigns each type of medical specialty to the equipment used. The government sets parameters of a minimum amount of secondary health care facilities F and equipment q P according to its financial conditions and to the desired service level, it aims at providing to its population. The cost parameters of the average annual cost of Medical Centers operations, acquisition of equipment, and patients' displacement between municipalities guide the objective function that aims at minimizing the sum of government and population costs.
We propose the use of cost functions (Horngren et al., 2002) to estimate fixed and variable costs of the annual cost of medical center operations. Let S be a set of support team that enables the operations of health centers, s N , the number of professionals of each team S, and s S , the salary of each professional of support (Brasil, 2019b). The fixed costs FC include a periodic time t payment of salaries of multifunctional teams, as nurses, pharmacists, psychologists, social workers, and, administrative support, their corresponding social taxes T , and a General Costs (GC), therefore, The data for parameters above are base for answering some questions, translated into model variables: Which municipality should receive a secondary level healthcare unit? How many physicians of specialty e should be hired to such municipalities and how many equipment of type q should be acquired for each of them? The demand of patients from municipality i for medical specialty e and their demand for exams on equipment q should be satisfied in which municipality? Table 2 presents the variables from the current questions.
The objective function 1.0 minimizes the total annual cost composed of three classes of costs shared by the government and the society, that is (i) the maintenance of medical centers, (ii) the depreciation costs of newly acquired equipment of different types for each unit, and a social cost of patients' displacement for having medical care. This function is subject to constraints 1.1-1.17.
In Equation 1.1 and Equation 1.2, the demand for medical services and the demand for exams based on equipment, respectively, of each district must be satisfied by only one medical center. Thus, the patient municipality does not have to assign the demand of every medical specialty nor every equipment to a single destination municipality, but it can be assigned to different municipalities, however, the split of demand of a patient municipality is not allowed for each medical specialty or equipment.
, , (1.4) Following, the demand for medical specialty, on the Constraint 1.3, or for medical equipment, on the Constraint 1.4, can only be assigned to a selected municipality, that is, the demand assignment is conditioned to the installation of a Medical Center on municipality j. (1.5) The Equation 1.5 establishes that the demand of a candidate municipality that was selected to receive a Medical Center must be locally satisfied since patients would not displace to different municipalities if its own municipality has healthcare units. Even if its health unit is overcrowded, the patient may not know nor may not accept being moved to different municipalities for healthcare services.
(1.7) Constraint 1.6 link the medical specialty demand to the type of equipment each specialty depends on, such that the patient does not have to move to different municipalities to take exams. On Constraint 1.7, the number of equipment of type q is limited by an upper bound on selected municipalities or equal zero, if the municipality is not selected to receive a healthcare unit.
The healthcare service level desired by the government is presented on Constraints 1.8 and 1.9 which defines the minimum number of Medical Centers and equipment, respectively. The more healthcare units, the higher is the healthcare service level, but the higher is the operating cost with administration, physicians, and equipment. Besides, the number of medical centers and equipment is dependent on the state's long-term financial conditions. (1.10) Sometimes, an optimal solution may yield to municipalities with no infrastructure, which may be very close to other municipalities with available infrastructure and idle capacity. For such cases, Constraint 1.10 is interesting for decision-makers to force the selection of some candidates with better infrastructure, which may reduce the healthcare unit installation cost.

( )
, , (1.14) On solving the optimization problem, we obtain the number of municipalities, medical specialists, and equipment that satisfy all described constraints at a minimum overall cost. This solution, however, is static; therefore, we suggest a sequence for installing medical centers guided by inequality indexes. The inequality indexes are the Social Vulnerability Index (SVI) (Costa & Marguti, 2015), the Inequality-adjusted Human Development Index (IHDI) from the United Nations Development Programme (2018), or the Slope Index of Inequality (SII), and the Concentration Index (CIX), based on demographic health survey (Silva et al., 2018).
In this study, the authors propose the above methodology to aid Minas Gerais (MG) health care policy. MG is the Brazilian state with the largest number of municipalities, 853, the second most populous, with more than 20 million inhabitants, and fourth biggest, with 586,528 km 2 . Solving this health care problem to MG indicates that the method can solve the health care problem for the remaining states of the country. Minas Gerais proposed in 2015 advances in health care including a decentralization process in the second level to increase the population access to specialized centers. The government inaugurated the first medical center, in this new arrangement, in 2016 (Jornal Estado de Minas, 2016). Some constraints were established. Candidate municipalities should satisfy a combination of requirements of least 30 thousand inhabitants to receive nine medical specialties services and equipment, which reveals physicians' preference on minimal infrastructure and on working with similar professionals. The patients' preference for minimal displacement establishes a maximal allowed distance from patients' municipalities to medical centers' municipalities.

Results and discussion
The authors adopted open route services (Neis & Zipf, 2007) to obtain time and distance between the 853 municipalities (i,j) yielding 727,609 registries for each parameter. Following, we defined the lower bound parameter of the maximal allowed distance; however, 4.2% of municipalities do not meet these both requirements (more than 30 thousand inhabitants, and at most 77 km from candidate municipalities). Those municipalities that do not meet these requirements are an exception to a general rule and require a practical solution with special conditions. In general, we established a displacement ranging from one to three hours according to the accessibility need to health care services, setting 80 km for cardiology, and 120 km to 180 km to the remaining specialties resulting in the selection of 121 candidates.
We set the demand of patients from 853 municipalities for physicians of specialties ie HE and exams iq HQ on equipment as stochastic parameters ( ) , a b  . Demands are uniformly distributed ranging from a to b. The parameters , a b range from a central value (central value -range, central value + range) which is selected on Ministerial Degree 1,631 (Brasil, 2015). We adopt a range variability of . 0 26 based on indicators of Mexico physicians' density, which is a country included in Organisation for Economic Co-operation and Development (2017) and comparable to Brazil in indicators, like GDP.
The OEM information of standard equipment set base parameters of the nominal capacity of MRI scanners, Mammograph, CT scanners, Ultrasound, and Doppler. DATASUS database provides the municipality capacity on offering the service of exams for each type of equipment and the number of machines available on 853 municipalities (Departamento de Informática do SUS 2018a, b). We obtained 10 years of the historical production of physicians. We reached the information of availability of FTE physicians' specialists in angiology, cardiology, endocrinology, gynecology, mastology, nephrology, ophthalmology, pediatrics, and urology by dividing the average values of physicians' historical production from 2008 to 2017 on each municipality by a standard period of work of 40 hours of FTE physician.
The government sets the minimum number of secondary health care facilities and equipment according to its financial conditions and the desired service level it aims at providing to its population. This study evaluates the optimal strategy with minimum annual overall costs of heal care on the secondary level. The method considers simultaneously the installation of medical centers in different municipalities of MG, the provision of nine medical specialties services, and exams on five types of equipment.
The annual cost function of medical center operations considers the salary of multifunctional teams of nurses, pharmacists, psychologists, social workers, and, administrative support include fixed costs. The variable costs consist of the average number of hours provided by each medical specialty and their category's average salary, available at PDET (Brasil, 2019b). Following, we include the annual cost of new equipment, traditionally depreciated in 10 years. The values are based on average prices of catalogs of the standard equipment of each type, therefore, we set depreciation costs of R$300,000 for MRI scanner, R$41,000 for CT scanners, R$5,600 for mammograph, R$4,500 for ultrasound, and R$850 for doppler. Finally, the annual patients' displacement costs between municipalities of MG considered: (i) the average rate of Reals per kilometer (R$0.50/km) of bus companies for two (round trip) inter-municipal trips; (ii) the opportunity cost spent by patients; and (iii) the average number of trips, for health care with physicians, in a year. The number of trips is a stochastic parameter  We evaluated eight optimized scenarios. The scenarios [1-3] consider the maximal patients' displacement, highlighting municipalities that need special treatment. The scenarios [4][5][6] takes the government budget availability into account. Following, the scenario [7] proposes a re-design of MG's network of medical centers, and finally, on scenario [8] we suggest a sequence for setting medical centers, based on equity principles. We implemented in MathProg (GLPK) and run the MILP models in a Linux Mint 17.3 64-bit, RAM of 8 GB, Intel Core I5 2.50 GHz x 2 processor. The mathematical programming problem comprises 1,445,574 constraints and 709,193 variables, all of which are integers, being 683,586 of them, binary. Instances were optimally solved within two hours.
The recent disclosure of MG State Secretary of Health shows that the health care policy of installing new medical specialties centers units has not been implemented yet (Ricardo, 2019). The state government has recently faced a serious financial crisis (Agência Minas, 2019), and much of the health resource has been used for working capital expenditures rather than investments (Assembleia Legislativa do Estado de Minas Gerais, 2018). The State government has transferred financial resources (R$19,350,970.03) to the existing 26 Specialized Care Centers (with Brazilian Portuguese initials CEAE) and 1 Medical Center (with Brazilian Portuguese initials CEM) (Minas Gerais, 2018). Since the 27 health units are dispersed over the state (see Figure 2), we adopt the plausible assumption that the model must capacitate the existing health care centers before creating new units.

Scenarios [1-3] based on patients' displacement
Scenarios based on patients' displacement provides the minimum number of municipalities, equipment, and specialists that satisfy the maximum patients' displacement constraints. Therefore, we deactivate constraints 1.8 and 1.9 since parameter eij DM imposes the limitation. We present three scenarios (see Figure 3) based on patients that move distances ranging from 180 km to 80 km (or take from 3 hours to approximate 1 hour). We describe the third scenario in more detail; evaluating the number of selected municipalities, the additional amount of each type of equipment, and the suggestion of hiring extra FTE specialists. Besides, we evaluate optimal overall costs and compare the state's costs with social costs.
In the scenario [3], patients should move at most 80 km, or approximate 1 hour and 20 minutes to reach a health unit. In this scenario, 64 municipalities (in black and green, see Figure 4) offer secondary care at medical centers. Since 27 municipalities (in black, see Figure 4) already have a basic infrastructure (health units), the government should install 37 additional medical centers (in green, see Figure 4).
The analysis revealed each medical center serving an average of 14 districts patients traveling on average 57 km and a standard deviation of 9 km from their municipality to the nearest medical center. The maximal distance is 80 km, limited by the scenario, while the shortest distance, excepting patients with medical center on their municipality, is 33 km. The Appendix A provides the number of equipment and specialist per municipality for this scenario in detail. Figure 7 presents the financial results of the overall annual costs of 64 medical centers, equipment maintenance, and a comparison with social cost. The medical centers' costs include a maintenance cost for existing 27 health care units (SES Resolution 6563, 2018), the costs for expanding its capacity to meet patients' requirements, and the costs of additional 37 medical centers. Following, the results present the annual depreciation costs of equipment. The overall state annual costs are approximate R$250 M, which is comparable to the overall annual social opportunity cost, of R$259 M.

Scenarios [4-6] based on budget availability
Scenarios based on budget availability focus on service offers. The government sets the number of municipalities and the minimum number of equipment it desires to provide according to its financial conditions. The model satisfies the patients' demand and government offer constraints. Therefore, we activate constraints 1.8 and 1.9. We present three scenarios (see Figure 8) based on requirements of 70, 60, and 50 medical centers and at least the number of equipment selected for the first scenario (maximum 180 km), that is 125 MRIs, 62 CTs, 134 mammograms, 952 ultrasounds, and 43 ultrasound Doppler. We describe the financial implications for scenario 4 [70 CEM]; evaluating the average patients' displacement, and the optimal overall costs and compare the state's costs with social costs.
With 70 medical centers of scenario 4, MG patients' displacement follows a normal distribution with ( ) , 156 50  km. As presented in Figure 9 and Figure 10, the number and cost of additional equipment are relatively similar to scenario 1, with 131 MRIs, 78 CTs, 147 mammograms, 961 ultrasounds, and 70 ultrasound Doppler. Figure 10 also demonstrates that when the MG state offers 70 medical centers, its costs are higher than the social opportunity cost.

Scenario [7] based on patients' displacement with no fixed infrastructure
So far, we have evaluated scenarios considering the available infrastructure. In the following scenario, we propose a maximum patient's displacement of 120 km (or two hours) and we relax the constraint 1.10 that fixes medical centers to existing infrastructure. The goal in this scenario is to evaluate a re-design option of the MG secondary care network. Under this strategy, the MG state requires 41 municipalities with medical centers (see Figure 11). The solution requires the procurement of 119 MRIs, 61 CTs, 136 mammograms, 946 ultrasounds, and 41 ultrasound Doppler. Besides, the government should hire 60 angiologists, 124 cardiologists, 59 endocrinologists, 176 gynecologists, 61 mastologists, 90 nephrologists, 152 ophthalmologists, 199 pediatrics, and 80 urologists. MG patients' average displacement takes 1.5 hours, following a normal distribution with ( ) , 95 31  km. From the 41 municipalities selected to receive medical centers, 12 municipalities already have the available infrastructure, that is Capelinha, Frutal, Governador Valadares, Janaúba, Januária, Leopoldina, Patos de Minas, Patrocínio, Pirapora, São João del Rei, São Lourenço, and Teófilo Otoni. Municipalities without services are assigned to one of the 41 municipalities. The model generates this strategy for all scenarios. The strategy is aligned with the Intermunicipal Health Consortiums and provided for in the organic law of the SUS (Law 8,080/1990) to assure integral care to the population of the connected municipalities (Brasil, 1990).

Priority-base schedule of medical centers installation
A time-based schedule for creating new medical specialties centers depends on the government budget and its priorities, therefore, for the future health units, we suggest the priority-based schedule adopting the SVI index (Instituto de Pesquisa Econômica Aplicada, 2019) for the selected municipalities. We designated 70 municipalities of the previous scenarios from which 43 are new medical centers, and 27 are available health units. Table 3 presents the proposed sequence based on SVI for installing new medical centers and improving the existing health units' capacity. The authors and the IT lab team developed a system to provide flexibility and automate the analysis of the secondary care planning process. The system provides the results of optimizations run, with the location of medical centers, the number of medical specialties, the assignment of patients' demand to medical centers on municipalities. Simultaneously, the analyst can evaluate the assignment of patients' demand for exams to medical centers and the number of the equipment for procurement within a maximum distance. The system, presented in Figure 12, enables scenario investigations by changing the quantity of new equipment and a maximum distance of coverage.

Conclusion
This study proposed an optimization method for simultaneously setting out the location of medical centers and the equipment allocation for secondary care taking into account the tradeoff between patients' desire for minimum displacement, and physicians' inclination for working on settled MG municipalities. The proposed method considered existing infrastructure on 27 municipalities of MG and developed analysis based on patients' displacement, government budget availability.
For patients' displacement scenarios, we observed that it is possible to implement a solution where a patient may meet secondary care at most one hour and a half from its residence. For this scenario, the social opportunity cost is equivalent to government costs. For budget availability scenarios, we observed that the service level can increase in quality (with numerous medical centers and equipment), however, with 70 medical centers, for example, patients would take longer to reach such services, besides, the government costs would be higher than the citizens' opportunity costs. Following, we presented a scenario that re-designs the secondary care network of MG and demonstrated that 41 municipalities would be enough to satisfy the state's demand. Finally, the authors presented a table with 70 municipalities and a sequence for installing medical centers and increasing the capacity of the available infrastructure based on the SVI index. The table took into account 43 similar municipalities of optimization runs and the 27 municipalities of MG that already offer specialized care.
In general, the outcomes suggested that this integrated optimization modeling approach is a prospective method to endorse the integration of professionals of public health, medicine, and engineering, to support the decision-making process. We consulted specialists offer for decades and provided a deep financial evaluation of a wide system projected to ensure universality, equity, and long-term sustainability.
The developed system is a potential tool to provide knowledge-based policymaking. It is also fundamental to highlight that both descriptive models, as cost functions, as the mathematical model do not replace managers. Its' contribution resides on aiding decision-makers interacting to find the best solution by consensus by evaluating scenarios.
Some limitations of the study are worth mentioning. First, physicians move among municipalities, therefore, the offer of medical specialties is not accurate when we evaluate the values by municipalities. Micro or macro-regions describes better medical specialties' offer. Second, we did not consider the possibility of physicians of a municipality meet the requirements of other municipalities. This strategy would increase the dynamics of patients' and physicians' movement increasing considerably the model complexity. Such an analysis would have provided answers to questions of political aspects. Finally, a common adoption is to consider population coverage in static computations. Although it is desirable, we did not consider such a parameter since it trades-off maximum distance, but both parameters can be evaluated simultaneously in future works.
For future works, we recommend (i) a study about municipalities' efficiency on providing secondary care using Data Envelopment Analysis (DEA) scores (Mitropoulos et al., 2013); (ii) evaluating further decades' projection of both, population (Instituto Brasileiro de Geografia e Estatística, 2019), and specialists' offer. For this last, we recommend systems dynamics, since it is dependent on several causal variables with data not readily available. Finally, (iii) the decision of installing medical centers can be modelled by two-stage stochastic programming (Errarhout et al., 2016;Zarrinpoor et al., 2017), where the first stage decision is the installation of medical centers, and on the second stage, the uncertain demand and offer of medical specialties are revealed.