Every building must have effective maintenance processes and methods that should be systematically planned throughout its lifespan. For public buildings, this need is even more important because, in addition to being public property, they are the nerve center where civil servants perform their duties.

Buildings have a profound impact on people's quality of life, providing the basic structure required for conducting their productive activities, in addition to playing an important social role (Brandão and Santos 2020).

According to Barbosa and Pusch (2011), depending on the form of use, conservation, and especially the frequency of adequate maintenance, a building may exhibit early signs of wear and performance capacity loss.

According to Fontoura et al. (2019), maintenance should always be regarded as one of the precepts for ensuring the safety, health, and well-being of a building's users. Non-compliance with legislation on the compulsory maintenance of equipment and systems can lead to putting people's safety at risk and holding managers accountable.

In a study of public buildings conducted by the researchers, it was found that 66% of the probable causes of accidents involving equipment are related to inadequate maintenance, early loss of performance, and rapid deterioration. All of these aspects can be directly addressed with good and correct maintenance. Meanwhile, the cause and origin of the remaining 34% of accidents are related to so-called constructive defects, or endogenous anomalies.

Figure 1 Source: Poli (2017)

From a legal viewpoint, it is important to highlight the NBR 14037 technical norms (ABNT, 2011) regarding the drafting of manuals on the use and maintenance of buildings, in addition to norm NBR 5674 (ABNT, 2012), which addresses the requirements for building maintenance management systems.

The definition of maintenance, identifying its different forms, is that of Dos Santos (2010), with the basic purpose of keeping equipment running most of the time at a low cost. The author also understands that maintenance, depending on how it is done, can be divided into corrective, preventive, and predictive aspects.

Meanwhile, technical norm NBR 5674 (ABNT, 1999) divides maintenance management into three groups: routine maintenance (characterized by a constant flow of standardized and cyclical services); corrective maintenance (characterized by services that require immediate action or intervention to enable the continuous use of the systems, elements, or components of buildings, or to avoid serious risks to people and/or the property of users or owners); and preventive maintenance (characterized by services that are scheduled in advance, prioritizing user requests, estimates of the expected durability of systems, the elements or components of buildings in use, severity and urgency, and periodic reports to gauge their state of disrepair).

In addition to the maintenance services themselves, there is also a need to optimally define the regions of Paraná State that each building maintenance team must serve. When defining this aspect, it must be considered that teams should travel the minimum distance from their place of origin to the locations that require maintenance, in order to guarantee a rapid service for users, which is the aim of this article.

Within the broad scope of the territorial partitioning problem, several mathematical models have been presented to group smaller territorial units into larger units with application to different domains. Although mixed integer linear programming (MILP) is the most suitable approach to deal with these problems, it has proven to be inadequate for large real-world problems. Therefore, different metaheuristic procedures have been developed for the territorial partitioning problem. Some of these approaches, which have been reported in the specialized literature over the last decade, are presented below in chronological order.

D'Amico et al. (2002) applied the simulated annealing metaheuristic to the problem of police management in the city of Buffalo (USA), with the goal of elaborating a map with optimized regions to minimize the disparity between the extreme workloads of police officers, specifically in the process of determining the number of patrol vehicles in use at various times of the day. Muyldermans et al. (2002) studied the salt spreading operation in the province of Antwerp (Belgium) through multi-purpose heuristic search procedures, where the road network was partitioned to minimize the distance traveled, as well as the number of trucks needed for the operation. Applying a genetic algorithm with multiple objectives to partition the electricity grid of the Republic of Ghana into economically viable districts, Bergey et al. (2003) identified some fundamental characteristics to model and solve a district's electric power problem.

Bozkaya et al. (2003) proposed an approach using the tabu search metaheuristic with an adaptive memory heuristic procedure for a districting problem in the city of Edmonton (Canada). Galvão et al. (2006) presented a Voronoi diagram for a logistics districting problem applied to parcel delivery in the city of São Paulo (Brazil), which resulted in a more balanced time/capacity use when compared to other approaches. A delivery planning system was studied by Haugland et al. (2007) in two steps using the tabu search and multi-start metaheuristics with the regions defined in the first stage, while the vehicle routing problem was solved for each district in the second stage. Combining a local search procedure with a multipurpose genetic algorithm, Tavares-Pereira et al. (2007) analyzed the public transport system in the region of Paris to suggest a reform of its pricing system.

A multipurpose genetic algorithm was studied by Datta et al. (2007) for the problem of land use management in Baixo Alentejo (Portugal), aiming to achieve the natural balance of the environment and financial profit. Ricca and Simeone (2008) compared several heuristic and metaheuristic procedures for the electoral districting problem in a set of regions of Italy with multiple goals, seeking to achieve population equality, compactness, and administrative compliance. Ricca et al. (2008) compared two methods for locating Italian polling stations using weighted Voronoi regions with two aims: population equality and compactness.

Also using Voronoi diagrams, but associated with continuous approximation models, Novaes et al. (2009) solved location-districting problems related to transport and logistics applied to an urban distribution service covering part of the city of São Paulo (Brazil). Salazar-Aguilar et al. (2011) solved the distribution of beverages in the city of Monterrey (Mexico) via a bi-objective programming model, aiming to achieve an equilibrium between dispersion and balance in territories regarding the number of customers and the sales volume. Datta et al. (2012), applied a multi-objective genetic algorithm to a problem with census tracts in order to aggregate census units to obtain better compactness and population/area uniformity in the Metropolitan Census area of London (Ontario, Canada).

Shirabe (2012) used a heuristic procedure to solve a MILP model with several illustrative school bus instances. Benzarti and Dallery (2013) addressed home care as a districting problem, modeling it as a MILP to achieve, among other things, compactness, and workload balance. Ríos-Mercado and López-Pérez (2013) used a branch-and-bound approach to solve the problem of a bottled beverage distribution company located in the city of Monterrey (Mexico). A GRASP procedure was used by Assis et al. (2014) to solve a territorial partitioning problem applied to the reading of energy meters in the city of São Paulo (Brazil) while striving for compactness and homogeneity.

Aiming to optimize truck routes for urban waste collection, Vecchi et al. (2016) presented a sequential approach involving three phases: the clustering of arcs based on a model adapted from the p-median problem, formulated as a binary integer linear programming (BILP) problem; the development of a model for the solution of the capacitated arc routing problem, formulated as a MILP problem; and, finally, the application of an adapted Hierholzer algorithm to sequence the arcs obtained in the previous phase. Ferreira et al. (2017) presented a vehicle routing problem solved in two steps: the definition of demand point clusters as a facility location problem (PLF) and the definition of routes within each cluster, i.e., an asymmetric traveling salesman problem (ATSP). For the ATSP, the simulated annealing and tabu search metaheuristics were used, as well as a hybrid algorithm.

Corn and soy production, which has grown continuously in Brazil, was also researched in the context of territorial partitioning with the intention of analyzing how to increase the number of grain silos. To this effect, Steiner Neto et al. (2017) proposed aggregating the municipalities of Paraná State into regions for the effective transport of grains. Their work organized the storage regions, aggregating the municipalities as a multi-objective graph (territories) partitioning problem, aiming to maximize the homogeneity of the storage deficit and minimize the cost of inter-region transport from the production sources to the silos. The authors made use, among other procedures, of genetic algorithms. Endler et al. (2018) evaluated the spatial distribution of public daycare centers in the city of Curitiba (Paraná, Brazil), based on three types of analysis: the analysis of the current location of daycare centers, the analysis of the possible expansions for existing units, and the analysis of opening new units. The authors solved the problem using mathematical modeling.

Franco et al. (2021) used the concept of the FLP to find the best configuration for clusters of municipalities (consortia) for the municipal solid waste (MSW) landfill network in Paraná State, Brazil. To this effect, they used the BILP mathematical model. In 2022, Franco et al. used a MILP mathematical model to analyze the current and future scenarios (2021 and 2033) for the location of MSW landfills in the State of Rio Grande do Sul (Brazil).

This work, in comparison with those analyzed above, deals with territorial partitioning aimed at improving the performance of a common building maintenance service contract in Paraná State. To this effect, following data collection and processing, a BILP mathematical model was used for several scenarios, the results of which were analyzed.

The PRCJ experienced problems due to the retirement of civil servants from its own staff who worked in the maintenance services of its buildings in Paraná State.

With no interest in holding public tenders for the hiring of operational maintenance workforce, the administration opted to hire a company that would supply outsourced labor to establish service stations for preventive and corrective maintenance operations.

As the solution required outsourced labor, the project was led by the technical engineering and maintenance sector, together with the outsourced services sector, as they had detailed knowledge of policies and management related to outsourcing services.

The following criteria were used as the basic premises with regard to labor laws in order to optimize the human resources that were hired: 1) employees could work a maximum of nine hours and 48 minutes per day, with a 1 h break; 2) activities would be carried out exclusively during daytime; 3) any necessary action on the weekends would be compensated for on working days; and 4) it was not possible to pay expenses or fund overnight stays for employees.

Due to legal requirements, it was determined that the buildings occupied by the PRCJ should be divided into regions, with the outsourced work teams headquartered in one of the counties belonging to the region in which they operated. The division of Paraná State (which comprises 399 municipalities) into 11 regions (a number obtained through preliminary experiments), is shown in Figure 2. These regions are duly named as follows, beginning on the right: the state capital, Curitiba; Santo Antônio da Platina; Ponta Grossa; Telêmaco Borba; Londrina; Guarapuava; Francisco Beltrão; Maringá; Campo Mourão; Umuarama; and Cascavel.

Figure 2 Source: Authors

Based on the areas and the number of buildings to be serviced in each region, the number of service stations for the outsourced teams (headquarters/maintenance offices) in each region was determined. Each work team was made up of four people: one supervisor and three building maintenance officers with multiple specialties. These data are presented in Table 2.

Table 2

Source: Authors

Due to the substantial number of buildings and areas served, it was decided that six work teams would be allocated to the Curitiba regional office. There are also outsourced professionals distributed in pairs to permanently service buildings that have a high number of requests for service. It was also determined that it would be necessary to have three administrative supervisors from the outsourced company for direct and daily dialogue with the management and inspection team.

Each work team has a company vehicle for traveling between cities, as well as tools, equipment, and smartphones. These smartphones are used for timekeeping, displacement, and service provision through the company's own application for team management regarding outsourced maintenance services, mainly for evaluating and monitoring displacement and the duration of services.

Services are provided primarily through visits for scheduled preventive maintenance. Emergency services are also provided based on local demand and on assessments by the contract management team regarding their urgency. If the demand is not identified as an emergency, the problem will be recorded in the building's list, to be addressed during the next technical visit by the team for preventive and corrective maintenance services.

The scheduled service cycle of each maintenance team is 45 to 60 consecutive days at its regional office, which means that a service routine is conducted in each building six to eight times a year. As these are services of a common nature, with no need for specialized knowledge to evaluate them, but rather only the identification of their performance and result, the documentation regarding the performance of the activities is done directly by the requesting customers, in other words, the claimants of the services, represented by the Assistants of the Court Directors or the supervisors of administrative units.

To determine and select the offices of the work teams in each of the 11 regions, an investigation was conducted with the Court Boards to consider the possibility of using a small room (headquarters/office) to accommodate the structure of the team, simply furnished with lockers and shelves for storing tools and equipment. The initial objective was to allocate the teams to the district after which the region is named. However, due to insufficient space in some buildings, this goal was not achieved.

It should be highlighted that a county is defined as the territory over which the first instance judge has jurisdiction, and it may cover one or more municipalities depending on the number of residents and voters, the caseload, and the territorial extension of the municipalities, among other aspects (CNJ, 2016).

Some counties were chosen to function as the headquarters/ offices of maintenance teams according to the availability of office space and distributed as shown in Table 2. Other counties were chosen as host municipalities, so that the teams' maintenance facilities could be properly located, in such a way that the travel distance to the buildings to be served did not exceed 161 km. It is important to note that, due to the large built area, the cities of Londrina, Maringá, Cascavel, and Guarapuava have a maintenance team in permanent operation, i.e., they do not have to travel to other cities.

Figure 3 shows the territorial division of the state into municipalities, indicating, in the colored areas, those that house the county offices, i.e., the locations that have at least one building occupied by the PRCJ which must be serviced within the framework of the building maintenance contract. Thus, the problem in question consists of 161 clients to be served (the counties, represented in color in Figure 3). The other 238 municipalities in the state do not have judiciary structures for maintenance.

Figure 3 Source: Authors

Early work on the scope of the contract indicated that the region of Santo Antônio da Platina should have two offices instead of one, and the chosen ones were Ribeirão Claro and Cornélio Procópio. The Cascavel region had an additional team allocated to Foz do Iguaçu to better serve a large local building. Cruzeiro do Oeste was chosen as the headquarters for the Umuarama region, and Pato Branco was chosen as the location of the office in Francisco Beltrão. Finally, União da Vitória was chosen to host an additional team in the Ponta Grossa region due to its considerable distance to the main city, which would require long travel times. All these 14 cities for the maintenance teams are represented in the fourth column of Table 2 and the map in Figure 3, consequently generating 14 areas of action. It should be mentioned that the configuration of the map in Figure 3 is the same as in Figure 2, only with the inclusion of the areas of operation of the regions' additional host municipalities. Figure 3 highlights the data necessary for this work: the 161 counties and the 14 offices.

The contracting process was managed by the PRCJ, following a bidding procedure in the electronic trading modality, in February 2021, and services began in April of the same year.

For this scenario, as contracted by the PRCJ, the data regarding the work teams' displacement are presented in Table 3.

Table 3

Source: Authors

Table 3 shows that the sum of all the distances traveled by the maintenance teams from their offices to the serviced buildings is 11 150 km. Out of the distances traveled, the longest was 161 km, and the average was 69,3 km.

Analyzing the clusters in the current solution, we have five offices in Curitiba, Cruzeiro do Oeste, Maringá, Londrina, and Pato Branco, serving around 53% of the counties. The complete distribution of counties regarding the offices in the current solution is presented in Table 4.

Table 4

Source: Authors

Table 4 also shows that the five offices that serve the fewest counties are Ribeirão Claro, União da Vitória, Cornélio Procópio, Ponta Grossa, and Foz do Iguaçu. Combined, they serve approximately 19,3% of the counties. Regarding the maintenance area, the highest numbers are those in Curitiba, Londrina, and Cascavel, which represent 62,0% of the total area.