Evolutionary multicriteria planning of bus stops location in smart cities

1 Introduction

The role of public transportation in modern cities is multifaceted, serving as a cornerstone of mobility services and a crucial tool for urban development, sustainability, and social equity (Litman, 2011). Public bus services are instrumental in enhancing accessibility and the quality of life for citizens, by providing a flexible and affordable means of transportation to a large portion of the population (Risso et al., 2023b). Nowadays, intelligent public transportation systems are a key component of sustainable smart cities (Elassy et al., 2024) and their successful planning and management are crucial for maximizing the overall efficacy.

A critical task in bus network planning is the strategic determination of bus stop locations across the service area (Wu and Levinson, 2021). This challenge, formulated in the Bus Stop Location Problem (BSLP), involves complex trade-offs between competing interests (Ibarra et al., 2015). Traditionally, the primary goal for network design has been achieving cost-effectiveness for system operators, often relegating user quality of service (QoS)—such as travel time and accessibility—to a secondary, surrogate objective. Recent modeling efforts have shifted focus toward user-oriented objectives to align with and bolster the principles of Transit-Oriented Development (TOD) (Litman, 2021; Nesmachnow and Hipogrosso, 2022).

A poor location plan for bus stops can lead to several inconveniences, negatively impacting system performance metrics like efficiency, accessibility, and ridership, also increasing operational costs and negatively affecting safety, land use, and environmental impact. Optimizing bus stop locations is critical for improving both system performance and user satisfaction. In realistic scenarios, the BSLP is inherently complex, especially when applied to large-scale, real-world instances. The formal definition of the problem is typically presented as a multiobjective combinatorial optimization problem that requires the simultaneous optimization of conflicting objectives. Typical conflicting goals include minimizing user inconvenience (like the walking distance to a stop) and system costs (such as minimizing the number of stops and their installation costs), all while maximizing the coverage of passenger demand (Fabbiani et al., 2018).

Traditional optimization methods (e.g., mathematical programming, heuristics) struggle with scalability to handle the BSLP complexity on practical city-wide scenarios. Multiobjective Evolutionary Algorithms (MOEAs), however, are well-suited for the BSLP and other complex transportation planning problems. Their strength lies in their ability to approximate the Pareto front by exploring a broad solution space independent of gradient information or specific problem knowledge (Deb, 2001). This process yields a diverse set of solutions, offering planners many trade-off options for informed decision-making.

In this line of work, this article introduces a multiobjective evolutionary approach to solve realistic instances of the BSLP. The proposed methodology explicitly models three critical conflicting objectives for effective urban public transportation planning: i) maximize service coverage, measured by the total number of demand-weighted covered trips, ii) minimize travel times, approximated by reducing the total number of stops, which, in turn, minimizes bus dwell time: and iii) minimize operational costs, calculated as the cost of stop installation and operation, properly weighted considering the travel demand. To ensure a realistic optimization framework for city-scale instances, the problem formulation is enhanced by incorporating spatial constraints and models for simulating passenger behavioral patterns regarding system and bus stops utilization, considering a coverage neighborhood N ( i ) for each candidate stop location i.

The algorithmic implementation uses a customized NSGA-II framework for multiobjective optimization. An integer encoding considering four stop types defined based on different (better/costlier) facilities for bus stops is applied. In turn, tailored evolutionary operators (including ad hoc two-point crossover, integer polynomial mutation, and binary tournament selection with crowding-distance diversity preservation) are developed to efficiently explore trade-offs between accessibility, efficiency, and infrastructure cost.

The efficacy of the proposed MOEA is demonstrated through applications on problem instances defined in real-world scenarios in Montevideo, Uruguay, and Buenos Aires, Argentina. The case-study in Montevideo illustrates the application on a mid-sized Latin American city characterized by rapid urban growth, socioeconomic diversity, and heavy reliance on public transportation. The Montevideo scenario covers an area of 187 km² and 987 candidate locations for bus stops. In turn, the case-study in Buenos Aires allows evaluating the scalability of the resolution approach on a significantly large scenario. The Buenos Aires scenario covers an area of 205 km² and 118,677 candidate locations for bus stops. Detailed public transportation mobility data from both studied cities are used for instances definition. The considered case studies properly model the real challenges associated to public transportation planning in Latin America, including non.uniform mobility demand patterns, limited investment on infrastructure, and the incipient shift towards sustainable transportation options (Bocarejo et al., 2013; Hidalgo and Huizenga, 2013).

The experimental analysis shows that the proposed MOEA reliably generated high-quality Pareto fronts, providing policymakers with a clear visualization of the potential trade-offs and a diverse set of practical solutions that tend to prioritize high-demand central areas. The empirical comparison with two baseline methods, a local search (LS) and a single-objective evolutionary algorithm (EA), the proposed MOEA notably reduced travel times and operational costs. Improvements over LS were up to 46.9% in the coverage-to-cost ratio, whereas the improvements over the single-objective EA were up to The multiobjective approach also computed several diverse location plans that offer different alternatives to policymakers. All results were computed with high computational efficiency, in overall execution times shorter than half an hour.

The main contributions of the research to the field of transportation planning include:

• A novel multicriteria BSLP formulation, based on a robust, integer mathematical model that balances the conflicting objectives of demand coverage, operational cost, and travel time; and a discussion of modeling choices (i.e., coverage neighborhood, demand normalization, and optional constraints).
• A custom multiobjective evolutionary solution approach, based on a practical MOEA implementation (a customized version of the NSGA-II developed in the jMetal library), tailored to the integer stop-type encoding and to city-scale instances, including algorithmic parameterization and representation-specific operators to effectively solve city-scale BSLP instances.
• An extensive experimental study on real-world urban public transportation scenarios defined in Montevideo and Buenos Aires, providing a quantitative analysis of the trade-offs between problem objectives, compares the MOEA against LS and single-objective EA, reports computational demands, and provides sensitivity insights for key modeling parameters.
• Managerial insights for planning and policy support, via a set of guidelines for decision-makers to identify important factors (e.g., handling heterogeneous demand, robustness of solutions, multimodal integration) to facilitate the use of the proposed algorithmic approach for effective urban planning.

The article is organized as follows. Next section describes the considered multicriteria optimization problem. Section 3 reviews relevant related literature on bus network optimization and multi-objective approaches. The proposed MOEA and baseline metaheuristic methods for results comparison are described in Section 4. Methodological details about development and reproducibility of the software are described in Section 5. The real-world case studies are described in Section 6. Results are reported and discussed in Section 7. Section 8 presents recommendations for planning and the application of the proposed approach. Finally, Section 9 presents the conclusions and formulates the main lines for future work.

2 The bus stops location problem: description and mathematical formulation

This section describes the considered optimization problem and presents its mathematical formulation.

2.1 BSLP: conceptual definition

The efficacy of public transportation systems is based on a proper strategic planning Risso et al. (2022). As proposed by Ceder (2016), the public transportation planning process is structured hierarchically, beginning with high-level strategic planning decisions that set the foundation for all subsequent operational activities. Correctly locating bus stops is critical for improving system efficiency; it directly translates to reduced travel times, minimal delays, and higher service reliability. Well-located stops are also essential for user-centric goals, ensuring high accessibility for all passengers, including those with disabilities, thus making public transport a truly inclusive and equitable mobility option (Litman, 2021).

Achieving the balance between system- and user-related metrics involves managing complex trade-offs between the optimization objectives. Optimal bus stop placement is crucial for cost-effectiveness, minimizing operational expenses from fuel and maintenance, and at the same time it also boosts pedestrian safety by reducing inconvenient or risky access paths for riders. Overall, the optimization of bus stop networks is central to modern urban development, aligning directly with principles of sustainability and livability, such as those promoted by the TOD paradigm (Nesmachnow and Hipogrosso, 2022). By fostering greater public transport ridership and lessening dependence on private cars, optimized networks reduce carbon emissions and enhance air quality across the city.

To address the inherent complexity of bus stops location, this article focuses on a variant of the BSLP formulated as a multiobjective problem. The proposed framework aims to find the ideal balance by simultaneously optimizing three key conflicting objectives: maximizing service coverage, minimizing travel times, and minimizing operational and installation costs.

2.2 Mathematical formulation

The Bus Stop Location Problem (BSLP) is modeled as a multiobjective mixed-integer program defined over the following sets, parameters, and decision variables.

Sets:

• S = { 1 , … , | S | }: Set of census segments, which are the candidate locations for bus stops.
• T = { 0,1,2,3 }: Set of bus stop types, where 0 denotes no stop, 1 a basic stop (no shelter), 2 a simple shelter, and 3 a double shelter (higher-cost facility).
• N ( s ) ⊆ S: coverage neighborhood of census segment s ∈ S, i.e., the set of census segments from which users are considered able to access a bus stop located at s ∈ S. The neighborhood N ( s ) may be defined using a walking distance radius, network-based distances, or topological adjacency of census.

Parameters:

• t r i p s o d: the number of trips from census segment o ∈ S to census segment d ∈ S analyzed time window, i.e., an origin–destination (OD) matrix.
• c t: cost associated with a bus stop of type t.
• D max = max { t r i p s o d + t r i p s d o | ∀ o , d ∈ S }: maximum joint demand between any pair of census segments.

Variables:

• y t s: (binary) 1 if a bus stop of type t is located in census segment s ∈ S, 0 otherwise.
• z s: (binary) 1 if census segment s ∈ S is covered (by a bus stop located in any o ∈ N ( s )), 0 otherwise.

On one hand, a census segment i ∈ S is considered covered if there exists at least one bus stop located in any neighborhood census segment s ∈ N ( i ). On the other hand, a trip ( i , j ) is considered covered if both its origin and destination segments are covered, i.e., that is when z i = 1 and z j = 1 , i , j ∈ S.

Considering the defined sets, parameters and variables, the BSLP is formulated as the mixed-integer mathematical programming model defined in Equation 1:

max f cov = ∑ o ∈ S ∑ d ∈ S t r i p s o d + t r i p s d o z o z d , (1a)

min f stops = ∑ i ∈ S ∑ t ∈ T y i t , (1b)

min f cost = ∑ i ∈ S ∑ t ∈ T c t ∑ o ∈ S t r i p s o i + ∑ o ∈ S t r i p s i o D max y i t . (1c)

Subject to: ∑ t ∈ T y i t = 1 , ∀ i ∈ S , (1d)

z i ≤ ∑ s ∈ N i ∑ t ∈ T \ 0 y s t , ∀ i ∈ S , (1e)

y , z ∈ 0,1 (1g)

Equations 1a–c define the three objectives of the BSLP. Equation 1a maximizes demand-weighted coverage by counting all trips which are covered, i.e., segments whose origin and destination lie in covered segments. Equation 1b minimizes the total number of installed stops as a proxy for reducing dwell events and thus travel times of the buses. Equation 1c minimizes the installation cost of the selected stop types, weighted by the demand associated with each census segment. Regarding constraints, Equation 1d ensures that exactly one stop type (including type 0) is assigned to each segment. Equations 1e, f jointly enforce that a segment is covered if and only if at least one non-zero-type stop exists in its coverage neighborhood. Finally, binary restrictions (Equation 1g) ensure the integrality of all decision variables.

3 Literature review

The strategic positioning of infrastructure has emerged as a central subject in contemporary public transportation research, with a substantial body of literature dedicated to the spatial optimization of bus stop locations. The proposed contributions have addressed many operational variants of the problem, ranging from the optimization of isolated public transportation corridors to the management of large-scale, interconnected urban networks. By exploring diverse scenarios, researchers have sought to account for the heterogeneous nature of urban mobility demands, the complexities of varying road geometries and city plannings, and the inherent conflict between the objectives of different interests,—primarily commuters seeking accessibility and operators aiming for efficiency.

Cui et al. (2015) introduced a multi-objective Genetic Algorithm (GA) for exploring the conflicts between stakeholders in urban public transportation planning. The addressed problem variant focused on locating bus stops to balance three conflicting objectives: the accessibility for passengers, the operational efficiency of bus operators, and the flow of general traffic. The applied fitness function weighted the average walking distance for passengers, the travel time delays for buses during pull-in and pull-out maneuvers, and the delay of private vehicles that share (or cross) the road space with public bases. Through these factors, the model attempted to consider the broader social cost of public transportation infrastructure rather than focusing solely on passenger convenience. From a methodological point of view, although the problem was presented as a multi-objective optimization, the resolution approach relied on a weighted sum scalarization technique, aggregating the considered objectives in a linear formulation. This approach has a significant limitation for complex urban planning, as it requires an a priori decision on the relative importance of each stakeholder and collapses the multidimensional objective space into a single solution. Consequently, the model does not generate a true Pareto front of non-dominated solutions, which would more accurately reflect the inherent trade-offs of the problem. The proposed GA was designed to identify a single stop location relative to intersections and demand centers. The applicability of the model was demonstrated through a case study on the Xizhimen bus terminal in Beijing, with the main goal of evaluating how shifts in location influenced traffic queuing and walking distances. A sensitivity analysis to varying weight coefficients was performed. The main findings indicated that the GA was highly sensitive to the prioritization of objectives: increasing the weight of passenger walking distance tended to pull the stop closer to high-demand intersections, whereas prioritizing automobile flow moved the stop further away to reduce bottlenecking. The study confirmed that the GA computed reasonable accurate trade-off solutions for a single-stop scenario, but the analysis was not extended to realistic, large-scale urban scenarios involving interdependent network effects or multiple competing stop locations across a city-wide grid.

Guo et al. (2019) introduced an integrated bi-objective model for the Feeder Bus Route Design Problem, which addressed the selection of bus stop locations and finding vehicle trajectories as a coupled decision-making process. Bus stops were identified as the critical entry points of the public transportation system, and the mathematical model aimed at capturing the trade-off between passenger accessibility and operational efficiency. The competing optimization objectives were maximizing the total travel demand served within a maximum walking distance threshold and minimizing the total operational costs, measured by the aggregate length of the feeder bus routes. The formulation ensured that the resulting network design prioritizes the last-mile connectivity to public transportation hubs and maintains sustainable resource allocation for the operator. A specific methodological strength of the study is the focus on the spatial relationship between demand points and candidate stop locations. Unlike traditional models that assume predetermined stop locations, Guo et al. modeled stop selection as a variable that directly influences the coverage area of the public transportation service. The model was validated on a synthetic network comprising 81 demand points and 88 candidate stop locations, to analyze the sensitivity of the system to different service levels. A custom algorithm was applied to solve the bi-objective problem, allowing for a proper exploration of the Pareto front. Results provided relevant insights into the behavior of feeder systems under different constraints: mobility demands served was highly sensitive to the maximum walking distance permitted for passengers, but the total route length tended to remain relatively stable across different trade-off scenarios. The stability suggests that once the backbone of the feeder network is established, improvements in service coverage are primarily achieved by the strategic placement of stops within the existing route rather than by increasing the total mileage of the bus fleet. Although the study demonstrated the applicability of the bi-objective approach in a synthetic environment, further extensions are needed to incorporate complex real-world variables, such as varying vehicle capacities and dynamic traffic conditions in dense metropolitan areas.

Perugia et al. (2011) proposed an Adaptive Large Neighborhood Search (ALNS) metaheuristic approach for the Home-to-Work Bus Service Design Problem, focused on the transportation of employees from residential neighborhoods to a central workplace or industrial hub under strict temporal constraints. The mathematical model simultaneously addresses the selection of bus stop locations, the assignment of passengers to those stops, and the routing of vehicles to ensure timely arrival. A bi-objective optimization is considered: minimizing the total operational costs (evaluated by the number of vehicles in the fleet) and simultaneously minimizing user dissatisfaction, quantified through a weighted sum of walking distances to stops and total in-vehicle travel time. The ALNS was based on modifying and rebuilding candidate solutions using removal and insertion heuristics. The method was evaluated on a case study from a large company in Milan, Italy, considering 1,300 employees and 2,735 potential stop locations, subject to real-world constraints such as vehicle capacity limits and maximum allowable walking distances. The evaluation studied the sensitivity of service quality relative to operational investment. The Pareto front revealed that significant reductions in walking distances were achievable with marginal increases in the number of routes, up to a certain saturation point. The spatial distribution of demand and the time windows for workplace arrival heavily influenced the efficiency of the generated routes. The model computed high-quality solutions for a specific corporate context, but the authors acknowledged that extending the approach to more general, multi-destination public transportation systems would require further refinements to manage the increased complexity of overlapping time windows and transfer synchronization.

In Latin America, the TransMilenio Bus Rapid Transit (BRT) system in Bogotá was studied by Bocarejo et al. (2013), shifting the analytical focus from traditional operational metrics to the long-term impacts on urban design. The authors presented a critical urban planning perspective to close the gap between pure transportation engineering and the BLSP. While traditional optimization models consider bus stops as static nodes intended to minimize immediate walking distances or operational costs, Bocarejo et al. argued that the placement of these stops acts as a powerful catalyst for long-term urban transformation. The spatial coordinates of a bus stop define the center of a catchment area, within which the most significant shifts in population density and land value occur. Therefore, the decision-making process in the BSLP is not only about service efficiency; it is about determining where future urban growth and commercial activity will be concentrated. Another difference with standard BSLP formulations is the role of accessibility as a driver of dynamic demand. Instead of considering demand as a fixed input based on mobility and census data, the phenomenon of densification effect suggests a feedback loop: placing a bus stop in a location increases accessibility, which in turn attracts more residents and businesses, thereby increasing the future travel demand at that location. Thus, robust optimization models should consider the densification potential as an objective function, rather than just serving population clusters. By selecting stop locations that maximize the potential for TOD, planners can guarantee the long-term sustainability and patronage of the public transportation system. The socioeconomic heterogeneity of land value was also recognized as a relevant factor of the multi-objective nature of the BLSP, suggesting that the social equity should also be considered among the optimization objectives, to prioritize stop locations in areas where the land value and accessibility improvement provide a higher socioeconomic impact for vulnerable populations.

Medina et al. (2013) developed an analytical framework for the spatial and temporal configuration of public transportation systems, focusing on the simultaneous determination of bus stop spacing and service headways. Instead of solving a pure geographic BLSP formulation, the optimal distance between stops is determined considering the interplay with service headways, since each new bus stop imposes a time penalty due to bus deceleration/acceleration and dwell time, considering passenger boarding and alighting rates. In the context of the BSLP, this implies that increasing accessibility by reducing walking distance directly degrades the speed of the fleet, which requires a larger number of vehicles to maintain a constant headway. A continuous multiperiod optimization model was proposed to minimize total social costs, including operator costs and passenger travel time components, accounting for the dynamic nature of mobility demand across different times of the day to optimize the infrastructure not just for a single peak hour, but for a representative set of demand profiles. The model was evaluated on a case study in the Pajaritos Avenue corridor in Santiago, Chile, using empirical demand patterns and operational data. The model computed stop distributions closely aligned with the existing infrastructure, thereby validating the historical planning. However, the potential for significant efficiency gains through the strategic consolidation of stops that severely penalizes travel times was also found. The findings suggested that by fine-tuning stop locations and headways in response to multiperiod demand fluctuations, a better balance between passenger convenience and the economic sustainability of the service is feasible.

Murat and Kutluhan (2014) proposed a hybrid optimization framework for urban bus line organization, to balance the conflicting interests of authorities and passengers. The approach integrated fuzzy optimization to handle the inherent uncertainty and imprecision of transit data with Linear Goal Programming to prioritize multiple objectives, such as minimizing total travel time and reducing operational costs. the optimization model was applied to a real-world transportation network in Denizli, Turkey, with 6 main bus lines in its backbone and 117 individual bus stops. The results demonstrate that the hybrid method successfully identified high-quality route frequencies and fleet assignments that outperformed traditional single-objective models in terms of efficiency and service quality. The article contributed with a flexible multi-criteria decision-making tool that provided a realistic and mathematically robust foundation for urban transportation planning compared rigid linear models. Later, Murat et al. (2023) studied the trade-off between environmental sustainability and passenger service quality using a decision-making framework to optimize daily bus trips. The proposed multiobjective approach simultaneously evaluated carbon footprint (CF), calculated based on fuel consumption and emission factors, and passenger comfort, quantitatively modeled through bus occupancy rates and standing densities. The model was applied to a larger instance of the public transportation network in Denizli, Turkey, considering a fleet of 230 buses operating in 50 bus lines, and performing more thar 3000 daily trips. The results indicated that increasing the number of daily trips significantly enhanced comfort by reducing onboard crowding, but a diminishing returns point is achieved, where incremental carbon emissions outweigh the marginal gains in passenger satisfaction. The optimum trip frequency balanced the conflicting goals, providing planners with a data-driven threshold to minimize the ecological impact of urban mobility without sacrificing the quality of the commuter experience.

Akgol et al. (2024) proposed the Flow Direction Method (FDM) to enhance the efficiency of multi-route networks in public transportation systems. The network was considered as an interconnected flow system, and a heuristic algorithm was applied to determine optimal vehicle assignments and scheduling to minimize total travel time and operational delays. The method was evaluated over benchmarks and real-world network data, showing improvements over EAs and standard Linear Programming techniques. FDM emerged as a robust, scalable flow-based heuristic to address the curse of dimensionality in large-scale urban networks, providing a practical tool for transit agencies to optimize resources in dynamic, high-traffic environments.

Chen et al. (2018) optimized bus stop locations within an urban grid applying a network-level optimization model to balance the conflicting interests of passengers and operators. The mathematical formulation was based on minimizing the social cost, (including the passenger walking time and the on-board travel time), and the operational costs associated with bus deceleration and acceleration at stops. A global optimization algorithm was applied in a continuous space model to solve an instance on a synthetic rectangular grid network (but similar to networks in cities like Chicago, Barcelona, or parts of Beijing). Results showed that the proposed method identified a proper stop spacing that significantly reduced overall travel delays compared to traditional local-search heuristics. The approach moved from a single-route stop placement to a comprehensive network-wide perspective to enhance the efficiency of entire public transportation system.

Taplin and Sun (2020) proposed a stops-first approach for public transportation network design to prioritize bus stop placement during the initial layout phase rather than as an afterthought. A mathematical optimization approach over continuous-space model minimized the total walking distance for all residents in a developmental area, accounting for the geometric constraints of street networks. The approach was applied to a case study in a residential neighborhood in Perth, Australia, to design a feeder bus route that maximizes accessibility. The results showed that by integrating transit access into the early urban design process, walking distances can be significantly reduced compared to standard post hoc route planning. The research contributed towards the shift in the urban planning paradigm from route-first to stops-first, providing developers with a quantitative tool to enhance the TOD potential of new suburban neighborhoods.

Our previous research (Fabbiani et al., 2018) explored the use of big data analytics for urban mobility characterization and planning, within the context of smart cities. A parallel/distributed computing architecture was used to process massive streams of public transportation data, enabling the extraction of high-resolution mobility metrics. The study generated demand profiles and OD matrices for the public transportation system in Montevideo, Uruguay. Results were applied to a complex multiobjective BSLP variant focused on identifying an optimal spatial reconfiguration of public transportation nodes that simultaneously improved the QoS, measured by total passenger travel time, and reduced the operational costs of the bus system. The empirical evaluation revealed that strategic stop relocation could lead to substantial gains over the existing configuration. Compared to a Business-as-Usual baseline, the proposed method achieved a 16.7% reduction in travel times and a 33.9% decrease in operational costs. These results underscore the effectiveness of using big data analysis to advance beyond static planning, offering a scalable methodology for enhancing the efficiency of public transportation in dense urban centers.

Overall, the reviewed literature (Table 1) covers a wide spectrum of multi-criteria and network-planning approaches, ranging from preference-based techniques that aggregate multiple objectives into a single one to methods that explicitly combine decision-making paradigms such as fuzzy optimization and goal programming, as well as network-level efficiency models that optimize system-wide spacing or social-cost surrogates. Despite the reported advances, there is still a practical gap for city-scale bus stop planning when decision-makers require a discrete configuration over many candidate locations and stop types, and a proper representation of trade-offs through a set of non-dominated alternatives rather than a single compromise dictated by a-priori targets, membership functions, or weights. This article addresses the gap by developing a Pareto-based multiobjective evolutionary framework tailored to the BSLP that explicitly explores the tri-objective trade-off space and supports policy-driven selection of solutions. Moreover, the experimental design of this article combines validation in real scenarios, by applying the proposed framework on two large real-world networks (Montevideo and Buenos Aires) and method-level benchmarking, by comparing the MOEA against two complementary baselines, i.e., a local search strategy and a single-objective EA.

TABLE 1. Summary of related studies.

4 The proposed multiobjective evolutionary approach

This section describes the proposed MOEA to solve the BSLP and the baseline methods used in the experimental evaluation.

4.1 MOEA

This section presents the main features of the proposed MOEA.

4.1.1 Solution representation

The solution is represented by an integer vector y = { y 0 , y 1 , … , y | S | } of length | S |. Each position in the vector corresponds to a census segment, and its associated value indicates the type of bus stop assigned to the segment, as defined in Section 2.2.

4.1.2 Evolutionary algorithm

The algorithmic framework adopted uses the well-established Non-dominated Sorting Genetic Algorithm II (NSGA-II), a state-of-the-art method in the field of multiobjective evolutionary optimization (Deb et al., 2002). The basic procedure of the NSGA-II organizes candidate solutions by ranking them according to Pareto dominance through an efficient sorting procedure. Solution diversity is promoted by crowding-distance. At each iteration, new individuals are generated via crossover and mutation operators and evaluated with respect to all objectives. The population update follows an elitist strategy: non-dominated solutions are preferentially retained. Pareto fronts are incorporated sequentially until the population capacity is approached, and remaining positions are filled by selecting individuals with the largest crowding distance. This selection mechanism allows the algorithm to simultaneously advance convergence toward optimal trade-offs and maintain a diverse set of solutions regarding the trade-off among the different objectives.

Algorithm 1 outlines the proposed MOEA. The initial population is generated and evaluated. At each generation, parent solutions are selected using a tournament based on Pareto rank and crowding distance, ensuring a balance between convergence and diversity. New offspring are created through crossover and mutation operators, followed by a repair to enforce feasibility when required. Offspring are evaluated and merged with the current population, and an environmental selection process based on non-dominated sorting and crowding distance truncation is applied to form the next-generation. The algorithm returns the final set of non-dominated solutions, which approximates the Pareto front of the problem.

Algorithm 1 MOEA procedure.

4.1.3 Evolutionary operators

The evolutionary operators that were used in the MOEa used are:

• Selection and replacement. Parents are selected via binary tournament selection using dominance rank and crowding distance for tie breaking. Replacement is carried out according to the NSGA-II frontwise inclusion scheme, with crowding-distance truncation applied as needed.
• Crossover and mutation. Two-point crossover for integer-encoded vectors is used. Two mutation operators are considered: (i) random gene replacement, and (ii) a greedy-like replacement, i.e., a gene is preferentially replaced by an adjacent type, promoting smoother exploration of stop-quality levels.
• Initialization. The initial population is generated by sampling integer-encoded vectors from a configurable categorical distribution that biases selection toward empty locations (type 0) and assigns a small proportion of higher-quality stops to seed initial coverage. The runnable configuration (implemented in Main) exposes the probabilities and allows initialization from ad hoc solutions.
• Evaluation optimizations. To improve computational efficiency, the implementation precomputes coverage masks for each candidate location and employs incremental update strategies to rapidly evaluate the demand-weighted coverage objective. Demand normalization and any scaling of cost coefficients are performed once during initialization and reused throughout the evolutionary process.

4.2 Baseline methods for results comparison

This subsection describes two metaheuristic optimization methods implemented as baseline for the results comparison: a LS metaheuristic and a single-objective EA.

4.2.1 Local search

The proposed LS is based on exploring a neighborhood structure from an initial low-quality, low-cost, solution and applying a movement that adds bus stops. Candidate solutions are evaluated using an aggregation function that combines the considered optimization objectives into a scalar fitness function. The description of the LS metaheuristic is presented in Algorithm 2.

Algorithm 2 Local search metaheuristic for the Bus Stop Location Problem.

The LS method starts from an initial solution in which no stops are installed (all census segments are assigned the value 0). At each iteration, the algorithm scans the set of remaining candidate segments and, for each segment, evaluates the effect of assigning stop levels 1, 2, and 3. For every tentative modification, the value of the aggregation function of the resulting solution is computed and compared against the value for the current solution. The pair (segment, stop level) that yields the largest improvement in the aggregation function with respect to the current solution is applied. This decision is permanent, the corresponding segment is removed from the candidate set, and the process is repeated. The iteration ends when no candidate modification improves the aggregation function value or when all segments have been evaluated.

4.2.2 Single-objective evolutionary algorithm

A second metaheuristic was implemented to be used as baselines for the results comparison of the proposed MOEA: a single-objective evolutionary approach that explores the search space considering an aggregation fitness function that combines the multiple objectives of the BLSP. In the applied formulation, the original multiobjective problem was transformed into a single-objective optimization problem by combining the objective functions using predefined weights. Formally, the aggregated fitness function was defined as F ( x ) = ∑ k = 1 K w k, where f k ( x ) represent the k-th objective function and w k ≥ 0 is the corresponding weight for objective function k-th. Each objective function defined in Section 2.2 was normalized to a scale between [ 0,10 ] before aggregation. The weights are fixed using an a priori technique, according to the relative importance assigned to each objective. This approach simplifies the optimization process and allows the direct application of an standard (single-objective) EA.

Function f cov measures the total passenger demand satisfied by the proposed bus stop configuration It is normalized using the theoretical maximum coverage (sum of all demands) ( C O V max ) (Equation 2).

f cov ′ = f cov C O V max × 10 (2)

Function f stops, which counts the number of segments with at least one bus stop (Equation 1b), is normalized using | S | which is the total number of census segments as expressed in Equation 3.

f stops ′ = f stops − | S | | S | × 10 (3)

Finally, function f cost, which considers the demand-weighted placement cost (Equation 1c), is normalized considering the minimum and maximum achievable costs (i.e., C max and C min) as expressed in Equation 4). The factor v i ranges from − 1 (high-demand areas, favorable) to + 1 (low-demand areas, penalized).

f cost ′ = C max − ∑ i ∈ S c i C max − C min × 10 , c i = ∑ t ∈ T c t y i t 2 1 − ∑ o ∈ S t r i p s o i + ∑ o ∈ S t r i p s i o D max − 1 . (4)

The final fitness function used for the single-objective EA combines the defined components with empirically determined weights w cov, w stops, and w cost Equation 5. The distribution of weights reflects the primary goal of maximizing passenger coverage while balancing infrastructure costs and encouraging strategic placement in high-demand areas. A standard grid search was applied, defining the Cartesian product of five candidate values for each aof the three weights, for a total number of 125 combinations.

F s = w cov f coverage ′ + w stops f stops ′ + w cost f cost ′ (5)

The implemented EA applies the same solution encoding and variation operators than the studied MOEA: binary tournament selection (but without considering Pareto dominance), two-point crossover for integer-encoded vectors, random gene replacement mutation, and greedy-like replacement mutation.

5 Development, reproducibility, and execution

This section describes specific features of the developed implementation regarding the bridge between the high-level theoretical model and the specific software that implement them, and reproducibility details, according to the reproducible research paradigm.

5.1 Code-to-concept mapping

To guarantee the transparency and rigorous reproducibility of the research and the main findings, the computational framework is designed to provide a direct mapping between theoretical constructs and their software implementation.

The primary entry point for the system is the Java class com.paradas.Main, which serves as the orchestrator for the optimization process. This class is responsible for loading the input dataset, specifically the OD matrices and neighborhood definitions, configuring the MOEA parameters, and ensuring the persistence of resulting Pareto sets and statistical traces. The Main class (and associated utilities) also pre-computes demand normalization and coverage masks, sets RNG seeds for reproducibility, and writes output Pareto sets, metric traces and log files for post-processing and plotting. By centralizing the execution logic within this runner, the framework maintains a clear separation between problem definition and algorithm control, allowing for exact replication of the experimental conditions described in this article.

The representation of candidate solutions is managed by a custom problem class integrated with the jMetal framework, which evaluates each solution against the three objectives: coverage maximization, stop-count minimization, and demand-weighted cost reduction. To explore the search space effectively, the implementation employs variation operators specifically tailored for integer representations, including a two-point integer crossover and a per-gene random-reset mutation. Furthermore, the framework incorporates repair operators that enforce feasibility during the offspring generation phase, ensuring that all candidate solutions adhere to hard constraints such as budget limits or per-type facility caps.

The protocol for the experimental evaluation follows a structured generational loop, consistent with the NSGA-II architecture. In each generation, the system logs standard multiobjective performance indicators. To account for the stochastic nature of the evolutionary process, the protocol requires performing a number of independent executions. The resulting data is then processed using included aggregation scripts to compute summary statistics, including medians and interquartile ranges, which form the statistical basis of the comparative analysis presented in the results section.

All the developed code is publicly available at https://github.com/SantiagoLopezDeharo/Optimizacion-de-transporte-publico. Beyond simple reproduction and validation of experiments, the modularity of the implementation, which includes discrete problem classes, variation operators, and the execution runner, allows for an easy extensibility for future research. One primary extension point involves the augmentation or replacement of the considered objective functions to incorporate more granular transportation metrics, such as dwell-time models or social equity indicators. Similarly, the framework is designed to integrate diverse constraint types or alternative spatial accessibility models, such as complex network-based distances instead of standard Euclidean calculations. Because the problem wrapper is decoupled from the evolutionary search engine, researchers can easily change the optimization algorithm for other metaheuristics available in the jMetal library, such as MOEA/D or SPEA2, to evaluate and perform comparative analysis. Additionally, the structure of the evaluation routine facilitates JVM-level parallelization to address large-scale instances, enabling the concurrent evaluation of individuals within a population to reduce total execution time on multi-core hardware.

5.2 Practical reproduction notes

The associated repository provides a standardized environment built upon the Maven framework to facilitate the reproduction or extension of the experimental evaluation. Researchers can compile the project and execute the optimization process via the entry point com.paradas.Main, which uses the pom.xml configuration to manage all necessary dependencies and Java sources. The execution logic is exposed through a flexible Command Line Interface, allowing users to modify invocation patterns and experiment parameters dynamically without requiring major modifications to the source code.

The computational model relies on specific data structures, requiring preprocessed OD matrices and clearly defined segmentation or neighborhood parameters. The repository includes detailed documentation for the data formats utilized in the case studies defined in Montevideo and Buenos Aires, providing a template for researchers to either format new datasets accordingly or adapt the existing input-parsing routines. Proper data alignment is critical, as the evaluation routines are optimized for specific matrix structures to ensure computational efficiency during the fitness assessment phase.

Statistical rigor in the optimization process is maintained through the explicit management of stochastic variables and algorithmic parameters. The system provides robust default settings within the Main runner and reproducibility is further guaranteed by the ability to define specific RNG seeds. This is standard practice within frameworks implementing EAs to perform multiple independent executions, recording each unique seed to allow for the exact reconstruction of evolutionary trajectories and the calculation of statistically sound aggregate results.

Finally, the implementation is designed to produce high-fidelity output files that serve as the basis for performance analysis. The generated files include the final non-dominated solution sets (Pareto fronts) and a temporal trace of multiobjective metrics (see next subsection). The structured logs are designed for immediate compatibility with the provided post-processing and visualization scripts, enabling the seamless generation of the comparative plots and tables presented in this study.

For researchers seeking to verify the exact experimental configuration or inspect the default parameter values used in the published results, the project repository is the definitive source of provenance. The main entry point, com.paradas.Main, contains the specific settings and configurations used in all optimization runs discussed in this study. In addition to the core source code, the repository hosts the final technical reports and specialized plotting utilities required to generate the figures and tables referenced throughout this paper. This centralized documentation strategy ensures a transparent and auditable link between the raw computational logic and the final analytical outcomes presented in the manuscript.

5.3 Metrics and experimental protocol

The software architecture integrates specialized routines for the calculation of multi-objective performance indicators, which are logged at each generation to provide a temporal view of algorithm convergence and diversity. The implemented indicators specifically include Relative Hypervolume (RHV), spread, and Generational Distance (GD) (Deb, 2001; Coello et al., 2007), formally defined in Section 7.2.

Following established experimental practices in evolutionary computation, the protocol involves performing R independent executions for each problem instance to mitigate the impact of stochastic variation. This approach allows for the collection of detailed per-run traces, which are subsequently processed to derive robust summary statistics, including medians and interquartile ranges. Furthermore, non-parametric statistical tests are employed to calculate p-values, ensuring the significance of the observed performance differences across different configurations. The repository includes automated scripts to replicate this metric aggregation and visualization pipeline directly from the raw execution logs.

6 Real world-case studies

This section describes the two real world-case studies considered for the evaluation of the proposed MOEA.

6.1 Public transportation system in montevideo

The public transportation system in Montevideo, Uruguay (Metropolitan Transportation System, STM) is the backbone of urban mobility in Montevideo, Uruguay, serving as a critical artery for the daily commute of residents and visitors alike. As a medium-sized capital city, Montevideo heavily relies on its extensive, comprehensive bus network. For a city the size of Montevideo, the system boasts remarkable infrastructure, featuring 145 principal bus lines and well over a thousand specific line variants across the city. This dense coverage is facilitated by an extensive physical network including approximately five thousand bus stops, providing a high degree of accessibility across the metropolitan area (Perera et al., 2025). The commented numbers highlight the vital operational scale of STM and its centrality to the transportation infrastructure in Montevideo. The network of bus routes within the STM is represented in Figure 1, using data from sig.montevideo.gub.uy. The number of bus stops locations considered in the problem instance defined over Montevideo is 987.

6.2 Public transportation system in Buenos Aires

The public transportation system in Buenos Aires, Argentina (Capital Federal and surrounding metropolitan area) is one of the most extensive and complex in Latin America, serving a daily ridership of millions of people across the vast metropolitan region.

Unlike the bus-centric system of Montevideo, Buenos Aires operates a fully integrated multimodal network, with components including buses, subway (the oldest metro system in Latin America), and commuter rail. Public buses (colectivos) are the primary transportation mode. The bus network is massive, comprising nearly 300 bus lines that cover a route extension of over 25,000 km. This comprehensive system utilizes a fleet of approximately 15,000 units and runs 24 h a day, ensuring access to virtually every part of the city and its suburbs. The total number of individual bus stops across the entire metropolitan area is very large (approximately 6,850), with a high density of stops. Furthermore, dedicated Metrobus corridors, a Bus Rapid Transit (BRT) system, have been implemented across several major avenues to drastically reduce travel times for multiple bus lines. The network of bus routes within the STM is represented in Figure 2 using data from https://datos.transporte.gob.ar/. The number of bus stops locations considered in the problem instance defined over Buenos Aires is 12,247.

7 Experimental analysis

This section describes the experimental validation of the proposed MOEA for solving the BLSP over the two real-world case studies described in the previous section.

7.1 Evaluation metrics

To evaluate the proposed MOEA, relevant multiobjective optimization metrics were used (Coello et al., 2007). The applied metrics evaluated the two main goals of multiobjective optimization methods: convergence to the real Pareto front and diversity of the computed solutions.

Because the true Pareto front was unknown for the considered case studies, the common practice for real-world multiobjective problems suggested by Coello et al. (2007) was applied. A reference Pareto front was constructed by aggregating all non-dominated solutions found by the studied algorithm across the 30 independent experimental executions performed for each scenario and problem instance.

7.1.1 Pareto optimal solutions

The Pareto optimal (PO) metric quantifies the number of non-dominated solutions in the approximated Pareto Front that belong to the true or reference Pareto Front. A higher PO value indicates a better contribution of the approximated Pareto front to the reference Pareto front.

7.1.2 Error ratio

The Error Ratio (ER) measures the proportion of solutions computed in the approximated Pareto front that are not truly Pareto-optimal when compared to the known true or reference Pareto front. The ideal value of ER is 0, which corresponds to a perfect Pareto Front that computes all non-dominated solutions that are optimal, and the worst possible value is 1, indicating that none of the computed solutions belongs to the true Pareto front. ER is defined by ER = | { s ∈ P ∣ s ∉ P * } | | P | where P is the approximated Pareto front obtained by the algorithm, P * is the true or reference Pareto front, and | ⋅ | is the cardinality (number of elements) in the set.

7.1.3 Generational distance

The Generational Distance (GD) metric evaluates how well the approximated Pareto front converges to the true (or reference) Pareto front. GD is calculated as the average Euclidean distance from each solution in the computed Pareto front to the closest solution in the true Pareto front (Equation 6, where P is the set of solutions in the computed Pareto front, P * is the set of solutions in the true (reference) Pareto front, d ( s , P * ) is the minimum distance from solution s ∈ P to any solution in P * and p is the parameter typically set to 2 (Euclidean norm)).

GD = 1 | P | ∑ s ∈ P d s , P * p 1 / p (6)

7.1.4 Spread

The spread metric evaluates the diversity and the distribution of non-dominated solutions across the approximated Pareto front. To accurately capture the distribution, the spread calculation (Deb, 2001) incorporates the extreme points of the true Pareto front. A lower value for the spread metric is preferred, as it signifies a more uniform and well-distributed set of non-dominated solutions, with the optimal value being zero for a perfectly equidistance distribution.

The definition of the spread metric is presented in Equation 7, where d i denotes the distance between the i-th solution in the computed Pareto front and its nearest neighbor, whereas d ̄ is the average of all d iand d h e is the distance between the extreme of the h-th objective function in the true Pareto front and the nearest point in the computed Pareto front.

spread = ∑ h = 1 k d h e + ∑ i = 1 N D d ̄ − d i 2 / ∑ h = 1 k d h e + N D × d ̄ (7)

7.1.5 Relative Hypervolume

The Relative Hypervolume (RHV) is a comprehensive metric that assesses both the quality and distribution of the approximated Pareto front. RHV is calculated as the ratio of the volume in the objective space dominated by the computed Pareto front ( P ) to the volume dominated by the true or real Pareto front ( P * ). Both volumes are considered as bounded by a predefined reference point ( r ) located in the objective space opposite the Pareto fronts (usually, the nadir point in the objective functions space). A higher RHV value indicates superior results in both convergence and distribution. The ideal value for RHV is 1.0, achieved when the computed front perfectly coincides with the true Pareto front. The RHV metric is defined as RHV = HV ( P , r ) HV ( P * , r ) where H V ( P , r ) is the hypervolume dominated by P with respect to r, and H V ( P * , r ) is the hypervolume dominated by the true Pareto front with respect to the same point r.

7.1.6 Problem-specific solution quality

In addition to the standard multiobjective optimization metrics studied (ER, GD, spread, and RHV), the evaluation of the proposed multiobjective evolutionary approach also incorporated the problem-specific objectives that are crucial for public transportation planning: demand coverage, travel times, and operational costs. These metrics are essential for directly measuring the trade-offs between user-centric QoS goals (e.g., maximizing demand served and minimizing travel time) and system-centric efficiency goals (e.g., minimizing operational expenditures).

7.2 Parameter configuration analysis

Parameter configuration analysis are mandatory for stochastic optimization methods, considering that using different parameter configurations lead to different results (Nesmachnow, 2014). In order to perform the parametric tuning of the proposed MOEA, a different problem instance than the ones used for evaluation was used, to avoid biased results. A problem instance of smaller size than the real-world scenarios in Montevideo and Buenos Aires was selected, to allow all possible parameter combinations to be evaluated within a reasonable execution time. Specifically, the problem instance corresponds to the public transportation system in Pittsburgh, USA. This scenario covers 101 km² includes 4,100 possible bus stop locations (Xu et al., 2024) and the corresponding OD matrix was estimated using commuting information from the LODES dataset, available online (https://lehd.ces.census.gov/data/lodes).

The three main parameters with the higher influence on the MOEA results were studied: population size ( P S ), crossover probability ( p C ), and mutation probability ( p M ). The candidate values for the studied parameters were P S ∈ { 50,100,200 }, p C ∈ { 0.6 , 0.7 , 0.8 }, and p M ∈ { 0.1 , 0.01 , 0.001 }. The 27 orthogonal combinations of parameter values were studied, and 30 independent executions of the proposed MOEA were performed for each configuration. The RHV metric was used to assess the solution quality and diversity of the computed solutions. The statistical analysis of the results distributions included performing the Kolmogorov-Smirnov statistical test to check normality, and after concluding that most of the results did not follow a normal distribution (with a confidence level given by the p-value 0.05), the Analysis of Variance (ANOVA) parametric test was applied to determine the statistical significance of the differences on the RHV metric for the studied parameter configurations. The boxplots of the RHV results for the studied configurations are reported in Figure 3.

Parameter configuration #27 (P S = 200, p C = 0.8, and p M = 0.1) allowed computing the best RHV results. This parameter configuration was applied in all validation experiments, reported next.

7.3 Computational experimentation

This section reports the computational results of the proposed MOEA and the comparison with the considered baseline optimization methods.

7.3.1 Multiobjective optimization metrics

Table 2 reports the values of multiobjective optimization metrics for the proposed MOEA and the single-objective reference EA for the Montevideo and Buenos Aires instances. For each studied metric, ↗ indicates that higher values are better and ↘ indicates that lower values are better.

TABLE 2. Multiobjective evaluation metrics for Montevideo and Buenos Aires instances.

For both instances, the MOEA clearly outperformed the EA across all metrics. In particular, the MOEA attained zero GD and ER, indicating that all obtained solutions belong to the reference non-dominated set and lie on the best-known Pareto front. In contrast, the EA exhibited substantially higher GD and ER values, reflecting poorer convergence and a larger proportion of dominated solutions.

Regarding diversity, the spread metric showed significantly lower values for the MOEA in both Montevideo and Buenos Aires, suggesting a more uniform and well-distributed approximation of the Pareto front. The improvement on diversity was further supported by the RHV results, where the MOEA dominated the EA, indicating a larger dominated objective space. Finally, the ND was markedly higher for the MOEA in both instances, confirming its superior ability to generate a richer and more informative set of trade-off solutions for decision-makers. Overall, the MOEA consistently achieved better convergence, diversity, and dominance than the baseline EA.

Regarding the robustness across independent runs, the distribution of the results of the metrics ER and RHV across all the independent runs for the instance of Montevideo are presented in Figures 4, 5 respectively. The RHV presented a symmetric distribution with an approximately linear Q–Q pattern, with only minor deviations in the tails, which is consistent with an approximately normal behavior. On the other hand, the ER exhibited a marked concentration near its upper bound with a left tail, indicating a bounded, non-normal behavior.

7.3.2 Pareto fronts

Figure 6 presents the Pareto front computed by the proposed MOEA for Montevideo (3D visualization) and the corresponding 2D projections (pairwise objective functions).

The 3D graphic in Figure 6a clearly shows that the proposed MOEA computed a significantly wider range of Pareto optimal solutions, properly covering different regions of the solution space. Different options are provided to the decision maker, with different trade-offs between the problem objectives. The decomposition.based EA used as reference baseline computed accurate solutions with high cost and coverage, but did not properly sampled other regions of the solution space. The 2D projections show that solutions computed by the MOEA dominate solutions found by the EA with similar values of cost, coverage, and number of stops.

Figure 7 presents the Pareto front computed by the proposed MOEA for Buenos Aires (3D visualization) and the corresponding 2D projections (pairwise objective functions).

Table 3 presents the comparison between the baseline methods -LS and EA- and the MOEA for the Montevideo instance. In the case of the MOEA and EA, the compromising solution is used for comparison as it provides a good trade-off among different objectives. The Table reports the value of each objective and the relative improvement based on the ratio between the coverage objective and the cost objective. This metric enables a compact and interpretable comparison across methods since the ratio captures how efficiently coverage is achieved per unit of cost, providing a synthetic indicator of cost-effectiveness that is particularly useful for planning purposes.

TABLE 3. Objective function values and comparison of the MOEA (compromise solution) with the baseline methods.

Results in Table 3 show that the MOEA compromise solution achieved a substantially better balance between coverage and cost. Although the MOEA covered fewer trips in absolute terms than LS and EA, it obtained a strong reduction in both the number of stops and operational cost. In particular, compared to the LS solution, the MOEA reduces the number of stops by more than two thirds (317 vs. 987) and cuts operational cost by approximately 66%, while still preserving a significant fraction of demand coverage.

The trade-off between coverage and cost was captured by the improvement of the ratio, where the MOEA attained the highest value (23,352), clearly outperforming EA (12,395) and LS (16,677). In relative terms, the MOEA improved this ratio by 46.9% with respect to LS and 28.6% with respect to EA, indicating a much more efficient conversion of infrastructure cost into covered demand. On the other, hand the EA also outperformed the LS with an improvement of 34.5%.

Overall, the results of Table 3 confirmed that the MOEA compromise solution provides a cost-effective and operationally efficient configuration, whereas LS prioritized coverage at the expense of excessive infrastructure and EA remained dominated by the MOEA in terms of coverage–cost efficiency. These results highlight the advantage of multiobjective optimization for supporting balanced and scalable planning decisions in real-world urban transport systems.

7.3.3 Illustrative examples

To demonstrate the practical application of the proposed MOEA, Figure 8 illustrates the best trade-off Pareto-optimal solution for the Montevideo instance, mapped across various census segments. This solution is representative of a compromise between the considered problem objectives. The visualization confirms that the resulting distribution of bus stops aligns closely with the actual passenger demand and demographic density of the city.

The MOEA demonstrated a sophisticated spatial logic for resource allocation: i) it prioritized central districts and high-traffic corridors, deploying premium or higher-capacity stops in zones with higher public transportation volume; ii) peripheral zones with sparse populations or lower public transportation demand were assigned less infrastructure; and iii) rather than placing a stop in every individual census block, the model smartly grouped lower-demand segments, which are serviced by anchor stops in adjacent segments, ensuring full coverage while maintaining operational efficiency. The strategic distribution is a direct consequence of the demand-sensitive constraints and objective functions integrated into the mathematical model. By weighting stop quality against local transit needs, the model ensured that the urban infrastructure is both fiscally responsible and socially equitable.

Figure 9 shows that the resolution approach properly scale up to a megacity like Buenos Aires, having a significantly larger public transportation demand and a different spatial distribution.

Figure 9 shows a clear concentration of high quality stops in central and coastal areas. The MOEA correctly identified hotspots with higher passenger demands, likely representing the central business district and major transit corridors. By locating the highest quality infrastructure in hotspots, the MOEA maximized the utility-per-stop, ensuring that a larger number of users benefit from superior and higher capacity bus stops. There is a clear transition from high-quality low-quality or even unassigned segments in the interior and peripheral suburbs. Instead of spreading a rather uniform distribution, a hierarchical network is build, which strategically prioritizes the main backbone of the public transportation system. This is a higly desirable design pattern when building hierarchical, backbone-shaped urban networks (Risso et al., 2023a).

In both case studies, the proposed MOEA computed accurate bus stops configurations, correctly reflecting the modeling assumption that prioritizes operational cost and efficiency over concentrated placement. The solutions guarantee that residents are within a reasonable coverage of a nearby bus stop and resources are smartly managed to provide accurate trade-offs between the problem objectives. Solutions are sensitive to street grid and connectivity, meaning that the MOEA properly integrated geographic demand data with the existing road topology, rather than just treating the city as an non-structured set of points.

8 Managerial insights

To bridge the quantitative results presented in this article with practice, this section summarizes the main managerial implications derived from the observed trade-offs and solution patterns. The contributions presented in this study provide several actionable insights for urban planners, transportation agencies, and policymakers seeking to design or update public bus networks.

First, users demand geographic heterogeneity must be explicitly incorporated into planning decisions. Solutions that prioritize dense and highly transited areas lead to significant efficiency gains, but they should be complemented with context-specific rules to avoid systematically under-serving peripheral zones where public transportation may be the only mobility alternative. In this line, the proposed multiobjective approach directly contributes to achieving a careful balance: placing too many stops in highly populated areas while neglecting low-demand regions lead to a cost-efficient but socially inequitable configuration, whereas deploying an excessive number of stops in sparsely populated zones improve accessibility but unnecessarily increases installation and operational costs.

Second, robustness of the system against demand uncertainty is essential. Cities evolve dynamically, and commuting patterns fluctuate due to demographic changes, new services, or emerging mobility habits. Planners should periodically reassess the spatial distribution of stops and maintain the capability to reoptimize solutions, using algorithmic tools to evaluate different growth or disruption scenarios. In this regard, the proposed MOEA provides an efficient computational tool to reoptimize the current network configuration and adapt it to new travel patterns in reasonable computing times, enabling data-driven adjustments that preserve service quality as the city evolves.

Third, the use of multiobjective optimization enables transparent, evidence-based decision-making. By offering a set of non-dominated alternatives rather than a single prescriptive solution, the proposed approach allows decision-makers to select a configuration that best aligns with financial constraints, political priorities, or service expectations. This flexibility is particularly valuable in real-world contexts, where trade-offs between coverage, cost, and efficiency cannot be ignored. In addition, the explicit availability of extreme solutions—those that maximize a single objective at the expense of others—provides valuable strategic perspective, helping planners understand the boundaries of what is attainable and assess how much must be sacrificed in one dimension to gain improvements in another.

Fourth, from an implementation perspective, transit agencies can deploy the proposed approach as a repeatable planning workflow: (i) compile candidate stop locations (existing and potential) and their attributes/costs, (ii) prepare demand inputs and define coverage neighborhoods (e.g., walking catchments or network-based accessibility), (iii) set policy constraints (budget caps, maximum stops, minimum service rules), and (iv) run the MOEA to obtain a Pareto set and select a preferred alternative (e.g., via a compromise solution or agency-defined priorities). In practice, the main requirements are:

• Data requirements: candidate stops (GIS), demand representation (OD matrix or demand weights), distance/travel-time information to evaluate coverage and efficiency, and unit costs/constraints by stop type.
• Scalability: runtime primarily depends on the number of candidate locations and the number of objective evaluations (population size × generations); agencies can control this through time/iteration budgets and leverage parallel evaluation when available.
• Computational considerations: while the underlying selection problem is combinatorial, the proposed metaheuristic is designed for city-scale scenario analysis, enabling periodic re-optimization and “what-if” studies (growth, disruptions, budget changes) within practical computing times.

9 Conclusion and future work

This article presented a multiobjective evolutionary approach for solving the optimization problem that proposes finding optimal locations for stops in a public transportation system. A specific problem variant was formulated, aiming at determining the locations and bus stop types for the simultaneous optimization of three objectives: the maximization of the system coverage, the minimization of travel times, and the minimization pf operational costs. The considered problem objectives are very relevant for the practical implementation of efficient, equitable, and affordable public transportation systems in modern smart cities.

The proposed MOEA was designed to provide an efficient exploration pattern of the intricate solution space of the problem. An ad hoc Pareto based evolutionary search was implemented, including several features to guarantee replicability and the practical application of the developed software.

Two real-world case studies were considered for the validation of the proposed multiobjective approach, built using real data from the public transportation systems in Montevideo, Uruguay, and Buenos Aires, Argentina. The proposed MOEA was compared against two baseline optimization methods, namely a LS technique and a single-objective EA. The computational results confirmed the effectiveness of the proposed MOEA for the bus stop location problem. Compared with the baseline methods, the MOEA consistently delivered more balanced solutions across conflicting objectives. In particular, the compromise solution of the MOEA improved the coverage-to-cost ratio by 28.6% with respect to the EA and by 46.9% compared to local search, while simultaneously reducing both the number of stops and the associated operational costs. Although the local search achieved the highest absolute demand coverage, it did so at the expense of significantly higher infrastructure deployment and cost, resulting in less efficient configurations overall. Consequently, the results highlighted the advantage of explicitly handling trade-offs among objectives, as opposed to relying on single-criterion or sequential decision strategies, since the MOEA identified solutions that are not only competitive in terms of coverage but also substantially more cost-effective and operationally efficient. Finally, the inspection of illustrative solutions showed that the MOEA exhibited a sophisticated spatial logic for resource allocation: (i) it prioritized central districts and high-traffic corridors, deploying higher-capacity stops in areas with greater public transport demand; (ii) peripheral zones with sparse populations or lower demand are assigned lighter infrastructure; and (iii) instead of placing a stop in every census block, the model strategically groups lower-demand segments and serves them through anchor stops in adjacent areas, ensuring full coverage while maintaining operational efficiency.

Several promising future research directions emerge across modeling, algorithmic development, and practical implementation of the proposed framework. From a modeling perspective, future extensions include, firstly, incorporating demand uncertainty through stochastic or robust optimization frameworks, allowing planners to assess solution stability under evolving mobility patterns. Secondly, another research line is to include additional constraints and objectives—such as equity indicators, explicit budget limits, or environmental metrics—to further enrich the decision space and better reflect real-world planning requirements. Finally, further developments in modeling focus on refining the coverage definition by integrating walking distances, network-based accessibility measures, and physical barriers, thus improving the realism of accessibility assessment. From an algorithmic standpoint, future extensions include studying computational scalability of the algorithms by means of parallelization, hybrid heuristics, or adaptive operator selection, to enable the application of the approach to larger metropolitan areas. Regarding practical implementation, the future research line is to integrate the proposed model and algorithms into interactive decision-support tools, coupled with field validation and pilot studies to enhance usability in urban transport planning processes. Regarding practical sensitivity of the results to input parameters, future work includes a systematic sensitivity analysis of the proposed framework. Conducting such a study would constitute a dedicated contribution on its own, as it requires defining realistic parameter ranges and uncertainty levels based on operational practice (e.g., walking-catchment thresholds, demand representations/weights, and budget or stop-cap limits), as well as analyzing their impact on the resulting Pareto fronts and on recommended planning decisions.

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Footnotes

Edited by: Furkan Ahmad, Hamad bin Khalifa University, Qatar

Reviewed by: Gang Cheng, Tibet University, China

Yetis Murat, Pamukkale University, Türkiye