Mathematical Models Applied to the Localization of Park-and-Ride Systems: A Systematic Review

Abstract

Vehicle congestion and the environmental problems associated with the increasing vehicle fleet have led stakeholders to create solutions to these problems. Park-and-Ride (P&R) facilities are provided as a solution for public transportation to avoid increasing vehicular flow and using private vehicles. However, the optimal location of these facilities is still a challenge to be considered. Therefore, this article aims to present a systematic review of the mathematical models applied for P&R localization, using the PRISMA protocol to ensure a comprehensive analysis. A total of 44 articles between 2002 and 2025 were identified into four categories: decision support models, econometric models, optimization models, and other models. The review also examines the term distribution of urban contexts where the mathematical models are applied, distinguishing between Global North versus Global South urban contexts. The results showed the efficiency of mathematical models within the decision support models category due to their integration with multiple criteria. The econometric models analyze factors influencing user behavior, while the optimization models improve and optimize the efficiency of transport networks despite facing computational challenges. Finally, other models, such as multilevel programming and fuzzy logic, offer adaptive solutions for highly variable urban environments. The primary contribution of this study is its comprehensive application of the mathematical models used for the location of P&R facilities. This offers a systematic approach for anticipating future urban situations, developing supporting policies, and analyzing their effects.

1. Introduction

Nowadays, the expansion of the global automotive industry and the substantial reliance on automobiles have produced considerable challenges in human transportation [1,2]. These issues are mostly seen during traffic congestion or increased greenhouse gas emissions from heightened vehicle use [3,4]. Many recommendations have been implemented in response, including implementing Park-and-Ride (P&R) systems. P&R systems are proposed as an efficient means to enhance public transport use and alleviate urban traffic congestion [5,6,7].

The P&R systems are a kind of public transportation in which users drive their vehicles to a central point and leave them there before boarding a bus or train [8]. Most of the time, P&R lots are situated close to highway interchanges, making it convenient for drivers to enter the parking space specifically intended for their vehicles [9,10]. After parking their vehicles, users can use either a bus or a train to reach their ultimate destination [11,12]. However, the optimal location of P&R facilities is a complex problem that needs to be addressed through mathematical modeling.

Over the past few decades, various mathematical models have also been developed to address the problem of the location of P&R facilities [13]. These mathematical models seek to improve transportation efficiency and reduce the costs associated with the use of transportation modes [14,15]. Among the most widely used approaches are decision support models, econometric models, and optimization models. Moreover, innovative methodologies have emerged that integrate techniques such as fuzzy logic and multilevel programming to solve specific design problems in complex urban environments [16,17].

Although there are advances in the field, there is still a need to improve the adaptability, accuracy, and scalability of these models. Transportation networks are also evolving with new technologies, such as the inclusion of intelligent transportation systems (ITS) or autonomous vehicles [18,19], as well as social factors such as the increasing demand for sustainable solutions [20]. Therefore, mathematical models that adapt to all these innovations and changes are essential to remain relevant and practical.

Although different forms of P&R system applications have been studied and explored in the previous literature, there is no comprehensive review or framework explicitly focused on the mathematical models used for P&R system localization. This study addresses this gap by systematically evaluating the body of knowledge in the literature on mathematical models used for P&R system localization, using the PRISMA protocol to ensure a comprehensive and detailed analysis. The properties, applications, and characteristics of each model are also examined. Thus, a strategic framework for future research and its application in different urban scenarios is proposed. This paper thus aims to improve the theoretical understanding and decision-making applied to the localization of P&R systems.

In order to support this review, the following research questions have been proposed:

• RQ1: What are the mathematical models that have been applied to the localization of Park-and-Ride (P&R) facilities?
• RQ2: What are the main characteristics, strengths, and limitations of each type of model when applied to the location of P&R systems?

The rest of the systematic review is organized as follows: The first section provides an overview of the many mathematical models utilized in location P&R. Section 2 describes the approach utilized to conduct the systematic review. Section 3 presents the findings, while Section 4 presents the analysis and recommendations for further study. Finally, Section 5 reveals this study’s conclusions.

2. Materials and Methods

The Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) method clarifies what other authors did [21,22]. This method might help researchers summarize the current literature in a continuous, step-by-step, complete, and straightforward manner [23,24]. The iterative nature of the PRISMA method means that the research process is looked over and improved, which is very important for ensuring that the results are reliable and valid [25]. This method was first created for evidence-based medical studies but has since been used in other areas [26]. It is essential to use this tool because it allows the data to be analyzed in a systematic and repeatable way, making the data more accurate and reliable [27]. In general, this methodological technique is fundamental to guaranteeing the results can be trusted and used again by other researchers [28,29,30].

Using a systematic review and applying the PRISMA protocol, Abelha et al. [31] examined 69 publications to assess graduates’ employability and competency development. Mason et al. [32] performed a systematic study of 6 sources (CINAHL, EMBASE, EMCARE, Medline, PsycINFO, and Web of Science) to find out what steps are taken to help parents whose babies need to be transported from one hospital to another. Out of the 831 papers that were picked, 61 were chosen. The PRISMA protocol can also be used to determine the features, instructional strategies, technological advancements, and shortcomings of immersive virtual reality for teaching foreign and second languages [33]. Henriques et al. [23] collected 74 articles published between 2018 and 2023 to determine the current state of artificial intelligence (AI) technologies in the tourism industry and their potential for future application in order to improve the consumer experience across a variety of tourism sectors. The PRISMA protocol was used to conduct an analysis of 158 articles in the study that was developed by Yigitcanlar et al. [34]. The purpose of this study was to address important knowledge gaps concerning digital technologies and technological advancements in the transportation industry.

This systematic review covers several components: (1) search strategies detailing advanced queries in databases such as Scopus, Web of Science, Science Direct, and Google Scholar; (2) admissibility criteria outlining inclusion and exclusion variables; and (3) study selection.

2.1. Search Strategies

The databases Scopus, Web of Science, Google Scholar, and Science Direct were used to gather 44 articles. An advanced search was conducted utilizing logic gates to precisely identify the articles. In this search, Boolean phrases like AND, and OR were utilized. The words used in the various databases are displayed in

Table 1. A list of the most effective term phrases used in these databases.

Database	Term
Scopus Science Direct Google Scholar	TITLE-ABS ((“Park and Ride” OR “P&R”) AND (“location” OR “localization”)) AND “mathematical” AND (“Program” OR “model”)
Web of Science	ALL = (Park and Ride) AND (ALL = (Mathematical) OR ALL = (Mathematics)) AND ALL = (models)

. Searches in Scopus, Google Scholar, and Science Direct yielded identical terms. Web of Science, on the other hand, used a different term.

2.2. Inclusion and Exclusion Criteria

There are two parts to the eligibility requirements: those that must be met and those that cannot. All the publications were peer-reviewed, written in English, and dealt with mobility constraints. Excluded from this search were encyclopedias, editorials, books, conference proceedings, and chapters from reviews. In addition, papers that had been duplicated were removed. Finally, it looked at the titles and abstracts of all the publications to see if they were eligible for the systematic review. A total of 44 studies were included in the systematic review after the publications were located, re-evaluated, and admitted.

2.3. Study Selection

Scopus yielded 89 items, Science Direct 1183, Google Scholar 4, and Web of Science 80 during the initial part of the study. As a result, 1356 records were generated. After an initial search of 1356 records, duplicate papers were identified. A total of 13 duplicate articles were detected, reducing the number of duplicate articles from 1356 to 1343. Phase 2 showed the rejection of 720 records due to non-English content in the following articles: reviews, chapters, books, conference proceedings, editorials, and encyclopedias. As a result, 623 papers met the criteria for a thorough evaluation. A total of 44 full papers remained after adding 2 additional papers that met the established criteria and discarding 581 based on the abstracts and titles. A flowchart displaying the outcomes at each stage is shown in Figure 1.

3. Results

3.1. Features of the Study

A total of 44 papers were found that focused on mathematical models utilized for the localization of P&R facilities. Consequently, the current body of research conducted from 2002 to 2025 was analyzed. The article distribution comprises many mathematical models, with the predominant model being the optimization model (n = 9). Five articles use the equilibrium model, while four articles utilize the Bi-Level programming approach and the simulation model. The traffic congestion and fuzzy logic forecast models are used in the two articles, respectively. The rest of the mathematical models are used only once. The two papers use traffic congestion and fuzzy logic prediction models, respectively. The remaining mathematical models are used just once (See

Table 2. Articles classified by the mathematical model.

Mathematical Model	Number of Publications	Percentage %
Optimization Model	9	20.45
Equilibrium Model	5	11.36
Bi-Level Programming Model	4	9.09
Simulation Model	4	9.09
Traffic Congestion Model	2	4.55
Fuzzy Logic Forecast Model	2	4.55
Multinomial Logistic Regression	1	2.27
Paired Combinatorial Weibit (PCW) Model and Mixed Integer Linear Programming (MILP)	1	2.27
Best Worst Method (BWM)	1	2.27
Fixed-Point Problem Formulation and Logit Model	1	2.27
Simulated Annealing Algorithm	1	2.27
P-Hub Location Model and Logit Model	1	2.27
Weibit-Based Choice Model and Mixed Integer Linear Programming (MILP)	1	2.27
Stochastic User Equilibrium Model	1	2.27
Logit Choice Model	1	2.27
Analytic Hierarchy Process—Best Worst Method	1	2.27
Grey Analytic Hierarchy Process	1	2.27
Haversine Distance Formula	1	2.27
Analytical Method and Traffic Algorithm	1	2.27
Branch-and-Bound Algorithm and Multinomial Logit	1	2.27
Bi-Level Programming Model and Multinomial Logit Model	1	2.27
Mixed-Integer Programming, Lagrangian Relaxation	1	2.27
Decision Support Model	1	2.27
Fuzzy Analytic Hierarchy Process (FAHP)	1	2.27
Total	44	100%

).

3.2. Overview of the Results

The main papers used in this investigation are summarized in Appendix A. The findings of this study are categorized into divisions based on the mathematical model. Similarly, the papers pertaining to the mathematical model and the findings derived from each article are elaborated upon.

The following is a comprehensive analysis of each of the articles that are included in this systematic review in order to provide the mathematical model that was employed.

3.2.1. Decision Support Models

A decision support model is an informational framework that enhances decision-making across several fields of study. These models are interactive information systems that provide multi-criteria assessments and organized decision frameworks, particularly for P&R infrastructure locations. Models such as the Analytic Hierarchical Process (AHP) and its fuzzy variations are prevalent in this domain and are used by several studies [35,36,37,38,39].

3.2.2. Econometric Models

An econometric model is a mathematical or statistical representation of the relationship between variables that allows estimates and predictions. With it, the variables that affect P&R users can be identified, improving and helping to forecast the use of these facilities under different conditions. The study conducted by Islam et al. 2015 [10] determined that variables such as public transport vehicle travel and transfer time primarily affect the location of P&R systems.

3.2.3. Optimization Models

An optimization model is a mathematical depiction of a real-world issue that aims to maximize or minimize an objective subject to constraints. One example of its use is in addressing complex problems related to the location and design of P&R facilities. These models aim to enhance the efficiency of urban transportation and improve traffic flow management. The flexibility of this model is evident in its integration with multiple models, including Bayesian Optimization (BO), branch and bound (B&B), Trust Region Sequential Quadratic Programming (TRSQP), modal splitting and traffic assignment (CMSTA), and other models to determine optimal P&R facility location configurations [40,41,42]. Also, through optimization models, optimization algorithms, stochastic equilibrium models, continuous models, and simulation models, the software (MATSim 12.0) can be used to identify optimal P&R locations in urban networks, multimodal transport networks, linear monocentric cities, or in linear corridors [43,44,45,46,47,48,49,50,51].

3.2.4. Other Models

Within these models, several approaches and innovations are included, such as advanced techniques in fuzzy logic or adaptive random rounding, mixed programming models, or multilevel programming, which help to solve specific problems related to the planning and design of P&R facilities. Furthermore, the complexity of location P&R in urban areas requires the use of advanced computational mathematical approaches. These methodologies go beyond traditional methods, approaching the design and optimization problems from different perspectives [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77].

3.3. Urban Classification

The studies reviewed were classified according to the urban environment in which they were applied, distinguishing between Global North and Global South urban regions. This distinction allows for the observation of the distribution of mathematical models according to the degree of socioeconomic development and urban infrastructure of the area of application. A total of 14 studies were identified in more than 20 different locations (See

Table 3. Grouped urban settings by global context.

Context Classification	Urban Setting	Number of Studies
Global North	Lyon, France	2
	Perth, Australia	2
	Amsterdam, Netherlands	1
	Halle, Germany	1
	Melbourne, Australia	1
	Rome, Italy	1
Global South	Brussels, Belgium	1
	New York City, USA	2
	Chicago, USA	1
	Dallas, USA	1
	Los Angeles, USA	1
	Nashville, USA	1
	Cuenca, Ecuador	8
	Beijing, China	1
	Chengdu, China	1
	Chiang Mai, Thailand	1
	Iquique, Chile	1
	Isfahan, Iran	1
	Suzhou, China	1
Global	Not specified	14

). The majority of these cities are located in the Global North, including Lyon (France), Perth (Australia), Halle (Germany), Amsterdam (The Netherlands), Melbourne (Australia), and New York City, Chicago, Dallas, Los Angeles, Nashville (USA). These cities are distinguished by their consolidated multimodal transportation networks, the availability of high-quality data, and the institutional capacity to implement advanced solutions. In these contexts, the prevalence of optimization models, integer programming, simulation, and other approaches with high computational and technical demands reflects intentional design.

Conversely, studies classified within the Global South were predominantly developed in countries such as China, Ecuador, Iran, Thailand, and Chile, where urban conditions present unique challenges, including limited data availability, informal or expanding transportation systems, and budgetary constraints. In such environments, there is a notable prevalence of decision support models, multi-criteria approaches such as AHP or BWM, and adaptations of classical models to contexts characterized by lower data accuracy.

4. Discussion

This section provides a comprehensive examination of the results from research on the use of mathematical models for the localization of P&R systems. The analysis is based on a comprehensive systematic examination of the relevant literature.

The previous section showed that the examined studies provide a comprehensive overview of the various mathematical models used to place P&R facilities. A total of 44 papers were discovered focusing on the siting of P&R facilities. The systematic study identifies four primary categories: decision support models, econometric models, optimization models, and other models. Among these categories, optimization models are primarily distinguished by those that address complex urban contexts, as they are particularly well-suited to such environments and possess the capacity to simulate large-scale scenarios. However, it should be noted that these advanced analysis capabilities may impose limitations on the computational capacity of these models. The AHP and BWM variants of decision support models are preferred in contexts that require participatory planning and multi-criteria analysis. These models are valuable and useful for local governments and urban planners working with qualitative data or based on expert opinion. The utilization of economic models remains comparatively less prevalent in comparison to alternative models. Nevertheless, these platforms offer insights into user behavior and demand sensitivity, making them valuable resources for understanding and responding to market trends. Finally, hybrid models (other models) demonstrate a promising capacity for dynamic and adaptive network planning, particularly in scenarios characterized by uncertainties such as fluctuating demand or the emergence of new technologies.

Each category examines planning and design complexities from several aspects, yielding precise insights for enhancing urban transport networks. With these viewpoints, stakeholders might stimulate new discussions, future trajectories, and potential domains for research and development. This may assist researchers, practitioners, and decision-makers in urban planning concerning P&R facilities.

The systematic review also revealed no single model for performing or determining the location of P&R systems. Combinations of mathematical models are essential for developing robust and sustainable solutions. A clear example is the mathematical models applied in the optimization models category. This category is mainly known for handling large volumes of data and simulating various scenarios. Adding this to another category, such as decision support models, can offer better P&R location planning results. Also, integrating new advanced technologies such as autonomous vehicles and shared mobility platforms should be a priority for future research to further improve the sustainability and optimization of P&R system facilities in the face of the advent of intelligent transportation systems (ITS).

Practical Implications for P&R Planning

The findings of this review also allow us to derive practical recommendations for policymakers and urban planners. From the comparative analysis of studies applied in different contexts, five key criteria are identified that should be considered in the selection and design of P&R facilities:

• Accessibility of mass transit: Locating facilities near high-frequency corridors such as metro trains or BRT systems significantly improves their potential use.
• Integration with land use: It is essential that P&R stations are aligned with local urban dynamics and trip generation patterns.
• Capacity optimization: The size of facilities should consider the estimated demand, as well as the availability of complementary multimodal connections.
• User behavior and cost sensitivity: Factors such as parking fees, public transport fare structure, and perceived travel times affect system acceptance.
• Availability of ITS technologies and real-time data: The integration of intelligent transportation systems allows for occupancy management, user information, and dynamic operating schemes.

These recommendations can be used as the basis for local or national planning guidelines. Their degree of application may vary according to the context, being more demanding in Global North cities and more adaptive in Global South urban contexts, but in all cases, they are essential elements to improve the effectiveness and sustainability of P&R solutions.

Overall, mathematical models should not only be considered as technical tools, but also as a method of decision-making that is in line with transport infrastructure planning and urban sustainability objectives. Understanding and analyzing the strengths and limitations of each type of applied model allows for customized applications. Future work should focus on combining complementary modeling approaches to address the complexity of urban mobility systems.

5. Conclusions

The location of the P&R system was analyzed since its success is contingent upon this factor. Mathematical models provide a framework of categories to assist researchers and transportation managers in optimal placement. This research concentrated on the many mathematical methodologies used via a systematic review to enhance comprehension of the application and utilization of these categories.

The research facilitated the formulation of a PRISMA protocol to identify the mathematical models mostly used to evaluate the location of P&R systems. This systematic methodology is essential since it offers a coherent and standardized framework for analyzing and comparing diverse categories used in distinct papers. This is particularly crucial in mathematical models, where several features and variables, including accessibility, cost, influence on other transportation modes, journey time, and user demand, must be considered.

A variety of methods used to establish localization were also observed, highlighting the mathematical models established in the category of optimization models due to their flexibility, adaptability, and ability to handle large volumes of data and simulate various scenarios.

The research enhances comprehension of the many mathematical approaches and systems used to localize P&R systems. It creates a robust basis for future research by thoroughly examining current methodologies. Future research may enhance this work by improving and expanding the mathematical models used to discover, investigate, and assess suitable sites for P&R systems. This foundation will enhance site selection tactics and promote the development of novel methods to address the changing difficulties of urban transport planning. The predominant study findings indicate that public transportation is critical in establishing a P&R system. Future research should focus on examining public transit as an integrated system, including the P&R system.