Simulation Study on Freeway Toll Optimization Considering Bounded Rationality and Dynamic Relationships Among Toll Rates, Travel Demand, and Revenue

Abstract

As an essential component of China’s comprehensive transportation network, freeways play an irreplaceable role in promoting regional economic integration, improving logistics efficiency, and serving public travel. However, the development of freeways faces challenges such as the underutilization of road resources, significant financial pressure for construction and maintenance, and imbalanced revenue and expenditure leading to heavy debt burdens, which severely impact the sustainable development of freeways. Optimizing freeway toll rates is an effective measure to alleviate these issues, playing a crucial role in enhancing the operational efficiency of the road network and increasing the revenue of freeway operating enterprises. Existing studies have focused on finding the optimal toll rates for freeways based on bi-level programming models, neglecting the dynamic relationships among individual travel behavior preferences, toll rates, travel demand, and toll revenue. Grounded in bounded rationality theory, the research employs microscopic traffic simulation technology to analyze the dynamic relationships among freeway toll rates, travel demand, and toll revenue. The results confirm that travel demand decreases as toll rates increase, while toll revenue exhibits asymmetric “synchronization” and “asynchronization” phases, peaking at CYN 58.9 thousand (USD 8246) when the toll rate reaches CYN 0.45/km (USD 0.06/km). Additionally, users’ rationality levels significantly affect the stabilization time of toll revenue, and the speed difference between freeways and parallel roads demonstrates a threshold effect on travel demand and revenue. These findings provide theoretical and technical support for optimizing freeway toll strategies, enhancing operational efficiency, and promoting sustainable transportation development.

1. Introduction

Freeways are a crucial component of the transportation infrastructure, serving as rapid channels for point-to-point movement. At the end of 2023, China’s total road network reached 5.44 million kilometers, with 184,000 km of high-speed freeways operational, ranking first in the world [1]. However, alongside the rapid expansion of highway construction and traffic volume, two prominent challenges have emerged. Firstly, there is an imbalance between traffic supply and demand, resulting in congestion on some links while others remain underutilized [2]. According to data from the Ministry of Transport, 11.3% of the national high-speed highway network flow and capacity factor exceeds 0.95, while 47.5% of links have flow and capacity factors below 0.33, making localized congestion during holidays a common occurrence. Secondly, there is a significant financial strain on highway operations, with a current funding gap of approximately CYN 300 billion in China’s road network [3,4].

Leveraging economic levers for traffic demand management is an effective approach to mitigate the aforementioned issues. In the context of highway transportation, the economic lever refers to highway tolls, where the toll rate directly influences user travel costs, thereby adjusting the overall traffic demand and affecting the operational revenue of the highway system. Conducting targeted research on highway toll rate optimization is a crucial means to optimize traffic flow distribution within the road network and promote the efficient utilization of road resources. This approach provides methodologies for implementing differentiated highway toll schemes, traffic guidance within the road network, and variable information dissemination. It holds significant theoretical and practical value for enhancing the operational benefits, efficiency, and service levels of freeways.

This study comprehensively considers multiple factors, including the socio-economic attributes of freeway users, route choice preferences, travel distance distribution, toll rates, and design speeds, to design an agent-based simulation model, simulation parameters, and update mechanisms. By adjusting the input variables, this study analyzes the changes in total system revenue and user proportions under various scenarios. This research aims to investigate the dynamic relationships among toll rates, travel demand, and revenue, which can effectively address issues such as insufficient evaluation data before and after adjustments to freeway toll policies across multiple scenarios, as well as the lack of numerical validation of these relationships. It analyzes the impacts of various factors, including toll rates, travel speeds, and levels of rationality, on freeway travel demand and toll revenue. Additionally, it reveals behavioral pattern differences among freeway users under different conditions, providing data support for the subsequent exploration of constructing and solving optimization models for freeway toll rates. The main contributions of this research are as follows: (1) integrating users’ bounded rationality and perfect rationality to construct a path choice behavior model; (2) utilizing agent-based simulation to update path choice preferences and dynamically analyze the relationships among toll rates, travel demand, and revenue; and (3) providing data support for the construction and solution of optimization models for freeway toll rates, laying the foundation and technical tools for traffic demand inducement, toll rate scheme formulation, optimization, and evaluation.

The remainder of this paper is organized as follows: Section 2 summarizes the literature review; Section 3 describes the methods and the related processing procedures of the research; Section 4 summarizes the simulation results and the policy implications of the experiments; Section 5 describes a conclusion about the findings; followed by the discussion, which summarizes the limitations, the theoretical and practical implications, and the possible extensions of our work in Section 6.

2. Literature Review

The mathematical foundation for researching highway toll rate optimization schemes is the traffic network equilibrium model, which is typically formulated as a bi-level programming model. Generally, the upper level aims to maximize the benefits of the highway network or toll revenue, while the lower level aims to minimize the travel time costs for highway users. For instance, Xie et al. [5] and Cheng et al. [6] established a bi-level programming model with the upper level maximizing highway revenue and the lower level minimizing user travel impedance, based on the perspectives of highway managers and travelers. Li et al. [7] proposed a bi-level programming model to solve optimal tolls based on the equilibrium of heterogeneous household residential location choices, with the upper-level objective being the minimization of the total city system travel time and the lower-level planning problem combining residential location choice and traffic distribution using fixed-point theory. Li et al. [8] proposed optimizing dynamic credit strategies and constructed a bi-level programming model based on the Stackelberg game between road managers and travelers, adjusting the path choice behavior of users with high travel times to achieve traffic network flow allocation and carbon reduction targets. Wang et al. [9] combined parking fees with congestion pricing to construct an elastic demand user equilibrium model under dual constraints, designing a pattern search algorithm for the solution. The results indicated that combined pricing promotes a more balanced distribution of road network traffic, alleviates congestion on tolled links, and suppresses some travel demand. Lots of research [10,11,12,13,14,15] has established bi-level programming models to solve road pricing schemes. In recent years, with the development of theoretical and practical aspects of differentiated highway tolls, various studies have explored differentiated pricing methods based on time links, vehicle types, and road links [16]. Additionally, similar to the bi-level programming modeling approach, there is the multi-objective optimization pricing method. The main difference is that the former separates objectives into different levels, while the latter sets the objectives of different stakeholders at the same level, optimizing with different weights. For example, Li [17] conducted real-vehicle tests to study the driving conditions and fuel consumption characteristics of six types of trucks and established a carbon emission calculation model based on driving speed for different road types. The study also established a multi-objective optimization model for carbon reduction benefits, operational benefits, and travel impedance, determining the optimal toll scheme.

The pricing method based on bi-level programming models provide theoretical and empirical support for highway toll rate optimization schemes. However, two main issues persist. On one hand, the lower-level model in bi-level programming describes user path choice behavior but remains based on generalized travel cost, ignoring the influence of individual behavioral preference. On the other hand, bi-level programming models struggle to capture the dynamic interrelationships among highway toll rate, travel demand, and toll revenue.

Highway path choice behavior is a key variable linking toll rate, demand volume, and toll revenue, and it is a crucial aspect of exploring their relationships. The existing research on path choice behavior is predominantly based on random utility theory, assuming that travelers are perfectly rational, have access to all travel information, and make decisions according to the principle of utility maximization [18,19,20]. For instance, Erhardt et al. [21] utilized household travel survey data from Texas to establish a nested multinomial Logit choice model, investigating the influencing factors on the use of toll roads and High-Occupancy Vehicle lanes. Lu [22] established a nested Logit-cumulative prospect theory model for travel time period selection, testing how users adjust their travel times in response to toll rate changes and evaluating the distribution of travel demand under different schemes. Wang et al. [23] proposed that driver path choice behavior is composed of multiple dimensions and devised the following four scenarios: transferring costs to the recipient, reducing delivery frequency, shortening the driving distance on toll roads, and changing travel times. They established a multivariable Probit model to explore the factors influencing driver path choice behavior. Kong et al. [24] used survey data to segment truck travelers and built a model for travel path and time period selection, analyzing the impact of different factors on these choices. Tsirimpa et al. [25] established a discrete choice model to study the influence of tolls, dedicated lanes, and travel time on route selection, finding that tolls have a negative impact on the use of toll roads, as truck drivers may choose non-toll roads to avoid costs.

However, the ‘fully rational person’ assumption is an ideal state that differs from reality, as actual travel behavior is influenced by preferences, psychology, personality, and environmental factors, leading to the development of alternative behavioral theories such as bounded rationality [26,27], prospect theory [28], cumulative prospect theory [29], and random regret theory [30,31,32]. Bounded rationality, introduced into traffic science in the 1980s, has gained attention for its more realistic modeling of travel behavior. Incorporating preferences can effectively explain decision-making behavior under uncertain conditions. Xu et al. [28] integrated multiple behavioral mechanisms, including the shortest path, bounded rationality, asymmetric preferences, and time maximization, into a path choice model. They assumed that travelers compare travel costs with the current path’s travel costs before deciding whether to change paths, demonstrating the influence of adaptive time value and path choice inertia on path decision-making behavior. González et al. [33] surveyed 496 students from the University of Lyon, France, using a mobility decision game (MDG) experiment solved with a mixed Logit model. The study revealed that path choice behavior deviates from fully rational conclusions, as not all travelers choose the shortest path, supporting the bounded rationality assumption. Previous studies [34,35,36,37] incorporated travel preference characteristics into their path choice models to more comprehensively depict micro-individual travel behavior.

The existing research has shown that, influenced by bounded rationality, users do not necessarily choose the shortest path for travel. However, due to the complexity of models and preferences, current studies typically set preference influences as fixed parameter values, which deviate from real-world situations. Moreover, the mathematical analytical models used to assess the impact of toll rates on travel behavior often rely on survey data, where the reliability of the results depends on the scale of the data and the quality of the surveys. These models cannot address the dynamic relationships among toll rates, traffic volume, and revenue, making it difficult to achieve precise analysis of large-scale complex traffic systems. Simulation models, utilizing computer simulation technology to mimic traveler behavior, offer new opportunities for precise analysis of large-scale user behavior, toll rate optimization, and traffic demand management [38,39]. For instance, Ihab et al. [40] employed an agent-based simulation framework oriented towards real-world networks to study optimal congestion pricing strategies based on complex user travel behavior. He et al. [41] used an open-source multi-agent simulation platform to evaluate congestion pricing strategies for different population groups in New York over a day. Gu et al. [42] focused on the discrete decision-making problem of heterogeneous individuals in large-scale traffic simulations and proposed a Two-Stage Ensemble Reinforcement Learning method, effectively solving the discrete decision-making problem of traveling individuals in large-scale traffic simulations. Multi-agent simulation models can establish interactions with the environment without prior knowledge by adjusting simulation parameters to control intelligent agents, studying individual behavioral decisions and collective characteristics. This not only overcomes the limitations of small-scale data, but also achieves dynamic model updates, efficiently considering the long-term effects of control and providing irreplaceable research theories and methods for individual travel behavior decision making [43].

Given the background, this paper interactively considers users’ bounded rationality and perfect rationality, incorporating them as constraints to construct a path choice behavior model by utilizing agent-based simulation to update path choice preferences, which provides a foundation and technical tools for traffic demand inducement, toll rate scheme formulation, optimization, and evaluation.

3. Methodology

Microscopic traffic simulation is an important research method used to help understand the formation mechanism of traffic flow distribution and congestion by simulating users’ behaviors of route selection on freeways and to provide a basis for the establishment of traffic demand management strategies. As a major part of microsimulation, agent-based simulation enables the establishment of behavioral decision-making models for the interaction between the agents and the environment and controls the complex behavior of agents by adjusting the simulation parameters to study the behavioral decision-making mechanism and the set of behavioral characteristics of individuals under different environments and strategies. NetLogo6.4.0 provides an intuitive interface and a simple programming language to support the modeling of multi-agents, which can easily simulate complex interactions between agents, making it easy to create agents’ simulation models, and its powerful visualization function can help users to intuitively observe the behavior of agents and the interaction with control parameters. In addition, NetLogo6.4.0 has a large user community and a rich resource library for rapid modeling to accelerate the research process.

Accordingly, combining the deficiencies of existing research, this study builds a simulation model of freeway toll rate, travel demand, and toll revenue based on Netlogo6.4.0, an open-source simulation platform for agents. Leveraging one month (June 2023) of toll collection data from the Guangxi Zhuang Autonomous Region’s freeway network—comprising 13.8 million toll records, which analyze key indicators such as vehicle travel frequency, average travel distance, travel duration, travel speed, and the most frequently used entry and exit points as individual travel behavior characteristics—this method analyzes the correlation between the three through simulation. The specific simulation technical route is shown in Figure 1.

The simulation model architecture is as follows:

Step 1: Set basic parameters of the simulation system, as follows:

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Freeway toll rate (charge rate, CR);

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Time value (TV);

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Rational level of the population (RLP);

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Preference reward decay (PRD);

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Design speeds of the freeway and parallel national/provincial roads (speed of freeway, SF; speed of other roads, SOR).

Step 2: Define the population information, as follows:

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Personal social-economic features (PSEF, as determined by user travel behavior survey questionnaire);

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Personal travel distance distribution (PTDD, as determined through freeway toll records);

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Travel route preference (TRP, as determined by user travel behavior survey questionnaire).

Step 3: Select rational routes for each individual, as follows

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Determine the travel distance (TD) for the current trip based on PTDD;

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Calculate the travel costs for the freeway and parallel national/provincial roads (cost for freeway, CF; cost for other roads, COR) based on SF, SOR, TV, and CR;

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Compare CF and COR to determine the route in ration (RIR).

Step 4: Select travel routes for each individual, as follows:

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Determine the preferred route (route in preference, RIP) based on TRP;

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Determine the final selected route (route in actual, RIA) based on RIP, RIR, and RLP.

Step 5: Aggregate freeway travel demand and revenue and update individual attributes, as follows:

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Calculate the number and proportion of travelers, travel mileage, and total revenue of the freeway system based on RIA, TD, and CR for both the freeway and the parallel national/provincial roads.

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Update TRP based on RIP, RIA, and PRD.

Step 6: Repeat steps 3 to 5.

First, initialize the fundamental attributes of the agents, including income (Inc), travel distance (Dis), and route preference (P). In particular, according to the income of Guangxi, the income is divided into five levels of CYN 5000, CYN 10,000, CYN 15,000, CYN 20,000, and CYN 25,000, and, once determined, it does not change with the number of simulation iterations. For the travel distance attribute, considering that the individual travel distance varies from time to time, based on the analysis of the average travel distance metric derived from one month of freeway toll collection data in Guangxi, in which the largest travel distance is 500 km, a random number within 500 km is defined as the maximum travel distance of the individual, and the travel distance in each simulation iteration is a uniformly distributed random number within the range of the maximum travel distance. The initial route preference attribute is a uniformly distributed random number between 0 and 1, which represents the probability of individuals choosing to travel via the freeway or parallel national/provincial roads, with the larger values indicating that individuals tend to use freeways for travel. The initial path preference attribute does not change with the number of simulation iterations.

Then, depending on the simulation demand in each round, the degree of rationality of the agents, the speed of the freeway, and the speed of the national and provincial arterial roads are set by the control. In this study, the degree of rationality is set in the interval of 0–1, with 0 indicating that individuals are completely irrational, and the travel route selection only considers the route preference; and 1 indicating that individuals are completely rational, and the travel route selection only considers the travel cost. The freeway speed has been set between 70 km/h and 120 km/h and the parallel road speed has been set between 50 and 80 km/h.

The most critical part is to execute the route selection behavior of the agent to determine whether it chooses to take the freeway or the parallel road. The route selection behavior is divided into two parts, with the first based on the distance of each trip of the agents, where we estimate the cost required to take the freeway or the parallel road, respectively. The cost formulas for the two routes are shown in Equations (1) and (2).

$$C_F=Dis\times Rate+{\displaystyle \frac{Dis}{V_F}}\times {\displaystyle \frac{\gamma Inc}{21\times 8}}$$

(1)

$$C_P={\displaystyle \frac{Dis}{V_P}}\times {\displaystyle \frac{\gamma Inc}{21\times 8}}$$

(2)

where C_F and V_F denote the cost and speed of the freeway and C_p and V_p denote the cost and speed of the parallel road. Rate represents the toll rate of the freeway, which we set between CYN 0 and 1 per kilometer. Dis is the travel distance and Inc refers to the monthly income. γ is the modifying factor, as the time spent on the journey might be more “expensive” than the average value denoted by ${\displaystyle \frac{Inc}{21\times 8}}$, that is, the ratio of monthly income to monthly working hours.

After computing the travel costs of the two roads, the route with the lower travel cost is viewed as the rationally chosen road (RR), which is 1 if the lower-cost road is a freeway and 0 if the lower-cost road is a parallel road. Then, we find the following:

$$R=Rationality\times RR+(1-Rationality)\times RP$$

(3)

where Rationality is the degree of rationality, RP is the road chosen in preference, and the value equals the route preference. The routes chosen for the two scenarios may or may not match each other. An individual with a degree of rationality of 1 has a route choice outcome in the formula that depends entirely on RR. Whereas an individual with a degree of rationality of 0 has a route choice outcome in the formula that depends entirely on RP, i.e., route preference (P). Thus, when the variable R has a value of less than 0.5, the route selection turns out to be a parallel road, and, when the value of R is greater than 0.5, the route selection turns out to be a freeway.

The final step is to update the individual’s route preferences, as the outcomes that result from each actual behavior enhance or attenuate this preference. To achieve the goal, we define the reward (Rew_t) for agents at the current moment t. This is calculated as follows:

$$Re{w}_{\mathrm{t}}=\chi Re{w}_{\mathrm{t}-1}+{\displaystyle \frac{abs(c_1-c_2)}{\max (C)}},Re{w}_0=0$$

(4)

where $\chi$ is the reward decay, based on the agents travel behavior, which we set as 0.1, and the $Re{w}_{\mathrm{t}-1}$ is the previous reward. c₁ and c₂ are the cost of the actual road chosen and the cost of the unchosen road, respectively, and C is the current cost of all individual trips. The individual’s route preference attribute at moment t is then updated to the following:

$$P_{\mathrm{t}}=\left\{\begin{array}{l}{P}_{\mathrm{t}-1}+\beta \times Re{w}_{\mathrm{t}},\mathrm{actual}\mathrm{route}\mathrm{choice}\mathrm{is}\mathrm{consistent}\mathrm{with}\mathrm{preferred}\mathrm{route}\mathrm{choice}\\ P_{\mathrm{t}-1}-\beta \times Re{w}_{\mathrm{t}},\mathrm{actual}\mathrm{route}\mathrm{choice}\mathrm{is}\mathrm{consistent}\mathrm{with}\mathrm{preferred}\mathrm{route}\mathrm{choice}\end{array}\right\},0\le P_{\mathrm{t}}\le 1$$

(5)

where β is the reward preference correction coefficient, with reference to other literature and the context of Guangxi, taken as 0.1.

4. Results and Discussion

As Figure 2 and Figure 3 shows, freeway travel demand decreases as toll rates increase, and tolls show “synchronous phases” and “asynchronous phases” as toll rates increase, which is basically consistent with the findings of existing theoretical studies. In the lower toll rate “synchronous phase,” the increase in toll rates reduces travel demand to a certain extent, but the impact of the increase in toll rates outweighs the impact of the decrease in travel demand, so the overall tolls show an upward trend. In the higher toll rate “asynchronous phase,” further toll rate increases also reduce travel demand, but the impact of increased toll rates is weaker than the impact of decreased travel demand, resulting in a decrease in overall tolls. In the results from the simulation, the experimental scenario when the toll rate of CYN 0.45 per kilometer is used, the average freeway toll taken to reach the maximum value is CYN 58.9 thousand. Obviously, in theory, there exists a toll rate interval that makes the freeway tolls the highest. In addition, different toll rates can lead to different travel demand; however, there could be situations with the same freeway toll, such as when the toll rate is CYN 0.3 per kilometer and CYN 0.6 per kilometer, respectively. Here, the freeway user share is 55.94% and 28.47%, respectivley, but the corresponding freeway tolls are very similar to each other, at CYN 53.7 thousand and 53.6 thousand, respectively.

Notably, the “synchronous phase” and “asynchronous phase” of changes in freeway tolls in response to toll rate changes are not symmetrical, with the “synchronous phase” being relatively more dramatic and the “asynchronous phase” being relatively smoother.

The rationality level varies in the interval of 0–1 while the tolls remain basically unchanged, as shown in Figure 4, which shows that the rationality level does not have a significant direct effect on the tolls. The main reason for this lies in the fact that the tolls in this study are defined as the average of the freeway charging in the last 600 s of the simulation to ensure that the value is stabilized. However, as shown in Figure 5, the degree of rationality affects the velocity at which tolls gradually reach a steady state from an initial state, i.e., the higher the degree of rationality, the less time it takes to reach a steady state. At the toll rate of CYN 0.45/km, the speed of the freeway is 120 km/h, and the speed of the parallel national and provincial arterial roads is 80 km/h; moreover, when the degree of rationality is 0.25, it takes 100 iterations of simulation for the toll to reach the steady state from the initial state, while, when the degree of rationality is 0.5, it takes only 10 iterations or less to reach the steady state. As a result, the degree of rationality does not affect the value of tolls in the steady state, but instead affects the time required for tolls to reach the steady state.

This also brings to light the practice that the evaluation of the effects of freeway rate adjustment programs needs to be carried out after a specific time period, which depends on the rationality of local users’ travel to a certain extent. In the “synchronous phase,” raising toll rates will lead to an increase in tolls, and, if an evaluation of the effectiveness of the rate scheme is carried out before a steady state is reached, it will result in an effectiveness evaluation that is lower than the actual benefits. In the “asynchronous phase,” raising toll rates will lead to a decrease in tolls, and, if an evaluation is carried out before a steady state is reached, an effectiveness evaluation that is higher than the actual benefits will result in a greater loss for the operation of the enterprise. Therefore, before determining the rate adjustment program or evaluating the effectiveness of the rate adjustment, it is necessary to adequately ascertain the rationality level of local freeway users and to pre-estimate the appropriate timing for evaluating the effectiveness of the implementation of the rate program.

Theoretically, due to the competitive relationship between freeways and parallel national and provincial arterial roads, given all other conditions alike and the higher speeds of freeways and the lower speeds of parallel national and provincial arterial roads, the number of users choosing to travel on the freeway will increase, and, thus, freeway tolls generally increase. As shown in Figure 6, Figure 7, Figure 8 and Figure 9, The simulation results are consistent with the above basic understanding. Remarkably, increasing freeway speeds beyond 110 km/h will have an attenuating effect on the freeway user share and the total tolls for freeway trips. Similarly, when we increase the speed of parallel state and provincial arterials over 70 km/h, the depressing effect on the freeway user share and total freeway tolls decays. From this, in conditions where the toll rate and other conditions remain unchanged, the freeway tolls can be increased through speed management and other means (such as limiting low speeds) to reduce the cost of freeway travel. For heavily congested road sections, we can increase the parallel national and provincial arterial roads’ speed to absorb part of the highway travel demand, thus achieving the goal of congestion relief.

Moreover, the difference in speed between the freeway and the parallel national and provincial arterial roads is also correlated with the freeway travel demand and tolls. Theoretically, under the same conditions, the larger the speed difference between the freeway and the parallel road, the more attractive the freeway is, and its travel demand and tolls will be larger. The simulations shown in Figure 10 and Figure 11 verify this theoretical result. Furthermore, we find that when the speed difference between the freeway and the parallel road is less than 5 km/h, the growth of the speed difference will not have a positive contribution to the freeway toll collection. This shows that the parallel national and provincial arterial roads still have an advantage in attracting passenger traffic. In practice, it can be understood that, although freeway speeds are slightly higher than those of parallel roads, the accessibility of the freeway is relatively low due to its full closure nature, and the cost of getting to the freeway and entering and exiting the toll booths overrides the speed advantage of the freeway. Hence, in addition to improving the speed advantage of freeways, improving the accessibility of freeways can somehow increase the demand for freeway travel, which in turn can increase the toll revenues of freeway operators.

5. Conclusions

Traditional toll rate optimization models typically assume perfect rationality in travel behavior, often neglecting users’ subjective preferences, and lack real-world toll data in multi-rate scenarios. Relying on the Netlogo6.4.0 platform, this study constructs an agent-based simulation model, innovatively incorporating factors such as individual attributes, route preferences, travel distance, toll rates, and operating speeds as inputs. By setting up a more realistic simulation environment, the research dynamically analyzes the relationships among toll rates, operating speeds, user rationality levels, toll revenue, and freeway travel demand under various simulation scenarios. This study identifies a theoretically optimal toll rate and provides a new theoretical framework and technical tools for freeway toll rate optimization.

This research reveals that freeway travel demand declines as toll rates increase, and toll collection exhibits a “synchronous phase” and an “asynchronous phase” as toll rates increase, so we conclude that there is a theoretical toll rate range (in this study, the toll rate is CYN 0.45 per kilometer) that maximizes freeway toll collection. The effectiveness of the proposed model in accurately identifying toll rate adjustment schemes or evaluating the effectiveness of toll rate adjustments is demonstrated through a case study in Guangxi. Currently, the toll rate for Type I passenger vehicles on Guangxi’s freeways is CYN 0.5 per kilometer. When a 10% discount is applied to the toll rate for this vehicle type, reducing it to CYN 0.45 per kilometer, the system revenue reaches its maximum value. We further observed that the “synchronous phase” and “asynchronous phase” are not symmetrical, with the “synchronous phase” having relatively more dramatic changes and the “asynchronous phase” having relatively more dramatic changes. The rationality level does not have a direct impact on toll collection, but it affects the time needed for toll collection to reach a steady state after conducting a toll rate scheme. Therefore, prior to determining the toll rate adjustment scheme or evaluating the effectiveness of the toll rate adjustment, it is necessary to fully understand the rationality level of the local freeway users and to estimate the appropriate time to evaluate the effectiveness of the implementation of the rate plan in advance. In addition, due to the competitive relationship between freeways and parallel state and provincial arterials, the higher the freeway speed and the lower the parallel road speed, the higher the freeway travel demand and toll collection. However, after the freeway speed increases to 110 km/h, continuing to raise the speed of the freeway, the promotion effect on the freeway user share and the total toll of the freeway is attenuated. Similarly, after parallel road speeds grow to 70 km/h, continuing to increase their speeds attenuates the attraction of the users traveling on the freeway, constraining the effectiveness of adopting parallel roadways to shed freeway travel demand. Furthermore, the speed difference between the freeway and the parallel road also has a certain correlation with the freeway travel demand and toll collection. When the speed difference between the freeway and the parallel road is less than 5 km/h, the widening of the speed difference will not positively promote the freeway toll collection. The results could help the government to formulate more effective transportation policies to guide rational transportation planning, improve the efficiency and sustainability of the transportation system, and balance the traffic demand and toll revenues to ensure the stability of freeway tolls while satisfying the freeway travel demand.

6. Discussion

This study contributes to the field by integrating bounded rationality into the analysis of freeway toll rate optimization, addressing a gap in traditional models that rely solely on utility maximization. By incorporating user preferences and perceptual demand, the proposed agent-based simulation model provides a more realistic representation of route selection behavior, offering a novel approach to understanding the dynamic relationships among toll rates, travel demand, and revenue. Compared to the existing literature, which often assumes perfect rationality or relies on static models, this study’s dynamic and behaviorally grounded framework enhances the accuracy and applicability of toll rate optimization strategies.

The findings have significant theoretical and practical implications. Theoretically, this study advances the understanding of how bounded rationality influences travel behavior and toll revenue, providing a foundation for future research on complex decision-making processes in transportation systems. Practically, the model offers actionable insights for policymakers and freeway operators to design toll rate schemes that maximize revenue while maintaining travel demand. For instance, the identification of a theoretical toll rate range and the analysis of speed differentials between freeways and parallel roads can inform more effective traffic management strategies.

However, this study has several limitations. First, the agent-based simulation model is relatively simplistic. In the actual scenario, route selection behavior is not only affected by the travel cost and preference, such as traffic conditions, but weather, travel purpose, and other factors will also affect the results of route selection. Moreover, the traffic condition also affects the travel cost, and the complex relationship of agents’ decision making is not well reflected, which leads to the discrepancy between the results of the model analysis and the actual situation. Due to the sensitivity of travel demand and toll collection data of the freeway system, this study lacks side-by-side comparison data and historical cumulative data, therefore, the model results are not verified by empirical data, which makes the model not that convincing. Therefore, a natural next step is to further improve the agents’ decision-making model to make it more consistent with the actual decision-making process. At the same time, the simulation model should be verified and improved by carrying out cooperative programs with freeway enterprises to accumulate the changes in traffic demand and toll collection before and after the adjustment of rate schemes in different regions, making the analysis results more reliable.

In conclusion, this study provides valuable insights into freeway toll rate optimization by incorporating bounded rationality and dynamic relationships among toll rates, travel demand, and revenue. While the findings are promising, further refinement and empirical validation are necessary to fully realize the model’s potential in guiding transportation policy and improving system efficiency.