Day-to-Day Traffic Assignment Model considering Information Fusion and Dynamic Route Adjustment Ratio

Li, Manman; Lu, Jian; Sun, Jiahui

doi:https://doi.org/10.1155/2020/5751349

Discrete Dynamics in Nature and Society

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Abstract Introduction Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5751349 | https://doi.org/10.1155/2020/5751349

Day-to-Day Traffic Assignment Model considering Information Fusion and Dynamic Route Adjustment Ratio

Manman Li,^1,2,3Jian Lu ,^1,2,3and Jiahui Sun⁴

Academic Editor: Adrian Petrusel

Received17 Feb 2020

Revised19 Apr 2020

Accepted04 May 2020

Published30 Jun 2020

Abstract

A new day-to-day traffic assignment model is proposed to describe travelers’ day-to-day behavioral changes with advanced traffic information system. In the model, travelers’ perception is updated by a double exponential-smoothing learning process combining experience and traffic information that is explicitly modelled. Route adjustment ratio is dynamically determined by the difference between perceived and expected utilities. Through theoretical analyses, we investigate the existence of its fixed point and the influence factors of uniqueness of the fixed point. An iterative-based algorithm that can solve the fixed point is also given. Numerical experiments are then conducted to investigate effects of several main parameters on its convergence, which provides insights for traffic management. In addition, we compare the system efficiencies under the static route adjustment ratio and dynamic route adjustment ratio and show the application of the model.

1. Introduction

Equilibrium assignment has been providing theoretical basis for the transport planners and operators during the past few decades. While useful, it ignores traffic adjustment process. To analyze it, a stream of research about traffic dynamics has been developed. A variety of day-to-day traffic assignment (DTD) models have been proposed including deterministic process (DP) models, derived from the discrete-time nonlinear dynamic systems theory, introduced for transportation by [1], and stochastic process (SP) models considering random fluctuation of demand and (or) supply, introduced for transportation systems by [2]. Some DP models are established on the assumption that travelers’ perception is consistent with the reality and converge to user equilibrium [3, 4] while others incorporate the randomness of perception and converge to stochastic user equilibrium [1, 2]. Cantarella and Cascetta [5] have unified DP and SP models. Apart from these models based on route flow variables, link-based DTD models recently are developed to deal with route overlapping and numeration problem [6].

Nowadays, with the development of information and communication technologies and the popularity of navigation services, it is easy for travelers to access various traffic information. The information might affect travelers’ behavior significantly, which arises the interest to study traffic dynamics with traffic information [7–16]. Some of them focus on traffic dynamics with the traffic information obtained from social network [10–12, 15, 16], while others focus on traffic dynamics with the publicly available information provided by advanced traveler information system (ATIS) [7–9, 13, 14]. We focus on the latter.

In the majority of DTD models that aim to capture traffic dynamics with traffic information provided by ATIS, travelers are divided into two categories based on their dependent information resources. Information dependent travelers choose the route according completely to the information, while experience dependent travelers do it according entirely to the perceived cost learned from experience [7–9, 14]. However, Bogers has evidenced that travelers comprehensively use traffic information and experience in the route choice scenario [17].

In addition, these studies generally assume that the proportion of travelers to adjust route every day is fixed [7–9]. Although the fixed value has been changed from 1 to less than 1 to capture the inertia [18], this way is too simple to demonstrate that inertia is controlled by the difference between perceived utilities of alternatives, which is found by Mahmassani and Liu using a set of laboratory experiments [19] and by Wang et al. using stated preference survey data [20].

This study simultaneously deals with the above two problems to establish a more realistic day-to-day traffic assignment model with ATIS and reinvestigates traffic dynamics. In our model, travelers learn from experience to form their initial perception and the perception is then modified by the traffic information provided by ATIS, for example, Liu et al. [13]. However, different from them, we use the single day information to adjust travelers’ perceived route cost rather than the difference between information in consecutive two days because it is nearly impossible for travelers to clearly recall traffic information in the previous day. Besides, route adjustment ratio is dynamically determined by the difference between perceived and expected utilities. We also explicitly model traffic information, especially, the predictability of traffic information, for example, Liu et al. [13].

The rest of the paper is organized as follows. Section 2 describes our day-to-day traffic assignment model. Section 3 theoretically analyses the existence of its fixed point and influence factors of uniqueness and stability of the fixed point. An iterative algorithm to solve the fixed point is also given. Section 4 conducts numerical experiments to specifically illustrate how several main parameters affect convergence of the proposed model. The system efficiencies under static route adjustment ratio and dynamic route adjustment ratio are also compared. Furthermore, the proposed model is applied to examine traffic dynamics when there are changes on link capacities. Section 5 concludes the paper and gives some future directions.

2. Day-to-Day Dynamic Traffic Assignment Model

2.1. Travelers’ Perceived Travel Time

Experienced travel time and descriptive traffic information provided by traffic agency both affect travelers’ perception for route cost. When travelers finish a travel, they can obtain route cost and thus update their perception. The process can be built up through an exponential smooth approach. When the pre-trip descriptive information is provided by ATIS, travelers will further modify their perception, which can also be described through an exponential smooth approach. Hence, a double exponential smoothing approach is constructed to describe travelers’ perception updating as follows:where is learning rate and describes the extent that travelers put an emphasis on experience and is information fusion rate; the rest of notations are listed in Appendix A.

In the case with , equation (1) deteriorates into single exponential smoothing approach and travelers just learn from experience as typically done in the literature.

2.2. Predictive Information

Traffic information is mainly represented in two ways in the DTD model. The first one views traffic information to be fully accurate and without an explicit model. However, due to the uncertainty of traffic supply and demand, there is no fully accurate information. Therefore, the second way aims at modelling imperfect traffic information. Generally, it directly adopts free-flow travel time or the updated value based on historical data. These methods ignore travelers’ behavior and traffic information generated lacks predictability. Inspired by Liu et al. [13], we generate predictive traffic information based on travelers’ behavior.

Travelers’ perceived travel time is firstly predicted through an exponential smooth learning process as travelers do, but a different learning parameter is introduced, , because traffic agency does not exactly grasp the extent of travelers valuing single day’s experience in long perception:

Traffic agency then distributes travelers over traffic network based on perceived route cost. Specifically, a random utility model is adopted here. Through the process, new route travel time is obtained. It is predictive traffic information and provided to travelers. The whole process can be described as follows:where reflects perception variation. A higher -value means a smaller perception variation for travelers.

2.3. Route Adjustment Ratio

For each trip, travelers have an expected utility. If the perceived utility of previously chosen route is higher than their expectation, they will continue to travel on the route due to inertia; otherwise, it is possible for them to adjust route. The adjustment ratio has a positive relationship with the difference between perceived utility and expected utility. Based on the work of the literature [21], the adjustment ratio of travelers previously choosing route is expressed as follows:where is a constant that represents the maximal adjustment ratio; is a parameter to adjust the sensitivity of route adjustment behavior for utility difference; is similar with and reflects perception variation; and is expected utility on day through OD pair and expressed as follows:

Because travelers differently rely on traffic information, stress single-experience, and so on, their perceived utilities for traveling in a route are not the same. Assuming random residuals of perceived utilities for each route comply the same Gumbel distribution and the random residuals for different routes are independent, equation (6) is derived from equation (7).

2.4. Link Traffic Flow

Route adjustment travelers will rechoose route based on their perceived utilities that negatively depend on perceived travel times. Similarly, there are random residuals for perceived utilities. If random residuals of perceived utilities are the same independent Gumbel distributions, route adjustment travelers will choose route in a logit-based formula:

The rest will continue to travel on previous routes. Therefore, the total chosen probability of a route is

The corresponding route flow and link flow pattern are

3. Model properties and Solution Algorithm

3.1. Model properties

3.1.1. Existence and State of Fixed Point

Suppose a fixed point is formed at day , we have , , , and for . Furthermore, we can infer and .

Then, the fixed-point of our model can be written as follows:where , ,

The feasible link flow set of our model is closed, bounded, convex, and nonempty [22]. According to Brouwer’s fixed-point theorem [23], if the self-map about link flow vector is continuous, there is at least a fixed point.

Assuming that is continuous with , is a continuous function with . Furthermore, is continuous with . With the route choice function and route adjustment ratio function, it can be proved that route choice vector is continuous with perceived travel time vector . Therefore, the self-map of are continuous and there is at least a fixed point.

In addition, we can find that the parameters , , , and affect the state of the fixed point. However, does not affect the state of fixed point because it can be eliminated in the . As well, does not affect the state of fixed point.

3.1.2. Uniqueness of Fixed Point

relies on travelers’ perceived travel times which are determined by traffic information and experience. Traffic information depends on perceived travel times predicted by traffic agency. Therefore, we can rewrite our model in an abstract way: , where covers all relations formulated in Section 2. Given is continuously differentiable, we can infer that is continuously differentiable. Let denotes the Jacobian matrix of the function . If , is invertible according to Hadamard’s global inverse theorem [21]. Because is invertible, it also is injective and is invertible and injective. is exclusive.

Specifically,where

Therefore, we can infer that if link performance function is a constant without traffic flow on it, , and the fixed point of our model is uniqueness; otherwise, , , , , and all affect the uniqueness of the fixed point except .

3.1.3. Stability of Fixed Point

Our model will converge to a fixed point from its attractor region if eigenvalues of the matrix (Appendix B) are within the unit circle [24]. Hence, we can know that , , , , , and all affect the stability of a fixed point. In Section 3.2, we will conduct sensitive analyses to specifically investigate how these parameters affect the convergence of the model as to provide insights for traffic management.

3.2. Solution Algorithm

The fixed point of our model is identical to the fixed point of the dynamic system established in the following [25]:where parameter is iterative step length; the superscript represents the iterative time.

Based on it, the fixed point of our model can be numerically computed by executing an iteration-based algorithm [26].

Step 1. Initialization: assign a value for the iterative step length . Set .

Step 2. Choose a feasible initial vector of link flow , where represents the feasible link flow set which satisfies both the nonnegative constraint of link flow and total demand constraint.

Step 3. Renew the link flow vector using the following iterative formula: .

Step 4. Check the iterative convergence:. represents the convergence criterion. If the convergence condition is met, and stop; otherwise, set and go to Step 2.
According to the work of Cascetta and Cantarella [5], the algorithm can converge to a fixed point when is small enough.

4. Numerical Experiments and Analyses

4.1. Small Network

We firstly use a small network with an OD pair to show the effects of and on the convergence of the model. The traffic demand of the OD pair is 500 (pcu·h⁻¹). As shown in Figure 1, the small network has two links, whose free-flow travel time and the capacity are, respectively, (2 min, 300 pcu·h⁻¹) and (2 min, 400 pcu·h⁻¹). Bureau of public roads (BPR) link performance function is used to compute the link travel time as do most of relevant literature studies [7, 8, 21, 25, 27]. The function is established based on many field surveys and thus reflects the real relationship between link travel time and link flow:where and are parameters with typical value and .

The specific numerical settings are given in Table 1.

When the initial route choice is set as (0.4, 0.6), the fixed point we can get under the above parameters is (0.225, 0.775). Based on it, we can calculate the eigenvalues of under different -values and -values, which are displayed in Tables 2 and 3, respectively.

We can know from Table 2 that the system can converge to an equilibrium when is 0.4, 0.6, or 0.8 (all eigenvalues are smaller than 1). Figure 2 displays the same result through evolutionary processes of link 1 chosen probabilities under different values. These results indicate that if traffic agency adopts an underestimated learning rate to predict travelers’ perceived cost, the system is still able to converge to the equilibrium, but the error should be smaller than a certain value. As well, Figure 2 shows a larger can make the system more quickly converge to the equilibrium.

Table 3 shows that as increases, the system starts not to converge to an equilibrium, suggesting that the route adjustment tendency has a negative effect on the stability of the system. We also depict that evolutionary processes of link 1 chosen probabilities under different values in Figure 3. It shows the same result. Furthermore, it shows has a positive relationship with the convergence speed.

4.2. Nguyen-Dupuis Network

We then conduct numerical experiments on Nguyen-Dupuis network [28] displayed in Figure 4 to further illustrate the other nature of the proposed DTD model and its application. Nguyen-Dupuis network consists of 13 nodes, 19 links, 4 OD pairs, and 25 routes. The characteristics of 19 links are displayed in Table 4. The traffic demand pattern between the four OD pairs is assumed to be (pcu·h⁻¹) [27]. All alternative routes connecting these OD pairs are shown in Table 5. Likewise, we adopt Bureau of Public Roads’ (BPR) link performance function as the link travel time function. The specific numerical settings are given in Table 6.

4.2.1. Sensitivity Analysis

We then explore how the other parameters might affect the convergence of the system. Figure 5 displays chosen probabilities of route 1 over time under different parameter values.

(a)

(b)

(c)

(d)

It can be clearly observed from Figure 5 that the travelers’ learning rate, information fusion rate, and perception variations adopted by travelers and traffic agency all affect the convergence of the proposed system, which is consistent with our theoretical analyses.

We further find from Figure 5 that when travelers’ learning rate and perception variation parameter are beyond certain values, the system will not be able to converge to an equilibrium. The reason can be, respectively, explained as follows:(1)The higher travelers’ learning rate is, the more travelers put emphasize on single day experience. Travelers’ route choice will lack consistency, leading to an unstable system.(2)A lower perception variation (a higher value) means that more travelers choose the same route to travel. As a result, they will frequently adjust the route and the system oscillates.

When increases to 0.8, the system starts to converge to an equilibrium. The implication is that appropriate information dependence could help stabilize the system. Furthermore, from the fact that the system with is convergent, we can infer that more dependence on a stable information system might contribute to the system’s convergence.

In addition, when is equal to 0.01 (an underestimated value), the traffic system cannot converge to an equilibrium, while the traffic system still is convergent when is equal to 0.4 (an overestimated value). These results indicate that the system can converge to an equilibrium when an inaccurate perception variation is adopted by traffic agency to predict traffic information. As well, when traffic agency cannot obtain the real travelers’ perception variation, a relatively larger adopted is more beneficial.

4.2.2. Comparison Analysis of Static Route Adjustment Ratio and Dynamic Route Adjustment Ratio

In our model, travelers decide whether to adjust route based on the difference between perceived and expected utilities, leading to a dynamical adjustment ratio (DR). However, as mentioned above, most relevant studies consider that a fixed proportion of travelers adjust the route every day, called static adjustment ratio (SR). For simulating it, we keep the maximal route adjustment ratio unchanged in the whole evolutionary process. The efficiencies of the systems under the two different adjustment ratios are displayed in Figure 6. It clearly shows that the travelers’ total travel times with DR is higher than that with SR, demonstrating the model with SR overestimates the efficiency of the system. This might be because in dynamic route adjustment scenario travelers ignore the small difference among route travel times and keep travelling in routes with slightly high travel times.

It should be noted that, for the network with single route connecting each OD pair, the efficiencies of the systems under the two different adjustment ratios are the same because no alternative route can be used. Therefore, we can conclude that the model with SR overestimates the efficiency of the system with alternative routes.

4.2.3. Day-to-Day Dynamic after the Change of Road Capacity

The proposed model can be further used to analyze traffic dynamics when there are changes in road capacity, such as, drops in link capacity caused by road maintenance. The scenario can be assumed as at a day, such as 50^th day, segment of link 7 is closed due to maintenance and its capacity drops into 200 (pcu·h⁻¹); after road maintenance is finished, its capacity returns into the original value at the 70th day. Corresponding evolutionary process of the system is displayed in Figure 7.

As shown in Figure 7, when the capacity of link 7 drops, traffic equilibrium immediately breaks down because traffic agency informs travelers of road maintenance through descriptive information. The whole traffic system is affected, such as route 1 and route 24 that do not include link 7, because links and routes in a traffic system interact with each other. During the whole period of maintenance, the system always oscillates and does not achieve an equilibrium. After recovery of the capacity, the traffic system takes 5 days to return to its initial equilibrium. This demonstrates that the change of temporary link capacity does not cause permanent change.

5. Conclusions

This study extends the day-to-day model by simultaneously considering the combined effect of traffic information and experience on travelers’ perception and dynamic route adjustment ratio. Then, we theoretically validate the existence of its fixed point. We find that except the learning rate adopted by traffic agency, travelers’ learning rate, information fusion rate, perception variation, and maximal route adjustment ratio in the model all affect uniqueness of its fixed point when link performance functions are the function with link flows. An iteration-based algorithm can be used to solve these fixed points. Finally, we conduct numerical experiments on a small network with two links and Nguyen-Dupuis network to investigate effects of several parameters on the convergence of the model. The results show the system can converge to an equilibrium with traffic information predicted by traffic agency using inaccurate parameters. Also, dependence on traffic information provided by a stable information system contributes to the stability of the system. Furthermore, maximal route adjustment rate has a negative effect on the stability of the fixed point and a positive effect on convergence speed. The results provide insights for ATIS design and traffic management.

An experiment is also conducted to analyze the difference between static adjustment ratio and dynamic adjustment ratio. The results show that the model with static route adjustment ratio overestimates the efficiency of the system with alternative routes. Finally, we show the application of our model by analyzing traffic dynamics when there are drops in link capacities.

In the future, various research directions ensuing from this work can be explored. One of them would be to calibrate and validate the model with empirical data. Another line of research would be to analytically derive the sufficient condition that assures the stability of the dynamic system. The third direction would be to design management measures to improve its stability. Last but not least, the model should be further extended into a link-based day-to-day model to avoid numerating all routes among O-D pairs.

Appendix

A. Notations

A traffic network is a directed graph (N, L), where N represents the node set and L corresponds to the link set. The notation used in this paper is = link index, = origin set, = destination set, = origin and destination (OD) pair set = OD pair index, = route set between OD pair = route index, = link-route index whose value is 1 if link belongs to route , zero otherwise = demand of OD pair = free-flow travel time of link = flow on link on day predicted by traffic agency = traffic flow on link on day = flow on the route on day predicted by traffic agency = traffic flow on the route on day = link performance function = route travel time on day = perceived travel time of route on day = perceived travel time of route on day predicted by traffic agency = route travel time on day predicted by traffic agency = expected utilities through OD pair = difference between perceived and expected utilities = adjustment ratio of travelers on day that choose route on day = probability that route adjustment travelers choose route = route chosen probability predicted by traffic agency =route chosen probability.

B. Jacobian Matrix

The Jacobian matrix iswhere

Data Availability

All data, models, and code generated or used during the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 51478110), Science and Technology Program of Jiangsu Province (no. BY2016076-05), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0139), and China Scholarship Council.

References

J. L. Horowitz, “The stability of stochastic equilibrium in a two-link transportation network,” Transportation Research Part B: Methodological, vol. 18, no. 1, pp. 13–28, 1984.
View at: Publisher Site | Google Scholar
E. Cascetta and G. E. Cantarella, “A day-to-day and within-day dynamic stochastic assignment model,” Transportation Research Part A: General, vol. 25, no. 5, pp. 277–291, 1991.
View at: Publisher Site | Google Scholar
T. L. Friesz, D. Bernstein, N. J. Mehta, R. L. Tobin, and S. Ganjalizadeh, “Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems,” Operations Research, vol. 42, no. 6, pp. 1120–1136, 1994.
View at: Publisher Site | Google Scholar
A. Nagurney and D. Zhang, “Projected dynamical systems in the formulation, stability analysis, and computation of fixed-demand traffic network equilibria,” Transportation Science, vol. 31, no. 2, pp. 147–158, 1997.
View at: Publisher Site | Google Scholar
G. E. Cantarella and E. Cascetta, “Dynamic processes and equilibrium in transportation networks: towards a unifying theory,” Transportation Science, vol. 29, no. 4, pp. 305–329, 1995.
View at: Publisher Site | Google Scholar
R.-Y. Guo, H. Yang, H.-J. Huang, and Z. Tan, “Link-based day-to-day network traffic dynamics and equilibria,” Transportation Research Part B: Methodological, vol. 71, pp. 248–260, 2015.
View at: Publisher Site | Google Scholar
H.-J. Tan, T.-L. Liu, and H. Yang, “Modeling the evolutions of day-to-day route choice and year-to-year ATIS adoption with stochastic user equilibrium,” Journal of Advanced Transportation, vol. 42, no. 2, pp. 111–127, 2008.
View at: Publisher Site | Google Scholar
L. Han, H. Sun, and J. Wu, “Day-to-day evolution of the traffic network with advanced traveler information system,” Chaos, Solitons & Fractals, vol. 44, no. 10, pp. 914–919, 2011.
View at: Publisher Site | Google Scholar
G. N. Zhu, G. E. Cantarella, and F. Simonelli, “Advanced traveller information systems under recurrent traffic conditions: network equilibrium and stability,” Transportation Research Part B: Methodological, vol. 92, pp. 73–87, 2016.
View at: Publisher Site | Google Scholar
T. Velonà, “Day-to-day dynamical model incorporating an explicit description of individuals' information collection behaviour,” Transportation Research Part B: Methodological, vol. 92, pp. 88–103, 2016.
View at: Publisher Site | Google Scholar
F. Wei, N. Jia, and S. Ma, “Day-to-day traffic dynamics considering social interaction: from individual route choice behavior to a network flow model,” Transportation Research Part B: Methodological, vol. 94, pp. 335–354, 2016.
View at: Publisher Site | Google Scholar
Y. Xiao and H. K. Lo, “Day-to-day departure time modeling under social network influence,” Transportation Research Part B: Methodological, vol. 92, pp. 54–72, 2016.
View at: Publisher Site | Google Scholar
W. Liu, X. Li, and F. Zhang, “Interactive travel choices and traffic forecast in a doubly dynamical system with user inertia and information provision,” Transportation Research Part C: Emerging Technologies, vol. 85, pp. 711–731, 2017.
View at: Publisher Site | Google Scholar
B. Yang, M. Xu, Q. Meng, and Z. Huang, “A day-to-day route flow evolution process towards the mixed equilibria,” Transportation Research Part C: Emerging Technologies, vol. 82, pp. 210–228, 2017.
View at: Publisher Site | Google Scholar
C. Huang, T.-L. Liu, and H.-J. Huang, “A cumulative prospect theory approach to commuters' day-to-day route-choice modeling with friends' travel information,” Transportation Research Part C: Emerging Technologies, vol. 86, pp. 527–548, 2018.
View at: Publisher Site | Google Scholar
Y. Chen, K. Han, and W. Ochieng, “Day-to-Day dynamic traffic assignment with imperfect information, bounded rationality and information sharing,” Transportation Research Part C: Emerging Technologies, vol. 114, pp. 59–83, 2020.
View at: Google Scholar
E. A. I. Bogers, Traffic Information and Learning in Day-To-Day Route Choice, TRAIL Research School, Delft, Netherlands, 2009.
X. Di and H. X. Liu, “Boundedly rational route choice behavior: A review of models and methodologies,” Transportation Research Part B: Methodological, vol. 85, pp. 142–179, 2016.
View at: Publisher Site | Google Scholar
H. S. Mahmassani and Y.-H. Liu, “Dynamics of commuting decision behaviour under advanced traveller information systems,” Transportation Research Part C: Emerging Technologies, vol. 7, no. 2-3, pp. 91–107, 1999.
View at: Publisher Site | Google Scholar
W. Wang, X. Zhao, X. Li et al., “Empirical study and modeling of variable message signs on route choice behavior,” Journal of Transportation Systems Engineering and Information Technology, vol. 13, no. 3, pp. 60–64, 2013.
View at: Google Scholar
X. Zhao, C. Wan, H. Sun, and Z. Xie, “Dynamic rerouting behavior and its impact on dynamic traffic patterns,” IEEE Transactions on Intelligent Transportation Systems, vol. 18, no. 10, pp. 2763–2779, 2017.
View at: Publisher Site | Google Scholar
R. D. Connors and A. Sumalee, “A network equilibrium model with travellers' perception of stochastic travel times,” Transportation Research Part B: Methodological, vol. 43, no. 6, pp. 614–624, 2009.
View at: Publisher Site | Google Scholar
L. E. J. Brouwer, “Über Jordansche Mannigfaltigkeiten,” Mathematische Annalen, vol. 71, no. 4, p. 598, 1912.
View at: Publisher Site | Google Scholar
P. A. Cook, Nonlinear Dynamical Systems, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.
L. Cheng, X.-m. Lou, and J. Zhou, “A mixed stochastic user equilibrium model considering influence of advanced traveller information systems in degradable transport network,” Journal of Central South University, vol. 25, no. 5, pp. 1182–1194, 2018.
View at: Publisher Site | Google Scholar
J. R. Ma, “Approximation methods which converge with probability one,” The Annals of Mathematical Statistics, vol. 25, no. 2, pp. 382–386, 1954.
View at: Publisher Site | Google Scholar
F. F. Wei, S. Ma, and N. Jia, “A day-to-day route choice model based on reinforcement learning,” Mathematic Problem in Engineering, vol. 2014, Article ID 646548, 19 pages, 2014.
View at: Publisher Site | Google Scholar
S. Nguyen and C. Dupuis, “An efficient method for computing traffic equilibria in networks with asymmetric transportation costs,” Transportation Science, vol. 18, no. 2, pp. 185–202, 1984.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Manman Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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