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This paper studies the distance-based congestion pricing in a network considering the day-to-day dynamic traffic flow evolution process. It is well known that, after an implementation or adjustment of a new congestion toll scheme, the network environment will change and traffic flows will be nonequilibrium in the following days; thus it is not suitable to take the equilibrium-based indexes as the objective of the congestion toll. In the context of nonequilibrium state, prior research proposed a mini–max regret model to solve the distance-based congestion pricing problem in a network considering day-to-day dynamics. However, it is computationally demanding due to the calculation of minimal total travel cost for each day among the whole planning horizon. Therefore, in order to overcome the expensive computational burden problem and make the robust toll scheme more practical, we propose a new robust optimization model in this paper. The essence of this model, which is an extension of our prior work, is to optimize the worst condition among the whole planning period and ameliorate severe traffic congestions in some bad days. Firstly, a piecewise linear function is adopted to formulate the nonlinear distance toll, which can be encapsulated to a day-to-day dynamics context. A very clear and concise model named logit-type Markov adaptive learning model is then proposed to depict commuters’ day-to-day route choice behaviors. Finally, a robust optimization model which minimizes the maximum total travel cost among the whole planning horizon is formulated and a modified artificial bee colony algorithm is developed for the robust optimization model.

Congestion toll is generally regarded as a potent economic instrument for transportation demand management (TDM) to alleviate the traffic congestion and improve the system performance in urban areas and also has received more and more attention both academically and practically. Since the successful implementation of congestion pricing in Singapore from 1975, many countries and cities (such as Norway, London, Stockholm and Milan) have implemented a road congestion pricing policy, which has achieved remarkable success in terms of easing urban traffic congestion [

The distance-based toll scheme receives more and more attention recently in both the academic community and industrial circles. References [

Generally, the total travel cost is regarded as the optimization objective of the congestion toll design problem. Most of the literatures studying the congestion pricing problem calculate the total travel cost (TTC) based on the equilibrium flows and make an evaluation based on the calculated TTC. However, when a toll pattern is implemented, the route flows will be totally different from day to day, because the implemented toll policy is an important component which will influence travelers’ route choice behaviors. The system cannot reach an equilibrium state overnight. Therefore, for the optimal toll design problem during the whole planning period, the day-to-day dynamics models can better describe the network flow conditions, rather than the final equilibrium state. Besides, to avoid the complicated implementation of governments and confusions of travelers on the toll in practice, it is necessary to levy an unchanged toll in the whole period

During the whole period of

The day-to-day dynamic flow evolution process is the foundation for the day-to-day DCP problem; lots of research work focus on the day-to-day dynamic flow evolution process [

The contributions of this paper are twofold. On the one hand, a finite learning process model named logit-type Markov adaptive learning model is proposed to depict commuters’ day-to-day route choice behaviors. On the other hand, a mini–max total travel cost model, which can overcome the expensive computational burden problem in previous work and make the congestion toll scheme more practical, is proposed to solve the congestion toll problem considering nonequilibrium flow evolution processes. This paper is structured as follows. The next section first introduces the nonlinear distance toll which can be approximated with a piecewise linear toll function. A logit-type Markov adaptive learning model is then proposed in Section

A strongly connected network, denoted by

List of notations.

| |
---|---|

| The total planning period for one toll pattern. |

| The number of days after the toll implementation, |

| The set of OD pairs. |

| The set of routes connecting an OD pair |

| The travel cost on route |

| The route flows over the entire network, |

| The traffic flow on route |

| The travel demands, |

| The travel demand connecting OD pair |

| The link travel time functions, |

| The link flows, |

| The travel time function of link |

| The traffic flow on link |

| |

| The vertex values, |

| The lower and upper bound of the distance-based toll. |

| The toll charge function. |

| The total number of the intervals in the toll function |

| Column vector of the travel distance in a cordon, |

| Column vector of the distance-based toll |

| Number of the colony size, the employed bees, and the onlookers. |

| Parameters used in the day-to-day dynamics model. |

A nonlinear-type distance-toll function is preferred according to Liu et al. [

Piecewise linear toll.

It is practical to define a nondecreasing distance-toll function in real life; thus we should have

From (

A reasonable day-to-day dynamics model should well reflect the realistic route adjustment process and learning behavior of commuters [

Erev et al. [

A very clear and concise finite learning process named logit-type Markov adaptive learning model is proposed to depict commuters’ day-to-day route choice behaviors. Specifically, the baseline probability of choosing route

Then, the flow of route

According to the baseline model, the following model is proposed to estimate the actual attraction of route

As discussed in Introduction, the whole network environment will be changed after a period of days, and the days

When a toll pattern

It is clear that model (

All of the existing solution methods (such as sensitivity analysis method, system optimal relaxation method, and gap function method) are not suitable to solve the proposed bilevel robust model due to the complexity of the flow evolution process

The ABC algorithm was originally proposed by Karaboga [

Flowchart of the solution algorithm.

This paper studies the nonlinear distance-based toll with day-to-day dynamic traffic flow evolution. After an implementation or adjustment of a new congestion toll scheme, the network environment will change and traffic flows will be nonequilibrium in the following days, which makes it not suitable to take the equilibrium-based indexes as the objective of the congestion toll. A mini–max total travel cost model, which can overcome the expensive computational burden problem in previous work and make the congestion toll scheme more practical, is then proposed to solve the congestion toll problem considering nonequilibrium flow evolution processes. The essence of the mini–max total travel cost model is to optimize the worst condition among the whole planning period and ameliorate severe traffic congestions in some bad days, and this takes into consideration the network performance on each day of the study horizon rather than the final equilibrium state.

No external data were used to support this study.

The authors declare that they have no conflicts of interest.

This study is supported by the National Natural Science Foundation of China (nos. 71501038 and 71601045), the Key Project of National Natural Science Foundation of China (no. 51638004), and the Scientific Research Foundation of Graduate School of Southeast University (no. YBPY1885).