Bi-level programming model for differentiated toll discounts for expressway trucks

Shengyu YAN, Jiaqi ZHAO, Wenbo YOU, Yang LIU, Shijie HAO, Fuwei WU

Journal of Tsinghua University(Science and Technology) ›› 2025, Vol. 65 ›› Issue (7) : 1347-1358.

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Journal of Tsinghua University(Science and Technology) ›› 2025, Vol. 65 ›› Issue (7) : 1347-1358. DOI: 10.16511/j.cnki.qhdxxb.2025.21.022
Traffic and Transportation

Bi-level programming model for differentiated toll discounts for expressway trucks

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Abstract

Objective: Implementing differentiated toll discounts for expressway trucks can lead to a more balanced traffic flow across the road network. Expressway operators hope to make profits by charging truck tolls, while truck groups aim to maximize profits through toll charges, whereas truck groups focus on minimizing travel costs in terms of economics and time. A balance exists between the benefits of both parties; however, determining differentiated toll discounts for expressways to reach this balance is difficult. Methods: (1) Based on consumer surplus theory, key factors affecting freight route selection are identified using the preference survey of traveling behavior, and a bi-level programming model is proposed for determining differentiated toll discounts, incorporating assumptions and constraints. (2) The upper model, considering truck cost and travel time, is a surplus maximization model for expressway operators. It is solved using a novel algorithm enhanced by combining a genetic algorithm with simulated annealing. In the upper model, a lower limit on the financial revenue targets of highway operating enterprises is included the constraints to avoid overflow of lower bound returns during the iteration process. (3) The lower model leverages a logit-based stochastic user equilibrium allocation model for multiple vehicle types under elastic demand, solved using the Frank Wolfe algorithm. A generalized impedance function considering economic and time costs is established in the lower model to demonstrate the impacts of road conditions on truck travel. Cost weighting coefficients are introduced, and calculation methods and recommended values are proposed to integrate economics and time costs. (4) Detailed execution steps are provided for solving algorithms of the upper and lower models. The model also introduces model convergence criteria to optimize the iteration efficiency of the solving algorithm. A fitness function is proposed based on the financial lower bound target, and the upper model is transformed into a minimum value problem, eliminating the constraint of discounted rates. Results: The feasibility of the model is validated using toll collection data of expressway and link traffic data of highways, with three instance highway sections. A reasonable range suitable for implementing differentiated toll discounts can attract trucks back to the expressway, and increasing the daily average traffic volume for each vehicle type. After 43 iterations, the upper model achieves a stable function value. The toll discount rates for small trucks, medium trucks, heavy trucks, and extra-heavy trucks on the instance expressway fall within the ranges of 78.68%-86.27%, 55.82%-65.82%, 47.90%-54.81% and 47.52%-48.31% respectively; consequently, the average truck flow on the expressway increases by 12.24%. Conclusions: The conclusion demonstrates that the bi-level programming model can accurately determine the toll discount range for trucks on expressways; however, even with a discount rate of 4.7% for oversized trucks on nearly 100 km of the actual expressway, attracting all oversized trucks to return to the expressway remains challenging. Fuel and toll fees remarkably impact travel path selection within the generalized impedance function; moreover, the same toll discount produces notable differences in implementation effects across truck types. The research provides support for developing differentiated toll policies for expressways, as well as their subsequent optimization and adjustment.

Key words

transportation planning / expressway / toll discounts for trucks / bi-level programming model / consumer surplus theory / solving algorithm

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Shengyu YAN , Jiaqi ZHAO , Wenbo YOU , et al . Bi-level programming model for differentiated toll discounts for expressway trucks[J]. Journal of Tsinghua University(Science and Technology). 2025, 65(7): 1347-1358 https://doi.org/10.16511/j.cnki.qhdxxb.2025.21.022

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