Optimal multi-step toll design under general user heterogeneity |
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| Affiliation: | 1. Department of Civil and Environmental Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA;2. Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL 32611, USA;1. Industry Solutions (Logistics, T&T and BAO), IBM Research – China, Beijing 100193, China;2. Department of Industrial & Manufacturing Engineering, Pennsylvania State University, University Park, PA 16802, USA;1. SUPA, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK;2. Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK;3. Kaggle, Inc., 188 King St. Suite 502, San Francisco, CA 94107, USA;4. Erasmus University, Rotterdam, The Netherlands;5. Department of Mathematics and CEMAT, Instituto Superior Técnico, University of Lisbon, Av. Rovisco, Pais, 1049-001 Lisboa, Portugal;1. University of California, Berkeley, 416 McLaughlin Hall, Berkeley, CA 94720, USA;2. The Hong Kong Polytechnic University, CF612, Hunghom, Kowloon, Hong Kong;1. Key Laboratory of Road and Traffic Engineering, Tongji University, Shanghai, China;2. Department of Civil and Environmental Engineering, Utah State University, USA;3. Faculty of Engineering, Chiangmai University, Chiangmai, Thailand;4. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong |
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| Abstract: | ![]()
This paper studies the optimal multi-step toll design problem for the bottleneck model with general user heterogeneity. The design model is formulated as a mathematical program with equilibrium constraints (MPEC), which is NP-hard due to non-convexity in both the objective function and the feasible set. An analytical method is proposed to solve the MPEC by decomposing it into smaller and easier quadratic programs, each corresponding to a unique departure order of different user classes. The quadratic programs are defined on a polyhedral set, which makes it easier to identify a local optimum. Importantly, each quadratic program is constrained by a set of linear feasibility cuts that define the presence of each user class in the arrival window. We prove that the proposed method ensures global optimality provided that each quadratic program can be solved globally. To obviate enumerating all departure orders, a heuristic method is developed to navigate through the solution space by using the multipliers associated with the feasibility cuts. Numerical experiments are conducted on several small examples to validate the proposed methodology. These experiments show that the proposed heuristic method is effective in finding near-optimal solutions within a relatively small number of iterations. |
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| Keywords: | Step toll General user heterogeneity Mathematical program with equilibrium constraint Quadratic program Bottleneck model |
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