Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 858731, 14 pages
Research Article

Convergence Study of Minimizing the Nonconvex Total Delay Using the Lane-Based Optimization Method for Signal-Controlled Junctions

Department of Civil and Architectural Engineering, City University of Hong Kong, Tat Chee Road, Kowloon, Hong Kong

Received 12 January 2012; Revised 8 March 2012; Accepted 20 March 2012

Academic Editor: Beatrice Paternoster

Copyright © 2012 C. K. Wong and Y. Y. Lee. 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.


This paper presents a 2D convergence density criterion for minimizing the total junction delay at isolated junctions in the lane-based optimization framework. The lane-based method integrates the design of lane markings and signal settings for traffic movements in a unified framework. The problem of delay minimization is formulated as a Binary Mix Integer Non Linear Program (BMINLP). A cutting plane algorithm can be applied to solve this difficult BMINLP problem by adding hyperplanes sequentially until sufficient numbers of planes are created in the form of solution constraints to replicate the original nonlinear surface in the solution space. A set of constraints is set up to ensure the feasibility and safety of the resultant optimized lane markings and signal settings. The main difficulty to solve this high-dimension nonlinear nonconvex delay minimization problem using cutting plane algorithm is the requirement of substantial computational efforts to reach a good-quality solution while approximating the nonlinear solution space. A new stopping criterion is proposed by monitoring a 2D convergence density to obtain a converged solution. A numerical example is given to demonstrate the effectiveness of the proposed methodology. The cutting-plane algorithm producing an effective signal design will become more computationally attractive with adopting the proposed stopping criterion.