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Design of Transit Networks: Global Optimization of Continuous Approximation Models via Geometric Programming

Haoyang Mao, Weihua Gu, Wenbo Fan, Zhicheng Jin, Xiaokuan Zhao

Abstract

Continuous approximation (CA) models have been widely adopted in transit network design studies due to their strong analytical tractability and high computational efficiency. However, such models are typically formulated as nonconvex optimization problems, and existing solution approaches mainly rely on iterative algorithms that exploit first-order optimality information or nonlinear programming solvers, whose solution quality lacks stability guarantees under complex demand conditions. This paper proposes a geometric programming (GP)-based CA method for transit network design, which can be efficiently solved to global optimality. Numerical experiments are conducted on both homogeneous and heterogeneous network settings to evaluate the effectiveness of the proposed approach. Comprehensive tests are performed under the combinations of six heterogeneous demand distributions, four levels of total passenger demand, and three value-of-time parameters. The results indicate that the GP approach consistently outperforms the coordinate descent method across all test cases, achieving cost reductions of approximately 1%-4%, even when the latter converges to identical solutions under different initializations. In comparison, nonlinear programming solvers, with fmincon as a representative example, are able to obtain globally optimal solutions comparable to those of the GP approach in low-demand heterogeneous networks; however, their performance becomes unstable under high-demand conditions. These findings demonstrate that GP provides an efficient and robust optimization framework for solving CA-based transit network design problems, especially in high-demand and highly heterogeneous network environments.

Design of Transit Networks: Global Optimization of Continuous Approximation Models via Geometric Programming

Abstract

Continuous approximation (CA) models have been widely adopted in transit network design studies due to their strong analytical tractability and high computational efficiency. However, such models are typically formulated as nonconvex optimization problems, and existing solution approaches mainly rely on iterative algorithms that exploit first-order optimality information or nonlinear programming solvers, whose solution quality lacks stability guarantees under complex demand conditions. This paper proposes a geometric programming (GP)-based CA method for transit network design, which can be efficiently solved to global optimality. Numerical experiments are conducted on both homogeneous and heterogeneous network settings to evaluate the effectiveness of the proposed approach. Comprehensive tests are performed under the combinations of six heterogeneous demand distributions, four levels of total passenger demand, and three value-of-time parameters. The results indicate that the GP approach consistently outperforms the coordinate descent method across all test cases, achieving cost reductions of approximately 1%-4%, even when the latter converges to identical solutions under different initializations. In comparison, nonlinear programming solvers, with fmincon as a representative example, are able to obtain globally optimal solutions comparable to those of the GP approach in low-demand heterogeneous networks; however, their performance becomes unstable under high-demand conditions. These findings demonstrate that GP provides an efficient and robust optimization framework for solving CA-based transit network design problems, especially in high-demand and highly heterogeneous network environments.
Paper Structure (14 sections, 19 equations, 3 figures, 3 tables)

This paper contains 14 sections, 19 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Optimal Network Design Model
  3. Basic concepts and assumptions
  4. Network design objective function
  5. Model constraints
  6. Model specialization for heterogeneous and homogeneous networks
  7. Solution method via geometric programming
  8. Numerical case studies
  9. Set-up
  10. Numerical results
  11. Summary and Conclusions
  12. Nomenclature
  13. Heterogeneous demand distribution generation
  14. Coordinate descent algorithm for model solution

Figures (3)

  • Figure 1: Illustration of homogeneous and heterogeneous grid-based transit networks in a square city.
  • Figure 2: Illustration of per-passenger system cost under six demand distributions in heterogeneous network: GP approach vs. two benchmark algorithms.
  • Figure 3: Sum of origin and destination demand densities ($D=50,000$).