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Bilevel Optimization and Heuristic Algorithms for Integrating Latent Demand into the Design of Large-Scale Transit Systems

Hongzhao Guan, Beste Basciftci, Pascal Van Hentenryck

TL;DR

The paper presents a generic bilevel optimization model, namely the Transit Networks Design with Adoptions (TN-DA), that considers the network design decisions in the leader problem, and routing of the riders in the follower problem under the given network design, while allowing a black-box choice function for representing the adoption behavior of latent demand.

Abstract

Capturing latent demand has a pivotal role in designing transit services as omitting these riders can lead to poor quality of service and/or additional costs. This paper explores this topic in the design of transit networks by considering the perspectives of both the transit agencies and riders. The paper presents a generic bilevel optimization model, namely the Transit Networks Design with Adoptions (TN-DA), that considers the network design decisions in the leader problem, and routing of the riders in the follower problem under the given network design, while allowing a black-box choice function for representing the adoption behavior of latent demand. The paper then identifies structural properties of the optimal solution of the TN-DA problem, which are desirable for transit agencies for capturing adoption behavior of the riders. The paper further provides guideline metrics for the transit agencies based on these desired adoption properties. Due to the computational complexity of this bilevel problem, the paper proposes five efficient heuristic algorithms to solve large-scale instances, which leverage an iterative procedure by solving a simpler version of the TN-DA problem and integrating the evaluation of rider choices. These algorithms either satisfy the desired properties of the optimal solution or provide fast approximations. The paper presents extensive large-scale case studies on two different transit systems by utilizing real datasets: (i) On-demand Multimodal Transit Systems (ODMTS) and (ii) Scooters-Connected Transit Systems (SCTS). The results demonstrate that the heuristic algorithms can find high-quality solutions much faster than exact approaches over various instances, while satisfying key adoption properties of the optimal solutions.

Bilevel Optimization and Heuristic Algorithms for Integrating Latent Demand into the Design of Large-Scale Transit Systems

TL;DR

The paper presents a generic bilevel optimization model, namely the Transit Networks Design with Adoptions (TN-DA), that considers the network design decisions in the leader problem, and routing of the riders in the follower problem under the given network design, while allowing a black-box choice function for representing the adoption behavior of latent demand.

Abstract

Capturing latent demand has a pivotal role in designing transit services as omitting these riders can lead to poor quality of service and/or additional costs. This paper explores this topic in the design of transit networks by considering the perspectives of both the transit agencies and riders. The paper presents a generic bilevel optimization model, namely the Transit Networks Design with Adoptions (TN-DA), that considers the network design decisions in the leader problem, and routing of the riders in the follower problem under the given network design, while allowing a black-box choice function for representing the adoption behavior of latent demand. The paper then identifies structural properties of the optimal solution of the TN-DA problem, which are desirable for transit agencies for capturing adoption behavior of the riders. The paper further provides guideline metrics for the transit agencies based on these desired adoption properties. Due to the computational complexity of this bilevel problem, the paper proposes five efficient heuristic algorithms to solve large-scale instances, which leverage an iterative procedure by solving a simpler version of the TN-DA problem and integrating the evaluation of rider choices. These algorithms either satisfy the desired properties of the optimal solution or provide fast approximations. The paper presents extensive large-scale case studies on two different transit systems by utilizing real datasets: (i) On-demand Multimodal Transit Systems (ODMTS) and (ii) Scooters-Connected Transit Systems (SCTS). The results demonstrate that the heuristic algorithms can find high-quality solutions much faster than exact approaches over various instances, while satisfying key adoption properties of the optimal solutions.
Paper Structure (53 sections, 7 theorems, 22 equations, 12 figures, 14 tables, 6 algorithms)

This paper contains 53 sections, 7 theorems, 22 equations, 12 figures, 14 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Transit Network Design with Adoptions Problem ( TN-DA)
  4. Problem Description
  5. Bilevel Optimization Framework
  6. Adoption Choice Model
  7. Subproblem
  8. Heuristic Algorithms and Design Properties
  9. Key Concepts for Design Evaluation
  10. Trip-based Iterative Algorithms
  11. Greedy Adoption Algorithm (\ref{['alg:rho_GRAD']})
  12. Greedy Rejection Algorithm (\ref{['alg:eta_GRRE']})
  13. Greedy Adoption Algorithm with Greedy Rejection Subproblem (\ref{['alg:rho_GAGR']})
  14. Additional Properties of Trip-based Algorithms
  15. Arc-based Iterative Algorithms
  16. ...and 38 more sections

Key Result

Proposition 1

Assume $\mathbf{z}^*$ is the optimal solution of the TN-DA problem (Figure fig:tn_da_formulation). There exists an input trip set $\hat{T}^* \subseteq T$ such that $\mathbf{z}^*=\hbox{\sc TN-DFD}(\hat{T}^*)$ and $\mathbf{z}^*$ satisfies both Correct Rejection and Correct Adoption properties (Prop

Figures (12)

  • Figure 1: The Generalized Optimization Model for the Transit Network Design with Adoption ( TN-DA) Problem.
  • Figure 2: The Generalized Formulation for the Transit Network Design with Fixed Demand $\hat{T}$: TN-DFD($\hat{T}$).
  • Figure 3: Picture that illustrates the On-demand Multimodal Transit Systems (ODMTS). The solid circles represent transit hubs. The solid line represents a typical ODMTS path offered by the transit agency.
  • Figure 4: The ODMTS designs found by the Algorithms for the extra-large case study. Black polylines and green straight lines are utilized to represent opened bus arcs and opened non-direct shuttle legs, respectively. Thickness of the black polylines indicates the number of users using the arcs. The rail systems are in red, yellow, dark green, and blue.
  • Figure 5: The Designs Found by the Heuristic Algorithms and modified P-Path for the SCTS-DA problem in Atlanta. Light green lines represent scooter connections.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • Remark 3
  • Proposition 3
  • Corollary 1
  • Proposition 4
  • Corollary 2
  • Proposition 5