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Integrated demand-side management and timetabling for an urban rail transit line: A Benders decomposition approach

Lixing Yang, Yahan Lu, Jiateng Yin, Shadi Sharif Azadeh

TL;DR

This work develops an integrated optimization framework that jointly optimizes urban rail timetabling, passenger directing, and trip booking via pricing. It introduces a novel Benders decomposition embedded in a branch‑and‑cut algorithm, which leverages partial passenger information to efficiently solve a line‑level INLP that minimizes the weighted sum of waiting time $F^{t}$ and subsidies $F^{s}$. Extensions cover peak/off‑peak pricing, elastic demand from other modes, and network scalability, with validation on proof‑of‑concept lines and Beijing's Batong line showing substantial improvements in fleet efficiency and service fairness. The approach delivers high‑quality, implementable timetables and passenger strategies, offering practical implications for urban transit operators and policy makers and paving the way for network‑level extensions under uncertainty.

Abstract

The intelligent upgrading of metropolitan rail transit systems has made it feasible to implement demand-side management policies that integrate multiple operational strategies in practical operations. However, the tight interdependence between supply and demand necessitates a coordinated approach combining demand-side management policies and supply-side resource allocations to enhance the urban rail transit ecosystem. In this study, we propose a mathematical and computational framework that optimizes train timetables, passenger flow control strategies, and trip-shifting plans through the pricing policy. Our framework incorporates an emerging trip-booking approach that transforms waiting at the stations into waiting at home, thereby mitigating station overcrowding. Additionally, it ensures service fairness by maintaining an equitable likelihood of delays across different stations. We formulate the problem as an integer linear programming model, aiming to minimize passengers' waiting time and government subsidies required to offset revenue losses from fare discounts used to encourage trip shifting. To improve the computational efficiency, we develop a Benders decomposition-based algorithm within the branch-and-cut method, which decomposes the model into train timetabling with partial passenger assignment and passenger flow control subproblems. We propose valid inequalities based on our model's properties to strengthen the linear relaxation bounds at each node of the branch-and-bound tree. Computational results from proof-of-concept and real-world case studies on the Beijing metro show that our solution method outperforms commercial solvers in terms of computational efficiency.

Integrated demand-side management and timetabling for an urban rail transit line: A Benders decomposition approach

TL;DR

This work develops an integrated optimization framework that jointly optimizes urban rail timetabling, passenger directing, and trip booking via pricing. It introduces a novel Benders decomposition embedded in a branch‑and‑cut algorithm, which leverages partial passenger information to efficiently solve a line‑level INLP that minimizes the weighted sum of waiting time and subsidies . Extensions cover peak/off‑peak pricing, elastic demand from other modes, and network scalability, with validation on proof‑of‑concept lines and Beijing's Batong line showing substantial improvements in fleet efficiency and service fairness. The approach delivers high‑quality, implementable timetables and passenger strategies, offering practical implications for urban transit operators and policy makers and paving the way for network‑level extensions under uncertainty.

Abstract

The intelligent upgrading of metropolitan rail transit systems has made it feasible to implement demand-side management policies that integrate multiple operational strategies in practical operations. However, the tight interdependence between supply and demand necessitates a coordinated approach combining demand-side management policies and supply-side resource allocations to enhance the urban rail transit ecosystem. In this study, we propose a mathematical and computational framework that optimizes train timetables, passenger flow control strategies, and trip-shifting plans through the pricing policy. Our framework incorporates an emerging trip-booking approach that transforms waiting at the stations into waiting at home, thereby mitigating station overcrowding. Additionally, it ensures service fairness by maintaining an equitable likelihood of delays across different stations. We formulate the problem as an integer linear programming model, aiming to minimize passengers' waiting time and government subsidies required to offset revenue losses from fare discounts used to encourage trip shifting. To improve the computational efficiency, we develop a Benders decomposition-based algorithm within the branch-and-cut method, which decomposes the model into train timetabling with partial passenger assignment and passenger flow control subproblems. We propose valid inequalities based on our model's properties to strengthen the linear relaxation bounds at each node of the branch-and-bound tree. Computational results from proof-of-concept and real-world case studies on the Beijing metro show that our solution method outperforms commercial solvers in terms of computational efficiency.

Paper Structure

This paper contains 39 sections, 1 theorem, 47 equations, 15 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Train timetabling problem combined with passenger flow control
  4. Trip reservation and congestion pricing
  5. Paper contributions
  6. Problem description
  7. Line structure and time discretization
  8. Time-dependent passenger demand and time equivalization
  9. Categorization of passengers and demand-side management strategies
  10. Interactions among the government, the operator, and passengers
  11. Mathematical formulation
  12. Notations used in the formulation
  13. INLP model for the BDTT problem
  14. Model extensions
  15. Solution methodology
  16. ...and 24 more sections

Key Result

Proposition 1

We define the number of passengers without reservations who head to station $v$ and are ensured to board station $u$ as $\tilde{b}_{iuv}$. Besides, we introduce the number of passengers without reservations who are on board at train $i$, who depart from train $i$ at station $u$, and the total number Recall that we require at least $\varrho$ percent of passengers without reservations to be served b

Figures (15)

  • Figure 1: Illustration of timetables before and after the adjustment of the time dimension.
  • Figure 2: Illustration of passengers' types, travel time, priority, and fare characteristics. Notes: Passengers with reservations are never detained.
  • Figure 3: Interactions among the government, passengers, and the operator.
  • Figure 4: Illustration of departure and headway indicators.
  • Figure 5: Number of waiting passengers within each headway.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Example 1
  • Remark 1
  • Remark 2
  • Proposition 1