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Vehicles
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Models and Algorithms for Railway Line Planning Problems
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Models and Algorithms for Railway Line Planning Problems

A special issue of Vehicles (ISSN 2624-8921).

Deadline for manuscript submissions: 30 June 2026 | Viewed by 3431

Special Issue Editors

Dr. Yuan Gao

E-Mail Website
Guest Editor
School of Management, Beijing Institute of Technology, Beijing 100081, China
Interests: railway traffic organization; logistics; scheduling; integer programming; online optimization
Prof. Dr. Shuguang Zhan

E-Mail Website
Guest Editor
School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 23009, China
Interests: train scheduling; intelligent dispatching; optimization algorithm; passenger behavior analysis
Special Issues, Collections and Topics in MDPI journals
Dr. Chuntian Zhang

E-Mail Website
Guest Editor
School of Systems Science, Beijing Jiaotong University, Beijing 100044, China
Interests: railway traffic organization; integer programming; robust optimization; exact and heuristic algorithm

Special Issue Information

Dear Colleagues,

In the complex and ever-evolving landscape of railway transportation, effective line planning is crucial to ensuring operational efficiency, safety, and customer satisfaction. This Special Issue aims to bring together cutting-edge research and innovative solutions addressing the challenges associated with railway line planning problems.

We invite submissions that delve into various aspects of this domain, including but not limited to optimization models, exact algorithms, heuristic and meta-heuristic approaches, machine leaning and reinforcement learning, and case studies demonstrating practical applications. Contributions should explore how these models and algorithms can be tailored to specific railway systems, taking into account factors such as track capacity, train scheduling, passenger demand, and operational constraints.

The Special Issue seeks to foster a comprehensive understanding of the latest advancements in railway line planning, promoting interdisciplinary collaboration and knowledge sharing. We encourage researchers, practitioners, and policymakers to contribute their insights, sharing experiences and best practices that can inspire further innovation and improvement in this vital area of transportation planning.

With a focus on rigorous methodology, real-world applicability, and forward-thinking solutions, this Special Issue promises to be a valuable resource for anyone involved in the planning, operation, and optimization of railway systems. We look forward to receiving contributions that will help shape the future of railway line planning.

Dr. Yuan Gao
Prof. Dr. Shuguang Zhan
Dr. Chuntian Zhang
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Vehicles is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • railway system
  • line planning
  • train scheduling
  • mathematical model
  • algorithm

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (4 papers)

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Research

15 pages, 839 KB  
Article
Optimizing a Heavy-Haul Railway Train Formation Plan for Maximized Transport Capacity
by Shichao Han, Yun Bai and Yao Chen
Vehicles 2026, 8(3), 45; https://doi.org/10.3390/vehicles8030045 - 28 Feb 2026
Viewed by 427
Abstract
Heavy-haul railways are important for bulk freight transport, and improving their transport capacity is essential for railway operators to enhance operational efficiency. This study develops an integer linear programming model for train formation planning that maximizes transport capacity, incorporating key practical constraints such [...] Read more.
Heavy-haul railways are important for bulk freight transport, and improving their transport capacity is essential for railway operators to enhance operational efficiency. This study develops an integer linear programming model for train formation planning that maximizes transport capacity, incorporating key practical constraints such as section headway, station capacity, and locomotive matching. This study makes two main contributions: (1) explicit formulation of transport-capacity maximization as the primary objective; and (2) incorporation of specific train formation rules through linear resource-flow coefficients that characterize the combination and decomposition operations. The model is applied to the Shuozhou–Huanghua Railway in a case study. Experimental results show that the optimized train formation plan increases total freight volume from 2810.4 thousand tons to 3080.0 thousand tons, representing a capacity improvement of approximately 9.6%. This result is achieved by adjusting the mix of train tonnage levels, increasing combination operations for medium-capacity trains, and reallocating locomotive types in accordance with traction requirements. The study demonstrates that a capacity-oriented optimization framework can effectively support train-formation plan decisions under practical operational constraints, providing railway operators with a systematic tool to enhance line utilization without expanding infrastructure. Full article
(This article belongs to the Special Issue Models and Algorithms for Railway Line Planning Problems)
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17 pages, 1285 KB  
Article
Joint Optimization of Dynamic Pricing and Flexible Refund Fees for Railway Services
by Wuyang Yuan, Zhen Ren, Zhongrui Zhou and Yu Ke
Vehicles 2026, 8(2), 31; https://doi.org/10.3390/vehicles8020031 - 6 Feb 2026
Viewed by 541
Abstract
This study explores strategies for dynamic pricing and flexible refund fee setting in railway line services, aiming to optimize ticket sales revenue by integrating refund mechanisms into the revenue management framework. By introducing a consistent concept of opportunity cost applicable to both passengers [...] Read more.
This study explores strategies for dynamic pricing and flexible refund fee setting in railway line services, aiming to optimize ticket sales revenue by integrating refund mechanisms into the revenue management framework. By introducing a consistent concept of opportunity cost applicable to both passengers and railway operators, we propose an integrated approach that combines dynamic pricing with flexible refund fees grounded in the demand-driven opportunity cost of seat resources. A dynamic programming model is constructed to quantify the opportunity cost of seat resources. To address the computational challenges arising from the model’s scale, state and time dimension compression methods are applied to develop an approximate linear programming model with fewer constraints. The proposed model is solved using a turning point search algorithm and a constraint generation algorithm. Numerical experiments and ticket sales simulations are conducted to verify the feasibility of the proposed methods and to explore the application effects of different pricing strategy combinations. The results demonstrate that the integration of dynamic pricing and flexible refund fees can significantly enhance ticket sales revenue, particularly in scenarios of supply shortfall. Full article
(This article belongs to the Special Issue Models and Algorithms for Railway Line Planning Problems)
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19 pages, 882 KB  
Article
Line Planning Based on Passenger Perceived Satisfaction at Different Travel Distances
by Xiaoqing Qiao, Li Xie, Yun Yang and Chao Luo
Vehicles 2026, 8(1), 10; https://doi.org/10.3390/vehicles8010010 - 5 Jan 2026
Viewed by 506
Abstract
The rapid development of China’s high-speed railways (HSRs) and the implementation of revenue management policies have promoted the marketization of railway passenger transport, which is mainly reflected in the gradual transformation from a seller’s market dominated by operating companies to a buyer’s market [...] Read more.
The rapid development of China’s high-speed railways (HSRs) and the implementation of revenue management policies have promoted the marketization of railway passenger transport, which is mainly reflected in the gradual transformation from a seller’s market dominated by operating companies to a buyer’s market dominated by passenger demand. Passenger travel needs are becoming increasingly diverse. In order to improve the quality of HSR services and attract more passengers, this paper starts from passenger satisfaction and considers the heterogeneity of travel preferences of passengers with different travel distances. Based on the passenger travel data of the Nanning-Guangzhou (NG) HSR line, the K-means clustering method is used to classify passengers into three categories: short-distance, medium-distance, and long-distance travel. A structural equation modeling–multinomial logit (SEM-MNL) model integrating both explicit and latent variables was constructed to analyze passenger travel origin-destination (OD) choices. Stata software was used to estimate passenger preferences for perceived satisfaction functions across different travel distances. Finally, considering constraints such as load factor, departure capacity, and spatiotemporal passenger flow demand, a line planning optimization model was constructed with the goal of minimizing train operating costs and maximizing passenger travel satisfaction. An improved subtraction optimizer algorithm was designed for the solution. Using the NG Line as a case study, the proposed method achieved a reduction in train operating costs while enhancing overall passenger satisfaction. Full article
(This article belongs to the Special Issue Models and Algorithms for Railway Line Planning Problems)
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16 pages, 1982 KB  
Article
Joint Optimization of Full-Length and Short-Turning Plan and Schedule: Case Study of Nanchang Metro Airport Line
by Jian Peng, Cong Huang, Hui Fei, Zhaozhi Liu, Zhen Di and Jungang Shi
Vehicles 2025, 7(4), 132; https://doi.org/10.3390/vehicles7040132 - 19 Nov 2025
Viewed by 874
Abstract
This study addresses the joint optimization of full-length and short-turning operations for the Nanchang Metro Airport Line, aiming to balance operational efficiency and passenger service quality. A novel mathematical model is proposed, which integrates train schedule design, capacity allocation, and passenger flow assignment [...] Read more.
This study addresses the joint optimization of full-length and short-turning operations for the Nanchang Metro Airport Line, aiming to balance operational efficiency and passenger service quality. A novel mathematical model is proposed, which integrates train schedule design, capacity allocation, and passenger flow assignment into a linear programming framework. The model features three key innovations: (1) precise calculation of passenger waiting times under strict capacity constraints by incorporating dynamic passenger flow distribution and train occupancy thresholds; (2) implicit treatment of train numbers as decision variables, enabling flexible adjustments to service frequency based on time-varying demand patterns; and (3) a linear formulation for direct optimal solution computation, avoiding the complexity of nonlinear constraints through variable substitution and constraint relaxation. The model is validated through a case study of the Nanchang Metro Line 1 (Airport Line), where passenger demand is derived from historical data and flight schedules. Numerical experiments demonstrate that the optimized strategy reduces the number of full-length trains by 53%, achieves a 22% power cost saving, and decreases the waiting time for all passengers by 3.4%. The relevant findings and recommendations can offer valuable guidance to metro companies in making operational decisions related to the full-length and short-turning service plans and schedules. Full article
(This article belongs to the Special Issue Models and Algorithms for Railway Line Planning Problems)
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