Research on an Optimization Method for Metro Train Formation Based on Virtual Coupling Technology

Comments 1:

Literature Review: The current literature review is insufficient.  Referencing only 22 sources is not comprehensive enough to provide a solid foundation for the study.More importantly, the article fails to clearly identify the research gap in the existing literature that this work aims to fill.A thorough review is needed to properly position the contribution of this paper.

Response 1: [We sincerely thank the reviewer for this important comment. We fully agree that the literature review in the original submission was too limited and did not sufficiently highlight the research gap. In the revised manuscript, we have therefore enriched and reorganized the literature review section. Specifically, we have added recent studies on virtual coupling control methods, operational optimization strategies, and lightweight vehicle design, including the comprehensive reviews by Felez and Vaquero-Serrano (2023) and Xun et al. (2022), as well as additional research on flexible formation and capacity optimization. These additions not only broaden the scope of the review but also provide a more comprehensive context for our work.

Furthermore, we have made the research gap more explicit. While most existing studies emphasize control strategies and technical feasibility, few works focus on demand-driven optimization of train formation. Our study addresses this gap by proposing a load-factor-driven virtual formation optimization method that directly links passenger demand patterns with train formation strategies through a mixed-integer linear programming model. This positions our contribution more clearly within the current state of the art.

The corresponding revisions can be found in the Introduction (page 2, lines 70–77; page 3, lines 113–141) of the revised manuscript.

Once again, we sincerely thank the reviewer for this helpful suggestion, which has allowed us to strengthen the foundation of our paper and clarify its academic contribution.]

“[In addition to these technical contributions, recent review papers have also provided a comprehensive overview of virtual coupling research. For example, Felez and Vaquero-Serrano错误!未找到引用源。 summarized the state of the art and identified key challenges for implementing virtual coupling in practice, while Xun et al.错误!未找到引用源。 surveyed control methods and operational frameworks for railway virtual coupling. These reviews highlight the rapid development of virtual coupling technology but also indicate that most existing work emphasizes control strategies and technical feasibility. This gap motivates the present study to focus on demand-driven optimization of train formation.]”

“[To address this gap, this study proposes a dynamic virtual formation optimization method based on a full load rate threshold, aiming to enhance the real-time adaptability of train formations and provide more precise dynamic scheduling solutions for urban metro systems. Building on this objective, the contributions of this paper are threefold: it introduces a novel load-factor-driven optimization framework that explicitly connects passenger demand dynamics with formation strategies; it develops a mixed-integer linear programming (MILP) model that simultaneously coordinates unit allocation, turnover scheduling, and virtual coupling/decoupling operations; and it validates the proposed approach through a real-world case study, demonstrating significant improvements in both passenger waiting time and operating costs compared with conventional fixed-formation schemes. Together, these contributions extend the literature on virtual coupling by shifting the focus from technical feasibility to demand-oriented optimization, thereby providing new insights for sustainable metro operations.]”

Comments 2:

Methodology: The manuscript currently lacks a schematic diagram of the proposed method.  Adding a clear, well-labeled diagram is essential for readers to understand the methodology and for the potential reproducibility of the study.

Response 2:

We are truly grateful for the reviewer’s thoughtful and constructive suggestion. We fully agree that a schematic diagram is essential for enhancing readers’ understanding of the methodology and for ensuring reproducibility. In line with this valuable advice, we have carefully prepared and added a new flowchart (now included as Figure 2) that presents the methodological framework of our study.

As shown in Figure 2 (attached below), the framework consists of five main steps: (1) Input data, including passenger demand (OD matrix and time distribution), timetable, and operational constraints; (2) Passenger flow distribution, where spatiotemporal passenger arrivals are analyzed based on OD assignments; (3) Load factor calculation, in which carriage load factors are computed and categorized into comfort, medium, and crowded zones, triggering corresponding decoupling modes; (4) MILP model construction and optimization, where the objectives (minimizing passenger waiting time and operating cost) and constraints (unit conservation, turnover scheduling, passenger capacity, and coupling feasibility) are formulated and solved with commercial solvers such as CPLEX or Gurobi; and (5) Output results, including optimized train unit allocation, virtual coupling/decoupling strategies, and corresponding performance improvements.

Figure 2. Methodological framework of the proposed optimization approach for virtual coupling in urban rail transit.

The corresponding revision has been made on page 5, lines 170–181 of the revised manuscript. We sincerely believe that this addition makes the methodology clearer, improves readability, and facilitates reproducibility for future research.

Once again, we sincerely thank the reviewer for this valuable comment, which has guided us in making an important improvement to the manuscript.

Comments 3:

Conclusion and Discussion: The conclusion needs to be improved and expanded.  A significant omission is the lack of a discussion section.  This section is crucial for interpreting the results in a broader context and, most importantly, for transparently addressing the limitations of the proposed method.

Response 2:

We sincerely thank the reviewer for this thoughtful and constructive suggestion, which we find extremely helpful for improving the manuscript. We fully agree that the paper required both a dedicated discussion section and an expanded conclusion to better interpret the results, acknowledge limitations, and highlight the significance of our contributions.

In response, we have revised the manuscript by creating a combined section entitled “4. Discussion and Conclusion.” In the new Discussion part, we interpret the findings in the context of related research, highlight the novelty of integrating segmented load-factor thresholds into a MILP framework, and explicitly acknowledge the limitations of the study, including the focus on a single-line case, the assumption of stable passenger demand, and the limited scope of optimization objectives. We also present directions for future work, such as extending the framework to multi-line networks, integrating real-time passenger flow forecasting and disruption management, and adopting advanced optimization methods to enhance computational efficiency.

In the Conclusion part, we have expanded the summary of our findings by incorporating the numerical results of the case study. Specifically, compared with the fixed 6-car consist scheme, passenger waiting time and operating costs were reduced by 4.2% and 11.6%, respectively; compared with the fixed 8-car scheme, operating costs were reduced by 48.4% with only a 4.3% increase in waiting time. These results demonstrate the efficiency, sustainability, and practical applicability of the proposed approach. The revised conclusion further emphasizes the broader implications of this work for improving unit utilization, optimizing energy use, and supporting sustainable urban rail operations, as well as its potential deployment in multi-depot, multi-line, and real-time scheduling scenarios.

The corresponding revisions can be found in Section 4. Discussion and Conclusion (pages 18–20, lines 491–562) of the revised manuscript.

We would once again like to sincerely thank the reviewer for this valuable comment. It has helped us substantially improve the clarity, completeness, and academic contribution of the paper.

Comments 1:

line 15–18: the sentence “train unit allocation and turnover scheduling as core decision variables” could be simplified for clarity. Consider rephrasing to “with train unit allocation and turnover scheduling as the main decision variables” to avoid redundancy in technical descriptions.

Response 1:

We sincerely thank the reviewer for this helpful suggestion. We agree that the original phrasing was somewhat redundant. Following the reviewer’s advice, we have revised the sentence to:
“with train unit allocation and turnover scheduling as the main decision variables.”

This modification improves readability while preserving the original meaning. The revision can be found in the Abstract (page 1, lines,14-15) of the revised manuscript.

Once again, we are very grateful to the reviewer for this valuable comment, which has helped us improve the clarity of the paper.

Comments 2:

line 9–12: the expression “by considering the coupling between the temporal-spatial characteristics of passenger demand and the real-time dynamic formation of train units” is lengthy and difficult to parse. A clearer phrasing could be “by integrating temporal-spatial passenger demand with real-time dynamic train formation.”

Response 2:

We sincerely thank the reviewer for this constructive suggestion. We fully agree that the original phrasing was unnecessarily long and could be simplified for better readability. Following the reviewer’s advice, we have revised the expression to:
“by integrating temporal-spatial passenger demand with real-time dynamic train formation.”

This change improves clarity and conciseness without altering the intended meaning. The revision can be found in the Abstract (page 1, lines 9–12) of the revised manuscript.

Once again, we are very grateful to the reviewer for this valuable comment, which has helped us improve the clarity and accessibility of the paper.

Comments 3:

line 33–36: The sentence “the traditional train formation scheme cannot flexibly adjust the train formation according to the passenger flow demand when coping with the peak and tidal passenger flow” is repetitive. Consider simplifying to “traditional train formations cannot flexibly adapt to peak and tidal passenger flows.”

Response 3:

We sincerely thank the reviewer for this helpful suggestion. We fully agree that the original sentence was repetitive and could be simplified for clarity. Following the reviewer’s advice, we have revised the sentence to:
“traditional train formations cannot flexibly adapt to peak and tidal passenger flows.”

This modification improves conciseness while preserving the original meaning. The revision can be found in the Introduction (page 1, lines 33–34) of the revised manuscript.

Once again, we are very grateful to the reviewer for this valuable comment, which has improved the readability of our paper.

Comments 4:

line 45–48: The description “the Spanish CAF company realized the virtual grouping operation of two trams at a speed of 20km/h” could be expressed more precisely as “CAF successfully implemented a virtual coupling of two trams operating at 20 km/h.”

Response 4:

We sincerely thank the reviewer for this precise and constructive suggestion. We agree that the revised phrasing is clearer and more accurate. Following the reviewer’s advice, we have modified the sentence to:
“CAF successfully implemented a virtual coupling of two trams operating at 20 km/h.”

This change improves the precision of the expression while keeping the original meaning intact. The revision can be found in the Introduction (page 2, lines 44–45) of the revised manuscript.

Once again, we are very grateful to the reviewer for this valuable comment, which has enhanced the clarity and accuracy of the paper.

Comments 5:

line 115–123: The explanation of the operating structure is somewhat fragmented. Consider merging sentences for smoother flow, for example: “This study focuses on a bidirectional urban rail line with depots at both terminal stations, where trains can rapidly perform virtual decoupling and coupling procedures at designated turnback stations.”

Response 5:

We sincerely thank the reviewer for this constructive suggestion. We agree that the original description was fragmented and could be improved for smoother readability. Following the reviewer’s advice, we have revised the text to:
“This study focuses on a bidirectional urban rail line with depots at both terminal stations, where trains can rapidly perform virtual decoupling and coupling procedures at designated turnback stations.”

This revision merges the original sentences into a more concise and fluent statement while preserving the original meaning. The modification can be found in the Methods section (page 4, lines 144–146) of the revised manuscript.

Once again, we are very grateful to the reviewer for this valuable comment, which has improved the clarity and readability of the paper.

Comments 6:

line 235–244: The explanation of the decoupling rules based on load factor thresholds is overly detailed in prose form. Consider condensing the text and referring directly to the table or figure to improve readability.

Response 6:

We sincerely thank the reviewer for this constructive suggestion. We fully agree that the original description was unnecessarily detailed and could be streamlined for better readability. Accordingly, we have revised the text by condensing the explanation and referring directly to Figure 3, which clearly presents the decoupling rules based on segmented load factor thresholds.

In the revised version, the prose has been shortened to briefly explain the logic—that lower load factors correspond to more units being decoupled, medium load factors result in fewer units being decoupled, and higher load factors lead to no decoupling—while the detailed thresholds are left for the reader to interpret directly from the figure.

This modification improves the conciseness and clarity of the manuscript, ensuring that the rules are both accessible and easy to understand. The revision can be found in the Methods section (page 10, lines 279–284) of the revised manuscript.

Once again, we are very grateful to the reviewer for this valuable comment, which has helped us enhance the readability and presentation of our work.

Comments 7:

line 387–394: The description of Train 10 and Train 11 is technically correct but stylistically complex. Simplifying to “Train 10 detached two units at Station 6, which were then virtually coupled to Train 11, increasing its capacity” would improve clarity.

Response 7:

We sincerely thank the reviewer for this valuable suggestion. We fully agree that the original description was somewhat lengthy and could be simplified for better readability. Following the reviewer’s advice, we have revised the sentence to:
“Train 10 detached two units at Station 6, which were then virtually coupled to Train 11, increasing its capacity from 4 to 6 units (see dashed boxes in Figures 5 and 6).”

This modification improves clarity and conciseness while retaining the technical accuracy of the description. The revision can be found in the Results section (page 16, lines 438–439) of the revised manuscript.

Once again, we sincerely thank the reviewer for this constructive comment, which has helped us enhance the clarity of the manuscript.

Comments 8:

line 452–459: In the conclusion, the phrase “reduces operating costs by 48.4% compared to the fixed 8-car consist scheme with only a 4.3% increase in waiting time” is strong but would be clearer if the comparison was introduced earlier in the sentence, e.g., “Compared with the fixed 8-car consist scheme, the proposed method reduces operating costs by 48.4% while increasing waiting time by only 4.3%.”

Response 8:

We sincerely thank the reviewer for this valuable suggestion. We fully agree that placing the comparison at the beginning of the sentence makes the expression clearer and more straightforward. Following the reviewer’s advice, we have revised the sentence in the conclusion to:
“Compared with the fixed 8-car consist scheme, the proposed method reduces operating costs by 48.4% while increasing waiting time by only 4.3%.”

This revision improves clarity while preserving the original meaning and emphasis on the efficiency of the proposed method. The modification can be found in the Conclusion (page 18, lines 500–502) of the revised manuscript.

Once again, we sincerely thank the reviewer for this constructive comment, which has enhanced the readability and precision of our conclusion.

Comments 9:

line 129–134: The text states that “virtual coupling and decoupling operations are implemented at eligible stations, enabling dynamic adjustment of capacity resources across both spatial and temporal dimensions.” However, the criteria for selecting “eligible stations” are not explained. It would be helpful to clarify what engineering, operational, or demand-based factors make a station suitable for these operations.

Response 9:

We sincerely thank the reviewer for this insightful comment. We fully agree that the criteria for selecting “eligible stations” required further clarification to enhance the rigor and transparency of the manuscript.

Accordingly, we have revised the text to explicitly describe the factors considered in station selection. Eligible stations are now defined based on three main criteria: (i) engineering feasibility, meaning sufficient track length and signaling capacity to support safe operations; (ii) operational feasibility, ensuring that coupling and decoupling can be scheduled without disrupting train services; and (iii) demand relevance, where the station is located in a section with significant passenger flow imbalances, making such operations beneficial.

This revision provides a clearer rationale for why certain stations are suitable for virtual coupling and decoupling operations and ensures that the methodology is both transparent and reproducible. The modification can be found in the Methods section (page 4, lines 160–164) of the revised manuscript.

Once again, we sincerely thank the reviewer for this valuable suggestion, which has helped us improve the completeness and academic quality of the paper

Comments 10:

line 349–357: The case study describes the chosen simulation horizon (5:00–15:00), but it is not clear why the afternoon peak period was excluded. Since tidal flows often occur in both morning and evening peaks, the authors should justify this choice or discuss whether the method is equally valid under evening peak demand conditions.

Response 10:

We sincerely thank the reviewer for this thoughtful comment. We agree that the choice of the simulation horizon required further clarification. In the revised manuscript, we have explained that the afternoon peak was not included because the studied metro line exhibits more concentrated and representative tidal flows in the morning, which makes this period sufficient for validating the effectiveness of the proposed method.

At the same time, we have emphasized that the proposed optimization framework is not restricted to morning operations. Since it is designed to adapt dynamically to any tidal demand scenario, it can be equally applied to evening peak conditions or other demand patterns.

The corresponding revision can be found in the Case Study section (page 14, lines 396–400) of the revised manuscript.

Once again, we sincerely thank the reviewer for this valuable suggestion, which has helped us improve the completeness and applicability of our work.

Comments 1:

References and quotations of sources style across text needs to be standarized

Response 1:

We sincerely thank the reviewer for pointing out this important detail. We fully agree that reference formatting and in-text citation style must be consistent throughout the manuscript. In response, we have carefully checked all references and standardized the formatting in accordance with the journal’s guidelines. Specifically, all in-text citations have been revised to the numerical style (e.g., [1], [2–4]), and the reference list has been reformatted to ensure consistency in style, punctuation, and capitalization.

This revision ensures that the manuscript now fully complies with the required reference style, improving its readability and professionalism.

Once again, we sincerely thank the reviewer for this helpful suggestion, which has improved the overall quality and presentation of the paper.

Comments 2:

Urban rail transit is a very general term, including trams, light rail, subways, commuter rail and more. The modes listed above do not operate on the same principle, e.g. it's rather impossible to find a light rail trains which is formed of 6-8 units, as hard it is to point a subway or commuter railway in which all cars are powered and therefore can be independently uncoupled.  It would improve the clarity of the paper, to name the example system more precisely.

Response 2:

We sincerely thank the reviewer for this insightful comment. We fully agree that the term “urban rail transit” is too broad and may cause ambiguity, as it covers multiple modes with different operating principles and formation characteristics.

In response, we have carefully revised the manuscript and replaced the general expression with the more precise term “metro systems” to accurately reflect the research object of this study. In addition, the title of the paper has also been updated accordingly to ensure clarity and consistency.

Once again, we sincerely thank the reviewer for this valuable suggestion, which has greatly improved the precision and clarity of our manuscript.

Comments 1:

Contributions can be relevant  witin several innovations and, overall, the article is well structured and presents interesting results applied to virtual coupling.  The proposed methodology is adequate.

The conclusions are sufficiently well justified.

However, the innovation is not well justified, and there are some improvements that should be considered.

On the one hand, the main innovations and contributions of this work should be better highlighted.

Response 1:

We sincerely thank the reviewer for this valuable comment. We fully agree that the innovations and contributions of the paper should be made more explicit to improve the clarity and impact of the work. In the revised manuscript, we have strengthened the end of the Introduction by clearly highlighting the main contributions of the study. Specifically, we emphasize that this paper: (i) introduces a novel load-factor-driven optimization framework that explicitly connects passenger demand dynamics with train formation strategies; (ii) develops a mixed-integer linear programming (MILP) model that simultaneously coordinates unit allocation, turnover scheduling, and virtual coupling/decoupling operations; and (iii) validates the proposed method through a real-world case study, demonstrating significant improvements in both passenger waiting time and operating costs compared with conventional fixed-formation strategies.

The corresponding revision has been made at the end of the Introduction (page 3, lines 113–141) of the revised manuscript. By presenting these contributions in a concise and structured way, the novelty and relevance of the work are now much clearer.

Once again, we sincerely thank the reviewer for this constructive suggestion, which has helped us improve the positioning and clarity of the paper.

Comments 2:

On the other hand, there are other studies that should be mentioned and analysed in the section on the current state of the art.  In particular, there are several review articles on the type of virtual coupling that can serve as a reference.

The main references for comparison are listed below:

Felez, J., & Vaquero-Serrano, M. A. (2023).  Virtual coupling in railways: A comprehensive review.  Machines, 11(5), 521.

Xun, J., Li, Y., Liu, R., Li, Y., & Liu, Y. (2022).  A survey on control methods for virtual coupling in railway operation.  IEEE Open Journal of Intelligent Transportation Systems, 3, 838-855.

Based on these references, the state of the art section should be improved and the main contributions of the paper clarified.

Response 2:

We sincerely thank the reviewer for pointing out these important references. We fully agree that the state-of-the-art section should be strengthened by incorporating recent comprehensive reviews on virtual coupling. In the revised manuscript, we have therefore added a discussion of these works. Specifically, we note that Felez and Vaquero-Serrano (2023) provide a broad survey of virtual coupling research and identify key challenges for its implementation, while Xun et al. (2022) review control methods and operational frameworks. These reviews highlight that most existing studies emphasize control strategies and technical feasibility, which contrasts with the demand-driven optimization perspective adopted in our work.

The corresponding revision has been made in the literature review section of the Introduction (page 2, lines 70–77) of the revised manuscript.

Once again, we sincerely thank the reviewer for this helpful suggestion, which has enabled us to strengthen the positioning of our study within the broader research landscape.