Digitalizing Railway Operations: An Optimization-Based Train Rescheduling Model for Urban and Interurban Disrupted Networks

Alireza Eslami; Alireza Yazdiani; Shayan Bafandkar; Yousef Shafahi

REVIEW 5 major objections 6 minor 25 references

Digitalizing Railway Operations: An Optimization-Based Train Rescheduling Model for Urban and Interurban Disrupted Networks

T0 review · 5 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that a two-stage optimization approach can reschedule an entire disrupted rail network, with an adjusted version cutting computation time by up to 88%.

desk verdict The two-stage aggregation idea is legitimate, but the printed IP model is internally inconsistent, so the reported results stand on sand. read the letter →

arxiv 2411.14671 v1 pith:FWTHFAW5 submitted 2024-11-22 math.OC

Shayan Bafandkar , Yousef Shafahi , Alireza Eslami , Alireza Yazdiani This is my paper · ORCID

classification math.OC MSC 90B0690C1090B20

keywords trainreschedulingrailwaydisruptionmanagementintegerprogrammingnetworkaggregationcriticalnodeidentificationopposite-trackoperationspace-timetrajectorytransportationdigitalization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that a two-stage digitalization method can solve the train rescheduling problem across a whole rail network after a disruption, not just on one block. In the first stage it identifies important stations by combining network topology and passenger demand, then compresses the network around them. In the second stage it solves an integer programming model that minimizes total travel time, allows trains to use opposite tracks during partial block failures, and respects station and block capacity. The authors report that on a sparse real network (the Iranian railway) the model produces small delays, and on a crowded test network an adjusted version that only reschedules directly affected trains cuts computation time by up to 88%. If the claim holds, dispatchers could receive a new network-wide timetable within minutes of a disruption instead of waiting for a slow block-by-block recovery process.

What carries the argument

The load-bearing object is the space-time trajectory network with binary arc-occupancy variables $x_{ij}^{k,tt'}$, where each time interval of the horizon is a layer and each block is a possible occupation. On this network the model enforces flow conservation at stations, per-block headways and travel times, dwelling-time and capacity limits, and a conflict constraint $x_{ij}^{k+,tt'}+x_{ij}^{k-,ll'} \le 1 + b_{ij}-dis_{ij}^{gg'}$, which is the mechanism that permits opposite-track running when a double-track block is partially disrupted. The compression step is carried by the criticality index built from normalized node degree and normalized passenger demand, and the runtime-reduction adjustment is a constraint replacement that fixes the movements of every train outside the directly affected set $R^*$, shrinking the problem to the only trains whose paths intersect the disruption.

What would settle it

Run the adjusted model on a network that includes every intermediate station, then compute the optimal schedule for the full network under the same disruption; if the aggregated schedule requires an omitted station for a passing or overtaking maneuver, or its total delay is substantially worse than the full-network optimum, the central claim is false. A simpler observable version: on a two-track line with one passing siding at a low-demand station, disrupt the block just before that siding and test whether the compressed network can still produce a feasible plan without the siding.

Watch

Extended reading notes

Core claim

The central discovery is that network-wide train rescheduling after a block disruption can be decomposed into a network-compression step and an optimization step without sacrificing operational realism. The importance index $C_{ri}=(\bar{D}_i)^{\alpha_1}(\bar{P}_i)^{\alpha_2}\times 100$ combines normalized node degree with normalized passenger demand, and keeping only the highest-scoring nodes yields a medium-scale network on which an integer program with space-time arc variables finds a new timetable. The model lets a train on a double-track block use the opposite track during a partial disruption, and the adjusted variant freezes all trains not directly affected while optimizing only the affected set, which is what produces the reported runtime reduction. On the Iranian case, a two-hour closure of the critical Tehran–Qom block adds 155 minutes of delay relative to the original timetable, and on the crowded test network the adjusted model solves the disruption scenario in 224 seconds.

Load-bearing premise

The whole method rests on the assumption that removing non-important stations, and the passing and overtaking capacity they provide, does not materially change the best rescheduling plan; the paper acknowledges but never tests this.

Editorial extensions

If this is right

If the claim is correct, operators can produce a network-wide secondary timetable in minutes rather than hours after a disruption, because the adjusted model solved the crowded test scenarios in 153 to 224 seconds.
Partial block failures need not stop traffic on a double-track line, since the model reroutes trains onto the opposite track; in the sparse Iranian case a partial closure of the Tehran–Qom link left the original timetable unchanged.
The importance-index compression makes whole-network rescheduling tractable when the original network has many low-demand stations, because only high-demand, high-degree nodes need to be modeled explicitly.
On sparse networks with low trip frequencies, the results suggest that headway assumptions and disruption duration have only modest effects, while the worst delays occur on the busiest block, indicating where operators should focus capacity investment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct consequence the paper leaves untested is that the aggregation step's fidelity determines whether the reported runtime gains are real; if omitted low-demand stations are needed for passing and overtaking, the compressed-network schedule could be infeasible or substantially suboptimal in the actual network.
The same two-stage compression idea could be applied to crew and rolling-stock rescheduling during disruptions, since those problems also shrink when the modeled node set is reduced.
A testable extension is to make the affected-train set adaptive: instead of permanently freezing all unaffected trains, let trains re-enter the optimization when delays propagate to them, which would trade some runtime for robustness on longer disruptions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version. This paper combines two established things—important-node scoring and an integer-programming rescheduling model—and adds opposite-track use and station capacity. That combination, as a single network-wide formulation, is new relative to the cited literature, and the authors say so without overselling. The Iranian case study is a real network, the sensitivity analysis on headway and disruption duration is reasonable, and they openly acknowledge in Section 4.1 that deleting nodes loses overtaking capacity. Those are genuine positives.

The trouble is the model itself. As printed, Constraint 2 only limits a train to one block per identical time interval, so overlapping intervals let a train occupy two blocks at once. Constraint 5 is a time-aggregated flow balance that never ties arrival at a node to departure from it, so the 'path' need not be time-respecting. Constraints 12 and 19 use self-loop variables x_{ii} and x_{jj} that are never defined and are excluded by i≠j elsewhere. These aren't cosmetic typos; the optimization problem stated in Table 2 is not the problem that would produce the reported schedules. With no code or erratum, there is no way to attribute the delay numbers or the 88% runtime figure to the printed equations.

The aggregation-fidelity issue—comparing the aggregated solution against the full network—is a real and acknowledged limitation, but it's secondary to the formulation problem. Also, the runtime reduction in the adjusted model is largely a consequence of shrinking the problem to directly affected trains, which is expected.

So where does that leave us? The core idea has merit, and a careful rewrite could make it a decent contribution. But the current text is not publishable, and a reviewer would need the authors to fix the formulation and supply the actual model code or a precise notation for intervals and self-loops. I would not desk-reject it out of hand—the topic is relevant and the combination is worth a look—but I would send it back for major revision with a clear list of the technical issues. For my own work, I wouldn't cite it in its current form.

Referee Report

5 major / 6 minor

Summary. The paper proposes a two-stage methodology for train rescheduling after disruptions. In the first stage, important nodes are identified using a normalized degree/demand importance index, and the network is aggregated around those nodes. In the second stage, an integer programming model is formulated and solved with CPLEX to reschedule trains, with features such as use of opposite tracks, minimum headways, and dwelling/capacity limits. The method is tested on the Iranian railway network and on a synthetic crowded test network; the adjusted model is reported to cut computational time by up to 88%. The central claim is that this pipeline provides a practical network-wide rescheduling tool for disruption management.

Significance. The practical problem is relevant, and the paper honestly lists several limitations in Section 8, including the lack of validation for the aggregation step. If the formulation were correct, the two-stage idea would be a useful contribution to railway disruption management. However, as printed, the integer programming model is not a valid time-respecting train-path formulation: the key constraints permit overlapping block occupations, the flow-balance constraint is time-aggregated, and two constraints use undefined self-loop variables. Because the reported delays and runtimes are presented as solutions of this model, the empirical claims are not supported by the equations. The aggregation-fidelity concern is real but secondary to the formulation errors.

major comments (5)

[Table 2, Constraint 2] Constraint 2 only restricts the sum of x_{ij}^{k,tt'} over blocks for one and the same indexed interval tt'. Since Constraint 6 explicitly quantifies over overlapping intervals with tt'∩ll'∩gg'≠∅, the model as written allows the same train k to occupy one block in interval [t1,t2] and another block in an overlapping interval [t3,t4] with t1<t3<t2<t4. This violates the stated assumption that a train occupies at most one block at any instant and makes the space-time trajectory ill-defined. The constraint must be written for all pairs of intervals with nonempty intersection, or the interval set must be defined as a partition of the horizon.
[Table 2, Constraint 5] Equation 5 is a time-aggregated flow balance: the left-hand side sums departures and arrivals over all time intervals, while the right-hand side for origins and destinations is specified only at a single time t'=d̄_{O_k}^k. There is no constraint relating the arrival of train k at a node to a later departure from that node, and no temporal ordering of traversed blocks. As written, the auxiliary variables a_i^k and d_i^k, and therefore the objective function, are not tied to a feasible chronological trajectory. A time-expanded network formulation or explicit arrival-before-departure sequencing constraints are needed.
[Table 2, Constraints 12 and 19] Equations 12 and 19 use variables x_{jj}^{k,tt'} and x_{ii}^{k,tt'} for self-loop blocks. These variables are never defined, and Constraint 3 explicitly excludes i=j for block variables, so the dwelling-time and capacity constraints cannot be instantiated for any arc in the model. Equation 12 also has an indexing inconsistency: the first term sums over t' without a matching index in the multiplicand, and the second term uses a loop variable t in the multiplier. These constraints must be corrected before the model can be solved or the reported results reproduced.
[Sections 4.1 and 8] The aggregation step is not validated against the full network. Section 4.1 explicitly states that omitted nodes 'may provide additional capacity for meet-pass and overtaking maneuvers,' and Section 8 repeats that omitting demand from unimportant nodes may negatively affect results, but no experiment compares the rescheduling solution on the aggregated network with the solution on the original network. Without such a comparison, the Iranian case-study delays and the claim that the model can reschedule trains 'across the entire network' are unsupported. This is a load-bearing external-validity gap, albeit secondary to the formulation errors above.
[Section 7, Tables 12 and 13] The reported runtime reduction of up to 88% is obtained by fixing the schedules of trains not in R* and omitting Constraint 20 for affected trains. The feasible regions of the basic and adjusted models are therefore different, so the objective values in Tables 12 and 13 are not directly comparable and the runtime comparison is not an algorithmic speedup on the same problem. The paper should present this as a model-reduction trade-off, with an explicit statement of what is sacrificed (optimality for unaffected trains and the β bound on affected trains), rather than as a pure computational improvement.

minor comments (6)

[Section 4.1, Equation 26] The normalization for nodes with d_i=d_min assigns 0.5/(d_max-d_min) rather than 0; this is an unusual choice that should be justified, since Table 3 shows all degree-1 nodes receiving D_i=0.25.
[Table 1 and Table 2] The notation t,t'∈T is overloaded: T denotes both the horizon length and a set of time points, and the interval [t,t'] is never formally defined. The paper should define the set of interval endpoints, the relation t'=t+δ (or the general interval set), and the interpretation of overlapping intervals in Constraints 2 and 6.
[Figures 5 and 6] The text says the orange hatch indicating the disruption period is in Figure 5, but Figure 5 is captioned as the undisrupted network and Figure 6 is the disrupted one; please clarify which figure contains the hatched disruption interval.
[Table 9] The footnote states that one scenario was initially infeasible because of β and was solved after relaxing the corresponding constraint, yet the sensitivity discussion treats all runs in Tables 8 and 9 uniformly; the infeasibility and relaxation should be addressed in the text.
[Section 5.1] The dummy node at Kashan is introduced manually after the important-node algorithm; please clarify whether this node participates in the importance ranking and how its capacity and dwelling parameters are set.
[Throughout] Several typos and wording issues appear, e.g., 'occupying the opposite tacks' in Section 5.3 and 'to encounter the run time issue' in Section 5.4; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rescheduling model and the 88% runtime comparison are empirical claims, not reductions of the outputs to the inputs.

full rationale

No circular step is present in the derivation chain. The paper formulates an integer programming rescheduling model whose objective is total travel time and whose constraints encode safety, headway, dwelling, and disruption restrictions; the reported delays and runtimes are outputs of solving that model, not quantities fitted into the model. The 88% runtime reduction is an empirical comparison between the basic model and the adjusted model on a test network, not a fitted parameter disguised as a prediction. The important-node procedure is explicitly defined: Equations 26–28 compute a criticality index as a weighted combination of normalized node degree and passenger demand, so selecting high-index nodes is a transparent definitional calculation rather than a hidden reproduction of the result. Hand-set parameters such as the 9-node cutoff, alpha weights, and beta threshold are modeling choices acknowledged in the text, and the paper explicitly notes the possible loss of meet-pass and overtaking capacity from omitted nodes as a limitation. No load-bearing self-citation is used: prior works cited, including the space-time trajectory concept from Yang et al., are external to the present authors, and no uniqueness theorem or equivalent authority from the authors is invoked to force the model choice. Technical inconsistencies in the printed equations, such as the undefined self-loop variables appearing in Constraints 12 and 19, are correctness concerns rather than evidence that the derivation reduces to its inputs. The derivation chain is therefore self-contained.

Assumptions & free parameters 7 free parameters · 6 assumptions · 2 invented entities

The central claim rests on several hand-chosen parameters (alpha weights, cutoff, beta, headway, capacity) and on domain assumptions about train behavior. The aggregation step is an explicit ad hoc assumption. The dummy node and undefined x_{ii} variables are additional modeling constructs with no independent evidence.

free parameters (7)

alpha1 and alpha2 (importance power weights) = 0.6 and 0.4 for Iran case
Chosen by hand based on network structure; they control the relative weight of degree vs passenger demand in Eq. 28.
Number of important nodes (cutoff) = 9 for Iran, implied 10 for test network
Chosen so the aggregated Iran network spans all geographic directions; no principled rule is given.
beta (delay threshold) = 480 min for Iran, 15 min for test network
Set to ensure feasibility and bound delays; scenario-specific.
Time interval length delta = 5 min
Discretization step for the space-time model.
Coefficient 0.8 in Eq. 27 = 0.8
Arbitrary shift in demand normalization; no justification is provided.
Minimum headway h_ij = 5 min, with sensitivity to 10 and 20 min
Assumed constant across all blocks; sensitivity analysis shows low impact on the Iran case.
Capacity Cap_ij = 3 for test network; not specified for Iran
Station/block capacity parameter; chosen ad hoc.

assumptions (6)

domain assumption An initial, satisfactory timetable for undisrupted conditions is available.
Section 3.1 assumption; all rescheduling is measured as deviation from this timetable.
domain assumption All trains move at a constant speed that is the same among all of them.
Section 3.1; used in constraint 7 to fix block travel times.
domain assumption All trains are passenger trains and no freight trains are considered.
Section 3.1; excludes freight-train objectives and capacity interactions.
domain assumption Trains can only overtake at stations, and railway switches are available at all stations.
Section 3.1; simplifies conflict handling.
ad hoc to paper Removing non-critical nodes preserves enough fidelity for a valid network-wide rescheduling.
Section 4.1 and conclusions; relies on omitted nodes being low-demand, but no full-network validation is given.
ad hoc to paper The importance index formula (Eq. 28) weights degree and demand as specified.
Section 4.1; no theoretical justification for the multiplicative form or normalization.

invented entities (2)

Dummy node 'Kashan'
purpose: Added to the aggregated network to separate overlapping Qom-Isfahan and Qom-Kerman routes so trips can be split.
Section 5.1; an artificial node not present in the physical railway network, used solely to make the aggregated graph handle a junction.
Undefined x_{ii} variables
purpose: Appear in constraints 12 and 19 supposedly representing station capacity/occupancy, but never defined in the decision variables.
Section 3.2 and Table 2; a modeling artifact that makes the formulation incomplete.

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Cite this review

Pith. "Pith review of Digitalizing Railway Operations: An Optimization-Based Train Rescheduling Model for Urban and Interurban Disrupted Networks." pith.science (2026). https://pith.science/paper/FWTHFAW5

@misc{pith2026241114671,
  author       = {Pith},
  title        = {Pith review of: Digitalizing Railway Operations: An Optimization-Based Train Rescheduling Model for Urban and Interurban Disrupted Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWTHFAW5}},
  note         = {Machine review of arXiv:2411.14671}
}

read the original abstract

This study introduces a novel methodology for managing train network disruptions across the entire rail network, leveraging digital tools and methodologies. The approach involves two stages, taking into account possible and practical features such as allowing trains to occupy opposite tracks and considering infrastructure capacity for train stops. In the first stage, important nodes within the train network are identified, considering both a topological feature and passenger demand. Subsequently, the network is aggregated based on these important nodes, employing a digital approach to reduce problem complexity. In the second stage, we develop an Integer Programming model for train rescheduling. We then solve this model using the CPLEX solver to evaluate its efficiency. The first case study applies this methodology to the Iranian railway, which is known as a sparse rail network. The results show minimal deviation from the initial train timetable due to the low frequency of trips in each block. Although the approach successfully addresses the train rescheduling problem for various disruption scenarios on the Iranian railway, the excessive computational time required by the optimization model prompts us to make adjustments. Finally, the second case study demonstrates the implementation of the adjusted model in a busy test network. This adaptation significantly reduces computational time by up to 88%. It can be effectively utilized for disruption management in busy networks, where trains need to receive a secondary timetable promptly when facing disruptions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

Poliński, J., & Ochociński, K. (2020). Digitization in rail transport. Problemy Kolejnictwa, (188), 137-148

work page 2020
[2]

Zhuravleva, N., Kliestik , T. (2023). Railway Transport Digitalization: Development Methodology and Effects of Digital Implementation Processes. In: Bencsik, A., Kulachinskaya, A. (eds) Digital Transformation: What is the Company of Today?. Lecture Notes in Networks and Systems, vol 805. Springer, Cham

work page 2023
[3]

Chechenova, L., & Batalova, N. (2023). Digitalization of freight rail transportation as a factor in improving their efficiency. E3S Web of Conferences, 460, 06042

work page 2023
[4]

Montero, J. (2020). The digitalization dilemma in the railway industry. In Handbook on Railway Regulation (pp. 379-396). Edward Elgar Publishing

work page 2020
[5]

W. Jing, X. Xu, Y. Pu, Route redundancy -based approach to identify the critical stations in metro networks: A mean-excess probability measure, Reliability Engineering & System Safety 204 (2020)107

work page 2020
[6]

L. Wang, S. Zheng, Y. Wang, L. Wang, Identification of critical nodes in multimodal transportation network, Physica A: Statistical Mechanics and its Applications 580 (2021) 126170

work page 2021
[7]

L. Sun, Y. Huang, Y. Chen, L. Yao, Vulnerability assessment of urban rail transit based on multi-static weighted method in Beijing, China, Transportation Research Part A: Policy and Practice 108 (2018) 12–24

work page 2018
[8]

Beˇsinovi´c, R

N. Beˇsinovi´c, R. F. Nassar, C. Szymula, Resilience assessment of railway networks: Combining infrastructure restoration and transport management, Reliability Engineering & System Safety 224 (2022) 108538

work page 2022

Show all 25 references

[9]

J. Tang, L. Xu, C. Luo, T. S. A. Ng, Multi -disruption resilience assessment of rail transit systems with optimized commuter flows, Reliability Engineering & System Safety 214 (2021) 107715

2021
[10]

L. Xu, T. S. A. Ng, A robust mixed -integer linear programming model for mitigating rail transit disruptions under uncertainty, Transportation Science 54 (5) (2020) 1388 –1407

2020
[11]

L. Yang, X. Zhou, Z. Gao, Credibility-based rescheduling model in a double-track railway network: a fuzzy reliable optimization approach, Omega 48 (2014) 75–93

2014
[12]

C. Gong, J. Shi, Y. Wang, H. Zhou, L. Yang, D. Chen, H. Pan, Train timetabling with dynamic and random passenger demand: A stochastic optimization method, Transportation Research Part C: Emerging Technologies 123 (2021) 102963

2021
[13]

B¨armann, P

A. B¨armann, P. Gemander, L. Hager, F. N¨oth, O. Schneider, EETTlib—energy- Bafandkar et al. 24 efficient train timetabling library, Networks 81 (1) (2023) 51–74

2023
[14]

P. G. Sessa, V. De Martinis, A. Bomhauer -Beins, U. A. Weidmann, F. Corman, A hybrid stochastic approach for offline train trajectory reconstruction, Public Transport 13 (2021) 675–698

2021
[15]

Y. Wang, U. A. Weidmann, H. Wang, Using catastrophe theory to describe railway system safety and discuss system risk concept, Safety Science 91 (2017) 269–285

2017
[16]

N. Leng, U. Weidmann, Discussions of the reschedule process of passengers, train operators and infrastructure managers in railway disruptions, Transportation Research Procedia 27 (2017) 538–544

2017
[17]

Herrigel, M

S. Herrigel, M. Laumanns, J. Szabo, U. Weidmann, Periodic railway timetabling with sequential decomposition in the PESP model, Journal of rail transport planning & management 8 (3-4) (2018) 167–183

2018
[18]

Toletti, M

A. Toletti, M. Laumanns, U. Weidmann, Coordinated railway traffic rescheduling with the resource conflict graph model, Journal of Rail Transport Planning & Management 15 (2020) 100173

2020
[19]

G. C. Caimi, Algorithmic decision support for train scheduling in a large and highly utilised railway network, ETH Zurich, (2009)

2009
[20]

Ghaemi, O

N. Ghaemi, O. Cats, RMP. Goverde, Railway disruption management challenges and possible solution directions, Public Transport 9 (2017) 343–364

2017
[21]

Kecman, F

P. Kecman, F. Corman, A. D’Ariano, Rescheduling models for railway traffic management in large-scale networks, Public Transport 5 (2013) 95–123

2013
[22]

Goverde, Railway timetable stability ana lysis using max -plus system theory, Transportation Research Part B: Methodological 41(2) (2007) 179–201

RMP. Goverde, Railway timetable stability ana lysis using max -plus system theory, Transportation Research Part B: Methodological 41(2) (2007) 179–201

2007
[23]

Goverde, A delay propagation algorithm for large -scale railway traffic networks, Transportation Research Part C, Emerging Technologies 18(3) (2010) 269 –287

RMP. Goverde, A delay propagation algorithm for large -scale railway traffic networks, Transportation Research Part C, Emerging Technologies 18(3) (2010) 269 –287

2010
[24]

Beˇsinovi´c, Resilience in railway transport systems: a literature review and research agenda, Transport Reviews 40 (4) (2020) 457–478

N. Beˇsinovi´c, Resilience in railway transport systems: a literature review and research agenda, Transport Reviews 40 (4) (2020) 457–478

2020
[25]

L. Ge, S. Voß, L. Xie, Robustness and disturbances in public transport. Public Transport 14 (2022) 191–261

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Poliński, J., & Ochociński, K. (2020). Digitization in rail transport. Problemy Kolejnictwa, (188), 137-148

work page 2020

[2] [2]

Zhuravleva, N., Kliestik , T. (2023). Railway Transport Digitalization: Development Methodology and Effects of Digital Implementation Processes. In: Bencsik, A., Kulachinskaya, A. (eds) Digital Transformation: What is the Company of Today?. Lecture Notes in Networks and Systems, vol 805. Springer, Cham

work page 2023

[3] [3]

Chechenova, L., & Batalova, N. (2023). Digitalization of freight rail transportation as a factor in improving their efficiency. E3S Web of Conferences, 460, 06042

work page 2023

[4] [4]

Montero, J. (2020). The digitalization dilemma in the railway industry. In Handbook on Railway Regulation (pp. 379-396). Edward Elgar Publishing

work page 2020

[5] [5]

W. Jing, X. Xu, Y. Pu, Route redundancy -based approach to identify the critical stations in metro networks: A mean-excess probability measure, Reliability Engineering & System Safety 204 (2020)107

work page 2020

[6] [6]

L. Wang, S. Zheng, Y. Wang, L. Wang, Identification of critical nodes in multimodal transportation network, Physica A: Statistical Mechanics and its Applications 580 (2021) 126170

work page 2021

[7] [7]

L. Sun, Y. Huang, Y. Chen, L. Yao, Vulnerability assessment of urban rail transit based on multi-static weighted method in Beijing, China, Transportation Research Part A: Policy and Practice 108 (2018) 12–24

work page 2018

[8] [8]

Beˇsinovi´c, R

N. Beˇsinovi´c, R. F. Nassar, C. Szymula, Resilience assessment of railway networks: Combining infrastructure restoration and transport management, Reliability Engineering & System Safety 224 (2022) 108538

work page 2022