Quantification and evaluation for resilience of railway transportation system based on the cloud model

The railway transportation system comprises infrastructure such as tracks, stations, traction power supply, signaling and communication systems, rolling stock and personnel from various disciplines including operations, locomotive maintenance, track maintenance, signaling, rolling stock maintenance, power supply and building maintenance. The system involves numerous components and complex operational processes, exposing it to various external natural disaster risks. Natural disasters encompass hazard-prone environments, disaster-inducing factors and vulnerable entities, posing significant threats to human society. These include meteorological hazards such as strong winds, heavy rain, floods, snowstorms, ice, dense fog and extreme temperatures, as well as geological hazards such as collapses, landslides, mudflows, ground fissures, subsidence, sinkholes and earthquakes. These hazards are characterized by complexity, suddenness and randomness, posing substantial risks to railway transportation operations.

Railway transportation resilience refers to the system’s ability to resist, adapt to and recover from adverse factors such as natural disasters. Resilience is widely applied in disaster risk research, offering a systemic perspective that emphasizes resistance, adaptability and recovery under disaster conditions, providing a novel approach to enhancing system safety. Given the inevitability of natural disaster risks under current conditions, preemptive analysis and evaluation of railway transportation resilience – identifying key influencing factors and vulnerabilities – can improve the system’s resilience and disaster resistance, thereby contributing significantly to railway safety.

In the field of railway and transportation system resilience evaluation, numerous scholars worldwide have conducted research in recent years. Internationally, Adjetey et al. (2016) proposed a simulation model to systematically analyze subsystems such as transportation, power, communication and organization, along with their interdependencies. They studied resilience measurement methods for large-scale railway systems based on passenger delays and capacity, with a case study of the Paris railway system. Azadeh et al. (2018) used data envelopment analysis and decision-making unit evaluation methods to propose a new framework integrating health, safety and ergonomics with resilience engineering, assessing the performance of the Tehran-Karaj railway traction power supply system. Janić (2018) developed a comprehensive analysis model to evaluate the resilience of high-speed rail networks under large-scale disruptive scenarios, focusing on Japan’s high-speed rail network under earthquake conditions. Gu et al. (2020) proposed that network resilience encompasses the ability to resist, absorb, adapt to disturbances and recover from disruptions. Besinovic (2020) reviewed railway transportation resilience research, noting that system-based metrics better capture impacts on transportation services and demand. Mathematical optimization holds great potential for evaluating and enhancing railway resilience, while data-driven methods are promising for detailed post-disruption analysis. The study also identified emerging scientific topics, including learning from historical data, interdependent critical systems and community resilience. Wang et al. (2022) used complex network theory to deconstruct high-speed rail networks into topological, functional and service layers, evaluating comprehensive performance from a traffic accessibility perspective. Ilalokhoin et al. (2023) proposed a network topology model to assess the resilience of the traction power supply system in the UK’s southern regional railway network. Fisher et al. (2023) reviewed resilience-related research and enhancement methods for nonmainline railways, including high-speed rail, mass rapid transit and light rail systems under climate change.

Domestically, Bai Ying (2020) constructed a resilience evaluation index system, applied the Analytic Network Process (ANP) for resilience indicator correlation analysis, and evaluated the resilience of the Changchun-Jilin high-speed rail operational system under severe weather using an ANP-extension cloud comprehensive evaluation model. Ke Yuhao (2022) proposed a dual-layer static network model and a multi-window dynamic network resilience evaluation model to assess the static resilience of high-speed rail networks and their dynamic resilience under rainstorms and earthquakes. Yang et al. (2022) analyzed the characteristics of public transportation network resilience literature, reviewed the application and research status of complex networks in resilience evaluation and optimization and discussed future research trends, including innovative evaluation methods, improved disruption modeling and recovery model exploration. Liu Yuming et al. (2023) viewed major railway projects in complex and challenging regions as project organizational systems composed of heterogeneous entities, extracted key organizational resilience factors from internal and external dimensions, analyzed their logical relationships and hierarchical structures using an Interpretive Structural Model (ISM), and proposed recommendations for enhancing organizational resilience. Fan Yanyan et al. (2023) established a resilience evaluation index system for plateau railway operations, incorporating human, material, environmental, and management factors. They linked resilience dynamics—adaptability, resistance, and recovery – and conducted combined weighting, determining resilience levels based on Euclidean distance. Fan Yanyan et al. (2025) also constructed a safety resilience evaluation index system for railway operations in sandstorm-prone areas, integrating pressure, state and response resilience, and proposed enhancement measures. Liu Enze (2024) used mobile data to identify dynamic passenger flows, developed multiobjective bi-level programming models and proposed multimodal emergency response methods for multimodal coordinated emergency scheduling and railway operation adjustments using data-driven hierarchical deep reinforcement learning. Wu Peng et al. (2024) established a spatiotemporal-weighted urban agglomeration railway passenger network model, simulated link disruptions, proposed dynamic resistance resilience evaluation metrics and conducted case studies based on resistance resilience evaluation models and simulation methods.

In summary, existing research covers resilience concepts, connotations, characteristics, frameworks, evaluation methods and analytical models. However, most studies conflate adaptability and recovery, overlooking the dynamic adjustment of resources in the former and the efficiency of functional reconstruction in the latter. There is a lack of comprehensive evaluation of railway transportation resilience that integrates resistance, adaptability and recovery capabilities.

Against this backdrop, this paper examines railway transportation systems, systematically analyzes resilience factors, establishes a resilience evaluation index system and a cloud model-based evaluation method and constructs an interdimensional correlation analysis model based on cloud model parameters. The expected value reflects system exposure, entropy represents resistance stability and hyper-entropy describes response uncertainty. Quantitative analysis using the cloud model aims to identify vulnerabilities in resilience enhancement, improving the system’s resistance, adaptability and recovery capabilities under disaster risks, thereby elevating overall resilience and providing theoretical and methodological support for building a “high-resilience railway transportation system.”

In this paper, “key influencing factors” refer to those decisively impacting railway transportation resilience, selected through AHP weight analysis, cloud model parameter validation and expert scoring. Screening criteria include high weight (top 20%), strong stability (low entropy) and expert consensus (mean score ≥7.5).

Under natural disasters, railway transportation resilience enables rapid recovery to normal operations, mitigating disaster impacts and ensuring system stability and safety. It embodies the integrated capabilities of resistance, adaptability and recovery. Specifically:

Resistance reflects the ability to withstand natural disasters (e.g. earthquakes, floods and mudflows), maintaining stable operations during disasters.

Adaptability involves adjusting operations and resource allocation to address disaster-induced challenges.

Recovery ensures prompt restoration post-disaster, minimizing disruptions.

Resilience encompasses physical, managerial and operational aspects. Physically, infrastructure robustness, redundancy and disaster protection measures (e.g. earthquake- and flood-resistant bridges and tunnels) are critical. Managerially, organizational methods, emergency response mechanisms and personnel training play key roles.

Thus, resilience factors under natural disasters can be analyzed in terms of exposure, resistance, response capability, adaptability and recovery (Figure 1). Each factor requires quantification based on scoring criteria, data sources and context-specific analysis. Due to space constraints, Table 1 uses flood resistance as an example, detailing scoring methods for design flood frequency, culvert dimensions, foundation depth and embankment slope stability.

The construction of a resilience analysis indicator system for railway transportation serves as the critical foundation for model algorithm application, with its core objective being the establishment of a scientifically quantifiable evaluation framework that systematically reveals railway system performance under natural disaster conditions. Based on the core characteristics of railway transportation resilience in disaster scenarios, this study integrates literature research findings with actual operational data to develop a multi-level assessment system comprising 5 criterion-layer indicators, 23 secondary indicators and 54 tertiary indicators (see Figure 1), forming evaluation dimensions that cover the entire “prevention-response-recovery-adaptation” cycle.

To achieve scientific evaluation of railway transportation resilience levels, this study innovatively combines three methodological tools: First, the analytic hierarchy process (AHP) is used to establish the weight distribution of the indicator system, addressing the prioritization of importance among multi-level indicators. Subsequently, the fuzzy comprehensive evaluation (FCE) method is applied to process fuzzy information during evaluation, enabling holistic quantitative assessment of resilience levels. Particularly, cloud model theory is introduced, using its three digital characteristics – Expectation (Ex), Entropy (En) and Hyper-Entropy (He) – to transform uncertain elements such as expert experience and historical data into computable probability distributions, effectively resolving the pain point of traditional methods where qualitative indicators and quantitative data are difficult to integrate. This three-stage progressive modeling approach ensures both the scientificity of weight allocation and enhances the representation capability of evaluation results for complex system uncertainties.

The AHP method is suitable for multi-level, multicriteria decision-making problems. Its characteristic lies in constructing judgment matrices through expert scoring and calculating relative weights of indicators through hierarchical analysis. In this study, the AHP method is used to calculate weights for primary, secondary and tertiary indicators. By quantifying the relative importance between dimensions through judgment matrices (Table 2), such as the 3:1 weight relationship between “recovery capability” and “adaptation capability,” it demonstrates the significant impact of recovery effectiveness on dynamic adjustment capacity.

After calculating weights through the AHP method, this study defines primary indicators with weight values ≥0.1 (e.g. recovery capability) and their subordinate high-weight tertiary indicators (e.g. recovery quality) as key influencing factors.

Example: Create the following judgment matrix.

Calculation of weights: For each judgment matrix, we need to calculate the eigenvector corresponding to its maximum eigenvalue, and then normalize the eigenvector to obtain the weight of each factor.

Consistency check: To ensure the rationality of the judgment matrix, a consistency check is required. This is typically done by calculating the Consistency Index (CI) and Consistency Ratio (CR). If CR is less than 0.1, the consistency of the judgment matrix is considered acceptable:

Step 1: Calculate the Consistency Index (CI):

Step 2: Find the corresponding Average Random Consistency Index (RI).

Step 3: Calculate the Consistency Ratio (CR), and the results are shown in Table 3:

In fuzzy comprehensive evaluation, it is necessary to quantitatively process the evaluation objects of each factor, determine their membership degree values in the fuzzy evaluation subset, so as to form a fuzzy relation matrix. First, the membership degree of the single-factor fuzzy evaluation subset needs to be calculated:

Among them:

$rij$ represents the membership degree of the i-th index corresponding to the j-th grade;

$vij$ represents the number of people who choose the j-th grade among the i-th index; and

$∑j = 1n vij$ represents the total number of people participating in the evaluation.

By calculating the single-factor evaluation membership degrees of each index, the fuzzy evaluation matrix R is established. On this basis, the membership degree values are further improved through normalization processing. Finally, the fuzzy membership degrees of each first-level index and other relevant indices are combined to form the fuzzy relation matrix:

After determining the fuzzy evaluation matrix R and the weight vector W, the fuzzy comprehensive evaluation vector can be further calculated. For this purpose, a weighted average-type fuzzy operator is usually adopted, and the specific calculation steps are as follows.

First, calculate the comprehensive evaluation vector Z:

Grade assignment and score calculation: Form a row vector $U={u1,u2,⋯,um}$ based on the weights set for different evaluation grades. Finally, calculate the comprehensive score F:

The cloud model parameters（$Ex$⁠, $En$⁠, $He$）establish a cross-level association mechanism through weighted fusion of indicators in each dimension. By applying the cloud model, it can precisely characterize the fuzziness and randomness of influencing factors on the resilience of the railway transportation system, weaken subjective cognitive biases and objectively reflect the actual state of the system from multiple dimensions, thus significantly improving the accuracy and reliability of evaluation results.

According to expert scoring data, combined with literature and actual conditions, the evaluation results and intervals for railway transportation resilience are divided. The calculation formula for the digital characteristics of the standard cloud is as follows:

Among them:

$xmax$ represents the maximum value of the expert scoring interval;

$xmin$ represents the minimum value of the scoring interval; and

k is a constant mainly reflecting the cloud thickness.

Based on expert scores, the cloud model is used to determine the evaluation clouds of each secondary indicator. The calculation of numerical characteristics for the secondary indicator cloud model is as follows:

Among them:

N represents the number of evaluation indicators;

$xi$ represents the score given by experts for each evaluation;

$Exi$⁠, $Eni$⁠, $Hei$ represent the cloud numerical characteristics of each evaluation indicator derived from expert scores.

The calculation formula for comprehensive cloud numerical characteristics is as follows:

Among them:

Ex, En, He represent the cloud numerical characteristics of comprehensive project evaluation indicators calculated based on each evaluation indicator; $wai$ represents the comprehensive weight of each evaluation indicator.

The calculation formula for cloud membership degree is as follows:

Among them: $u$ represents the cloud membership degree; $xu$ represents the actual score given by experts for a certain evaluation indicator.

We invited 21 professional railway practitioners and experts in related fields to score and evaluate the railway operation status, equipment and emergency management system of a certain railway line in 2023, combining historical data and actual operation data. The specific evaluation results are shown in Figure 2 and Figure 3.

Due to the subjectivity and fuzziness of expert scores, this study adopts cloud model theory for data processing to more accurately handle these scoring data and transform qualitative evaluations into quantitative results. The cloud model can effectively address the uncertainty and randomness of linguistic evaluation values, describing the distribution characteristics of evaluation results through three numerical features: Expectation (Ex), Entropy (En) and Hyperentropy (He), thus providing more reliable data support for subsequent comprehensive evaluations. The entropy (En) and hyperentropy (He) parameters of the cloud model can quantify the interactions between dimensions—the more stable the resistance capacity (lower entropy value), the more controllable the volatility of response capacity (hyperentropy value).

Taking Recovery Capacity as an example, its comprehensive weight reaches 0.247 (Table 4), and the recovery quality score is 8.62 (out of 10). Combined with the entropy value En = 1.30 (Table 5), this indicates that the index is a key leverage point for enhancing railway resilience.

Expectation (Ex): Calculated by the weighted sum of FCE evaluation results and AHP weights.

Entropy (En): Calculated based on the deviation between samples and the expectation.

Hyperentropy (He): Derived through the relationship between the second moment and entropy.

After calculating the comprehensive evaluation cloud, it is compared with the evaluation standard cloud diagram to find the standard cloud diagram most similar to the comprehensive evaluation cloud diagram, so as to determine the resilience evaluation grade of railway transportation (see Table 6 for details).

Figures 4 to 8 intuitively reflect the distribution characteristics of resilience levels in each dimension by superimposing the actual cloud model with the standard cloud diagram. The exposure degree cloud of Figure 4 shows a wide distribution between the Qualified and Good intervals, indicating significant variation in risk exposure levels across different sections; it is necessary to enhance overall risk controllability through improved hazard identification and dynamic monitoring. The resistance capacity of Figure 5 shows actual cloud center (Ex = 7.74) is close to the Qualified interval (Ex = 7.5), but the cloud is wider than the standard cloud (En = 1.41 > 0.15), indicating significant differences in the stability of sub-indicators of resistance capacity (such as flood control and seismic resistance). It is necessary to reduce dispersion by strengthening weak links (e.g., upgrading lightning protection facilities). The comparison between the narrow entropy distribution (En = 1.41) of Figure 5 (Resistance Capacity Cloud Model) and the hyperentropy (He = 0.44) of Figure 6 (Response Capacity Cloud Model) shows that the stability of resistance capacity can reduce the uncertainty of the response process, and the two jointly affect the overall system resilience through weight distribution (AHP method). The adaptability cloud of Figure 7 is mainly concentrated in the Good interval, indicating that the line demonstrates strong self-adjustment and optimization capability under changing operating conditions, maintaining high stability when facing uncertainties. The recovery ability cloud of Figure 8 is clearly right-skewed toward the Excellent interval, with a narrow cloud width and low dispersion, indicating high stability and consistency in the post-disaster recovery process. Both recovery speed and repair quality remain at high levels.

According to the principle of maximum membership, in the comprehensive evaluation of railway transportation resilience for a certain railway line, both Recovery Capacity and Adaptability are rated as Good, while Exposure Degree, Resistance Capacity and Response Capacity are Qualified.

The comprehensive cloud model in Figure 9 is right-skewed to the Good interval (Ex = 8.60), indicating that the high efficiency of recovery capacity makes a prominent contribution to overall resilience. This suggests:

In the comprehensive evaluation system of a certain railway line’s transportation resilience, the construction of recovery capacity has formed an institutionalized emergency response mechanism, equipped with an intelligent facility condition monitoring system.

In terms of adaptability construction, the line has significantly improved system collaborative operation efficiency by establishing a dynamic scheduling optimization model and multimodal transportation connection schemes.

At the aspect of exposure degree management, the operation agency has initially constructed a risk geographic information system (GIS), achieving digitized documentation and hierarchical management and control of disaster hazard points.

For resistance capacity cultivation, it has gradually improved the infrastructure health diagnosis system, forming a preventive maintenance system covering rail flaw detection and catenary system inspection.

In response capacity optimization, through improving the emergency plan drill mechanism and establishing a regional joint response platform, the timeliness of emergency response has been significantly enhanced compared with the benchmark value.

Enhancing the resilience of railway transportation is of great significance for ensuring railway transportation safety and promoting social and economic development. This paper studies the influencing factors, evaluation indicators and methods of railway transportation resilience under natural disasters. By analyzing the impact of disaster characteristics on railway transportation resilience, it expounds the influencing factors from multiple aspects, including Exposure Degree, Resistance Capacity, Response Capacity, Adaptability and Recovery Capacity. To scientifically evaluate the resilience level of railway transportation, this paper constructs a correlation analysis model among dimensions based on cloud model characteristic parameters (Expectation, Entropy, Hyperentropy) and AHP weight matrix. The Expectation reflects the exposure degree of the railway system, the Entropy characterizes the stability of resistance capacity and its inhibitory effect on response capacity, and the Hyperentropy describes the uncertain fluctuations of response capacity. The overlay diagram of the cloud model, through visual comparison, not only clarifies the grade attribution of each dimension but also reveals the fuzziness (En) and randomness (He) of index distribution, providing a basis for “targeted optimization” in the construction of “high-resilience railways”.

This study can provide theoretical support and practical guidance for railway transportation emergency management, and the results can be applied to the construction and operation of actual railway transportation systems. For example, in railway engineering construction, the requirements of railway transportation resilience should be fully considered to improve the seismic and disaster resistance capabilities of railway facilities; in railway operation management, scientific and reasonable emergency plans should be formulated to enhance the emergency response capacity of the railway system.

In the future, the evaluation indicators and methods for railway transportation resilience can be further optimized. It is necessary to study the impact mechanisms of different disaster types on railway transportation resilience, and how to improve the accuracy and reliability of resilience evaluation through technical means such as big data and artificial intelligence. Meanwhile, research and practical applications of railway transportation resilience should be further strengthened to continuously improve the disaster resistance and recovery capabilities of railway transportation systems.