An Improved Optimal Cloud Entropy Extension Cloud Model for the Risk Assessment of Soft Rock Tunnels in Fault Fracture Zones

Abstract

Existing risk assessment approaches for soft rock tunnels in fault-fractured zones typically employ single weighting schemes, inadequately integrate subjective and objective weights, and fail to define clear risk. This study proposes a risk-grading methodology that integrates an enhanced game theoretic weight-balancing algorithm with an optimized cloud entropy extension cloud model. Initially, a comprehensive indicator system encompassing geological (surrounding rock grade, groundwater conditions, fault thickness, dip, and strike), design (excavation cross-section shape, excavation span, and tunnel cross-sectional area), and support (initial support stiffness, support installation timing, and construction step length) parameters is established. Subjective weights obtained via the analytic hierarchy process (AHP) are combined with objective weights calculated using the entropy, coefficient of variation, and CRITIC methods and subsequently balanced through a game theoretic approach to mitigate bias and reconcile expert judgment with data objectivity. Subsequently, the optimized cloud entropy extension cloud algorithm quantifies the fuzzy relationships between indicators and risk levels, yielding a cloud association evaluation matrix for precise classification. A case study of a representative soft rock tunnel in a fault-fractured zone validates this method’s enhanced accuracy, stability, and rationality, offering a robust tool for risk management and design decision making in complex geological settings.

1. Introduction

With the rapid development of transportation infrastructure in China [1,2,3,4,5,6,7,8,9], soft rock tunneling projects have grown in scale and have encountered increasingly complex geological conditions. In particular, during construction through fault fracture zones, the superposition of low-strength surrounding rock, fractured fault structures, and rich water characteristics has significantly elevated tunneling risks. Geological hazards—such as water inrush, collapse, and severe deformation—occur frequently, not only severely impeding construction progress but also posing serious threats to personnel safety. Therefore, establishing a risk classification assessment method that is both scientifically rigorous and practically operable onsite is of significant theoretical value and can be used in engineering applications for pre-construction early warning, design optimization, and dynamic monitoring and control during construction.

Extensive research on tunnel risk assessment methodologies has been conducted both domestically and internationally, yielding significant advancements. Sun et al. [10] employed the Analytic Hierarchy Process method within an enhanced two-dimensional cloud model to objectively quantify risks associated with shield tunneling in karst regions. Guo et al. [11] developed a risk evaluation framework for shield tunnel crossings beneath existing structures by integrating variable weight theory into the cloud model. Xun et al. [12] combined Building Information Modeling (BIM) with intuitionistic fuzzy Dempster–Shafer evidence theory to perform safety assessments of subsea tunnels using multi-source information fusion. Chai et al. [13] proposed an FANP–FGRA–cloud model framework to enhance the precision of multi-attribute fire risk discrimination in metro stations. Liang et al. [14] applied game theoretic combined weighting alongside a two-dimensional cloud model to categorize construction risks in void-area tunnels, demonstrating the visualization benefits of their methodology.

However, these models remain limited by their reliance on single weighting schemes and fixed parameters. For example, the method proposed by Chen et al. [15], which integrates a trapezoidal cloud model with a Bayesian network to enhance the risk transmission mechanism for shield tunneling beneath existing tunnels, still depends on empirically determined initial values. The weights were reassigned by Guo et al. [16] using an improved nonlinear fuzzy analytic hierarchy process (FAHP), improving consistency but leaving boundary stability susceptible to subjective bias. Reliability-based weighting was combined with an Extension cloud model by Jiang et al. [17] to assess water-related hazards in karst tunnels. A multi-damage indicator cloud model framework was introduced by Shen et al. [18] for seismic resilience evaluation, offering fresh perspectives on tunnel seismic behavior; however, cloud entropy and hyper-entropy parameters still require manual specification. Additionally, a two-dimensional cloud model dynamic early warning system was developed by Sun et al. [19], and a cloud model evidence-theory-based fire assessment for municipal culverts was presented by Niu et al. [20], both underscoring the need for the cloud model parameters and weights to enhance accuracy. Finally, a real-time updating model for large soft rock deformations was proposed by Bai et al. [21], a multi-source intelligent fusion approach to tunnel collapse risk prediction was developed by Wu et al. [22], and a variable weight Extension cloud theory for shield tunnel risk evaluation was advanced by Sun et al. [23], collectively highlighting the importance of integrating subjective and objective weighting with cloud parameters. Gray Relational Analysis (GRA) [24] and the cloud model risk evaluation approach each emphasize different aspects: GRA is adept at revealing the closeness of each alternative to the ideal target via sequence geometric similarity; the cloud model, employing forward and reverse cloud generators, converts qualitative assessments into stochastic, fuzzy “cloud droplets,” thereby fully capturing the uncertainty inherent in judgments.

In summary, an innovative risk-grading methodology is urgently needed to adaptively optimize cloud model parameters and deeply integrate subjective and objective weightings, thereby improving the accuracy and robustness of soft rock tunnel risk evaluations in fault-fractured zones. To address this need, this study proposes an enhanced evaluation framework that combines a game theoretic subjective–objective weighting scheme with an optimal entropy extension cloud model. The framework comprises the following: (1) the development of a multi-source indicator system; (2) the optimization of subjective and objective weights; (3) Cloud parameter optimization; (4) the computation of the extension cloud correlation degree. Applications to representative engineering cases demonstrate that the proposed approach surpasses existing models in scientific weight allocation, boundary delineation, and grading stability and exhibits strong concordance with in situ monitoring data, thus providing robust technical support for risk management in tunnels.

2. Framework for Risk Grade Evaluation of Soft Rock Tunnels Crossing Fault Fracture Zones

2.1. Overview of Risk Assessment Process

To address common challenges in constructing soft rock tunnels through fault fracture zones—namely, ambiguous index boundaries, high data randomness, and insufficient model adaptability—this study proposes a comprehensive, dynamically traceable framework for risk grade evaluation. The framework comprises three core components: the development of a multi-source indicator system; the acquisition and optimization of subjective and objective weights; and risk grade assessment based on an improved, optimal cloud entropy extension cloud model. This framework enables pre-construction risk warning and provides quantitative support for design optimization while also facilitating real-time monitoring and rapid response to anomalies during construction, thereby ensuring enhanced tunnel safety.

• Construction of a multi-source indicator system: From the three dimensions of geological environment, design characteristics, and support systems, key risk indicators affecting the safety of soft rock tunnels in fault fracture zones are screened and extracted to establish a comprehensive evaluation indicator system.
• Weight acquisition and integration: Subjective and objective weights are obtained separately using the analytic hierarchy process (AHP), entropy weighting, variation coefficient, and CRITIC methods, and they are then optimally combined within an improved game theoretic framework.
• Extension cloud model evaluation: Under adaptive adjustment by an entropy optimization algorithm, the Extension cloud model is used to compute each indicator’s cloud correlation degree for different risk levels; a comprehensive evaluation matrix is then constructed, yielding the final risk-grading results.

2.2. Construction of Indicator System

The indicator system is shown in Figure 1. To comprehensively capture the risks associated with constructing soft rock tunnels in fault fracture zones, this study classifies indicators into three main categories, based on both risk causation and critical construction stages:

• Geological characteristics: The geological environment fundamentally influences soft rock tunnel stability. Key indicators include surrounding rock classification, groundwater conditions, fault thickness, fault dip, and fault strike, which jointly reflect stability.
• Design characteristics: Tunnel design parameters substantially alter the stress field and failure mechanisms of the surrounding rock. Indicators include excavation cross-sectional shape, excavation span, and tunnel cross-sectional area, which quantify disturbance and stress distribution within the rock mass.
• Support characteristics: The support system is critical for maintaining tunnel cross-section stability. An appropriate support scheme effectively constrains rock deformation. Indicators include initial support stiffness, support timing, and construction advance length, which characterize the strength and timing effects of the support system on the surrounding rock.

2.3. Weight Determination Method

In this study, subjective weights were first determined using the analytic hierarchy process (AHP) to fully respect expert judgment, and objective weights were subsequently derived by combining the entropy weight method, coefficient of variation method, and CRITIC method to capture the data’s intrinsic distribution characteristics. Finally, an improved game theoretic model was applied to optimally fuse subjective and objective weights, yielding a more reliable set of weights for risk assessment.

2.3.1. Subjective Weights via AHP

To leverage expert experience in assessing the risks of soft rock tunnels in fault fracture zones, pairwise comparisons were conducted among the three main indicator groups—geological, design, and support—to construct a hierarchical structure and calculate weights [25]:

Construction of the judgment (pairwise comparison) matrix: For any two indicators i and j at the same level, experts provided relative importance scores on a 1–9 scale, forming the matrix A = [a_ij].

$$\begin{gathered} a_{ij}={\displaystyle \frac{1}{a_{ji}}} \\ {a}_{ii}=1 \end{gathered}$$

(1)

Initial weight estimation: The product of each row’s elements was computed, the geometric mean was taken, and then normalization was conducted:

$${\mathrm{w}}_i={\displaystyle \frac{{M_i}^{1/n}}{{\displaystyle \sum _{k=1}^n{M_k}^{1/n}}}}$$

(2)

Consistency check: The maximum eigenvalue λ_max, the consistency index (CI), and the consistency ratio (CR) were determined:

$$\begin{gathered} CI={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}} \\ CR={\displaystyle \frac{CI}{RI}} \end{gathered}$$

(3)

In this context, CI represents the random consistency index. A judgment matrix is deemed to satisfy the acceptable consistency criterion when the consistency ratio (CR) is below 0.10. If this threshold is exceeded, the pairwise comparisons must be iteratively adjusted until the required consistency is obtained.

After multiple rounds of expert scoring and consistency verification, the resulting subjective weight vectors for the three indicator categories and their sub-indicators are obtained:

$$W^{(s)}=\left[w_1,w_2,\mathrm{\dots },w_{11}\right]$$

(4)

W^(s) is the vector of subjective weights obtained from expert scoring, where each wi denotes the weight assigned to the i-th sub-indicator.

2.3.2. Objective Weight Determination via the Entropy–CV–CRITIC Method

To maximize the objectivity of characterizing each indicator’s informational properties, three objective weighting methods were employed: the entropy method, which quantifies information content; the coefficient of variation, which measures relative dispersion; and the CRITIC approach, which assesses inter-criterion correlation. The resulting weights were subsequently compared, consolidated, and refined using the coefficient of concordance test and an information analysis to bolster the objectivity and robustness of the final weight distribution [26,27].

• Data preprocessing and standardization

An original decision matrix X(m × n) was constructed, where m is the number of samples, and n is the number of indicators.

In directional transformation, indicators for which “smaller is better” were inverted, ensuring that higher values consistently denote superior performance.

• Ratio normalization

Ratio normalization was applied for the entropy weight method and CRITIC method.

The entropy value of each indicator was calculated as follows:

$$e_j=-k{\displaystyle \sum _{i=1}^m{p}_{ij}}In(p_{ij}),k={\displaystyle \frac{1}{In\mathrm{m}}}$$

(5)

where e_j denotes the entropy of the jth indicator, p_ij is the normalized proportion contributed by the ith sample to that indicator, m is the total number of samples (evaluation objects), and k = 1/lnm is a scaling constant that confines the entropy value to the interval [0, 1].

The degree of divergence was computed and normalized to yield entropy weights:

$${w_j}^{(E)}={\displaystyle \frac{d_j}{{\displaystyle \sum _{j=1}^n{d}_j}}}$$

(6)

• Coefficient of variation method

The mean and standard deviation of each normalized indicator were computed. The coefficient of variation for each indicator was determined as follows:

$$C{V}_j={\displaystyle \frac{{\sigma }_j}{{\mu }_j}}$$

(7)

Here, CV_j denotes the coefficient of variation of indicator j, computed as the ratio of its standard deviation (σ_j) to its mean (μ_j).

Coefficient of variation weights were then normalized:

$${w_{\mathrm{j}}}^{(\mathrm{v})}={\displaystyle \frac{C_j}{{\displaystyle \sum _{j=1}^n{C}_j}}}$$

(8)

• CRITIC method

The standard deviation of the normalized data matrix was computed. A correlation degree matrix R = [rᵢⱼ] was constructed. The information for each indicator was calculated as follows:

$$C_j={\delta }_j{\displaystyle \sum _{k=1}^n(1-r_{jk}})$$

(9)

Here, C_j denotes the information weight of j; δ_j is the standard deviation of that criterion’s normalized scores, capturing its contrast intensity; r_jk is the correlation degree between criteria j and k, quantifying their mutual conflict; and n is the total number considered.

The CRITIC weights were then normalized:

$${w_j}^{(C)}={\displaystyle \frac{C_j}{{\displaystyle \sum _{j=1}^n{C}_j}}}$$

(10)

• Composite objective weight calculation

To mitigate biases or limitations inherent to any single objective weighting method, all three sets of objective weights underwent consistency testing and information analysis and were then optimally combined to produce more robust objective weights, as detailed below:

The objective weight matrix W was constructed:

$$W=\left[w^{(E)},W^{(V)},W^{(C)}\right]$$

(11)

Consistency test: Pairwise τ coefficients were computed among the three weight vectors to assess inter-method agreement.

Conflict analysis: If the global divergence S between any two methods meets the conditions, the arithmetic mean of their weights is adopted directly; otherwise, the information for each method is calculated and normalized:

$$H_{\mathrm{i}}=-{\displaystyle \sum _{j=1}^3{a}_{ij}In}{a}_{ij}, a_{ij}={\displaystyle \frac{{\gamma }_{ij}}{{\displaystyle \sum _{i=1}^3{\gamma }_{ij}}}}$$

(12)

where ${\gamma }_{ij}$ denotes the weight assigned by method i to indicator j.

$$W(O)={\left[{w_1}^{(\mathrm{comb})},\mathrm{\dots },{w_n}^{(comb)}\right]}^T$$

(13)

Finally, the objective weights for each indicator within the “Geological,” “Design,” and “Support” categories were computed.

2.3.3. Game Theoretic Integration of Subjective and Objective Weights

To reconcile the advantages of subjective weights derived from the analytic hierarchy process (AHP) with objectively derived weights (entropy, coefficient of variation, and CRITIC), we employed a game theoretic framework that models the two weight sets as players and optimizes their linear combination coefficients to achieve an integrated fusion [28,29].

• Linear combination: The composite weight vector W is defined as follows:

$$K={\displaystyle \sum _{j=1}^n({\alpha }_j{w_j}^{(s)}}+{\beta }_j{w_j}^{(O)}), {\alpha }_j+{\beta }_j=1$$

(14)

n is the number of evaluation criteria; w_j^(s) and w_j^(o) are the subjective AHP-derived and objective weights for j, respectively; α_j and β_j are their combination coefficients constrained by α_j + β_j = 1; and the summation yields K, the composite weight that fuses expert judgment with data objectivity.

This formulation preserves expert judgment from the AHP while incorporating data objectivity from entropy, CV, and CRITIC.

• Game objective: A Lagrangian function was constructed to minimize the sum of squared differences between subjective and objective weight vectors, yielding first-order optimality conditions:

$${\min }_{\left\{{\alpha }_j,{\beta }_j\right\}}{\displaystyle \sum _{j=1}^n({\alpha }_j{w_j}^{(S)}}+{\beta }_j{w_j}^{(O)}-\overline{w_{\mathrm{j}}}{)}^2$$

(15)

where $\overline{w_{\mathrm{j}}}$ may be taken as the mean of the subjective and objective weight vectors.

• Solution and normalization: The optimization problem was solved to obtain the optimal coefficient vector K*, which was then normalized to produce the final composite weight vector:

$$W^{*}={[{w_1}^{*},{w_2}^{*},\mathrm{\dots },{w_n}^{*}]}^T$$

(16)

The optimized composite weights W* are employed as the core parameters in the enhanced optimal entropy extension model, providing a balanced weighting scheme that integrates expert knowledge with data objectivity for the risk grading of soft rock tunnels in fault fracture zones.

3. Evaluation Model Based on Improved Extension Cloud Theory

As shown in Figure 2, building on the subjective–objective weighting results, an improved extension cloud theory is introduced to develop a risk assessment model for tunnels that accommodates both qualitative and quantitative uncertainties. The model follows a three-step process:

(1)

The evaluation element, traditionally represented by a fixed value characteristic V, is transformed into cloud digital characteristics—expectation (Ex), entropy (En), and hyper-entropy (He)—to capture the inherent fuzziness and randomness of each indicator.

(2)

Cloud entropy is nonlinearly optimized according to the “3En rule” and the “50% correlation rule” to determine optimal entropy values.

(3)

A comprehensive cloud correlation matrix is constructed using the cloud correlation function, from which risk grades and credibility factors are derived, achieving a classification that balances clear boundaries with inherent uncertainty.

3.1. Construction of Extension Cloud Model

To address the inability of the object R = (N, C, V) to express the uncertainty inherent in tunnel surrounding rock indicators, this study replaces its fixed value feature with the cloud digital characteristics. Let the original sample values of an indicator be {xᵢ}; then we obtain the following metrics:

• Expectation (Ex) represents the central tendency of the sample distribution:

$$E_{xi}={\displaystyle \frac{{\mathrm{c}}_{\max }+{\mathrm{c}}_{\min }}{2}}$$

(17)

• Entropy (En) quantifies the fuzziness of the overall cloud correlation distribution:

$$E_{ni}={\displaystyle \frac{c_{\max }-c_{\min }}{6}}$$

(18)

• Hyper-entropy (He) measures the dispersion of the entropy:

$$H\mathrm{e}=s$$

(19)

Here, c_max and c_min denote the upper and lower bounds of a given risk grade for the indicator, and s is an empirically determined constant. Each indicator forms a cloud that simultaneously captures correlation fuzziness and statistical randomness.

3.2. Optimal Entropy Determination

3.2.1. The “3En” Rule and the 50% Association Degree” Rule

• The 3En Rule: The cloud model should cover 99.7% of the sample values within the interval [Ex − 3En, Ex + 3En] to ensure clear grade delineation:

$$E_{{\mathrm{n}}^{\prime }1}={\displaystyle \frac{c_{\max }-c_{\min }}{6}}$$

(20)

• The “50% Association Degree” Rule: The boundaries between adjacent grades are defined where the cloud correlation degree equals 0.5, ensuring a fuzzy transition:

$$E_{{\mathrm{n}}^{\prime }1}={\displaystyle \frac{c_{\max }+c_{\min }}{2.3548}}$$

(21)

These two rules determine the cloud entropy from complementary perspectives, each with its strengths. However, they may yield conflicting correlation assessments when applied independently.

3.2.2. Optimal Cloud Entropy Determination

To improve the reliability of the assessment while balancing the discreteness and fuzziness of risk level classification, an optimal cloud entropy computation method is proposed. First, for a given indicator score x, k cloud models corresponding to the predetermined risk levels are constructed. For each level, the expectation, hyper-entropy, and the entropy computed by the “3En” rule and the “50% association degree” rule are recorded. On this basis, the optimal cloud entropy is obtained via combinatorial optimization, enabling the adaptive adjustment of entropy values across cloud models of different risk levels [30,31,32].

In the standard extension cloud model, the entropy En′ is regarded as a normally distributed random variable with mean En and variance He; approximately 99.7% of the generated cloud droplets lie within the interval [En − 3He, En + 3He] (the “3En” criterion). Based on this, the inner and outer correlation degree curves are defined as follows:

$$l_1=\exp (-{\displaystyle \frac{{(x-Ex)}^2}{2{(En-3He)}^2}})$$

(22)

$$l_2=\exp (-{\displaystyle \frac{{(x-Ex)}^2}{2{(En+3He)}^2}})$$

(23)

Here, x is the value under evaluation, Ex is the cloud expectation (central value), En denotes the cloud entropy, and He is the hyper-entropy that quantifies the dispersion of the entropy itself.

The intersection of these two curves yields the minimum association degree produced by the “3En rule” and the maximum association degree produced by the “50% association degree ” rule. The expected association degree based on the optimal cloud entropy is denoted as ud. The formula for the deviation of the maximum association degree among the three is given by the following:

$$\Delta u_{\max }={({u^{″}}_{\max }d-ud)}^2+{(ud-{u^{\prime }}_{\min }d)}^2$$

(24)

In Equation (24), u″_maxd denotes the upper-bound (maximum) association degree for level d obtained with the 50% association degree (“50% relevance”) rule, u′_mind denotes the lower-bound (minimum) association degree for that level derived via the 3En rule, and ud is the expected association degree for level d calculated with the optimal cloud entropy algorithm.

To combine the advantages of both entropy determination rules, the optimal cloud entropy is defined to minimize the maximum association degree deviation across all risk levels. A nonlinear programming model involving End and Hed is formulated and solved, resulting in the optimal cloud entropy for a single evaluation indicator.

3.3. Integrated Cloud Correlation Matrix and Risk Level Output

3.3.1. Integrated Cloud Correlation Degree

First, the cloud correlation matrix U for every indicator at each grade is multiplied by the established indicator weight vector; the resulting product yields the integrated cloud correlation degree for each grade.

3.3.2. Repeated Monte Carlo Sampling to Enhance Reliability

Because the entropy employed in computing the cloud correlation degree is generated as a normally distributed random variable with the prescribed En and He as parameters, stochastic perturbations are unavoidable during the evaluation. Consequently, the Monte Carlo procedure is executed repeatedly so that the overall central tendency and dispersion of the assessment results can be characterized.

3.3.3. Definition and Significance of Credibility Factor

Based on the expectation and entropy obtained from the multiple simulations, a credibility factor CF is defined as follows:

$$\theta ={\displaystyle \frac{ERn}{ERx}}$$

(25)

ERn is defined as the central tendency of the final rating obtained from repeated sampling, whereas ERx is defined as the variance, which quantifies the degree of dispersion among the samples.

This factor quantifies the reliability of the assessment. A value approaching zero indicates highly concentrated results with minimal variability, reflecting high credibility. Conversely, a larger CF signifies greater dispersion among simulation outcomes and increased uncertainty, warranting cautious interpretation.

4. Case Verification

The AHP incorporates expert knowledge; the entropy weight method and coefficient of variation capture the dispersion of data; CRITIC further integrates conflict–contrast information among indicators; and finally, the game theory offsets the biases of different weighting methods to achieve a solution. This establishes a “subjective–objective–conflict–game” chain. Specifically, we first employed the “3 En” Rule and the “50% association degree ” rule to dynamically optimize the cloud entropy. Next, we conducted 1000 times via Monte Carlo resampling and computed the association degree distributions for each grade to establish discrimination windows. Through these improvements, our method achieves significantly enhanced discrimination accuracy and stability even when the association degrees of adjacent grades are extremely close.

4.1. Determination of Risk Indicators

By integrating geological survey reports, construction monitoring data, and expert review feedback for soft rock tunnels traversing fault fracture zones, this study systematically validates the enhanced optimal cloud entropy extension cloud model combined with the game theoretic subjective–objective weighting scheme in order to assess its applicability and risk discrimination accuracy under complex geological conditions. To establish a scientific and rational tunnel risk-grading system, the widely adopted five-level classification method from international tunnel risk management guidelines and the five-level framework of the ITA-AITES “Guidelines for Tunnel Risk Assessment” were fully referenced. Existing tunnel collapse risk models were then integrated and refined—based on the principle of maximizing the comprehensive cloud correlation of the score vector—to detail the stability characteristics and recommended countermeasures for each risk grade. Ultimately, risk levels were categorized into five grades (I–V), corresponding to “Low risk,” “Relatively low risk,” “Moderate risk,” “Relatively high risk,” and “High risk,” with the key parameter thresholds for each grade clearly defined. The grading ranges for the indicators were determined by referencing the pertinent literature and adhering to relevant industry standards [33,34,35].

Given the pronounced uncertainty of soft rock geology conditions and the dynamic evolution of tunnel construction, the grading thresholds of all key parameters were defined within three indicator systems—geology, design, and support—so that the resulting classes are both operationally practical and suitable for stage-specific risk control and decision support.

Table 1. Comprehensive evaluation grades and characteristic descriptions.

Grade	Risk Level	Risk Characteristics	Corresponding Control and Support Measures
I	Low risk	Overall stability of surrounding rock; negligible plastic deformation; controllable seepage; minimal fluctuation in monitoring data.	Conventional primary support with secondary lining; routine monitoring; maintain monitoring frequency for early detection and response.
II	Relatively low risk	Risk of slight plastic deformation of surrounding rock; localized, uniform seepage; overall within controllable limits.	Enhanced local drainage (efficient drainage channels + collection pipes); local rock bolts with wire mesh shotcrete primary support; dynamic monitoring with periodic indicator evaluation.
III	Moderate risk	Noticeable extension of plastic zone; increased likelihood of intermittent water inflow or small-scale mud bursts.	Full-section rock bolts with wire mesh shotcrete primary support; advance small-diameter drainage or pre-drainage grouting; install 24 h online displacement monitoring alarms.
IV	Relatively high risk	Sharp increase in surrounding rock instability; continuous development of plastic failure zones; frequent water inrush, mud bursts, or flowing sand.	Advanced high-pressure forceful grouting (multistage pressurization, multipoint layout); combined support of pipe shed + anchor cables + shotcrete; high-density monitoring network (displacement, stress, water pressure).
V	High risk	Extreme surrounding rock instability; uncontrollable water inrush and mud such as collapse posing severe threats to construction and safety.	High-pressure pre-grouting of water-rich surrounding rock + multilayer pipe shed + anchor cables + wire mesh + composite shotcrete support; full work stoppage, alignment adjustment, or tunnel bypass if necessary.

and

Table 2. Comprehensive evaluation levels and feature descriptions.

Objective Level	Criterion Level	Factor Level	Grade
			I	II	III	IV	V
	Geological factors	Surrounding rock grade	I [80, 100]	II [60, 80]	III [40, 60]	IV [20, 40]	V [0, 20]
		Groundwater conditions	Capillary water [0, 20]	Pore water [20, 40]	Fracture water [40, 60]	Karst water [60, 80]	Confined water [80, 100]
		Fault thickness	[0, 0.5]	[0.5, 2]	[2, 5]	[5, 8]	[8, 100]
		Fault dip	[80, 90]	[65, 80]	[45, 65]	[30, 45]	[0, 30]
		Fault strike	[0, 10]	[10, 30]	[30, 45]	[45, 80]	[80, 90]
	Design factors	Excavation Cross-section shape	Circular (or near-circular) cross-section [0, 20]	Elliptical cross-section [20, 40]	Semi-circular arch with flat invert [40, 60]	Horseshoe-shaped cross-section [60, 80]	Rectangular (box-culvert-type) cross-section [80, 100]
		Excavation span	[0, 6]	[6, 9]	[9, 12]	[12, 15]	[15, 50]
		Tunnel cross-sectional area	[0, 30]	[30, 50]	[50, 80]	[80, 120]	[120, 300]
	Support Factors	Initial support stiffness	High stiffness [80, 100]	Moderately high stiffness [60, 80]	Moderate (engineering-optimal) stiffness [40, 60]	Moderately low stiffness [20, 40]	Very low stiffness [0, 20]
		Support installation timing	Highly appropriate [80, 100]	Appropriate [60, 80]	Moderately appropriate [40, 60]	Inappropriate [20, 40]	Highly inappropriate [0, 20]
		Construction step length	Highly appropriate [80, 100]	Appropriate [60, 80]	Moderately appropriate [40, 60]	Inappropriate [20, 40]	Highly inappropriate [0, 20]

presents the composite score ranges and the characteristic thresholds of the geological, design, and support indicators for each risk grade, thereby offering a clear and systematic basis for subsequent project implementation.

Figure 3 shows the standard interval of the optimal cloud-entropy Extension cloud model. A weighting scheme based solely on expert judgment may introduce subjective bias. Consequently, the outcomes may reflect individual experience or preference rather than objective criteria. Therefore, when establishing weights, data-driven methodologies should be integrated to correct for potential biases and enhance overall reliability. Next, we use the data from

Table 3. Standard indicators used for selecting engineering case studies.

Tunnel	Surrounding Rock Grade	Groundwater Conditions	Fault Thickness	Fault Dip Angle	Fault Strike	Excavation Cross-Section Shape	Excavation Span	Tunnel Cross-Sectional Area	Initial Support Stiffness	Support Installation Timing	Construction Step Length
1	10	50	20	75	5	10	5	78.5	50	30	50
2	10	30	10	70	5	10	15.63	191.77	30	30	30
3	10	70	30	85	5	70	17	239.6	30	30	10
4	50	70	15	75	5	70	13.1	180.8	30	10	30
5	10	30	10	30	3	50	4.16	15.85	10	10	10
6	10	50	50	10	8	70	11.8	141.6	30	30	30
7	30	50	10	50	0	70	11.34	103	30	30	30

to conduct the case analysis. To overcome the limitations inherent in using solely subjective or objective weighting methods, an improved game theoretic approach was incorporated into the weight determination process. Subjective weights derived via the AHP and objective weights obtained through the entropy weight, coefficient of variation, and CRITIC methods are treated as players in a game model. The objective function minimizes the deviation between the composite weight and the original subjective and objective weights, thereby achieving optimal synergy. This method not only balances expert judgment and data-driven results but also significantly enhances the objectivity and robustness of the assessment process. Figure 4 presents the subjective, objective, and composite weights for each evaluation indicator as calculated by the AHP, the Entropy–CV–CRITIC methods, and the improved game theoretic model. Both the optimal cloud entropy-based Extension cloud model proposed in this paper and the conventional Extension cloud model are based on the same weight analysis.

By integrating the evaluation data with the improved extension cloud evaluation model, the tunnel under evaluation and its corresponding cloud models are obtained.

Table 4. Numerical characteristics (Ex, En, He) of grade level cloud models.

	I	II	III	IV	V
1	(90, 3.333, 1.111)	(70, 3.333, 1.111)	(50, 3.333, 1.111)	(30, 3.333, 1.111)	(10, 3.333, 1.111)
2	(10, 3.333, 1.111)	(30, 3.333, 1.111)	(50, 3.333, 1.111)	(70, 3.333, 1.111)	(90, 3.333, 1.111)
3	(0.25, 0.083, 0.028)	(1.25, 0.25, 0.083)	(3.5, 0.5, 0.167)	(6.5, 0.5, 0.167)	(54.5, 15.333, 5.111)
4	(85, 1.667, 0.556)	(72.5, 2.5, 0.833)	(55, 3.33, 1.111)	(37.5, 2.5, 0.833)	(15, 5, 1.667)
5	(5, 1.667, 0.556)	(20, 3.333, 1.111)	(37.5, 2.5, 0.833)	(62.5, 5.833, 1.944)	(85, 1.667, 0.556)
6	(10, 3.333, 1.111)	(30, 3.333, 1.111)	(50, 3.333, 1.111)	(70, 3.333, 1.111)	(90, 3.333, 1.111)
7	(3, 1, 0.333)	(7.5, 0.5, 0.167)	(10.5, 0.5, 0.167)	(13.5, 0.5, 0.167)	(32.5, 5.833, 1.944)
8	(15, 5, 1.667)	(40, 3.333, 1.111)	(65, 5, 1.667)	(100, 6.6667, 2.222)	(210, 30, 10)
9	(90, 3.333, 1.111)	(70, 3.333, 1.111)	(50, 3.333, 1.111)	(30, 3.333, 1.111)	(10, 3.333, 1.111)
10	(90, 3.333, 1.111)	(70, 3.333, 1.111)	(50, 3.333, 1.111)	(30, 3.333, 1.111)	(10, 3.333, 1.111)
11	(90, 3.333, 1.111)	(70, 3.333, 1.111)	(50, 3.333, 1.111)	(30, 3.333, 1.111)	(10, 3.333, 1.111)

presents the numerical characteristics (Ex, En, and He) of the cloud models at each grade level.

4.2. Comparison of Model Performance

In light of the aforementioned necessity and sufficiency analysis, as well as common practice in the literature, the present implementation of the optimal cloud entropy extension cloud model likewise employs one thousand Monte Carlo iterations to ensure the stable convergence of the cloud entropy estimates while maintaining computational efficiency [36,37,38]. In the field of tunnel and underground engineering, the optimum cloud entropy Extension cloud model is widely employed precisely because of its inherent suitability for “small-sample, high-uncertainty” scenarios. To comprehensively contrast the performance of the improved optimal cloud entropy extension cloud model with that of the conventional extensional cloud model in grading the risk of soft rock tunnels traversing fault fracture zones, seven representative tunnel engineering cases were selected. Using a unified five-level framework (Grades I–V), the cloud correlation degree associated with each grade was calculated for both models. Given the intrinsic random disturbance effects of the cloud model, the composite evaluation score of every case at every grade was recalculated through 1000 Monte Carlo simulation iterations, thereby yielding robust mean correlation degrees, fluctuation intervals, and corresponding credibility indices. Emphasis is placed on large sample sizes and repeated sampling in Monte Carlo simulations; similarly, extensive statistical testing is required to validate the reliability of cloud model applications. The stability and reproducibility of the simulated cloud model numerical characteristics are only attained when the sample size is sufficiently large, thus avoiding inconsistent evaluation results arising from random fluctuations. More reliable and stable evaluation results are thus produced.

To comprehensively and systematically evaluate the performance gap between the improved optimal cloud entropy extension cloud model (“improved model”) and the conventional extensional cloud model (“baseline model”) in the risk level assessment of soft rock tunnels intersecting fault fracture zones, the present section analyzes seven identical tunnel cases under uniform indicator weights and 1000-run Monte Carlo simulations. Figure 5 shows the risk assessment results of the two models. Two analytical layers are adopted—key quantitative metrics and assessment consistency—to dissect each model’s strengths and weaknesses in terms of classification clarity, stability, evaluation accuracy, and engineering applicability and to explore methodological innovations and prospects for wider adoption. Overall, the improved model achieves significant optimization in five indices—mean correlation degree, maximum correlation degree fluctuation, mean credibility index, credibility index fluctuation, and grade agreement rate—thereby delivering more reliable and fine-grained decision support for the early warning of soft rock tunnel risks in fault fracture zones.

After comparing with actual onsite conditions, we found that Tunnel 1 experienced leakage (Grade III); Tunnels 2, 3, 4, 6, and 7 were subjected to sudden surges of water and mud (Grade IV); and Tunnel 5 collapsed (Grade V). Tunnel 1 was categorized as moderate-risk; Tunnels 2, 3, 4, 5, and 7 were categorized as relatively high risk; and Tunnel 5 was classified as high-risk. Figure 6 below presents images of the water–mud inrush in Tunnel 2.

Table 5. A comparison of key performance indicators between the traditional extension cloud model and the optimal cloud entropy extension cloud model.

Performance Indicator	Traditional Extension Cloud Model	Optimal Cloud Entropy Extension Cloud Model	Relative Change
Mean maximum composite cloud association degree	0.3968	0.6123	+54.31%
Standard deviation of maximum composite cloud association degree	0.1013	0.0771	−23.89%
Mean credibility factor	0.0022	0.0008	−63.64%
Standard deviation of credibility factor	0.0017	0.0003	−82.35%
Evaluation grade consistency rate (1000 times)	85.714%	100%	+14.286%

illustrates the superiority of the improved model in performance metrics.

4.2.1. Classification Clarity and Boundary Sample Discrimination

(1)

Overall Improvement in Clarity

The minimum maximum composite cloud correlation degree among the seven samples was 0.5364 (Tunnel 2) when using the improved model—an increase of 106.15% compared with the 0.2602 value obtained with the original model (Tunnel 2). Correspondingly, the average maximum composite cloud correlation degree rose from 0.3968 to 0.6123 (+54.31%), thereby significantly enhancing the overall clarity of risk grade delineation and effectively mitigating classification challenges posed by samples with low composite cloud correlation degrees.

(2)

Comparison of Representative Boundary Samples

Tunnel 2 (a boundary-ambiguous sample)

The original model yielded a maximum composite cloud correlation degree of only 0.2598 for Grade V—slightly below the 0.2602 threshold. In the improved model, the Grade IV composite cloud correlation degree surged to 0.5364 (+106.15%), accurately capturing high-risk features and enhancing the accuracy of actual risk level identification.

Tunnel 1 (a boundary-critical sample)

The original model produced a maximum composite cloud correlation degree of 0.3016 (Grade V). The improved model achieved a coefficient of 0.5386 for Grade III—78.58% increase, nearly doubling—and yielded classification results that closely matched the actual high-risk level, demonstrating superior discriminative ability for critical samples.

(3)

Enhanced Convergence of Integrated Cloud Correlation Distribution

The standard deviation of the maximum composite cloud correlation degree decreased from 0.1013 to 0.0771, indicating that the sensitivity of the classification boundary to random perturbations was substantially reduced. For boundary-critical samples (e.g., Tunnel 2), the original model’s maximum coefficient of 0.2602 fell within Grade V, whereas the improved model achieved a value of 0.5364 (Grade IV), representing a 106.15% increase. This enhancement enables consistent risk grade output under minor input fluctuations, thereby preventing the grade jumps observed in the original model due to slight metric variations.

4.2.2. Stability and Consistency Evaluation

(1)

Significant Reduction in Mean Credibility Factor

As a credibility factor for assessing the dispersion of multiple simulation results, the improved model’s mean decreased from 0.0022 to 0.0008—a reduction of 63.64%—indicating that the improved model produces more consistent outputs across repeated evaluations. Its standard deviation decreased from 0.0017 to 0.0003—a reduction of 82.35%—further confirming the high robustness of the evaluation results.

(2)

Dramatic Increase in Grade Consistency Rate

Across 1000 independent simulations, the evaluation grade concordance rate of the improved model increased from 85.714% to 100% (+14.286%), with nearly all runs producing the same risk grade. The original model’s confidence factor fluctuated within ±0.0017 over time, whereas the improved model’s fluctuations were constrained to approximately ±0.0003, demonstrating its markedly enhanced resilience to random perturbations.

4.2.3. Assessment Accuracy and Practical Applicability

In the high-risk (Grades IV–V) and moderate-risk (Grade III) zones, the improved model further demonstrated superior evaluation accuracy and enhanced cross-sample consistency:

(1)

Reduced Individual Sample Error

For Tunnel 1, for example, the Grade III composite cloud correlation degree increased from 0.3016 in the original model to 0.5386 (+78.58%), while the corresponding confidence factor decreased from 0.0036 to 0.0013 (−63.89%), yielding a marked reduction in evaluation error magnitude.

(2)

Engineering Practicality

In fault-induced fractured soft rock zones, complex fracture and heterogeneous stress fields are commonly encountered. Optimized cloud entropy weighting effectively balances the uncertainties across multiple indicators. When combined with repeated Monte Carlo simulations, this enhancement not only preserves the precision of the assessment results but also improves stability in continuous monitoring scenarios.

In fault-induced fractured soft rock zones, complex fracture and heterogeneous stress fields are commonly encountered. Optimized cloud entropy weighting effectively balances the uncertainties across multiple indicators. When combined with repeated Monte Carlo simulations, this enhancement not only preserves the precision of the assessment results but also improves stability in continuous monitoring scenarios.

In summary, the optimized entropy cloud extensible cloud model outperforms the traditional extensible cloud model across classification clarity, evaluation stability, accuracy, and engineering applicability. It is recommended as the preferred technical approach to the risk grade assessment of fractured soft rock tunnels, and it shows promise for broader application in geotechnical engineering.

5. Conclusions

This study proposed and validated a risk-grading method for soft rock tunnels in fault fracture zones based on an improved optimal cloud entropy extension cloud model. Compared to the conventional extension cloud model, the enhanced model achieved significant improvements in key performance metrics and demonstrated superior feasibility for engineering applications. The main conclusions are as follows:

(1)

Fusion of Subjective and Objective Weighting Enhances Scientific Rigor

By combining the analytic hierarchy process (AHP) with the entropy weight, coefficient of variation, and CRITIC methods, and optimizing the composite weights via an improved game theoretic approach, effective synergy between expert judgment and objective data was achieved. This weighting mechanism retains deep expert insights while mitigating sensitivity to random data deviations, substantially enhancing the robustness and credibility of the evaluation system.

With the model’s improvement, the mean maximum composite cloud correlation was raised from 0.3968 to 0.6123 (+54.31%), and the standard deviation was reduced from 0.1013 to 0.0771 (−23.81%). The higher mean indicates a more sensitive differentiation among varying risk levels, while the lower variability ensures the consistent discrimination of boundary-critical samples, thereby providing clearer demarcations for tunnel risk stratification.

The mean confidence factor was reduced from 0.0022 to 0.0008 (−63.64%), and the standard deviation was narrowed from 0.0017 to 0.0003 (−82.35%), indicating that the improved model’s outputs are more tightly concentrated across repeated evaluations. Moreover, the consistency rate over 1000 simulations increased from 85.714% to 100% (+14.286%), thereby demonstrating the method’s enhanced robustness and repeatability.

(2)

Engineering Applications

The algorithm maintains a concise workflow and can rapidly perform risk grade predictions once cloud-based digital features are extracted from field survey data. Compared with the conventional extension cloud model, the improved optimal cloud entropy extension cloud exhibits superior stability and robustness when processing high-dimensional, multi-source, and coupled information.

(3)

Dynamic Indicator System Optimization and Future Research Directions

Although the constructed indicator system covers the primary controlling factors for soft rock tunnels in fault fracture zones, further refinement is needed to address more complex geological heterogeneity, extreme hydrogeological conditions, and multi-disturbance environments. Continued validation and enhancement through large-scale case studies drive the method toward intelligent and automated implementations.

In summary, the optimal cloud entropy extension cloud model outperforms the conventional extension cloud model in classification clarity, assessment stability, result reliability, and practical applicability. It is recommended as the preferred technical approach to the risk grading of soft rock tunnels traversing fault fracture zones.