Wildland-urban interface fire risk assessment based on gray linguistic 2-tuple relational analysis

1 Introduction

The wildland-urban interface (WUI) represents the critical convergence zone between anthropogenic activities and natural ecosystems, where fuel spatial distributions and evolutionary trajectories are intrinsically linked to global wildfire risk patterns. Radeloff et al. (2018) categorized the WUI into two distinct typologies: the Intermix WUI and the Interface WUI. The Intermix WUI is characterized by a housing density exceeding 6.17 units/mi² and a wildland vegetation cover greater than 50%. Conversely, the Interface WUI maintains a similar housing density but features a vegetation cover below 50%, provided that the target area lies within 1.5 miles (2.4 km) of a large, contiguous patch of wildland vegetation. This classification framework has been widely used in WUI identification and related research.

The global WUI has expanded substantially in recent decades. Since 2000, the global WUI area has increased by more than one-third, with new areas formed between 2010 and 2020 accounting for 85% of this total growth. As of 2020, the global WUI covered approximately 1.93 million square kilometers (about 1.44% of the land area) and accommodated nearly 1.2 billion residents. WUI areas are concentrated in densely populated regions such as the Southeastern United States, Western Europe, and Southeastern China (Guo et al., 2024). Several factors have contributed to this expansion. First, urban sprawl has extended urban fringes into natural landscapes such as forests and grasslands. Second, amenity-led development has promoted the construction of vacation and residential homes in scenic natural areas. Finally, rural-to-WUI migration has created new settlement edges in developing countries, where informal settlements at forest boundaries often lack essential fire infrastructure and planning.

Climate change has further exacerbated the WUI fire risk profile. Rising temperatures and shifting precipitation are driving global fire risks into high-latitude and high-altitude regions, rapidly converting low-risk boreal forests and alpine meadows into vulnerable WUI areas (Mockrin et al., 2022). The distinctive hazard of WUI fires stems from their unique double-layered fuel-bed structure: the lower layer consists of natural vegetation such as shrubs, herbs, and litter, while the upper layer is composed of man-made structures, including wooden buildings, furniture, and oil storage facilities. When fire spreads from vegetation to buildings, the Heat Release Rate (HRR) increases sharply, generating temperatures that far exceed the fire resistance limits of standard building materials. Critically, spotting (ember attack) has emerged as the dominant spreading mechanism. Burning debris can be transported by strong winds up to 20 kilometers, creating new fires that render traditional linear firebreaks ineffective. Under extreme conditions, a single burning residence can ignite 30% of neighboring buildings within a 500-meter radius, potentially leading to rapid cascading fire spread (Mell et al., 2010).

Losses from WUI fires are multidimensional and long-lasting. The catastrophic 2019–2020 Australian bushfire season serves as a critical case study: the fires consumed over 10 million hectares, caused 33 direct deaths, and destroyed over 3,000 homes, with economic losses estimated by Westpac Bank at AUD 5 billion. Beyond direct damage, the loss of ecosystem services is substantial. For instance, the decline in carbon sink function resulted in the emission of 430 million tons of CO₂, while weakened water conservation capacity affected the drinking water safety of 12 million people. Biodiversity recovery in these regions is estimated to require 50 to 100 years. Furthermore, the economic multiplier effect is significant, with post-disaster reconstruction costs typically reaching 2.5 to 3.5 times the direct losses. Additionally, the long-term decline in key industries such as tourism and agriculture further destabilizes regional economies (Driscoll et al., 2024).

Current WUI fire risk assessment methods face significant data challenges characterized by incompleteness, inconsistency, and uncertainty. Key risk factors are often compromised by missing data, while mismatches in the spatiotemporal scales of multi-source information can lead to ecological fallacies. Furthermore, qualitative indicators are inherently difficult to quantify, often resulting in the loss of substantial semantic information during the evaluation process. Gray System Theory provides a mathematical framework for analyzing systems characterized by small samples and incomplete information. Its core methodology involves extracting information from limited data through gray generation and measuring sequence similarity using the gray relational degree. In the context of WUI fire risk assessment, gray relational analysis (GRA) offers several useful characteristics. It does not impose strict requirements on data distributions or require large samples, making it potentially suitable for risk assessment problems involving limited or incomplete information.

However, traditional GRA has limitations when processing linguistic evaluation information. In WUI fire risk assessment, experts frequently use linguistic variables such as “high,” “moderate,” and “low” to describe indicators that are difficult to quantify. The linguistic 2-tuple model provides a useful means of representing such linguistic evaluations within a predefined linguistic term set U = {u₀, u₁,..., u_*g*} (Herrera and Martínez, 2000). A linguistic 2-tuple is represented as (u_*i*, v_*i*), where u_*i* denotes the closest linguistic label and v_*i* represents the corresponding symbolic translation. Integrating this representation with GRA therefore provides a potential approach for incorporating linguistic evaluation information into gray relational analysis.

Given the complexity of WUI environments and the limited availability of complete evaluation information, risk assessments often involve heterogeneous data types, including crisp values, interval values, and fuzzy linguistic variables. To integrate these different forms of information within a unified framework, this study develops a gray linguistic 2-tuple model for WUI fire risk assessment. Crisp values, interval values, and fuzzy linguistic variables are transformed into gray linguistic 2-tuple representations, and multi-expert evaluation information is subsequently aggregated to determine the relative fire risk of the evaluated areas through gray relational analysis. This framework provides a unified basis for processing heterogeneous and incomplete evaluation information in WUI fire risk assessment.

2 Literature review

Comprehensive risk assessment is a prerequisite for mitigating fire hazards within the WUI. Over the past few decades, researchers have developed a diverse array of methodologies to capture the intricate, non-linear, and multidimensional nature of WUI fire risks. To facilitate a systematic understanding of the existing research landscape and highlight the methodological evolution, this section provides a critical review of the dominant paradigms in risk assessment.

2.1 Risk assessment methods based on fuzzy sets and linguistic variables

Fuzzy set theory and its extensions are widely used to capture subjective uncertainty in complex systems. Xu and Peng (2021) integrated intuitionistic fuzzy sets with TOPSIS to tackle the fuzziness and multi-attribute group consensus issues in high-rise building fire safety assessments. Similarly, addressing dynamic transport risks, Zhang et al. (2024a) proposed an improved failure mode and effects analysis (FMEA) model using normal wiggly hesitant fuzzy sets (NWHFS) to capture expert hesitation, achieving a structured integration of group opinions via clustering algorithms.

In fire risk applications, fuzzy logic has been widely employed to handle uncertainty in various aspects, including spatial analysis, system reliability, and causal reasoning. Kumi-Boateng et al. (2021) combined fuzzy sets with the analytic hierarchy process (AHP) and integrated geographic information systems (GIS) to model forest fire risks. Addressing facility risk assessment under data-scarce scenarios, Masalegooyan et al. (2022) proposed a fuzzy fault tree analysis (FFTA) framework to quantify risk probabilities in a landfill fire case. Furthermore, Da Rocha et al. (2023) utilized fuzzy Petri nets to handle vague parameters in residential electrical fire assessments, converting questionnaire data into network configurations to establish a rule-based reasoning path for complex causal chains.

However, the aforementioned fuzzy set-based methods have several limitations in the context considered in this study. First, although fuzzy linguistic variables can represent semantic vagueness, they do not explicitly characterize the degree of information incompleteness associated with an evaluation. Second, the representation of linguistic terms may not fully reflect differences in the information available to individual experts. Finally, when evaluation information includes multiple data types, such as crisp values, interval values, and linguistic variables, integrating them within a common representation remains challenging.

2.2 Risk assessment methods based on gray system theory

Gray system theory has been widely applied in disaster risk assessment because of its capability to handle small-sample and incomplete-information problems. Huang et al. (2022) conducted a fire risk assessment and early warning study based on satellite monitoring data in Yunnan Province, using GRA to investigate the quantitative correlation between high-density fire zones and key meteorological elements. Xu et al. (2022) focused on coal mine spontaneous combustion risk by incorporating interval gray number theory into the classic decision-making trial and evaluation laboratory (DEMATEL) method, thereby effectively mitigating the inherent fuzziness and epistemic uncertainty in expert evaluations. Furthermore, Wang et al. (2023) combined the Entropy Weight Method with GRA to quantify the fire hazard of mine cables under different thermal radiation conditions. Lv et al. (2025) addressed the challenges of incomplete information and epistemic uncertainty in urban production safety risk assessment by constructing an advanced extension gray clustering model, thereby quantifying the safety system's structural resilience to catastrophic accidents.

Although these studies have expanded the application of gray system theory, traditional gray models still have limitations in representing semantic and linguistic evaluation information. Classic GRA is primarily designed for numerical sequences and does not directly represent linguistic variables. Interval gray numbers can characterize numerical intervals and incomplete information, but they do not explicitly preserve the semantic information contained in linguistic assessments such as “high” or “moderate.” Consequently, integrating gray information and linguistic evaluations within a unified representation remains an issue that warrants further investigation.

2.3 Probabilistic risk assessment methods

Probabilistic methods assess risk by quantifying event likelihoods and consequence severities, grounded in probability theory and mathematical statistics. Key approaches include Monte Carlo simulation, fault tree analysis (FTA), Bayesian networks (BN), and Markov Chains.

Monte Carlo simulation estimates the expected values and confidence intervals of risk indicators by simulating the distributions of risk variables through random sampling. Widodo et al. (2022) proposed a comprehensive fire and explosion risk assessment method for coal recovery tunnels, using Monte Carlo simulation to handle the probability distribution of volatile risk factors. In forest fire risk assessment, Liu et al. (2024) integrated multi-source data including remote sensing and meteorology, combining Cellular Automata with Monte Carlo methods to simulate fire spread and assess potential burned areas.

FTA maps accident causal chains using logic gates to determine top-event probabilities for risk assessment and failure prediction. To overcome data scarcity, Ramezanifar et al. (2023) coupled FTA with set pair analysis (SPA) for storage tank fire risk assessment. Similarly, Masalegooyan et al. (2022) constructed a multi-level fault tree for landfill fire risk assessment, transforming linguistic descriptions into fuzzy numbers to enhance model feasibility under limited information.

BNs leverage conditional probability graphs to dynamically update posterior risk probabilities for active decision support. Chen et al. (2023) built a BN fire risk assessment model for historical buildings based on Chinese fire codes, incorporating variable weight synthesis theory to adaptively adjust node weights in emergency scenarios. To assess fire and explosion risks in bulk carriers, Aydin et al. (2024) employed Fuzzy BN, utilizing fuzzy sets to resolve uncertainty and semantic ambiguity in expert judgments.

Markov Chains describe the dynamic evolution of risks through state transition probability matrices and have been applied in various disaster and accident assessments. Zhang et al. (2024b) validated a Spatial Markov Chain model using Chi-square tests to evaluate high-rise building fire risks across probability and consequence dimensions. Additionally, Feng et al. (2024) addressed the cascading nature of urban gas pipeline leaks by constructing a Markov-based disaster chain model to quantify failure probabilities and identify key causative factors.

Although probabilistic methods provide a well-established framework for risk analysis, their application generally requires sufficient information to characterize probability distributions, transition probabilities, or conditional relationships. In data-limited WUI settings, such probabilistic information may be difficult to estimate reliably, particularly when the available evaluation information is heterogeneous or incomplete.

2.4 Data-driven risk assessment methods based on machine learning and deep learning

Driven by advancements in pattern recognition, non-linear modeling, and high-dimensional feature processing, machine learning (ML) and deep learning (DL) have been increasingly applied to risk assessment, hazard identification, and dynamic early warning.

Traditional ML methods are widely used for structured data due to their strong interpretability and robust performance on small-to-medium datasets. Researchers often combine multidimensional features (e.g., geographic, meteorological, and socio-economic) using algorithms such as random forest (RF), classification and regression trees (CART), gradient boosting trees (XGBoost, LightGBM), support vector machines (SVM), and naive bayes (NB). For example, Sheriff et al. (2025) integrated google earth engine data with RF and CART models to map forest fire susceptibility. Pacheco et al. (2021) compared k-nearest neighbor and random forest classifiers for mapping forest-fire-affected areas using multisource satellite imagery. Lei et al. (2023) combined Entropy-TOPSIS with XGBoost to evaluate monthly fire risks in the Palace Museum, while Sun et al. (2022) used LightGBM for forest fire mapping, achieving 88.8% accuracy. To optimize dynamic evaluations, Li et al. (2021) enhanced an SVM framework using an electrostatic discharge algorithm and an improved hybrid kernel for warehouse fire assessment. Similarly, Chen et al. (2021) used the Relief algorithm and NB to model wildfire risks along transmission line corridors.

DL methods are widely used for unstructured, high-dimensional data such as imagery, video, and sensor sequences. Zheng et al. (2022) integrated transfer learning with a dynamic convolutional neural network (CNN) and principal component analysis (PCA) to identify core forest fire risk zones. For temporal tracking, Yin et al. (2025) combined spatial interpolation with a long short-term memory (LSTM) model to reconstruct historical fire danger indices from 1921 to 1960. For early wildfire detection, Zhao and Ban (2022) proposed a gated recurrent unit (GRU) framework that leverages high-temporal-resolution satellite sequences to achieve efficient and robust time-series modeling.

Despite their strong predictive capabilities, data-driven methods may face several challenges in the assessment context considered in this study. Their performance often depends on the availability and quality of training data, and complex models may provide limited interpretability for decision-making. Moreover, heterogeneous evaluation information involving crisp values, interval values, and linguistic judgments generally requires additional encoding or preprocessing before it can be incorporated into conventional machine learning or deep learning models. These characteristics may limit their direct applicability to WUI fire risk assessment when labeled data are limited and expert evaluations constitute an important source of information.

2.5 GIS-based spatial multi-criteria assessment methods

GIS integrates spatial data management with advanced spatial analysis, enabling the precise visualization and interpretation of geographic information. In fire risk research, GIS serves as a critical tool for delineating hazard zones and optimizing emergency response strategies. For example, Aksoy et al. (2023) combined GIS and remote sensing data to evaluate fire risks in vulnerable WUI areas across Istanbul, applying machine learning (RF, XGBoost, and LightGBM) to construct regional risk maps by intersecting factor importance weights with geospatial layers. Similarly, Nuthammachot and Stratoulias (2021) coupled GIS with the AHP to map fire susceptibility in a Thai peat swamp forest; their framework synthesized 55 historical fire records (2012–2016) with topographic, meteorological, and anthropogenic indicators, generating risk classifications highly correlated with empirical fire occurrences. Addressing cascading multi-hazard scenarios, Lyu and Yin (2023) developed an enhanced multi-criteria decision-making (MCDM) framework integrating interval values with Fuzzy AHP (FAHP). They validated this methodology within a GIS software environment against classic AHP, interval-FAHP, and the analytic network process (ANP) to evaluate and map the spatial risks of floods, mudflows, and landslides in Hong Kong.

Although GIS-based spatial multi-criteria assessment methods provide effective tools for integrating, visualizing, and analyzing spatial information, the representation and aggregation of heterogeneous expert evaluation information can remain challenging. In particular, when evaluation data include fuzzy linguistic judgments or interval-valued information, conventional overlay-based assessment procedures may require additional transformation or modeling mechanisms to integrate these data within a common analytical framework.

Addressing the aforementioned research gaps, this study proposes a WUI fire risk assessment model based on gray linguistic 2-tuple relational analysis. The main methodological contributions of this study are summarized as follows:

• (1) Development of a gray linguistic 2-tuple representation framework for WUI fire risk assessment: the proposed framework transforms crisp values, interval values, and fuzzy linguistic variables into a unified gray linguistic 2-tuple representation. By incorporating the information gray degree into the linguistic 2-tuple representation, the framework simultaneously represents the evaluation value and its associated degree of information incompleteness, providing a unified basis for the subsequent aggregation and analysis of heterogeneous WUI fire risk information.
• (2) Formulation of a gray 2-tuple relational degree for WUI fire risk assessment: the proposed relational degree measures the similarity between the reference and comparison sequences by considering both their linguistic 2-tuple values and information gray degrees. This formulation extends conventional gray relational analysis to accommodate gray linguistic 2-tuple information and provides a quantitative basis for the comparative ranking of WUI fire risk.
• (3) Integration of gray system theory and the linguistic 2-tuple model for heterogeneous and incomplete evaluation information: the proposed method combines the capability of gray system theory to analyze systems with limited and incomplete information with the linguistic 2-tuple model for representing linguistic evaluations. This integration enables heterogeneous evaluation information from multiple sources and experts to be aggregated within a unified framework and incorporated into the subsequent WUI fire risk assessment.
• (4) Adaptation of the indicator system to WUI characteristics in China: in Southern China, bamboo-dominated landscapes constitute a distinctive type of WUI environment. The thick litter layer typical of bamboo forests is highly flammable, and the thermal rupture of bamboo nodes under extreme heat can trigger explosive fire behavior. The proposed indicator system incorporates indigenous vegetation characteristics together with the intense human-land use conflicts characteristic of China, thereby allowing these region-specific factors to be considered within the WUI fire risk assessment framework.

3 Basic concepts

3.1 The linguistic 2-tuple

The concept of the linguistic 2-tuple was originally introduced by Herrera and Martínez (2000) to address multi-attribute decision-making problems involving linguistic assessment information. Formally, a linguistic 2-tuple is expressed as a dual pair (u_*i*, v_*i*), where u_*i* and v_*i* represent the following operational semantics:

• (1) The term u_*i* denotes the i-th linguistic variable within a predefined evaluation set U, where comprises g+1 fuzzy linguistic variables, such that U = { u₀, u₁, ..., u_*g* } . This evaluation set systematically satisfies three fundamental axiomatic properties, namely ① Ordered Relation: u_*i* ≥ u_*j* for i ≥ j; ② Negation Operator: Neg(u_*i*) = u_*j* for j = g−i; ③ Max and Min Operators: max(u_*i*, u_*j*) = u_*i* and min(u_*i*, u_*j*) = u_*j* given u_*i* ≥ u_*j*.
• (2) The parameter v_*i* represents the symbolic translation value bounded within the interval [−0.5, 0.5), which quantitatively measures the deviation of the numerical evaluation result from its closest linguistic label u_*i*.

For WUI fire risk assessment, the linguistic evaluation term set is defined as follows:

U = { u₀: extremely low risk, u₁: low risk, u₂:slightly low risk, u₃:moderate risk, u₄:slightly high risk, u₅:high risk, u₆:extremely high risk }

Assuming that the membership functions overlap only between adjacent linguistic terms, the corresponding membership functions are defined as follows:

μ0(x)={1,0≤x≤0.1(0.2−x)/0.1, 0.1<x<0.20,x≥0.2

μ1(x)={0,x≤0.1 or x≥0.4(x−0.1)/0.1, 0.1<x<0.21,x=0.2(0.4−x)/0.2, 0.2<x<0.4

μ2(x)={0,x≤0.2 or x≥0.6(x−0.2)/0.2, 0.2<x<0.41,x=0.4(0.6−x)/0.2, 0.4<x<0.6

μ3(x)={0,x≤0.4 or x≥0.7(x−0.4)/0.2, 0.4<x<0.61,x=0.6(0.7−x)/0.1, 0.6<x<0.7

μ4(x)={0,x≤0.6 or x≥0.8(x−0.6)/0.1, 0.6<x<0.71,x=0.7(0.8−x)/0.1, 0.7<x<0.8

μ5(x)={0,x≤0.7 or x≥0.9(x−0.7)/0.1, 0.7<x<0.81,x=0.8(0.9−x)/0.1, 0.8<x<0.9

μ6(x)={0,x≤0.8(x−0.8)/0.1, 0.8<x<0.91,0.9≤x≤1

Several definitions concerning the transformation and comparison of linguistic 2-tuples are presented below.

Definition 1: let u_*i* ∈ U be a fuzzy linguistic variable. Its corresponding linguistic 2-tuple representation is obtained through the mapping function φ, defined as follows:

φ:U→U×[-0.5,0.5) φ(ui)=(ui,0),ui∈U

Definition 1 establishes a direct mapping from each linguistic term u_*i* ∈ U to its corresponding linguistic 2-tuple (u_*i*, 0).

The following definitions describe the bidirectional transformations between numerical values and linguistic 2-tuples.

Definition 2: let U = { u₀, u₁, ..., u_*g* } be a predefined linguistic term set, and γ ∈ [0, g] be a real number representing the result of a symbolic aggregation operation. The value γ can be converted into an equivalent linguistic 2-tuple through the function ψ :

ψ:[0,g]→U×[-0.5,0.5),ψ(γ)=(ui,vi)

where i=round(γ), v_*i* = γ−i, such that v_*i* ∈ [−0.5, 0.5), and round(·) denotes the standard rounding function. Conversely, for any given linguistic 2-tuple (u_*i*, v_*i*), there exists a unique inverse mapping ψ⁻¹ that returns its equivalent numerical value γ ∈ [0, g], expressed as follows:

ψ-1:U×[-0.5,0.5)→[0,g] ψ-1(ui,vi)=i+vi=γ

Definition 2 operates on the conventional scale [0, g]. For normalized data in [0, 1], Zhang (2013) proposed an alternative transformation that enables direct conversion between normalized numerical values and linguistic 2-tuples.

Definition 3: let U = { u₀, u₁, ..., u_*g* } be a predefined linguistic term set, and γ ∈ [0, 1] be a real number representing the result of a symbolic aggregation operation. The value γ can be mapped into an equivalent linguistic 2-tuple through the function ψ, defined as

ψ:[0,1]→U×[-0.5,0.5),ψ(γ)=(ui,vi)

where i = round(γ·g), v_*i* = γ·g−i, such that v_*i* ∈ [−0.5, 0.5), and round(·) represents the standard rounding function. Conversely, for any given linguistic 2-tuple (u_*i*, v_*i*), there exists a unique inverse function ψ⁻¹ that returns its equivalent numerical value γ ∈ [0, 1], expressed as follows:

ψ-1:U×[-0.5,0.5)→[0,1] ψ-1(ui,vi)=i+vig=γ

For interval-valued data, the following transformation is introduced to obtain the corresponding linguistic 2-tuple representation.

Definition 4: let U = { u₀, u₁, ..., u_*g* } be a predefined linguistic term set, and H = [x_*L*, x_*R*] be an interval value. The membership function of H = [x_*L*, x_*R*] is defined as:

μH(x)={1, x∈[xL,xR]0, otherwise

The degree of intersection between the interval value H = [x_*L*, x_*R*] and each linguistic term u_*i* ∈ U = { u₀, u₁, ..., u_*g* } is calculated as ci=maxxmin{μH(x),μui(x)}. A representative crisp value for H, denoted by H_γ, is then obtained as the weighted average of the indices of the linguistic terms, with the intersection degrees c_*i* serving as weights:

Hγ=∑i=0gi·ci∑i=0gci.

The resulting crisp value H_γ can subsequently be transformed into its equivalent linguistic 2-tuple using the transformation defined in Definition 2.

For any two linguistic 2-tuples (u_*i*, v_*i*) and (u_*j*, v_*j*), the following comparison rules apply:

• (1) If i>j, then (u_*i*, v_*i*)>(u_*j*, v_*j*), indicating that the ranking is primarily determined by the linguistic hierarchical index.
• (2) If i = j, the ranking is determined by the symbolic translation values such that:

• ① if v_*i* = v_*j*, then (u_*i*, v_*i*) = (u_*j*, v_*j*);
• ② if v_*i* > v_*j*, then (u_*i*, v_*i*) > (u_*j*, v_*j*);
• ③ if v_*i*<v_*j*, then (u_*i*, v_*i*) < (u_*j*, v_*j*).

3.2 Gray linguistic 2-tuple

In WUI fire risk assessment, evaluators' judgments are frequently constrained by environmental complexity, spatiotemporal limitations, and differences in expertise and experience. Consequently, the resulting attribute evaluations may exhibit varying degrees of information incompleteness, which are characterized by information gray degrees. In this study, the degree of information sufficiency is classified into seven linguistic levels: extremely sufficient, sufficient, relatively sufficient, moderate, relatively incomplete, incomplete, and extremely incomplete.

The information gray degree and its corresponding quantification scheme are defined as follows.

Definition 5: let E = { e₁:extremely sufficient, e₂: sufficient, e₃:relatively sufficient, e₄:moderate, e₅:relatively incomplete, e₆:incomplete, e₇:extremely incomplete } be a linguistic term set describing the degree of information sufficiency. Each linguistic state e_*i* is associated with a corresponding interval of the information gray degree e ∈ [0, 1], defined as follows:

e1∈[0,0.15),e2∈[0.15,0.3),e3∈[0.3,0.45),e4∈[0.45,0.6),e5∈[0.6,0.75),e6∈[0.75,0.9),e7∈[0.9,1]

To provide a unified representation of heterogeneous evaluation information, the linguistic 2-tuple is extended by incorporating an information gray degree. The resulting gray linguistic 2-tuple is defined as follows.

Definition 6: let (u_*i*, v_*i*) be a linguistic 2-tuple and e ∈ [0, 1] denote its associated information gray degree. Then d = ((u_*i*, v_*i*), e) is defined as a gray linguistic 2-tuple, where e represents the gray component of d. When e = 0, d reduces to a standard linguistic 2-tuple.

For any two gray linguistic 2-tuples d₁ = ((u_*i*, v_*i*), e₁) and d₂ = ((u_*j*, v_*j*), e₂), d₁ = d₂ if and only if (u_*i*, v_*i*) = (u_*j*, v_*j*) and e₁ = e₂. Based on this representation, the following algebraic operations are defined.

Definition 7: let d₁ = ((u_*i*, v_*i*), e₁) and d₂ = ((u_*j*, v_*j*), e₂) be two arbitrary gray linguistic 2-tuples. The fundamental algebraic operations are defined as follows:

• (1) d1+d2=(ψ(ψ-1(ui,vi)+ψ-1(uj,vj)),(e1+e2)∧1);
• (2) kd1=(ψ(kψ-1(ui,vi)),(ke1)∧1), k≥0

where ∧ denotes the minimum operator.

Definition 8: let d₁ = ((u_*i*, v_*i*), e₁) and d₂ = ((u_*j*, v_*j*), e₂) be two gray linguistic 2-tuples. The distance between d₁ and d₂ is defined as follows:

l(d1,d2)=12(|ψ-1(ui,vi)-ψ-1(uj,vj)|+|e1-e2|)

The proposed distance measure satisfies the following three fundamental properties:

• (1) l(d₁, d₂) ≥ 0 and l(d₁, d₂) = 0⇔d₁ = d₂;
• (2) l(d₁, d₂) = l(d₂, d₁);
• (3) l(d₁, d₃) ≤ l(d₁, d₂)+l(d₂, d₃)

Based on the preceding definitions and algebraic operations, the gray linguistic 2-tuple weighted averaging (GLTWA) operator is defined as follows.

Definition 9: let d_*i* = ((u_*i*, v_*i*), e_*i*)(i = 1, 2, ⋯ , n) be a collection of n gray linguistic 2-tuples, and let W = (w₁, w₂, ⋯ , w_*n*) be the associated weight vector, satisfying 0 ≤ w_*i* ≤ 1 and ∑i=1nwi=1. The gray linguistic 2-tuple weighted averaging (GLTWA) operator is defined as follows:

GLTWAW(((u1,v1),e1),((u2,v2),e2),⋯,((un,vn),en)) =(ψ(∑i=1nwiψ-1(ui,vi)),∑i=1nwiei∧1)

The GLTWA operator satisfies the following fundamental properties:

• (1) Permutation invariance: let ((u′1,v′1),e′1), ((u′2,v′2),e′2), ⋯ , ((u′n,v′n),e′n) be any permutation of the gray linguistic 2-tuple sequence ((u₁, v₁), e₁), ((u₂, v₂), e₂), ⋯ , ((u_*n*, v_*n*), e_*n*), and let W′=(w′1,w′2,...,w′n) be the corresponding permutation of the weight vector W = (w₁, w₂, ..., w_*n*). Then the following identity holds:

GLTWAW(((u1,v1),e1),((u2,v2),e2),⋯,((un,vn),en))=GLTWAW′(((u′1,v′1),e′1),((u′2,v′2),e′2),⋯,((u′n,v′n),e′n))

• (2) Idempotency: if ((u₁, v₁), e₁) = ((u₂, v₂), e₂) = ⋯ = ((u_*n*, v_*n*), e_*n*) = ((u, v), e), then

GLTWAW(((u1,v1),e1),((u2,v2),e2),⋯,((un,vn),en))=((u,v),e)

• (3) Monotonicity: let d_*i* = ((u_*i*, v_*i*), e_*i*) and di=((ui,vi),ei) be two sequences of gray linguistic 2-tuples. If (ui,vi)≥(ui,vi) and ei≤ei* for all i, then:

GLTWAW(d1,d2,⋯,dn)≥GLTWAW(d1,d2,⋯,dn*)

• (4) Boundedness: let d_min = min{d₁, d₂, ⋯ , d_*n*} and d_max = max{d₁, d₂, ⋯ , d_*n*}. For any weight vector W = (w₁, w₂, ⋯ , w_*n*) subject to ∑i=1nwi=1, the aggregated result is bounded as follows:

dmin≤GLTWAW(d1,d2,⋯,dn)≤dmax

4 WUI fire risk assessment model

This section delineates the comprehensive architecture of the proposed WUI fire risk assessment model, which systematically operationalizes the gray linguistic 2-tuple aggregation framework. Implementing this decision-support framework requires a multidimensional indicator system to characterize the complex risk environment. Accordingly, the specific dimensions of these evaluation indicators are detailed in Section 4.1.

4.1 Assessment indicator system

The WUI is characterized by complex interactions between heterogeneous biophysical conditions and intensive anthropogenic activities, resulting in high susceptibility to severe fire events. To quantitatively assess this complex risk environment, 15 evaluation indicators were identified based on a systematic review of the literature on WUI fire behavior and risk factors, with consideration given to their relevance to WUI fire processes and their applicability to the assessment context. The resulting indicator system characterizes WUI fire risk across four primary dimensions: fuel conditions, surrounding environment, meteorological conditions, and topographic characteristics.

4.1.1 Fuel conditions: the material basis for fire occurrence

As a fundamental component of the fire triangle, combustible fuels provide the material basis for fire ignition and subsequent spread. Within the WUI environment, fuel-associated risks are characterized by three core indicators:

Fuel load (I_**1**) denotes the dry weight of biomass per unit area, encompassing both natural vegetation and anthropogenic fuels such as wooden structures and vehicles. A high fuel load provides a substantial energy reservoir and can contribute to increased heat release and fire intensity. The resulting increase in fire intensity can facilitate the transition from surface fires to crown fires or the spread of fire to nearby structures.

Moisture content (I_**2**) is a key factor influencing fuel ignitability. Low moisture content reduces the energy required for water evaporation, thereby accelerating fuel preheating and pyrolysis. Consequently, drier fuels are more readily ignited and can facilitate fire spread from surface fuels to canopies and nearby structures.

Fuel spatial distribution (I_**3**) characterizes the spatial continuity and three-dimensional configuration of fuels. Continuous fuel corridors linking wildlands directly to residential boundaries can reduce the effectiveness of natural firebreaks. Concurrently, a continuous vertical fuel structure establishes a laddering effect, facilitating rapid vertical fire spread from surface fuels toward canopies and structures.

4.1.2 Surrounding environment: anthropogenic disturbance and vulnerability

The defining characteristic of the WUI lies in the profound coupling of natural and social systems. Particularly in regions characterized by intense human-land use interactions, such as the bamboo-dominated WUI areas in Southern China, anthropogenic activities can strongly influence both ignition probability and the socioeconomic vulnerability of exposed communities. The surrounding environment dimension is evaluated using four distinct indicators:

Uncontrolled fire from agricultural, forestry, and pastoral activities (I_**4**) refers to fires that escape control during agricultural, forestry, or pastoral production activities, such as land clearing and vegetation burning. Such fires may become important anthropogenic ignition sources for WUI fire events, particularly when production areas are adjacent to wildland vegetation or residential communities.

Domestic fire use (I_**5**) refers to fire-use behaviors associated with daily human activities, such as ritual paper burning, cooking, and residential heating. Improper management of these activities may generate ignition sources within or near residential areas, thereby increasing the potential for fire occurrence in densely populated WUI environments.

Population density (I_**6**) serves as an indicator of both anthropogenic disturbance intensity and disaster exposure. Elevated population density substantially increases potential human-caused ignition sources. Furthermore, it increases the complexity of emergency evacuation and may magnify potential socioeconomic losses.

Road network density (I_**7**) influences spatial accessibility for emergency response. From the perspective of disaster resilience, a dense road network provides critical infrastructure for rapid firefighting deployment and multi-path evacuation, thereby counteracting community vulnerability. Consequently, within the multi-criteria evaluation framework, it is operationalized as a cost-type indicator, where higher values contribute to a lower net risk score.

4.1.3 Meteorological conditions: dynamic drivers of fire potential

Meteorological conditions serve as pivotal exogenous drivers governing both fire ignition and behavioral escalation. This driving role has become increasingly critical amid recurrent extreme weather events. Specifically, precipitation, relative humidity, air temperature, and wind velocity collectively influence fuel desiccation and combustion potential. Consequently, variations in these meteorological conditions strongly influence temporal changes in fire potential.

Precipitation (I_**8**) and relative humidity (I_**9**), although mathematically correlated, represent distinct physical mechanisms in fire dynamics. Precipitation influences longer-term moisture conditions and the moisture retention of deeper soil and fuel layers. In contrast, relative humidity affects short-term moisture exchange in fine surface fuels, such as dead grass and litter, thereby influencing their immediate susceptibility to ignition.

Air temperature (I_**10**) accelerates fuel dehydration by intensifying evaporative demand. Under extreme heat, elevated temperatures can intensify vegetation stress and fuel desiccation, thereby increasing the potential for rapid fire spread. Concurrently, extreme temperatures may increase infrastructure vulnerability and anthropogenic ignition risks through heat-induced failures (e.g., power line faults).

Wind velocity (I_**11**) is a major factor influencing fire behavior and spread. High-velocity winds affect flame tilt and enhance convective oxygen supply, potentially increasing fire spread rates substantially. Crucially, wind-driven spotting can allow firebrands to cross fuel breaks and other defensive barriers. Consequently, such spotting can facilitate fire spread into densely populated residential areas and substantially reduce the time available for emergency evacuation.

4.1.4 Topographic characteristics: geographical drivers of fire behavior

Topography strongly influences fuel distribution, energy transfer, and moisture conditions and can either facilitate or constrain fire spread. Topographic indicators are represented by either linguistic or crisp variables to quantify their effects on fire behavior.

Slope gradient (I_**12**) can accelerate fuel preheating by increasing the exposure of uphill fuels to radiative and convective heat flux. This enhanced heat transfer can increase fire spread rates and intensity on slopes. Consequently, rapid upslope fire spread may substantially reduce the time available for emergency evacuation in WUI communities located on or above steep slopes.

Slope aspect (I_**13**) influences fuel moisture conditions by regulating local solar radiation. Sunny aspects, such as south-facing slopes, generally receive greater solar radiation, which can increase evaporative demand and reduce fuel moisture, thereby creating conditions more conducive to combustion. In contrast, shady aspects tend to retain higher moisture levels but may also support greater biomass accumulation. Consequently, the influence of slope aspect on fire risk depends on the combined effects of fuel moisture and fuel availability, particularly under drought conditions.

Topographic position (I_**14**) identifies key geomorphological features such as ridges and canyons. While physically correlated with absolute elevation, it specifically characterizes terrain-driven wind fields and localized heat accumulation. Canyons promote the chimney effect and intense heat retention, whereas ridges serve as channels for prevailing macro-winds and optimal launch points for airborne firebrands.

Elevation (I_**15**) influences vegetation distribution and the duration of the fire season. In typical Chinese WUI scenarios, lower-elevation areas are often characterized by relatively high ambient temperatures, substantial vegetative fuel loads, and intensive anthropogenic activities, thereby contributing to relatively high fire risk in these areas.

4.1.5 Summary and methodological justification of the indicator system

The WUI fire risk indicator system integrates natural and social dimensions by coupling quasi-static spatial variables (Fuel Conditions and Topographic Characteristics) with dynamic meteorological indicators. The framework incorporates human-geographic factors (Surrounding Environment) to address uncertainties associated with anthropogenic activities. These interrelated indicators jointly influence the overall risk propensity and potential intensity of WUI fire events.

Potential collinearity among indicators, notably between the meteorological pair (I₈, I₉) and the topographic pair (I₁₂, I₁₄), is intentionally retained because these indicators represent distinct physical mechanisms in WUI fire dynamics. Although correlated, the meteorological pair captures differences between longer-term moisture conditions and short-term moisture-related ignition potential, whereas the topographic pair distinguishes slope-induced effects on fire spread from broader terrain-related influences. Unlike traditional regression models, the GRA framework does not require strict independence among indicators and is therefore less sensitive to multicollinearity. This property allows correlated indicators representing distinct physical mechanisms to be retained in the assessment framework.

4.2 Risk assessment model

4.2.1 Problem description

The WUI fire risk assessment problem is formulated as a gray linguistic 2-tuple MCDM problem. Let Y = { Y₁, Y₂, …, Y_*m*} denote the set of evaluated geographic areas and I = { I₁, I₂, …, I_*n*} denote the set of evaluation indicators, with the weight vector W = (w₁, w₂, …, w_*n*), satisfying 0 ≤ w_*i* ≤ 1 and ∑i=1nwi=1. Let P = {P₁, P₂, …, P_*r*} represent the panel of experts, with the corresponding expert weight vector σ = (σ₁, σ₂, …, σ_*r*), where 0 ≤ σ_*i* ≤ 1 and ∑i=1rσi=1. The initial evaluation matrix provided by expert P_*k* is formulated as:

Ak=(aij(k),eij(k))m×n,i=1,2,...,m,j=1,2,...,n,k=1,2,...,r

where aij(k) denotes the evaluation value of indicator I_*j* for area Y_*i*, and eij(k) represents the corresponding information gray degree, which characterizes the degree of information incompleteness associated with aij(k).

Let N₁ = {1, 2, ..., n₁}, N₂ = {n₁+1, n₁+2, ..., n₂}, and N₃ = {n₂+1, n₂+2, ..., n} be three mutually exclusive indicator index sets. According to the data type of each indicator, the evaluation value is represented in different forms: if j ∈ N₁, then aij(k)∈ℝ (crisp value); if j∈N₂, then aij(k)=[aij(k)L,aij(k)R] (interval value); and if j ∈ N₃, then aij(k) is a fuzzy linguistic variable. The objective of the proposed framework is to determine the comprehensive fire risk ranking of all evaluated areas based on the initial evaluation matrices provided by the expert panel.

4.2.2 Evaluation matrix standardization

To integrate heterogeneous evaluation data into a unified framework, the initial evaluation matrices A₁, A₂, ..., A_*r* are standardized according to the mathematical characteristics of different data types, thereby ensuring compatibility with the subsequent gray relational analysis. The standardization procedures are as follows:

• (1) If j ∈ N₁ = {1, 2, ..., n₁}, then aij(k)∈ℝ represents a crisp value. For cost-type indicators, the standardized value bij(k) is formulated as:

bij(k)=maxlalj(k)-aij(k)maxlalj(k)-minlalj(k),i=1,2,...,m(1)

• For benefit-type indicators, the standardized value bij(k) is formulated as:

bij(k)=aij(k)-minlalj(k)maxlalj(k)-minlalj(k),i=1,2,...,m(2)

• (2) If j ∈ N₂ = {n₁+1, n₁+2, ..., n₂}, then aij(k)=[aij(k)L,aij(k)R] is characterized as an interval value. We employ a vector normalization approach to preserve the intrinsic geometric properties of the interval boundaries. For cost-type indicators, the lower and upper bounds of the standardized interval value, denoted as bij(k)=[bij(k)L,bij(k)R], are calculated as follows:

bij(k)L=1aij(k)R∑l=1m(1alj(k)L)2,bij(k)R=1aij(k)L∑l=1m(1alj(k)R)2, i=1,2,...,m(3)

• For benefit-type indicators, the lower and upper bounds of the standardized interval value, denoted as bij(k)=[bij(k)L,bij(k)R], are calculated as follows:

bij(k)L=aij(k)L∑l=1m(alj(k)R)2,bij(k)R=aij(k)R∑l=1m(alj(k)L)2, i=1,2,...,m(4)

• (3) If j ∈ N₃ = {n₂+1, n₂+2, ..., n}, then aij(k) represents a fuzzy linguistic variable. The qualitative assessment scale is mapped into a linguistic term set U = {u₀, u₁, u₂, u₃, u₄, u₅, u₆} as follows:

extremely low risk=u0, low risk=u1,slightly low risk=u2,moderate risk=u3,slightly high risk=u4,high risk=u5,extremely high risk=u6(5)

Through the standardization framework, the initial evaluation matrices A₁, A₂, ..., A_*r* are transformed into standardized evaluation matrices B₁, B₂, ..., B_*r*, where Ak=(aij(k),eij(k))m×n and Bk=(bij(k),eij(k))m×n, with eij(k) denoting the information gray degree. Specifically, the standardized entry bij(k) is defined across the hybrid data domain as:

bij(k)={γ∈[0,1]⊂ℝj∈N1[bij(k)L,bij(k)R]j∈N2uq∈U={u0,u1,u2,u3,u4,u5,u6}j∈N3

Based on Definitions 1–4 and Definition 6, the standardized evaluation values in matrix B_*k*, together with their associated information gray degrees, are converted into gray linguistic 2-tuple representations. Consequently, the individual gray linguistic 2-tuple evaluation matrix for expert P_*k* is constructed as:

Ck=((cij(k),vij(k)),eij(k))m×n,k=1,2,...,r

To comprehensively evaluate the WUI fire risk, the GLTWA operator defined in Definition 9 is employed to aggregate the individual gray linguistic 2-tuple evaluation matrices from the expert panel into a comprehensive evaluation matrix C:

C=((cij,vij),eij)m×n

where the aggregated entries are calculated as:

((cij,vij),eij)=GLTWAσ(((cij(1),vij(1)),eij(1)),((cij(2),vij(2)),eij(2)),...,((cij(r),vij(r)),eij(r)))=(ψ(∑k=1rσkψ−1(cij(k),vij(k))),(∑k=1rσkeij(k))∧1)(6)

Here, σ = (σ₁, σ₂, …, σ_*r*) represents the weight vector assigned to the r experts.

4.2.3 WUI fire risk ranking based on gray 2-tuple relational degree

With the comprehensive evaluation matrix established, the gray 2-tuple relational degree is used to measure the proximity of each evaluated area to the positive ideal reference sequence, which represents the reference state of highest assessed fire risk. Accordingly, the positive ideal reference sequence is constructed as the comparison benchmark.

Definition 10: let C = ((_{c_*ij*, v_*ij*), e_*ij*)m×n} be the comprehensive evaluation matrix obtained from Equation (6). The positive ideal reference sequence, denoted as ε+=(ε1+,ε2+,...,εn+), is constructed by selecting the ideal element for each indicator from C:

ε+=(ε1+,ε2+,...,εn+) =((maxi(ci1,vi1),miniei1),(maxi(ci2,vi2),miniei2), ...,(maxi(cin,vin),miniein))(7)

where max denotes the maximum operation for linguistic 2-tuples, and min denotes the minimum operation for information gray degrees. Correspondingly, the n elements in the i-th row of matrix C constitute the comparison sequence for the i-th evaluated geographic area, expressed as:

εi=(((ci1,vi1),ei1),((ci2,vi2),ei2),...,((cin,vin),ein)), i=1,2,...,m(8)

It should be noted that using the minimum operator (min_*i*e_*ij*) for the information gray degree in Equation (7) implies that a lower gray degree corresponds to a more ideal reference state. In the gray linguistic 2-tuple representation, the linguistic 2-tuple and the information gray degree characterize two distinct dimensions: the evaluated risk level and the associated degree of information incompleteness, respectively. Therefore, the positive ideal reference sequence combines a high risk evaluation with a low information gray degree, representing a reference state with both high assessed risk and a high degree of information completeness. Here, minimizing the information gray degree does not imply any reduction in the actual WUI fire risk; rather, it reflects the construction of an ideal reference sequence characterized by minimal information incompleteness.

Definition 11: let C = ((_{c_*ij*, v_*ij*), e_*ij*)m×n} be the comprehensive evaluation matrix. Given the i-th comparison sequence ε_*i* and the positive ideal reference sequence ε⁺, the distance l(εj+,εi(j)) between their corresponding j-th elements is defined as follows:

l(εj+,εi(j))=12(|ψ−1[maxi(cij,vij)]−ψ−1(cij,vij)| +|minieij−eij|)

Based on this distance metric, the gray relational coefficient R(εj+,εi(j)) between the comparison sequence and the positive ideal reference sequence is calculated as:

R(εj+,εi(j))=ρmaximaxjl(εj+,εi(j))l(εj+,εi(j))+ρmaximaxjl(εj+,εi(j)),(9)

where ρ ∈ (0, 1) denotes the distinguishing coefficient and is conventionally set to 0.5. Finally, the overall gray 2-tuple relational degree R(ε+,εi) for the i-th evaluated geographic area (i = 1, 2, ⋯ , m) is aggregated by:

R(ε+,εi)=∑j=1nwjR(εj+,εi(j))(10)

where w_*j* denotes the weight of the j-th evaluation indicator.

The proposed gray 2-tuple relational degree satisfies the four fundamental properties of GRA: normality, integrality, symmetry, and proximity.

The systematic procedure for the proposed WUI fire risk assessment model is summarized through the following steps:

• (1) Establish the initial evaluation matrices A₁, A₂, ..., A_*r* from the panel of r experts for the m investigated geographic areas.
• (2) Standardize the initial matrices A₁, A₂, ..., A_*r* using Equations (1–5) to obtain the standardized evaluation matrices B₁, B₂, ..., B_*r*.
• (3) Map the elements within B₁, B₂, ..., B_*r* into their equivalent gray linguistic 2-tuple representations based on Definitions 1–4 and Definition 6, yielding the individual gray linguistic 2-tuple evaluation matrices C₁, C₂, ..., C_*r*.
• (4) Aggregate the individual evaluation matrices C₁, C₂, ..., C_*r* using Equation (6) to construct the comprehensive evaluation matrix C = ((_{c_*ij*, v_*ij*), e_*ij*)m×n}.
• (5) Extract the reference sequence ε⁺ and the comparison sequences ε₁, ε₂, ..., ε_*m* from the comprehensive evaluation matrix C = ((_{c_*ij*, v_*ij*), e_*ij*)m×n} using Equations (7) and (8), respectively.
• (6) Calculate the gray relational coefficient R(εj+,εi(j)) and the gray 2-tuple relational degree R(ε+,εi)(i = 1, 2, ..., m) using Equations (9) and (10).
• (7) Determine the WUI fire risk ranking of the evaluated geographic areas according to their gray 2-tuple relational degrees. Since the positive ideal reference sequence represents the highest assessed fire risk, a larger value of R(ε+,εi) indicates greater proximity to this reference sequence and, consequently, a higher WUI fire risk level for the corresponding geographic area Y_*i*.

To further translate the quantitative output into operational decision-making support, the comprehensive WUI fire risk assessment is categorized into seven discrete qualitative levels: extremely low risk, low risk, slightly low risk, moderate risk, slightly high risk, high risk, and extremely high risk. Let the linguistic rating set for the WUI fire risk assessment be denoted as U = {u₀, u₁, u₂, u₃, u₄, u₅, u₆}, with corresponding quantized core values within the normalized numerical spectrum defined as V = {0.1, 0.2, 0.4, 0.6, 0.7, 0.8, 0.9}. These reference values are derived from the core regions (i.e., regions with a membership degree of 1) of the linguistic membership functions defined in Section 3.1, thereby maintaining consistency between the initial linguistic representation and the final risk classification.

Considering the potentially severe consequences of underestimating WUI fire risk, a conservative upward classification rule is adopted when the calculated risk value falls between two adjacent reference values. Specifically, the risk value is assigned to the higher of the two adjacent risk levels. For example, if the calculated risk value is 0.81, which lies above the reference value of 0.8 for the high-risk level, it is classified as extremely high risk. This conservative rule is intended to provide an additional safety margin for fire risk management.

5 Numerical example analysis

5.1 Scenario-based numerical experiment

In this section, a scenario-based numerical experiment is conducted to illustrate the application of the proposed model and establish a quantitative baseline for the subsequent robustness analysis. A controlled synthetic dataset is constructed to represent heterogeneous WUI risk scenarios that may be difficult to capture within a single empirical case. To enhance the realism of the numerical experiment, the characteristics of the synthetic dataset are informed by representative WUI settings reported in the literature and historical fire contexts (Chuvieco et al., 2010; Ganteaume et al., 2013; Radeloff et al., 2018).

Five representative WUI areas (designated as Area 1 through Area 5) are designed to represent diverse combinations of geographical, meteorological, and socioeconomic risk factors. Their baseline profiles are described as follows:

• (1) Area 1 (Urban Edge Zone): located adjacent to the main urban center, this area is characterized by dense residential development, high population density, and a well-developed road network. In contrast, it has a relatively low natural fuel load. Fire risk in this scenario is therefore mainly associated with human exposure and anthropogenic ignition-related factors, while its relatively favorable fuel and accessibility conditions limit the overall risk level.
• (2) Area 2 (Interlaced Transition Zone): this area represents a typical suburban fringe where scattered rural settlements are interspersed with unmanaged forest land. This scenario is characterized by a heterogeneous landscape in which fire risk arises from the combined effects of natural fuel conditions and localized anthropogenic activities.
• (3) Area 3 (Continuous Mountainous Forest Zone): situated within continuous mountainous forests, this area is characterized by high fuel loads, low fuel moisture, steep and complex terrain, relatively low road network density, and high seasonal wind velocities. This scenario represents comparatively severe natural fire conditions and limited accessibility, resulting in a high overall fire risk.
• (4) Area 4 (Scenic and Recreational Area): this area represents a forested recreational setting characterized by relatively high fuel loads, substantial vegetation coverage, and moderately complex terrain. Anthropogenic fire-use activities are also considered in the scenario, although the overall combination of risk factors is less severe than in the higher-ranked areas.
• (5) Area 5 (Agriculture–Forest Edge): this area represents an ecological boundary where agricultural land directly adjoins forested areas. Compared with the other scenarios, it is characterized by a relatively high level of fire-use activity associated with agricultural production. This scenario therefore represents seasonal fire risk associated with agricultural burning activities, such as straw burning during the harvest season.

By defining these five distinct scenarios, the numerical experiment captures a range of heterogeneous WUI conditions and provides a controlled setting for illustrating the proposed aggregation and ranking procedures. The evaluation indicators are presented in Table 1.

Table 1. Index system for assessing fire risk in WUI.

The indicator weights are determined using the scale-extending AHP method proposed by Huang and Zheng (2003). Unlike the conventional 1–9 scale, this method constructs the judgment matrix using a base scale and its logical extensions, thereby facilitating pairwise comparisons and improving the consistency of the judgment matrix. Accordingly, the weight vector for the 15 evaluation indicators is obtained as:

W=(0.1458,0.1303,0.1114,0.0666,0.0612,0.0585,0.0559,0.0650,0.0542,0.0451, 0.0376,0.0450,0.0430,0.0411,0.0392)

Additionally, a predefined weight vector representing the relative importance of the three simulated expert profiles is specified as σ = (σ₁, σ₂, σ₃) = (0.2, 0.3, 0.5).

To improve the reproducibility of the numerical experiment, the procedures used to construct the synthetic dataset and simulated expert inputs are explicitly described. First, the evaluation values assigned to the five WUI areas are specified with reference to plausible ranges reported in the relevant literature. For quantitative indicators (crisp and interval values), the data ranges are selected to remain within plausible physical ranges reported in the relevant fire literature (e.g., fuel loads between 12.5 and 45.2 t/ha and slope gradients not exceeding 60°). For qualitative indicators, fuzzy linguistic variables and their associated information gray degrees are assigned according to the predefined characteristics of the five scenario prototypes. Second, the three initial evaluation matrices do not represent judgments elicited from an actual panel of experts. Instead, they are simulated expert profiles constructed for methodological illustration. The profiles are designed to incorporate controlled variation among evaluators as well as different degrees of information incompleteness, thereby providing heterogeneous inputs for evaluating the aggregation and ranking procedures of the proposed model.

Given the synthetic baseline conditions across the five investigated areas, the initial evaluation matrices Ak=(aij(k),eij(k))5×15(k = 1, 2, 3) corresponding to the three simulated expert profiles are presented in Tables 2–4.

Table 2. Initial evaluation matrix A₁.

Table 3. Initial evaluation matrix A₂.

Table 4. Initial evaluation matrix A₃.

The initial evaluation matrices A₁, A₂, and A₃ are standardized using Equations (1–5), yielding the standardized matrices B₁,B₂, and B₃, as presented in Tables 5–7.

Table 5. Standardized evaluation matrix B₁.

Table 6. Standardized evaluation matrix B₂.

Table 7. Standardized evaluation matrix B₃.

Following Definitions 1–4 and Definition 6, we uniformly project the crisp values, interval values, and fuzzy linguistic variables in the standardized evaluation matrices B₁, B₂, and B₃ into equivalent gray linguistic 2-tuples. The resulting matrices C₁, C₂, and C₃ are presented in Tables 8–10.

**Gray linguistic 2-tuple evaluation matrix C₁.**

Table 8.

Table 9.

**Gray linguistic 2-tuple evaluation matrix C₂.**

Table 10.

Table 11.

**Gray linguistic 2-tuple evaluation matrix C₃.**

Table 12.

Table 13.

Aggregating the evaluation matrices C₁, C₂, and C₃ corresponding to the three simulated expert profiles via Equation (6) yields the comprehensive evaluation matrix C, as shown in Table 11.

**Comprehensive evaluation matrix C.**

Table 14.

Table 15.

Applying Equation (7) to the comprehensive evaluation matrix C yields the reference sequence ε⁺.

ε+=(((u6,0.00),0.1),((u6,0.00),0.1),((u5,−0.20),0.1),((u6,−0.50),0.1),((u5,0.20),0.1),((u6,0.00),0.1),((u6,0.00),0.1),((u6,0.00),0.1),((u3,0.43),0.1),((u2,0.48),0.1),((u4,−0.01),0.1),((u6,0.00),0.1),((u5,−0.20),0.1),((u5,−0.20),0.1),((u6,0.00),0.1))

Applying Equation (8) to matrix C yields the comparison sequences ε₁, ε₂, ε₃, ε₄, ε₅ for the five WUI areas as follows:

ε1=(((u0,0.00),0.1),((u0,0.00),0.1),((u2,−0.20),0.1),((u1,−0.20),0.1),((u5,0.20),0.1)((u6,0.00),0.1),((u0,0.00),0.1),((u0,0.00),0.1),((u2,−0.14),0.1),((u2,0.10),0.1)((u1,0.46),0.1),((u0,0.00),0.1),((u2,−0.20),0.1),((u1,−0.20),0.1),((u6,0.00),0.1))

ε2=(((u3,−0.08),0.1),((u4,−0.18),0.1),((u4,−0.20),0.2),((u3,−0.20),0.2),((u4,0.20),0.2),((u2,−0.37),0.1),((u4,−0.40),0.1),((u5,−0.20),0.1),((u2,0.23),0.1),((u2,0.30),0.1),((u3,−0.21),0.1),((u3,−0.40),0.1),((u4,−0.20),0.1),((u3,−0.20),0.2),((u4,−0.31),0.1))

ε3=(((u6,0.00),0.2),((u6,0.00),0.2),((i5,−0.20),0.3),((u5,−0.20),0.3),((u2,0.20),0.3),((u6,0.00),0.2),((u0,0.00),0.2),((u6,0.00),0.2),((u3,0.43),0.2),((u2,0.48),0.2),((u4,−0.01),0.2),((u6,0.00),0.2),((u5,−0.20),0.2),((u5,−0.20),0.3),((u0,0.00),0.1))

ε4=(((u4,0.13),0.1),((u2,0.18),0.1),((u4,−0.20),0.2),((u2,−0.20),0.1),((u4,0.20),0.1),((u1,−0.26),0.1),((u4,0.05),0.1),((u2,0.00),0.1),((u2,0.04),0.1),((u2,0.22),0.1),((u2,−0.06),0.1),((u3,0.40),0.1),((u3,−0.20),0.1),((u5,−0.20),0.2),((u2,0.14),0.1))

ε5=(((u1,0.38),0.1),((u4,0.36),0.1),((u3,−0.20),0.2),((u6,−0.50),0.2),((u3,0.20),0.2),((u0,0.40),0.1),((u5,−0.05),0.1),((u4,0.00),0.1),((u3,−0.24),0.1),((u2,0.38),0.1),((u3,0.31),0.1),((u1,0.40),0.1),((u4,−0.20),0.1),((u2,−0.20),0.1),((u5,−0.11),0.1))

Applying Equation (9) yields the gray relational matrix G between the comparison sequences and the reference sequence as follows:

G=[0.350.350.520.411.001.000.350.350.680.900.570.350.520.451.00 0.520.600.670.500.670.430.580.730.730.950.730.490.770.560.59 0.850.850.730.630.440.330.850.850.850.850.850.850.850.730.35 0.640.460.670.470.770.390.630.450.700.930.620.560.620.670.46 0.420.670.560.850.560.370.750.620.830.970.830.420.770.520.75 ]

For presentation purposes, the elements of G are rounded to two decimal places, whereas the subsequent gray 2-tuple relational degrees are calculated using the original unrounded relational coefficients. Subsequently, applying Equation (10) yields the gray 2-tuple relational degrees between the comparison sequences ε₁, ε₂, ε₃, ε₄, ε₅ and the reference sequence ε⁺ as follows:

R(ε+,ε1)=0.5411,R(ε+,ε2)=0.6199,R(ε+,ε3)=0.7407,R(ε+,ε4)=0.6018,R(ε+,ε5)=0.6279

Sorting these relational degrees in descending order gives the final evaluation ranking:

R(ε+,ε3)>R(ε+,ε5)>R(ε+,ε2)>R(ε+,ε4)>R(ε+,ε1).

Therefore, the WUI fire risk ranking is Area 3≻ Area 5≻ Area 2≻ Area 4≻ Area 1. According to the fire risk classification criteria in Section 4.2, Area 3 is classified as high risk, Area 5 and Area 2 as slightly high risk, and Area 4 and Area 1 as moderate risk.

5.2 Rationality of gray degree initialization and sensitivity analysis

The initial information gray degrees (0.1, 0.2, and 0.3) are assigned to represent different levels of information incompleteness associated with heterogeneous evaluation data. For instance, a lower value (0.1) represents evaluation information with a relatively high degree of completeness, such as numerical data obtained from relatively reliable sources or linguistic judgments supported by sufficient information, whereas a higher value (0.3) represents evaluation information with a greater degree of incompleteness due to factors such as limited data availability or insufficient supporting information.

A sensitivity analysis is conducted to examine the robustness of the evaluation results with respect to the initialization of the information gray degrees. By varying the gray degree parameters while keeping the indicator and expert weights unchanged, the stability of the risk ranking among the evaluated areas is examined. Three perturbation scenarios are considered:

• Scenario A: gray degree e increases from 0.1 to 0.15.
• Scenario B: gray degree e increases from 0.2 to 0.25.
• Scenario C: gray degree e increases from 0.3 to 0.35.

The gray 2-tuple relational degrees for the five WUI areas are recalculated under these perturbed conditions (Figure 1).

As shown in Figure 1, although the gray 2-tuple relational degrees vary with changes in the information gray degrees (e.g., the value for Area 3 ranges from 0.7107 to 0.7909), the relative ranking of the five areas remains unchanged across the three perturbation scenarios: Area 3≻ Area 5≻ Area 2≻ Area 4≻ Area 1. These results indicate the following:

• (1) Ranking robustness: the unchanged ranking across all perturbation scenarios indicates that the relative risk ranking remains stable under the tested variations in the initial information gray degrees. More importantly, the model consistently preserves the relative risk differentiation among the five WUI areas despite changes in the assumed levels of information incompleteness. This result indicates that, within the tested perturbation ranges, variations in information incompleteness affect the numerical relational degrees without altering the overall risk hierarchy. Consequently, the identification of higher- and lower-risk areas is not unduly dependent on a particular initialization of the information gray degrees, providing greater confidence in the use of the proposed model for comparative WUI fire risk assessment when evaluation information is incomplete.
• (2) Representation of information incompleteness: the observed changes in the gray 2-tuple relational degrees across the perturbation scenarios indicate that variations in the information gray degree are reflected in the evaluation results. By explicitly characterizing the degree of information incompleteness associated with evaluation information, the proposed model allows differences in information completeness to be incorporated into the resulting relational degrees. Together with the ranking stability discussed above, these results suggest that the model is responsive to variations in information incompleteness at the numerical level while maintaining stable relative risk differentiation within the tested perturbation ranges. This capability is particularly relevant to WUI fire risk assessment, where evaluation information is often heterogeneous and incomplete.

To quantitatively evaluate ranking stability, Spearman's rank correlation coefficient (r_*s*) is calculated between the baseline ranking and each perturbed scenario. The Spearman rank correlation coefficient is r_*s* = 1.00 for all three scenarios, indicating complete rank-order agreement with the baseline ranking. This quantitative result further supports the conclusion that the relative risk differentiation among the five WUI areas remains stable under the tested variations in the information gray degrees.

5.3 Weight sensitivity analysis

To examine the sensitivity and robustness of the ranking results to variations in indicator weights, five weight-perturbation scenarios are designed using a one-to-one compensation mechanism. In each scenario, the weight of one selected indicator is increased or decreased by 10% of its baseline value. Since the sum of the indicator weights must remain equal to 1, the resulting change is offset by a corresponding adjustment to another indicator. The specific adjustment rules are presented in Table 12.

Table 12. Operational definitions of weight perturbation scenarios.

The gray 2-tuple relational degrees for the five WUI areas are recalculated under these perturbed conditions, with the results presented in Figure 2.

As shown in Figure 2, the gray 2-tuple relational degrees vary slightly across the five indicator-weight perturbation scenarios, while the relative ranking of the five evaluated areas remains unchanged. Two main observations can be made:

• (1) Stability of the overall risk structure: across the baseline and all weight-perturbed scenarios, the top-ranked area (Area 3: Continuous Mountainous Forest Zone, with relational degrees ranging from 0.7376 to 0.7407) and the bottom-ranked area (Area 1: Urban Edge Zone, with relational degrees ranging from 0.5404 to 0.5436) remain unchanged in their respective positions. Areas 5, 2, and 4 also consistently maintain their relative ranking positions. Thus, the ranking remains Area 3≻ Area 5≻ Area 2≻ Area 4≻ Area 1 across all tested weight-perturbation scenarios. The persistence of these positions indicates that the overall risk structure remains stable under the tested variations in indicator weights.
• (2) Numerical responsiveness to weight perturbations: although the relative ranking remains unchanged, the gray 2-tuple relational degrees exhibit modest variations across the five perturbation scenarios. For example, the relational degree of Area 5 varies from 0.6279 at baseline to 0.6342 in Scenario 3, while that of Area 2 varies within a narrower range of 0.6188–0.6222. These changes indicate that the model remains numerically responsive to variations in indicator weights while preserving the overall rank structure within the tested perturbation ranges.

To quantitatively evaluate ranking stability, Spearman's rank correlation coefficient (r_*s*) is calculated between the baseline ranking and each weight-perturbed scenario. The Spearman coefficient is r_*s* = 1.00 for all five scenarios, indicating complete rank-order agreement with the baseline ranking. This result provides quantitative evidence that the relative ranking of the five evaluated areas remains stable under the tested indicator-weight perturbations.

Overall, the weight sensitivity analysis indicates that the proposed model maintains a stable relative risk ranking under the tested indicator-weight perturbations while remaining numerically responsive to changes in indicator importance. The unchanged rank order across all five perturbation scenarios suggests that the evaluation results are not unduly dependent on the specific baseline weight configuration within the tested ranges.

6 Discussion

6.1 Distinctive methodological capabilities and comparability

In MCDM and risk assessment research, methods such as FAHP, classical GRA, and TOPSIS are widely used. However, a direct numerical comparison between these conventional methods and the proposed gray linguistic 2-tuple relational analysis model would require additional data transformation or methodological adaptation because the present study involves heterogeneous evaluation information, including crisp values, interval values, and fuzzy linguistic variables.

In WUI fire risk assessment, evaluation information may differ substantially in both form and completeness. Standard implementations of classical GRA and TOPSIS typically operate on homogeneous numerical decision matrices. Applying such methods to the heterogeneous dataset used in this study would therefore require the conversion of interval and linguistic information into compatible numerical forms, which may alter some of the information contained in the original representations.

In contrast, the proposed framework transforms crisp values, interval values, and fuzzy linguistic variables into a unified gray linguistic 2-tuple representation and incorporates the associated information gray degree into the evaluation process. This provides a consistent basis for subsequent aggregation and gray relational analysis without requiring all heterogeneous inputs to be reduced to a single crisp form at the initial stage. The sensitivity analyses further show that the resulting rankings remain stable under the tested perturbations, while the numerical relational degrees remain responsive to variations in information gray degrees and indicator weights. Nevertheless, because the present study does not include a direct numerical benchmark against alternative MCDM approaches, a systematic comparative evaluation remains an important direction for future research.

6.2 Computational complexity and scalability analysis

To examine the computational scalability of the proposed model, a computational complexity analysis is conducted.

Let m, n and r denote the numbers of evaluated WUI areas, risk evaluation indicators, and participating experts, respectively. The computational procedure primarily consists of data standardization [O(r·m·n)], multi-expert information aggregation [O(r·m·n)], gray relational degree computation [O(m·n)], and final ranking [O(mlogm)]. Consequently, the overall asymptotic time complexity of the algorithm is O(r·m·n+mlogm).

Since n and r can be treated as fixed within a given regional assessment framework, the overall computational cost scales approximately linearly with the number of evaluated WUI areas m. In addition, the proposed model can be implemented using vectorized matrix operations and does not require iterative model training or large labeled historical datasets. These characteristics suggest that the framework has relatively low computational requirements and may be suitable for assessments involving a larger number of WUI areas. Nevertheless, further computational benchmarking using large empirical datasets would be valuable for evaluating its practical scalability and runtime performance.

6.3 Practical implications and WUI resilience

Beyond providing comparative WUI fire risk rankings, the proposed framework can support the identification of areas requiring greater management attention under conditions of heterogeneous and incomplete evaluation information. Such risk differentiation may provide a quantitative basis for prioritizing further investigation, mitigation planning, and resource allocation.

In this context, the practical interpretation of the assessment results can be combined with the underlying indicator profiles of each evaluated area. For example, an area characterized by a relatively low Road Network Density (I₇) and dense Fuel Spatial Distribution (I₃) may warrant greater attention to emergency accessibility and fuel management, respectively. Information on individual indicators can complement the comprehensive risk ranking and help decision-makers identify potential areas for targeted investigation and mitigation planning. In this sense, the proposed framework can serve as one component of a broader resilience-oriented decision-support process rather than as a standalone prescription for specific interventions. This perspective is consistent with the broader emphasis on resilience and performance-based wildfire management discussed by Tampekis et al. (2023).

7 Conclusion

7.1 Main conclusions

This study develops a comprehensive WUI fire risk assessment framework based on gray linguistic 2-tuple relational analysis. By integrating heterogeneous evaluation data and explicitly representing the associated degree of information incompleteness, the proposed model is applied to evaluate and rank WUI fire risk across a set of representative synthetic scenarios. The main conclusions are summarized as follows:

• (1) Unified representation and relational analysis of heterogeneous evaluation information: the proposed gray linguistic 2-tuple framework provides a unified mathematical representation for heterogeneous evaluation data, including crisp values, interval values, and fuzzy linguistic variables. By incorporating the information gray degree into the linguistic 2-tuple representation, the framework simultaneously represents the evaluation value and its associated degree of information incompleteness. On this basis, the proposed gray 2-tuple relational degree measures the proximity of each area to the positive ideal reference sequence by jointly considering the linguistic 2-tuple values and information gray degrees, thereby providing a quantitative basis for comparative WUI fire risk ranking.
• (2) Multidimensional WUI fire risk indicator system: the established 15-indicator evaluation system characterizes WUI fire risk from four dimensions: fuel conditions, surrounding environment, meteorological conditions, and topographic characteristics. The indicator system incorporates both natural and anthropogenic factors relevant to WUI fire risk and includes region-specific characteristics associated with Chinese WUI environments, such as indigenous vegetation conditions and intensive human-land use interactions. It also retains correlated indicators when they represent distinct physical mechanisms. Since the GRA framework does not require strict independence among indicators, these correlated indicators can be incorporated without imposing the independence assumptions commonly required by some statistical models.
• (3) Robustness and sensitivity of the risk ranking: the sensitivity analyses indicate that the proposed model maintains a stable relative risk ranking while remaining numerically responsive to variations in information gray degrees and indicator weights within the tested perturbation ranges. Variations in the information gray degrees affect the numerical gray 2-tuple relational degrees without altering the relative ranking of the five evaluated areas. Similarly, the indicator-weight perturbations produce modest changes in the relational degrees while preserving the complete rank order across all five tested scenarios. These results indicate that the relative risk ranking is not unduly dependent on the specific parameter settings within the tested perturbation ranges.
• (4) Computational simplicity and scalability: the computational complexity analysis indicates that the asymptotic computational cost of the proposed model scales approximately linearly with the number of evaluated WUI areas m, with a computational complexity of O(r·m·n+mlogm). When the numbers of evaluation indicators and experts are fixed, this complexity implies an approximately linear increase in computational cost with the number of evaluated areas. In addition, the framework does not require iterative model training or large labeled historical datasets. These characteristics suggest that the proposed approach has relatively low computational requirements and may be suitable for comparative WUI fire risk assessment involving a larger number of areas.

7.2 Limitations and future work

Despite the methodological contributions of this study, several limitations remain and provide directions for future research:

• (1) Information resolution in heterogeneous data transformation: although the linguistic 2-tuple representation retains symbolic translation information during numerical-to-linguistic transformation, some information loss may still arise in other stages of heterogeneous data processing. For example, the transformation of interval-valued information into representative values and the use of predefined linguistic membership functions may simplify the original information structure. Future research could investigate more flexible multi-granular linguistic representations and interval-preserving transformation procedures to retain more information from the original data.
• (2) Subjectivity in parameter initialization: the current numerical implementation relies on a common indicator weight vector, predefined information gray degrees, and predetermined weights for the simulated expert profiles. Although the sensitivity analyses indicate that the overall ranking remains stable within the tested perturbation ranges, the initialization of these parameters still involves a degree of subjective judgment. Future research could investigate more data-informed or adaptive procedures for estimating indicator weights and information gray degrees. In addition, more objective procedures for determining expert weights could be explored to reduce the subjectivity associated with their assignment. Expert clustering mechanisms could also be considered for larger and more heterogeneous expert groups.
• (3) Static assessment and spatio-temporal dynamics: the current implementation is primarily a cross-sectional assessment framework and is therefore more suitable for comparative risk evaluation at a given stage or time point. However, WUI fire risk can change over time in response to short-term meteorological conditions, medium-term fuel dynamics, and longer-term land-use change. Future research could integrate the proposed framework with time-series models, dynamic sensing data, or internet of things (IoT) observations to support temporally updated risk assessment. Such integration could provide a basis for extending the current framework toward dynamic WUI fire risk assessment and early warning.
• (4) Synthetic evaluation and the need for empirical validation: the current numerical experiment is based on a controlled synthetic dataset designed to represent heterogeneous WUI scenarios and to illustrate the aggregation, ranking, and sensitivity properties of the proposed method. Although this design provides a controlled setting for methodological evaluation, it cannot fully reproduce the physical, spatial, and social complexity of a specific real-world WUI environment. Future research should therefore apply the proposed framework to empirical WUI cases using data sources such as satellite remote sensing, high-resolution GIS information, local meteorological observations, and historical fire records. Such empirical applications would provide further evidence regarding the practical applicability and regional adaptability of the proposed framework.

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Footnotes

Edited by: Aseesh Pandey, Govind Ballabh Pant National Institute of Himalayan Environment and Sustainable Development, India

Reviewed by: Hao Xu, Anhui University of Technology, China

Stergios Tampekis, Agricultural University of Athens, Greece