The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation

Abstract

Accurate and objective evaluation of existing bridges is critical for ensuring the bridge’s safety and optimizing maintenance strategies. This study proposes an integrated Fuzzy Best and Worst Method and fuzzy comprehensive evaluation (FBWM-FCE) model to evaluate uncertainties in expert judgments and complex decision-making. A four-layer evaluation indicator system and five distinct grades for bridges were established, aligned with the JTG 5120-2004 and JTG/T H21-2011 standards. The FBWM innovatively employs triangular fuzzy numbers (TFNs) to reduce linguistic uncertainties and cognitive bias in bridge evaluation. Subsequently, by integrating FCE for multi-level fuzzy comprehensive operations, the method translates qualitative evaluations into quantitative evaluations using membership matrices and weights. A case study of Ding Jia Bridge and Jigongling Bridge validated the FBWM-FCE model, revealing Class III Bridge (fail condition), consistent with on-site inspections in the 2020 Bridge Inspection and Evaluation Report (Highway Administration of Hubei Provincial Department of Transportation). Comparative analysis demonstrated FBWM’s operational efficiency, requiring 20% fewer pairwise comparisons than AHP while maintaining higher consistency than BWM. The model’s reliability stems from its systematic handling of epistemic uncertainties, offering a high reduction in procedural complexity compared to standardized methods. These advancements provide a scientifically rigorous yet practical tool for bridge management, balancing computational efficiency with evaluation accuracy to support maintenance decisions.

1. Introduction

A variety of factors, including loading, environmental conditions, and material degradation, contribute to the progressive decline in bridge performance throughout its service life, posing significant risks to structural safety and durability [1]. To mitigate catastrophic failures and socioeconomic losses caused by structural damage [2,3], the implementation of scientifically rigorous evaluation models has become imperative. Such models must not only diagnose current conditions but also inform targeted maintenance strategies to extend service life and ensure safety.

Bridge evaluation has long been a focal point in bridge engineering, with evolving methodologies reflecting advancements in the field. Traditional evaluation techniques, while foundational, increasingly integrate modern approaches such as Artificial Neural Networks [4], and ensemble learning techniques such as Random Forest [5], which have demonstrated remarkable predictive capabilities in controlled environments. However, their practical implementation faces insurmountable barriers in real-world bridge management scenarios. These limitations primarily stem from the prohibitive costs of large-scale sensor deployment [6], the computational complexity of model training, and the chronic data scarcity characterizing small to medium-span bridges that form the backbone of regional transportation systems [7].

Within this context, multi-criteria decision-making (MCDM) paradigms have emerged as vital tools for reconciling quantitative data limitations with expert engineering judgment. Conventional approaches such as the Analytic Hierarchy Process (AHP) [8], Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [9], and Cloud Model [10]. Among these methods, the AHP is particularly noteworthy for its capacity to represent complex issues in structured hierarchical formats [11]. Nevertheless, AHP’s requirement for n(n − 1)/2 pairwise comparisons in n-dimensional problems introduces exponential complexity in bridge systems with multiple interdependent criteria, frequently resulting in inconsistent priority rankings that undermine evaluation reliability [12]. The Best-Worst Method (BWM), introduced by Rezaei [13] as a streamlined alternative, reduces comparison requirements to 2(n − 1) while maintaining robust consistency ratios, as evidenced in seismic resilience evaluations of coastal bridges and lifecycle evaluations of aging infrastructure [14]. However, the conventional BWM, which relies on crisp numerical scales, fails to adequately address the linguistic ambiguities and epistemic uncertainties inherent in expert judgments. This limitation becomes particularly significant in the context of emerging measurement technologies, which often generate fuzzy data and require probabilistic risk modeling for accurate interpretation [15].

The integration of fuzzy set theory with MCDM methodologies has catalyzed significant advancements in uncertainty quantification, particularly through the Fuzzy Best-Worst Method (FBWM) employing triangular fuzzy numbers (TFNs) to model human cognitive ambiguity [16,17]. Sina Bahrami et al. [18] pioneered the FBWM-TOPSIS hybrid model for critical feeder identification in power distribution networks, achieving a significant improvement in prioritization accuracy compared to conventional BWM through optimized fuzzy reference comparisons. Their work demonstrated the method’s capacity to handle conflicting expert opinions with a consistency ratio significantly below the 0.1 acceptability threshold. Mahdi Malakoutikhah et al. [19] developed an innovative FBWM-Fuzzy Cognitive Mapping (FCM) framework for construction safety management, systematically quantifying multiple risk factors through expert surveys and reducing subjective bias through iterative cognitive map refinement. Mandana Irannezhad et al. [20] further extended FBWM’s applicability to supply chain blockchain readiness evaluation, achieving an exceptional consistency ratio through probabilistic fuzzy scaling of 22 blockchain adoption indicators. Sajad Jahangiri et al. [21] enhanced sustainability management through three-stage FBWM-Fuzzy Cognitive Mapping (FCM), Decision-Making Trial and Evaluation Laboratory (DEMATEL), demonstrating faster consensus-building in multi-stakeholder environments compared to conventional Delphi techniques. In manufacturing quality control, Muhammet Gul et al. [22] synergized FBWM with Fuzzy Bayesian Networks (FBNs) for failure mode analysis, attaining higher diagnostic precision than traditional FMEA methods through probabilistic risk propagation across multiple failure pathways.

Despite these cross-domain successes, bridge engineering remains conspicuously underdeveloped in systematic FBWM applications, particularly in synthesizing multi-level fuzzy relationships for comprehensive condition evaluation.

This study addresses these critical knowledge gaps through the development of an innovative FBWM-FCE integrated bridge evaluation model, establishing four fundamental advancements in bridge condition evaluation methods. The model introduces a comprehensive bridge evaluation indicator system compliant with JTG 5120-2004 [23] specifications, incorporating safety, durability, suitability, and reinforcement economics, along with 46 indicators derived from extensive field surveys and historical maintenance records. By employing fuzzy triangular numbers to quantify expert linguistic preferences, the model reduces cognitive bias compared to the conventional BWM. The integrated FCE method converts qualitative evaluations into quantitative metrics, enabling comprehensive evaluations of objects influenced by multiple factors based on the actual situation, thereby implementing graded output for bridge condition evaluation. Furthermore, the system’s compatibility with legacy inspection records ensures practical deployability across uninstrumented aging bridges. These innovations carry profound implications for sustainable infrastructure management by providing maintenance engineers with a scientifically robust yet operationally feasible decision-support tool.

2. Fuzzy Best and Worst Method-Fuzzy Comprehensive Evaluation Model

This study marks the first application of the FBWM in the field of bridge condition evaluation. The proposed FBWM-FCE model synergistically integrates the strengths of the FBWM and the FCE method, effectively combining qualitative analysis with quantitative evaluation. Compared to traditional standalone evaluation methods, this hybrid method significantly mitigates the inherent uncertainties and ambiguities in the evaluation process, providing a more accurate and scientifically robust framework for bridge condition evaluation.

2.1. Fuzzy Set and Membership Function

Fuzzy theory was pioneered by professor L.A. Zadeh, a distinguished American cybernetics expert, in 1965 [24]. Its primary objective is to provide an effective method for analyzing and addressing complex problems that are challenging to describe using traditional precise mathematical models. Fuzzy set theory, which forms the foundation and core of fuzzy theory, represents an extension of classical set theory. Specifically, fuzzy sets generalize the membership degree of elements to sets from a binary value (either 0 or 1) to a continuous interval (between 0 and 1). This generalization enables fuzzy sets to provide a more accurate and flexible representation of objects with ambiguous or uncertain characteristics. The mathematical formulation of a fuzzy set is expressed as Equation (1):

$$\widehat{P}=\left\{\left(\sigma ,{\mu }_{\widehat{P}}\left(\sigma \right)\right)|\sigma \in U\right\}$$

(1)

where U represents the universe of discourse; $\widehat{P}$ represents the fuzzy set; $\sigma$ signifies an element within the universe; and ${\mu }_{\widehat{P}}\left(\sigma \right)$ defines the membership function of $\sigma$ with respect to $\widehat{P}$.

Triangular and trapezoidal functions are the most widely utilized membership functions in fuzzy theory, with their mathematical expressions presented in Equation (2) and Equation (3), respectively.

Triangular Function:

$${\mu }_f(t)\left\{\begin{array}{cc}0,& t\le a_i\\ {\displaystyle \frac{t-a_i}{b_i-a_i}},& a_i\le t\le b_i\\ {\displaystyle \frac{c_i-t}{c_i-b_i}},& b_i\le t\le c_i\\ 0,& t\ge c_i\end{array}\right.$$

(2)

A triangular membership function ${\mu }_f\left(t\right)$ is defined by three line segments forming either a symmetric or asymmetric triangle. The vertex of the triangle is located at coordinates (b_i, 1), while the base endpoints are positioned at (a_i, 0) and (c_i, 0).

Trapezoidal Function:

$${\mu }_f\left(t\right)=\left\{\begin{array}{cc}0,& t\le a_i\\ {\displaystyle \frac{t-a_i}{b_i-a_i}},& a_i\le t\le b_i\\ 1,& b_i\le t\le c_i\\ {\displaystyle \frac{d_i-t}{d_i-c_i}},& c_i\le t\le d_i\\ 0,& x\ge d_i\end{array}\right.$$

(3)

A trapezoidal membership function ${\mu }_f\left(t\right)$ is characterized by four line segments forming a trapezoid. The top plateau spans the interval [b_i, c_i], while the base endpoints are located at coordinates (a_i, 0) and (d_i, 0).

A triangular fuzzy number (TFN) is a specialized type of fuzzy number derived from fuzzy set theory, which employs a triangular membership function to characterize the variation in an element’s degree of membership within a set. The triangular structure of TFNs provides a precise and quantitative extension of the fuzzy set concept, making it particularly suitable for modeling fuzzy phenomena that exhibit a distinct central tendency and ambiguous boundaries. This characteristic renders TFNs a valuable and practical tool for addressing uncertainty in a wide range of applications. The TFN ${\widehat{T}}_F$ can be mathematically represented as ${\widehat{T}}_F=\left({\widehat{l}}_F,{\widehat{m}}_F,{\widehat{u}}_F\right)$, and its corresponding triangular membership function ${\mu }_{T_F}\left(t\right)$ is illustrated in Figure 1. Where ${\widehat{l}}_F\le {\widehat{m}}_F\le {\widehat{u}}_F$, the parameters ${\widehat{l}}_F$, ${\widehat{m}}_F$ and ${\widehat{u}}_F$ denote the lower bound, the most likely value, and the upper bound of the ${\widehat{T}}_F$, respectively.

Given two triangular fuzzy numbers ${\widehat{T}}_{F1}=\left({\widehat{l}}_{F1},{\widehat{m}}_{F1},{\widehat{u}}_{F1}\right)$ and ${\widehat{T}}_{F2}=\left({\widehat{l}}_{F2},{\widehat{m}}_{F2},{\widehat{u}}_{F2}\right)$, the fundamental principles of arithmetic operations on TFNs are demonstrated in Equations (4)–(6) [25].

$${\widehat{T}}_{F1}\pm {\widehat{T}}_{F2}=\left({\widehat{l}}_{F1}\pm {\widehat{l}}_{F2},{\widehat{m}}_{F1}\pm {\widehat{m}}_{F2},{\widehat{u}}_{F1}\pm {\widehat{u}}_{F2}\right)$$

(4)

$${\widehat{T}}_{F1}\times {\widehat{T}}_{F2}=\left({\widehat{l}}_{F1}\times {\widehat{l}}_{F2},{\widehat{m}}_{F1}\times {\widehat{m}}_{F2},{\widehat{u}}_{F1}\times {\widehat{u}}_{F2}\right)$$

(5)

$${\displaystyle \frac{1}{{\widehat{T}}_{F1}}}=\left({\displaystyle \frac{1}{{\widehat{u}}_{F1}}},{\displaystyle \frac{1}{{\widehat{m}}_{F1}}},{\displaystyle \frac{1}{{\widehat{l}}_{F1}}}\right)$$

(6)

The triangular fuzzy number ${\widehat{T}}_F=\left({\widehat{l}}_F,{\widehat{m}}_F,{\widehat{u}}_F\right)$ can be converted into a crisp value R(T_F) through the Graded Mean Integration Representation (GMIR) formula [26], as expressed in Equation (7).

$$R\left(T_F\right)={\displaystyle \frac{{\widehat{l}}_F+4{\widehat{m}}_F+{\widehat{u}}_F}{6}}$$

(7)

2.2. Fuzzy Best and Worst Method

The BWM is widely employed for determining indicator weights. However, it often struggles to accurately capture relative preferences due to the inherent vagueness and intangibility of human qualitative judgments, as well as the uncertainties present in real-world data. To address these limitations, the FBWM replaces crisp numerical values with TFNs, thereby offering a more robust and flexible comparative framework [27]. This method effectively accounts for uncertainties in the comparison process, namely fuzzy reference comparison, reduces the influence of subjective factors on evaluation indicators, and ensures that the resulting evaluation more accurately reflects the complexities of real-world conditions.

Unlike the traditional 1–9 scale used in BWM [28], the FBWM employs a five-level importance scale to perform fuzzy reference comparisons, as shown in

Table 1. Five-level importance scale and its associated TFN [30].

Five-Level Importance Scale	TFN
Identically Important (II)	$\left(1,1,1\right)$
Slightly Important (SI)	$\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right)$
Relatively Important (RI)	$\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}\right)$
Highly Important (HI)	$\left({\displaystyle \frac{5}{2}},3,{\displaystyle \frac{7}{2}}\right)$
Extremely Important (EI)	$\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right)$

. The procedural steps of the FBWM, along with their detailed descriptions, are based on the method developed by Guo et al. [29].

The steps of the FBWM are as follows:

Step1. Identifying the best indicator K_B and the worst indicator K_W.

In this step, the best indicator, K_B and the worst indicator, K_W should be determined in general, without making any comparisons.

Step2. Performing the fuzzy reference comparisons for the best indicator.

The best indicator K_B is compared with all other indicators using a five-level importance scale provided in Table 2. The resulting fuzzy comparison vector is derived as Equation (8):

$${\widehat{P}}_B=\left({\widehat{\beta }}_{B1},{\widehat{\beta }}_{B2},\dots ,{\widehat{\beta }}_{Bn}\right)$$

(8)

where ${\widehat{\beta }}_{Bj}$ represents the fuzzy reference comparisons of the best indicator over all other indicators, ${\widehat{\beta }}_{Bj}=(l_{Bj},m_{Bj},u_{Bj})$, j = 1, 2, …, n.

Step3. Performing the fuzzy reference comparisons s for the worst indicator.

All indicators are compared with the worst indicator K_W using a five-level importance scale provided in Table 2. The resulting fuzzy comparison vector is derived as Equation (9):

$${\widehat{P}}_W=\left({\widehat{\beta }}_{W1},{\widehat{\beta }}_{W2},\dots ,{\widehat{\beta }}_{Wn}\right)$$

(9)

where ${\widehat{\beta }}_{jW}$ represents the fuzzy reference comparisons of all indicators over the worst indicator, ${\widehat{\beta }}_{jW}=\left(l_{jW},m_{jW},u_{jW}\right)$, j = 1, 2, …, n.

Step4. Calculating the optimal fuzzy weights vector ${\widehat{v}}^{\ast }=\left({{\widehat{v}}_1}^{\ast },{{\widehat{v}}_2}^{\ast },\cdots ,{{\widehat{v}}_n}^{\ast }\right)$.

The obtained weights are optimal when ${\displaystyle \frac{{\widehat{v}}_B}{{\widehat{v}}_j}}={\widehat{\beta }}_{Bj}$ and ${\displaystyle \frac{{\widehat{v}}_j}{{\widehat{v}}_B}}={\widehat{\beta }}_{jW}$. In addition, in order to obtain the optimal weights, it should minimize the maximum absolute differences of the $\left|{\displaystyle \frac{{\widehat{v}}_B}{{\widehat{v}}_j}}-{\widehat{\beta }}_{Bj}\right|$ and $\left|{\displaystyle \frac{{\widehat{v}}_j}{{\widehat{v}}_W}}-{\widehat{\beta }}_{jW}\right|$ (${\widehat{v}}_B$, ${\widehat{v}}_j$ and ${\widehat{v}}_W$ are TFNs). In practical applications, it is often necessary to convert fuzzy weight vectors into crisp values for each indicator. Consequently, after obtaining the fuzzy weights represented by TFNs, the GMIR formula in Equation (7) can be applied to determine the optimal weights of the indicators. Equation (10) formulates the optimization problem designed to determine the fuzzy weight vectors of the indicators.

$$\begin{array}{l}\begin{array}{cc}\min & \underset{j}{\max }\end{array}\left\{\left|{\displaystyle \frac{{\widehat{v}}_B}{{\widehat{v}}_j}}-{\widehat{v}}_{Bj}\right|,\left|{\displaystyle \frac{{\widehat{v}}_j}{{\widehat{v}}_W}}-{\widehat{v}}_{jW}\right|\right\}\\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{j=1}^{n}Q\left({\widehat{v}}_j\right)}=1\\ l_j^v\le m_j^v\le u_j^v\\ l_j^v\ge 0\\ j=1,2,\cdots ,n\end{array}\right.\end{array}$$

(10)

Equation (10) can be further transformed into Equation (11), which represents a nonlinear constrained optimization problem. By solving Equation (11), the fuzzy weight vectors $\left({{\widehat{v}}_1}^{\ast },{{\widehat{v}}_2}^{\ast },\cdots ,{{\widehat{v}}_n}^{\ast }\right)$ as well as the optimal target vector ${\widehat{\phi }}^{\ast }$, can be obtained.

$$\begin{array}{l}\min {\widehat{\phi }}^{\ast }\\ s.t.\left\{\begin{array}{l}\left|{\displaystyle \frac{\left(l_B^v,m_B^v,u_B^v\right)}{\left(l_j^w,m_j^w,u_j^w\right)}}-\left(l_{Bj},m_{Bj},u_{Bj}\right)\right|\le {\widehat{\phi }}^{\ast }=({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast })\\ \left|{\displaystyle \frac{\left(l_j^v,m_j^v,u_j^v\right)}{\left(l_W^v,m_W^v,u_W^v\right)}}-\left(l_{jW},m_{jW},u_{jW}\right)\right|\le {\widehat{\phi }}^{\ast }=({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast })\\ {\displaystyle \sum _{j=1}^{\mathrm{n}}Q\left({\widehat{v}}_j\right)=1}\\ l_j^v\le m_j^v\le u_j^v\\ l_j^v\ge 0\\ j=1,2,\cdots ,n\end{array}\right.\end{array}$$

(11)

The consistency ratio (CR) serves as the core metric for evaluating the logical consistency of the FBWM and expresses the degree of consistency in fuzzy reference comparisons. The CR is calculated using Equation (12). If the CR is less than or equal to 0.1, it is considered acceptable; otherwise, it is recommended to repeat the pairwise comparisons until the CR is less than or equal to 0.1 [31]. The closer the CR value is to 0, the fewer logical contradictions exist in expert judgments, and the stronger the methodological consistency becomes, thereby ensuring the scientific validity and reliability of weight calculation results.

$$CR={\displaystyle \frac{{\phi }^{\ast }}{CI}}$$

(12)

where CI represents the Consistency index and Table 2 shows the CI corresponding to different fuzzy numbers in the FBWM.

Table 2. Consistency index (CI) for FBWM. TFN [32].

Five Levels of Importance	Identically Important (II)	Slightly Important (SI)	Relatively Important (RI)	Highly Important (HI)	Extremely Important (EI)
${\widehat{\beta }}_{BW}$	$\left(1,1,1\right)$	$\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right)$	$\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}\right)$	$\left({\displaystyle \frac{5}{2}},3,{\displaystyle \frac{7}{2}}\right)$	$\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right)$
CI	3.00	3.8	5.29	6.69	8.04

Table 2. Consistency index (CI) for FBWM. TFN [32].

Five Levels of Importance	Identically Important (II)	Slightly Important (SI)	Relatively Important (RI)	Highly Important (HI)	Extremely Important (EI)
${\widehat{\beta }}_{BW}$	$\left(1,1,1\right)$	$\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right)$	$\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}\right)$	$\left({\displaystyle \frac{5}{2}},3,{\displaystyle \frac{7}{2}}\right)$	$\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right)$
CI	3.00	3.8	5.29	6.69	8.04

The optimal weights of evaluation indicators are calculated based on the FBWM. Firstly, a comprehensive evaluation indicator system is constructed. Subsequently, through consultations with experts in design and management, the best and worst indicators are determined based on the consensus of expert discussions. Fuzzy preference comparison vectors are then established using a five-level importance scale. Finally, the fuzzy weights of each indicator are computed through a nonlinear programming model and subjected to defuzzification. After consistency verification, a weight set for the comprehensive condition indicators is established. The detailed procedural workflow is illustrated in Figure 2.

2.3. Fuzzy Comprehensive Evaluation

The FCE method, grounded in the membership degree theory of fuzzy set theory, transforms qualitative evaluations into quantitative values, enabling a comprehensive evaluation of objects influenced by multiple factors [33]. This method effectively processes fuzzy information, thereby improving the reliability of evaluations [34]. The procedural steps of the FCE method are outlined as follows:

Step1. Constructing the factor set:

It is assumed that O is a factor set consisting of all the indicators in each layer together, which can be expressed as $O=\left\{o_1,o_2,\dots ,o_m\right\}$, where o_i (i = 1, 2, …, m) is the i-th evaluation indicator and m is the number of evaluation indicators in the factor set.

Step2. Determining the evaluation set:

The proposed evaluation set Z is the set of all possible evaluation results for the evaluation object. The set can be defined as $Z=\left\{z_1,z_2,\cdots ,z_n\right\}$, where z_j (j = 1, 2, …, n) denotes the grade corresponding to the evaluation grade of the object in question. Furthermore, to accurately differentiate between the evaluation grades, the value of z_j can be defined in ascending order from the smallest to the largest value, represented as 1, 2, …, n.

Step3. Determining the weight set of the indicators:

The weight v_i, corresponding to the indicator o_i, can be determined through historical data analysis, expert evaluation methods, or fuzzy methods. In this study, the weights were calculated using the FBWM, as detailed in Section 2.2. Consequently, the weight set for the indicators is expressed as $V=\left\{v_1,v_2,\cdots ,v_m\right\}$.

Step4. Constructing the membership matrix

To determine the evaluation set for a single indicator, it is essential to define the membership degree from the factor set O to the evaluation set Z. Assuming that the single indicator membership matrix can be represented as $Q_i=\left\{q_{i1},q_{i2},\dots ,q_{in}\right\}$. Then, the membership matrix based on the entire single-indicator matrix could be established as follows:

$$Q={(q_{ij})}_{m\times n}=\left[\begin{array}{cccc}{q}_{11}& q_{12}& \cdots & q_{1n}\\ q_{21}& q_{22}& \cdots & q_{2n}\\ & \cdots & \cdots & \\ q_{m1}& q_{m2}& \cdots & q_{mn}\end{array}\right]$$

(13)

where q_ij is the membership degree from o_i to z_j.

In this study, semi-trapezoidal and trapezoidal membership functions are used to calculate the membership degree of the indicators [35,36], as illustrated by Equations (14)–(16).

$$q_z(t)=\left\{\begin{array}{ll}1,& t\le a_i\\ {\displaystyle \frac{b_i-t}{b_i-a_i}},& a_i<t\le b_i\\ 0,& x>b_i\end{array}\right.$$

(14)

$$q_z(t)=\left\{\begin{array}{ll}0,& t\le a_i\\ {\displaystyle \frac{t-a_i}{b_i-a_i}},& a_i\le t<b_i\\ 1,& t\ge b_i\end{array}\right.$$

(15)

$$q_z(t)=\left\{\begin{array}{ll}0,& t<a_i\\ {\displaystyle \frac{t-a_i}{b_i-a_i}},& a_i\le t\le b_i\\ 1,& b_i\le t\le c_i\\ {\displaystyle \frac{d_i-t}{d_i-c_i}},& c_i\le t<d_i\\ 0,& t>d_i\end{array}\right.$$

(16)

The functional characteristics of the semi-trapezoidal membership function are illustrated in Figure 3a,b, whereas the trapezoidal membership function is depicted in Figure 4.

Step5. Calculating the evaluation results

A comprehensive evaluation vector is constructed by applying fuzzy comprehensive operations to the weight set and the membership matrix:

$$E=V\times Q$$

(17)

where E is the current layer’s comprehensive evaluation vector, V is the set of weights of the lower layer, and Q is the membership matrix of the lower layer.

To determine the comprehensive evaluation result of the bridge’s condition, Equation (18) is utilized to compute the comprehensive evaluation score:

$$\overline{E}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j=1}^n{V}_i{z}_j}}}{{{\displaystyle \sum }}_{i=1}^m{V}_i}}$$

(18)

where $\overline{E}$ represents the comprehensive evaluation score.

To compute the comprehensive evaluation score, the evaluation set is quantized as Z = {z₁, z₂, …, z_n} = {1, 2, …, n}. Furthermore, according to Equation (18), the result intervals should fall within ranges such as (0, 1), (1, 2), (2, 3), and so on. The evaluation grade is determined based on the interval to which the final score belongs. This forms the foundation for subsequent bridge evaluations, enabling the identification of the optimal plan for repair, reinforcement, or reconstruction.

In practical applications, it is essential to construct a factor set O for evaluation indicators and an appropriate evaluation set Z, tailored to the specific context. A membership matrix Q is then established through expert evaluations or other suitable methods. Finally, E = V × Q is applied to perform comprehensive evaluation operations, yielding the final comprehensive evaluation result. The detailed procedural workflow is illustrated in Figure 5.

3. Condition Evaluation Indicator System for Bridges

Prior to the establishment of a comprehensive bridge evaluation indicator system, this study first introduces the application of fuzzy theory in the evaluation model. Additionally, a thorough review of relevant methodologies is provided, detailing their procedural steps and the key computational techniques employed in the evaluation process. To enhance the scientific rigor of the evaluation, this research proposes a comprehensive bridge evaluation indicator system. The system is elaborated on in the subsequent sections and further validated through a case study.

3.1. Principles for Selecting Indicators

The condition evaluation indicator system for bridges serves as a foundational component of the bridge evaluation process, significantly influencing the selection and direction of subsequent evaluation procedures. It provides a scientifically robust and effective framework for decision-making regarding bridge maintenance, repair, or reconstruction. Consequently, when establishing an evaluation indicator system for bridges, it is imperative to meet the requirements of standardization and authority [23,37] while adhering to the principles of systematicity and completeness.

3.2. Establishing the Indicator System

The proposed comprehensive bridge evaluation indicator system integrates four core dimensions: safety (B1), durability (B2), suitability (B3), and reinforcement economy (B4). These dimensions were derived through systematic analysis of structural integrity and functional criticality across diverse bridges. Among these, safety, durability, and suitability are identified as the three critical indicators that warrant prioritization. Specifically, safety refers to a bridge’s ability to maintain overall structural stability under normal operational conditions; durability reflects its capacity to fulfill its original functional requirements over prolonged use; and suitability emphasizes the bridge’s ability to sustain optimal operational performance throughout its service life. Furthermore, the reinforcement economy is also recognized as a significant indicator within the evaluation system. Based on these considerations, the bridge evaluation indicator system is established, as illustrated in Figure 6.

4. Case Study

4.1. Case Study 1

The Ding Jia Bridge, located in Xiangyang, Hubei Province, China, is a T-beam structure with a total length of 18 m and a width of 6.2 m (as shown in Figure 7).

A comprehensive condition evaluation of the Ding Jia Bridge has been conducted due to identified structural deficiencies in the deck system, superstructure, and substructure. As illustrated in Figure 8, the bridge exhibits multiple types of damage and structural defects, including longitudinal and transverse cracks at the base of the T-beam and within the deck pavement, water infiltration in multiple sections of the superstructure, and efflorescence deposits associated with chronic water seepage.

4.1.1. Weight Calculation for Dingjia Bridge

The FBWM was employed to determine weight coefficients within the novel bridge condition evaluation model. Specifically, the weighting calculation process for the criteria layer is demonstrated using safety (B1), durability (B2), suitability (B3), and reinforcement economy (B4) as representative indicators. Within this set, safety (B1) was selected as the best indicator, while reinforcement economy (B4) served as the worst indicator.

Table 3. The fuzzy reference comparisons of the best and the worst indicator.

Indicator	B1	B2	B3	B4
Best Indicator B1	II	SI	HI	EI
Worst Indicator B4	EI	HI	RI	II

summarizes the expert-derived fuzzy reference comparisons between the best indicator (B1) and other indicators, as well as comparisons of all indicators against the worst indicator (B4).

The fuzzy vector of comparing the best indicator to all other indicators is shown in Equation (19):

$${\widehat{P}}_B=\left(\left(1,1,1\right),\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right),\left({\displaystyle \frac{5}{2}},3,{\displaystyle \frac{7}{2}}\right),\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right)\right)$$

(19)

The fuzzy vector of comparing all other indicators to the worst indicator is shown in Equation (20):

$${\widehat{P}}_W=\left(\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right),\left({\displaystyle \frac{5}{2}},3,{\displaystyle \frac{7}{2}}\right),\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}\right),\left(1,1,1\right)\right)$$

(20)

Following the identification of the best and worst indicators through the FBWM, Equation (11) was mathematically reformulated as Equation (21) through algebraic manipulation. This reformulated equation was subsequently solved numerically via constrained nonlinear optimization implemented in MATLAB (2022), yielding optimized fuzzy weight vectors for the four evaluation indicators (safety, durability, suitability, and reinforcement economy), along with the derived optimal target vector: ${\widehat{v}}_{B1}^{\ast }$ = (0.4045, 0.4197, 0.4372); ${\widehat{v}}_{B2}^{\ast }$ = (0.2840, 0.3213, 0.3494); ${\widehat{v}}_{B3}^{\ast }$ = (0.1293, 0.1540, 0.1966); ${\widehat{v}}_{B4}^{\ast }$ = (0.0956, 0.1027, 0.1125); ${\widehat{\phi }}^{\ast }$ = (0.2893, 0.2893, 0.2893).

After determining fuzzy weight vectors for one hierarchical layer via the FBWM, subsequent indicator weights were systematically calculated according to the established protocol. Criteria layer fuzzy weight vectors were defuzzified through the application of Equation (7). This procedure yielded optimal weights for the four indicators, namely “Safety (B1), Durability (B2), Suitability (B3), and Economic Reinforcement (B4)”, are, respectively, 0.4201, 0.3198, 0.1570, and 0.1032, with the corresponding optimal target value calculated as 0.2893.

$$\begin{array}{l}\begin{array}{cc}\min & {\widehat{\phi }}^{\ast }\end{array}\\ s.t.\left\{\begin{array}{c}\left|{\displaystyle \frac{\left(l_1^v,m_1^v,u_1^v\right)}{\left(l_2^v,m_2^v,u_2^v\right)}}-\left(l_{12},m_{12},u_{12}\right)\right|\le \left({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast }\right)\\ \left|{\displaystyle \frac{\left(l_1^v,m_1^v,u_1^v\right)}{\left(l_3^v,m_3^v,u_3^v\right)}}-\left(l_{13},m_{13},u_{13}\right)\right|\le \left({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast }\right)\\ \left|{\displaystyle \frac{\left(l_1^v,m_1^v,u_1^v\right)}{\left(l_4^v,m_4^v,u_4^v\right)}}-\left(l_{14},m_{14},u_{14}\right)\right|\le \left({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast }\right)\\ \left|{\displaystyle \frac{\left(l_2^v,m_2^v,u_2^v\right)}{\left(l_4^v,m_4^v,u_4^v\right)}}-\left(l_{24},m_{24},u_{24}\right)\right|\le \left({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast }\right)\\ \left|{\displaystyle \frac{\left(l_3^v,m_3^v,u_3^v\right)}{\left(l_4^v,m_4^v,u_4^v\right)}}-\left(l_{34},m_{34},u_{34}\right)\right|\le \left({\tau }^{\ast },{\tau }^{\ast },{\tau }^{\ast }\right)\\ {\displaystyle \sum _{j=1}^{4}Q\left({\widehat{v}}_j\right)=1}\\ l_j^v\le m_j^v\le u_j^v\\ l_j^v\ge 0\\ j=1,2,3,4\end{array}\right.\end{array}$$

(21)

Given that ${\widehat{\beta }}_{BW}={\widehat{\beta }}_{14}=\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right)$, it can be seen from Table 2 that the CI for this case is CI = 8.04.

The Consistency index (CI) and its corresponding optimal target value ${\widehat{\phi }}^{\ast }$, derived from the FBWM, are systematically incorporated into Equation (12) to compute the consistency ratio (CR). The detailed computational procedure is mathematically formulated in Equation (22).

$$CR={\displaystyle \frac{{\phi }^{\ast }}{CI}}={\displaystyle \frac{0.2893}{8.04}}=0.0360<0.1$$

(22)

The computed consistency ratio (CR = 0.0360) falls below the critical threshold of 0.1, confirming satisfactory consistency in the evaluation results. Furthermore, the CR value closely approximates zero, demonstrating exceptional consistency in the pairwise comparison process.

As in the previous process, the optimal weights of each indicator at each layer can be derived, as shown in

Table 4. The optimal weights of each layer’s indicators based on FBWM.

Criteria Layer	Final Weights	Primary Indicator Layer	Final Weights	Set of Weights for Secondary Indicators Layer
B1	0.4201	C1	0.3823	{0.3152, 0.2304, 0.2478, 0.1186, 0.0787}
		C2	0.1793	{0.4287, 0.3244, 0.2469}
		C3	0.2987	{1}
		C4	0.1397	{1}
B2	0.3198	C5	0.5510	{0.0860, 0.0657, 0.1337, 0.2660, 0.2047, 0.2439}
		C6	0.2551	{0.0878, 0.2686, 0.2171, 0.1388, 0.0707, 0.2171}
		C7	0.1939	{0.5510, 0.2551, 0.1939}
B3	0.1570	C8	1	{0.3452, 0.2140, 0.1125, 0.0892, 0.2392}
B4	0.1032	C9	0.5265	{1}
		C10	0.1265	{0.3359, 0.6642}
		C11	0.3471	{0.2509, 0.7491}

.

Table 4 details the optimized weight distribution across hierarchical evaluation layers. The FBWM was systematically implemented across all evaluation strata: target layer, criteria layer, primary indicator layer, and secondary indicator layer. Within the criteria layer, safety demonstrated the highest weighting (0.4201), followed sequentially by durability (0.3198), suitability (0.1570), and reinforcement economy (0.1032) in decreasing significance. The weight distribution analysis identifies safety as the critical determinant in bridge condition evaluation, while durability maintains considerable influence within the evaluation framework. These derived weights were subsequently incorporated into the fuzzy comprehensive evaluation method to systematically determine the bridge’s operational condition.

4.1.2. Condition Rating of Dingjia Bridge

The systematic construction of evaluation factor sets constitutes a critical foundation for ensuring the validity and precision of the FCE method. By systematically analyzing the bridge condition determinants, as schematically represented in Figure 1, the evaluation factor set O is formally defined as: $O_A=\left\{o_{B1},o_{B2},o_{B3},o_{B4}\right\}$, $O_{B1}=\left\{o_{C1},o_{C2},o_{C3},o_{C4}\right\}$, $O_{C1}=\left\{o_{D1},o_{D2},o_{D3},o_{D4},o_{D5}\right\}$, and so forth.

In accordance with the Specifications for Maintenance of Highway Bridges and Culverts (JTG 5120-2004) [23] and the Standards for Technical Condition Evaluation of Highway Bridges (JTG/T H21-2011) [37], the comprehensive condition of bridge’s is classified into five distinct grades. Accordingly, the evaluation set is formally defined as:

Z = {z₁, z₂, z₃, z₄, z₅} = {Class I bridge, Class II bridge, Class III bridge, Class IV bridge, Class V bridge} = {Excellent, Good, Fail, Poor, Critical}.

Following the membership function determination method detailed in Section 2.3 and leveraging data from the 2020 Bridge Inspection and Evaluation Report issued by the Highway Administration of the Hubei Provincial Department of Transportation [38], the membership matrices were systematically constructed, as shown in

Table 5. The membership matrices for the secondary indicator layer.

Criteria Layer	Primary Indicator Layer	Secondary Indicator Layer	Membership Matrix
			Excellent	Good	Fail	Poor	Critical
B1	C1	D1	0	0.4	0.6	0	0
		D2	0	0.5	0.5	0	0
		D3	0	0.3	0.7	0	0
		D4	0	0	0.4	0.6	0
		D5	0	0	0.7	0.3	0
	C2	D7	0	0.5	0.5	0	0
		D7	0	0.4	0.6	0	0
		D8	0	0.6	0.4	0	0
	C3	D9	0	0.4	0.6	0	0
	C4	D10	0	0	0.7	0.3	0
B2	C5	D11	0	0.6	0.4	0	0
		D12	0	0.5	0.5	0	0
		D13	0	0.3	0.7	0	0
		D14	0.8	0.2	0	0	0
		D15	0	0.7	0.3	0	0
		D16	0	0.4	0.6	0	0
	C6	D13	0	0.3	0.7	0	0
		D14	0.8	0.2	0	0	0
		D15	0	0.7	0.3	0	0
		D16	0	0.4	0.6	0	0
		D17	0	0	0.6	0.4	0
		D18	0	0	0.5	0.5	0
	C7	D19	0	0.7	0.3	0	0
		D20	0	0.5	0.5	0	0
		D21	0	0.6	0.4	0.6	0
B3	C8	D22	0.2	0.8	0	0	0
		D23	0	0.4	0.6	0	0
		D24	0	0.7	0.3	0	0
		D25	0	0.6	0.4	0	0
		D26	0	0.7	0.3	0	0
B4	C9	D27	0	0.5	0.5	0	0
	C10	D28	0	0.7	0.3	0	0
		D29	0.8	0.2	0	0	0
	C11	D30	0	0.8	0.2	0	0
		D31	0	0.4	0.6	0	0

.

The fuzzy comprehensive operation is performed through the matrix multiplication by using Equation (17), where the weight vector V and membership matrix Q of the secondary indicator layer generate the comprehensive evaluation vector E for primary indicators. Specifically, the computational procedures for the E_C1–E_C4 are mathematically expressed as in Equations (23)–(26):

$$E_{C1}=V_{D1-D5}\times Q_{C1}={\left(\begin{array}{l}0.3152\hfill \\ 0.2304\hfill \\ 0.2478\hfill \\ 0.1186\hfill \\ 0.0787\hfill \end{array}\right)}^{{\rm T}}\times \left(\begin{array}{lllll}0\hfill & 0.4\hfill & 0.6\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0.5\hfill & 0.5\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0.3\hfill & 0.7\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.4\hfill & 0.6\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.7\hfill & 0.3\hfill & 0\hfill \end{array}\right)=\left(\begin{array}{ccccc}0& 0.32& 0.58& 0.10& 0\end{array}\right)$$

(23)

$$E_{C2}=V_{D6-D8}\times Q_{C2}={\left(\begin{array}{c}0.4287\\ 0.3244\\ 0.2468\end{array}\right)}^{{\rm T}}\times \left(\begin{array}{ccccc}0& 0.5& 0.5& 0& 0\\ 0& 0.4& 0.6& 0& 0\\ 0& 0.6& 0.4& 0& 0\end{array}\right)=\left(\begin{array}{ccccc}0& 0.49& 0.51& 0& 0\end{array}\right)$$

(24)

$$E_{C3}=V_{D9}\times Q_{C3}={\left(1\right)}^{{\rm T}}\times \left(\begin{array}{ccccc}0& 0.4& 0.6& 0& 0\end{array}\right)=\left(\begin{array}{ccccc}0& 0.40& 0.60& 0& 0\end{array}\right)$$

(25)

$$E_{C4}=V_{D10}\times Q_{C4}={\left(1\right)}^{{\rm T}}\times \left(\begin{array}{ccccc}0& 0& 0.7& 0.3& 0\end{array}\right)=\left(\begin{array}{ccccc}0& 0& 0.70& 0.30& 0\end{array}\right)$$

(26)

The comprehensive evaluation vectors of the primary indicators, C1–C4, constitute the membership matrix for the criteria layer indicator B1, as expressed in

Table 6. The comprehensive evaluation vectors of the primary indicators, C1–C4.

Criteria Layer	Primary Indicator Layer	Optimal Weight	Comprehensive Evaluation Vector
B1	C1	0.3823	(0, 0.32, 0.58, 0.10, 0)
	C2	0.1793	(0, 0.49, 0.51, 0, 0)
	C3	0.2987	(0, 0.40, 0.60, 0, 0)
	C4	0.1397	(0, 0, 0.70, 0.30, 0)

.

Thus, the membership matrix for B1 can be obtained as expressed in Equation (27).

$$Q_{B1}=\left(\begin{array}{ccccc}0& 0.32& 0.58& 0.10& 0\\ 0& 0.49& 0.51& 0& 0\\ 0& 0.40& 0.60& 0& 0\\ 0& 0& 0.70& 0.30& 0\end{array}\right)$$

(27)

Following the fuzzy comprehensive operation of each indicator in the primary indicator layer, the fuzzy vectors for each indicator in the criteria layer can be derived. The calculation process for the criteria layer indicator B1 is presented as follows:

$$E_{B1}=V_{C1-C4}\times Q_{B1}={\left(\begin{array}{l}0.3823\\ 0.1793\\ 0.2987\\ 0.1397\end{array}\right)}^T\times \left(\begin{array}{lllll}0\hfill & 0.32\hfill & 0.58\hfill & 0.10\hfill & 0\hfill \\ 0\hfill & 0.49\hfill & 0.51\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0.40\hfill & 0.60\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.70\hfill & 0.30\hfill & 0\hfill \end{array}\right)=\left(0,0.32,0.59,0.08,0\right)$$

(28)

Following the process outlined above, the comprehensive evaluation vectors for each layer can be obtained. Finally, by applying the weights from the criteria layer and the membership matrices, the comprehensive evaluation vector for the target layer is calculated, as shown in Equation (28), with the specific fuzzy computation results for the target layer indicator “Bridge Comprehensive Condition (A)” presented in

Table 7. The comprehensive evaluation vectors of the target layer.

Target Layer	Criteria Layer	Optimal Weight	Comprehensive Evaluation Vector
A	B1	0.4201	(0, 0.33, 0.59, 0.08, 0)
	B2	0.3198	(0.17, 0.36, 0.37, 0.10, 0)
	B3	0.1570	(0.07, 0.66, 0.27, 0, 0)
	B4	0.1032	(0.07, 0.48, 0.45, 0, 0)

.

Thus, the membership matrix and the comprehensive evaluation vector of the target layer A can be obtained as expressed in Equations (29) and (30).

$$Q_A=\left(\begin{array}{ccccc}0& 0.33& 0.59& 0.08& 0\\ 0.17& 0.36& 0.37& 0.10& 0\\ 0.07& 0.66& 0.27& 0& 0\\ 0.07& 0.48& 0.45& 0& 0\end{array}\right)$$

(29)

$$E_A=V_{B1-B4}\times Q_A={\left(\begin{array}{l}0.4201\\ 0.3198\\ 0.1570\\ 0.1032\end{array}\right)}^T\times \left(\begin{array}{lllll}0\hfill & 0.33\hfill & 0.59\hfill & 0.08\hfill & 0\hfill \\ 0.17\hfill & 0.36\hfill & 0.37\hfill & 0.10\hfill & 0\hfill \\ 0.07\hfill & 0.66\hfill & 0.27\hfill & 0\hfill & 0\hfill \\ 0.07\hfill & 0.48\hfill & 0.45\hfill & 0\hfill & 0\hfill \end{array}\right)=\left(0.007,0.41,0.46,0.06,0\right)$$

(30)

A multi-level fuzzy comprehensive evaluation method was adopted to systematically evaluate bridge conditions, progressing from the bottom to the top layer. This method incorporates a hierarchical structure comprising distinct evaluation levels, each assigned to evaluate specific performance indicators. The evaluation procedure begins at the secondary indicator layer, progresses through primary indicator and criteria layers, and culminates in a comprehensive evaluation at the target layer. This tiered framework enables systematic performance analysis by simultaneously considering localized structural details and comprehensive operational performance. The comprehensive evaluation score ${\overline{E}}_A$ was computed through weighted mean analysis, with bridge condition grades assigned numerical values from 1 (excellent) to 5 (critical). Thus, the comprehensive evaluation score is mathematically expressed as:

$${\overline{E}}_A={\displaystyle \frac{0.07\times 1+0.41\times 2+0.46\times 3+0.06\times 4+0\times 5}{0.07+0.41+0.46+0.06+0}}=2.51$$

(31)

According to Equation (31), the comprehensive score of Ding Jia Bridge is 2.51. This score falls within the third interval (2, 3), as detailed in Section 2.3; the Ding Jia Bridge was classified as a Class III bridge, consistent with on-site inspections in the 2020 Bridge Inspection and Evaluation Report (Highway Administration of Hubei Provincial Department of Transportation) [38].

The comprehensive evaluation of Ding Jia Bridge based on BWM can be obtained from Li and Zhang [39]. The optimal weights of each layer’s indicators based on BWM are shown in

Table 8. The optimal weights of each layer’s indicator based on BWM [39].

Criteria Layer	Final Weights	Primary Indicator Layer	Final Weights	Set of Weights for Secondary Indicators Layer
B1	0.4873	C1	0.5333	{0.4484, 0.2651, 0.1558, 0.0765, 0.0542}
		C2	0.2667	{0.5714, 0.2857, 0.1429}
		C3	0.1333	{1}
		C4	0.0667	{1}
B2	0.3057	C5	0.5714	{0.3716, 0.2548, 0.1740, 0.0928, 0.0634, 0.00434}
		C6	0.2857	{0.3630, 0.2547, 0.1968, 0.0801, 0.0619, 0.0435}
		C7	0.1429	{0.6338, 0.2274, 0.1388}
B3	0.1272	C8	1	{0.3940, 0.3034, 0.1576, 0.0819, 0.0631}
B4	0.00798	C9	0.6338	{1}
		C10	0.2274	{0.5, 0.5}
		C11	0.1388	{0.5, 0.5}

.

Following the determination of the membership matrix and optimal weight vectors, the comprehensive evaluation vector E_A for target layer A is computed through the fuzzy comprehensive operation: E_A = (0.0315, 0.2247, 0.5148, 0.2290, 0). So, the comprehensive score ${\overline{E}}_A$ based on BWM is expressed as:

$${\overline{E}}_A={\displaystyle \frac{0.0315\times 1+0.2247\times 2+0.5148\times 3+0.2290\times 4+0\times 5}{0.0315+0.2247+0.5148+0.2290+0}}=2.9413$$

(32)

As calculated by Equation (32), the comprehensive evaluation score for Ding Jia Bridge was determined to be 2.9413. This value falls within the third classification interval (2–3), corresponding to a Class III bridge according to the evaluation set established in Section 2.3.

Comparative analysis of Equations (31) and (32) reveals that both FBWM and BWM produce consistent Class III bridge classifications, aligning with the standardized evaluation results stipulated in the Highway Bridge and Culvert Maintenance Code [23]. The implementation process for Ding Jia Bridge evaluation following this code has been detailed in reference [39]. As a fuzzy extension of BWM, the proposed FBWM demonstrates enhanced robustness through systematic quantification of linguistic uncertainties using TFN theory, reducing cognitive bias compared to conventional BWM implementations.

4.2. Case Study 2

The Jigongling Bridge, located on the S457 highway alignment in Xingshan County, Yichang City (as shown in Figure 9), features a total deck width of 9 m, an overall length of 46.04 m, and a span arrangement of 2 × 16 m. The orthogonally paved concrete decks are complemented by reinforced concrete crash barriers on both sides. The superstructure consists of reinforced concrete prestressed hollow slabs, while the substructure features gravity-type bridge piers and enlarged foundations with double-column piers supported by pile foundations. The plan view of the bridge structure is presented in Figure 10.

The bridge has undergone years of service during which various structural components, including the superstructure, substructure, and deck system, exhibited varying degrees of deterioration. A field survey conducted by professional engineers identified the primary defects, as illustrated in Figure 11.

4.2.1. Weight Calculation for Jigongling Bridge

Comprehensive field investigations conducted on the Jigongling Bridge revealed significant structural anomalies within the deck system, including transverse cracking, localized spalling, and expansion joint blockages. These defects, characterized by accelerated material degradation kinetics and service life expectancy reduction risks, necessitate a systematic contributory factor analysis. This research integrates certified engineering measurements (acquired through laser scanning and sampling techniques) with FBWM-based weight determination to establish a multi-criteria decision-making framework. The resultant priority ranking of five secondary indicators under the Bridge Deck System (C8)—including bridge deck pavement (D22), expansion and contraction joint (D23), pedestrian walk and handrail (D24), illuminated sign (D25), and drainage systems (D26)—provides evidence-based guidance for cost-effective rehabilitation strategies that balance short-term maintenance expenditures with long-term structural sustainability.

Among these, “drainage systems (D22)” and “illuminated sign (D25)” are, respectively, determined to be the best indicator and the worst indicator based on the Bridge Deck System (C8). The linguistic terms of decision-maker for fuzzy reference comparisons of the best indicator over all the indicators and all the indicators over the worst indicator are listed in

Table 9. The fuzzy reference comparisons of the best and the worst indicators.

Indicator	D22	D23	D24	D25	D26
Best Indicator D22	II	SI	RI	EI	RI
Worst Indicator D25	EI	HI	SI	II	SI

.

The fuzzy vector of comparing the best indicator against all other indicators is shown in Equation (33):

$${\widehat{P}}_B=\left(\left(1,1,1\right),\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right),\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}\right),\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right),\left({\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}\right)\right)$$

(33)

The fuzzy vector of comparing all other indicators against the worst indicator is shown in Equation (34):

$${\widehat{P}}_W=\left(\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right),\left({\displaystyle \frac{5}{2}},3,{\displaystyle \frac{7}{2}}\right),\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right),\left(1,1,1\right),\left({\displaystyle \frac{2}{3}},1,{\displaystyle \frac{3}{2}}\right)\right)$$

(34)

After solving Equation (21) using MATLAB, the optimal fuzzy weight vectors for the five indicators D22–D26 and the optimal target vector can be determined: ${\widehat{v}}_{B1}^{\ast }$ = (0.2881, 0.3384, 0.3485); ${\widehat{v}}_{B2}^{\ast }$ = (0.2393, 0.2943, 0.3022); ${\widehat{v}}_{B3}^{\ast }$ = (0.1249, 0.1453, 0.1560); ${\widehat{v}}_{B4}^{\ast }$ = (0.0845, 0.0963, 0.0967); ${\widehat{\phi }}^{\ast }$ = (0.1249, 0.1453, 0.1560).

Following the optimal weights of the five indicators can be obtained: $v_{D22}^{\ast }=0.3317,$ $v_{D23}^{\ast }=0.2865,$ $v_{D24}^{\ast }=0.1437,$ $v_{D25}^{\ast }=0.0944,v_{D26}^{\ast }=0.1437$, and the optimal target values are denoted by ${\phi }^{\ast }=0.3605$.

Given that ${\widehat{\beta }}_{BW}={\widehat{\beta }}_{14}=\left({\displaystyle \frac{7}{2}},4,{\displaystyle \frac{9}{2}}\right)$, It can be seen from Table 2 that the CI for case study 2 is CI = 8.04.

The CI and corresponding optimal target value, obtained through the FBWM, are systematically substituted into Equation (12) to calculate the CR.

The detailed computational procedure is mathematically formulated in Equation (22).

$$CR={\displaystyle \frac{{\phi }^{\ast }}{CI}}={\displaystyle \frac{0.3605}{8.04}}=0.0448<0.1$$

(35)

The value of CR (0.0448) is below 0.1, signifying that the outcomes have successfully passed the consistency examination.

To validate the scientific validity and accuracy of the FBWM-derived results, this case study also employed the AHP and BWM methods to calculate the optimal weights and the CRs. This comparative verification process, whose quantitative findings are systematically summarized in

Table 10. Comparative analysis of the optimal weights for D22–D26 indicators and the CR derived from AHP, BWM, and FBWM.

Method	Secondary Indicators Layer	Optimal Weights	CR	Number of Comparisons
AHP	D22	0.3811	0.0625	${\displaystyle \frac{n(n-1)}{2}}$
	D23	0.2869
	D24	0.1203
	D25	0.0820
	D26	0.1297
BWM	D22	0.4664	0.0608	$2(n-1)$
	D23	0.2291
	D24	0.1161
	D25	0.0570
	D26	0.1314
FBWM	D22	0.3317	0.0448	$2(n-1)$
	D23	0.2865
	D24	0.1437
	D25	0.0944
	D26	0.1437

Where n represents the number of indicators, with n = 5 in Table 3 of the case study.

.

The FBWM employs TFNs to quantify linguistic uncertainties in bridge decision-making contexts. A comparative analysis among the AHP, BWM, and FBWM models revealed smaller discrepancies in optimal weight allocations; however, significant differences emerged in their consistency ratios (CRs): FBWM achieved a CR value of 0.0448 < 0.05, which was 19.7% lower than that of the BWM (0.0608) and 22% lower than that of the AHP (0.0625). This result demonstrates that the fuzzy extension strategy effectively alleviates subjective cognitive biases arising from the limitations of rigid scales in traditional decision models. Furthermore, the comparative analysis reveals that FBWM achieves a 20% reduction in required pairwise comparisons relative to the AHP while maintaining decision consistency. The FBWM significantly reduces decision-making complexity by decreasing the number of pairwise comparisons required. The specific reduction percentage depends on the number of indicators, typically ranging from 20% to 60%, with the reduction percentage increasing as the number of indicators grows.

These methodological advancements collectively validate the robustness and practical applicability of FBWM in sustainable bridge management, providing bridge practitioners with a standardized computational framework that complies with specifications, significantly enhancing infrastructure decision-making efficiency. Therefore, the FBWM can be systematically applied to compute the optimal weights and target values for each layer’s indicators, with the optimal weights of each layer’s indicators are shown in

Table 11. The optimal weights of the indicators for each layer.

Criteria Layer	Final Weights	Primary Indicator Layer	Final Weights	Set of Weights for Secondary Indicators Layer
B1	0.4133	C1	0.3776	{0.3034, 0.2256, 0.2783, 0.1066, 0.0861}
		C2	0.1852	{0.3998, 0.3145, 0.2857}
		C3	0.3002	{1}
		C4	0.1367	{1}
B2	0.3246	C5	0.5498	{0.0886, 0.0701, 0.1249, 0.2578, 0.1998, 0.2588}
		C6	0.2610	{0.0902, 0.2685, 0.2208, 0.1383, 0.0711, 0.2111}
		C7	0.1892	{0.5487, 0.2539, 0.1974}
B3	0.1620	C8	1	{0.3317, 0.2865, 0.1437, 0.0944, 0.1437}
B4	0.1001	C9	0.5263	{1}
		C10	0.1307	{0.3405, 0.6595}
		C11	0.3430	{0.2496, 0.7504}

.

4.2.2. Condition Rating of Jigongling Bridge

The membership matrices for both the primary indicator and secondary indicator layers are presented in

Table 12. The membership matrices for the primary indicator and secondary indicator layers.

Criteria Layer	Primary Indicator Layer	Secondary Indicator Layer	Membership Matrix
			Excellent	Good	Fail	Poor	Critical
B1	C1	D1	0	0.3	0.6	0	0
		D2	0	0.82	0.5	0	0
		D3	0	0.12	0.7	0	0
		D4	0	0.5	0.4	0	0
		D5	0	0	0.7	0.35	0
	C2	D7	0	0.2	0.5	0	0
		D7	0	0.16	0.6	0	0
		D8	0	0.6	0.4	0	0
	C3	D9	0	0.7	0.6	0	0
	C4	D10	0	0.1	0.7	0	0
B2	C5	D11	0	0	0.4	0.25	0
		D12	0	0.7	0.5	0	0
		D13	0	0.5	0.7	0	0
		D14	0.8	0.2	0	0	0
		D15	0	0.7	0.3	0	0
		D16	0	0.45	0.6	0	0
	C6	D13	0	0.4	0.7	0	0
		D14	0	0.8	0	0	0
		D15	0	0.5	0.3	0	0
		D16	0	0.6	0.6	0	0
		D17	0	0	0.6	0.3	0
		D18	0	0.1	0.5	0	0
	C7	D19	0	0.7	0.3	0	0
		D20	0	0.6	0.5	0	0
		D21	0	0.7	0.4	0	0
B3	C8	D22	0	0.34	0	0	0
		D23	0	0.25	0.6	0	0
		D24	0	0.4	0.3	0	0
		D25	0.11	0.85	0.4	0	0
		D26	0.15	0.80	0.3	0	0
B4	C9	D27	0	0.4	0.5	0	0
	C10	D28	0	0.5	0.3	0	0
		D29	0	0.8	0	0	0
	C11	D30	0	0.3	0.2	0	0
		D31	0	0.5	0.6	0	0

.

Therefore, the comprehensive evaluation vectors E_C1, E_C2, E_C3, and E_C4 can then be obtained as follows:

$$E_{C1}=V_{D1-D5}\times Q_{C1}={\left(\begin{array}{l}0.3034\hfill \\ 0.2256\hfill \\ 0.2783\hfill \\ 0.1066\hfill \\ 0.0861\hfill \end{array}\right)}^{{\rm T}}\times \left(\begin{array}{lllll}0\hfill & 0.3\hfill & 0.7\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0.82\hfill & 0.18\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0.12\hfill & 0.88\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0.5\hfill & 0.5\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.65\hfill & 0.35\hfill & 0\hfill \end{array}\right)=\left(\begin{array}{ccccc}0& 0.36& 0.61& 0.03& 0\end{array}\right)$$

(36)

$$E_{C2}=V_{D6-D8}\times Q_{C2}={\left(\begin{array}{c}0.3998\\ 0.3145\\ 0.2857\end{array}\right)}^{{\rm T}}\times \left(\begin{array}{ccccc}0& 0.2& 0.8& 0& 0\\ 0& 0.26& 0.74& 0& 0\\ 0& 0.6& 0.4& 0& 0\end{array}\right)=\left(\begin{array}{ccccc}0& 0.33& 0.67& 0& 0\end{array}\right)$$

(37)

$$E_{C3}=V_{D9}\times Q_{C3}={\left(1\right)}^{{\rm T}}\times \left(\begin{array}{ccccc}0& 0.7& 0.3& 0& 0\end{array}\right)=\left(\begin{array}{ccccc}0& 0.70& 0.30& 0& 0\end{array}\right)$$

(38)

$$E_{C4}=V_{D10}\times Q_{C4}={\left(1\right)}^{{\rm T}}\times \left(\begin{array}{ccccc}0& 0& 0.1& 0.9& 0\end{array}\right)=\left(\begin{array}{ccccc}0& 0& 0.10& 0.90& 0\end{array}\right)$$

(39)

Following the process outlined above, the membership matrix and the comprehensive evaluation vector of the target layer A can be obtained as Equations (40) and (41).

$$Q_A=\left(\begin{array}{ccccc}0& 0.40& 0.46& 0.14& 0\\ 0.12& 0.47& 0.39& 0.02& 0\\ 0.03& 0.44& 0.53& 0& 0\\ 0& 0.46& 0.54& 0& 0\end{array}\right)$$

(40)

$$E_A=V_{B1-B4}\times Q_A={\left(\begin{array}{l}0.4133\\ 0.3246\\ 0.1620\\ 0.1001\end{array}\right)}^T\times \left(\begin{array}{ccccc}0& 0.40& 0.46& 0.14& 0\\ 0.12& 0.47& 0.39& 0.02& 0\\ 0.03& 0.44& 0.53& 0& 0\\ 0& 0.46& 0.54& 0& 0\end{array}\right)=\left(0.04,0.42,0.47,0.07,0\right)$$

(41)

So, the ${\overline{E}}_A$ is expressed as:

$${\overline{E}}_A={\displaystyle \frac{0.04\times 1+0.42\times 2+0.47\times 3+0.07\times 4+0\times 5}{0.04+0.42+0.47+0.07+0}}=2.57$$

(42)

According to Equation (42), the comprehensive score of Jigongling Bridge is 2.57. This score falls within the third interval (2, 3), as detailed in Section 2.3, the Jigongling Bridge was classified as a Class III bridge.

The inspection and evaluation results for the Jigongling Bridge, provided in the “Yichang City Ordinary Highway Bridge ‘Three-Year Elimination of Hazards’ Project, Xingshan County S457 High Water Line Jigongling Bridge Maintenance and Reinforcement Project”of the 2020 Bridge Inspection and Evaluation Report (Highway Administration of Hubei Provincial Department of Transportation [38]) are presented in

Table 13. The inspection and evaluation results for Jigingling Bridge.

Bridge Comprehensive Condition	Bridge Components	Weight	Technical Condition Rating	Technical Condition Grade	Evaluation Result
	Superstructure	0.40	73.5	3	Dr = 77.4 Class III bridge
	Substructure	0.40	82.0	2
	Bridge Deck System	0.20	76.0	3

below.

As demonstrated in Table 13, the standard method produces outcomes consistent with the FBWM. However, the standard method exhibits procedural complexity and operational inefficiency, whereas the FBWM demonstrates superior applicability in bridge condition evaluation through effectively balancing computational efficiency and rating accuracy.

5. Conclusions

• Model Introduction and Comparative Analysis: This study presents a novel integrated evaluation model that synergistically combines the FBWM and FCE methods to address uncertainties and enhance operational efficiency in bridge condition evaluation. The proposed FBWM-FCE model firstly establishes a four-layer indicator system, ensuring the system’s alignment with the structural characteristics of bridges and regulatory requirements. Subsequently, by introducing TFNs to quantify linguistic ambiguities in the bridge’s expert judgments, the model reduces cognitive bias compared to the conventional BWM, achieving a CR of 19.7%, which is 22% lower than BWM and AHP, respectively. This innovation streamlines decision-making, requiring 20% fewer pairwise comparisons than AHP while maintaining robust methodological consistency.
• Case Validation: The practical efficacy of the FBWM-FCE model was validated through case studies of Ding Jia Bridge and Jigongling Bridge in Hubei Province, China. Evaluations of both bridges according to the FBWM-FCE demonstrated full alignment with on-site inspections documented in the 2020 Bridge Inspection and Evaluation Report issued by the Highway Administration of Hubei Provincial Department of Transportation. The model’s reliability was further corroborated by its consistent outcomes with conventional standardized methods, while overcoming their limitations in procedural complexity and operational inefficiency.
• Practical Application Significance, Limitations, and Future: A distinctive contribution of this research lies in its pioneering application of FBWM to bridge condition evaluation. The hybrid methodology reduces reliance on rigid numerical scales, decreasing subjective bias in weight determination processes, particularly in handling ambiguous expert judgments and multi-criteria interactions. Furthermore, the developed indicator system demonstrated exceptional adaptability to diverse bridge typologies, particularly small-to-medium span structures, as evidenced by its successful implementation in the Hubei Provincial Highway Bridge maintenance project. However, the current research on the real-time monitoring of bridge condition and the evaluation of long-span bridges is still not sufficient, and improvements can be made to the following aspects in the future: extending the model to long-span bridges, incorporating real-time sensor data for dynamic condition monitoring, and developing AI-driven automation for large-scale network-level evaluations. These advancements promote the proposed methodology as a transformative solution for modern infrastructure management while balancing theoretical innovation with engineering pragmatism.