Reliability Assessment of Highway Bridges Based on Combined Empowerment–TOPSIS Method

Abstract

(1) In recent years, with the continuous increase of the state’s investment in infrastructure, the construction of highways and bridges has developed rapidly, which has brought great convenience to people’s lives. At the same time, with the increase of bridge service time, the reliability of bridges declines. In order to meet the requirements of sustainable development, it is necessary to accurately evaluate the reliability of bridges. However, most of the existing evaluation methods have single-weighting and one-sidedness. There are problems such as strong subjectivity and overly simple evaluation procedures. Therefore, it is urgent to establish a new scientific bridge reliability evaluation system. (2) Methods. In this paper, a bridge superstructure is taken as the research object, and the “Technical Condition Assessment Standard for Highway Bridges” (JTG/T H21-2011) is used as the criterion to establish a bridge reliability evaluation index system. The subjective and objective weights of the evaluation indicators are based on minimum discriminant information. Each evaluation indicator is combined and weighted; then, the closeness of each evaluation object to the positive ideal solution is determined according to Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). Finally, the reliability level of each evaluation object is determined. (3) Results. Reliability evaluation of the three-span superstructure of the bridge was carried out, and final reliability evaluation results of “Grade 2, Grade 2, Grade 2” were obtained, which are consistent with the actual working state of the bridge. (4) Conclusions. The evaluation results of this paper are consistent with the results obtained by the traditional AHP–Extenics method, but the evaluation model of this paper adopts combined weighting, which avoids the one-sidedness of the weighting of a single method—thus, the comprehensive weight obtained not only reflects the subjective intention of decision makers, but also reflects the objective properties of the data.

1. Introduction

Since the reform and opening up, China’s bridge construction has made rapid achievements, and a series of technically complex and difficult-to-construct bridges have been built. According to statistics, there are 832,500 highway bridges in China, with a total of 52,256,200 m, and highway bridges have become an important part of national infrastructure. For the highway bridges constructed early in this process, due to the rapid growth of traffic volume, the bridges enter the period of concentrated decay after 20~30 years of operation [1]. By the end of 2015, about 16% of the bridges on national and provincial trunk lines had been in service for more than 25 years, and about 31% of the bridges had been in service for more than 15 years. As the service time of bridges increases under the combined effects of external factors such as environment and load, material performance gradually deteriorates and component damage accumulates, and their technical condition tends to decline, which affects the normal operation of bridges and even causes serious safety accidents, resulting in huge economic losses and casualties. Domestic bridges have collapsed one after another, such as the Tianzhuangtai Bridge in Liaoning and the Fusong Jinjiang Bridge in Jilin. The causes are numerous, involving design, construction, maintenance and other aspects, but in general, lack of reliability is still the main reason. Current evaluation of bridge reliability is mostly studied from the perspective of structural reliability and expressed by the probability of failure of the structural system; however, in the process of judging structural failule, some states cannot be clearly given boundaries, and there are many factors affecting bridge reliability. If reliability is judged from a single or a few indicators, accuracy will be greatly reduced [2]. Therefore, it is necessary to establish a more perfect scientific evaluation system of highway bridge reliability, reasonably evaluate the reliability of highway bridges, and take corresponding technical measures to improve bridge reliability in time, which is of great significance to improve the overall reliability of highway bridges.

At present, scholars at home and abroad have done a lot of research on reliability assessment of highway bridges, and many bridge reliability assessment models have been established. Jingjing Cui et al. [2] used an improved hierarchical analysis (FAHP) and entropy weight to study the reliability of concrete bridges in service in the dry-cold region of northwest China in order to evaluate reliability objectively and accurately. In order to reasonably evaluate the reliability of the superstructure of a concrete hyperbolic arch bridge, Yannian Zhang et al. [3] established an object element model, introduced the correlation function and the correlation degree in the topologizable set, and calculated the correlation to the actual reliability. Yu et al. [4] applied a global beta decompression method and an improved differential equivalent recursive algorithm to the system reliability assessment of complex structures such as self-anchored suspension bridges and achieved good results; Liu et al. [5] conducted a system reliability assessment of assembled concrete hollow slab bridges based on an improved AHP method and compared it with the traditional method, verifying that their method is more suitable for assessing the system reliability of such bridges and providing maintenance decision makers with more reasonable bridge condition information; Leander. J [6] conducted a reliability assessment using a Eurocode fatigue assessment model for steel bridges. The study showed that the assessment model needs to be improved and pointed out the parameters that the model needs to focus on; Xiaoya Bian et al. [7] proposed a response surface bounds method based on non-probabilistic reliability theory; Wang Lei et al. [8] proposed a reliability assessment method for existing reinforced concrete (RC) bridges with small sample data by combining inspection results with expert experience and using the influence parameters of structural resistance as fuzzy variables in response to the problem that it is sometimes difficult to obtain sufficient samples to accurately determine the statistical characteristics of parameters for reliability assessment of existing reinforced concrete (RC) bridges. Rui-Feng Nie et al. [9] proposed a temporary structural condition assessment method based on system reliability analysis that can better solve the problem of resistance and load uncertainty in the current design and also ensures that the structure has consistent reliability; Yanweerasak et al. [10] proposed a new probabilistic method to assess whole-life reliability of existing reinforced concrete (RC) bridges under multiple hazards. In the actual assessment process, existing bridge reliability assessment methods have disadvantages such as incomplete index systems, index weights that do not match the engineering reality and rough assessment results. Therefore, based on the summary of the existing assessment methods, this paper innovatively proposes a highway bridge reliability assessment method based on a combined assignment–TOPSIS method.

This paper firstly establishes the bridge reliability evaluation index system based on the normative standard [11,12], which determines subjective and objective weights of each evaluation index by applying hierarchical analysis and entropy weight; it then combines subjective and objective weights based on minimum discriminative information, adopts the Spearman consistency coefficient to characterize the consistency between the weights calculated by hierarchical analysis and entropy weight, and verifies the reasonableness of this combined weighting method. Finally, TOPSIS is used to calculate the relative closeness of each evaluation object to the positive ideal solution and to determine the reliability evaluation grade of each evaluation object. In this paper, this evaluation method is successfully applied to the reliability evaluation of a reinforced concrete bridge, and the results show that the reliability rating evaluation results of the bridge are consistent with the bridge’s actual working condition, which verifies that the combined assignment–TOPSIS method is scientific and effective when used for reliability evaluation of highway bridges. The AHP–Extenics method was also used to evaluate the reliability of this bridge, and, according to the results of the comparative study, the method proposed in this paper is more reasonable.

2. Selection of Evaluation Indicators

Reliability assessment of highway bridges is a very complex problem; there are many factors affecting the reliability of bridges, and the interactions between factors is very complex [13]. Therefore, the scientificity of road bridge reliability evaluations will directly affect their accuracy.

According to the Specification for Maintenance of Highway Bridges and Culverts [14] (JTG5120-2021), there are five grades of girder bridge reliability, denoted in this paper as: X = {X₁,X₂,X₃,X₄,X₅} = {grade 1,grade 2,grade 3,grade 4,grade 5} Grade 1 is a beam bridge with intact function; Grade 2 is a beam bridge with minor defects; Grade 3 is a beam bridge with moderate defects; Grade 4 is a beam bridge with major defects in the main components for which normal use cannot be guaranteed; Grade 5 is a beam bridge with serious defects in the main components, with the bridge in a dangerous condition.

In this paper, taking bridge superstructure as an example, 10 bridge superstructure reliability evaluation indexes were selected with reference to the Technical Condition Assessment Standard for Highway Bridges [11] (JTG/T H21-2011) and the Bearing Capacity Assessment Regulations for Highway Bridges [12] (JTG/T J21-2011): reinforcement corrosion potential level, electrical resistivity, chloride ion content, concrete presumed strength homogeneity coefficient, average concrete carbonation depth/average measured protective layer thickness, characteristic value of reinforcement protective layer thickness/design value, crack width of main girder, deformation of main girder, height of wrong platform and crack width of bridge deck. The grading criteria of each index are shown in

Table 1. Reliability index rating standard.

Indicators	Evaluation Grade
	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$
Steel corrosion potential level $u_1/\mathrm{mV}$	≥−200	−300~−200	−400~−300	−500~−400	<−500
Resistivity $u_2/\left(\mathsf{\Omega }·\mathrm{cm}\right)$	≥20,000	15,000~20,000	10,000~15,000	5000~10,000	<5000
Chloride ion content (% of cement content) $u_3/\%$	<0.15	0.15~0.4	0.4~0.7	0.7~1.0	≥1.0
Homogeneity factor of presumed strength of concrete $u_4$	≥0.95	0.9~0.95	0.8~0.9	0.7~0.8	<0.7
Average value of concrete carbonation depth/average value of measured protective layer thickness $u_5/\mathrm{mm}$	<0.5	0.5~1.0	1.0~1.5	1.5~2.0	≥2.0
Reinforcing steel protective layer thickness characteristic value/design value $u_6/\mathrm{mm}$	>0.95	0.85~0.95	0.7~0.85	0.55~0.7	≤0.55
Crack width of main beam $u_7$/mm	<0.05	0.05~0.10	0.10~0.15	0.15~0.20	>0.20
Main beam deformation (maximum deflection in span/1/1000 of calculated span diameter) $u_8$	<0.83	0.83~1	1~1.67	1.67~2	≥2
Wrong platform height $u_9$/cm	<1	1~2	2~5	5~8	≥8
Width of bridge deck cracks $u_{10}$/mm	<1	1~3	3~5	5~7	≥7

.

3. Rating Method

3.1. Hierarchical Analysis

Hierarchical analysis is a subjective assignment method; it is a simple, flexible and practical multi-criteria decision-making method for quantitative analysis of qualitative problems [15]. The calculation steps for hierarchical analysis to determine the weights of evaluation indicators are as follows:

(1)

Construction of judgment matrix

After stratifying the research problem according to objectives, criteria and solutions, the relative importance of each indicator at the same level is compared by a two-by-two comparison to determine the relative importance lower-level to upper-level indicators so as to obtain the weights of each evaluation indicator [16]. If there are n evaluation indicators in a certain layer, the judgment matrix can be constructed as shown in Equation (1).

$$A={(a_{ij})}_{n\times n}=\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{n1}& \cdots & a_{nn}\end{array}\right]$$

(1)

where i = 1, 2, …, n; j = 1, 2, …, n; $a_{ij}$ indicates the judgment result of the comparative importance of two evaluation indicators i and j in criterion A. In this paper, we use expert scoring to determine the base data based on two-by-two comparison of factor indicators based on a 1–9 scale to quantify the relative importance between two factors [17]. The numerical values and importance relationships are shown in

Table 2. Relationship between numerical value and importance.

Numerical Value	Importance
1	Equal importance
3	Moderate importance of one over another
5	Essential or strong importance
7	Very strong importance
9	Extreme importance

.

(2)

Calculation of evaluation index weights

According to judgment matrix A obtained from two-by-two comparisons of evaluation indicators, the eigenvector β corresponding to its maximum eigenvalue ${\lambda }_{\max }$ is calculated; then, the problem of calculating the weights of evaluation indicators is also transformed into the problem of solving eigenvector β of judgment matrix A. The equation is as follows.

$$A\beta ={\lambda }_{\max }\beta$$

(2)

In this paper, the square root is used for weight calculation as follows [18]:

Step 1: Judgment matrix A rows are multiplied by rows to obtain $u_i$; see Equation (3).

$$u_i={\displaystyle \prod _{j=1}^n{a}_{ij}}, i=1,2,\dots ,n$$

(3)

Step 2: The $u_i$ is squared n times separately to obtain $u_i^{\prime }$; see Equation (4).

$$u_i^{\prime }=\sqrt[n]{u_i}$$

(4)

Step 3: Regularize $u_i^{\prime }$ to obtain the feature vector ${\beta }_i$; see Equation (5).

$${\beta }_i=\frac{u_i^{\prime }}{{\displaystyle \sum _{i=1}^n{u}_i^{\prime }}}$$

(5)

Step 4: Calculate the maximum characteristic root ${\lambda }_{\max }$ of judgment matrix A; see Equation (6).

$${\lambda }_{\max }={\displaystyle \sum _{i=1}^n\frac{{(A\beta )}_i}{n{\beta }_i}}$$

(6)

(3)

Consistency test

In order to verify the reasonableness of the above weight calculation, a consistency test of judgment matrix A is required, and the specific steps are as follows [19].

Step 1: Calculate the consistency index $C.I.$; see Equation (7).

$$C.I.=\frac{{\lambda }_{\max }-n}{n-1}$$

(7)

where n is the order of judgment matrix A; λ_max is the maximum characteristic root of judgment matrix A.

Step 2: Calculate the consistency ratio $C.R.$; see Equation (8).

$$C.R.=\frac{C.I.}{R.I.}$$

(8)

where R.I. is the average random consistency index, which can be determined by checking

Table 3. Average random consistency index R.I.

n	1	2	3	4	5	6	7	8	9	10
R.I.	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46	1.49

[20].

According to the consistency test criteria, a consistency ratio $C.R.<0.10$ indicates that the constructed judgment matrix satisfies the consistency test; otherwise, it is necessary to rescore by experts and construct new judgment matrixes until one meets the consistency test.

3.2. Entropy

Entropy is a measure of system uncertainty in information theory. The greater the amount of information, the smaller the uncertainty and the lower the entropy; conversely, the smaller the amount of information, the greater the uncertainty and the higher the entropy. Entropy can be used as an objective weighting method to determine the weight of decision indicators based on the amount of information contained in each indicator [21].

There are m evaluation objects, denoted as $M=\left(M_1,M_2,\dots ,M_m\right)$, n evaluation indicators, denoted as $N=\left(N_1,N_2,\dots ,N_n\right)$, and the value of evaluation object $M_i$ on evaluation indicator $N_j$ is denoted as $r_{ij}\left(i=1,2,\dots ,m;j=1,2,\dots ,n\right)$. The original data matrix R is formed by $r_{ij}$, where $r_{ij}$ is the evaluation value of the ith object under the jth evaluation indicator [22].

The steps for calculating the weights of each evaluation index using entropy weighting are as follows:

(1)

Data standardization

Different evaluation indexes with different dimensional units impact the evaluation results in the calculation [23], so the original data need to be dimensionlessly processed to obtain the dimensionless matrix $B={\left(b_{ij}\right)}_{m\times n}$.

For metrics for which larger values are more favorable, the dimensionless treatment expressions is [24]:

$$b_{ij}=\frac{r_{ij}-{\min }_j(r_{ij})}{{\max }_j(r_{ij})-{\min }_j(r_{ij})}$$

(9)

For indicators for which smaller values are more favorable, the dimensionless treatment expression is:

$$b_{ij}=\frac{{\max }_j(r_{ij})-r_{ij}}{{\max }_j(r_{ij})-{\min }_j(r_{ij})}$$

(10)

where ${\max }_j(r_{ij})$ is the maximum value of $r_{ij}$ under the jth evaluation index and ${\max }_j(r_{ij})$ is the minimum value of $r_{ij}$ under the jth evaluation index.

(2)

Calculate the weight of the ith evaluation object under the jth indicator

$$p_{ij}=\frac{b_{ij}}{{\displaystyle \sum _{i=1}^m{b}_{ij}}}$$

(11)

(3)

Calculate the entropy of the jth index

$$e_j=-k{\displaystyle \sum _{i=1}^m{p}_{ij}\ln p_{ij}}$$

(12)

$$K=\frac{1}{\ln m}$$

(13)

The greater the variability of an indicator in the evaluation system, the smaller the $e_j$; the smaller the variability, the larger the $e_j$; $e_j$ = 1 means that indicator i has no influence on the evaluation system at this time [25].

(4)

Calculate the coefficient of variation of the information entropy of the jth index

$$d_j=1-e_j$$

(14)

(5)

Determine the weights of each evaluation index

$${\gamma }_j=\frac{d_j}{{\displaystyle \sum _{j=1}^n{d}_j}}$$

(15)

3.3. Calculation of Portfolio Assignment Weights

Hierarchical analysis requires less quantitative information and is concise and practical, but it is influenced by subjective factors and does not fully reflect the connection of each evaluation index [26]. As an objective weighting method, the greater the number of samples collected, the closer the weight calculation results are to the objective facts. In order to avoid the one-sidedness of a single method, this paper combines two methods to calculate the weights, reflecting both the subjective intention of decision makers and the objective properties of data [27].

In this paper, combination weights are determined based on minimum discriminative information [28], and combination weights are as close as possible to the subjective and objective weights. The constructed objective function is shown in Equation (16); the Lagrange multiplier is used to solve the objective function, which yields Equation (17):

$$\begin{array}{l}\min F={\displaystyle \sum _{j=1}^n({\omega }_j\ln \frac{{\omega }_j}{{\beta }_j})}+{\displaystyle \sum _{j=1}^n({\omega }_j\ln \frac{{\omega }_j}{{\gamma }_j})}\\ s.t.{\displaystyle \sum _{j=1}^n{\omega }_j=1;}{\omega }_j>0\end{array}$$

(16)

$${\omega }_j=\frac{{({\beta }_j+{\gamma }_j)}^{0.5}}{{\displaystyle \sum _{j=1}^n{({\beta }_j+{\gamma }_j)}^{0.5}}}$$

(17)

where ${\omega }_j$ is the weight vector obtained from combination assignment, ${\beta }_j$ is the weight vector determined by hierarchical analysis, and ${\gamma }_j$ is the weight vector determined by entropy weight.

3.4. Reasonableness Analysis of Portfolio Empowerment

It is unknown whether the combination of hierarchical analysis and entropy weighting used in this paper to calculate weights is reasonable, so we analyzed the reasonableness as follows [23]:

(1)

Summary of weights

The results of calculating the weights of each index by hierarchical analysis and entropy weight are shown in

Table 4. Evaluation index weight result.

Evaluation Method	Indicator 1	Indicator 2	⋯	Indicator n
Hierarchical analysis	${\beta }_1$	${\beta }_2$	⋯	${\beta }_n$
Entropy	${\gamma }_1$	${\gamma }_2$	⋯	${\gamma }_n$

, where ${\beta }_j$ is the weight of the jth evaluation index determined by hierarchical analysis and ${\gamma }_j$ is the weight value of the jth evaluation index determined by entropy weight.

(2)

Weight ordering

The results obtained from the calculation of evaluation index weights by the two methods are ranked in

Table 5. Ranking result of evaluation index weight.

Evaluation Method	Indicator 1	Indicator 2	⋯	Indicator n
Hierarchical analysis	${\beta }_1^{\prime }$	${\beta }_2^{\prime }$	⋯	${\beta }_n^{\prime }$
Entropy	${\gamma }_1^{\prime }$	${\gamma }_2^{\prime }$	⋯	${\gamma }_n^{\prime }$

, where ${\beta }_j^{\prime }$ is the ranked value of subjective weight value ${\beta }_j$ and ${\gamma }_j^{\prime }$ is the ranked value of objective weight value ${\gamma }_j$. The ranked values are expressed by 1~n (n is a positive integer), with 1 and n being the maximum and minimum weight value of the evaluation index, respectively.

(3)

Spearman Consistency Coefficient

The Spearman consistency coefficient [29] reflects the correlation between two sets of variables and is denoted by $\rho$. In this paper, it is used to reflect the consistency between the weights obtained from hierarchical analysis and entropy weighting and is calculated as follows:

$$\rho =1-\frac{6}{n(n^2-1)}{\displaystyle \sum _{j=1}^n{({\beta }_j^{\prime }-{\gamma }_j^{\prime })}^2}$$

(18)

where ${\beta }_j^{\prime }$ is the ranking value of subjective weight value ${\beta }_j$, ${\gamma }_j^{\prime }$ is the ranking value of objective weight value ${\gamma }_j$, and $\rho$ is in the range of $\left[-1,1\right]$. When $\rho \in \left[-1,0\right)$ there is no consistency between the weights calculated by the two methods; when $\rho =0$, the correlation between the weights calculated by the two methods is 0; $\rho =\left(0,1\right]$ indicates that there is consistency between the weights calculated by the two methods, meaning they can be combined to assign a weight.

3.5. TOPSIS Evaluation Model

TOPSIS is a multi-attribute decision-making method proposed by C. L. Hwang and K. Yoon in 1981 to rank a limited number of preferred solutions according to their proximity to the idealized solution [30]. The positive ideal solution is a virtual optimal solution whose index values are optimized among the evaluation objects; the negative ideal solution is a virtual worst solution whose index values reach the worst values among the evaluation objects [31].

The basic principle of TOPSIS is to calculate the positive and negative ideal solutions of the multi-attribute decision problem and then rank solutions according to their distances from the positive and negative ideal solutions—if the preferred solution is closest to the positive ideal solution and farthest from the negative ideal solution, then it is the best solution; conversely, if the preferred solution is closest to the negative ideal solution and farthest from the positive ideal solution, then it is the worst solution [32].

The comprehensive evaluation process of the combined assignment–TOPSIS method is shown in Figure 1 [33].

(1)

Initial decision matrix

Let the set of schemes $F=\left(F_1,F_2,\dots ,F_m\right)$, the set of evaluation indicators $D=\left(D_1,D_2,\dots ,D_n\right)$ for each scheme, and evaluation indicator $e_{ij}$ denote the indicator value of scheme $F_i$ under indicator $D_j$ [34], where $i=1,2,\dots ,m$, and $j=1,2,\dots ,n$; the initial decision matrix formed by the evaluation indicator values is:

$$E={(e_{ij})}_{m\times n}=\left[\begin{array}{ccc}{e}_{11}& \cdots & e_{1n}\\ ⋮& \ddots & ⋮\\ e_{m1}& \cdots & e_{mn}\end{array}\right]$$

(19)

(2)

Standardized decision matrix

The evaluation indicators are divided into benefit-based indicators and cost-based indicators. For benefit-based indicators, the larger the value the better; for cost-based indicators, the smaller the value the better. The evaluation indicators have different scales and scale units and are not comparable. In order to eliminate the incommensurability of the indicators, it is necessary to dimensionlessize the evaluation indicators [35] and obtain the standardized decision matrix $C={\left(c_{ij}\right)}_{m\times n}$.

For benefit-based indicators, the expressions are calculated as:

$$c_{ij}=\frac{e_{ij}-{\min }_j(e_{ij})}{{\max }_j(e_{ij})-{\min }_j(e_{ij})}$$

(20)

For cost-based metrics, the expression is calculated as:

$$c_{ij}=\frac{{\max }_j(e_{ij})-e_{ij}}{{\max }_j(e_{ij})-{\min }_j(e_{ij})}$$

(21)

where ${\max }_j(e_{ij})$ is the maximum value of column j in the initial decision matrix $E$, and ${\min }_j(e_{ij})$ is the minimum value of column j in the initial decision matrix $E$.

(3)

Weighted standardized decision matrix

Using the combination weight values ${\omega }_j$ derived from the combination assignment above, the data in the standardized decision matrix are multiplied by the weights corresponding to each indicator [13] to obtain the weighted standardized decision matrix $Z={\left(z_{ij}\right)}_{m\times n}$.

$$Z={(z_{ij})}_{m\times n}=\left[\begin{array}{ccc}{z}_{11}& \cdots & z_{1n}\\ ⋮& \ddots & ⋮\\ z_{m1}& \cdots & z_{mn}\end{array}\right]=\left[\begin{array}{ccc}{\omega }_1{e}_{11}& \cdots & {\omega }_n{e}_{1n}\\ ⋮& \ddots & ⋮\\ {\omega }_1{e}_{m1}& \cdots & {\omega }_n{e}_{mn}\end{array}\right]$$

(22)

(4)

Determine the positive ideal solution and negative ideal solution

The positive ideal solution $Z^{+}$ consists of the maximum value of Z in each column

$$Z^{+}=(Z_1^{+},Z_2^{+},\dots ,Z_n^{+})=(\max Z_{i1},\max Z_{i2},\dots ,\max Z_{in})$$

(23)

The negative ideal solution $Z^{-}$ consists of the smallest value of Z in each column

$$Z^{-}=(Z_1^{-},Z_2^{-},\dots ,Z_n^{-})=(\min Z_{i1},\min Z_{i2},\dots ,\min Z_{in})$$

(24)

(5)

Calculate Euclidean distance

The expression for the distance $S_i^{+}$ between each object to be evaluated and the positive ideal solution is:

$$S_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^n{(z_{ij}-z_j^{+})}^2}}$$

(25)

The expression for the distance $S_i^{-}$ between each object to be evaluated and the negative ideal solution is:

$$S_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^n{(z_{ij}-z_j^{-})}^2}}$$

(26)

where $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

The relative closeness of each evaluation object to the positive ideal solution is:

$$G_i=\frac{S_i^{-}}{S_i^{-}+S_i^{+}}, i=1,2,\dots ,\mathrm{m}$$

(27)

3.6. The Advantages and Disadvantages of the Combined Weighting–TOPSIS Method

The combined weighting–TOPSIS method proposed in this paper provides a new method for reliability evaluation of highway bridges. This method has certain advantages and disadvantages, as follows:

Advantages:

(1)

This paper adopts AHP and entropy weight to give subjective and objective weights to bridge reliability evaluation indicators, which fully avoids the one-sidedness of single-method weighting so that the obtained comprehensive weight does not only reflect the subjective intention of decision makers but also reflects the objective properties of the data.

(2)

TOPSIS is a commonly used comprehensive evaluation method that makes full use of the original data to accurately reflect the relative closeness of each evaluation scheme to the optimal scheme and the worst-case scheme as the basis for evaluating the pros and cons, and this method has no strict restrictions on data distribution and sample size.

Disadvantages:

(1)

When AHP is used for subjective weighting, expert scoring is used to construct the judgment matrix. There are too many qualitative components, which are unconvincing. When there are many evaluation indicators, statistical data is too complicated, weight is difficult to determine, and the consistency test is more complicated.

(2)

When using TOPSIS for comprehensive evaluation, the general weighted standard decision-making matrix is more complex, and it is not easy to solve the positive ideal solution and the negative ideal solution, making the calculation is difficult.

3.7. Related Work

At present, a large number of experts and scholars have done in-depth research on reliability evaluation of highway bridges, with many results having very practical value. Highway bridge reliability evaluation based on the right method combined with TOPSIS can scientifically and effectively evaluate the reliability of bridges. Comparison of different reliability assessment methods is shown in

Table 6. Comparison of reliability assessment methods.

Author	Related Work
Y. N. Zhang	Y. N. Zhang established a matter–element model for reliability evaluation of the superstructure of a concrete double-curvature arch bridge, introduced a correlation function and correlation degree in the extension set, and calculated the degree of correlation to the actual reliability.
H. B. Liu	H. B. Liu evaluated the system reliability of fabricated concrete hollow slab bridges based on an improved AHP method and compared it with the traditional method to verify that this method is more suitable for evaluating the system reliability of such bridges.
J. Leander	J. Leander used the steel bridge fatigue assessment model in the Eurocode for reliability assessment. The study showed that the assessment model needs to be improved and pointed out the parameters that the model should focus on.
X. Y. Bian	X. Y. Bian proposed a response-surface bound method based on non-probabilistic reliability theory to solve a non-probabilistic reliability index and evaluate the reliability of bridges in service.
L. Wang	L. Wang proposed a reliability assessment method for existing reinforced concrete (RC) bridges with small sample datasets using the influence parameters of structural resistance as fuzzy variables.
R. F. Nie	R. F. Nie proposed a temporary structure state assessment method based on system reliability analysis that can better solve the problem of resistance and load uncertainty and can ensure consistent structural reliability.
B. S. Xu	B. S. Xu proposed a bridge reliability assessment method based on a combined weighting–TOPSIS method. This method can avoid the one-sidedness of the weighting of a single method, and, when using TOPSIS for evaluation, it can make full use of the information of the original data and accurately evaluate bridge reliability grades.

.

4. Project Example Analysis

4.1. Project Overview

A reinforced concrete bridge has been in use for 15 years; it has a net width of 11 m and a span of 5 × 20 m. The design load rating of the bridge is highway class I, and there is no structural maintenance history. The reliability evaluation of the superstructure $\left\{S_1,S_2,S_3\right\}$ of three of the spans was carried out separately, and the measured values of the indexes [36] are shown in

Table 7. Measured value of reliability evaluation index.

Span Number	Evaluation Indicators
	u1/mV	u2/Ω·cm	u3/%	u4	u5/mm	u6/mm	u7/mm	u8	u9/cm	u10/mm
$S_1$	−245	18,273	0.16	0.92	0.40	0.80	0.08	0.90	1.6	2.0
$S_2$	−281	16,483	0.12	0.85	0.56	0.88	0.05	0.88	1.4	1.6
$S_3$	−253	16,769	0.17	0.84	0.65	0.82	0.03	0.84	1.1	1.1

.

4.2. Hypothesis and Limitations

In bridge reliability evaluation, the hypothesis and limitations need to be made according to the actual situation of the project. In the reliability assessment of reinforced concrete bridges in this paper, the following hypothesis are made:

(1)

The 10 reliability evaluation indicators selected in this paper are independent of each other, there is no significant correlation, and each indicator can be quantified by a certain measurement method.

(2)

This paper uses analytic hierarchy to carry out subjective empowerment and adopts expert scoring to construct the judgment matrix. Experts must be familiar with bridge reliability and be able to score objectively and accurately.

(3)

In this paper, entropy weight is used for objective weighting, which requires the measured data of each indicator to be known, and for different indicators, different numerical standardization formulas need to be used for dimensionless processing.

(4)

When using TOPSIS for reliability evaluation, there must be two or more research objects.

4.3. Bridge Superstructure Reliability Evaluation

(1)

Calculation of subjective weights based on hierarchical analysis

According to the reliability evaluation indexes in Table 6, expert scoring is used to analyze the 10 evaluation indexes in a two-by-two comparison based on a 1-9 scale, and the relative importance of the two indexes is quantified to obtain judgment matrix $A={\left(a_{ij}\right)}_{n\times n}$ of the evaluation indexes. The values in judgment matrix A are obtained after the comprehensive balance based on the data, expert opinions and decision makers’ understanding as follows:

$$A={(a_{ij})}_{n\times n}=\left[\begin{array}{cccccccccc}1& 0.5& 0.5& 0.333& 1& 1& 1& 1& 1& 0.5\\ 2& 1& 1& 1& 2& 1& 2& 1& 2& 2\\ 2& 1& 1& 1& 2& 1& 1& 1& 1& 2\\ 3& 1& 1& 1& 2& 1& 1& 1& 2& 3\\ 1& 0.5& 0.5& 0.5& 1& 1& 1& 1& 2& 1\\ 1& 1& 1& 1& 1& 1& 3& 0.5& 1& 1\\ 1& 0.5& 1& 1& 1& 0.333& 1& 1& 1& 2\\ 1& 1& 1& 1& 1& 2& 1& 1& 1& 1\\ 1& 0.5& 1& 0.5& 0.5& 1& 1& 1& 1& 1\\ 2& 0.5& 0.5& 0.333& 1& 1& 0.5& 1& 1& 1\end{array}\right]$$

The subjective weight vector ${\beta }_j$ = (0.071, 0.138, 0.120, 0.139, 0.085, 0.101, 0.087, 0.104, 0.079, 0.076) of each evaluation index can be calculated according to Equations (3)–(5). From Equations (6) and (7), we get ${\lambda }_{\max }=10.514, C.I.=0.057$. Checking Table 2, we know that when judgment matrix A is of order $n=10$, the average random consistency index is $R.I.=1.49$; using Equation (8), $C.R.=0.038<0.10$—the consistency test meets the requirements, so subjective weight vector ${\beta }_j$ is acceptable.

(2)

Calculation of objective weights based on entropy weight

The measured data of the reliability indexes in Table 6 are dimensionlessized using Equations (9) and (10), and the dimensionlessization matrix is obtained as

$$B={(b_{ij})}_{m\times n}=\left[\begin{array}{cccccccccc}1& 1& 0.2& 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0.125& 0.36& 1& 0.6& 0.333& 0.4& 0.444\\ 0.778& 0.160& 0& 0& 0& 0.25& 1& 1& 1& 1\end{array}\right]$$

For dimensionless matrix B, the entropy $e_j$ = (0.624, 0.365, 0.410, 0.318, 0.526, 0.455, 0.602, 0.512, 0.545, 0.562) of each evaluation index can be calculated according to Equations (11)–(13), and then the difference coefficient $d_j$ = (0.376, 0.635,0.590, 0.682, 0.474, 0.545, 0.398, 0.488, 0.455, 0.438) of the information entropy of each index is derived according to Equation (14). Finally, the objective weight vector ${\gamma }_j=\left(0.074,0.125,0.116,0.134,0.093,0.107,0.078,0.096,0.090,0.086\right)$ of each evaluation index based on the entropy weight is derived from Equation (15).

(3)

Portfolio empowerment and rationalization analysis

Based on the index weights calculated by the above hierarchical analysis and entropy weighting, the weight values are ranked as shown in

Table 8. Ranking of evaluation index weights.

Method	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
Hierarchical analysis	10	2	3	1	7	5	6	4	8	9
Entropy	10	2	3	1	6	4	9	5	7	8

.

The consistency coefficient $\rho =0.915\in \left(0,1\right]$ can be obtained from Equation (18) and shows consistency between the weights calculated by hierarchical analysis and the entropy weighting; thus weights can be combined. Based on the weights ${\beta }_j$ and ${\gamma }_j$ calculated by the above hierarchical analysis and entropy weighting, the combined weight vector ${\omega }_j=\left(0.086,0.115,0.109,0.118,0.095,0.103,0.091,0.101,0.092,0.091\right)$ can be obtained using Equation (17).

(4)

Combination of weighting and TOPSIS

First, the initial decision matrix E₀ is established with the bridge reliability index rating criteria, and the initial decision matrix is obtained according to Table 5 and Equation (19).

$$E_0=\left[\begin{array}{cccccccccc}-200& 20,000& 0& 0.95& 0& 0.95& 0& 0& 0& 0\\ -300& 15,000& 0.15& 0.9& 0.5& 0.85& 0.05& 0.83& 1& 1\\ -400& 10,000& 0.4& 0.8& 1& 0.7& 0.1& 1& 2& 3\\ -500& 5000& 0.7& 0.7& 1.5& 0.55& 0.15& 1.67& 5& 5\\ -600& 0& 1& 0.6& 2& 0.4& 0.2& 2& 8& 7\end{array}\right]$$

From Equations (20) and (21), the normalized decision matrix is

$$C_0=\left[\begin{array}{cccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 0.75& 0.75& 0.85& 0.857& 0.75& 0.818& 0.75& 0.585& 0.875& 0.857\\ 0.5& 0.5& 0.6& 0.571& 0.5& 0.545& 0.5& 0.5& 0.75& 0.571\\ 0.25& 0.25& 0.3& 0.286& 0.25& 0.273& 0.25& 0.165& 0.375& 0.286\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

Then, based on Equation (22) and combined weights ${\omega }_j$ = (0.086, 0.115, 0.109, 0.118, 0.095, 0.103, 0.091, 0.101, 0.092, 0.091), the weighted decisionalization matrix is obtained:

$$Z_0=\left[\begin{array}{cccccccccc}0.086& 0.116& 0.110& 0.118& 0.095& 0.097& 0.093& 0.101& 0.093& 0.090\\ 0.065& 0.087& 0.093& 0.101& 0.072& 0.079& 0.070& 0.059& 0.081& 0.077\\ 0.043& 0.058& 0.066& 0.068& 0.048& 0.053& 0.047& 0.051& 0.070& 0.051\\ 0.022& 0.029& 0.033& 0.034& 0.024& 0.026& 0.023& 0.017& 0.035& 0.026\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

The positive ideal solution $Z^{+}$ = (0.086, 0.116, 0.110, 0.118, 0.095, 0.097, 0.093, 0.101, 0.093, 0.090) and negative ideal solution $Z^{-}=\left(0,0,0,0,0,0,0,0,0,0\right)$ of weighted decision matrix A are obtained according to Equations (23) and (24). Then, the distance between each evaluation grade and the positive and negative ideal solutions is calculated according to Equations (25) and (26) as $S_i^{+}$ and $S_i^{-}$, respectively. Finally, the relative closeness $G_i$ of each evaluation grade to the positive ideal solution is calculated by Equation (27) as listed in Table 8, and the bridge reliability classification criteria can be obtained from

Table 9. Evaluation target calculation results.

Durability Grade	${\mathit{S}}_{\mathit{i}}{}^{+}$	${\mathit{S}}_{\mathit{i}}{}^{-}$	${\mathit{G}}_{\mathit{i}}$
$X_1$ (Grade 1)	0	0.318	1
$X_2$ (Grade 2)	0.073	0.251	0.775
$X_3$ (Grade 3)	0.144	0.177	0.552
$X_4$ (Grade 4)	0.233	0.087	0.271
$X_5$ (Grade 5)	0.318	0	0

as listed in

Table 10. Bridge reliability class.

	Indicators
$X_1$ (Grade 1)	$0.775<G_i\le 1$
$X_2$ (Grade 2)	$0.552<G_i\le 0.775$
$X_3$ (Grade 3)	$0.271<G_i\le 0.552$
$X_4$ (Grade 4)	$0<G_i\le 0.271$
$X_5$ (Grade 5)	0

.

The measured data of the reliability evaluation indexes in Table 6 are dimensionlessized using Equations (20) and (21) to obtain the standard decision matrix as

$$C_1=\left[\begin{array}{cccccccccc}1& 1& 0.2& 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0.125& 0.36& 1& 0.6& 0.333& 0.4& 0.444\\ 0.778& 0.16& 0& 0& 0& 0.25& 1& 1& 1& 1\end{array}\right]$$

Then, Equation (22) is used to multiply the data in the standardized decision matrix by the weight ${\omega }_j$ of each corresponding indicator to obtain the weighted decisionalization matrix as

$$Z_1=\left[\begin{array}{cccccccccc}0.086& 0.115& 0.022& 0.118& 0.095& 0& 0& 0& 0& 0\\ 0& 0& 0.109& 0.015& 0.034& 0.103& 0.055& 0.034& 0.037& 0.040\\ 0.067& 0.018& 0& 0& 0& 0.026& 0.091& 0.101& 0.092& 0.091\end{array}\right]$$

Then, the positive ideal solution $Z^{+}$ = (0.086, 0.115, 0.109, 0.118, 0.095, 0.103, 0.091, 0.101, 0.092, 0.091) and negative ideal solution $Z^{-}=\left(0,0,0,0,0,0,0,0,0,0\right)$ of the weighted decision matrix can be obtained using Equations (23) and (24). Then, the distance $S_i^{+}$ and $S_i^{-}$ between each object to be evaluated and the positive and negative ideal solutions plus the relative closeness $G_{\mathrm{i}}$ between each object and the positive ideal solution can be calculated by Equations (25)–(27). Finally, the results of the evaluation of the reliability grade of the three-span upper structure are listed in

Table 11. Evaluation results of bridge reliability grade.

Span Number	${\mathit{S}}_{\mathit{i}}{}^{+}$	${\mathit{S}}_{\mathit{i}}{}^{-}$	${\mathit{G}}_{\mathit{i}}$	Evaluation Results
$S_1$	0.231	0.290	0.556	Grade 2
$S_2$	0.215	0.276	0.562	Grade 2
$S_3$	0.225	0.302	0.573	Grade 2

.

(5)

Analysis of evaluation results

From the evaluation results in Table 10, it can be seen that the reliability grade of the bridge for the three spans is evaluated as Grade 2, Grade 2 and Grade 2, indicating that the bridge has slight deficiencies, which is consistent with the reliability grade evaluation results in the literature [36]. Based on TOPSIS, the closer the relative closeness $G_i$ of each evaluation object to the positive ideal solution is to 1, the closer the solution is to the optimal level, so the superiority ranking of the reliability of the superstructure of the three spans of this bridge is: $S_3>S_2>S_1$. When maintenance resources are limited, span 1 and span 2 can be given priority.

In summary, the combined assignment–TOPSIS method is well-tested in the reliability assessment of actual bridge projects and reflects the accuracy and applicability of the method, which provides a new method for the reliability assessment of highway bridges.

4.4. Comparative Study

In order to illustrate the rationality and scientificity of the method, this paper does a comparative study using the AHP–Extenics method. AHP calculates subjective weights, as was explained in detail in the previous section; thus we focus on Extenics here.

Extenics was first founded as a new discipline by scholars such as Wen Cai from Guangdong University of Technology and adopts a formal model to study the possibility of expansion of things and the laws and methods of pioneering innovation. It is widely used in multi-objective comprehensive determination problems, and is mainly divided into the objects, extensible sets and extensible logic, among which objects have a greater advantage in solving uncertainty and fuzziness of things [37]. Objects mainly include the following three parts: determination of classical domain and nodal domain, determination of objects to be evaluated, and calculation of the correlation degree of the evaluation index [38].

Let the highway bridge reliability assessment study be the object element R with m evaluation indicators under it, and also divide n evaluation grades for each evaluation indicator $u_i$, the evaluation indicator set $U=\left\{u_1,u_2,\dots ,u_m\right\}$ and the evaluation grade set $V=\left\{v_1,v_2,\dots ,v_n\right\}$ [39].

(1)

The object element classical domain is denoted as:

$$R_j=(V_j,u_i,x_i{}_j)=\left[\begin{array}{ccc}{V}_j,& u_1,& x_{1j}\\ & u_2& x_{2j}\\ & ⋮& ⋮\\ & u_m& x_{mj}\end{array}\right]=\left[\begin{array}{ccc}{V}_j,& u_1& \langle a_{1j},b_{1j}\rangle \\ & u_2& \langle a_{2j},b_{2j}\rangle \\ & ⋮& ⋮\\ & u_m& \langle a_{mj},b_{mj}\rangle \end{array}\right]$$

(28)

where $V_j$—the jth $(j=1,2,\dots ,\mathrm{n})$ grade of the evaluation object;

$u_i$—the ith $(i=1,2,\dots ,\mathrm{m})$ evaluation indicator of the evaluation object;

$x_{ij}$—the value of the corresponding $u_i$ indicator when the evaluation object belongs to the jth grade;

$\langle a_{ij},b_{ij}\rangle$—the range of values when evaluation object i belongs to the jth evaluation grade $V_j$, i.e., the classical domain; $a_{ij}$ is the lower limit of the value of the quantity of the ith indicator $u_i$; and $b_{ij}$ is the upper limit of the value of the quantity of the ith indicator $u_i$ [40].

The nodal domain of the object element is expressed as:

$$R_P=[P,u_i,x_{iP}]=\left[\begin{array}{ccc}P,& u_1,& x_{1P}\\ & u_2& x_{2P}\\ & ⋮& ⋮\\ & u_m& x_{mP}\end{array}\right]=\left[\begin{array}{ccc}{V}_j,& u_1,& \langle a_{1P},b_{1P}\rangle \\ & u_2& \langle a_{2P},b_{2P}\rangle \\ & ⋮& ⋮\\ & u_m& \langle a_{mP},b_{mP}\rangle \end{array}\right]$$

(29)

where $P$—the whole of the evaluation grade;

$x_{ip}=\langle a_{ip},b_{ip}\rangle$—the range of all values of evaluation index $u_i$, i.e., the section field; $a_{ip}$ is the minimum value of the lower limit of the ith evaluation index $u_i$ in all evaluation grades; and $b_{ip}$ is the maximum value of the upper limit of the ith index $u_i$ in all evaluation grades.

(2)

Determination of the elements to be evaluated

$$R_W=\left[\begin{array}{ccc}W& u_1& x_1\\ & u_2& x_2\\ & ⋮& ⋮\\ & u_n& x_n\end{array}\right]$$

(30)

where W—an object to be evaluated;

$x_i$—$W$ on the evaluation of the value range of indicator $u_i$ is the specific indicator data of the object to be evaluated.

(3)

Calculation of the correlation of evaluation indicators

Correlation of reliability grade j of the road bridge to be evaluated is as follows:

$$\rho (x_i,x_{ij})=\left|x_i-\frac{a_{ij}+b_{ij}}{2}\right|-\frac{b_{ij}-a_{ij}}{2}$$

(31)

$$\rho (x_i,x_{ip})=\left|x_i-\frac{a_{ip}+b_{ip}}{2}\right|-\frac{b_{ip}-a_{ip}}{2}$$

(32)

$$K_j(x_i)=\frac{\rho (x_i,x_{ij})}{\left[\rho (x_i,x_{ip})-\rho (x_i,x_{ij})\right]}$$

(33)

where $K_j(x_i)$—the correlation degree of the object to be evaluated when for evaluation grade j.

(4)

Determine the reliability evaluation grade of highway bridges

Combining of the weight coefficients of each index with the calculated correlation function value synthesized to get the evaluation object’s grade correlation is shown in Equation (34). If $K_{\max }=K_j$, then the evaluation object belongs to grade j.

$$K_j(W)={\displaystyle \sum _{i=1}^n{\omega }_i{K}_j(x_i)}$$

(34)

where ${\omega }_i$ is the weight value of evaluation index $u_i$; the larger ${\omega }_i$ is, the greater the degree of influence of the evaluation index on the evaluation object.

The classical domain and section domain in this bridge reliability evaluation index are in

Table 12. Reliability evaluation index classical domain and node domain.

Indicators	Classic Domain					Nodal Domain
	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$
$u_1$	$\left[-200,-100\right)$	$\left[-300,-200\right)$	$\left[-400,-300\right)$	$\left[-500,-400\right)$	$\left[-600,-500\right)$	$\left[-600,-100\right)$
$u_2$	$\left[20,000,25,000\right)$	$\left[15,000,20,000\right)$	$\left[10,000,15,000\right)$	$\left[5000,10,000\right)$	$\left[0,5000\right)$	$\left[0,25,000\right)$
$u_3$	$\left[0,0.15\right)$	$\left[0.15,0.4\right)$	$\left[0.4,0.7\right)$	$\left[0.7,1.0\right)$	$\left[1.0,1.3\right)$	$\left[0,1.3\right)$
$u_4$	$\left[0.95,1\right)$	$\left[0.9,0.95\right)$	$\left[0.8,0.9\right)$	$\left[0.7,0.8\right)$	$\left[0.6,0.7\right)$	$\left[0.6,1\right)$
$u_5$	$\left[0,0.5\right)$	$\left[0.5,1\right)$	$\left[1,1.5\right)$	$\left[1.5,2\right)$	$\left[2,2.5\right)$	$\left[0,2.5\right)$
$u_6$	$\left[0.95,1.05\right)$	$\left[0.85,0.95\right)$	$\left[0.7,0.85\right)$	$\left[0.55,0.7\right)$	$\left[0.4,0.55\right)$	$\left[0.4,1.05\right)$
$u_7$	$\left[0,0.05\right)$	$\left[0.05,0.10\right)$	$\left[0.10,0.15\right)$	$\left[0.15,0.20\right)$	$\left[0.20,0.25\right)$	$\left[0,0.25\right)$
$u_8$	$\left[0,0.83\right)$	$\left[0.83,1\right)$	$\left[1,1.67\right)$	$\left[1.67,2\right)$	$\left[2,2.5\right)$	$\left[0,2.5\right)$
$u_9$	$\left[0,1\right)$	$\left[1,2\right)$	$\left[2,5\right)$	$\left[5,8\right)$	$\left[8,10\right)$	$\left[0,10\right)$
$u_{10}$	$\left[0,1\right)$	$\left[1,3\right)$	$\left[3,5\right)$	$\left[5,7\right)$	$\left[7,9\right)$	$\left[0,9\right)$

, and the measured values of each evaluation index are shown in

Table 13. Measured values of reliability evaluation index.

Evaluation Indicators	$\mathbf{Span}{\mathit{S}}_1$	$\mathbf{Span}{\mathit{S}}_2$	$\mathbf{Span}{\mathit{S}}_3$
$u_1$	−245	−281	−253
$u_2$	18,273	16,483	16,769
$u_3$	0.16	0.12	0.17
$u_4$	0.92	0.85	0.84
$u_5$	0.4	0.56	0.65
$u_6$	0.8	0.88	0.82
$u_7$	0.08	0.05	0.03
$u_8$	0.9	0.88	0.84
$u_9$	1.6	1.4	1.1
$u_{10}$	2	1.6	1.1

.

From Equations (31)–(33), we can obtain the correlation degree of highway bridge reliability evaluation indexes to be evaluated, and the calculation results are as follows:

$$\begin{gathered} \mathrm{span}{S}_1: K_j(x_i)=\left[\begin{array}{ccccc}-0.237& 0.450& -0.275& -0.517& -0.638\\ -0.204& 0.345& -0.327& -0.552& -0.664\\ -0.059& 0.067& -0.200& -0.771& -0.840\\ -0.273& 0.333& -0.200& -0.600& -0.733\\ 0.333& -0.200& -0.600& -0.733& -0.800\\ -0.375& -0.167& 0.250& -0.615& -0.500\\ -0.273& 0.333& -0.200& -0.467& -0.600\\ -0.072& 0.084& -0.100& -0.461& -0.550\\ -0.273& 0.333& -0.200& -0.680& -0.800\\ -0.333& 1.000& -0.333& -0.600& -0.714\end{array}\right] \\ \mathrm{span}{S}_2: K_j(x_i)=\left[\begin{array}{ccccc}-0.309& 0.117& -0.095& -0.397& -0.548\\ -0.292& 0.211& -0.148& -0.432& -0.574\\ 0.333& -0.200& -0.400& -0.829& -0.880\\ -0.400& -0.250& 0.500& -0.250& -0.500\\ -0.097& 0.120& -0.440& -0.627& -0.720\\ -0.292& 0.214& -0.150& -0.738& -0.660\\ 0& 0& -0.455& -0.625& -0.714\\ -0.054& 0.060& -0.120& -0.473& -0.560\\ -0.222& 0.400& -0.300& -0.720& -0.825\\ -0.273& 0.600& -0.467& -0.680& -0.771\end{array}\right] \\ \mathrm{span}{S}_3: K_j(x_i)=\left[\begin{array}{ccccc}-0.257& 0.443& -0.235& -0.490& -0.618\\ -0.282& 0.274& -0.177& -0.451& -0.588\\ -0.105& 0.133& -0.150& -0.757& -0.830\\ -0.407& -0.273& 0.333& -0.200& -0.467\\ -0.188& 0.300& -0.350& -0.567& -0.675\\ -0.361& -0.115& 0.150& -0.646& -0.540\\ 2& -0.400& -0.700& -0.800& -0.850\\ -0.012& 0.012& -0.160& -0.497& -0.580\\ -0.083& 0.100& -0.450& -0.780& -0.863\\ -0.083& 0.100& -0.633& -0.780& -0.843\end{array}\right] \end{gathered}$$

Note: The correlation degree represents the distance between the point and the interval and can have a positive or negative value. Positive values indicate that the degree of conformity of the point with the interval is large. correlation degree reflects the degree of connection between each evaluation index and each evaluation grade—the greater the correlation degree, the better the conformity between the evaluation index and the evaluation level.

The values of subjective weights calculated by hierarchical analysis are ${\omega }_i$ = (0.071, 0.138, 0.120, 0.139, 0.085, 0.101, 0.087, 0.104, 0.079, 0.076). According to Equation (34), the comprehensive correlation of the reliability of this highway bridge regarding evaluation grade j can be calculated as:

$$\begin{gathered} K_j(S_1)=(-0.178,0.240,-0.211,-0.601,-0.685) \\ {K}_j(S_2)=(-0.160,0.094,-0.169,-0.562,-0.665) \\ {K}_j(S_3)=(-0.019,0.043,-0.188,-0.571,-0.668) \end{gathered}$$

Based on maximum affiliation, the reliability level of this highway bridge is span $S_1$ (Grade 2), span $S_2$ (Grade 2), span $S_3$ (Grade 2), which indicates that all three spans of the bridge are in a slightly deficient condition.

Throughout the text, the comparison reveals that the reliability evaluation of the bridge using two different methods is consistent, and the evaluation grade is two, which is also consistent with the actual working condition of the bridge. In the control study, the traditional AHP method was used to calculate the weights, and it was difficult to ensure scientific and accurate weights due to the uneven level of experts. In contrast, this paper uses AHP and entropy to assign subjective and objective weights to each evaluation index separately, avoiding the one-sidedness of a single assignment method and taking into account the subjective intention of decision makers and the objective properties of the data itself. Further, this paper uses TOPSIS to calculate the relative closeness of each span and then determine its reliability level. The obtained evaluation results are consistent with those determined by AHP–Extenics. The comparative study highlights the rationality of the proposed method.

5. Conclusions

Reliability evaluation of highway bridges is an important link in the normal use of bridges. Proposing reasonable and effective reliability evaluation methods for highway bridges is of great significance for the normal operation and maintenance of bridges. Based on evaluation after consulting a large amount of research, this paper uses combined weighting–TOPSIS to evaluate the reliability of a certain bridge in use. The main conclusions are as follows:

(1)

This article refers to the “Technical Condition Evaluation Standards of Highway Bridges” (JTG/T H21-2011) and “Regulations for the Evaluation of Bearing Capacity of Highway Bridges” (JTG/T J21-2011) to select the corrosion potential level of steel bars, resistivity, chloride ion content, estimated strength homogeneity coefficient of concrete, average value of concrete carbonation depth/average measured protective layer thickness, characteristic value of steel protective layer thickness/design value, main beam crack width, main beam deformation, staggered height and bridge deck crack width, etc. A total of 10 indicators are used to evaluate the reliability of the bridge superstructure.

(2)

There are uncertainties in the indicators that affect the reliability of highways and bridges, and it is difficult to quantitatively describe them. This paper uses AHP and entropy weight to carry out subjective weighting and objective weighting, respectively, for each evaluation index, and then uses minimum discriminant information to combine subjective and objective weighting to obtain a combined weighting, which fully avoids single-method weighting. The one-sidedness of the data makes the obtained comprehensive weights not only reflect the subjective intentions of decision makers, but also reflect the objective attributes of the data.

(3)

In this paper, combined weighting and TOPSIS are used for highway bridge reliability evaluation, and a complete highway bridge reliability evaluation model is established. In order to determine the reliability level of the upper structure of the three spans of a reinforced concrete bridge, the relative closeness $G_i$ of each span is calculated using TOPSIS, and the order is: 0.556, 0.562, 0.573. According to the size of the obtained relative closeness $G_i$, the reliability level evaluation results of the superstructure of the three spans of the bridge are: Grade 2, Grade 2 and Grade 2. It can also be obtained that the reliability of the three spans of the bridge is ranked as follows: $S_3>S_2>S_1$. At the same time, a comparative study was done with the AHP–Extenics method. The evaluation results are consistent with the results obtained by the method used in this paper and with the actual bridge working conditions. The comparison results show that the combined weighting–TOPSIS method is reliable for highway bridges while being more reasonable and effective.

(4)

This paper evaluates the reliability of highway bridges. Since there are many factors that affect the reliability of bridges, this paper only selects 10 of the more common and impactful evaluation indicators. There are deficiencies in many aspects, such as the importance. Therefore, more in-depth exploration and research are needed for the reliability assessment of highways and bridges in order to move forward in the direction of sustainable development.

The research flow of this paper is shown in Figure 2.