# Life Cycle Assessment of Bridges Using Bayesian Networks and Fuzzy Mathematics
## 3 Methodology
### Theoretical Model Bridge Process  
The introduction of some important definitions used in FMT structural hierarchy theory helps to fully understand the subsequent modelling applications (a single factor influences the weight matrix: the FMT comprehensive evaluation principle is based on the analysis and evaluation of each single impact factor (\(\mathrm{u_{1}}\)) in U, and after the analysis is completed, it is summarized into a set form (Figure 4). To solve the maximum eigenvalue \(\lambda_{\mathrm{max}}\) and corresponding eigenvector \(\mathrm{v}\) of the judgement matrix at this level, the judgement matrix needs to be normalized as the impact factor of this level on the previous level. Consistency index:  
\[\mathrm{CI}=\ (\lambda_{\mathrm{max}}-\mathrm{n})/(\mathrm{n}-1) \tag{1}\]  
where CI is the consistency index; \(\lambda_{\mathrm{max}}\) is the maximum eigenvalue of the judgement matrix; and n is the number of levels. \[\mathrm{CR}{=}\ \mathrm{CI}/\mathrm{AI} \tag{2}\]  
where CR is the consistency ratio; and AI is the average consistency index. Equation (1) is substituted into Equation (2), giving  
\[\mathrm{CR}{=}\ (\lambda_{\mathrm{max}}-\mathrm{n})/(\mathrm{n}-1)/\mathrm{AI} \tag{3}\]  
Figure 3: The basic methods and steps of FMT comprehensive evaluation. Figure 2: Schematic diagram of BN principle. Judgement basis: if CR \(\geq\) 0.1, the consistency of the impact weight matrix is not acceptable; if CR \(<\) 0.1, the consistency of the impact weight matrix is acceptable. Hypothesis set U=\(\{\text{u}_{1},\text{u}_{2},\cdots\cdots,\text{u}_{\text{m}}\}\), weight set A=\(\{\text{a}_{1},\text{a}_{2},\cdots\cdots,\text{A},\text{a}_{\text{m}}\}\), and comment set V=\(\{\text{v}_{1},\text{v}_{2},\cdots\cdots,\text{A},\text{v}_{\text{m}}\}\) are established according to the fuzzy synthetical assessment theory. To perform quantitative results analysis, the comment set is divided into five levels, as shown in Figure 5:  
Description: There are a total of 117 environmental impact factors for cable-stayed bridges. The amount of data information is huge--see the attached table for relevant data. Figure 4: Schematic diagram illustrating the single factor influence weight matrix. Figure 5: Schematic diagram of tree-shaped BN of environmental impact contribution of cable-stayed bridge. \[\mathrm{V}=\left\{ \begin{array}{ll}\mathrm{v}_{1}=(-\infty,\ 0]&\mathrm{Not\ effect}\\ \mathrm{v}_{2}=(0,+0.5]&\mathrm{Slight\ effect}\\ \mathrm{v}_{3}=(0.5,\ 1.0]&\mathrm{Affect}\\ \mathrm{v}_{4}=(1.0,\ 1.5]&\mathrm{Moderate\ effect}\\ \mathrm{v}_{5}=(1.5,+\infty)&\mathrm{Great\ effect}\end{array} \right.\]  
The i-th impact factor is evaluated by the i the factor \(\mathrm{u}_{\mathrm{i}}\) of the assessment object (factors at each phase), and the degree of membership of the jth element \(\mathrm{v}_{\mathrm{j}}\) in the comment set is \(\mathrm{r}_{\mathrm{i}\bar{\mathrm{y}}}\) (\(\in\)\(v\)\(\
ot\in\) cannot be used) because the fuzzy set has no strict demarcation. The degree of membership, that is, \(\mathrm{r}_{\mathrm{i}\bar{\mathrm{y}}}\) is introduced to represent the degree of belonging of the element \(\mathrm{j}\) to the fuzzy set \(\mathrm{v}_{\mathrm{j}}\)[50]; \(\mathrm{r}_{\mathrm{i}\bar{\mathrm{y}}}\) is any number between 0 and 1. The fuzzy set \(\mathrm{u}_{\mathrm{i}}\) of factors \(\mathrm{i}\) can be expressed as:  
\[\mathrm{R}_{\mathrm{i}}\mathrm{=}\left\{\frac{\mathrm{r}_{\mathrm{i}1}}{ \mathrm{v}_{1}}+\frac{\mathrm{r}_{\mathrm{i}2}}{\mathrm{v}_{2}}+\dots\dots+ \Lambda+\frac{\mathrm{r}_{\mathrm{i}m}}{\mathrm{v}_{\mathrm{m}}}\right\} \tag{4}\]  
where \(\mathrm{R}_{\mathrm{i}}\) is the assessment set of a single factor; \(\mathrm{r}_{\mathrm{i}m}\) is the membership of \(\mathrm{m}\) kinds of elements; and \(\mathrm{v}_{\mathrm{m}}\) is the comment set of \(\mathrm{m}\) kinds of elements. All the single-factor fuzzy assessment sets are integrated into an impact weight matrix:  
\[\mathrm{R}\mathrm{=}\left[