# BN-RA: A Hybrid Model for Risk Analysis of Overload-Induced Early Cable Fires
## 4 Numerical Analysis of the Fire Risk Assessment
### Determination of the Risk-Factor Weights  
Because different categories of risk factors contribute unevenly to the overall fire risk, different priorities (i.e., weights) of these risk factors need to be considered. To determine the weights, three popular methods are usually used: (1) difference-driven, (2) function-driven, and (3) comprehensive set assignment methods [47]. The difference-driven weight determination method was used in this study because it not only retains the data characteristics perfectly but also adapts to the data changes automatically. At the same time, this method can avoid subjectivity in determining the weights, which makes it more scientific and reasonable. Consequently, the entropy weight method based on the difference-driven approach was used for weight analysis. The steps of the entropy weight method are described below.  
(i) Data evaluation matrix  
The weight judgment of the entropy weight was based on the input data. Before evaluating the data, an evaluation matrix \(X_{m\times n}\) must be constructed, consisting of \(m\) cases and \(n\) indicators, as in:  
\[X_{m\times n}=\left[\begin{array}{ccc}x_{11}&\cdots&x_{1n}\\ \vdots&\ddots&\vdots\\ x_{m1}&\cdots&x_{mn}\end{array}\right] \tag{17}\]  
(ii) Standardization of the evaluation matrix  
Firstly, indicators in the matrix need to be standardized using the following equation:  
\[y^{\prime}_{ij}=\frac{x_{ij}-\min\bigl{(}x_{ij}\bigr{)}}{\max\bigl{(}x_{ij} \bigr{)}-\min\bigl{(}x_{ij}\bigr{)}}\text{, }(i=1,\,2,\,\ldots,\,m;\,j=1,\,2\ldots,\,n) \tag{18}\]  
Then, each element is normalized by:  
\[y_{ij}=\frac{y^{\prime}_{ij}}{\sum_{i=1}^{m}y^{\prime}_{ij}} \tag{19}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Risk Factors** & **Grade** & **Meaning** & **Value** \\ \hline \multirow{4}{*}{Cable heat source risk} & 1 & Normal working condition, Cable heat source temperature 0–40 \({}^{\circ}\)C & 0.25 \\  & 2 & Less than the maximum temperature of long-term operation, 40–70 \({}^{\circ}\)C & 0.5 \\  & 3 & Less than the maximum temperature of short-time operation, 70–90 \({}^{\circ}\)C & 0.75 \\  & 4 & More than the maximum temperature of long-term operation, over 90 \({}^{\circ}\)C & 1 \\ \hline \multirow{2}{*}{Insulation breakdown risk} & 1 & Damage or heat causes less than 50\% of local core exposed & 0.3 \\  & 2 & Damage or heat causes 50–75\% of local core exposed & 0.6 \\  & 3 & Damage or heat causes 75–100\% of local core exposed & 1 \\ \hline \multirow{2}{*}{Material pyrolysis risk} & 1 & There is no obvious thorn in the cable insulation material & 0.3 \\  & 2 & Obvious pungent gas appeared during pyrolysis & 0.6 \\  & 3 & Strong pungent smell and continuous smoke & 1 \\ \hline \end{tabular}
\end{table}
Table 5: Classification of risk factors with severities.  
The standardized evaluation matrix \(Y_{m\times n}\) is given by:  
\[Y_{m\times n}=\left[\begin{array}{ccc}y_{11}&\cdots&y_{1n}\\ \vdots&\ddots&\vdots\\ y_{m1}&\cdots&y_{mn}\end{array}\right] \tag{20}\]  
(iii) Computation of the information entropy  
Given the evaluation matrix \(Y_{m\times n}\), the information entropy of each evaluation factor is calculated as:  
\[H_{j}=-k\sum_{i=1}^{m}y_{ij}\log y_{ij},\ \ (i=1,\ 2,\ldots,m;\ j=1,\ 2\ldots,n) \tag{21}\]  
where \(H_{j}\) represents the information entropy of factor \(j\), and 0 < \(H\) < 1, \(k=1/\log m\). It should be noted that the index value of the logarithmic function is not zero; thus, \(y_{ij}\) log\(y_{ij}\) = zero when \(y_{ij}\) = 0.  
(iv) Calculation of weights  
The weight of each factor is calculated according to the information entropy of each index, and is given by:  
\[w_{j}=\frac{1-H_{j}}{n-\sum_{j=1}^{n}H_{j}} \tag{22}\]