# A Comprehensive Failure Risk Assessment Method of
## 2 Comprehensive Risk Assessment of the Machining Center Component Based on Topological Analysis
### Analysis of the Topological Characteristics of the Machining Center Component
#### 2.2.1 Analysis of Multi-Dimensional Topological Structure of Machining
Because there is only one level in the index system in this paper, the single-level ranking is the final ranking. Commonly used judgment matrix calculation methods are the square root method or sum product method. This paper uses the sum product method to calculate a judgment matrix and conduct a consistency test. (d) Determine the weight, and calculate the relative weight of the index based on the criterion according to the judgment matrix and the weight of each index can be obtained as \(\omega^{\prime}=\left(\omega^{\prime}1,\omega^{\prime}2,\cdots,\omega^{\prime }g\right)\). (2) Objective weighting  
This paper uses the entropy method [32] to objectively weight each index. Suppose the weight is \(\omega^{\prime\prime}=\left(\omega^{\prime\prime}1,\omega^{\prime\prime}2, \cdots,\omega^{\prime\prime}g\right)\), for matrix \(P\), the information entropy of the \(j\)-th evaluation index is:  
\[e\big{(}d_{j}\big{)}=-\frac{1}{\ln n}\sum\limits_{i=1}^{n}\left(\frac{p_{ij}}{ p_{j}}\ln\frac{p_{ij}}{p_{j}}\right), \tag{11}\]  
where \(p_{ij}\) is the normalized value corresponding to the \(j\)-th index of the \(i\)-th component, \(p_{j}=\sum\limits_{i=1}^{n}p_{ij}\). According to Equation (11), the information entropy of each index can be obtained, and substituting it into Equation (12), the objective weight \(\omega^{\prime\prime}\) of each index can be obtained. \[\omega^{\prime\prime}{}_{i}=\frac{1}{g-E_{e}}\big{[}1-e\big{(}d_{j}\big{)} \big{]}, \tag{12}\]  
where \(E_{e}=\sum\limits_{j=1}^{g}e\big{(}d_{j}\big{)}\), \(0\leq\omega^{\prime\prime}\leq 1\), \(\sum\limits_{i=1}^{g}\omega^{\prime\prime}{}_{i}=1\). Assume that the consistency test is performed on \(k\) kinds of weights. When \(k=2\), the degree of consistency of the weights can be indicated by the Spearman rank correlation coefficient or by the distance function [33]:  
\[d\left(\omega^{(1)}\omega^{(2)}\right)=\left[\frac{1}{2}\sum_{j=1}^{m}\left( \omega^{\prime}-\omega^{\prime\prime}\right)^{2}\right]^{\frac{1}{2}}, \tag{13}\]  
where \(0\leq d\left(\omega^{(1)}\omega^{(2)}\right)\leq 1\). When \(d\left(\omega^{(1)}\omega^{(2)}\right)\) is smaller, it indicates that the results of the two weights are closer. When \(k\geq 3\), the Kendall concordance test method [34] in the non-parametric statistical method is generally used to test the consistency of multiple weighting results. Combination weighting  
After passing the consistency check, the subjective and objective weights are combined according to equal weights to obtain a new weight \(\omega=\left[\omega_{1},\omega_{2},\cdots,\omega_{g}\right]\). \[DC_{i}=\frac{D_{i}}{n-1}. \tag{5}\]0  
Comprehensive structure importance model of the component of the machining center  
Combining the multi-dimensional structural importance index with the combined weight, the calculation equation for the importance of the component node of the machining center can be obtained, as shown in Equation (15):  
\[DC_{i}=\frac{D_{i}}{n-1}. \tag{5}\]1  
where \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) are the weight values of each structural index, respectively. 2.3 Calculation Importance of Connection Edges of Machining Center Component Based on Topological Characteristics  
When the component \(k\) fails, the failure will be transmitted to the component \(i\) with a certain value along the connecting edge, causing the component \(i\) to fail, and this value is related to the importance of the connecting edge where the two components are located. [PERSON] et al. (2014) uses the efficacy function to describe the contribution of influencing factors to the overall state evolution of the system and uses the coupling degree model to describe the coupling degree between the coupled factors.