# 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
The \(j\)-th importance index value of the \(i\)-th component is recorded as \(x_{ji}(i=1,2,\cdots,n;j=1,2,\cdots,g)\), and then the initial matrix \(\mathbf{X}=\left[x_{ji}\right]_{g\times n}\). (2) Normalization of evaluation index values  
In order to solve the incommensurability of each structural importance index and the contradiction of the measurement criteria, it is necessary to standardize the structural importance index value \(x_{ji}\) and convert it into a single index coefficient \(q_{ji}\) with the same value range and trend. Then, normalize the different states of the same indicator to form a normalized matrix \(\mathbf{P}=\left[p_{ji}\right]_{g\times n}\). The data normalization processing methods include range normalization, linear normalization, curve normalization [29] and fuzzy normalization [30]. In order to simplify the calculation, this paper uses the linear scale transformation method in the literature [29] to normalize the original data. For the positive indicators \(x_{ji}\) that are bigger and better, take  
\[q_{ji}=\frac{x_{ji}}{x_{j}^{\max}}(i=1,2,\cdots,n), \tag{8}\]  
where \(x_{j}^{\max}=\max\limits_{1\leq n}\{x_{ji}\}\), \((j=1,2,\cdots,g)\). For the negative indicators \(x_{ji}\) that are smaller and better, take  
\[q_{ji}=\frac{x_{j}^{\min}}{x_{ji}}(i=1,2,\cdots,n), \tag{9}\]  
where \(x_{j}^{\min}=\min\limits_{1\leq n}\{x_{ji}\}\), \((j=1,2,\cdots,m)\). The initial matrix \(\mathbf{X}\) is normalized to obtain the structural importance index coefficient matrix \(\mathbf{Q}=\left[q_{ji}\right]_{g\times n}\), \(0\leq q_{ji}\leq 1\). Normalize the structural importance index coefficient matrix \(\mathbf{Q}\) to obtain the normalized matrix \(\mathbf{P}=\left[p_{ji}\right]_{g\times n}\). \[p_{ji}=q_{ji}\Bigg{/}\sum_{i=1}^{n}q_{ji}\,(j=1,2,\cdots,g), \tag{10}\]Determination of combination weight  
There are two main types of weight determination methods, namely subjective weighting method and objective weighting method. The weight of the subjective weighting method reflects the intention of the decision maker and is greatly influenced by the evaluation subject. The objective weighting method mainly relies on mathematical theories and methods, has a certain dependence on sample data and practical problems, ignores the subjective information of decision makers, and cannot reflect the importance of decision makers to different structural importance indicators. Therefore, in order to obtain a reasonable weight value, the weighting results of the two methods need to be integrated to obtain the combined weight value. However, the index weights calculated by different methods will be different or even contradictory. Therefore, the consistency of the different weights must be checked before the combination of weights. Only after passing the consistency check can the combination weighting be carried out. At present, there are many related theories of combination weighting. This paper assumes that the importance of the two weighting methods is the same. Therefore, after the two weighting results have passed the consistency check, equal weights are used for combination weighting to obtain combined weights. Specific steps are as follows:  
1. Subjective weighting  
In this paper, the analytic hierarchy process (AHP) is used to subjectively weight each indicator [31]. The main steps are as follows:  
(a) Determine the evaluation factors, that is, the three centrality measures that characterize the topological characteristics of the model in this paper, degree centrality, betweenness centrality, and closeness centrality. (b) Construct a judgment matrix. The value of the element of the judgment matrix reflects the expert's understanding of the relative importance of each centrality measurement evaluation index of the machining center component. Generally, the scale method of 1-9 and its reciprocal is used to obtain the judgment matrix \(A\). (c) Hierarchical single sorting. Maximum eigenvalue and consistency tests were calculated.