## 2 The Proposed Approach
### Specific Steps of the Proposed Approach  
An AM machine selection problem can be described thus: \(m\) alternative AM machines \(A_{1}\sim A_{m}\), \(n\) attributes \(c_{1}\sim c_{n}\), and \(n\) weights of attributes \(w_{1}\sim w_{n}\) such that \(w_{j}\) is the weight of the attribute \(c_{j}\), \(w_{1}\sim w_{n}\in[0,\,1]\) and \(\sum_{j=1}^{n}w_{j}=1\), and a decision matrix \(M=\left[x_{ij}\right]_{m\times n}\) (\(i\in[1,\,m]\); \(j\in[1,\,n]\); \(x_{ij}>0\)) such that \(x_{ij}\) is the values corresponding to attribute \(c_{j}\) of alternative \(A_{i}\). Based on these components, the AM machine selection problem can be solved by evaluating each alternative \(A_{i}\) according to the weight \(w_{j}\) of each attribute and the decision matrix \(M\)[62]. The specific process mainly includes the following steps:  
1. Generate the values of attributes. According to the alternative AM machines and attributes of an AM machine selection problem, the values of attributes are automatically generated by the automatic reasoning of the developed DL ontology for AM machine selection. 2. Construct a decision matrix \(M\). According to the attribute values generated by the DL ontology, a decision matrix \(M\) is established as: \[M=\left[ \begin{matrix}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\ldots&x_{mm}\end{matrix} \right]\]
3. Construct a fuzzy decision matrix \(M^{{}^{\prime}}\). The decision matrix \(M\) is fuzzified using: \[x^{{}^{\prime}}_{ij}=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{m}x_{ij}^{2}}}\] (6) where \(x_{ij}\) indicates the attribute value of the \(j\)-th attribute of the \(i\)-th alternative. Then a fuzzy decision matrix \(M^{{}^{\prime}}\) is obtained as: \[M^{{}^{\prime}}=\left[ \begin{matrix}\left<x^{{}^{\prime}}_{11}\right>&\left< x^{{}^{\prime}}_{12}\right>&\cdots&\left<x^{{}^{\prime}}_{1n}\right>\\ \left<x^{{}^{\prime}}_{21}\right>&\left<x^{{}^{\prime}}_{22}\right>&\cdots& \left<x^{{}^{\prime}}_{2n}\right>\\ \vdots&\vdots&\ddots&\vdots\\ \left<x^{{}^{\prime}}_{m1}\right>&\left<x^{{}^{\prime}}_{m2}\right>&\cdots& \left<x^{{}^{\prime}}_{mm}\right>\end{matrix} \right]\]
4. Construct a normalized fuzzy decision matrix \(M^{{}^{\prime\prime}}\). In an AM machine selection problem, there are two types of attributes (i.e., positive attributes and negative attributes), in which a positive attribute has a positive impact on decision making, and a negative attribute has a negative impact on decision making. In order to eliminate such influences, a complement rule is needed to transform negative attributes into positive attributes [65]:  
\[x^{{}^{\prime\prime}}_{ij}= \begin{cases}\ x^{{}^{\prime}}_{ij},&\text{if $c_{j}$ is a positive criterion}\\ 1-x^{{}^{\prime}}_{ij}&\text{if $c_{j}$ is a negative criterion}\end{cases}  \tag{7}\]  
Thus, a fuzzy decision matrix \(M^{{}^{\prime}}\) is transformed into a normalized fuzzy decision matrix \(M^{{}^{\prime\prime}}\). \[M^{{}^{\prime\prime}}= \begin{bmatrix}\left\langle x^{{}^{\prime\prime}}_{11} \right\rangle&\left\langle x^{{}^{\prime\prime}}_{12}\right\rangle&\cdots& \left\langle x^{{}^{\prime\prime}}_{1n}\right\rangle\\ \left\langle x^{{}^{\prime\prime}}_{21}\right\rangle&\left\langle x^{{}^{ \prime\prime}}_{22}\right\rangle&\cdots&\left\langle x^{{}^{\prime\prime}}_{2n} \right\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \left\langle x^{{}^{\prime\prime}}_{m1}\right\rangle&\left\langle x^{{}^{ \prime\prime}}_{m2}\right\rangle&\cdots&\left\langle x^{{}^{\prime\prime}}_{mn }\right\rangle\end{bmatrix} \]
5. Calculate the relative loss function of the attributes considered for each alternative. The calculation method is shown in Table 3.
6. Calculate the comprehensive relative loss function of each alternative. The calculation method is shown in Table 4.
7. Calculate the comprehensive thresholds \(\alpha_{i}\) and \(\beta_{i}\) of each alternative. According to Equations (1) and (2), the comprehensive thresholds \(\alpha_{i}\) and \(\beta_{i}\) of the alternatives are calculated. 8.