## 3 Material and Methods  
### Fuzzy Operations for Supplier Evaluation and Selection  
In the universe of discourse \(X\), the fuzzy subset \(\widetilde{A}\) of \(X\) is defined by the membership function \(\mu_{\widetilde{A}}(x)\), which maps each element \(x\) in \(X\) to a real number between 0 and 1. The function value of \(\mu_{\widetilde{A}}(x)\) determines the grade of membership of \(X\) in set \(\widetilde{A}\). The higher the grade of membership of \(x\) in \(\widetilde{A}\), the nearer the value of \(\mu_{\widetilde{A}}(x)\) is to unity. [PERSON] et al. [46] pointed out that, in fuzzy theory, the most commonly used fuzzy number is a triangular number fuzzy (TNF) (see Figure 1), the membership function of which is given by the following equation:  
\[\mu_{\widetilde{A}}(x)=\left\{\begin{array}{ll}0&x\leq a\;or\;x\geq c\\ (x-a)/(b-a)&a\leq x\leq b\\ (c-x)/(c-b)&b\leq x\leq c\end{array}\right. \tag{1}\]  
Figure 1: Triangular number fuzzy (TNF).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **Fuzzy** & **Cognitive** & **Reasonable** & **Empirical** \\  & **Linguistics** & **Differences** & **Evaluation Criteria** & **Research** \\ \hline Reference [19] & ✓ & & & \\ Reference [43] & ✓ & & ✓ & \\ Reference [44] & & & ✓ & ✓ \\ Reference [45] & ✓ & & ✓ & \\ Reference [46] & ✓ & & ✓ & ✓ \\ Proposed method & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparative analysis.  
[PERSON] et al. [47] proposed that the needs of customers can be viewed as product perceptions or service perceptions. These form the input data for the following definition of the fuzzy \"if-then\" rule \(R_{j}\):  
\[R_{j}:If(X_{j1}\text{ is }x_{j1},\text{ and }X_{j2}\text{ is }x_{j2},\text{ }\dots,\text{ and }X_{jk}\text{ is }x_{jk}),\text{ then }Y_{j}\text{ is }y_{j} \tag{2}\]  
The above rule means that, when the fuzzy set \(x_{j1},\ x_{j2},\ \dots,\ x_{jk}\) meets the input semantic variables \(X_{j1},\ X_{j2},\ \dots,\ X_{jk}\), then fuzzy set \(y_{j}\) is a semantic variable for output \(Y_{j}\). The fuzzy inference process is performed when the input semantic variables \(X_{j1},\ X_{j2},\ \dots,\ X_{jk}\) become the fuzzy set \(x_{j1},\ x_{j2},\ \dots,\ x_{jk}\), with membership grades \(m_{j1},\ m_{j2},\ \dots,\ m_{jk}\).  
[PERSON] et al. [48] proposed the application of a fuzzy rule to determine fuzzy subsets using intersection. The membership function for this rule is defined as follows:  
\[\mu_{\overline{x}}(x)=\min\{\mu_{\overline{x}j1}(x),\ \mu_{\overline{x}j2}(x),\ \dots,\ \mu_{\overline{x}jk}(x)\} \tag{3}\]  
A fuzzy rule that uses union to determine subsets has the following membership function:  
\[\mu_{\overline{x}}(x)=\max\{\mu_{\overline{x}j1}(x),\ \mu_{\overline{x}j2}(x),\ \dots,\ \mu_{\overline{x}jk}(x)\} \tag{4}\]  
Therefore, the membership grades for input \(m_{j}\) and output \(o_{j}\) of the fuzzy inferences are  
\[m_{j}=\min\{m_{j1},m_{j2},\dots,m_{jk}\} \tag{5}\]  
\[o_{j}=\max\{o_{j1},o_{j2},\dots,o_{jk}\} \tag{6}\]  
Because \"\(Y_{j}\) is \(y_{j}\)\" is the result of its membership grade for \(R_{j}\), it can be equated to \(o_{j}\), as follows:  
\[Y_{j}\text{ is }y_{j1}:o_{j1};\ Y_{j}\text{ is }y_{j2}:o_{j2};\ \dots;\ Y_{j}\text{ is }y_{jn}:o_{jn} \tag{7}\]  
After the rules are established, all fuzzy relations and ambiguity semantic assessments are aggregated into a complete fuzzy area. In order to obtain a more comprehensive output, the center of gravity method is adopted to obtain crisp value \(\beta_{j}\). The center of gravity method is defined as follows:  
\[\beta_{j}=\int\limits_{a}^{b}xu(x)dx/\int\limits_{a}^{b}u(x)dx \tag{8}\]  
where \(a\) and \(b\) represent the boundary between the lower and upper bounds of \(Y_{j}\) (the output semantic variable).