# Fuzzy Supplier Selection Method Based on Smaller-The-Better Quality Characteristic
## 4 Numerical Example  
This study aims to demonstrate the efficacy of our methodology, and develops the observed value \(\left(x_{h,1},x_{h,2},\cdots,x_{h,36}\right)\) of random sample, with sample size \(n=36\) for each supplier. The upper specification limit for roundness of the parts is 0.02. According to Section 3, if we want to determine whether the quality levels of any two suppliers are identical, we should adopt the following null hypothesis \(H_{0}\):\(Q_{PUIi}=Q_{PUIj}\) versus the alternative hypothesis \(H_{1}\):\(Q_{PUIi}\
eq Q_{PUIj}\) for any \(i\
eq j\). The observed value of \(Q_{PUIi}^{*}\) and \(Q_{PUIj}^{*}\) can be described as follows:  
\(Q_{PUIi0}^{*}=3.81\) and \(Q_{PUIj0}^{*}=5.42\).  
Thus, based on Equations (17) and (19), the triangular fuzzy numbers of \(Q_{PUIi}^{*}\) and \(Q_{PUIj}^{*}\) can be shown as follows:  
\(\Delta\left(\begin{array}{ll}Q_{Li},Q_{Mi},&Q_{Ri}\end{array}\right)=\Delta (2.852,3.778,4.755)\),  
\(\Delta\left(\begin{array}{ll}Q_{Li},Q_{Mi},&Q_{Ri}\end{array}\right)=\Delta (4.047,5.365,6.771)\).  
Besides, observed from Equation (20), the membership function of fuzzy number \(\overline{Q}_{PUIi}^{*}\) is  
\[\eta_{i}(x)=\left\{\begin{array}{ll}0\ if\ x<2.852\ \  
based on Equations (21) and (22), where \(\alpha^{\prime}_{j}\) and \(\alpha^{\prime\prime}_{j}\) are determined by  
\[3.92\times\sqrt{\frac{\chi_{0.5+\sqrt{1-\alpha^{\prime\prime}_{j}/2\eta-1}}^{2}} {n}}+\frac{Z_{0.5-\sqrt{1-\alpha^{\prime\prime}_{j}/2}}}{\sqrt{n}}+1.5=x,\text { and }\]  
\[3.92\times\sqrt{\frac{\chi_{1-\alpha/2.35}^{2}}{36}}+\frac{Z_{\alpha/2}}{6}+1.5 =x.\]  
Obviously, when the confidence intervals \(Cl_{i}=[2.852,4.755]\) and \(Cl_{j}=[4.047,6.771]\), then \(Cl_{i}\)\(\cap\)\(Cl_{j}\)\(\
eq\)\(\phi\). According to statistical testing rules, \(H_{0}\) is not rejected, and \(Q_{PUI}=Q_{PUI}\) is concluded. Nevertheless, it is discovered that the observed values of \(Q_{PUI0}^{*}=3.81\) and \(Q_{PUI0}^{*}=5.42\). The quality levels of the two suppliers differ by at least one standard deviation, but the conclusions are not significantly different. This is because the small sample size (\(n=36\)) resulted in a large error. Next, this study performed the fuzzy testing method using the fuzzy selection model suggested in Section 3. Figure 2 exhibits a graph related to the membership functions \(\eta_{i}(x)\) and \(\eta_{j}(x)\) with \(Q_{PUI0}^{*}=3.81\) and \(Q_{PUI0}^{*}=5.42\). In Figure 3, when \(c_{ij}=4.446\), such that \(\eta_{i}(4.446)=\eta_{j}(4.446)\), the slashed area \(a_{ij}\) of \(\eta_{i}(x)\) can be calculated.  
Based on Figure 2 and Equations (25) and (26), the values of \(d_{ij}\) and \(d_{Ti}\) can be shown as follows:  
\[d_{ij}=Q_{Ri}-c_{ij}=4.755-4.446=0.309,\]  
\[d_{Ti}=Q_{Ri}-Q_{Li}=4.755-2.852=1.903.\]  
Therefore,  
\[d_{ij}/d_{Ti}\]  
As noted by [PERSON] et al. [12], \(\phi_{1}\) and \(\phi_{2}\) can be 0.2 and 0.4 in practice. Because \(d_{ij}/d_{Ti}=0.162<\phi_{1}\), \(H_{0}\) is rejected, and \(Q_{PUIi}<Q_{PUIj}\) is concluded. Then, the differences between the fuzzy testing method proposed in this study and the traditional statistical testing method are displayed in Table 1:Clearly, the small sample size (\(n=36\)) produced a large error. Statistical testing would demonstrate that there is no major difference between the quality levels of the two suppliers, even though their quality levels differ by at least one standard deviation. In contrast, fuzzy testing using the proposed fuzzy selection model shows significant differences between the quality levels of the two suppliers; that is, \(Q_{Pui}<Q_{Puj}\). It is thus clear that the proposed method is more accurate in practice.