# Fuzzy Supplier Selection Method Based on Smaller-The-Better Quality Characteristic
## 3 Research Design and Methods
### Developing a Fuzzy Supplier Selection Model  
If we let \((x_{h1},x_{h2},\cdots,x_{hm})\) be the observed value of \((X_{h1},X_{h2},\cdots,X_{hm})\) for supplier \(h\), then the observed value of \(Q_{PU\_{HU}}^{*}\) is  
\[Q_{PU\_{HU0}}^{*}=\frac{USL-\overline{x}_{h}}{s_{h}}+1.5, \tag{13}\]  
where \(\overline{x}_{h}=\left(n^{-1}\right)\times\sum_{i=1}^{n}x_{hi}\) and \(s_{h}=\sqrt{(n^{-1})\times\sum_{i=1}^{n}(x_{hi}-\overline{x}_{h})^{2}}\). Therefore, the observed values of the lower and upper confidence limits of \(Q_{PU\_{HU}}\) are functions of \(\alpha\), and can be shown as follows:  
\[LQ_{PU\_{HU0}}(\alpha)\Big{(}Q_{PU\_{HU0}}^{*}-1.5\Big{)}\times\sqrt{\frac{ \chi_{0.5-\sqrt{1-\alpha}/2\alpha-1}^{2}}{n}}-\frac{Z_{0.5-\sqrt{1-\alpha}/2} }{\sqrt{n}}+1.5, \tag{14}\]  
\[LQ_{PU\_{HU0}}(\alpha)\Big{(}Q_{PU\_{HU0}}^{*}-1.5\Big{)}\times\sqrt{\frac{ \chi_{0.5-\sqrt{1-\alpha}/2\alpha-1}^{2}}{n}}+\frac{Z_{0.5-\sqrt{1-\alpha}/2} }{\sqrt{n}}+1.5. \tag{15}\]  
This study examined the statistical testing process before developing the fuzzy supplier selection model. Based on [PERSON] et al. [12], when determining whether the quality levels of any two suppliers are equal, this study adopted the following null hypothesis \(H_{0}\):\(Q_{PU\_{HU}}=Q_{PU\_{U\_{U\_{U\_{U\_{U\_{U\_{U\_{U\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\_\\_\\(UQ_{PUIi}<LQ_{PUij}\), then \(H_{0}\) is rejected, and \(Q_{PUIi}<Q_{PUj}\) is concluded. 2. If \(Cl_{i}\cap Cl_{j}\
eq\phi\), then \(H_{0}\) is not rejected, and \(Q_{PUIi}=Q_{PUj}\) is concluded. 3. If \(LQ_{PUIi}>UQ_{PUj}\), then \(H_{0}\) is rejected, and \(Q_{PUIi}>Q_{PUj}\) is concluded. Next, this study developed the fuzzy supplier selection model based on the aforementioned statistical testing rules related to the fuzzy hypothesis testing method recommended by [PERSON] et al. [12]. This study first examined the case of \(Q^{*}_{PUj}\geq Q^{*}_{PUIi}\). If \(Q^{*}_{PUj}<Q^{*}_{PUIi}\), then the decision will not be made until the inequality is reversed. Applying the confidence interval \(CI\) in accordance with the method proposed by [PERSON] et al. [12], the \(\alpha-\)cuts of the triangular fuzzy number \(\widetilde{Q}^{*}_{PUIi0}\) are derived as follows:  
\[\widetilde{Q}^{*}_{PUIi0}[\alpha]=\left\{ \begin{array}{ll}\left[LQ_{PUIi0}( \alpha),UQ_{PUIi0}(\alpha)\right],\text{ for }0.01\leq\alpha\leq 1\\ \left[LQ_{PUIi0}(0.01),UQ_{PUIi0}(0.01)\right],\text{ for }0\leq\alpha\leq 0.01 \end{array} \right.. \tag{16}\]  
Thus, the triangular fuzzy number of \(Q^{*}_{PUIi0}\) is \(\widetilde{Q}^{*}_{PUIi0}=\Delta(Q_{Lh},Q_{Mh},\ Q_{Rh})\), where  
\[Q_{Lh}=\left(Q^{*}_{PUIi0}-1.5\right)\times\sqrt{\frac{\chi^{2}_{0.0025\pi-1} }{n}}-\frac{Z_{0.0025}}{\sqrt{n}}+1.5, \tag{17}\]  
\[Q_{Mh}=\left(Q^{*}_{PUIi0}-1.5\right)\times\sqrt{\frac{\chi^{2}_{0.5;n-1}}{n} }+1.5, \tag{18}\]  
\[Q_{Rh}=\left(Q^{*}_{PUIi0}-1.5\right)\times\sqrt{\frac{\chi^{2}_{0.9975\pi-1} }{n}}+\frac{Z_{0.0025}}{\sqrt{n}}+1.5. \tag{19}\]  
It is observed that the membership function of the fuzzy number \(\widetilde{Q}^{*}_{PUIi0}\) is  
\[\eta_{h}(x)=\left\{ \begin{array}{ll}0&\text{if }x<Q_{Lh}\\ \alpha^{\prime}&\text{if }Q_{Lh}\leq x<Q_{Mh}\\ 1&\text{if }x=Q_{Mh}\\ \alpha^{\prime\prime}&\text{if }Q_{Mh}<x\leq Q_{Rh}\\ 0&\text{if }Q_{Rh}<x\end{array} \right.. \tag{20}\]  
where \(\alpha^{\prime}\) and \(\alpha^{\prime\prime}\) are determined by  
\[\left(Q^{*}_{PUIi0}-1.5\right)\times\sqrt{\frac{\chi^{2}_{0.5-\sqrt{1-\alpha ^{\prime}}/2;n-1}}{n}}-\frac{Z_{0.5-\sqrt{1-\alpha^{\prime}}/2}}{\sqrt{n}}+1.5 =x, \tag{21}\]  
\[\left(Q^{*}_{PUIi0}-1.5\right)\times\sqrt{\frac{\chi^{2}_{0.5-\sqrt{1-\alpha ^{\prime}}/2;n-1}}{n}}+\frac{Z_{0.5-\sqrt{1-\alpha^{\prime}}/2}}{\sqrt{n}}+1.5 =x. \tag{22}\]  
Figure 2 is a diagram showing the membership functions of \(\eta_{i}(x)\) and \(\eta_{j}(x)\).