# Fuzzy Supplier Selection Method Based on Smaller-The-Better Quality Characteristic
## 3 Research Design and Methods
### Confidence Interval of Six Sigma Index \(Q_{pu}\)  
Assume that \(X\) is normally distributed; i.e., \(X\)\(\sim\)\(N\)\(\left(\mu,\sigma^{2}\right)\), where \(\mu\) represents the process average and \(\sigma\) represents the standard deviation of the process. Then, the six sigma index of STB quality characteristics can be illustrated below:  
\[Q_{PU}=\frac{USL-\mu}{\sigma}+1.5 \tag{1}\]  
where \(USL\) refers to upper specification limit. [PERSON] et al. [11] suggested that the target value of STB quality characteristics is 0. Nevertheless, due to cost considerations and technical issues, the process average is not likely to approach this target value. So, when \(USL-\mu\geq(k-1.5)\sigma\), then the process is considered to have reached the \(k-sigma\) quality level. That is  
\[Q_{PU}\geq Q_{PU}(k)=\frac{(k-1.5)\sigma}{\sigma}+1.5=k. \tag{2}\]  
It is obvious that when the process quality level is \(k-sigma\), then the value of \(Q_{PU}\) is equal to at least \(k\). In addition, it is assumed that there is a one-to-one mathematical relationship between index \(Q_{PU}\) and process yield \(p\), expressed as  
\[p=p[X\leq USL]=p[Z\leq Q_{PU}-1.5]=\Phi(Q_{PU}-1.5) \tag{3}\]  
where \(\Phi(\cdot)\) is a cumulative distribution function of standard normal distribution. For example, when \(Q_{PU}=4.5\), then we can derive the process yield \(p=\Phi(3.0)=99.865\%\). Suppose \((X_{1},X_{2},\cdots,X_{n})\) is a random sample of \(X\), then the estimator of six sigma index \(Q_{PU}\) can be written below:  
\[Q_{PU}^{*}=\frac{USL-\overline{X}}{S}+1.5, \tag{4}\]  
Figure 1: Research design flowchart. where \(\overline{X}=\left(n^{-1}\right)\times\sum_{i=1}^{n}X_{i}\) and \(S=\sqrt{\left(n^{-1}\right)\times\sum_{i=1}^{n}\left(X_{i}-\mu^{*}\right)^{2}}\) are the maximum likelihood estimates (MLEs) of \(\mu\) and \(\sigma\), respectively. Under the assumption of normality, we let  
\[Z=\frac{\overline{X}-\mu}{\sigma}\text{ and }K=\frac{nS^{2}}{\sigma^{2}}.\]  
Then \(Z\) and \(K\) are distributed as \(N(0,1)\) and \(\chi_{n-1}^{2}\), respectively. As a result,  
\[p\left\{-Z_{\alpha^{\prime}/2}\leq Z\leq Z_{\alpha^{\prime}/2}\right\}=\sqrt{1 -\alpha}\text{ and }p\left\{\chi_{\alpha^{\prime}/2n-1}^{2}\leq K\leq\chi_{1-( \alpha^{\prime}/2)n-1}^{2}\right\}=\sqrt{1-\alpha}\]  
where \(\alpha^{\prime}=1-\sqrt{1-\alpha}\), \(Z_{\alpha^{\prime}/2}\) is the upper \(\alpha^{\prime}/2\) quantile of the standardized normal distribution, and \(\chi_{a,n-1}^{2}\) is the lower \(a\) quantile of the chi-square distribution with \(n-1\) degrees of freedom, \(a=\alpha^{\prime}/2\) or \(1-\alpha^{\prime}/2\). \(\overline{X}\) and \(S^{2}\) both are independent, so are \(Z\) and \(K\). Resulting from these relationships, the following equation is expressed as  
\[p\left\{-Z_{\alpha^{\prime}/2}\leq Z\leq Z_{\alpha^{\prime}/2},\chi_{\alpha^{ \prime}/2n-1}^{2}\leq K\leq\chi_{\alpha^{\prime}/2n-1}^{2}\right\}=1-\alpha. \tag{5}\]  
Equivalently,  
\[p\left\{\overline{X}-\left(Z_{0.5-\sqrt{1-\alpha}/2}\right)\frac{\sigma}{ \sqrt{n}}\leq\mu\leq\overline{X}+\left(Z_{0.5-\sqrt{1-\alpha}/2}\right)\frac {\sigma}{\sqrt{n}},\sigma_{D}\leq\sigma\leq\sigma_{U}\right\}=1-\alpha \tag{6}\]  
where  
\[\sigma_{D}=\sqrt{\frac{n}{\chi_{0.5+\sqrt{1-\alpha}/2;n-1}^{2}}}S\text{ and }\sigma_{U}=\sqrt{\frac{n}{\chi_{0.5-\sqrt{1-\alpha}/2;n-1}^{2}}}S.\]  
Therefore, the confidence region can be shown as  
\[Q_{PU}\geq Q_{PU}(k)=\frac{(k-1.5)\sigma}{\sigma}+1.5=k. \tag{2}\]0  
Obviously, the six sigma index \(Q_{PU}\) is the function of \((\mu,\sigma)\). Based on [PERSON] et al. [11], the object function is \(Q_{PU}\)\((\mu,\sigma)\) and the confidence region \((CR)\) is the feasible solution area. First, to find the lower confidence limit of \(Q_{PU}\), an appropriate mathematical programming model is expressed as follows:  
\[\left\{ \begin{array}{l}LQ_{PU}=Min\left\{\left(USL-\mu\right)/\sigma+1.5 \right\}\\ \text{subject to }(\mu,\sigma)\in CR\end{array} \right.