## 3 Material and Methods
### Supplier Performance Index  
[PERSON] and [PERSON] [49] and [PERSON] et al. [50] indicated that the application of statistical techniques and methods can help managers in risk assessment and other decision-making processes. The proposed supplier performance index \(S_{PL}\) not only measures supplier performance, but also reduces cognitive differences among multiple decision-makers. In general, companies want suppliers to provide high-quality products or services [51; 52; 53; 54; 55]; that is, the higher the better. Therefore, under the assumption of normality, a supplier performance index \(S_{PL}\) can be derived based on the following concept proposed by [PERSON] [56]:  
\[S_{PL}=\frac{\mu-LPB}{3\sigma} \tag{9}\]  
where \(LPB\) denotes the lower performance bound of the supplier, \(\mu\) is the mean, and \(\sigma\) is standard deviation. In practice, the parameters \(\mu\) and \(\sigma\) are typically unknown and must be estimated using the sample data. Hence, the estimated index of \(S_{PL}\) can be given as  
\[\widetilde{S}_{PL}=\frac{\overline{X}-LPB}{3s} \tag{10}\]  
where \(\overline{X}\) estimates \(\mu\) and the sample standard deviation \(s\) estimates \(\sigma\). Since the natural estimator \(\widetilde{S}_{PL}\) is a biased estimator of \(S_{PL}\) and inappropriate for precisely evaluating the supplier performance, a correction factor \(b_{n}\) can be added to obtain the unbiased estimator of \(\widetilde{S}_{PL}\) for \(S_{PL}\) as follows [57]:  
\[\hat{S}_{PL}=(b_{n})\times\overline{S}_{PL} \tag{11}\]  
Figure 5: Fuzzy rule. where \(b_{n}\) is a function of \(n\) and is determined by  
\[b_{n}=\frac{\sqrt{2}\times\Gamma[(n-1)/2]}{\sqrt{n-1}\times\Gamma[(n-2)/2]},\ n>2 \tag{12}\]  
In fact, the unbiased estimator \(\hat{S}_{PL}\) is only the function of \(\left(\overline{X},\ s^{2}\right)\). Because the distribution of \(\left(\frac{3\sqrt{n}}{b_{n}}\right)\times\hat{S}_{PL}\) is a non-central \(t\) distribution with \(n-1\) degrees of freedom, the non-central parameter \(\gamma\) of this distribution is \(3\sqrt{n}\times S_{PL}\), often denoted as \(t_{n-1}^{\prime}(\gamma)\). Hence, \(\hat{S}_{PL}\) can be rewritten as follows:  
\[\hat{S}_{PL}=\left(\frac{b_{n}}{3}\right)\times\sqrt{\frac{n-1}{n}}\times(K)^{ -1/2}\times(Z) \tag{13}\]  
where \(Z=\sqrt{n}\times\frac{(\overline{X}-LPB)}{\sigma}\) obeys \(N(3\sqrt{n}\times S_{PL},\ 1)\) and \(K=\frac{(n-1)s^{2}}{\sigma^{2}}\) obeys \(\chi_{n-1}^{2}\). Under the assumption of normal distribution, \(\overline{X}\) and \(s^{2}\) are independent of each other, so  
\[E(\hat{S}_{PL})^{2}=\left(\frac{b_{n}}{3}\right)^{2}\times\left(\frac{n-1}{n} \right)\times E(K)^{-1}\times E(Z)^{2}=\left(\frac{b_{n}}{3}\right)^{2}\times \left(\frac{n-1}{n}\right)\times\left(\frac{\Gamma[(n-3)/2]}{2\Gamma[(n-1)/2] }\right)\times\left[9n(S_{PL})^{2}+1\right] \tag{14}\]  
\[Var(\hat{S}_{PL})=E(\hat{S}_{PL})^{2}-E^{2}(\hat{S}_{PL})=\left(\frac{\Gamma[( n-1)/2]\Gamma[(n-3)/2]}{\Gamma^{2}[(n-2)/2]}\right)[(1/9n)+(S_{PL})^{2}]-(S_{PL})^{2} \tag{15}\]  
For the further derivation of the probability density function (PDF) of \(\hat{S}_{PL}\), we let \(W=\frac{3\sqrt{n}}{b_{n}}\times\hat{S}_{PL}=\frac{Z}{\sqrt{K/(n-1)}}\) obey the distribution of \(w_{n-1}^{\prime}(\gamma)\) and \(Q=\hat{S}_{PL}=\frac{b_{n}}{3\sqrt{n}}\times W\). Then, there exists a one-to-one mathematical relationship between \(W\) and \(Q\). Hence,  
\[f_{Q}(q)=f_{W}(w)\left|\frac{d_{Q}}{d_{W}}\right|\text{, where }\left|\frac{d_{Q}}{d_ {W}}\right|=\frac{3\sqrt{n}}{b_{n}} \tag{16}\]  
and  
\[f_{W}(w)=\frac{2^{-(n/2)}}{\Gamma[(n-1)/2]}\int_{0}^{\infty}x^{(\frac{n-2}{2 })}\times\exp\left(-\frac{1}{2}\left[x+\left(w\sqrt{\frac{x}{(n-1)}}-\delta \right)^{2}\right]\right)dx,\ w\in R \tag{17}\]  
where \(R\) is a real number.