# Fuzzy Evaluation Model of Process Improvement Capability with Costs Consideration
## 3 Fuzzy Testing for Process Improvement Capability Index  
If we want to determine whether the supplier's improvement potential is above the average level, we can evaluate it through the fuzzy test of the process improvement capability index. If the value of the process improvement capability index is at least \(VL\), which indicates that its potential reaches the required level, then the null hypothesis is \(H_{0}\): \(C_{PIM}\geq VL\), versus the alternative hypothesis, \(H_{1}\): \(C_{PIM}<VL\). If the upper confidence limit \(UC_{PIM}\) with a significance level at 0.05 is applied, we can obtain the following statistical test rules:  
(1) If \(UC_{PIM}\geq VL\), then do not reject \(H_{0}\) and conclude \(C_{PIM}\geq VL\). (2) If \(UC_{PIM}<VL\), then reject \(H_{0}\) and conclude \(C_{PIM}<VL\). \(UC_{PIM}=UC_{PIM}(0.05)\) can be shown as follows:  
\[UC_{PIM}=\left\{ \begin{array}{ll}\left(3\gamma_{0}^{*}\sqrt{\frac{(n-1)R}{ \chi_{0.5-\sqrt{0.5/2n-1}^{2}}}}\right)^{-1}\text{ if }0\in CI\\ \left(3\gamma_{0}^{*}\sqrt{Min\left\{\left(\frac{\delta_{0}^{*}}{\gamma_{0}^ {*}}-E_{0}(0.05)\right)^{2},\left(\frac{\delta_{0}^{*}}{\gamma_{0}^{*}}+E_{0} (0.05)\right)^{2}\right\}+\frac{(n-1)R}{\chi_{0.5-\sqrt{0.5/2n-1}}^{2}}} \right)^{-1}\text{ if }0\
otin CI\end{array} \right.. \tag{27}\]  
According to [PERSON] et al. [22] and [PERSON] et al. [23], the \(\alpha\)-cuts of the half-triangle-like shaped fuzzy number \(\widetilde{UC}_{PIM}\) can be expressed below:  
\[\widetilde{UC}_{PIM}[\alpha]=\left[UC_{PIM}(1),UC_{PIM}(\alpha)\right],\text{ }0.01\leq\alpha\leq 1, \tag{28}\]  
where  
\[UC_{PIM}(1)=\left\{ \begin{array}{ll}\left(3\gamma_{0}^{*}\sqrt{\frac{(n-1)R }{\chi_{0.55-1}^{2}}}\right)^{-1}\text{ if }0\in CI\\ \left(3\sqrt{(\delta_{0}^{*})^{2}+\frac{(n-1)R}{\chi_{0.55-1}^{2}}\gamma_{0}^ {*2}}\right)^{-1}\text{ if }0\
otin CI\end{array} \right.. \tag{29}\]  
Recall the \(\alpha-\)cuts of \(\widetilde{UC}_{PIM}\)\([\alpha]\), for \(0\leq\alpha<0.01\) are all equal to \(\widetilde{UC}_{PIM}\)\([0.01]\). The half-triangle-like shaped fuzzy number is \(\Delta\widetilde{UC}_{PIM}=(UC_{M},\ U_{CR})\), where \(UC_{M}=UC_{PIM}(1)\) (see Equation (29)) and  
\[UC_{R}=\left\{ \begin{array}{ll}\left(3\gamma_{0}^{*}\sqrt{\frac{(n-1)R}{ \chi_{0.0025\alpha-1}^{2}}}\right)^{-1}\text{ if }0\in CI\\ \left(3\gamma_{0}^{*}\sqrt{Min\left\{\left(\frac{\delta_{0}^{*}}{\gamma_{0}^ {*}}-E_{0}(0.01)\right)^{2},\left(\frac{\delta_{0}^{*}}{\gamma_{0}^{*}}+E_{0}(0.01)\right)^{2}\right\}+\frac{(n-1)R}{\chi_{0.0025\alpha-1}^{2}}}\right)^{-1} \text{ if }0\
otin CI\end{array} \right.. \tag{30}\]Let \(x=UC_{PIM}\), then the fuzzy membership function of \(x\) is:  
\[\eta(x)=\left\{ \begin{array}{l}0\text{ {\it if }}x\ <\ UC_{M}\\ 1\text{ {\it if }}x\ =\ UC_{M}\\ \text{ {\it a if }}UC_{M}\ <\ x\ <\ UC_{R}\\ 0\text{ {\it if }}UC_{R}\ \leq\ x\end{array} \right., \tag{31}\]  
where \(\alpha\) is determined by either of the following equations:  
\[\left(3\gamma_{0}^{*}\sqrt{\frac{(n-1)R}{\chi_{0.5-\sqrt{1-\alpha}/2n-1}^{2}}} \right)^{-1}=x\text{ {\it if }}0\ \in\ CI \tag{32}\]  
or  
\[\left(3\gamma_{0}^{*}\sqrt{Min\left\{\left(\frac{\delta_{0}^{*}}{\gamma_{0}^{ *}}-E_{0}(\alpha)\right)^{2},\left(\frac{\delta_{0}^{*}}{\gamma_{0}^{*}}+E_{0 }(\alpha)\right)^{2}\right\}+\frac{(n-1)R}{\chi_{0.5-\sqrt{1-\alpha}/2n-1}^{2} }}\right)^{-1}=x\text{ {\it if }}0\
ot\in CI. \tag{33}\]  
Figure 2 presents a diagram of membership function \(\eta(x)\) with vertical line \(x=VL\). Based on [PERSON] [24], let set \(A_{T}\) be the area in the graph of \(A_{R}\) and set \(A_{R}\) be the area in the graph of \(\eta(x)\), but to the right of the vertical line \(x=VL\)., then  
\[A_{T}=\{\ (x,\alpha)|UC_{M}\leq x\leq UC_{PIM}(\alpha),0\leq\alpha\leq 1\}, \tag{34}\]  
and  
\[A_{R}=\{\ (x,\alpha)|VL\leq x\leq UC_{PIM}(\alpha),0\leq\alpha\leq a\}, \tag{35}\]  
where \(UC_{PIM}(a)=VL\).