# Fuzzy Evaluation and Improvement Decision-Making Model for Machining Operation Performance
## 4 Fuzzy Evaluation and Improvement Decision-Making Model  
As mentioned above, the greater the value of the machining operation performance index is, the better the machining operation performance shows. Suppose that the value of V is the required performance value of the operation performance index \(\theta_{OPI}\). This paper has taken into account the problem of the hypothesis test, the statistical null hypothesis \(H_{0}:\theta_{OPI}=V\) against the alternative hypothesis \(H_{1}:\theta_{OPI}\
eq V\), at the desired level of significance level \(\alpha\). Let \(T_{1},\ldots,T_{j},\ldots,T_{n}\) be a random sample from the distribution \(exp(\tau)\). Then, the unbiased estimator of operation performance index \(\theta_{OPI}\) is expressed as follows:  
\[\theta_{OPI}^{*}=\frac{T_{n}}{nL}=\frac{\overline{T}}{L} \tag{7}\]  
where \(T_{n}=\sum_{j=1}^{n}T_{j}\) and \(\overline{T}=T_{n}/n\). Then, the random variable \(T_{n}\) is distributed as \(Gamma(n,\tau)\), and the probability density function (p. d. f.) of random variable \(T_{n}\) is presented below:  
\[f_{T_{n}}(t)=\frac{1}{\Gamma(n/2)\tau^{n/2}}t^{\frac{\tau}{2}-1}\times e^{- \frac{t}{\tau}},t\geq 0 \tag{8}\]  
Furthermore, let random variable \(W\) be:  
\[W=\frac{T_{n}}{\tau}=\frac{n\theta_{OPI}^{*}}{\theta_{OPI}} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\theta_{OPI}\) & \(p_{r}\) & \(\theta_{OPI}\) & \(p_{r}\) & \(\theta_{OPI}\) & \(p_{r}\) \\ \hline
1.1 & 0.368 & 1.2 & 0.403 & 1.3 & 0.435 \\
1.4 & 0.463 & 1.5 & 0.490 & 1.6 & 0.513 \\
1.7 & 0.535 & 1.7 & 0.555 & 1.8 & 0.574 \\
1.9 & 0.591 & 2.0 & 0.607 & 2.1 & 0.621 \\
2.2 & 0.635 & 2.3 & 0.647 & 2.4 & 0.659 \\
2.5 & 0.670 & 2.6 & 0.681 & 2.7 & 0.690 \\
2.8 & 0.700 & 2.9 & 0.708 & 3.0 & 0.717 \\
3.1 & 0.724 & 3.2 & 0.732 & 3.3 & 0.739 \\
3.4 & 0.745 & 3.5 & 0.751 & 3.6 & 0.757 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Various value of index \(\theta_{OPI}\) and the corresponding value of \(p_{r}\). Figure 1: Curve of the standard ratio of machining operation performance \(p_{r}\). Then, random variable \(W\) is distributed as \(Gamma(n,1)\), and the probability density function of random variable \(W\) is expressed as follows:  
\[f_{W}(w)=\frac{1}{\Gamma(n/2)}w^{\frac{n}{2}-1}\times e^{-w},w\geq 0 \tag{10}\]  
Then, we derive the \(100(1-\alpha)\%\) confidence interval of the machining operation performance index \(\theta_{OPI}\). Based on Equations (9) and (10), random variable W is distributed as \(Gamma(n,1)\). Therefore, we have  
\[ \begin{array}{l}1-\alpha=p\Big{\{}G_{\alpha/2}(n,1)\leq\frac{n\theta_{OPI}^ {*}}{\theta_{OPI}}\leq G_{1-\alpha/2}(n,1)\Big{\}}\\ =p\Big{\{}\frac{n\theta_{OPI}^{*}}{G_{1-\alpha/2}(n,1)}\leq\theta_{OPI}\leq \frac{n\theta_{OPI}^{*}}{G_{\alpha/2}(n,1)}\Big{\}}\end{array}  \tag{11}\]  
Then, the \(100(1-\alpha)\%\) confidence interval of the machining operation performance index \(\theta_{OPI}\) can be expressed as follows:  
\[ \begin{array}{l}[L\theta_{OPI},U\theta_{OPI}]=\left[\frac{n\theta_{OPI}^{*} }{G_{1-\alpha/2}(n,1)},\frac{n\theta_{OPI}^{*}}{G_{\alpha/2}(n,1)}\right]\end{array}  \tag{12}\]  
Let \(t_{j}\) be the observed value of random variable \(T_{j}\), then the observed value of test statistical \(\theta_{OPI}^{*}\) is  
\[\theta_{O}^{*}=\frac{\tilde{t}}{L} \tag{13}\]  
where \(\tilde{t}=n^{-1}\times\sum_{j=1}^{n_{I}}t_{j}\). Similarly, the observed value of the lower confidence limit \(L\theta_{OPI}\) and the upper confidence limit \(U\theta_{OPI}\) are defined respectively as follows:  
\[ \begin{array}{l}\theta_{OPI}=\frac{n}{G_{1-\alpha/2}(n,1)}\theta_{0}^{*} \end{array}  \tag{14}\]  
and  
\[ \begin{array}{l}u\theta_{OPI}=\frac{n}{G_{\alpha/2}(n,1)}\theta_{0}^{*} \end{array}  \tag{15}\]  
where \(G_{\alpha/2}(n,1)\) is the lower \(\alpha/2\) quantile of \(Gamma(n,1)\), and \(G_{1-\alpha/2}(n,1)\) is the lower \(1-\alpha/2\) quantile of \(Gamma(n,1)\). Therefore, the statistical test rules can be made as follows [26]:  
1.