# Fuzzy Evaluation and Improvement Decision-Making Model for Machining Operation Performance
## 3. Machining Operation Performance Index  
As mentioned above, machining machines will not be able to continue to run due to the failures of components. For all the sold machining machines, the number of failures, N, per unit time obeys the Poisson distribution with the mean of \(\lambda\). As noted by some studies, from a management point of view, the administrator can set the required value, \(N_{0}\), for the number of failures per unit time, namely the requirement of \(N<N_{0}\)[20]. Meanwhile, the longer the average interval time between machine failures, the smaller the number of machine failures per unit time [32; 33]. It is clear that evaluating the operation performance of machining machines with the time interval of machine failures is more effective than collecting the number of failures, N, per unit time [27; 28]. Let the random variable, T, represent the time interval of the machine failures, then T obeys the exponential distribution with the average of \(\tau=1/\lambda\), denoted as \(T\sim exp(\tau)\). Obviously, the higher the average number \(\tau\), the lower the number of machine failures per unit time, which indicates that the machining operation performance gets better. Based on this concept, this paper defines the machining operation performance index as follows:  
\[\theta_{OPI}=\frac{\tau}{L} \tag{4}\]  
where \(L=1/N_{0}\) represents the minimum required value of the time interval between machine failures. When the value of index \(\theta_{OPI}\) is larger, then the machining operation performance gets better. Conversely, as the value of index \(\theta_{OPI}\) is smaller, the frequency of machine failures is higher, which means the machining operation performance gets worse. Additionally, index \(\theta_{OPI}\) has a one-to-one mathematical relationship with the standard ratio of machining operation performance, \(p_{r}\), as follows:  
\[p_{r}=1-p(T\leq L)=1-\int\limits_{0}^{L}\frac{1}{\tau}exp\left\{-\frac{t}{\tau }\right\}d_{t}=exp\left\{-\frac{1}{\theta_{OPI}}\right\} \tag{5}\]  
Based on Equations (4) and (5), partial differentiation for the standard ratio of machining operation performance \(p_{r}\) divided by \(\theta_{OPI}\) is expressed as follows:  
\[p_{r}^{\prime}=\frac{dp_{r}}{d\theta_{OPI}}=\theta_{OPI}^{-2}\times exp\left\{ -\frac{1}{\theta_{OPI}}\right\} \tag{6}\]  
Obviously, the value of \(p_{r}^{\prime}\) is positive (\(p_{r}^{\prime}>0\)). Thus, the standard ratio of machining operation performance \(p_{r}\) is an increasing function of \(\theta_{OPI}\). Based on Equation (5), the value of machining operation performance index \(\theta_{OPI}\) is taken from 1.0 to 3.6, and the corresponding value of the standard ratio of machining operation performance \(p_{r}\) is shown in Table 2.  
It can be clearly seen from Table 1 that when the value of index \(\theta_{OPI}\) is larger, the value of \(p_{r}\) is also larger. The standard ratio of machining operation performance \(p_{r}\) is an increasing function of machining operation performance index \(\theta_{OPI}\). As shown in Figure 1, the curve of the standard ratio of machining operation performance \(p_{r}\) plots function values versus the values of index \(\theta_{OPI}\).