# Fuzzy Quality Evaluation Model Constructed by Process Quality Index
###### Abstract  
Many studies have pointed out that the-smaller-the-better quality characteristics (QC) can be found in many important components of machine tools, such as roundness, verticality, and surface roughness of axes, bearings, and gears. This paper applied a process quality index that is capable of measuring the level of process quality. Meanwhile, a model of fuzzy quality evaluation was developed by the process quality index as having a one-to-one mathematical relationship with the process yield. In addition to assessing the level of process quality, the model can also be employed as a basis for determining whether to improve the process quality at the same time. This model can cope with the problem of small sample sizes arising from the need for enterprises' quick response, which means that the accuracy of the evaluation can still be maintained in the case of small sample sizes. Moreover, this  
is usually far away from the target value \(T_{V}\) and very close to \(USL\), that is, the parameter \(\delta\) is usually less than 1 but very close to 1. Therefore, \(\gamma\) is usually very small.  
Moreover, in the case of normality, a one-to-one mathematical relationship exists between process quality index \(P_{QI}\) and process yield _Yield\(\%\)_ as described below:  
\[Yield\%\ =\ p\{Y\leq 1\}\ =\ p\bigg{\{}Z\leq\frac{1-\delta}{\gamma}\bigg{\}}\ =\ \Phi\big{(}P_{QI}\big{)}, \tag{2}\]  
where \(Z\ =\ (Y-\delta)/\gamma\) is distributed as standard normal distribution and \(\Phi(\cdot)\) is the cumulative function of standard normal distribution. For instance, if \(P_{QI}\ =4.0\), then it is guaranteed that the process yield is \(Yield\%\ =\Phi(4.0)\ =99.9968\%\). The \(x\)-axis of Figure 1 is index \(P_{QI}\) and the \(y\)-axis is process yield _Yield\(\%\)_. Obviously, the larger the value of index \(P_{QI}\), the larger the corresponding process yield _Yield\(\%\)_.  
Not only can the process quality index \(P_{QI}\) measure the process quality level, but it also has a one-to-one mathematical relation with the process yield at the same time. Accordingly, this paper modified the fuzzy testing method created by [PERSON] [26] and [PERSON] [27; 28] to develop a fuzzy quality evaluation model. In addition to having a simpler calculation procedure, the model will employ this index to come up with a quality fuzzy evaluation model based on the upper confidence limit for the STB quality characteristics and use it as a basis to determine whether the process needs to improve.  
Obviously, the model proposed in this paper is not only applicable to the evaluation of the small sample size and correspondent with the needs of enterprises for quick responses, but it is also capable of reducing the risk of misjudgment caused by sampling error for it is based on the confidence interval. There are many industries that are very suitable for the model described in this paper, such as the process quality evaluation of the-smaller-the-better quality characteristics produced by the machining process and the waste discharge of the-smaller-the-better characteristics regulated by many environmental regulations.  
The other sections are arranged as follows. According to [PERSON]'s inequality and [PERSON]'s theorem, we obtain the upper confidence limit of the process quality index in Section 2. In Section 3, we construct \(a\)-cuts of the half-triangular shaped fuzzy number and fuzzy membership function for index \(P_{QI}\). In Section 4, we propose a fuzzy hypothesis testing method aiming to evaluate the process quality and determine whether the process needs to improve. In Section 5, an application example is taken to prove the applicability of the approach proposed in this study. Finally, conclusions are made in Section 6.  
Figure 1: The relation between index \(P_{QI}\) and process yield _Yield\(\%\)_.