# The Fuzzy Process Quality Evaluation Model for the STB Quality Characteristic of Machining
## 1 Introduction  
Many studies on process technology and management have indicated that machine tools are assembled from a large number of components mainly used to process a variety of machinery equipment. Machine tools play a critical role in the entire machinery manufacturing industry, so that they are called the \"Mother of Machinery\" [1, 2]. Whether it is important components of a machine tool itself or various important components processed by the machine tool, many important quality characteristics, including roundness, concentricity and squareness of the shaft and the bearing, are mostly of the smaller-the-better (STB) type. When the process quality levels of these quality characteristics are not good enough, friction loss may increase during the machine operation, which will affect the precision and accuracy of processing as well as the lifetime of the product. Many statisticians and quality engineers have pointed out that Process Capability Indices (PCIs) are used as a very common process quality evaluation and analysis tool in the manufacturing industry. PCIs are not only a tool for selecting outsourcers but also a convenient and effective communication tool for internal process engineers and quality control engineers, which can assist with handlingvarious problems concerning process technology or quality arising in the production process of products [3; 4; 5; 6; 7; 8]. Furthermore, the scheme of six-sigma quality improvement initiated by Motorola can assist corporations with their quality improvement as well as reduce the defect rate for their products [9; 10; 11; 12]. Therefore, the six-sigma quality improvement process contains significant implications for industry and is widely applied to manufacturing, aiming to enhance product quality levels as well as lower production defect rates [13; 14; 15; 16]. Obviously, both Process Capability Indices and six-sigma quality levels are two significant tools in industry. Many scholars have explored the correlation between process capability indices and six-sigma quality levels [17; 18; 19; 20]. Attempting to directly define quality level on the basis of an index value, [PERSON] et al. [21] proposed the STB six-sigma quality index to evaluate the process quality of badminton racket handles. This STB six-sigma quality index can fully display the process quality level and the process yield. Therefore, it aims to appraise the process quality levels for the abovementioned STB quality characteristics, including roundness, concentricity and squareness of the shaft and the bearing. Let random variable \(X\) represent the process distribution for the STB quality characteristics. Assume that \(X\) is distributed as a normal distribution using mean \(\mu\) and standard deviation \(\sigma\), i.e., \(X\)-\(N\)\(\left(\mu,\sigma^{2}\right)\). Therefore, the STB six-sigma quality index can be denoted as follows:  
\[Q_{\text{ISS}}=\frac{USL-\mu}{\sigma} \tag{1}\]  
where \(USL\) stands for the upper specification limit. Obviously, the relationship between process yield (_yield_%) and the value of quality index \(Q_{\text{ISS}}\) is one-to-one, illustrated as follows:  
\[ \begin{array}{l}\text{{yield}}\%=p[X\leq USL]\\ \quad=p\Big{\{}Z\leq\frac{USL-\mu}{\sigma}\Big{\}}\\ =\int\limits_{-\infty}^{Q_{\text{ISS}}}\frac{1}{\sqrt{2\pi}}\times exp\Big{\{} -\frac{\mu}{2}\Big{\}}d_{z}\\ \quad=\Phi(Q_{\text{ISS}})\end{array}  \tag{2}\]  
where \(\Phi(\cdot)\) refers to the cumulative function of standard normal distributions. According to [PERSON] et al. [21], when \(\mu+k\sigma\)=\(USL\), it means the process quality level for the STB quality characteristic attains to the quality level of \(k-sigma\), that is  
\[Q_{\text{ISS}}=\frac{USL-\mu}{\sigma}=\frac{USL-\left(\mu+k\sigma\right)}{ \sigma}=k. \tag{3}\]  
Obviously, when the value of \(Q_{\text{ISS}}\) equals \(k\), the quality level is \(k-sigma\), and the process yield is \(\Phi(k)\). For example, when \(Q_{\text{ISS}}\) =3, the quality level is \(3-sigma\), and it is guaranteed that the process yield will be 99.865%.