_applied sciences_  
_Article_  
Monitoring, Evaluation, and Improvement Model for Process Precision and Accuracy  
[PERSON] 1\({}^{\copyright}\), [PERSON] 2\({}^{\copyright}\), [PERSON] 1,3,4,+ and [PERSON] 1,*  
Footnote 10: foot  
\[C_{pm}=\frac{d}{3\sqrt{\sigma^{2}+(\mu-T)^{2}}} \tag{2}\]  
where \(d\) is the half-length from the lower specification limit (\(LSL\)) to the upper specification limit (\(USL\)), that is \(d=(USL-LSL)/2\); \(T=(USL+LSL)/2\) refers to the target value. In fact, the denominator of process capability index \(C_{pm}\) means the expected value of the Taguchi loss function. Clearly, process capability has two important factors: (1) as the process mean \(\mu\) gets closer to target value \(T\), the process accuracy gets higher; (2) as the process standard deviation \(\sigma\) gets smaller, the process precision gets better [14]. Boosting the process accuracy and precision will cut down process loss as well as level up process capability. In addition, the target values and tolerances of various processes are all different. To facilitate the process evaluation, the tolerance can be standardized by variable transformation. Let RV \(Y=(X-T)/d\). When \(X=LSL\), then \(Y=-1\). When \(X=T\), then \(Y=0\). When \(X=USL\), then \(Y=1\). Since different quality characteristics have different specification limits, this variable transformation can help all quality characteristics convert their different specification limits from (\(LSL\), \(T\), \(USL\)) to (\(-1\), \(0\), \(1\)). In fact, \(\delta\), which stands for the mean of RV \(Y\), is regarded as an accuracy index; \(\gamma\), which refers to the standard deviation of RV \(Y\), is viewed as a precision index. These two indices are written as  
\[\text{accuracy index}:\ \delta=\frac{\mu-T}{d} \tag{3}\]  
\[\text{precision index}:\ \gamma=\frac{\sigma}{d} \tag{4}\]  
Based on the aforementioned, \(C_{pk}=(1-|\delta|)/3\gamma\) and \(C_{pm}=1/3\sqrt{\delta^{2}+\gamma^{2}}\) stand for the functions of \(\delta\) and \(\gamma\). Accuracy index \(\delta\) is mainly applied to measure the degree to which the process mean \(\mu\) deviates from the process target value \(T\). When the value is closer to \(0\), then the process is more accurate. The greater the positive number of accuracy index \(\delta\) is, the more the process is shifted to the right. Conversely, as the negative number of accuracy index \(\delta\) is greater, the process is shifted to the left more. As to the precision index \(\gamma\), it mainly represents the extent of process variation. As the index is smaller, the process variation is smaller; that is to say, the process is more precise. Apart from process capability indices, the six sigma method, initiated by Motorola in 1986 [15; 16; 17], is also applied by this study. The six sigma method is also a tool prevalently employed by the industry to assess and enhance process quality [18; 19; 20]. According to the research of [PERSON] and [PERSON] [14], when the standard deviation of the process is \(\sigma=d/k\) and the process mean shifted from the target value falls within \(1.5\sigma\), it means that the process has reached the \(k\)-sigma quality level. When the quality level of the process reaches six sigma, then  
\[\gamma=\frac{\sigma}{k\sigma}=\frac{1}{k} \tag{5}\]  
\[|\delta|=\frac{|\mu-T|}{d}\leq\frac{1.5\sigma}{k\sigma}=\frac{1.5}{k} \tag{6}\]  
Subsequently, the statistical testing method is adopted to evaluate whether the process standard deviation is \(1/k\) and the process mean shifted from the target value falls within \(1.5\sigma\) (i.e., \(\gamma=1/k\) and \(|\delta|\leq 1.5/k\)). When \(\gamma=1/k\) and \(|\delta|\leq 1.5/k\), it means that the process meets the \(k\)-sigma quality level. Numerous researchers have addressed that the evaluation of process capability is usually performed when the statistical process is in control [7]. Hence, in this paper, the \(\delta-\gamma\) control charts are derived to control the process accuracy and precision.