# Process Quality Evaluation Model with Taguchi Cost Loss Index
###### Abstract  
Process Capability Indices (PCIs) are not only a good communication tools between sales departments and customers but also convenient tools for internal engineers to evaluate and analyze process capabilities of products. Many statisticians and process engineers are dedicated to research on process capability indices, among which the Taguchi cost loss index can reflect both the process yield and process cost loss at the same time. Therefore, in this study the Taguchi cost loss index was used to propose a novel process quality evaluation model. After the process was stabilized, a process capability evaluation was carried out. This study used [PERSON]'s inequality and [PERSON]'s theorem to derive the \((1-\alpha)\times 100\%\) confidence region of \(\left(\delta,\gamma^{2}\right)\) based on control chart data. The study adopted the mathematical programming method to find the \((1-\alpha)\times 100\%\) confidence interval of the Taguchi cost loss index then employed a \((1-\alpha)\times 100\%\) confidence interval to perform statistical testing and to determine whether the process needed improvement. process capability indices; Taguchi  
where \(\Phi(\cdot)\) is the cumulative distribution function of a standard normal distribution. For example, when \(C_{PK}\)= 1, we can guarantee the process yield is equal to at least \(2\Phi(3)-1=99.73\%\). [PERSON] et al. [16] proposed the Taguchi cost loss index \(C_{PM}\) as follows:  
\[C_{PM}=\frac{d}{3\sqrt{\left(\mu-T\right)^{2}+\sigma^{2}}} \tag{4}\]  
where \((\mu-T)^{2}+\sigma^{2}\) is the expected value of the Taguchi loss function. As noted by [PERSON] et al. [17] and [PERSON] [18], the relationship between the process yield and Taguchi cost loss index \(C_{PM}\) is described as follows:  
\[\textit{Yield}\%\geq\ 2\Phi(3C_{PM})-1,\ C_{PM}>1/3 \tag{5}\]  
Obviously, the process-yield-based capability index \(C_{PK}\) can reflect the process yield, while the Taguchi cost loss index can reflect not only the process yield but also the process loss, since its denominator is the expected value of the Taguchi cost loss function. [PERSON] et al. [19] and [PERSON] et al. [20] derived the confidence intervals of the Taguchi cost loss index and determined the sample size for this index. [PERSON] et al. [21] developed a new method for constructing confidence intervals for the Taguchi cost loss index. [PERSON] et al. [22] proposed some new methods for comparing the capabilities of two processes using the Taguchi cost loss index. In addition, many other scholars have done a lot of research on the Taguchi cost loss index \(C_{PM}\). Since the statistical distribution of the index estimation formula is relatively complicated, it is also relatively difficult to make statistical inferences. Thus, [PERSON] developed neutrosophic statistics that are more flexible than traditional statistics [23]. In addition, [PERSON] et al. [17] used a mathematical programming method to derive the confidence intervals of PCIs, discussed the results, and found accuracy as high as 95% or more. Subsequently, many articles also applied the mathematical programming method to derive the confidence intervals of various indicators [24; 25]. [PERSON] et al. [17] adopted the model to derive the confidence interval of the indices, and the confident interval was also extended [18] by a one-sided Fuzzy test to a two-sided Fuzzy test. Therefore, this study proposes a novel evaluation model for process quality with the Taguchi cost loss index, the advantages being: (1) to help quality management engineers in the industry understand better and apply more conveniently in practice; (2) to conform to an enterprise's quick response demand, and (3) to help the enterprise to meet the goal of intelligent manufacturing management. Numerous studies have pointed out that industries usually uses a statistical process control chart to monitor the quality of processes [26; 27; 28; 29] as well as evaluate process capability when the process is stabilized [30; 31; 32; 33; 34].