## 6 Research Limitations and Future Research  
As mentioned above, the model proposed in this paper can assist the machining manufacturers, which is the customer end of the machine tool industry chain, to evaluate the process capability of all important quality characteristics of the processed products so that the machining manufacturers can grasp the overall picture of the processed products and improve the poor quality in time quality characteristics. In addition, when the entire machine tool industry chain uses this model for a period of time, the machine tool industry can collect the improvement experience of all machining manufacturers, form an improvement knowledge basement, and share it with all machine tool manufacturers at the customer end of the industry chain. Feedback on the problematic parts and components for suppliers or outsourcers to help them improve the quality of supplied parts and components. Due to the itinerary and sharing of the knowledge basement, the problems and feedback of common components not only need a long time to collect and build but also a complex problem. This study focuses on the process capability evaluation and improvement model for multi-quality characteristic products. The establishment and application of the other issues can be the focus of future research. Conceptualization, K.-S.C. and K.-K.L. ; methodology, K.-S.C. and K.-K.L. ; software, S.-C.L. and W.-P.W. ; validation, S.-C.L. and W.-P.W. ; formal analysis, K.-S.C. and K.-K.L. ; resources, S.-C.L. ; data curation, S.-C.L. and W.-P.W. ; writing--original draft preparation, K.-S.C., S.-C.L., K.-K.L. and W.-P.W. ; writing--review and editing, K.-S.C. and K.-K.L. ; visualization, S.-C.L. ; supervision, K.-S.C.; project administration, K.-K.L. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. All data underlying the results are available as part of the article and no additional source data are required. The authors declare no conflict of interest. The measurement coordinate point \((x_{h},y_{h})\) of quality characteristic h as follows:  
\[(x_{h},y_{h})=\left\{ \begin{array}{l}(\delta_{Lh},\gamma_{Lh})=\left(\left( \delta^{\prime}_{h}-\frac{t_{h}/x_{h}-1}{\sqrt{n}}\gamma^{\prime}_{h}\right) \times d_{h1},\ \sqrt{\frac{n}{\chi^{2}_{x-4/4n-1}}}\gamma^{\prime}_{h}\times d_{h1} \right),\delta^{\prime}_{Lh}\geq 0\\ (\delta_{Uh},\gamma_{Lh})=\left(\left(\delta^{\prime}_{h}+\frac{t_{h}/x_{h}-1} {\sqrt{n}}\gamma^{\prime}_{h}\right)\times d_{h2},\ \sqrt{\frac{n}{\chi^{2}_{x/4n-1}}}\gamma^{\prime}_{h}\times d_{h2} \right),\delta^{\prime}_{Uh}<0\end{array} \right. \tag{1}\]  
**Proof:**  
Let \(X_{h,1},\ldots,X_{h,n}\) be a random sample derived from quality characteristic h with sample size \(n\), where \(h=1,2,\)\(\ldots,k\). Then estimators of \(\mu_{h}\) and \(\sigma_{h}\) are expressed as follows:  
\[\overline{X}_{h}=\frac{1}{n}\sum_{j=1}^{n}X_{h,j} \tag{11}\]  
and  
\[S_{h}=\sqrt{\frac{1}{n}\!\sum_{j=1}^{n}\!\left(X_{h,j}-\overline{X}_{h}\right) ^{2}}. \tag{12}\]  
Therefore, the estimators of index \(\delta^{\prime}_{h}\) and index \(\gamma^{\prime}_{h}\) are displayed below:  
\[\delta^{\prime}_{h}\!=\frac{\overline{X}_{h}-T_{h}}{d_{h}^{*}} \tag{13}\]  
and  
\[\hat{\gamma}^{\prime}_{h}\!=\frac{S_{h}}{d_{h}^{*}}. \tag{14}\]  
With an assumption of normality, \(\sqrt{n}\big{(}\delta^{\prime}_{h}-\delta^{\prime}_{h}\big{)}/\hat{\gamma}_{h}\) is distributed as t-distribution with n\(-\)1 degree of freedom, denoted by \(t_{n-1}\), and \(n\hat{\gamma}^{2}_{h}/\gamma^{2}_{h}\) is distributed as chi-square distribution with n\(-\)1 degree of freedom, denoted by \(\chi^{2}_{n-1}\).