## 4 An Empirical Example  
As mentioned above, the central part of Taiwan is an industrial cluster of machine tools, and the value of output and the volume of export of machine tools are both among the best. Usually, components processed by machine tools are simultaneously equipped with multiple quality characteristics. The quality of finished goods can only be ensured when each quality characteristic conforms to the required level of quality. It will contribute to smart manufacturing and intelligent management of the machine tool industry to apply a decision-making model of production data management for multi-quality characteristic products built in the second and third sections of this paper as well as conduct the process capability evaluation of each quality characteristic with the production data of the components processed by machining factories through the Internet of Things and cloud platforms. The components processed by a machining factory have a total of four important quality characteristics, including inner diameter, outer diameter, length, and weight. According to Equation (5), when the value of the process capability index \(C^{\prime\prime}_{PMT}\) is set to be at least 1.0 (\(C_{T}=1.0\)), then the required value of index \(C^{\prime\prime}_{pmh}\) is at least 1.133 (\(C=1.133\)) for each quality characteristic. Thus, the process capability accept zone \(Z^{\prime\prime}_{A}\) is defined as  
\[Z^{\prime\prime}_{A}=\Big{\{}\left(\delta_{h},\gamma_{h}\right)|\gamma^{2}_{h} +\delta^{2}_{h}=1/\left(4.5\right)^{2},\gamma_{h}\geq 0\Big{\}}\]  
In addition, the units and tolerance specifications for these four important quality characteristics are illustrated in Table 1 below:The sample mean and sample standard of random samples from four quality characteristics with sample size \(n=36\) can be displayed as follows:  
\[\delta=\left\{ \begin{array}{l l}\delta^{\prime}d_{1},\mu\geq T\\ \delta^{\prime}d_{2},\mu<T\end{array} ,\right.\]  
where \(\delta^{\prime}=(\mu-T)/d^{*}\), \(d_{1}=d^{*}/d_{U}\), and \(d_{2}=d^{*}/d_{L}\). Then,  
quality characteristic 1: (\(T_{1}=1.2\), \(d_{L1}=0.01\), \(d_{U1}=0.03\), \(d_{1}^{*}=0.01\), \(d_{11}=1/3\), \(d_{12}=1\)), \(\overline{X}_{1}=1.201\) and \(S_{1}=0.002\);  
quality characteristic 2: (\(T_{2}=1.8\), \(d_{L2}=0.03\), \(d_{U2}=0.02\), \(d_{2}^{*}=0.02\), \(d_{21}=1\), \(d_{22}=2/3\)), \(\overline{X}_{2}=1.796\) and \(S_{2}=0.005\);  
quality characteristic 3: (\(T_{3}=30\), \(d_{L3}=0.05\), \(d_{U3}=0.05\), \(d_{3}^{*}=0.05\), \(d_{31}=1\), \(d_{32}=1\)), \(\overline{X}_{3}=30.02\) and \(S_{3}=0.005\);  
quality characteristic 4: (\(T_{4}=12\), \(d_{L4}=0.1\), \(d_{U4}=0.1\), \(d_{4}^{*}=0.1\), \(d_{41}=1\), \(d_{42}=1\)), \(\overline{X}_{4}=12.01\) and \(S_{4}=0.01\). Therefore, the value of estimators for index \(\delta^{\prime}_{h}\) and index \(\gamma^{\prime}_{h}\) are expressed as follows: quality characteristic 1: \(\delta^{\prime}_{1}=0.2\) and \(\gamma^{\prime}_{1}=0.2\);  
quality characteristic 2: \(\delta^{\prime}_{2}=-0.2\) and \(\gamma^{\prime}_{2}=0.25\);  
quality characteristic 3: \(\delta^{\prime}_{3}=0.4\) and \(\gamma^{\prime}_{3}=0.167\);  
quality characteristic 4: \(\delta^{\prime}_{4}=0.2\) and \(\gamma^{\prime}_{4}=0.2\). In fact, the value of the upper 0.0025 quintile of \(t_{35}\) is 2.996 (\(t_{0.0025,35}=2.996\)). Then, based on Equation (A11), this paper will calculate the value of measurement coordinate point for four quality characteristics and show as follows: quality characteristic 1: The value of \(\left[\delta^{\prime}_{L1},\delta^{\prime}_{U1}\right]\) is \([-0.325\), \(-0.075]\). Thus, the value of \(\delta^{\prime}_{L1}\) is bigger than zero (\(\delta^{\prime}_{L1}=0.1\geq 0\)), and the measurement coordinate point is  
\[(x_{1},y_{1})=(\delta_{L1},\gamma_{L1})=(0.033,0.050).\]  
quality characteristic 2: The value of \(\left[\delta^{\prime}_{L2},\delta^{\prime}_{U2}\right]\) is [0.1, 0.3].