## 3 Methods
### Development of Multivariate Control Charts
 \tag{13}\]  
Substitute in Equation (6) to obtain the \(C_{1}\) value of Equation (14) and evaluate the case in Equation (5). \[C_{1}= \begin{bmatrix}[A]\times\begin{bmatrix}0.00021095&0.00082909&0.00016761\\ 0.00082909&0.00336724&0.00064279\\ 0.00016761&0.00064279&0.00013834\end{bmatrix}^{-1}\times[A]^{\prime}\end{bmatrix} ^{1 /2}\]  
\[C_{1}=0.5901 \tag{14}\]  
\[C_{1}>k;\,0.5901>0.5\]  
Equation (15) is solved, obtaining Equation (16). \[R_{1}=A\times\left(1-\frac{0.5}{0.5901}\right) \tag{15}\]  
\[R_{1}= \begin{bmatrix}0.0011&0.0045&0.0009\end{bmatrix}  \tag{16}\]  
Finally, the multivariate MCUSUM statistic is calculated by substituting in Equation (4) and the statistic for observation number is obtained; see Equation (17). \[M^{2}= \begin{bmatrix}[S_{1}]\times\begin{bmatrix}0.00021095&0.00082909&0.0001676 1\\ 0.00082909&0.00336724&0.00064279\\ 0.00016761&0.00064279&0.00013834\end{bmatrix}^{-1}\times[S_{1}]^{\prime}\end{bmatrix} ^{ 1/2}\]  
\[M^{2}=[0.0081]^{1/2}\]  
\[M^{2}=0.0901 \tag{17}\]  
Table 6 shows the MCUSUM statistics for each of the observations used to construct the control chart; the procedure is the same as shown above for each observation. Figure 3 shows the MCUSUM control chart resulting from plotting the statistics and the control limit. The MCUSUM chart shows that samples twenty-three and twenty-four generate an out-of-control signal in the welding application process; these samples should be analyzed to identify the nature of the variation and correct the statistical lack of control of the process. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{4}{c}{**Volume**} & \multicolumn{4}{c}{**Area**} & \multicolumn{4}{c}{**Height**} \\ \hline
**1** & 0.08568 & **13** & 0.08419 & **1** & 0.64758 & **13** & 0.64920 & **1** & 0.13429 & **13** & 0.13207 \\
**2** & 0.06342 & **14** & 0.06152 & **2** & 0.54432 & **14** & 0.55347 & **2** & 0.11940 & **14** & 0.11397 \\
**3** & 0.08120 & **15** & 0.07040 & **3** & 0.62873 & **15** & 0.59247 & **3** & 0.12952 & **15** & 0.12411 \\
**4** & 0.07921 & **16** & 0.07589 & **4** & 0.63411 & **16** & 0.61117 & **4** & 0.12850 & **16** & 0.12952 \\
**5** & 0.09614 & **17** & 0.07188 & **5** & 0.68196 & **17** & 0.56778 & **5** & 0.14573 & **17** & 0.12725 \\
**6** & 0.05896 & **18** & 0.06527 & **6** & 0.53613 & **18** & 0.57228 & **6** & 0.11616 & **18** & 0.11732 \\
**7** & 0.06080 & **19** & 0.08020 & **7** & 0.54702 & **19** & 0.63147 & **7** & 0.11704 & **19** & 0.12825 \\
**8** & 0.12039 & **20** & 0.09544 & **8** & 0.77509 & **20** & 0.69211 & **8** & 0.15590 & **20** & 0.14165 \\
**9** & 0.07521 & **21** & 0.05957 & **9** & 0.60214 & **21** & 0.52098 & **9** & 0.12761 & **21** & 0.11601 \\
**10** & 0.06058 & **22** & 0.08210 & **10** & 0.54761 & **22** & 0.64539 & **10** & 0.11184 & **22** & 0.12933 \\
**11** & 0.10462 & **23** & 0.09991 & **11** & 0.72132 & **23** & 0.76860 & **11** & 0.14571 & **23** & 0.14671 \\
**12** & 0.06999 & **24** & 0.11134 & **12** & 0.59840 & **24** & 0.77167 & **12** & 0.12060 & **24** & 0.13801 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Measurements to develop the MCUSUM chart. The structuring of the MEWMA control chart begins by defining the depth value for the chart memory; for this case, a value of (\(\lambda\) = 0.1) was taken. The estimation of the statistic starts with the substitution of the initial data in Equation (7), and the solution is shown in Equation (18)  
\[Z_{1}=0.1 \begin{bmatrix}0.08568\\ 0.64758\\ 0.13428\end{bmatrix} +(1-0.1) \begin{bmatrix}0.07856\\ 0.61838\\ 0.12829\end{bmatrix} = \begin{bmatrix}0.0793\\ 0.6213\\ 0.1289\end{bmatrix}  \tag{18}\]  
The MEWMA statistic for the first measurement is obtained by solving Equation (19). \[N_{1}^{2}= \begin{bmatrix}0.0793&0.07856\\ 0.6213-0.61838\\ 0.1289&0.12829\end{bmatrix} ^{\prime}\times \begin{bmatrix}8.1372&-1.0755&-4.8615 \\ -1.0755&0.1684&0.5205\\ -4.8615&0.5205&3.5441\end{bmatrix} 1\times 10^{7}\times \begin{bmatrix}0.0793&0.07856 \\ 0.6213-0.61838\\ 0.1289&0.12829\end{bmatrix}