## 5 Discussion  
The results of this research emphasize the significance of the proposed methodology in processing real data and estimating the process capability index of a given process through multivariate QCs. Generating special data that follow a specific distribution with specific parameters and a determined correlation is a very important research direction. Transforming this method into a software code would facilitate studying the effect of correlation in process capability analysis. This could be done by generating data from the same distribution with the same parameters and different levels of correlation. Also, combining the transformation method with generating data in one algorithm could be utilized to search for samples that could be transformed to normality easily. The part of skewness reduction in the algorithm could be combined in practice with other approaches that may enhance the other properties of normal distribution, such as bringing the mean, mode, and median close to each other. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Distribution** & \(n\) & **USL** & \(\alpha\) & \(\beta\) & **Correlation** & \multicolumn{3}{c}{**C\({}_{\mathcal{P}}\)**} \\ \cline{5-10}  & 50 & & & & & & & **Mean** & **Std** & **Actual** \\ \hline gamma & 100 & 13 & 1 & 2 & 1 & 0.49 & 0.932 & 0.000 & \\  & 500 & 26 & 2 & 3 & 0.49 & 1 & 0.895 & 0.000 & \\  & 1000 & & & & & & 0.872 & 0.006 & \\ \hline gamma & 50 & 130 & 5 & 7 & 1 & \(-\)0.37 & 0.58 & 1.155 & 0.001 & \\  & 100 & 58 & 6 & 3 & \(-\)0.37 & 1 & \(-\)0.28 & 1.2117 & 0.001 & \\  & 500 & 150 & 2 & 8 & 0.58 & \(-\)0.28 & 1 & 1.1806 & 0.001 & \\  & 1000 & & & & & & & 1.2050 & 0.000 & \\ \hline Beta & 50 & & & & & & & 0.957 & 0.000 & \\  & 100 & 1 & 2 & 5 & 1 & 0.79 & 1.088 & 0.000 & \\  & 500 & 1 & 4 & 4 & 0.79 & 1 & 1.041 & 0.000 & \\  & 1000 & & & & & & & 1.102 & 0.001 & \\ \hline Weibull & 50 & 7 & 2 & 2 & 1 & 0.28 & 0.58 & 1.685 & 0.010 & \\  & 100 & 9 & 4 & 3 & 0.28 & 1 & 0.49 & 1.125 & 0.000 & \\  & 500 & 10 & 6 & 6 & 0.58 & 0.49 & 1 & 1.278 & 0.000 & \\  & 1000 & & & & & & & 1.284 & 0.000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of proposed multivariate process capability indices. The presented results in the previous section show that the proposed methodology has the ability to estimate multivariate PCIs precisely, since almost all the estimated multivariate PCIs are very close to the actual. Moreover, the proposed methodology in this research has been applied to bivariate samples and samples with three variables. Both types of samples showed high performance. Furthermore, the samples investigated in this research have been studied in the literature using different methodologies. In this regard, [PERSON] [41] estimated the multivariate PCIs of the same 16 samples used in this research. They presented a good estimation comparing to the actual PCIs. However, the results in this research outperform their results in most cases. Table 5 presents a comparison between our research results and results from previous research. Both studies revealed little variation from the actual PCIs. These variations are due to either overestimating or underestimating the actual PCIs. To present a clear comparison between the results from this research and the results from previous research, the mean absolute percentage error (MAPE) is computed for each sample for both approaches. Mean absolute percentage error (MAPE) is a measure of estimation accuracy in statistics. It usually expresses the accuracy as a ratio defined by the formula:  
\[MAPE=\frac{|APCI-EPCI|}{APCI}*100\% \tag{5}\]  
where APCI is the actual MPCI, and EPCI is the estimated MPCI. MAPE can be interpreted as the average percentage error of the MPCI estimation compared to the actual values. A lower MAPE indicates a better estimation and a higher accuracy. MAPE is commonly used in model evaluation because of its very intuitive interpretation in terms of relative error.