Estimating Process Capability

In this section, simulation studies are carried out to check the accuracy of the proposed method to determine the specification limits of the residuals and the process capability indices calculated based on them. We considered the two-stage process, the quality characteristic for the stage 1 is $x$ and generated from a normal distribution as $x \sim N\left(\mu_{x}, \sigma_{x}\right)$ and the quality characteristic for the second stage is $y$ that is generated from the following equation:

$y_{i}=7.86+0.5 x_{i}+\varepsilon_{i}$

where $\varepsilon_{i}$ follows a normal distribution as $N\left(\mu_{e}, \sigma_{e}\right)$.

The specification limits for the quality characteristics $x$ and $y$ are set to [6.56, 19.73] and [9.46, 19.32], respectively. We assume that the process yields for the residuals and the quality characteristic $x$ are 0.9973. Then, the process yield of the quality characteristic $y$ is calculated from Eq. (11). The standard deviation values of the quality characteristics $x$ and $y$ are calculated as

$\sigma_{x}=\frac{19.73-13.145}{\Phi^{-1}(0.99865)}=\frac{19.73-13.145}{2.99998}=2.195$

$\sigma_{y}=\frac{19.32-14.39}{\Phi^{-1}(0.9973)}=\frac{19.32-14.39}{2.78215}=1.772$

The specification limits of the residuals are obtained by Eqs. (19) and (20). The lower and upper specification limits of the residuals are obtained as −4.1736 and 4.1736, respectively.

We calculate the $C_{\mathrm{p}}$, $C_{\mathrm{pk}}$, and $S_{\mathrm{pk}}$ and the process yields for the residuals and the quality characteristics $x$ and $y$ under different sample sizes, mean and variance values of the residuals and the quality characteristic $x$. For each simulated case, the true values of the process capability indices are calculated. The true values of the process capability indices are calculated based on the true values of the mean and variance of the quality characteristics in stage 1, stage 2 and the residuals as shown in Tables 1, 2, and 3. The process capability indices for each case are calculated for different sample sizes $n = 25, 50, 100, 200$. The mean values and variances of the quality characteristics in the first and second stages and the residuals are estimated from the random samples and then the process capability indices are calculated. These simulations were repeated 10,000 times and the process capability indices and process yields were calculated in each replicate. The mean and the true values of the $C_{\mathrm{p}}$, $C_{\mathrm{pk}}$, and $S_{\mathrm{pk}}$ indices for different cases under different sample sizes are shown in Tables 1, 2 and 3, respectively.

Table 1 $C_{\mathrm{p}}$ index for the residuals and the quality characteristics $x$ and $y$ under different sample sizes and simulated case

Table 2 $C_{\mathrm{pk}}$ index for the residuals and the quality characteristics $x$ and $y$ under different sample sizes and simulated case

Table 3 $S_{\mathrm{pk}}$ index for the residuals and the quality characteristics $x$ and $y$ under different sample sizes and simulated case

Tables 1, 2 and 3 show that when the sample size increases the mean values of process capability indices approach to the corresponding true values. Tables 1, 2 and 3 show the result of using the proposed method for different simulation cases. In the first two-stage process, the processes in the first and second stages are capable. In this condition, the process capability indices for residuals is greater than 1, therefore, the process in the second stage is capable. In the second process, the variance of the quality characteristic in the first stage is large and the process in the first stage is not capable and it affects the capability of the process in the second stage and makes it incapable. While the process capability indices for the residuals are greater than 1. It shows that the incapability of the process in the second stage is the effects of the incapability of the process in the first stage. In the third case, the process in the first stage is capable but the process in the second stage is not capable. The process capability indices for the residuals show that the process in the second stage is not capable. In the fourth process, the deviation of the mean value of the process in the first stage from the target value is large, therefore, the $C_{\mathrm{pk}}$ index for the $x$ and $y$ quality characteristics show that the process in the first stage and the process in the second stage are not capable. Moreover, $C_{\mathrm{pk}}$ index for the residuals are greater than 1. The other simulation cases can be interpreted similarly. Table 4 shows the accuracy of the second assumption.

Table 4 Process yield of the residuals and the quality characteristics $x$ and $y$ under different sample sizes and simulated case

Table 4 shows that the process yield of the quality characteristic $y$, that is obtained using Eq. (11), accurately estimates the true value of the process yield of the quality characteristic $y$ properly when the process in the first stage and the residuals are capable. When the sample size increases, the gap between the estimated value of the process yield of the quality characteristic $y$ decreases for all cases.