Estimating Process Capability

In the multistage processes, the process capability indices in each stage are affected by the capability of the processes in the previous stages. Therefore, in some cases the incapability of the process is the effect of incapability of its preceding processes. Hence, we attempt to estimate the process capability of each process independent of the capability of its previous processes. In our method, the specific process capability indices for each stage are calculated by measuring the process capability indices for the residuals. In the two-stage process, assumed that the parts in stage 1 feed into stage 2 and the quality characteristic in stage 1 is $x$ that follows a normal distribution $x \sim N\left(\mu_{x}, \sigma_{x}^{2}\right)$. The quality characteristic of the second stage is defined as:

$y_{i}=\beta_{0}+\beta_{1} x_{i}+\varepsilon_{i}$,                                                                                                                                                                      (2)

where $\varepsilon_{i}$ follows normal distribution $\varepsilon_{i} \sim N\left(0, \sigma_{\varepsilon}^{2}\right)$ and the parameters $\beta_{0}$ and $\beta_{1}$ re estimated based on the analysis on historical data using Eqs. (3) and (4)

$\hat{\beta}_{0}=\bar{y}-\hat{\beta}_{1} \bar{x}$                                                                                                                                                                                 (3)

$\hat{\beta}_{1}=\frac{\sum_{i=1}^{n} y_{i}\left(x_{i}-\bar{x}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}$                                                                                                                                                                           (4)

where $\bar{x}, \bar{y}$ are the mean values of the quality characteristics in the first and second stages, respectively.

The mean and variance of the quality characteristic in the second stage can be expressed as:

$\mu_{y}=\hat{\beta}_{0}+\hat{\beta}_{1} \mu_{x}$                                                                                                                                                                              (5)

$\sigma_{y}^{2}=\hat{\beta}_{1}^{2} \sigma_{x}^{2}+\sigma_{\varepsilon}^{2}$                                                                                                                                                                              (6)

The Eqs. (5) and (6) show that the mean and variance of the quality characteristic in the second stage are affected by the mean and variance of the quality characteristics in the first stage. Therefore, the process capability indices for the quality characteristic in the second stage are affected by the quality characteristic in the first stage and calculate the overall capability of that stage. In the multistage processes, the residuals have been used to omit the effects of the previous stages. The residuals are not affected by the previous processes. Therefore, the process capability indices for the residuals calculate the specific process capability of the process in the second stage. The residuals are obtained by:

$e_{i}=y_{i}-\hat{y}_{i}$                                                                                                                                                                                     (7)

where $\hat{y}_{i}$ is the prediction value for the quality characteristic $y_{i}$ and can be obtained as follows:

$\hat{y}_{i}=\hat{\beta}_{0}+\hat{\beta}_{1} x_{i}$                                                                                                                                                                              (8)

The specification limits are the allowable range of the quality characteristic that meet the customers' expectations. The lower and upper specification limits are $LSL$ and $USL$, respectively, and determined by design engineers or customers. The specification limits for the quality characteristics of the stages 1 and 2 are $\left[\operatorname{LSL}_{x} \mathrm{USL}_{x}\right]$ and $\left[\operatorname{LSL}_{y} \mathrm{USL}_{y}\right]$, respectively. To estimate the process capability of the second stage based on the residuals, we require the specification limits for the residuals. We used the specification limits of the quality characteristics in the first and second stages to obtain the specification limits for residuals. In our method, the specification limits for the residuals are determined using the process yield. The process yield for the quality characteristic $x$ can be obtained by:

$P_{x}=\operatorname{pr}\left(\mathrm{LSL}_{x} \leq x \leq \mathrm{USL}_{x}\right)=\Phi\left(\frac{\mathrm{USL}_{x}-\mu_{x}}{\sigma_{x}}\right)-\Phi\left(\frac{\mathrm{LSL}_{x}-\mu_{x}}{\sigma_{x}}\right)$                                                                                              (9)

where $\Phi(\cdot)$ is the cumulative density function of the standard normal distribution.

Similarly, the process yields of the quality characteristic $y$ can be computed. The process yield of the quality characteristic $y$ determines the process yield of the second stage when it is affected by the quality characteristic in the first stage. Since the residual and the quality characteristic $x$ are independent, the process yield of the quality characteristic $y$ can be calculated by Eq. (10) in the case that the specification limits of the residuals are known.

$P_{y}=P_{x} P_{e}-\left[1-\operatorname{pr}\left(\mathrm{LSL}_{y} \leq \beta_{0}+\beta_{1} x+\varepsilon \leq \mathrm{USL}_{y} \mid \mathrm{LSL}_{x} \leq x \leq \mathrm{USL}_{x}, \mathrm{LSL}_{e} \leq\right.\right. \left.\left.\leq \varepsilon \leq \mathrm{USL}_{e}\right)\right]$                              (10)

where $P_{x}$ and $P_{e}$ are the process yields of the quality characteristic $x$ and the residuals, respectively. The process yield of the residuals is calculated by Eq. (9).

In this paper, the specification limits of the residuals are determined under the following assumptions:

1. The mean values of the quality characteristics in the first and second stages are equal to target values.

2. The term $\operatorname{pr}\left(\mathrm{LSL}_{y} \leq \beta_{0}+\beta_{1} x+\varepsilon \leq \mathrm{USL}_{y} \mid \mathrm{LSL}_{x} \leq x \leq \mathrm{USL}_{x}, \mathrm{LSL}_{e} \leq \varepsilon \leq \mathrm{USL}_{e}\right)$ is equal to 1 when the process yields of the quality characteristic in the first stage and the residual are equal to 0.9973. Therefore, $P_{y}$ can be obtained by:

$P_{y}=P_{x} P_{e}$                                                                                                                                                                                      (11)

The second assumption represents that the quality characteristic in the second stage should be located between its specification limits if the residuals and the quality characteristic in the first stage lie between their corresponding specification limits. The proposed method for determining the specification limits of the residuals consists of two steps. The first step is calculating the variance of the residuals under the assumption that the mean of the residuals is equal to the target value. The second step is determining the specification limits of the residuals.

The process capability indices $C_{p}$, $C_{pk}$ and $S_{pk}$for the residuals can be obtained using Eqs. (12), (13) and (14), respectively

$C_{\mathrm{pe}}=\frac{\mathrm{USL}_{e}-\mathrm{LSL}_{e}}{6 \sigma_{e}}$,                                                                                                                                                                           (12)

$C_{\mathrm{pke}}=\min \left\{\frac{\mathrm{USL}_{e}-\mu_{e}}{3 \sigma_{e}}, \frac{\mu_{e}-\mathrm{LSL}_{e}}{3 \sigma_{e}}\right\}$,                                                                                                                                                 (13)

$S_{\mathrm{pke}}=\frac{1}{3} \Phi^{-1}\left\{\frac{1}{2} \Phi\left(\frac{\mathrm{USL}_{e}-\mu_{e}}{\sigma_{e}}\right)+\frac{1}{2} \Phi\left(\frac{\mu_{e}-\mathrm{LSL}_{e}}{\sigma_{e}}\right)\right\}$,                                                                                                                   (14)

where, $USL_{e}$ and $LSL_{e}$ are the specification limits of the residuals, $\mu_{e}$ and $\sigma_{e}$ are the mean and standard deviation of the residuals, respectively. The variance of the residuals is calculated by

$\sigma_{e}^{2}=\frac{\sum_{i=1}^{n}\left(Y_{i}-\hat{Y}_{i}\right)}{n-2}=\frac{\sum_{i=1}^{n} e_{i}^{2}}{n-2}$                                                                                                                                                         (15)

Similarly, $C_{p}$, $C_{pk}$and $S_{pk}$ indices can be calculated for the quality characteristics $x$ and $y$.

Variance of the residuals

The standard deviation values of the quality characteristics $x$ and $y$ are determined under the assumption that the corresponding mean values are equal to target values and the process yields of the quality characteristic in the first stage and the residuals are 0.9973. Therefore, the process yield of the quality characteristic in the second stage is calculated by Eq. (11). The standard deviation values of the quality characteristics are obtained by Eqs. (16) and (17), respectively

$\sigma_{x}=\frac{\mathrm{USL}_{x}-T_{x}}{\Phi^{-1}(0.99865)}=\frac{\mathrm{LSL}_{x}-T_{x}}{\Phi^{-1}(0.00135)}$.                                                                                                                                                    (16)

$\sigma_{y}=\frac{\mathrm{USL}_{y}-T_{y}}{\Phi^{-1}(0.9973)}=\frac{\mathrm{LSL}_{y}-T_{y}}{\Phi^{-1}(0.0027)}$.                                                                                                                                                       (17)

The variance of the residuals based on the standard deviations of the quality characteristics $x$ and $y$ that are calculated by Eqs. (16) and (17), respectively, and obtained as follows:

$\sigma_{e}^{2}=\sqrt{\sigma_{y}^{2}-\hat{\beta}_{1}^{2} \sigma_{x}^{2}}=\sqrt{\left(\frac{\mathrm{USL}_{y}-\mathrm{LSL}_{y}}{\Phi^{-1}(0.9973)}\right)^{2}-\hat{\beta}_{1}^{2}\left(\frac{\mathrm{USL}_{x}-\mathrm{LSL}_{x}}{\Phi^{-1}(0.99865)}\right)^{2}}$                                                                                                   (18)

The specification limits for the residuals

Under the assumptions that the mean of the residuals is equal to the target value and the process yield of the residuals is 0.9973, the specification limits of the residuals are obtained as follows:

$\mathrm{USL}_{e}=\sigma_{e} \Phi^{-1}(0.99865)$                                                                                                                                                              (19)

$\mathrm{USL}_{e}=\sigma_{e} \Phi^{-1}(0.00135)$                                                                                                                                                              (20)

The target value for the residuals is zero.