## Estimating Process Capability

Read this paper. The main topic is to estimate the capability of a particular process. How would you define a process capability?

### A real case

In this section, the proposed method is used to calculate the specific capability of the process in the second stage of the automobile brake system measurement process. This case was discussed by Constable et al.and Linn et al. In this case, the quality characteristics for the first and second stages are ROLLWT and BAKEWT, respectively. The sample size is 70 and the first 45 observations were used to estimate the parameters $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$ using Eqs. (2) and (3), respectively. Based on the parameters $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$, the relationship between the quality characteristics in the first and second stage is defined as:

$\hat{y}=104.325+0.460456 x$

where, $x=\text { RollWT }$ and $y=\text { BAKEWT }$

The specification limits for the quality characteristics in the first and second stages are [202.38, 218.10] and [194.91, 207.35], respectively. The mean and standard deviation values for the residuals and the quality characteristics in the first and second stages based on the last 25 observations are shown in Table 5.

Table 5 The mean and standard deviation values for the residuals and the quality characteristics in the first and second stages

Residuals Quality characteristic ROLLWT Quality characteristic BAKEWT
Mean −0.0003 210.24 201.131
Standard deviation 1.685 2.618524 2.072036

We assumed that the process yields of the residuals and the quality characteristic $x$ are 0.9973. Then, the process yield of the quality characteristic $y$ is calculated by Eq. (11). The lower and upper specification limits of the residuals were obtained using Eqs. (19) and (20), respectively. The specification limits of the residuals are [−5.67, 5.67]. The process capability indices $C_{\mathrm{p}}$, $C_{\mathrm{pk}}$ and $S_{\mathrm{pk}}$ for the residuals and the quality characteristics in the first and second stages are shown in Table 6.

Table 6 The process capability indices $C_{\mathrm{p}}$, $C_{\mathrm{pk}}$ and $S_{\mathrm{pk}}$ for the residuals and the quality characteristics in the first and second stages

Residuals Quality characteristic ROLLWT Quality characteristic BAKEWT
$C_{\mathrm{p}}$ 1.1217 1.0006 1.0006
$C_{\mathrm{pk}}$ 1.1216 1.0006 1.0005
$S_{\mathrm{pk}}$ 1.1168 1.0006 1.0006

Table 6 shows that the processes in the first stage and whole process are capable. The process capability indices for the residuals show the specific capability of the process in the second stage. It shows that the process in the first stage is less capable than the process in the second stage. This is an import result, as the manufacturer knows which process should be invested and improved.