## 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?

### Simulation studies

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

Simulated case $n$ True value of
$C_{px}$
Mean of $C_{px}$ True value of $C_{px}$ Mean of $C_{pe}$ True value of $C_{px}$ Mean of $C_{py}$
$\mu_{x}$ $\sigma_{x}^{2}$ $\mu_{e}$ $\sigma_{e}^{2}$
13.6 4 0 1.21 25 1.0975 1.1345 1.2647 1.2499 1.1054 1.1440
50 1.1155 1.2564 1.1221
100 1.1063 1.2621 1.1152
200 1.1019 1.2632 1.1100
13.6 6.76 0 1.21 25 0.8442 0.8721 1.2647 1.2492 0.9650 0.9953
50 0.8568 1.2594 0.9787
100 0.8503 1.2623 0.9724
200 0.8466 1.2632 0.9684
13.6 4 0 2.25 25 1.0975 1.1340 0.9272 0.9192 0.91160.9428
50 1.1140 0.9240 0.9261
100 1.1062 0.9255 0.9190
200 1.1018 0.9262 0.9149
14.42 4 0 1.21 25 1.0975 1.1342 1.2647 1.2513 1.1054 1.1404
50 1.1141 1.2581 1.1213
100 1.1065 1.2609 1.1137
200 1.1026 1.2631 1.1107
14.05 4.84 0 1.21 25 0.9977 1.0306 1.2647 1.2526 1.0564 1.0919
50 1.0126 1.2581 1.0722
100 1.0058 1.2615 1.0652
200 1.0014 1.2639 1.0603
13.6 4.84 0 1.96 25 0.9977 1.0319 0.9935 0.9819 0.9230 0.9521
50 1.0134 0.9889 0.9366
100 1.0060 0.9917 0.9299
200 1.0012 0.9925 0.9271

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

Simulated case $n$ True value of $C_{pkx}$ Mean of $C_{pkx}$ True value of $C_{pke}$ Mean of $C_{pke}$ True value of $C_{pky}$ Mean of $C_{py}$
$\mu_{x}$ $\sigma_{x}^{2}$ $\mu_{e}$ $\sigma_{e}^{2}$
13.6 4 0 1.21 25 1.0217 1.0467 1.2647 1.1976 1.0449 1.0674
50 1.0363 1.2193 1.0560
100 1.0295 1.2358 1.0536
200 1.0258 1.2446 1.0494
13.6 6.76 0 1.21 25 0.7859 0.7978 1.2647 1.1976 0.9121 0.9234
50 0.7925 1.2222 0.9184
100 0.7907 1.2360 0.9174
200 0.7878 1.2445 0.9150
13.6 4 0 2.25 25 1.0217 1.0472 0.9272 0.8676 0.8616 0.8735
50 1.0340 0.8872 0.8684
100 1.0291 0.8992 0.8666
200 1.0256 0.9262 0.8644
14.42 4 0 1.21 25 0.8850 0.9146 1.2647 1.1996 0.9530 0.9831
50 0.8995 1.2211 0.9674
100 0.8917 1.2344 0.9596
200 0.8895 1.2441 0.9579
14.05 4.84 0 1.21 25 0.8606 0.8886 1.2647 1.2011 0.9503 0.9796
50 0.8729 1.2204 0.9636
100 0.8674 1.2349 0.9578
200 0.8643 1.2454 0.9544
13.6 4.84 0 1.96 25 0.9288 0.9492 0.9935 0.9305 0.8724 0.9492
50 0.9395 0.9517 0.8774
100 0.9355 0.9654 0.8771
200 0.9321 0.9738 0.8761

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

Simulated case $n$ True value of $S_{pkx}$ Mean of $S_{pkx}$ True value of $S_{pke}$ Mean of $S_{pke}$ True value of $S_{pky}$ Mean of $S_{pky}$
$\mu_{x}$ $\sigma_{x}^{2}$ $\mu_{e}$ $\sigma_{e}^{2}$
13.6 4 0 1.21 25 1.0717 1.0922 1.2647 1.2305 1.0884 1.1093
50 1.0819 1.2457 1.0962
100 1.0766 1.2563 1.0940
200 1.0742 1.2602 1.0909
13.6 6.76 0 1.21 25 0.8318 0.8447 1.2647 1.2302 0.9533 0.9666
50 0.8367 1.2486 0.9585
100 0.8343 1.2566 0.9565
200 0.8322 1.2602 0.9546
13.6 4 0 2.25 25 1.0717 1.0926 0.9275 0.9043 0.9017 0.9168
50 1.0799 0.9158 0.9081
100 1.0763 0.9212 0.9049
200 1.0740 0.9240 0.9029
14.42 4 0 1.21 25 0.9592 0.9860 1.2647 1.2322 1.0203 1.0453
50 0.9723 1.2473 1.0322
100 0.9653 1.2550 1.0258
200 0.9633 1.2600 1.0246
14.05 4.84 0 1.21 25 0.9305 0.9529 1.2647 1.2825 1.0057 1.0330
50 0.9409 1.2726 1.0214
100 0.9351 1.2682 1.0161
200 0.9332 1.2667 1.0134
13.6 4.84 0 1.96 25 0.9778 0.9954 0.9934 0.9663 0.9127 0.9252
50 0.9853 0.9800 0.9177
100 0.9819 0.9871 0.9155
200 0.9793 0.9901 0.9147

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

Simulated case $n$ $\bar{P}_{x}$ $\bar{P}_{e}$ $\bar{P}_{x} \cdot \bar{P}_{e}$ Mean of $P_{y}$
$\mu_{x}$ $\sigma_{x}^{2}$ $\mu_{e}$ $\sigma_{e}^{2}$
13.6 4 0 1.21 25 0.9974 0.9991 0.9965 0.9977
50 0.9981 0.9996 0.9976 0.9983
100 0.9984 0.9997 0.9981 0.9986
200 0.9985 0.9998 0.9983 0.9988
13.6 6.76 0 1.21 25 0.9833 0.9991 0.9825 0.9932
50 0.9851 0.9996 0.9844 0.9947
100 0.9863 0.9997 0.9860 0.9951
200 0.9868 0.9998 0.9866 0.9954
13.6 4 0 2.25 25 0.9974 0.9890 0.9864 0.9901
50 0.9980 0.9920 0.9900 0.9914
100 0.9984 0.9933 0.9917 0.9923
200 0.9985 0.9940 0.9925 0.9927
14.42 4 0 1.21 25 0.9940 0.9992 0.9932 0.9963
50 0.9950 0.9996 0.9946 0.9970
100 0.9955 0.9997 0.9952 0.9974
200 0.9958 0.9998 0.9956 0.9976
14.05 4.84 0 1.21 25 0.9924 0.9992 0.9916 0.9958
50 0.9934 0.9996 0.9930 0.9967
100 0.9941 0.9997 0.9939 0.9972
200 0.9945 0.9998 0.9943 0.9974
13.6 4.84 0 1.96 25 0.9944 0.9932 0.9876 0.9907
50 0.9955 0.9953 0.9908 0.9922
100 0.9961 0.9963 0.9924 0.9930
200 0.9964 0.9967 0.9931 0.9934

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.