The proportion of conforming product (*C*) that gets rejected (*R*) by the specifications will be approximated by the volume under the bivariate model in the regions labeled *CR*. This is a misclassification that occurs due to measurement error.

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* X**Y**ICC*

And the intraclass correlation coefficient (*ICC*) defines that proportion of the variation in the stream of product measurements, *Var(X)*, that can be directly attributed to the variation in the product steam, *Var(Y)*:

With appropriate changes in the limits of integration, we can use the equation above to find volumes for the regions *CR*, NS*,* and *NR* for any given combination of the* P/T* and capability ratios.

The only way to avoid these misclassifications is to have a capability ratio large enough for the ellipse to fit within the square, or to have zero measurement error. So, if you depend upon inspection in order to ship conforming product, your target value for the *P/T *ratio has to be zero.

### Summary

Figure 3 shows the information of figure 2 in a different format and extends the coverage to lower capability ratios. Here each curve represents a fixed value for the capability ratio while the maximum process yields are shown on the vertical axis.

When a production process is not operated on-target, or when it is not operated predictably, it is not being operated up to its full potential. And when a process is not operated up to its full potential, then its actual yield will be less than the maximum process yield. Yet here the maximum process yield may still be used as a conservative estimate of the hypothetical process potential; it will give us an idea of what the process yield could be when operated up to its full potential.

So while none of these three ratios will tell the whole story by itself, any two of them may be used to determine the maximum process yield. Figure 2 shows how the *P/T* ratio and the capability ratio combine to determine the maximum process yield. The horizontal axis shows values of the *P/T* ratio ranging from 0.00 to 1.00, while the vertical axis shows capability ratios ranging from 0.50 to 1.20. The values in the graph are the maximum process yields, in parts per thousand. These are the number of conforming units shipped per thousand units produced.

The proportion of nonconforming product (*N*) that gets rejected (*R*) by the specifications will be approximated by the volume under the bivariate model in the regions labeled *NR.*

Figure 8 shows the minimum percentage of conforming product that will get rejected for a given capability and *P/T* ratio when the process is operated up to its full potential.

The maximum process yield depends upon three quantities: the specified tolerance; the variation in the measurement process, *SD(E)*; and the variation in the stream of product measurements, *SD(X)*.

Inspection places a premium on having a good measurement system. When a process is operated unpredictably or off target, the misclassifications will increase beyond the minimums given above. The costs of improving the measurement system, plus the costs of inspection, rework, and scrap, will lower productivity without increasing the quality of the product shipped. I have had clients who were so comfortable with inspection that they put on more inspectors, increased the size of the rework department, and slowly went out of business.

The regions labeled *CR* and *NS* represent misclassifications due to measurement error. *CR* represents conforming product that gets rejected, while *NS* represents nonconforming product that gets shipped. By integrating the bivariate normal distribution over the regions shown, we can estimate the *minimum* percentages for each type of misclassification. Figures 7 and 8 show these minimums as a function of the capability ratio and the *P/T* ratio.

So, the *P/T* ratio does not limit the maximum process yields. Neither does it characterize the relative usefulness of the measurements for keeping process behavior charts. But does it tell us anything about using the measurements to scrape the burnt toast?

### Inspection

*Var(X) = Var(Y) + Var(E) *= 1.0 + *Var(E)*

For the role of process improvement, we need a measurement system that allows us to detect those changes in our process that are large enough to be of economic impact. To see how a measurement system affects the ability of a process behavior chart to detect process changes, we will use the intraclass correlation coefficient.

So if you’re concerned with the value of your *P/T *ratio, you may not have progressed to the point of asking the right questions. The choice is yours. Are you going to continue to scrape the burnt toast, or are you going to learn how to stop burning the toast?

### Appendix: Finding the maximum process yields

For example, if your *P/T *ratio is 0.60 and your capability ratio is 0.80, your process will have a maximum process yield of about 980 *ppt*. Operating predictably and on-target will result in about 980 conforming units shipped per thousand units produced. Operating at less than full potential will result in a lower yield.

Published: Monday, September 4, 2023 – 12:03