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 1,000. These are the number of conforming units shipped per 1,000 units produced.

*P/T**ICC*

The following is provided for those interested in duplicating the results shown in the graphs. The relationship between the product values, *Y*, and the product measurements, *X*, can be modeled using a bivariate normal distribution as shown in figure 6. In this model, the minor axis of the ellipse is a function of the measurement error.

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.

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.*

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 1,000 units produced. Operating at less than full potential will result in a lower yield.

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)*.

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The flatness of the curves in figures 2 and 3 shows that the *P/T* ratio has very little impact upon maximum process yields. The yields are almost completely determined by the capability ratios. So, we don’t have to have a small *P/T *ratio in order to have a high maximum process yield. We can have very high yields even with a large *P/T* ratio. This means that knowing the *P/T *ratio doesn’t tell us anything useful about the maximum process yield.

### The usefulness of the measurement system

Figure 7 shows the minimum percentage of nonconforming product that will get shipped for a given capability and *P/T* ratio when the process is operated up to its full potential. (The bottom two curves are for capabilities of 0.80 and 0.90.)

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.

The product measurements, *X*, will consist of the product values, *Y*, plus measurement errors, *E*. The relationship between *X* and *Y* may be modeled with a bivariate normal distribution. The ellipse in figure 6 shows the three-sigma contour of a bivariate normal distribution having an *ICC* value of 0.80 and a capability ratio of 0.60.

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.

When the intraclass correlation is greater than 0.80, the measurement system is a first-class monitor for that production process. A process behavior chart using detection Rule 1 alone has better than a 99% chance of detecting a three-standard error shift.

The only way to maximize your process yield and minimize the costs of misclassifications is to operate your process predictably and on target. This requires that you use your measurements for process improvement. Fortunately, just about any old measurement process will serve to do this. Even second- and third-class monitors will work with process behavior charts. As you use these charts, you’ll discover assignable causes of exceptional variation. As these assignable causes are controlled, the process capability will increase, the process yields will increase, and the misclassifications will dwindle. This will reduce inspection, scrap, and rework costs, which will increase your productivity while improving product quality. I have clients who, when operating in this manner, have turned companies around, and others who have taken over markets by producing the best quality at the lowest cost.

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The region labeled *CS* represents the maximum process yield. As the capability ratio increases, the specifications get wider relative to the ellipse and more of the ellipse will fall within the specification square. As *Var(E)* gets smaller, the ellipse gets narrower and the minor axis shrinks.

Lean

## Using the Precision to Tolerance Ratio

### What does this ratio tell us?

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.

Three ratios are commonly used to compare the quantities on the corners of the triangle. The capability ratio, *Cp*, compares the specified tolerance with the (within subgroup) standard deviation of the product measurements:

When sorting nonconforming product from conforming product we will want perfect measurements. Any amount of measurement error will create the possibility of misclassifications where some conforming product gets rejected and some nonconforming product gets shipped. To illustrate how this happens, we return to the model for our measurements.

As we learned last month, the precision to tolerance ratio is a trigonometric function multiplied by a scalar constant. This means that it should never be interpreted as a proportion or percentage. Yet the simple *P/T* ratio is being used, and misunderstood, all over the world. So how can we properly make use of the *P/T* ratio?

The usual motivation behind the computation of the *P/T *ratio is a desire to determine if a measurement procedure is adequate for a given production process. And there are two ways that measurements support a production process: They can be used to scrape the burnt toast (inspection), or they can be used to learn how to stop burning the toast (process improvement). If the measurement system is adequate to allow us to improve the process, then we can often get to the point where we no longer have to depend upon inspection to ship conforming product. So, of these two ways that measurements support production, the role of process improvement is the more critical in the long term.

### Will the measurements support process improvement?

Thus, the complement of the *ICC* compares the variance of the measurement process, *Var(E)*, with the variance of the stream of product measurements, *Var(X)*.

* X**Y**ICC*

Using published tables of the power functions for process behavior charts, we can compute the probabilities of detecting various sized shifts for various values of the intraclass correlation. Here, we shall consider detecting a three-standard-error shift within 10 subgroups of when that shift occurs. Figure 4 shows two curves: The lower curve is for the use of detection rule one alone, while the upper is for the use of all four Western Electric zone tests. The interesting thing about both curves is how slowly they drop off as the intraclass correlation goes to zero.

Then, for a given *P/T* ratio and a given capability ratio, we may find the *ICC* value and compute the maximum process yield by integrating the resulting bivariate normal distribution with respect to both *X *and *Y *over the region labeled *CS* (i.e., between the specification limits).

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

When the intraclass correlation is between 0.50 and 0.20, the measurement system is a third-class monitor for that production process. Here, a process behavior chart using all four detection rules still has better than a 91% chance of detecting a three-standard error shift in the production process within 10 subgroups.

Published: Tuesday, September 5, 2023 – 12:03