Using the Precision to Tolerance Ratio

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).


Figure 5:
P/T does not determine ICC

The precision to tolerance ratio compares the standard deviation of the measurement process with the specified tolerance.

To assess the adequacy of the measurement process for the role of process improvement, we use the maximum process yield. This is that fraction of the product produced that is both conforming and gets shipped when the production process is operated predictably and on target.

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.

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.

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

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.

 


Figure 1: The maximum process yield depends upon three quantities

When the ICC falls below 0.20, the measurements contain very little information about the product. These fourth-class monitors are essentially useless for process improvement.

Mean(X) = Mean(Y) + Mean(E) = 0

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Figure 6: Relationship between X and Y when ICC = 0.80

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.

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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 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.

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

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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.

So that the ratio of SD(E) to SD(X) will be given by:

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 ten subgroups.


Figure 2: How maximum process yields change with P/T and capability

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

 Without loss of generality we may assume:


Figure 8: Minimum percentages of conforming product rejected

When we use probability models to characterize what to expect in practice, we should round off the results to parts per thousand. No probability model will ever describe reality out to parts per ten-thousand or beyond. While we may compute values out to more decimal places, and while the extra digits may be necessary for precise computations, these extra digits are meaningless when it comes to describing finite processes.