lot sample plan with two measures

[fms95032]

I've run into a problem that maybe this forum could help me.

I have a product that we would like to create a lot sampling plan. There are two measures, both variables, that are destructive and the product can fail by failing either measure's specification. The measures are of droplet size and flow rate, and are roughly correlated yet have unique failure causes.

We would like to use a sample plan based on variables, yet I'm not sure how to adjust the plan to account for the two measures? For the same set of alpha, beta, AQL and LTPD values, how do I achieve the same protection when using two specifications with different measures?

Cheers,

Fred

[Stan Hilliard]

Hi Fred,

I will add my two cents worth to this and I encourage others to do the same.

I think that the type of solution to this problem depends on whether the two variables are correlated -- as yours are. If so, then to measure one variable is, in a sense, to measure the other -- at least partially.

If the variables are at least partially correlated:
For two variables plans applied to one lot, you could adjust the risks of the two individual sampling plans such that the combined oc curve for the lot is your goal -- rather than the oc curves of the two individual variables plans. Each lot's producer and consumers risks will then be correct.

The principle is that you can multiply the two variables plans' oc curves together to get the lot oc curve.

For example, at the consumers point, for a rejectable (RQL) lot to unfortunately pass, it has to pass two sampling plans -- one for each variable. If you want the lot consumers risk to be beta=Pa=0.05 at RQL, then each variable plan beta risk should be the square root of 0.05 -- so that when multiplied together the lot's probability will be 0.05. Beta=SQRT(0.05)=0.22. Then the probability that an RQL lot would pass both variables plans is 0.22 X 0.22 = 0.05

Similarly at the producer's point, for an acceptable (AQL) lot to be unfortunately rejected, it has to be rejected by either or both variables plans. That is equivalent to saying it won't pass both plans. So if you want the lot producer's risk to be alpha=(1-Pa)=0.05 at AQL, then Pa is the product of the two variables plan Pa's. So Each variable plan alpha risk should be 1-SQRT(Pa)=1-SQRT(0.95)=(1-0.9025)=0.0975. Then the probability that an AQL lot would pass both variables plans is (0.975 X 0.975)=0.95.

Once you calculate the adjusted alpha and beta values for the variables sampling plans, the program TP414 can use them to produce the adjusted sample size and acceptance values. It will also produce the oc curve table of the combination from the original lot alpha and beta.

The above method doesn't apply if the two variables are not correlated. To imagine an extreme case, what if a rejectable condition of variable 1 never happens on the same lot as a rejectable condition of variable 2 -- They are mutually exclusive. Then to catch both conditions you would need to use the full lot oc curve for both variables sampling plans.

[fms95032]

Hi Stan,

Thanks - I'll take a closer look and see how to create the plans according to your direction.

Do you know of any reference or source for this approach?
cheers,

Fred

[Stan Hilliard]

Hi Stan,
Thanks - I'll take a closer look and see how to create the plans according to your direction.
Do you know of any reference or source for this approach?
cheers,
Fred

I do not know of another example because the approach might be unique. But it is based on the laws of probability.

You multiply two probabilities together when you want to know the probability that both events will occur. For example, that the lot will be accepted by both the sampling plan for droplet size and the sampling plan for flow rate.

The acceptabilities of the two variables -- droplet size and flow rate --are correlated. So the applicable probability rule for the lot Pa is the rule for the probability of non-independent events. That rule calls for using conditional probabilities.

Link to Rules of Probability

The conditional probability requirement is met in this case because the Pa's of the oc curves are conditional probabilities -- the condition being the lot percent nonconforming for that variable -- on the horizontal axis of the oc curve.

The probabilities of acceptance of the two variables plans are forced to be dependent by the way your specification limits define the effect of the variables on lot percent nonconforming.

So in my prior example when I took the square root and when I multiplied two equal Pa's, I assumed that the two variables made equal contributions to acceptability or rejectability of a lot. -- this is in terms of the variable's contribution to percent nonconforming. The relationships can be further visualized by showing the corresponding lot means next to % nonconforming on the horizontal axes of the oc curves. My program TP414 can generate oc curves for the mean.