From: Stan Hilliard
Date: 01 Aug 1999
Time: 22:05:05
Greetings John,
I do not know of a literature reference, but here is how to calculate the overall consumer risk by combining the consumer risks of the two individual plans. I am assuming that you will take the second sample if, and only if, the first one rejects.
In what follows, I use the simple addition and multiplication laws to calculate the overall consumer risk. Almost all introductory level statistics textbooks cover these laws.
There are two ways that a lot can be accepted under this scheme:
(1) Accept with the first sample (probability=Pa1) and don't take a second sample, or (2) Reject with the first sample (probability=Pr1) and accept with the second sample (probability=Pa2).
The probability that the first way of accepting (1) will occur is Pa1.
The probability that the second way of accepting (2) will occur is, by the multiplication law, equal to Pr1*Pa2. (where *=multiply)
The probability of accepting the lot is calculated by using the addition law to combine the probabilities of the two ways that acceptance.
Pa(lot) = Pa1 + Pr1*Pa2 , which becomes,
Pa(lot) = Pa1 + (1-Pa1)*Pa2 , which becomes,
Pa(lot) = Pa1 + Pa2 - Pa1*Pa2
You can use this method to calculate the consumer's risk (beta=Pa(lot)) at the consumer's point, or you can generate the complete oc-curve for the lot by calculating Pa(lot) at a sufficient number of lot percent defective points.
If you want to quickly evaluate the oc-curves of many combinations, the software program TP105 can write the oc curve columns to a file, which can then be imported into a spreadsheet to let it do the calculations. (To learn more about TP105, go to www.samplingplans.com, click "word search" and search for the word "TP105" - without the quotes.)
If this method doesn't bear fruit, you might try generating a sequential sampling plan having the oc-curve that you want, and then split it into two sample-size groups. Forming groups does not substantially change the oc-curve of a sequential, but it might increase the average sample number some because of overshoot. (To learn more about sequential sampling, go to www.samplingplans.com, click "word search" and search for the word "sequential" - without the quotes.)
Sincerely, Stan Hilliard