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Acceptance Sampling Tutorial

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BASICS OF ACCEPTANCE SAMPLING www.SixSigmaTutorial.com

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Acceptance Sampling Tutorial

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OP ER ATI NG C HAR AC TER IS TIC ( OC) C UR VE
The OC curve quantifies the α and β risks of an attribute sampling plan. Below is an ideal OC curve (the bold line) for asituation in which we might want to accept all lots that are, say, ≤ 1% defective and reject all lots that are > 1% defective:

1.0

Pa

.5

0 0 1 2 3
Lot % Defective With this ideal (no risks) curve, all batches with < 1% defective incoming quality level would have a probabilit y of acceptance (P a ) of 1.0. And, all lots with > 1% defective would have a Pa of 0. The Pa is the probabilitythat the sampling plan will accept the lot; it is the long-run % of submitted lots that would be accepted when many lots of a stated quality level are submitted for inspection. It is the probability of accepting lots from a steady stream of product having a fraction defective p.

TYP IC AL OC CUR VE
Since there will always be some risks, a more typical looking OC curve looks more like the onebelow. It is based on the Poisson distribution* (with the defective rate < 10% and n is relatively large compared to N).

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Acceptance Sampling Tutorial

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1.0 .9

Pa

.5

.1 0 0 AQL 1 2 3 4 5 6 RQL Lot % Def ect ive

The AQL (Acceptance Quality Level), the maximum % defective that can b e consid ered satis fact or y as a pro cess av erage for samplinginspection, here is 1%. Its corresponding Pa is about 89%. It should normally be at least that high. The RQL (Rejectable Quality Level) is the % defective, here at 5%, that is associat ed with the establis hed β risk (which is u s u a l l y s t a n d a r d i z e d a t 1 0 % ) . It i s a l s o k n o w n a s t h e Lo t Tolerance Percent Defective (LTPD). *The hypergeometric and binomial distns arealso used. The alpha risk is the probability of rejecting relatively good lots (at AQL). The beta risk is the probability of accepting relatively bad lots (at LTPD/RQL). It is the probability of accepting product of some stated undesirable quality; it is the value of Pa at that stated quality level. The OC curves are a means of quantifying alpha and beta risks for a given attribute sampling plan. ThePa value obtained assumes that the distribution of defectives among a lot is random – either the underlying process is in control, or the product was well mixed before being divided into lots. The samples must be selected randomly from the entire lot. The alpha risk is 1 – Pa. The shape of the OC curves is affected by the sample size (n) and accept number (c) parameters. Increasing both theaccept number and sample size will bring the curve closer to the ideal shape, with better discrimination.

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Acceptance Sampling Tutorial

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1.0 .9
n = 50 c=5

Pa
.5
n = 20 c=0 n = 10 c=2

.1 0 0 1 2 3 4 5 6 Lot % Def ect ive
If the lot size N changes, the above curves change very little. However, the curves will change quite a bit as sample size n changes.So, basing a sampling plan on a fixed percentage sample size will yield greatly different risks. For consistent risk levels, it is better to fix the sample size at n, even if the lot sizes N vary. If n = 10 & c = 2, what is the alpha risk for a vendor running at p = .02? Pa is about .55, so alpha is about .45. What is the beta risk if the worst-case quality the customer will accept is 3%? (about15%). To lower alpha & beta, you can increase n & c.

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Acceptance Sampling Tutorial

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Is c = 0 the best plan for the producer and the consumer?

1.0 .9 .8 .7

Pa

.6 .5 .4 .3 .2 .1 0 0 .5 (2) N = 500

(1) N = 500

n = 150 c = 1

n = 100 c = 0

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1.5

2.0

2.5

3.0

3.5

4.0

Lot % De fe ctiv e
At the 2.8% lot defect rate, both...