Get Email UpdatesOperating Characteristic Curve or The OC Curve
The OC curve quantifies the α and β risks of an attribute sampling plan. Below is an ideal OC curve (the bold line) for a situation in which we might want to accept all lots that are, say, ≤ 1% defective and reject all lots that are > 1% defective:
With this ideal (no risks) curve, all batches with ≤ 1% defective incoming quality level would have a probability of acceptance (Pa) of 1.0. And, all lots with > 1% defective would have a Pa of 0. The Pa is the probability that 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.
Typical OC Curve
Since there will always be some risks, a more typical looking OC curve looks more like the one listed in the next page. It is based on the Poisson distribution* (with the defective rate < 10% and n is relatively large compared to N).
Acceptance Quality Level (AQL), Rejectable Quality Level (RQL) and Lot Tolerance Percent Defective (LTPD)
AQL – Acceptance Quality Level
The AQL (Acceptance Quality Level), the maximum % defective that can be considered satisfactory as a process average for sampling inspection, here is 1%. Its corresponding Pa is about 89%. It should normally be at least that high.
RQL – Rejectable Quality Level
The RQL (Rejectable Quality Level) is the % defective, here at 5%, that is associated with the established β risk (which is usually standardized at 10%). It is also known as the Lot Tolerance Percent Defective (LTPD).
LTPD – Lot Tolerance Percent Defective
The LTPD of a sampling plan is a level of quality routinely rejected by the sampling plan. It is generally defined as that level of quality (percent defective, defects per hundred units, etc.) which the sampling plan will accept 10% of the time.
* The hyper geometric and binomial distance are also 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. The Pa 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 the accept number and sample size will bring the curve closer to the ideal shape, with better discrimination.
Sampling Plan Variation vs Lot Size Variation in Acceptance Sampling
If the lot size N changes, the below 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. Question: If n = 10 & c = 2, what is the alpha risk for a vendor running at p = .02? Answer: Pa is about .55, so alpha is about .45. Question: What is the beta risk if the worst-case quality the customer will accept is 3%? Answer: (about 15%). To lower alpha and beta, you can increase n and c.
Is c = 0 the best plan for the producer and the consumer?
At the 2.8% lot defect rate, both plans give the producer equal protection: Pa = 11%, or Prej = 89%. Which one gives better protection against rejecting relatively good lots, e.g., at the .5% lot defect rate, and why?
For (1), α = about 8% and for (2), α = about 30%.
(1) has a lower α error so less chance of rejecting good lots. With (2), you will reject any lot of 500 if there is even 1 defect in the sample, but it will lead to higher costs.
Increasing Protection from Rejecting Good Lots in Acceptance Sampling
Is c = 0 the best plan for the producer and the consumer?
At the 2.8% lot defect rate, both plans give the producer equal protection: Pa = 11%, or Prej. = 89%. Which one gives better protection against rejecting relatively good lots, e.g., at the .5% lot defect rate, and why? For (1), α = about 8%; for (2), α = about 30%. (1) has a lower α error so less chance of rejecting good lots. With (2), you will reject any lot of 500 if there is even 1 defect in the sample, but it will lead to higher costs.
Discrimination in Acceptance Sampling Plans
Discrimination is the ability of a sampling plan to distinguish between relatively good levels of Quality and relatively bad levels of quality. In other words, having
- A high Pa (e.g., 95%, 1-α) associated with a good level of quality P1 (e.g., .5% or better)
- A low Pa (e.g., 10%, β) associated with a bad level of quality P2 (e.g., 3% or worse)
The Operating Ratio is defined as
R = P2/P1 = Pβ/P1-αExample: R = .03/.005 = 6.0
Designing your own single acceptance sampling plan
Derive a plan that comes as close as possible to satisfying two points on the OC curve. The two points are (P1, 1-α) and (P2, β). The derived plan will contain an n and a c value.
Example
Desired α risk of .05 for a P1 of .005, along with a desired risk of .05 for a P2 of .03.
1. Determine R:
R = P2/P1 = .030/.005 = 6.0
2. Enter the Values of Operating Ratio Table with α and β and find the closest R to the calculated value in step 1.
For α = .05 and β = .05, the closest table value is 5.67. This is acceptable since it is slightly more discriminating than 6.0. Note the c value of 3 in the far left column.
3. Obtain the nP1 value in the far right column. Then calculate n from:
n = nP1/P1 = 1.366/.005 = 273.2 or 274.
The acceptance sampling plan is n = 274, c = 3.
AQL Based Sampling Plans
- ANSI/ASQ Z1.4-1993 (MIL-STD 105E was withdrawn in February 1995) sampling plans are based on the use of AQL – the percent defective that is considered acceptable as a process average for the purposes of acceptance sampling.
- With these plans, it is not necessary to assume the provision of 100% screening (with replacement of all defective units) of all rejected lots.
- To enter any of these tables, you must first decide on the AQL to use (and determine the sample size code letter); from the table you will get the acceptance number (Ac) and the rejection number (Re) for the plan.
- There are also separate tables to provide the AQL for different values of AQL and sample code letters
- More serious defects should have a lower AQL as the acceptance criterion, and less serious defects can have higher AQLs – using a Classification of Defects system.
- Tightened inspection should be used whenever the quality history is unsatisfactory or when there are other good reasons for being suspicious about quality – keeping the beta risk down.
- Reduced inspection can be used when the quality history is shown to be good enough through Normal inspection.
- These plans are generally chosen to protect the producer under normal conditions, i.e., not to reject submitted lots that are at the AQL or better when there are no reasons to be suspicious about quality.
Double Acceptance Sampling Plans
• There is usually less sampling than for a single sampling plan
• The OC curve is better than the c = 0 curve for the single sampling plan with a smaller R (for better discrimination)
Designing your own double acceptance sampling plan
Exercise
Desired α risk of .05 for a P1 of .008, along with a desired risk of .10 for a P2 of .06. Use Table 4-4.
1. Determine R:
R = P2/P1 = .060/.008= 7.5
2. Enter Duncan’s Double Sampling Tables and find the closest R to the calculated value in step 1.
The closest table value is 7.54 in Plan Number 2. This is very close to 7.5. Note the c1 value of 1 and the c2 value of 2.
3. For Pa1-α = .95, obtain the nP1 (Pn1 in the table) value. Then calculate n from:
n = nP2/P2 = 0.52/0.008 = 65.
The acceptance sampling plan is n1 = 65, c1 = 1; n2 = 65, c2 = 2.
Variables Sampling Plans
ANSI/ASQ Z1.9 (MIL-STD-414 is withdrawn) sampling plans are based on the use of variable data (from an assumed normal distribution).
Actual measurements are made on the samples : the sample data is used to calculate a statistic, such as X̄, R, or S and then the calculated statistic is compared to the critical value from a table.
Acceptance criteria must be applied separately to each quality characteristic (vs. overall lot accept vs. reject for attribute sampling), so it’s more expensive than attribute sampling for larger lots, so it’s generally best to use only on key characteristics, with attribute sampling on the rest.
Compared to attribute plans, these plans, for the same n, provide a greater quality protection in judging a single quality characteristic, or for the same amount of risk, a smaller n is OK.
The use of variable data can provide more information about the extent of nonconformity – How? (Hint: think frequency distribution and what you can do with this information).
These sampling procedures are based on the assumption that the quality characteristic is normally distributed (it is possible to use data transformation if it is not). Procedure:
To use any of these plans, you must first decide on the:
- Inspection level (II is the default for Z1.9)
- Method to use – S or R, with variability (population sigma) known or unknown (see Decision Tree)
- AQL
- Lot size
Determine the sample-size code letter from the table
Calculate the Q value
Enter the Master Table by sample size code letter and AQL, and look up the k value.
Compare the k and Q values – if the calculated Q value is ? than the critical k value from the table, accept the lot. If not, reject it.
Decision Tree for Selecting Type of Variables Sampling Plan
Determine the AQL to use in the Master Table: Since there are standard AQLs used in the Master Tables, you need to convert the AQL per table below:
Determine the Sample-Size Code Letter to use in the Master Table: ANSI/ASQ Z1.9 Table A.2 Formulas for the Q Value:
Formulas for the Q Value:
For Section B – Standard Deviation Method (σ Unknown):
Note: For 2-sided spec limits, an AQL can be assigned to both limits combined, or to each end of the spec limit separately.
Table B-1 Standard Deviation Method Master Table B-1 for Normal and Tightened Inspection for Plans Based on Variability Unknown (Single Specification Limit)
Obtain the k Value from the Master Table:
Designing your own Variable Sampling Plan
What would be the variables sampling plan (sigma unknown) for the following conditions? High Pa (α = .05) for a fraction non-conforming (P1) of .005, with a low Pa (β = .05) for a fraction conforming (P2) of .03.
Solution
Note: with the same operating characteristics, an attribute sampling plan would require n = 274.
Example (Method: Population Sigma Known)
A lot of 1500 bobbins is submitted for inspection. Inspection level II, normal inspection, with AQL = .65%, is to be used. The specified minimum yield value for the tensile strength is 25.0 lbs. The variability σ is known to be 2.4 lbs.
The sample size code letter from Table A.2 is K. In Table D-2 (p. 86 in ANSI/ASQ Z1.9-2003), for reduced inspection, the required sample size is 7 and the k value is 1.80. The 7 sample specimen’s tensile strengths are 25.7, 26.4, 26.1, 27.2, 25.8, 28.3, and 27.4. QL = (X̄-LSL)/σ = 24.6 – 25.0 / 2.4 = 0.63
Since QL < k, the lot does not meet the acceptability criterion and should be rejected.
What are the alpha and beta risks for sampling letter K (p. 23 in ANSI/ASQ Z1.9-2003), for an AQL of .65, for various incoming quality levels (P)?
| P | Pa | α | β | |
| Relatively Good Quality | .25 | 99.5 | .5 | |
| .50 | 96 | 4 | ||
| Marginal | .75 | 90 | 10 | 90 |
| 1.5 | 62 | 38 | 62 | |
| Poor | 2.00 | 46 | 46 | |
| 3.0 | 22 | 22 | ||
| 4.0 | 10 | 10 | ||
| 5.0 | 4.5 | 4.5 |
* All values are in Percentages
Why is the lot rejected even though none of the samples were out of spec? (Assuming this is a representative sample, a larger, +/- 3σ distribution would provide some product out of specification; in this case, in the left tail of the distribution.)
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