# Question about C=0 sampling method!(Help)

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• #31728

Jackey
Participant

Is there anybody who are familiary with C=0 sampling method (Zero defect)? Do you please tell me what the difference between C=0 sampling method and MIL-STD-105E? What’s the backgound theory of C=0 sampling?

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#83908

Charles H
Participant

This paper may help.
http://www.stochos.com/qcrep0698.pdf

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#83911

Gabriel
Participant

The MIL-STD is based on the binomial distribution for low defect rates and on a supplier’s risk of (I think it was) about 10%. That means that you won’t reject a batch unless the proportion of defectives in the sample would not come from a batch with a proportion of defectives = AQL in 90% of the trials. That’s why you see that the reject number / samnple size [(C+1)/N] is allways greater than the AQL.
Then you have the consummer risk, which is the risk to accept a batch that has more deffectives than a certain proportion.
For example, N=8 C=1. Batches containing 7% of defectives will be accepted 90% of the times, meaning that 90% of the times you take a sample of 8 from a batch with 7% of defectives you find 0 or 1 defective. What proportion of defective do you think that you need in a batch to reject it in 90% of the times with this sampling plan? 40%! That means that in 10% of the times you take a smaple of 8 from a batch with 40% of defectives you find 0 or 1 defectives.
So, as you see, to avoid the suppliers risk to reject a batch with less than 7% of defectives you are accepting a great risk to accept batches with 40% of defectives.
With C=0 it is conceptually the same, but the figures change. Wiith a sampling plan N=8 C=0 batches with 1.3% of defectives will be accepted 90% of the times and batches with 25% of defects will be rejected 90% of the times.
Several sampling plans in the MIL STD have C=0, but ussually for the same AQL you can find other sampling plans with C>0.
The problem is, no sampling plan (except a 100% effective 100% check) can ensure 0 defects. Further more, with the “infinite population” approximation (when the sample is small compared with the population) used in most sampling methods including MIL STD you can not give any confidence that you will reject a batch with anything more than than 0 defects. For example, with N=1000 C=0 you will still accept 90% of the batches with 100 PPM and reject 90% of the batches with 4500 PPM.
When Zero Defects (ZD) is wanted, it is somwhere where we want to aim, not somwhewe where we will get. You may get ZD in one month, or in one year, or even more. But you will never reach a level where a defect will be never produced again. The focus for ZD must be in the process (stable, capable, free of outliers), and not in the quality control. As shown before, ZD can not be assured with QC.
However, if you have some batch inspections (like incomming inspection), the only acceptable acceptance criteria in line with ZD will be C=0. For two reasons.
a) If the acceptable quality level ageed with the supplier is ZD, then the supplier is risk free. Regardless of which sampling plan you use, you will accept 100% of the batches with ZD. So using C=0 does not put the supplier in any risk of rejecting a batch that should have been accepted, but improves your chances to reject a batch with defects.
b) Any acceptance criteria other than C=0 will make a defect acceptable. C=0 does not ensures that any batch with 1 or more defectives will be rejected, but it ensures that if one defect is found in the sample (in which case that batch contains at least that defect) the batch will be rejected.
And the background theory is just the binomial distribution (as long as the infinite population aproximation applies). For C=0 or any other C.

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#83915

Mikel
Member

Gabriel is right. C = 0 plans were developed to send a message that defects were not acceptable. While the top end of the acceptance curve (producer’s risk) is about in the same place, the bottom end of the curve is very stretched elevating the consumers risk or what is called LTPD.
I believe these types of plans were first promoted by Chrysler in the late 80’s.
Just remember that a need to do an incoming inspection type activity is a breakdown of the supply chain (and a bottleneck).

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#83920

topher
Member

c=0 sampling plans are intended to provide equal or greater consumer protection with less inspection than MIL-STD-105. The c=0 plans were obtained by determining the smallest sample size for c=0 where the Operating Characteristic (OC) curve would approximately pass through the same Lot Tolerance Percent Defective (LTPD) point as the corresponding MIL-STD-105 plan. The LTPD is generally at the Pa=0.10 point on the OC curve.
This would provide equal or better consumer protection at the LTPD value. However, at the same time it increases the producer risk (rejecting a good lot) by reducing the probability of acceptance at the AQL level. Since the accept number remains at zero as the sample size number, the producer risk continues to increase the lot size increases.
If the process quality level is substantially better than the AQL the producer would probably not be adversely affected. If the process quality was at or near the AQL, the producer would be exposed to many more potential rejections (and the associated costs) than with the corresponding MIL-STD-105 plan.
A couple of items to note are:
1) The c=0 plans contain “associated AQLs”. The tables call these out as index values, since they are not AQLs.
2) Guidelines for c=0 plans use the terminology “withhold the lot”. The lot is accepted only if zero nonconformances are found. If 1 or more nonconformances are found the lot is withheld, not automatically rejected. The lot should then be reviewed and dispositioned by the relevant authorities (engineering, management, etc), since the nonconformance(s) are not necessarily a cause for rejection.
As mentioned earlier, if you have been accustomed to using MIL-STD-105E plans and your process quality level is not substantially better than the AQL, c=0 plans may significantly increase your costs due to lot rejections, rework, etc.
I’m not defending MIL-STD-105E or criticizing c=0 plans, just trying to advise you of the potential implications/ramifications associated with switching from the former to the later.
Hope this helps.

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#83929

Mikel
Member

All of the C=0 plans I have seen protect the producer. Can you give me an example of one that protects the consumer? Lot size and sample size should be adequate infomation.

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#83933

topher
Member

I haven’t seen the plans you mention, but I don’t understand how c=0 plans protect the producer. For a given level of quality, if the accept number remains at 0 and the sample size increases, the probability of accepting that lot decreases.
For example: The probability of accepting a lot that is 4.0% nonconforming with c = 0 plans are:
Lot size = 26 to 50              n = 5      Pa = 0.815
Lot size = 281 to 500          n = 11    Pa = 0.638
Lot size = 1201 to 3200      n = 18    Pa = 0.480
Let’s say I (the producer) am supplying you (the consumer) with lots of product that are at the agreed AQL of 4.0%. I assume a 18.5% risk of the lot being rejected when n = 5 and a 52.0% risk of rejection at n = 18. So as a producer, more the 50% of the  lots I’ve submitted that meet the specified AQL may be rejected at n = 18.
As my original post stated, c = 0 plans were devised to provide equal or greater consumer protection. In a single sampling scheme, this would result in increased producer risk. Also, remember that the “associated AQLs” in c = 0 plans are not the same as in MIL-STD-105.
Maybe the confusion is the terminology used for which party you consider the producer. The supplier of the product is the producer (me) and the party receiving the product is the consumer (you). The consumer isn’t necessarily the ultimate end user of the product (your customer).

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#83935

Mikel
Member

For example: The probability of accepting a lot that is 4.0% nonconforming with c = 0 plans are:
Lot size = 26 to 50              n = 5      Pa = 0.815
The probability of accepting a 36% defective lot is about 10% (LTPD)
Lot size = 281 to 500          n = 11    Pa = 0.638
The probability of accepting a 18% defective lot is about 10% (LTPD)
Lot size = 1201 to 3200      n = 18    Pa = 0.480
The probability of accepting a 12% defective lot is about 10% (LTPD)
This is not protecting the consumer – of course it is not doing a red hot job of protecting the producer either. C=0 plans are to make a point, they are not real good protection (but then again neither is incoming inspection).

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#83936

CSSBB
Participant

Jackey:
Best reference on c=0 sampling plans (in my opinion): Zero Acceptance Number Sampling Plans, Fourth Edition published by ASQ Press.
There’s an excellent discussion in this pamphlet about the relationship between c=0 sampling plans and 105E, as well.
http://qualitypress.asq.org/perl/catalog.cgi?item=H0862
Regards.

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#83940

topher
Member

Stan,
I appreciate your comments, but don’t understand what point you’re trying to make.
The probabilities of acceptance you listed are correct and just bear out what my earlier statements were. The c = 0 plans were devised to provide equal or greater consumer protection than MIL-STD-105 at the LTPD. If you calculate the Pa of the corresponding plans in MIL-STD-105E you will find them to be approximately equal to the ones you listed. (29.9 vs 36%, 18.5 vs 18%, and 12.3 vs 12%).
My first post was in response to Jackey’s original question as to the difference between the 2 plans and the background theory of c = 0 plans. I also stated that I was neither defending MIL-STD-105 or criticizing the c = 0 plans, merely calling out the differences.
I’m sorry if I struck a nerve with you, that certainly wasn’t my intent.
Topher

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#83941

Mikel
Member

My point is that they do not provide equal or better protection at the LTPD. The thought that they provide protection at all is an illusion.

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#83951

topher
Member

Stan,
I agree with you that the protection they provide can be an illusion. The only true acceptable quality level is zero defects. Sampling plans are no substitute for using the appropriate tools to investigate the root causes of problems, determine corrective/preventive solutions to improve quality and reduce costs, and put controls in place to maintain the gains.
However, the fact remains that c = 0 plans were originally developed by Squeglia in 1961 to favorably compare to MIL-STD-105C plans that were being used at that time.
The first sentence in the introduction of Zero Acceptance Number Sampling Plans 4th Ed. by Squeglia states:
“The zero acceptance number plans developed by the author were originally designed and used to provide equal or greater consumer protection with less inspection than the corresponding MIL-STD-104 sampling plans.”
Given the above quote and previous posts showing that the Pa at the LTPD is approximately equal between the two plans, how can you argue that they do not provide equal or better protection at the LTPD? That is exactly what the c = 0 plans were developed for.
You can post again to argue the above, but I think enough of my time (not to mention the others who may have been following this discussion) has been wasted on this.
Topher

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#102071

Rocky Wong
Member

I wonder if you mind hsaring the formula to obtain the OC curve for different N values? Im having some problem understading the basic for C=0 sampling plan. Thank you very much

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#102085

Gabriel
Participant

The formula for c=0 or for any other c value is the one for the  binomial distribution.
In Excel you can use the function =BINOMDIST(c,n,p,TRUE)
c is the acceptance number (the maximum number of defectives that can be found in the sample and still accept the lot). Obviously it is zero for c=0 sampling plans.
n is the sample size.
p is the actual lot quality (defectives rate in the lot).
If you want the OC curve for c=0 and a given n, just replace those values in the formula and try with different p. The outcome of the formula is the Pa (probability to accept the lot).
If you want the “original” formula used to compute the binomial distribution, use the Excel Help for that function.

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