# Designing a Sampling Plan - AQL

B

#### Black arrow

Hello,
We will soon recieve a shipment of products from a supplier and the quantity is 8000 pieces. The requirement from us is that they fullfill Cpk of 1.67. We have to take a sample size from this batch (lot) and verify whether they are ok or not. As i can see from Minitab 17 , a sampling plan can be made but i have some questions about he inputs deacribed there.

Measurement type: I choosed "Go/no go" is that correct?
AQL: How do i determine this value? AQL (Acceptable Quality Level) assumes no knowledge of the capability of the process that produced the part being sampled. What if the underlying performance capability is known? As in my case?
RQL: How do i determine this value? Do i have to fill in this data at all? The RQL describes what the sampling plan will reject, where as the acceptable quality level (AQL) describes what the sampling will accept. Perhaps i mix them (AQL & RQL) up?
Apha and Beta risks: Is Beta value always larger than Aplha? How do i determine those values?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
if you really want a Cpk of 1.67 (indicating a relatively Normal distribution with the bulk of the data near the nominal and only a few values in the tails with the observable data well within the specification) then you are mixing apples and oranges by going to a categorical (go, no-go) data sampling plan. If a 'real' Cpk of 1.67 is required, then you must use a continuous data sampling plan that will calculate the Cpk of the shipment and you should use the confidence intervals for the Cpk value to determine the confidence that your result is right. for example if you want a Cpk of 1.67, and a 95% confidence (alpha) with an acceptable error rate of .1 then your sample size is 408. (I will attach the spreadsheets for this so you can play with it. it will also give you the confidence intervals for your actual results. I do have to be transparent and apologize as one spreadsheet was posted here sometime ago and i don't remember the author. the other was posted by Barbara Bredner) If you actually get a Cpk of 1.5 in your sample of 408, the confidence intervals would indicate that the minimum actual Cpk might be as low as 1.39 or as high as 1.61.

If you are using Cpk of 1.67 as an indication only of the defect rate in the shipment, then you can use a categorical sample plan. and you can use a plan that is based only on the RQL. as the customer you are trying to protect yourself from receiving material that is worse than a Cpk of 1.67, hence the RQL approach. A Cpk of 1.67 is equivalent to a defect rate of 0.27 ppm or 0.00000027 and your sample size (using an accept number of 0 to get the smallest sample size possible) is 11,095,305 .

now you could simply say that you don't want any parts that are outside of the inner 60% of the tolerance zone on the drawing, you could simply inspect 100% of the lot and if no parts are outside the guardbanded limits you are good to go. you have a Cpk of 1.67 by the categorical data method.

So it depends on what you really want...the real question here is why are you requiring a Cpk of 1.67? and why can't you just have your supplier submit their data showing that they meet a Cpk of 1.67?

Staff member
Super Moderator

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B

#### Black arrow

Thanks,

If i look at this in another way. We have a product that have 10 different dimensions and 3 of them are important and the other 7 are not so important for the function. I would like determine some risk analysis that if we measure for example 100 of 9000 parts we can to accept the batch (9000 pieces) if for example x of the 3 important dimensions are ok and if y of the 7 less important dimensions are ok. Other wise reject whole batch.

I have seen som tables on internet for this but i do not know if those are correct way to follow.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
I am confused..all acceptance sampling plans are based on the need to either accept some defect rate or reject some defect rate or both. ALL of the tables you will have seen on the internet or in a book or wherever, are based on this fundamental concept.

you have said that teh requirement for your supplier is to have a Cpk of 1.67. if this IS what you are inspecting for you have no choice but to use an RQL plan. I have outlined several options for youin my last posts. obviously this is not what you are looking for? or are you simply appalled by the very large sample sizes that are required to inspect for aa Cpk of 1.67?

perhaps you are willign to accept a higher defect rate? what is the acceptabe rate for you? what is the rate that is unacceptable?

you cannot combine characteristics in an acceptance sampling plan unless the physics of the process guarantees that they are 'co-linear',meanign that if one goes high the other goes high. While this can happen (usually with material properties) it is rare for geometric features.

a sample size of 100 with zero defects could possibly have a defect rate as high as 3.6% (this is the upper 95% confidence limit). 3.6% is equivalent to a Cpk of 0.6

I am afraid that you are asking for something you cannot have...

#### Kronos147

Trusted Information Resource
It sounds to me like you want to have a sampling plan with different levels. Each characteristic can be inspected to a different level.

ANSI/ASQ Z1.4 Table IIA shows 4 levels, I, II, III and S-1. Since we have a sampling plan based on ANSI/ASQ, I would consider an inspection checklist that says something to the effect of inspect the three critical characteristics to Level 3 and the seven non-critical characteristics to Level 1 or even S-1.

Food for thought.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
yes of course these particular tables will work for applying different acceptance levels to different characteristics.

however, black arrow has a dilemma - what defect rate does he/she want to accept and/or reject?

the real problem here is that old bugaboo the capability index. imposing high Cpk values are intended to obviate incoming inspection. period, a Cpk of 1.67 represents a defect rate so low that anything less than 100% inspection will be useless and a complete waste of time.

the (original) intent is to have the supplier provide data that their products meet the Cpk requirement through continuous data analysis and not categorical pass/fail data because their should be no failures. 1.67 Cpk = 0.27 ppm there are NO sampling tables that cover that level of RQL....

B

#### Black arrow

Hello again,

Perhaps i have to be more clear of what i would like to have support from you all. At my job i have been told to make some research over existing AQL tables on different forums. The reason is that we in the future will have products send to us, products that supplier will made for us. First shipment is 8000 pieces so we had a meeting to discuss different approaches to this. Sample size, acceptance level,..But we need help here.

One colleague recomended me as a first step to follow the AQL tables (ISO 2859-1) on the internet, seems to be easy to follow but i am not sure. Sample size code letter, inspection level (I,II,II),...
So, what i understand ,from a AQL value in percent i will get sample size, the limits for major and minor defects.

So i was looking for some help to use those AQL tables, because it is the first time i see those kind of tables.

What about Cpk of 1.67? For the moment we do not not know the Cpk, we have not seen any data from the supplier. But can we anyway take some steps forward without the Cpk value?

As i mentioned before, the product have 10 dimensions and some of them are more important than other. How do i deal with that?

I'm a beginner at this topic...

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
Hello again,

Perhaps i have to be more clear of what i would like to have support from you all. At my job i have been told to make some research over existing AQL tables on different forums. The reason is that we in the future will have products send to us, products that supplier will made for us. First shipment is 8000 pieces so we had a meeting to discuss different approaches to this. Sample size, acceptance level,..But we need help here.

One colleague recomended me as a first step to follow the AQL tables (ISO 2859-1) on the internet, seems to be easy to follow but i am not sure. Sample size code letter, inspection level (I,II,II),...
So, what i understand ,from a AQL value in percent i will get sample size, the limits for major and minor defects.

So i was looking for some help to use those AQL tables, because it is the first time i see those kind of tables.

What about Cpk of 1.67? For the moment we do not not know the Cpk, we have not seen any data from the supplier. But can we anyway take some steps forward without the Cpk value?

As i mentioned before, the product have 10 dimensions and some of them are more important than other. How do i deal with that?

I'm a beginner at this topic...
Let me approach this from a slightly different angle. A Cpk of 1.67 is equivalent to 0.54 PPM based on a two-sided specification (0.27 PPM for a one-sided specification). If this is attribute feature (pass/fail), there is NO sampling plan that can guarantee that low a PPM with any real confidence. if it is a continuous feature (measured data), you can find what observed Cpk you would need to have a certain level of confidence that the true Cpk is no worse than 1.67. For example, a sample of 30 pieces would need to have a Cpk of 2.04 to have 95% confidence (assuming a two-sided specification).

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Perhaps i have to be more clear of what i would like to have support from you all. At my job i have been told to make some research over existing AQL tables on different forums. The reason is that we in the future will have products send to us, products that supplier will made for us. First shipment is 8000 pieces so we had a meeting to discuss different approaches to this. Sample size, acceptance level,..But we need help here.
OK, so first I recommend that you search this forum for AQL, Acceptance Sampling and LTPD, RQL. There is a wealth of information that has been posted here over the years that will help you out.

Next, I will disclose that I am not a fan of the standard AQL tables that have sampling levels and sample codes. These tables stem from the forties and fifties and while there is some statistical basis for them, they were the result of negotiations primarily aimed at reducing the burden of producers in the 'glory' days of US mass production when quantity superseded quality.

I much prefer and recommend a straightforward approach based on direct statistical calculations. I have posted a spreadsheet that has many different approaches to developing sample plans. It can be found here.

Some things you need to know.
• YOU must decide on the defect rate that you are willing to accept some percent of the time. This acceptable defect rate is known as the AQL. The percent of time that you are willing to accept it is known as the confidence and is typically set at 95%
• YOU should also decide on what defect rate you do NOT want to accept and the percentage of time that you want to detect and reject it. This defect rate is known as the RQL. It is also referred to as the LTPD (Lot Tolerance Percent Defective) in the standard AQL tables where only the AQL is specified. It is the defect rate that will be 'accepted' 10 % of the time - or rejected 90% of the time.
• There are many people who recommend a "c=0" plan. c is the letter used to indicate the maximum number of defects present in the sample in order to accept the lot. These plans provide the smallest sample size of all plans, but have the worst ability to discriminate between acceptable lots and non-acceptable lots. They are advocated by people who do not want to accept any lot that has a defect in the sample. This is a misguided belief - or misunderstanding - of how probability works.

If you look at the spreadsheet the tab called "Binomial AQL RQL" has the most sophisticated plan developer. It is annotated on what each term means...

Some good articles to read are listed below. The articles below are available at the ASQ website for \$10 if you are not a member and \$5 if you are. This price is a bargain. ASQ has a great archive of articles that can be found at www.asq.org – Knowledge center – publications – advanced search
http://asq.org/knowledge-center/search/

Mood’s Theorem, Deming’s kp Rule, and the Death of Acceptance Sampling”, Quality Engineering 8(1), 129-136 (1995-96)

Acceptance Sampling? The Enterprise Strikes Back! AS9100 c=0 Plans; When Slogans Supplant Science”, Quality Engineering 18:237-266, 2006

Selecting Statistically Valid Sampling Plans” Dr. Wayne Taylor

And even tho I am not a fan a great start at the standard old fashioned tables is Chapter 11 of a Quality Management Text by Sower. It has some excellent references on the origins of Acceptance Sampling.

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