# Translating AQL to a Production Line Inspection Plan

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#### Eric Rco

Hi guys, I'm looking for a way to translate our AQL sampling plan to a manufacturer so he can integrate our requirements to his quality control plan.

Giving an example: I have the product ABC with General Inspection II, AQL 0.25, single sampling. I'm purchasing 5000 units. Therefore, I should inspect 200 units and start rejecting at 2 defect units. The product has 5 criteria to be inspected. Each criteria is done at a different stage during production.

From the point of view of the manufacturer, considering the principles of Total Defect Per Unit (TDPU), if I decide to adopt the same sampling size, saying 200 units for 5000 units produced, if I get one defect per inspection stage, I'll get a TDPU of 5. Knowing the customer can only accept 2 defect and there are 5 inspected criteria, it also means that during my entire production, I cannot have more than 2 defects. By experience, chances are rare to have only 2 defect on an entire production line from start to finished product.

So this is where I am stuck since a couple of months now. I'm trying to determine what would be the ideal sampling plan to produce this product and how to determine my sampling plan per inspection stage? More than that, how to do the reversed calculation from the customer's AQL needs back to the production line?

If I decide to work with the AQL methodology, then I would say that I can have only 2 defect on my entire line and I need to inspect 200 units per inspection stage. However, what if I had a stronger AQL and I could only allow 1 defect? It is impossible to say that in other words, I could have 1/5 defect per inspection stage.

I tried to see this as a DPMO too. Certified Six Sigma Green Belt, my teacher sent me on that path. But since AQL and PPM don't mix because of the LTPD curve, correct me if I am wrong, but we can't extrapolate in that direction and simply raise the quantity to inspect thinking it will raise my quantity of acceptable defect.

So there is something I'm missing here. In a ideal world, inspection would allow zero defect and production lines would be stopped until the variation has been found, but since this product involves a lot of made by hand steps using old machinery not even allowing real time SPC, we are stuck with a need to define the proper sampling plan and an accurate Gauge R&R system.

In resume, if anyone can help me, it would be very appreciated:
1-How to determine the sampling methodology and rejection level per inspection stage considering the AQL needs from the customer?
2-If AQL is different from an inspected criteria to another, is this a proper method to apply AQL? If it is, how should we deal with this and should we keep in mind TDPU? (that would generate a monster calculation model!)
3-Is there a light at the end of the tunnel?

Thanks
BTW, I'm also looking for a Quality Engineer to coach me on different quality control questions. Anyone interested, please beep me.

#### Stijloor

Super Moderator
A Quick Bump!

Can someone help?

Thank you very much!!

#### somashekar

~~~ Welcome to the COVE Eric ~~~

Ask your manufacturer to ensure that his average percent defective level is below the AQL prescribed by you. What does this mean? It means the true percent defective level of the lots submitted for AQL based inspection must be less than the AQL. For this purpose an organisation has to measure its current average percent defective level (process average). This can be achieved by conducting sampling inspections of the lots before inspection by the customer. In such case, all the pieces in a sample drawn from the lot are inspected to arrive at percent defective level of respective lots. If an organisation does this for about 300 consecutive lots and calculates the average of the per cent defective of all lots inspected, it would give a good idea of the ‘process average’. Assuming his process average is lower than the AQL level, then there is a minimal chance (generally less than 5-10%) of his shipment getting rejected. If his process average is greater then AQL level, he need to work towards- if not eliminating-reducing the generation of defect level at source so that the process average becomes lower then the AQL. In case process coverage remains higher than the AQL, the chances of his shipments failing to pass AQL based inspection are higher, depending on the process average.
Makes sense ~~~
(Details taken from an article by Dr. Rajesh Bheda. He is a Professor at the GMT Department, NIFT, New Delhi,)

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#### Eric Rco

Thanks guys and thanks for the welcome. Very interesting website.

Regarding the answer, I understand the concept of Cp and Cpk where we need to see what are our requirements vs. what he can achieve. The thing is the supplier is ready to make the necessary to improve its quality level, but he is asking me how can he calculate the quantity of reject he can accept on the production line, taking in consideration the example of 5 different stages, 5 different inspection stages, at each stage. If my AQL is a sample of 200 units and a rejection at 2 units, how can he calculate what is the rejection level on each inspection stage? There is something I'm missing here considering that it is almost achievable to have only 1 defect on the entire production line.

If I could do a cross calculation to raise the 1 on 200 to DPM, it would be easy as we would simply ask to increase its inspected quantity, but DPM doesn't convert well with DPM because of normal distribution curve.

On the flip side, I could ask him to use a bigger sampling size than the 200 so the associated rejection quantity is bigger. But what happen if I'm producing a car and I have 1000 inspection points? How do I calculate the acceptance in terms of ratio? From Cpk? Plus, I'm limited by the quantity of samples I can inspect per day or hours. Of course, calculations are on the batch produced, not on x samples per hour. This is a common mistake I'm seeing with my suppliers: they inspect 7 samples per hour, not considering how many they have produced...

Not sure if I explain clearly where I'm trying to go with this.

Here's another concrete example: the part is a slide for making drawers. It's made in Southern China. The slide has 2 members and ball bearings. The members have dimensions, holes and some bending. We have determine that we have about 20 specifications that are CTQ.
Q1- Do I apply an AQL on each criteria or for the whole product?
Q2- We are presently applying an AQL per criteria and I believe, understanding the TDPU, that this is the wrong route. So let's say on this criteria, I reject at 2 defect. Could the supplier use this AQL for the inspection stage where they make this specific spec to measure it's rejection level? Writing this and I think I'm answering myself the question, but confirm me if I'm wrong.
Q3- Using TDPU, using AQL per criteria, having a sample of 200 units, having a rejection at 2 defect, can we say if each of the 20 criteria can accept 1 defect, then if nothing is rejected, I have a potential of 20 defect products on 200? Therefore, my targeted AQL is no more working?

Thanks a lot guys. BTW, is there a program you could recommend me to follow to become a QC specialist over production lines? The more I dig into this, the more fascinating it is.

#### Bev D

##### Heretical Statistician
Super Moderator
I think your issue here is that you are mish-mashing acceptance sampling with in-line inspection.

The concept of acceptance sampling is that it is done at the end of the line when the lot is complete. And remember that the AQL is the defect rate that will be ACCEPTED 95% of the time.

In line testing should be thought of and administered seperately from end of line acceptance testing. In general in-line testing is targeted at ensuring there are no defects that move forward so if a sample has a defect then the entire period it represents is reviewed. (of course given the product and industry and custoemr there are always exceptions). So the idea of in-line testing is to catch any process problems as early as possible while there is time to correct the process and redcue the numebr of defects created. The idea is that when the process begins making defects it does so at a high enough rate that small samples will catch it. So each time period is treated as it's own 'lot' or batch. Since sampling should be rather frequent to reduce the liability of defective material smaller sample sizes are usually quite effective. Of course this approach is not intended to catch random non-systemic defects. If the defect rate is 'high', such that every sampel contains at least one defect, 100% inspection is more effective.

If the defect rate is fairly low and can be measured using continuous (aka variable) data then SPC is usually a much better approach than using in-line inspection based on counts of defects.

Could you expand upon your comments regarding DPM and the Normal curve?

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#### Eric Rco

Thanks guys. It solved my questions.

Another one for you, if I can. I am trying to understand how a Cp can fall below 1 and what does it mean. I understand a Cpk below 1, but a Cp below one, this one, I'm kind of stuck muted.

I am running a Cp-Cpk study on QI Marco and getting Cp 0.51 and Cpk 0.46. Having a LSL 4.5, a USL 5.5 and target at 5, I should have a Cp of 1, right?

More than that, why do Cp are not always at 1? If I use the formulat (USL-LSL)/(6*sigma) where sigma=(USL-LSL)/6, it should always give me a Cp of 1, right? Is there something wrong in my formula?

Why are there Cp below zero?

Why are there Cp above zero?

Thanks guys

#### bobdoering

Trusted Information Resource
Thanks guys. It solved my questions.

Another one for you, if I can. I am trying to understand how a Cp can fall below 1 and what does it mean. I understand a Cpk below 1, but a Cp below one, this one, I'm kind of stuck muted.

They mean the same thing, only Cpk looks at each half of teh distribution (from the mean) and Cp looks at teh whole distribution. Cpk is used when centering teh distribution is important. It is almost always assumed to be important, withteh emphasis on assume.

I am running a Cp-Cpk study on QI Marco and getting Cp 0.51 and Cpk 0.46. Having a LSL 4.5, a USL 5.5 and target at 5, I should have a Cp of 1, right?

Cpk is almost always less than Cp because it is also considering centering. If the Cpk is less than 1, it is unlikely the Cp is 1.

More than that, why do Cp are not always at 1? If I use the formulat (USL-LSL)/(6*sigma) where sigma=(USL-LSL)/6, it should always give me a Cp of 1, right? Is there something wrong in my formula?

The Cp can be equal to 1, it can be more or less than 1. Your formula is wrong. sigma= (USL-LSL)/6*Cp

#### Bev D

##### Heretical Statistician
Super Moderator
Bob's last statement is spt on. Cp formula uses the same standard deviation as the one you used in Cpk. When you do this you will find that you will get Cp values ranging all over the place from infinity to zero (assymptotically)

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#### Eric Rco

Ok, I'm officially confused. Let's take a real example. I'm a visual person. Can you please demonstrate, calculation by calculation how you reach the Cp and how you reach the Cpk and how to interprete it? Thanks for sharing your knowledge. I'm realising that what I've learned and what you are saying are different and I believe you are right.

A cabinet door opening:
Target = 5mm
LSL = 4.5mm
USL = 5.5mm

Samples
Sample 1 = 5.3mm
Sample 2 = 4.6mm
Sample 3 = 5.5mm
Sample 4 = 5.3mm
Sample 5 = 5.1mm
Sample 6 = 5.0mm
Sample 7 = 4.7mm
Sample 8 = 5.5mm
Sample 9 = 5.2mm
Sample 10 = 5.7mm
Sample 11 = 5.1mm
Sample 12 = 5.8mm
Sample 13 = 5.2mm
Sample 14 = 5.1mm
Sample 15 = 5.1mm
Sample 16 = 5.1mm
Sample 17 = 4.8mm
Sample 18 = 4.9mm
Sample 19 = 5.4mm
Sample 20 = 5.0mm
Sample 21 = 5.5mm
Sample 22 = 5.2mm
Sample 23 = 4.9mm
Sample 24 = 4.7mm
Sample 25 = 5.1mm
Sample 26 = 4.6mm
Sample 27 = 4.3mm
Sample 28 = 4.5mm
Sample 29 = 5.1mm
Sample 30 = 5.0mm

Thanks.

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#### Eric Rco

Anyone to help me to solve the question above? Thanks