# The link between AQL, RQL and the reliability of a process

The problem is to distinguish between the definitions of AQL, RQL and reliability in the field of quality control. Let us just pose the context. We wish to calculate the reliability of a manufacturing process and/or the quality of the manufactured pieces by doing attributes testing. The particularity of attributes testing is that, for each piece tested, the outcome of the test will be either 'passed' or 'failed'.

I just remind here the definitions that I have at hand:

AQL

AQL stands for acceptance quality level. The AQL represents the poorest level of quality for the supplier's process that the consumer would consider to be acceptable in average.

RQL

RQL stands for risk quality level, it can be also denominated LTPD that stands for lot tolerance percent defective. The RQL is the poorest level of quality that the consumer is willing to accept in an individual lot.

Reliability The process reliability is the probability that a process will perform its intended function without failure for a specified time under stated conditions.

The following definitions will also be useful to go on:

$$\alpha$$ producer's risk:

$$\alpha$$ is the probability that a good product will be rejected as a bad product by the consumer.

$$\beta$$ consumer's risk:

$$\beta$$ is the probability that a product not meeting quality standards will enter the marketplace.

Now, suppose we will inspect $$n$$ manufactured samples out of $$N$$ and, as soon as a default is detected, we reject the lot of $$N$$ items. Let us note $$p_1$$ for the AQL and $$p_2$$ for the RQL (usual notations). Then, taking into account the above, we can link $$\alpha$$ and $$p_1$$ and $$\beta$$ and $$p_2$$ thanks to the followings equations:

$$1-\alpha = (1-p_1)^n = Prob(no \ defect \ are\ detected \ in\ the\ n \ samples | AQL = p_1),$$

$$\beta = (1-p_2)^n = Prob(no \ defect \ are\ detected \ in\ the\ n \ samples | RQL = p_2).$$ On the other hand, the reliability $$R$$ is linked to the confidence level $$C$$ by the following equation:

$$1-C = R^n.$$

I do not really understand what is the difference between the RQL and the reliability, and so, what formulas I should use to know how many samples I must choose to sustain with a certain confidence level that in $$p$$% of the cases, my pieces are not defective. In addition, we can note that the formulas relating RQL and $$\beta$$ and $$R$$ and $$C$$ are the same with different parameters.

The thing is that I receive pieces X from a producer and he gives me an AQL. These pieces X will then be merged with other components and I need to know what is the quality of the pieces X at the end of the process.

To calculate a (good) sample size you need the following inputs:

• RQL
• $$\alpha$$ risk of the test at the given RQL
• AQL
• power of the test at the given AQL

If you have these as inputs, you can calculate the optimal sample size for testing \begin{align} H_0: \hspace{1em} \mu &\ge \mu_{RQL}\\ H_1: \hspace{1em} \mu &< \mu_{RQL} \end{align} with the given $$\alpha$$ risk, and power. In this context the reliability is given by $$R=1-RQL$$, and the confidence is given by $$\gamma=1-\alpha$$. Thus, there is no simple/direct relationship between reliability and confidence -- this is why statements such as "we are 95% confident that the reliability is at least 99%" are meaningful. If these two terms had a simple/direct relationship such a statement would repeat its information.

On the one hand side we are testing against the $$RQL$$ value. Nevertheless, it's important to choose the AQL level correctly, because this affect the number of false negatives. E.g. suppose the customer sets $$RQL=1\%$$, and the producer is over confident and states that $$AQL = 0.01\%$$. Now, if we calculate a sampling plan using $$\alpha=5\%$$, and $$power= 99\%$$, we obtain a sample size of $$N=299$$. However, if the true quality level is only $$AQL=0.2\%$$, we are going to reject not only $$5\%$$ of the LOTs, but much more -- much to the dislike of the producer and the customer. This is why the operating characteristic curve is important: In the example the OC curve shows that we are going to reject approx 50% of the LOTs, if $$AQL=0.2\%$$ is the true quality value. The proper sampling plan for $$AQL=0.2\%$$ uses $$N=773$$ samples and rejects the LOT only if we obtain four or more failures.

An alternative to the hypothesis test described above is the use of tolerance intervals according to Wilks method -- which is often done in the medical industry.

• Thanks for the answer. My concern is more about how to define the reliability when we have only categorical data (i.e., passed or failed). In this case, the tolerance intervals are not appropriate. I understood what you said but why do you choose RQL as the reliability? My final goal is to ensure a certain reliability level of my manufacturing process. Commented May 8, 2023 at 6:34
• Why do you believe that we are unable to calculate a tolerance interval for binomial distributions? I recommend you google it, and you find that there are multiple methods. However, my answer does not really discuss tolerance intervals, but focuses onto hypothesis testing. Nevertheless, I agree, that most often a tolerance interval is more appropriate for the process validation. Commented May 8, 2023 at 16:43
• Also, I do not set $R=RQL$, but I use $R = 1 - RQL$. Why? Because if my reliability is at least 99%, I obtain at most 1% scraps. Furthermore, as we are testing the RQL(and not the AQL) value, I can only use this value for my "result statement". Commented May 8, 2023 at 16:44