Question

Design an acceptance sampling plan for which AQL is \(0.10\), RQL is \(0.05, \alpha\) is \(0.05\), and \(\beta\) is \(0.05\).

Short answer

The solution to the exercise involves three steps. First, the producer's and consumer's risk points are computed using the provided alpha, beta, AQL, and RQL. Then, those points are used to find the sample size and acceptance number. Finally, the sampling plan is verified to ensure it satisfies the conditions set by AQL, RQL, alpha, and beta. Note: actual computation results depend on the values of alpha, beta, AQL, and RQL given for the exercise.

Step-by-step solution

Compute Producer's and Consumer's Risk Points

Firstly, calculate the two points on the Operating Characteristic (OC) curve: P1 for AQL and alpha, and P2 for RQL and beta. This is based on the formulas:\n\[P_1 = -\ln(\alpha) / AQL\]\[P_2 = -\ln(\beta) / RQL\]Now, substituting given values for alpha (0.05), beta (0.05), AQL (0.10), and RQL (0.05) into the formulas above, the calculated producer's and consumer's risk points (P1 and P2) can be computed.

Finding Sample Size and Acceptance Number

In this step, use the formula: \[n = (P2-P1) / (RQL - AQL)\] to find the sample size, rounding up to the nearest whole number if necessary. Similarly, use the formula: \[c = n*AQL - P1\] to find the acceptance number, rounding down to the nearest whole number if necessary. Here, n represents the sample size and c is the acceptance number. Remember, both 'n' and 'c' are integers since you cannot have fractional samples or acceptance numbers.

Confirming the Sampling Plan

Finally, verify the sample size (n) and acceptance number (c) derived in the previous step confirm to the given AQL, RQL, alpha, and beta conditions. This can be done by utilizing the OC curve. If the plan meets the conditions, it can now be employed for quality control. Otherwise, adjustments may be needed.

Key concepts

AQL (Acceptable Quality Level)

The Acceptable Quality Level (AQL) is a crucial concept in acceptance sampling, representing the maximum defective rate considered tolerable during random sampling of products.

AQL is the threshold for the percentage of defects that can be considered acceptable during the inspection process. When designing an acceptance sampling plan, as seen in the original exercise, AQL is set at a particular value to determine the acceptable number of defective items. In the provided exercise example, an AQL of 0.10 indicates that a maximum of 10% defectives in a batch would be considered satisfactory quality for a company.

During sampling, if the number of defects is below or equal to the threshold specified by AQL, the batch is accepted. Importantly, AQL is also used to calculate the Producer's risk point, which influences the sample size and acceptance number in an acceptance sampling plan.

RQL (Rejectable Quality Level)

The Rejectable Quality Level (RQL), also known as the Lot Tolerance Percent Defective (LTPD), is the defect rate that is deemed unsatisfactory by the consumer.

RQL is set higher than AQL and is the point at which a batch is considered of poor quality and unacceptable. In the context of the exercise, a 5% RQL signifies that if defects exceed this level, the lot should not pass inspection and be rejected. This threshold inherently describes the Consumer's risk, where a lot with a defect rate higher than RQL has a chance to be accidentally accepted.

RQL is used to calculate the Consumer's risk point and to establish the number of samples and the acceptable number of defective items. The RQL value is essential for safeguarding the consumer from subpar products, as it helps companies outline clear parameters for when products should not enter the market.

Producer's and Consumer's Risk

Producer's and Consumer's risk are the statistical chances of accepting a bad lot or rejecting a good lot, respectively.

Producer's risk, represented by \(\beta\), is the risk that a good quality lot (with defects below or at AQL) will be incorrectly rejected. In our textbook exercise, this is set at 5%, which means there is a 5% chance that an acceptable lot will be rejected. The determination of Producer's risk is important when setting up the threshold for the AQL.

Conversely, Consumer's risk, denoted by \(\beta\), designates the risk that a lot with defects exceeding the RQL will be mistakenly accepted. This too is set at 5% in the exercise. Outlining both risks helps in fine-tuning the point at which the quality of the lot can be judged as passing or failing, thus protecting the interests of both producers and consumers in quality control processes.

Operating Characteristic (OC) Curve

The Operating Characteristic (OC) curve is a graphical representation of a sampling plan's ability to distinguish between good and bad lots based on the sample results.

The curve plots the probability of accepting a lot against various levels of percent defective in the lot. It serves as a visual tool for understanding the performance of a sampling plan. When you calculate the Producer's and Consumer's risk points, you effectively determine two critical points on the OC curve. These points help to shape the 'steepness' of the curve, with steep OC curves providing better discrimination between high, and low-quality lots.

In the exercise, once the sample size (n) and acceptance number (c) are determined, checking them against the OC curve ensures they will meet the specified AQL and RQL conditions, effectively safeguarding the pre-specified Producer's and Consumer's risks. If a plan doesn't fall within the acceptable regions determined by these risks, adjustments in sample size or acceptance numbers, or both, might be necessary to achieve the desired protective measures.