Quality Reporting for Acceptance Sampling
Acceptance Sampling of part lots is used to determine if parts that have already been produced and ready to be shipped to the customer are acceptable. It requires less than 100% inspection, therefore less inspectors are needed which makes the process more lean and cost effective. Acceptance sampling plans carry two types of risks: producer’s risk and consumer’s risk. The producer’s risk is the probability that a good lot will be rejected. This risk fits together with the maximum quality level the plan will accept (AQL) – where AQL is a general sampling plan meaning “acceptable quality level.” The consumer’s risk is the probability that a bad lot will be accepted, so it is clear that both of these scenarios should be avoided.
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An Operating Characteristics curve is a plot of the probability of accepting a lot vs. the percent of nonconforming parts within that lot. It is based on a probability distribution involving the total population of all parts produced, the sample size of the parts that will be inspected, and the maximum number of parts that are allowed to be defective within the chosen sample.
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For this example, a single acceptance sampling plan for attributes has been agreed on where the population (N) = 1,000,000, the same size (n) = 250, and the acceptance number (c) = 4. Management has asked to find the acceptable quality level for a 0.5% producer’s risk, and for a 4% consumer’s risk. This can be done graphically if that is all that is available, or by finding the corresponding values in the table that were used to plot the curve for more accuracy.
The table’s fraction defective values correspond to the x-values on the chart, while the probability of acceptance for hypergeometric distribution values correspond to the chart’s y-values.
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Producer’s risk corresponds to a probability of acceptance being: 1 – the probability of acceptance for the entire lot:
PR = 1-Pa
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The correct probability of acceptance value for the entire lot (Pa) is found by going to the 0.004 (0.4%) defective value in the table and getting the Pa value from the Hypergeometric column because this was the chosen distribution for this graphical analysis due to the population size. From here, the Pa value is approximately 0.996437 so the PR will be approximately 0.00356 (0.36%).
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To find the CR, consumer’s risk, a similar procedure is followed but instead of subtracting the probability of acceptance from 1 the risk is just the probability of acceptance. This goes back to the original definition of consumer’s and producer’s risk. Producer’s risk is the probability that a bad lot will be rejected, and consumer’s risk is the probability of accepting a bad lot so it makes sense. Therefore, correlating with the 0.04 (4%) value in the table, the consumer’s risk is approximately 0.026989 (2.7%).
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Another important point to note, is once the population gets large enough and there is a large enough sample size, the probability distribution chosen to use does not matter. Hypergeometric distribution was chosen for this example because it is always the most precise if the population size is available. Binomial, Poisson’s, and the Hypergeometric are the three different distributions that will be chosen from.
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