Sampling Plan Variation vs Lot Size Variation in Acceptance Sampling

If the lot size N changes, the below curves change very little. However, the curves will change quite a bit as sample size n changes. So, basing a sampling plan on a fixed percentage sample size will yield greatly different risks. For consistent risk levels, it is better to fix the sample size at n, even if the lot sizes N vary. Question: If n = 10 & c = 2, what is the alpha risk for a vendor running at p = .02? Answer: P_a is about .55, so alpha is about .45. Question: What is the beta risk if the worst-case quality the customer will accept is 3%? Answer: (about 15%). To lower alpha and beta, you can increase n and c.

Is c = 0 the best plan for the producer and the consumer?

At the 2.8% lot defect rate, both plans give the producer equal protection: P_a = 11%, or P_rej = 89%. Which one gives better protection against rejecting relatively good lots, e.g., at the .5% lot defect rate, and why?

For (1), α = about 8% and for (2), α = about 30%.

(1) has a lower α error so less chance of rejecting good lots. With (2), you will reject any lot of 500 if there is even 1 defect in the sample, but it will lead to higher costs.

Increasing Protection from Rejecting Good Lots in Acceptance Sampling

Is c = 0 the best plan for the producer and the consumer?

At the 2.8% lot defect rate, both plans give the producer equal protection: P_a = 11%, or P_rej. = 89%. Which one gives better protection against rejecting relatively good lots, e.g., at the .5% lot defect rate, and why? For (1), α = about 8%; for (2), α = about 30%. (1) has a lower α error so less chance of rejecting good lots. With (2), you will reject any lot of 500 if there is even 1 defect in the sample, but it will lead to higher costs.

Discrimination in Acceptance Sampling Plans

Discrimination is the ability of a sampling plan to distinguish between relatively good levels of Quality and relatively bad levels of quality. In other words, having

• A high P_a (e.g., 95%, 1-α) associated with a good level of quality P₁ (e.g., .5% or better)
• A low P_a (e.g., 10%, β) associated with a bad level of quality P₂ (e.g., 3% or worse)

The Operating Ratio is defined as
R = P₂/P₁ = P_β/P_1-αExample: R = .03/.005 = 6.0

Designing your own single acceptance sampling plan

Derive a plan that comes as close as possible to satisfying two points on the OC curve. The two points are (P₁, 1-α) and (P₂, β). The derived plan will contain an n and a c value.
Example
Desired α risk of .05 for a P₁ of .005, along with a desired risk of .05 for a P₂ of .03.
1. Determine R:

R = P₂/P₁ = .030/.005 = 6.0

2. Enter the Values of Operating Ratio Table with α and β and find the closest R to the calculated value in step 1.
For α = .05 and β = .05, the closest table value is 5.67. This is acceptable since it is slightly more discriminating than 6.0. Note the c value of 3 in the far left column.

3. Obtain the nP₁ value in the far right column. Then calculate n from:

n = nP₁/P₁ = 1.366/.005 = 273.2 or 274.

The acceptance sampling plan is n = 274, c = 3.