7.2 Acceptance Sampling

Acceptance Sampling Plans

Components and Types of Sampling Plans

Acceptance sampling is a method where you inspect a random sample from a larger lot and use the results to decide whether to accept or reject the entire batch. It's the practical middle ground between inspecting nothing and inspecting everything.

Every sampling plan is built around a few core components:

• Lot size (N): the total number of items in the batch
• Sample size (n): the number of items you actually inspect
• Acceptance number (c): the maximum number of defectives allowed in the sample for the lot to be accepted
• Rejection number (r): the number of defectives that triggers rejection (often $c + 1$)

Plans fall into two broad categories based on how you evaluate items. Attribute-based plans use simple pass/fail criteria (defective or not), while variable-based plans measure a specific characteristic (like weight or diameter) and compare it to a specification.

You'll also encounter different sampling strategies depending on the production context: lot-by-lot sampling (inspect each batch), continuous sampling (for assembly lines where items flow continuously), and skip-lot sampling (skip inspection on some lots when quality history is strong).

Quality Levels and Performance Measures

Two quality thresholds anchor every sampling plan:

• Acceptable Quality Level (AQL): the worst defect rate you'd still consider acceptable as a long-run process average. An AQL of 1% means you're okay with roughly 1 defective item per 100.
• Lot Tolerance Percent Defective (LTPD): the defect rate you consider clearly unacceptable. An LTPD of 5% means lots at that quality level should almost always be rejected.

Because sampling involves randomness, two types of risk are unavoidable:

• Producer's risk ($\alpha$): the probability of rejecting a lot that's actually at AQL quality. A 5% producer's risk means there's a 5% chance a good lot gets wrongly rejected.
• Consumer's risk ($\beta$): the probability of accepting a lot that's actually at LTPD quality. A 10% consumer's risk means there's a 10% chance a bad lot slips through.

Two additional measures evaluate long-term quality performance:

• Average Outgoing Quality (AOQ): the average defect rate of lots after the sampling process (accounting for lots that were rejected and screened). AOQ changes depending on the incoming quality level.
• Average Outgoing Quality Limit (AOQL): the worst-case AOQ across all possible incoming quality levels. This is the ceiling on how bad your outgoing quality can get under a given plan.

Finally, Average Sample Number (ASN) measures inspection efficiency. An ASN of 50 for a lot of 1,000 means you inspect 50 items on average per lot.

Single vs. Double vs. Multiple Sampling

Single Sampling Plans

With a single sampling plan, you draw one sample and make your decision right there. If the number of defectives is at or below the acceptance number, you accept the lot. If it exceeds that number, you reject.

For example, with $n = 100$ and $c = 2$: accept if you find 2 or fewer defects, reject if you find 3 or more.

Single plans are straightforward to administer and easy for inspectors to follow. The tradeoff is that they typically require larger sample sizes than double or multiple plans to achieve the same level of protection.

Double and Multiple Sampling Plans

Double sampling gives you a second chance before making a final call. You draw an initial sample and check the results against three possible outcomes: accept immediately, reject immediately, or take a second sample.

For example, suppose the first sample is 50 items. If you find 0 defects, accept the lot. If you find 4 or more, reject. If you find 1 to 3 defects, draw a second sample of 50 and combine the results from both samples to make your final decision.

Multiple sampling extends this logic to three or more stages, and sequential sampling takes it to the extreme by evaluating items one at a time until the cumulative evidence is strong enough for a clear accept or reject decision.

These multi-stage approaches tend to have a smaller average sample size because many lots get decided early (very good lots are accepted quickly, very bad lots are rejected quickly). Only borderline lots require additional sampling.

Factors Influencing Plan Selection

Choosing between single, double, and multiple plans depends on several practical considerations:

• Inspection cost: If testing each item is expensive (think destructive testing), double or multiple plans save money by reducing the average number of items inspected.
• Time constraints: Multi-stage plans require more administrative steps. If you need fast turnaround, single sampling is simpler to execute.
• Discrimination needs: If you need sharp separation between acceptable and unacceptable quality, you may need larger samples or multiple stages.
• ASN comparison: Compare the average sample number across plan types. A double plan with an ASN of 40 that provides equivalent protection to a single plan with an ASN of 60 is clearly more efficient.

Operating Characteristic Curves

Construction and Interpretation

An Operating Characteristic (OC) curve is a graph that shows how a sampling plan performs across different quality levels. The x-axis shows lot quality (usually percent defective), and the y-axis shows the probability that the plan will accept a lot at that quality level.

A few things to read from the curve:

• At the AQL on the x-axis, the probability of acceptance should be high ($1 - \alpha$). This confirms that good lots are usually accepted.
• At the LTPD on the x-axis, the probability of acceptance should be low ($\beta$). This confirms that bad lots are usually rejected.
• The steepness of the curve between these two points reflects the plan's discrimination power. A steep drop means the plan sharply distinguishes between acceptable and unacceptable quality. A gradual slope means there's a wide range of quality levels where the outcome is uncertain.

The region between AQL and LTPD is sometimes called the zone of indifference (or indecision zone). For a plan with AQL at 1% and LTPD at 5%, lots with defect rates in that 1%-5% range have moderate and uncertain acceptance probabilities.

Analysis and Comparison

OC curves are most useful when you plot multiple plans on the same graph. This lets you directly compare how different sample sizes, acceptance numbers, or plan types perform at the same quality levels.

For instance, you might overlay the OC curve for a single sampling plan against a double sampling plan to see which offers better protection at a specific defect rate. A near-vertical section of the curve at a particular quality level indicates the plan is very decisive in that range.

Increasing the sample size generally makes the OC curve steeper, improving discrimination but also increasing inspection cost. The goal is finding the plan whose OC curve best matches your quality requirements without over-inspecting.

Sampling Plan Selection

Quality Requirements and Cost Considerations

Selecting a plan means balancing quality protection against practical constraints. The key inputs are your target AQL, your LTPD, and the levels of producer's and consumer's risk you can tolerate.

Standardized reference tables make this process much easier. ANSI/ASQ Z1.4 (the civilian successor to Military Standard 105E, or MIL-STD-105E) provides pre-calculated sampling plans indexed by lot size and AQL. You look up your lot size, choose your AQL, and the table gives you the sample size and acceptance number.

It also helps to consider the process capability. A process with a capability index of $C_{pk} = 1.33$ (well-centered and capable) might justify reduced inspection, while a process with $C_{pk} = 1.0$ (barely meeting spec) warrants tighter sampling.

The broader cost of quality (COQ) framework ties this together. Higher sampling costs (appraisal costs) may be justified if they significantly reduce the cost of defective products reaching customers (failure costs).

Optimization Techniques

Once a plan is in place, switching rules let you adjust inspection intensity based on recent quality history:

1. Start with normal inspection.
2. Switch to tightened inspection if quality deteriorates (e.g., 2 out of 5 consecutive lots are rejected).
3. Switch to reduced inspection if quality has been consistently good (e.g., 10 consecutive lots accepted under normal inspection).

These rules are built into the ANSI/ASQ Z1.4 standard and keep inspection effort proportional to actual quality performance.

For specialized situations, Bayesian methods incorporate prior knowledge about process quality. If you have historical data showing seasonal variation or supplier-specific trends, you can adjust sampling parameters accordingly rather than treating every lot as if you know nothing about the process.