Acceptance Sampling and Supplier Quality

Introduction and context

Acceptance sampling is a quality-control procedure in which a decision is made to accept or reject a lot after inspecting a sample rather than every item. It is especially useful when 100% inspection is costly, destructive, slow, or less reliable than a controlled sample inspection. In mechanical engineering, typical applications include incoming bearings, fasteners, castings, machined shafts, heat-treated components, and purchased subassemblies.

Acceptance sampling is not a process-control method. Statistical process control attempts to prevent defects by monitoring and improving the production process. Acceptance sampling evaluates a completed lot and determines whether that lot is suitable for acceptance. A sampling plan therefore provides evidence for a decision; it does not prove that every item in an accepted lot is conforming.

Supplier quality management connects sampling decisions with supplier selection, incoming inspection, corrective action, supplier development, and contractual requirements. A good plan must balance inspection cost against the risks of accepting poor quality or rejecting good quality.

Learning objectives

After studying this chapter, a student should be able to:

1. Explain the purposes and limitations of acceptance sampling.
2. Distinguish AQL, LTPD, producer’s risk, and consumer’s risk.
3. Design and interpret single, double, and sequential sampling plans.
4. Construct and use operating characteristic (OC) curves.
5. Calculate acceptance probabilities for attribute sampling plans.
6. Apply sampling results to supplier qualification and incoming-quality decisions.
7. Evaluate sampling plans using risk, cost, engineering criticality, and ethical considerations.

1. Fundamental terminology

A lot is a defined quantity of product produced under sufficiently similar conditions. A sample is the subset selected for inspection. The sample size, n, is the number of units inspected. A defective unit fails one or more specified requirements; a defect is a specific nonconformity. These terms should not be confused: one unit may contain several defects but is counted as one defective unit for many attribute plans.

A plan is commonly written as (n,c), where n is sample size and c is the acceptance number. The lot is accepted if the number of defectives observed is at most c and rejected if it exceeds c.

The fraction defective, p, is the proportion of defective units in the lot or process. The corresponding percentage defective is 100p. For finite-lot sampling, the actual probability is hypergeometric. When the lot is large relative to the sample, the binomial model is usually appropriate.

The acceptable quality level (AQL) is a reference quality level associated with a high probability of acceptance under a sampling plan. It is not a guarantee that every lot at that quality will be accepted, nor is it necessarily the supplier’s target process average.

The lot tolerance percent defective (LTPD), also called limiting quality in some standards, is a poor quality level associated with a specified low probability of acceptance. It represents a quality level that the consumer wants to reject most of the time.

The producer’s risk, α, is the probability of rejecting a lot when its quality is at the AQL. The consumer’s risk, β, is the probability of accepting a lot when its quality is at the LTPD. Thus, a plan is designed to control two different decision errors:

The assumptions are that the quality level is defined consistently, the sampling method is unbiased, and the decision rule is followed exactly.

2. Probability models for acceptance sampling

For a single attribute plan (n,c), let X be the number of defectives found in the sample. Under independent sampling or sampling from a very large lot, X follows a binomial distribution:

where x = 0,1,…,n, p is the fraction defective, and C(n,x) = n!/[x!(n−x)!]]. The probability of accepting the lot is therefore:

The formula is dimensionless because probabilities and p are proportions. It applies to attribute inspection with independent observations and a stable definition of a defective unit.

For a finite lot of size N containing D defectives, sampling without replacement requires the hypergeometric model:

The binomial approximation is generally reasonable when the sampling fraction n/N is small, commonly less than about 0.10. When the sample is a substantial fraction of the lot, the hypergeometric model should be used.

Worked example 1: single sampling plan

A supplier plan uses n = 50 and c = 2. Estimate the probability of acceptance when the lot fraction defective is p = 0.02. Using the binomial model:

Using (0.98)⁵0 ≈ 0.3642, the three terms are approximately 0.3642, 0.3718, and 0.1858. Thus:

The probability of rejection at this quality level is approximately 0.0782. The result does not mean that 2% defective is automatically acceptable; it describes the operating behavior of this particular plan.

3. Operating characteristic curves

An OC curve plots P_a(p) against the fraction defective p. A strong plan has a relatively steep curve: good lots have high acceptance probability, while poor lots have low acceptance probability. The curve provides more information than a single AQL value because it shows the plan’s performance across a range of quality levels.

Figure: An illustrative OC curve showing decreasing acceptance probability as lot quality worsens.

The AQL and LTPD are points or regions used to specify desired risks, not necessarily points where the curve changes abruptly. Sampling plans always involve uncertainty. Two lots with the same actual quality may receive different decisions because different samples are observed.

A plan’s discrimination describes how effectively it separates acceptable from unacceptable quality. Increasing sample size generally makes the OC curve steeper, but it also increases inspection cost. Increasing the acceptance number generally makes acceptance easier and shifts the curve toward poorer quality.

4. Designing a single sampling plan

A single plan is selected by specifying:

• the lot definition and inspection unit;
• the quality characteristic and classification of defects;
• the target AQL and LTPD;
• producer’s and consumer’s risks;
• sample size n;
• acceptance number c;
• the sampling method and decision rule.

Design can be performed by consulting a recognized standard, such as an applicable national or international acceptance-sampling standard, or by solving the probability requirements directly. Standards should be used according to their stated scope; a plan for nonconforming items may not be suitable for defects per unit, variables data, or destructive testing.

For a specified n and c, check:

and

The first condition protects the producer at the reference good quality, while the second protects the consumer at the limiting poor quality. If both conditions cannot be met with a practical sample size, the quality requirements are too demanding for the proposed plan and must be renegotiated or supported by process controls.

Worked example 2: interpreting risks

Suppose a contract defines AQL = 1.0%, LTPD = 6.5%, α = 0.05, and β = 0.10. A compliant plan should have at least 0.95 probability of accepting a lot at 1.0% defective and no more than 0.10 probability of accepting a lot at 6.5% defective. If a candidate plan gives probabilities 0.96 and 0.14 respectively, it satisfies the producer-risk requirement but fails the consumer-risk requirement. It should not be approved without changing the plan or accepting the additional consumer risk through formal engineering and commercial authorization.

5. Double sampling plans

A double sampling plan can reduce the average amount of inspection when many lots are clearly good or clearly poor. It consists of a first sample, an immediate decision rule, and—if necessary—a second sample.

A plan may be represented by (n₁,c₁,r₁; n₂,c₂). After the first sample:

• accept if the first-sample count X₁ ≤ c₁;
• reject if X₁ ≥ r₁;
• continue if c₁ < X₁ < r₁.

If a second sample is taken, accept when X₁ + X₂ ≤ c₂ and reject otherwise. The samples should be independently selected from the lot, and the same defect definition must be used in both stages.

For independent binomial sampling, the overall probability of acceptance is:

Here, X₁ and X₂ are independent binomial counts with parameters n₁,p and n₂,p. The second term includes only continuation outcomes.

The average sample number (ASN) is the expected number inspected per lot:

This assumes the second sample is always of size n₂ whenever continuation occurs. A double plan can have a lower ASN than an equivalent single plan, but its administration and calculations are more complex.

6. Sequential sampling plans

In sequential sampling, units are inspected one at a time, or in small groups, and the cumulative evidence is compared with acceptance and rejection boundaries. Inspection stops as soon as a boundary is crossed. If neither boundary is crossed, inspection continues until a maximum sample size or a specified decision rule is reached.

For an approximate sequential probability ratio test comparing a good quality p₀ with a poor quality p₁, the cumulative number of defectives d after n observations can be compared with boundaries derived from likelihood ratios. For attribute observations, the log-likelihood ratio is:

The boundaries are selected to control risks approximately: an upper boundary supports rejection of the good-quality hypothesis, and a lower boundary supports acceptance. Exact boundary construction depends on the chosen hypotheses, risks, truncation rule, and whether indifference-zone or composite hypotheses are used. Sequential plans should therefore be obtained from validated statistical design procedures rather than improvised from a sketch.

Sequential plans often have a low ASN when quality is very good or very poor, but they require disciplined recording, randomization, and protection against stopping bias. They are less convenient when inspection is destructive, samples must be transported in batches, or production and receiving personnel cannot observe results promptly.

7. Attributes and variables sampling

Attribute sampling records conforming/nonconforming status or counts defects. It is suitable when requirements are categorical, inspection is rapid, or measurement precision is limited. Variables sampling uses measured values such as shaft diameter, hardness, or tensile strength and can provide more information per item when the measurement distribution and standard deviation assumptions are credible.

For a normally distributed characteristic with known standard deviation σ, a one-sided specification limit can be evaluated using a standardized distance such as:

for an upper specification limit USL, or

for a lower specification limit LSL. Here μ and σ are the process mean and standard deviation in the same measurement units as the specification. Variables plans may reduce sample size, but incorrect normality or variance assumptions can create serious risk. Attribute plans are often preferred for safety-critical characteristics unless a validated variables method is available.

8. Supplier quality and incoming inspection

Acceptance sampling should be part of a supplier-quality system, not the only supplier-control activity. A practical supplier decision process includes:

1. Define critical, major, and minor characteristics using engineering risk analysis.
2. Specify material, drawing, process, testing, traceability, and documentation requirements.
3. Qualify the supplier through capability evidence, audits, sample approval, and pilot lots.
4. Establish an incoming plan linked to product risk and supplier performance.
5. Record lot identity, sample selection, results, concessions, and corrective actions.
6. Periodically review performance and adjust inspection status through formal approval.

Supplier selection should consider total quality cost, not simply purchase price. Relevant factors include defect prevention, delivery reliability, calibration systems, process capability, responsiveness, traceability, and ability to control special processes such as heat treatment, welding, plating, and nondestructive testing.

An accepted lot may still contain defective units. For critical characteristics, acceptance sampling should be combined with supplier process audits, certificates, first-article inspection, mistake-proofing, and—where justified—100% automated inspection. Conversely, imposing unnecessarily severe sampling on a demonstrably capable supplier can create cost without proportional risk reduction.

Switching rules and supplier status

Organizations commonly use normal, tightened, and reduced inspection. Movement between these states should be governed by documented rules based on consecutive lot results, defect severity, and process evidence. A single critical nonconformance may require immediate containment regardless of the numerical sampling result. Reduced inspection should never be granted solely because a supplier has delivered several good lots if process changes, subcontracting, or traceability failures have occurred.

9. Illustrative case: machined steering shafts

Background. An automotive component manufacturer purchases machined steering shafts. The critical characteristic is a journal diameter with tight fit requirements; cosmetic marks are classified as minor defects. The supplier has recently changed a grinding subcontractor.

Problem. The receiving department proposes continuing the historical reduced-inspection plan. Engineering is concerned that a change in the grinding source may alter diameter variation. The purchasing department wants to avoid delivery delays.

Analysis. The diameter is safety-related and cannot be treated like a cosmetic characteristic. Historical acceptance results alone do not demonstrate continued capability after the subcontractor change. The team reviews calibration records, process capability, control charts, traceability, and measurement-system analysis. It also examines the OC curve of the existing plan and finds that its LTPD protection is inadequate for the new risk.

Approach. The company returns temporarily to normal inspection, requires a first-article submission from the new source, and uses a variables-based verification plan only after confirming measurement-system adequacy and approximate normality. Critical diameter results are recorded by lot and linked to furnace, machine, operator, and subcontractor identifiers. Any critical nonconformance triggers lot containment and supplier corrective action; it is not offset by good results on minor characteristics.

Outcome. The first two lots reveal a shifted process mean, although both would probably have passed the former reduced plan. The supplier corrects wheel dressing and establishes an automated diameter check. Capability and control evidence are reviewed over subsequent lots before a formal return to reduced inspection.

Learning points. Sampling plans must reflect current process risk, not historical convenience. Critical characteristics require layered controls. Supplier changes should trigger reassessment, and acceptance decisions must be traceable to the exact lot and quality requirement.

10. Limitations, ethics, and governance

Sampling results are probabilistic. A rejected lot may contain mostly conforming units, while an accepted lot may contain nonconforming units. Poor sampling practices—such as allowing the supplier to select only convenient items, inspecting only the first pieces, or replacing failed units—destroy the validity of the OC performance.

Ethically, organizations must not choose a plan merely because it produces favorable acceptance results. Safety-critical requirements should be based on hazard analysis, applicable regulations, customer contracts, and engineering judgment. A concession should be documented, technically justified, time-limited where appropriate, and approved by authorized personnel. Concealing rejected lots, altering inspection records, or shipping known nonconforming product can create legal liability and endanger users.

Digital inspection and AI can support image classification, anomaly detection, sampling optimization, and supplier trend analysis. However, an AI model does not replace a validated inspection method. It must be trained on representative data, monitored for drift, checked for false negatives, and controlled through configuration management. Automated decisions affecting safety should retain human oversight, auditable records, and a defined escalation route.

11. Best-practice checklist

• Define the lot before sampling begins.
• Use random or systematic-random selection that represents the entire lot.
• Separate critical, major, and minor nonconformities.
• Specify AQL, LTPD, α, and β in contractual or internal requirements.
• Verify the OC curve rather than relying on a plan number alone.
• Use finite-population calculations when the sampling fraction is substantial.
• Validate gauges and inspectors before relying on results.
• Record accepted, rejected, reworked, and conditionally released lots separately.
• Link inspection findings to supplier corrective action and process evidence.
• Reassess the plan after design, material, machine, subcontractor, or process changes.
• Use 100% inspection or preventive controls when the consequence of escape is unacceptable.

Chapter Summary

Acceptance sampling makes lot decisions from sample evidence and is distinct from process control. A single attribute plan (n,c) accepts a lot when the observed number of defectives does not exceed c. Its performance is described by the OC function P_a(p), calculated with the binomial model for large lots and the hypergeometric model when sampling without replacement from finite lots. AQL and LTPD define reference quality levels, while producer’s risk α and consumer’s risk β describe decision errors at those levels.

Double sampling can reduce average inspection through an intermediate decision, and sequential sampling can reduce the average sample number further by stopping when evidence is decisive. Their complexity requires carefully defined rules and validated statistical design. Supplier-quality decisions should combine sampling with process capability, audits, traceability, measurement assurance, corrective action, and risk-based treatment of critical characteristics. No sampling plan guarantees defect-free accepted lots; sound engineering judgment is required whenever the consequence of failure is high.

Key Terms

Lot: A defined quantity of product made or supplied under specified conditions.

Sample: A subset of a lot selected for inspection.

Defective unit: A unit that fails one or more specified requirements.

Defect: A specific nonconformity in a product or process.

Acceptance number: The maximum number of defectives permitted for acceptance.

Single sampling plan: A plan using one sample and one final decision.

Double sampling plan: A plan in which a second sample may be inspected after an inconclusive first sample.

Sequential sampling: Sampling in stages until cumulative evidence supports acceptance or rejection.

AQL: Acceptable quality level associated with a high probability of acceptance.

LTPD: Lot tolerance percent defective associated with a low probability of acceptance.

Producer’s risk: Probability of rejecting a lot at the AQL.

Consumer’s risk: Probability of accepting a lot at the LTPD.

OC curve: Plot of probability of acceptance against fraction defective.

ASN: Expected number of units inspected under a sampling plan.

Binomial model: Probability model for independent attribute observations with constant defective probability.

Hypergeometric model: Finite-lot model for sampling without replacement.

Incoming inspection: Evaluation of purchased product before release to production.

Supplier corrective action: Formal action addressing the cause of a supplier nonconformance.

Critical characteristic: A characteristic whose failure can create serious safety, regulatory, or functional consequences.

Measurement-system analysis: Evaluation of measurement accuracy, repeatability, reproducibility, and suitability.

Review Questions

Short Answer Questions

1. Why is acceptance sampling not equivalent to process control?
2. Define AQL and LTPD.
3. What is the difference between producer’s risk and consumer’s risk?
4. When is the hypergeometric model preferred over the binomial model?
5. State one advantage and one limitation of double sampling.

Long Answer Questions

1. Explain the construction and interpretation of an OC curve, including its relationship with AQL and LTPD.
2. Describe a systematic procedure for designing a single sampling plan for an incoming mechanical component.
3. Compare single, double, and sequential sampling plans in terms of inspection effort, complexity, and risk control.
4. Explain how acceptance sampling should be integrated with supplier qualification and corrective action.
5. Discuss why critical characteristics may require controls beyond acceptance sampling.

Analytical / Application Questions

1. For a plan with n = 50 and c = 2, calculate the approximate probability of acceptance at p = 0.02 using the binomial model. Interpret the result.
2. A proposed plan has P_a(AQL) = 0.96 and P_a(LTPD) = 0.14, while the contract requires α = 0.05 and β = 0.10. Determine which requirement is violated and recommend an action.
3. A supplier changes its heat-treatment subcontractor but requests reduced incoming inspection based on ten previously accepted lots. Develop a risk-based decision, identifying evidence required before granting reduced inspection.

Multiple Choice Questions with Answers

Question 1

For a single attribute plan (n,c), a lot is accepted when:

A. The number of defectives is greater than c

B. The number of defectives is less than or equal to c

C. The sample contains exactly c defectives

D. The lot contains no defects anywhere

Correct Answer: B. The number of defectives is less than or equal to c

Question 2

The producer’s risk α is the probability of:

A. Accepting a poor lot at the LTPD

B. Rejecting a good lot at the AQL

C. Accepting every inspected lot

D. Finding zero defects in a sample

Correct Answer: B. Rejecting a good lot at the AQL

Question 3

Which model is generally appropriate for sampling without replacement from a finite lot when the sampling fraction is substantial?

A. Normal

B. Poisson

C. Hypergeometric

D. Exponential

Correct Answer: C. Hypergeometric

Question 4

The OC curve primarily shows the relationship between:

A. Inspection cost and supplier price

B. Sample size and production time

C. Fraction defective and probability of acceptance

D. AQL and measurement resolution

Correct Answer: C. Fraction defective and probability of acceptance

Question 5

For a plan with n = 50 and c = 2, the acceptance probability at p = 0.02 is approximately:

A. 0.08

B. 0.36

C. 0.72

D. 0.92

Correct Answer: D. 0.92

Question 6

A double sampling plan proceeds to a second sample when the first sample result is:

A. Clearly acceptable

B. Clearly rejectable

C. In an intermediate continuation region

D. Always defect-free

Correct Answer: C. In an intermediate continuation region

Question 7

The consumer’s risk β is controlled at the:

A. AQL

B. LTPD

C. Mean specification limit

D. Inspection-cost maximum

Correct Answer: B. LTPD

Question 8

Which action best preserves the validity of an acceptance-sampling plan?

A. Allowing the supplier to select convenient items

B. Replacing failed units before recording results

C. Selecting samples randomly across the entire lot

D. Inspecting only the first production pieces

Correct Answer: C. Selecting samples randomly across the entire lot

Question 9

A principal advantage of sequential sampling is that it can:

A. Guarantee zero defects in accepted lots

B. Reduce average sample size when evidence is decisive

C. Eliminate the need for a sampling rule

D. Avoid defining producer’s and consumer’s risks

Correct Answer: B. Reduce average sample size when evidence is decisive

Question 10

After a supplier changes a subcontractor for a safety-critical process, the most appropriate initial response is to:

A. Automatically grant reduced inspection

B. Ignore the change if delivery history is good

C. Reassess process risk and require supporting qualification evidence

D. Remove the characteristic from inspection

Correct Answer: C. Reassess process risk and require supporting qualification evidence