4.5 Acceptance Sampling Plans & Standards

4.5 Acceptance Sampling Plans & Standards

Acceptance sampling is an audit procedure where a statistical sample of units is randomly drawn from a production lot and inspected to determine whether to accept or reject the entire lot. Unlike 100% inspection (which can be expensive, error-prone due to inspector fatigue, or impossible for destructive testing) or zero inspection, acceptance sampling provides a statistically defensible middle ground that balances quality assurance costs against risk.

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Operating Characteristic (OC) Curves and Risk Parameters

The performance of any acceptance sampling plan is fully described by its Operating Characteristic (OC) Curve, which plots the actual lot percent defective ($p$) on the horizontal axis against the probability of lot acceptance ($P_a$) on the vertical axis.

Key OC Curve Anchor Points & Risk Definitions

1. Acceptable Quality Limit (AQL): The worst tolerable process average percent defective that a consumer considers acceptable as a process average. At the AQL, the probability of lot acceptance $P_a = 1 - \alpha$.
2. Producer's Risk ($\alpha$): The probability of rejecting a good lot (a lot with quality equal to or better than the AQL). Conventionally set at $\alpha = 0.05$ (5%), representing a 95% chance of lot acceptance at AQL.
3. Lot Tolerance Percent Defective (LTPD) / Rejectable Quality Level (RQL): The unacceptable level of quality that the consumer wishes to reject most of the time. At the LTPD, the probability of lot acceptance $P_a = \beta$.
4. Consumer's Risk ($\beta$): The probability of accepting a bad lot (a lot with quality equal to or worse than the LTPD). Conventionally set at $\beta = 0.10$ (10%), representing a 10% chance of lot acceptance at LTPD.

  Probability of
  Acceptance (Pa)
    1.0 |-----\ (1 - alpha = 0.95 at AQL)
        |      \
        |       \  <--- OC Curve Slope depends on Sample Size (n)
        |        \
    0.1 |---------\ (beta = 0.10 at LTPD)
    0.0 +-----------------------
        0     AQL       LTPD   Lot Percent Defective (p)

Mathematical Formulation of $P_a$

For an attribute sampling plan defined by sample size $n$ and acceptance number $c$ (where the lot is accepted if the number of defective units $d \le c$):

• Exact (Hypergeometric Distribution): Used when sample size $n$ exceeds 5% of lot size $N$ ($n/N > 0.05$).
• Binomial Approximation: Used when lot size $N$ is large relative to sample size ($n/N \le 0.05$): $P_a = \sum_{d=0}^{c} \binom{n}{d} p^d (1-p)^{n-d}$
• Poisson Approximation: Used when sample size $n \ge 16$ and defect rate $p \le 0.10$, setting mean $\lambda = n p$: $P_a = \sum_{d=0}^{c} \frac{e^{-n p} (n p)^d}{d!}$

Effect of Plan Parameters ($n$ and $c$)

• Increasing sample size $n$ while holding $c/n$ constant steepens the OC curve, increasing discriminatory power (sharper distinction between good and bad lots).
• Decreasing the acceptance number $c$ shifts the entire OC curve to the left, making the plan stricter across all quality levels.

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Average Outgoing Quality (AOQ) & AOQL

When rejected lots undergo rectifying inspection (where 100% of units in rejected lots are inspected and all defective units are repaired or replaced with good units), the long-term quality delivered to the customer is quantified by the Average Outgoing Quality (AOQ).

AOQ Formula

For a lot size $N$, sample size $n$, incoming lot defect fraction $p$, and acceptance probability $P_a$:

$AOQ = \frac{P_a \cdot p \cdot (N - n)}{N} \approx P_a \cdot p$

• When incoming quality is extremely good ($p \approx 0$), $AOQ \approx 0$ because almost no defective units enter.
• When incoming quality is extremely poor ($p \gg 0$), $P_a \approx 0$, triggering 100% rectifying inspection on nearly all lots, so $AOQ \approx 0$.
• The AOQ curve rises to a maximum peak known as the Average Outgoing Quality Limit (AOQL), which represents the worst average quality that can possibly exit the inspection process regardless of incoming lot quality.

Average Total Inspection (ATI)

The average number of units inspected per lot under rectifying inspection is calculated as:

$ATI = n + (1 - P_a)(N - n)$

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ANSI/ASQ Z1.4 (Attribute Sampling)

ANSI/ASQ Z1.4 (aligned with ISO 2859-1 and MIL-STD-105E) is the standard for lot-by-lot attribute inspection. Attributes are discrete pass/fail or count data.

Inspection Levels & Sample Size Code Letters

Z1.4 provides seven inspection levels to determine the Sample Size Code Letter based on lot size:

Types of Sampling Plans

1. Single Sampling Plan: Decision made after evaluating one sample of size $n$.
2. Double Sampling Plan: Decision made after sample 1 ($n_1$); if nonconformities fall between acceptance $c_1$ and rejection $r_1$, a second sample ($n_2$) is inspected. Double plans reduce average sample size by ~20-30% for very good or very bad lots.
3. Multiple / Sequential Sampling Plans: Up to 7 sampling steps; minimizes total units inspected for extreme quality lots.

Dynamic Switching Rules

ANSI/ASQ Z1.4 requires mandatory switching between Normal, Tightened, and Reduced inspection based on historical performance:

1. Normal to Tightened: Mandatory when 2 out of 5 (or fewer) consecutive lots are rejected on initial Normal inspection.
2. Tightened to Normal: Mandatory when 5 consecutive lots are accepted on initial Tightened inspection.
3. Normal to Reduced: Optional/Permitted when:
   • 10 consecutive lots have been accepted under Normal inspection;
   • Total nonconformities in sample are less than specified limit numbers in Z1.4 tables;
   • Production is at a steady state;
   • Approved by the Responsible Authority.
4. Reduced to Normal: Mandatory if any 1 lot is rejected, 1 lot meets production irregularity, or sampling terminates without decision.
5. Discontinuance of Inspection: Mandatory if 10 consecutive lots remain on Tightened inspection without returning to Normal. Operations must stop and supplier processes audited.

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ANSI/ASQ Z1.9 (Variable Sampling)

ANSI/ASQ Z1.9 (aligned with ISO 3951-1 and MIL-STD-414) covers acceptance sampling by variables (continuous measurements such as diameter, tensile strength, or weight).

Key Prerequisites

1. The quality characteristic must be measurable on a continuous numerical scale.
2. The distribution of the metric must follow a Normal (Gaussian) distribution.

Methodologies: Quality Indices ($Q_U$ and $Q_L$)

To evaluate a lot under Z1.9 using the standard deviation ($s$) method with sample mean $\bar{X}$:

$Q_U = \frac{USL - \bar{X}}{s} \quad \text{and} \quad Q_L = \frac{\bar{X} - LSL}{s}$

• The calculated Quality Index ($Q_U$ or $Q_L$) is compared against a maximum allowable percentage nonconforming factor ($M$) obtained from Z1.9 tables based on AQL and sample size.
• If estimated lot percent nonconforming $p_{est} \le M$, the lot is accepted; otherwise, it is rejected.

Attribute (Z1.4) vs. Variable (Z1.9) Comparison

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Worked Numerical Example: AOQ & ATI Calculation

Problem: An incoming lot of size $N = 4,000$ electronic components is inspected using a single attribute sampling plan with sample size $n = 80$ and acceptance number $c = 1$. The historical lot defect rate is $p = 0.015$ (1.5%). Calculate:

1. The probability of lot acceptance $P_a$ using the Poisson approximation.
2. The Average Outgoing Quality (AOQ).
3. The Average Total Inspection (ATI).

Step 1: Calculate Poisson parameter $\lambda$ $\lambda = n \cdot p = 80 \cdot 0.015 = 1.2$

Step 2: Calculate $P_a$ for $c = 1$ $P_a = P(d=0) + P(d=1) = \frac{e^{-1.2} (1.2)^0}{0!} + \frac{e^{-1.2} (1.2)^1}{1!} = e^{-1.2} (1 + 1.2)$ $P_a = 0.3012 \cdot 2.2 = 0.6626 \quad (66.26\%)$

Step 3: Calculate AOQ $AOQ = \frac{P_a \cdot p \cdot (N - n)}{N} = \frac{0.6626 \cdot 0.015 \cdot (4000 - 80)}{4000} = \frac{0.6626 \cdot 0.015 \cdot 3920}{4000} = 0.00974 \quad (0.974\%)$

Step 4: Calculate ATI $ATI = n + (1 - P_a)(N - n) = 80 + (1 - 0.6626)(3920) = 80 + (0.3374 \cdot 3920) = 80 + 1322.6 = 1402.6 \text{ units}$