12.2 Acceptance Sampling: OC Curves, AQL, LTPD, AOQL, Z1.4/MIL-STD-105E & Dodge-Romig

12.2 Acceptance Sampling: OC Curves, AQL, LTPD, AOQL, Z1.4/MIL-STD-105E & Dodge-Romig

Acceptance sampling is a classical quality engineering methodology used to determine whether a submitted lot of incoming raw materials, fabricated components, or finished goods should be accepted or rejected based on the inspection of a representative sample. Developed at Bell Telephone Laboratories by Harold F. Dodge and Harry G. Romig beginning in the late 1920s, and adopted widely by the military during World War II, acceptance sampling remains a standard protocol for supplier qualification and receiving inspection.

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1. Acceptance Sampling Philosophy

The Purpose of Sampling: Lot Sentencing vs. Process Control

A critical distinction in industrial engineering is that acceptance sampling does not control or improve quality. Acceptance sampling merely inspects the output of an existing process and sentences the lot (accept or reject). It does not alter the underlying process distribution or prevent defective units from being manufactured in the first place.

Acceptance sampling is justified under specific operational circumstances:

• When testing is destructive (e.g., tensile pull testing to fracture, explosive ordnance detonator testing, fatigue lifespan cycling).
• When the cost of 100% inspection is prohibitively high compared to the financial consequence of passing a defective item.
• When 100% manual inspection is subject to inspector fatigue, boredom, and cognitive error, leading to screening errors where inspectors miss defective units or falsely reject conforming units.
• When supplier process capability is known and stable, but formal lot-by-lot verification is contractually mandated.

Deming's Economic Perspective: The $kp$ Rule

Quality pioneer W. Edwards Deming criticized traditional acceptance sampling, demonstrating that it represents an inefficient middle ground. Deming formulated an economic rule known as the $kp$ Rule (or $k_1/k_2$ Rule):

Let:

• $k_1$ = Cost to inspect one unit (in $ per unit).
• $k_2$ = Cost incurred if a defective unit slips past inspection and enters production or reaches a customer (damage, warranty, assembly line stoppage, rework) (in $ per unit).
• $p$ = Average incoming fraction defective of the submitted lots.

Deming proved that the total cost is minimized only at the extremes:

$\text{If } p < \frac{k_1}{k_2} \implies \text{Zero Inspection (Pass all lots without sampling)}$

$\text{If } p > \frac{k_1}{k_2} \implies \text{100\% Inspection (Screen all units in every lot)}$

Acceptance sampling is only economically optimal under Deming's model when $p$ hovers precisely at the breakeven threshold $p^* = \frac{k_1}{k_2}$, or when supplier quality fluctuates unpredictably between lots.

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2. Single Sampling Plans: Parameters & Mathematical Formulations

A Single Sampling Plan is fully specified by three parameters:

1. $N$ = Lot size.
2. $n$ = Sample size drawn at random from the lot.
3. $c$ = Acceptance number (maximum allowable defective units in the sample).

Operational Decision Rule

1. Draw a random sample of $n$ items from a lot of size $N$.
2. Inspect all $n$ items and count the number of defective (nonconforming) units, denoted as $d$.
3. If $d \le c$, Accept the entire lot.
4. If $d > c$, Reject the entire lot (diverting it to return, scrap, or 100% rectifying screening).

Underlying Probability Distributions

The probability of accepting a lot containing fraction nonconforming $p$, denoted $P_a(p)$, depends on the sampling model:

1. Hypergeometric Distribution (Exact Finite Lot Model - Type A OC Curve)

When sampling without replacement from a finite lot of size $N$ containing exactly $D = N \cdot p$ defective units:

$P_a = P(d \le c) = \sum_{d=0}^c \frac{\binom{N p}{d} \binom{N(1 - p)}{n - d}}{\binom{N}{n}}$

2. Binomial Distribution (Infinite Lot / Process Stream Model - Type B OC Curve)

When sampling from a continuous manufacturing stream, or from a finite lot where the sample size is small relative to the lot ($n/N < 0.10$):

$P_a = P(d \le c) = \sum_{d=0}^c \binom{n}{d} p^d (1 - p)^{n - d}$

3. Poisson Distribution Approximation

When the sample size is large and the fraction defective is small (a common rule of thumb is $n \ge 20$ with $p \le 0.05$), the binomial distribution is closely approximated by a Poisson distribution with parameter $\lambda = n \cdot p$:

$P_a = P(d \le c) \approx \sum_{d=0}^c \frac{e^{-n p} (n p)^d}{d!} = e^{-n p} \left( 1 + n p + \frac{(n p)^2}{2!} + \dots + \frac{(n p)^c}{c!} \right)$

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3. The Operating Characteristic (OC) Curve

The Operating Characteristic (OC) curve is the definitive performance signature of an acceptance sampling plan. It plots the probability of lot acceptance $P_a$ on the vertical axis against the incoming lot fraction defective $p$ on the horizontal axis.

Type A vs. Type B OC Curves

• Type A OC Curve: Calculated using the hypergeometric distribution to evaluate isolated, individual finite lots of size $N$. As lot size $N$ increases for a fixed $n$ and $c$, the Type A curve rapidly approaches the Type B curve.
• Type B OC Curve: Calculated using the binomial or Poisson distribution to evaluate lots produced by an ongoing, steady-state continuous manufacturing process.

Ideal vs. Actual OC Curves

• The Ideal OC Curve: A vertical step function that drops instantaneously from $P_a = 1.0$ to $P_a = 0.0$ at a defined quality threshold $p^*$. Any lot with $p \le p^*$ is accepted with 100% certainty; any lot with $p > p^*$ is rejected with 100% certainty. Achieving an ideal OC curve requires 100% inspection with zero inspection error.
• Actual OC Curve: A continuous, sigmoidal (S-shaped) curve. Because sampling inherently introduces random sampling error, there is always some probability of accepting bad lots or rejecting good lots.

Impact of Sample Size ($n$) and Acceptance Number ($c$)

• Increasing $n$ with fixed $c$: Shifts the OC curve to the left and makes it steeper, increasing the stringency and discrimination power of the plan.
• Increasing $c$ with fixed $n$: Shifts the entire OC curve to the right, loosening acceptance criteria and raising $P_a$ across all values of $p$.
• Increasing both $n$ and $c$ proportionally: Steepens the OC curve around the threshold $p = c/n$, minimizing both producer's and consumer's risks simultaneously.

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4. Key Quality Parameters: AQL, LTPD, $\alpha$, and $\beta$

An acceptance sampling plan balances four fundamental operational parameters:

  Probability
 of Acceptance (Pa)
   1.0 +-----------------------------------------
       |  * (AQL, 1 - alpha)   [Producer's Point]
1 - a  |---\ 
       |    \
       |     \
       |      \
       |       \
       |        \
       |         \
   beta|----------\-------------* (LTPD, beta) [Consumer's Point]
   0.0 +-----------+-------------+---------------> Incoming Fraction
                  AQL           LTPD               Defective (p)

1. Acceptable Quality Limit (AQL) & Producer's Risk ($\alpha$)

• Acceptance Quality Limit (AQL) (called the Acceptable Quality Level in MIL-STD-105E and older editions): The poorest quality level (maximum fraction nonconforming $p_1$) that the consumer considers satisfactory as a continuous process average. The producer expects lots of this quality to be accepted almost every time.
• Producer's Risk ($\alpha$): The probability that a lot of AQL quality will be mistakenly rejected by the sampling plan. This is a Type I error (rejecting a true null hypothesis $H_0: p = p_{\text{AQL}}$). In industrial practice, $\alpha$ is conventionally set to $0.05$ (5%), meaning $P_a(p_{\text{AQL}}) = 1 - \alpha = 0.95$.

2. Lot Tolerance Percent Defective (LTPD) & Consumer's Risk ($\beta$)

• Lot Tolerance Percent Defective (LTPD) (also called Rejectable Quality Level, RQL, or Limiting Quality, LQ): An unacceptable quality level ($p_2$) that the consumer demands be rejected almost every time.
• Consumer's Risk ($\beta$): The probability that an inferior lot of LTPD quality will be mistakenly accepted by the sampling plan. This is a Type II error (failing to reject a false null hypothesis). In industrial standards, $\beta$ is conventionally set to $0.10$ (10%), meaning $P_a(p_{\text{LTPD}}) = \beta = 0.10$.

3. Discrimination Ratio / Operating Ratio ($OR$)

The ratio of LTPD to AQL measures the discriminating power of the sampling plan:

$OR = \frac{p_{\text{LTPD}}}{p_{\text{AQL}}} = \frac{p_{0.10}}{p_{0.95}}$

A smaller Operating Ratio indicates a steeper OC curve, providing tighter discrimination between acceptable and unacceptable lots, but requiring a significantly larger sample size $n$.

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5. Rectifying Inspection Metrics: AOQ, AOQL, and ATI

In a rectifying inspection program, rejected lots are not returned to the vendor; instead, they undergo 100% screening inspection, where all defective units are removed and replaced with conforming items. Furthermore, all defective units identified in the sample $n$ from accepted lots are also replaced with conforming units.

1. Average Outgoing Quality (AOQ)

AOQ is the expected average fraction nonconforming in the outgoing product stream after rectifying inspection. It is a function of the incoming fraction nonconforming $p$:

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

• When incoming quality is pristine ($p \to 0$), $P_a \approx 1.0$, and $AOQ \approx 1.0 \times 0 = 0$.
• When incoming quality is terrible ($p \to 1.0$), $P_a \to 0.0$, so virtually all lots are rejected and 100% screened, replacing all defectives with good items, hence $AOQ \approx 0 \times p = 0$.
• In between, the AOQ curve rises to a peak and then descends.

2. Average Outgoing Quality Limit (AOQL)

The Average Outgoing Quality Limit (AOQL) is the mathematical maximum of the AOQ curve across all possible values of incoming fraction defective $p$:

$AOQL = \max_{0 \le p \le 1} \left[ AOQ(p) \right]$

The AOQL represents the absolute worst-case average quality level that can exit the rectifying inspection process over the long term, regardless of how poor incoming vendor quality might be.

Using the Poisson approximation for a given acceptance number $c$, the peak occurs at a dimensionless value $y_m = n \cdot p_m$:

$AOQL = y_m \cdot P_a(y_m) \left( \frac{1}{n} - \frac{1}{N} \right) \approx \frac{y_m \cdot P_a(y_m)}{n}$

3. Average Total Inspection (ATI)

ATI represents the average number of units inspected per lot under rectifying inspection:

• If a lot is accepted (probability $P_a$), only the sample of $n$ units is inspected.
• If a lot is rejected (probability $1 - P_a$), all $N$ units in the lot are inspected.

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

• As $p \to 0$, $P_a \to 1.0$, and $ATI \to n$ (minimum inspection cost).
• As $p \to 1$, $P_a \to 0.0$, and $ATI \to N$ (full screening of every lot).

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6. Double and Multiple Sampling Plans

A Double Sampling Plan introduces a second stage of sampling to resolve borderline cases, offering psychological benefits and reduced inspection costs for very clean or very dirty lots.

Plan Parameters

Specified by $(N, n_1, c_1, r_1, n_2, c_2, r_2)$, where typically $r_1 = c_2 + 1$ and $r_2 = c_2 + 1$:

• $n_1, n_2$: Sample sizes for the first and second samples.
• $c_1$: Acceptance number for sample 1.
• $r_1$: Rejection number for sample 1.
• $c_2$: Cumulative acceptance number after sample 2 ($c_2 \ge c_1$).
• $r_2$: Cumulative rejection number after sample 2 ($r_2 = c_2 + 1$).

                    [ Draw Sample n1 ]
                    [ Count Defects d1]
                           |
         +-----------------+-----------------+
         |                                   |
     d1 <= c1                             d1 >= r1
         |                                   |
     [ ACCEPT ]                          [ REJECT ]
         |                                   |
         +------------- c1 < d1 < r1 --------+
                           |
                    [ Draw Sample n2 ]
                    [ Count Defects d2]
                           |
                    d_total = d1 + d2
                           |
                 +---------+---------+
                 |                   |
           d_total <= c2       d_total >= r2
                 |                   |
             [ ACCEPT ]          [ REJECT ]

Average Sample Number (ASN)

The Average Sample Number for double sampling is:

$ASN = n_1 + n_2 \cdot P(\text{Decision deferred to 2nd sample}) = n_1 + n_2 \cdot P(c_1 < d_1 < r_1)$

When lot quality is either exceptionally good ($p \ll AQL$) or exceptionally bad ($p \gg LTPD$), a definitive accept/reject decision is reached on the first sample $n_1$. Because $n_1 < n_{\text{single}}$, a double sampling plan usually inspects fewer units on average than a single plan with a matching OC curve, although the ASN can exceed the single-plan sample size for lots near the AQL–LTPD region.

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7. ANSI/ASQ Z1.4 (Attributes) and Z1.9 (Variables) Standards

The most widely recognized acceptance sampling standards in industry are ANSI/ASQ Z1.4 (inspection by attributes) and ANSI/ASQ Z1.9 (inspection by variables), which directly descend from Military Standards MIL-STD-105E and MIL-STD-414.

Inspection Levels in ANSI/ASQ Z1.4

The standard provides two tiers of inspection levels that correlate lot size $N$ with a sample size code letter:

1. General Inspection Levels (I, II, III):
   • Level II: The default standard for normal operations.
   • Level I: Smaller samples than Level II; used when less discrimination is acceptable.
   • Level III: Larger samples than Level II; used when greater discrimination is required.
2. Special Inspection Levels (S-1, S-2, S-3, S-4): Utilize very small sample sizes; designated strictly for destructive testing, highly expensive test procedures, or large lots where substantial risk can be absorbed.

Dynamic Switching Rules

ANSI/ASQ Z1.4 is an adaptive system. It uses the supplier's recent quality history to switch between three operational states: Normal, Tightened, and Reduced inspection.

     +-------------------------------------------------------+
     |                                                       |
     |    [ TIGHTENED ] <====== 2 out of 5 lots rejected == [ NORMAL ]
     |          |                                                |
     |    5 consecutive                                    10 consecutive
     |    lots accepted                                    lots accepted
     |          |                                                |
     |          v                                                v
     |      [ NORMAL ] <=== 1 lot rejected, irregular === [ REDUCED ]
     |                      production, or authority
     |                      decision
     |
Z1.4: 5 lots not accepted while on Tightened
(MIL-STD-105E: 10 consecutive lots remain on Tightened)
     |
     v
[ DISCONTINUE INSPECTION ] (Suspend acceptance from vendor until corrective action verified)

• Normal to Tightened: Initiated when 2 out of 5 (or fewer) consecutive lots are rejected on original normal inspection. Tightened inspection keeps $n$ constant but lowers the acceptance number $c$, steepening the OC curve to penalize the supplier.
• Tightened to Normal: Initiated when 5 consecutive lots are accepted on original tightened inspection.
• Tightened to Discontinue: Under ANSI/ASQ Z1.4-2003 and later, inspection is discontinued when the cumulative number of lots not accepted in a sequence of consecutive lots on tightened inspection reaches 5. MIL-STD-105E used a different trigger: 10 consecutive lots remaining on tightened inspection. Acceptance resumes only after the supplier corrects the problem, starting on tightened inspection.
• Normal to Reduced: Permitted when 10 consecutive lots have been accepted under normal inspection, total nonconforming items across those lots falls below standard limit tables, production is steady, and quality management approves.
• Reduced to Normal: Triggered immediately if 1 lot is rejected, production becomes irregular, or customer authority mandates return.

Attributes (Z1.4) vs. Variables (Z1.9) Comparison

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8. Summary of Key Mathematical Formulas

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9. Dodge-Romig Tables and the MIL-STD-105E Lineage

Dodge-Romig Sampling Tables

Harold Dodge and Harry Romig published sampling tables for rectifying inspection, where rejected lots are 100% screened and defectives replaced. Their plans are consumer-oriented:

• LTPD plans guarantee a consumer's risk of 0.10 at a stated LTPD (lot tolerance percent defective). A lot that is only LTPD quality is accepted no more than 10% of the time.
• AOQL plans guarantee that average outgoing quality never exceeds a stated AOQL, whatever the incoming quality.

Within each family, the tables give single and double sampling plans indexed by lot size and process average. Among all plans that meet the protection requirement, the tabulated plan is the one that minimizes Average Total Inspection (ATI) at that process average. To use them, you need the lot size, the supplier's process average (fraction defective), and the required LTPD or AOQL. The plans apply only when rejected lots are actually screened.

MIL-STD-105E, ANSI/ASQ Z1.4, and MIL-STD-414

• MIL-STD-105E (1989) is the last edition of the military attribute sampling standard named in the NCEES specification. The Department of Defense canceled it in 1995 and adopted the civilian ANSI/ASQ Z1.4, which kept the same sample-size code letters and tables.
• ANSI/ASQ Z1.4 (and the international ISO 2859-1) is AQL-based. It protects the producer at the AQL and uses switching rules to raise protection when quality slips. The main differences from 105E are the term "Acceptance Quality Limit" and the discontinuation rule described above.
• MIL-STD-414, the variables-sampling counterpart, became ANSI/ASQ Z1.9.