Question 6 of 6: Acceptance Sampling — AQL/LTPD/AOQL, Sequential Sampling, and MIL-STD-105E

Question 6: Acceptance Sampling — AQL/LTPD/AOQL, Sequential Sampling, and MIL-STD-105E (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) AQL-based plans; Dodge-Romig LTPD plans; AOQL

An AQL-based sampling plan is designed around the Acceptable Quality Level — the worst tolerable process average nonconformance rate that, when actually submitted, should still be accepted most of the time (i.e. the plan is chosen so the producer's risk of rejecting AQL-quality lots is kept low, conventionally around 5% for many published plans). The plan's parameters ($n$, $Ac$) are selected to satisfy this producer-side condition first.

Dodge-Romig LTPD plans are not AQL-based. They are designed the opposite way, around the Lot Tolerance Percent Defective — the worst quality that should be accepted only rarely (fixing the consumer's risk, conventionally 10%, at the LTPD) — and then, subject to that consumer-protection constraint, chosen to minimize either the Average Outgoing Quality Limit or the Average Total Inspection, i.e. they are consumer-risk-anchored rather than producer-risk-anchored.

AOQL (Average Outgoing Quality Limit) is the maximum, over all possible incoming lot quality levels $p$, of the Average Outgoing Quality $AOQ(p)=p\cdot P_a(p)$ (assuming 100% inspection and correction of rejected lots) — i.e. no matter how bad the incoming quality gets, the long-run average outgoing quality after the sampling/rescreening plan cannot exceed the AOQL. It characterizes the worst-case long-run protection the plan gives the consumer under rectifying inspection, independent of any single LTPD point.

(b) Traditional vs. sequential sampling; MIL-STD-105E vs. MIL-STD-414

Traditional (fixed-size) sampling draws one predetermined sample of size $n$ and applies a single accept/reject rule. Advantages: simple to administer, fixed and predictable inspection workload/cost, easy to plan staffing around. Disadvantages: always inspects the full $n$ units even when the lot is obviously very good or very bad early in the count, so its expected sample size is larger than necessary in those clear-cut cases.

Sequential sampling inspects one unit (or small group) at a time and, after each, compares the cumulative nonconforming count against continuation/accept/reject boundaries (a sequential probability ratio test), stopping as soon as a decision is statistically justified. Advantage: substantially lower average sample number (ASN) for the same OC-curve protection, especially valuable when inspection is destructive, slow, or expensive per unit. Disadvantages: variable (unpredictable) sample size and inspection time per lot, more complex administratively, and it requires item-by-item testing/recording rather than a single batch count. Sequential sampling is preferable when unit inspection/test cost is high (e.g. destructive testing, expensive lab analysis) so minimizing the average number tested has real economic value, and when inspection results become available one at a time in a natural sequence.

MIL-STD-105E vs. MIL-STD-414: MIL-STD-105E is an attributes (go/no-go) AQL-indexed sampling system — each unit is simply classed conforming/nonconforming, needing only a count. MIL-STD-414 is a variables AQL-indexed system — it uses the actual measured value of a (assumed normally distributed) quality characteristic and its distance from the specification limit(s) in units of the sample standard deviation. For the same AQL protection, MIL-STD-414 achieves it with a substantially smaller sample size than the equivalent MIL-STD-105E attributes plan (because a measured value carries more information than a binary pass/fail), at the cost of requiring an assumption of normality and destructive/expensive measurement rather than simple gauging in some cases.

(c) MIL-STD-105E single sampling plan: lot=2,500, General II, AQL=0.25%

Given. Normal inspection, General Inspection Level II, lot size $N=2{,}500$, $AQL=0.25\%$; $LQL=2\%$ for the risk estimates.

Find. Sample size code letter, sample size $n$, acceptance/rejection numbers $Ac/Re$; producer's risk $\alpha=P(\text{reject}\mid p=AQL)$ and consumer's risk $\beta=P(\text{accept}\mid p=LQL)$.

1. Sample size code letter (Table 14-4). Lot size 2,500 falls in the row "1,201 to 3,200"; the General Inspection Level II column gives code letter K.
2. Sampling plan (Table 13-5). Code letter K → sample size $n=125$. Reading the K row at $AQL=0.25\%$: $Ac=3$, $Re=4$. $$\boxed{n=125,\ Ac=3,\ Re=4}$$
3. Producer's risk at $p=AQL=0.0025$. $\alpha=P(d>3\mid n=125,p=0.0025)=1-\sum_{k=0}^{3}\binom{125}{k}(0.0025)^k(0.9975)^{125-k}$. $$\boxed{\alpha=1-0.99970=0.030\%}$$
4. Consumer's risk at $p=LQL=0.02$. $\beta=P(d\le3\mid n=125,p=0.02)=\sum_{k=0}^{3}\binom{125}{k}(0.02)^k(0.98)^{125-k}$. $$\boxed{\beta=75.9\%}$$