Question 6 of 6: 100% Inspection vs. Acceptance Sampling, MIL-STD-105E, and a Single Sampling Plan

Question 6: 100% Inspection vs. Acceptance Sampling, MIL-STD-105E, and a Single Sampling Plan (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) 100% inspection vs. acceptance sampling; rectifying inspection, AQL, AOQL, LTPD

100% inspection checks every unit in a lot, in principle catching every nonconforming item and eliminating lot-disposition risk entirely; in practice it is expensive and slow for large lots, is impossible for destructive tests, and inspector fatigue/monotony on a large volume of repetitive inspection typically lets 100% inspection screen out only 80–90% of true defectives rather than 100% — the "100%" in the name describes coverage attempted, not accuracy achieved. Acceptance sampling inspects only a random sample and accepts/rejects the whole lot on that basis; it is far cheaper and faster, is the only option for destructive testing, and (counter-intuitively) can sometimes screen a lot MORE reliably than a fatigued 100%-inspection effort because a smaller, better-controlled sample gets more careful attention per unit — but it inherently accepts some bad lots (producer's/consumer's risk) and rejects some good ones, and provides no protection at all against a defect that happens to fall in the unsampled portion of an accepted lot.

Sampling plans that explicitly control BOTH producer's risk (rejecting a good lot at the AQL) and consumer's risk (accepting a bad lot at the LTPD/LQL) simultaneously are plans designed around two points on the OC curve — MIL-STD-105E (and its ANSI/ASQC Z1.4 civilian counterpart) is constructed this way, as are the Dodge–Romig LTPD-protection tables (though the latter emphasize the consumer point). Rectifying inspection is a sampling scheme in which every REJECTED lot is then 100% inspected (and all discovered defectives replaced with good units), so that the AVERAGE outgoing quality after the whole scheme is applied is bounded no matter how bad an individual incoming lot was — this bound is the AOQL (Average Outgoing Quality Limit), the worst-case long-run average fraction defective a customer will see under the rectifying scheme, maximized over all possible incoming quality levels. The AQL (Acceptable Quality Level) is the poorest quality level from the PRODUCER's process that the sampling plan still treats as acceptable most of the time (a high probability of acceptance) — it is a contractual target for the producer's own process, not a promise about any single lot. The LTPD (Lot Tolerance Percent Defective, also called LQL, Limiting Quality Level) is the poorest quality level the CONSUMER is willing to accept only rarely (a low, specified probability of acceptance, the consumer's risk point) — the plan is chosen so that a lot this bad is rejected with high probability.

(b) MIL-STD-105E features and comparison with Dodge–Romig

MIL-STD-105E is an AQL-indexed sampling SYSTEM (not just a single table): it provides single, double, and multiple sampling plans, organized around a chosen AQL and a sample-size code letter that depends on lot size and inspection level (I, II, III for general use, plus S-1–S-4 for special, small-sample situations). Its central feature is a built-in SWITCHING RULE among three inspection severities — normal, tightened, and reduced — that automatically escalates to tightened inspection (smaller acceptance numbers) after a run of rejected lots, and permits reduced inspection (smaller sample sizes) after a sustained run of good lots, so that a supplier's own recent track record continuously adjusts the inspection burden without a new contract negotiation. All published plans in a given AQL column are constructed to have (approximately) the same probability of accepting an AQL-quality lot regardless of code letter, giving a family of plans with a shared producer's-risk protection at the stated AQL.

The Dodge–Romig tables, by contrast, are indexed by LTPD (or by AOQL) rather than by AQL: they are built to guarantee the CONSUMER's protection point directly (a specified, usually low, probability of accepting a lot at the stated LTPD, or a guaranteed AOQL ceiling under rectifying inspection) and require knowledge of the incoming process average quality to select the most economical plan meeting that LTPD/AOQL target. In short, MIL-STD-105E is a producer-oriented, AQL-indexed system built around switching rules and requires no knowledge of the true incoming quality level to apply; Dodge–Romig is a consumer-oriented, LTPD/AOQL-indexed system that requires an estimate of the process average to select the most economical (smallest-sample) plan meeting the guaranteed protection.

(c) Single sampling plan for lots of 2,000, AQL = 0.5%

Given. Lot size $N=2{,}000$; required $AQL=0.5\%$; normal inspection, general inspection level II; $LQL=2\%$.

Find. The MIL-STD-105E single sampling plan ($n$, $Ac$, $Re$); producer's risk at the AQL; consumer's risk at the LQL.

1. Sample-size code letter. Table 10-34 (Sample Size Code Letters): lot size 2,000 falls in the $1{,}201$–$3{,}200$ row; general inspection level II gives code letter $\boxed{K}$.
2. Sample size. Table 13-5 (MIL-STD-105E, Table II-A): code letter $K\ \Rightarrow\ n=\boxed{125}$.
3. AQL conversion. Table 10-33 (AQL Conversion Table): the required $AQL=0.5\%$ falls in the "$0.440$ to $0.699$" band, which converts to the nearest STANDARD AQL value of $\boxed{0.65\%}$ — this is the AQL column actually used to read the plan.
4. Acceptance/rejection numbers. Reading Table 13-5 at code letter $K$ (row), AQL $=0.65\%$ (column) directly from the master table: $\boxed{Ac=3}$, $\boxed{Re=4}$.
5. Producer's risk, $\alpha$ (probability of rejecting a lot at the AQL). With $n=125$, $p=AQL=0.0065$, using the Poisson approximation ($\lambda=np=0.8125$): $$\alpha=P(D>3\mid\lambda=0.8125)=1-\sum_{d=0}^{3}\frac{e^{-0.8125}0.8125^d}{d!}=\boxed{0.0096\ (\approx1.0\%)}.$$
6. Consumer's risk, $\beta$ (probability of accepting a lot at the LQL). With $p=LQL=0.02$, $\lambda=np=125(0.02)=2.5$: $$\beta=P(D\le3\mid\lambda=2.5)=\sum_{d=0}^{3}\frac{e^{-2.5}2.5^d}{d!}=\boxed{0.758\ (\approx75.8\%)}.$$