Question 6 of 6: Acceptance Sampling Concepts and a MIL-STD-105E Single Sampling Plan

Question 6: Acceptance Sampling Concepts and a MIL-STD-105E 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.

Part (a) — 100% inspection vs. acceptance sampling; variables vs. attributes plans

100% inspection examines every unit in a lot. Its advantage is, in principle, complete screening — no nonconforming unit should pass, provided the inspection is perfect. In practice this advantage is often illusory: inspector fatigue and monotony on a large, repetitive task typically hold true screening effectiveness to 80–95%, so 100% inspection is neither free of escapes nor free of cost — it is also the slowest and most expensive option per lot, and is destructive testing's worst case (100% inspection of a destructively-tested characteristic destroys the entire lot). Acceptance sampling inspects only a sample and infers the lot's disposition statistically. Its advantages are much lower inspection cost and time, applicability to destructive testing, and (counter-intuitively) sometimes better overall reliability than a fatigued 100% inspection, because a smaller, well-designed sample can be inspected more carefully. Its disadvantage is sampling risk: some good lots will be rejected (producer's risk) and some bad lots accepted (consumer's risk) purely from sample-to-sample variation, and unlike a control chart, acceptance sampling provides no direct, ongoing feedback for improving the process itself — it only sorts already-produced lots.

Attribute sampling plans classify each inspected unit as simply conforming/nonconforming (or count its nonconformities) and accept/reject the lot by comparing the observed count against an acceptance number $Ac$ — simple to apply, requiring no assumption about the underlying distribution of the measured characteristic, but relatively inefficient (a large sample size is needed for a given level of protection, since each unit yields only one bit of information). Variables sampling plans instead measure the actual value of a continuous quality characteristic on each sampled unit and use the sample mean and standard deviation (assuming an underlying distribution, almost always normal) to estimate the fraction outside specification directly — this uses much more information per unit, so a variables plan achieves the same level of protection (the same OC curve) with a substantially smaller sample size than the equivalent attributes plan. The trade-off is that variables plans are more complex to administer (a separate plan is generally needed per specification limit and per characteristic, and the normality assumption must hold reasonably well or the risk calculations become unreliable), whereas one attributes plan can cover several different types of nonconformity on the same unit at once.

Part (b) — AQL, LQL, rectifying inspection, AOQ, AOQL; LTPD (Dodge–Romig) vs. MIL-STD-105E

The acceptable quality level (AQL) is the worst tolerable process average (fraction nonconforming) that, for purposes of acceptance sampling, is still regarded as satisfactory as a long-run process average — a sampling plan is designed so that lots at the AQL are accepted with high probability (the producer's risk of rejecting an AQL-quality lot is kept low, conventionally around 5%). The limiting quality level (LQL) (also called LTPD, lot tolerance percent defective, in the Dodge–Romig framework) is the worst quality in an individual lot that the consumer is willing to accept only rarely — the plan is designed so lots at the LQL are accepted with low probability (the consumer's risk, conventionally around 10%).

Rectifying inspection is an acceptance-sampling scheme under which every REJECTED lot is subjected to 100% inspection (and every nonconforming unit found is repaired or replaced with a conforming one), so that a rejected lot leaves the inspection station 100% conforming; only accepted lots pass through with their (unknown, sampled-but-not-corrected) fraction nonconforming intact. This creates a self-limiting quality-improvement mechanism, formalized by the average outgoing quality (AOQ): the expected fraction nonconforming in the outgoing (post-inspection) product, averaged over both accepted lots (which ship with their original fraction nonconforming $p$) and rejected-then-rectified lots (which ship at 0% nonconforming), i.e. $AOQ\approx p\cdot P_a(p)$ where $P_a(p)$ is the plan's probability of acceptance at incoming quality $p$. Because $AOQ\to0$ as $p\to0$ (almost everything is accepted, but there is little to find) and $AOQ\to0$ as $p\to1$ (almost everything is rejected and rectified to zero defects), $AOQ(p)$ has an interior maximum called the average outgoing quality limit (AOQL) — the worst-case average outgoing quality the rectifying scheme can produce, regardless of how bad the incoming quality actually is. The AOQL is therefore a guarantee to the consumer that is independent of the supplier's actual (possibly unknown or drifting) process quality, which no non-rectifying attributes plan can offer.

An LTPD plan such as Dodge–Romig is explicitly designed around a target consumer protection: given a stated LTPD and a fixed consumer's risk (typically 10%), the sample size and acceptance number are chosen to guarantee that risk at that specific quality level, and separate Dodge–Romig tables exist for minimizing average total inspection (AOQL tables) versus minimizing consumer risk directly (LTPD tables) — either way, the design target is the consumer's-risk point on the OC curve. A MIL-STD-105E plan, by contrast, is indexed by the AQL and is designed primarily around producer protection: for a stated AQL and lot-size/inspection-level combination, the plan guarantees a low producer's risk of rejecting AQL-quality lots, but it does not directly control what happens at any particular LQL — the actual consumer's risk that results at a given LQL is whatever falls out of the AQL-indexed plan's OC curve, which (as illustrated numerically in part (c)) can be much larger than the conventional 10% a Dodge–Romig LTPD plan would guarantee at the same LQL. MIL-STD-105E also includes a switching mechanism between normal, tightened, and reduced inspection based on recent lot history, which Dodge–Romig LTPD plans do not.

Part (c) — MIL-STD-105E single sampling plan, lot $=1000$, $AQL=1.5\%$, general level II, normal inspection

Given. Lot size $N=1000$; $AQL=1.5\%$; general inspection level II; normal inspection; $LQL=3\%$.

Find. The sample size $n$, acceptance number $Ac$ and rejection number $Re$; the producer's risk at $AQL$; the consumer's risk at $LQL$.

Approach. Look up the sample-size code letter from the lot-size/inspection-level table, then the $n$, $Ac$, $Re$ for that code letter and $AQL$ from the MIL-STD-105E master table for normal inspection; compute both risks from the binomial OC function of the resulting plan.

1. Sample-size code letter. $N=1000$ falls in the $501$–$1200$ lot-size row; at general inspection level II this gives code letter $\boxed{J}$.
2. Sampling plan. Code letter $J$ carries a sample size of $n=\boxed{80}$; at $AQL=1.5\%$ the master table (normal inspection, single sampling) gives $Ac=\boxed{3}$, $Re=\boxed{4}$ — accept the lot if 3 or fewer of the 80 sampled parts are nonconforming, reject if 4 or more are found.
3. Producer's risk ($\alpha$, at $p=AQL=0.015$). With $X\sim\text{Binomial}(80,0.015)$, the probability of (wrongly) rejecting an AQL-quality lot is $$\alpha=P(X\ge4)=1-P(X\le3)=1-0.9674=\boxed{0.0326\ (3.26\%)}.$$ This is comfortably close to the conventional $\sim5\%$ producer's-risk target MIL-STD-105E plans are built around.
4. Consumer's risk ($\beta$, at $p=LQL=0.03$). With $X\sim\text{Binomial}(80,0.03)$, the probability of (wrongly) accepting an LQL-quality lot is $$\beta=P(X\le3)=\boxed{0.7807\ (78.1\%)}.$$

The consumer's risk of 78.1% at $LQL=3\%$ is strikingly high — over three-quarters of lots at the stated "limiting" quality level would still be accepted by this AQL-indexed plan. This is exactly the structural difference discussed in part (b): the plan was designed to hold the producer's risk near 5% at the AQL, and the consumer's risk at any particular LQL is simply whatever falls out of that same OC curve, with no independent design control over it. A purchaser who actually needs a 10% (or lower) consumer's risk guarantee at $LQL=3\%$ would need a Dodge–Romig LTPD plan built around that target directly, not a MIL-STD-105E AQL-indexed plan.