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.

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

100% inspection examines every unit in a lot and, in principle, screens out every detectable nonconforming unit before shipment, which is attractive when a single escaped defect is very costly (safety-critical or destructive-failure items) or when the incoming quality is poor/unknown. Its disadvantages are substantial: it is expensive and slow for large lots, it is simply not possible for destructive tests, and — counterintuitively — it is NOT perfectly effective, since inspector fatigue and monotony typically let 100% inspection catch only about 80–95% of true nonconformances rather than 100%, so a large, tedious 100% inspection can in practice catch fewer defects per inspector-hour than a well-designed sample. Acceptance sampling inspects only a randomly drawn sample from each lot and accepts/rejects the whole lot based on the sample result; it is far cheaper and faster, is the only option for destructive testing, and (because inspectors examine fewer units more carefully) can sometimes achieve HIGHER effective quality than a fatigued 100% inspection. Its disadvantage is that it carries genuine sampling risk — a plan can accept a bad lot (consumer's risk) or reject a good lot (producer's risk) purely by chance, and it provides no protection against defects in individual units of an accepted lot beyond what "rectifying" (100%-inspecting rejected lots) provides.

Variables sampling plans measure a continuous quality characteristic (e.g. a dimension, a strength) on each sampled unit and use the sample mean and standard deviation, compared against the specification via a statistic like $Z$, to accept or reject the lot; they need FEWER sampled units than an equivalent attributes plan for the same statistical protection because a measured value carries far more information than a simple pass/fail classification, but they require the characteristic to follow a known distribution (usually normal) and a separate plan/statistic for every characteristic measured. Attributes sampling plans classify each sampled unit simply as conforming/nonconforming (or count total defects) and accept/reject based on the number of nonconforming units found versus the acceptance number $Ac$; they need no distributional assumption, can combine many different defect types into one plan, and are simpler to administer on the shop floor, at the cost of needing a larger sample to achieve the same discriminating power as a variables plan.

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

The acceptable quality level (AQL) is the poorest process-average fraction nonconforming that is still considered acceptable as a process average when submitted for acceptance sampling — a sampling plan is designed so lots at the AQL have a HIGH probability of acceptance (protecting the producer). The limiting quality level (LQL), also called LTPD (lot tolerance percent defective), is the poorest quality in an INDIVIDUAL lot that the consumer is willing to accept only occasionally — a plan is designed so lots at the LQL have a LOW probability of acceptance (protecting the consumer). Rectifying inspection refers to a sampling scheme in which every REJECTED lot is subjected to 100% inspection (with all discovered nonconforming units replaced or removed) before it may ship, so the final outgoing quality is a mixture of the (typically good) quality of lots that were sampled and passed, and the (perfect, by construction) quality of lots that were sampled, failed, and fully rectified.

The average outgoing quality (AOQ) is the expected fraction nonconforming in the outgoing (post-inspection) product stream as a function of the incoming lot quality $p$, under a rectifying scheme: $AOQ(p)=\dfrac{P_a(p)\cdot p\cdot(N-n)}{N}$, where $P_a(p)$ is the plan's probability of acceptance at incoming quality $p$. Because $P_a(p)$ falls as incoming quality worsens (more lots get 100%-inspected and rectified to near-zero defects), $AOQ(p)$ rises from zero, reaches a maximum, and then falls back toward zero as $p\to1$ — that maximum is the average outgoing quality limit (AOQL), the worst-case long-run average outgoing quality the rectifying scheme can produce REGARDLESS of how bad the incoming lots are, and it is a key design/marketing guarantee of a rectifying sampling plan.

An LTPD (Dodge–Romig) plan is designed around a SINGLE target: guarantee a fixed, low probability (commonly 10%) of accepting a lot at the stated LTPD, using the sample size and AOQL as secondary design constraints — it is optimized for minimum inspection given LTPD protection, and it requires knowledge of (or an assumption about) the process average to select the tightest plan. A MIL-STD-105E plan is instead built around the AQL as its primary index, organized into a switching system among Normal, Tightened and Reduced inspection based on recent lot history, so that sustained good quality is rewarded with LESS inspection (Reduced) while a run of rejected lots automatically triggers MORE inspection (Tightened) — it protects the producer at the AQL directly, gives only indirect (weaker, and switching-dependent) consumer protection compared with an LTPD plan, and is the standard chosen when ongoing supplier-relationship management (not a single worst-case guarantee) is the goal.

(c) MIL-STD-105E single sampling plan for $N=1000$, AQL$=1.5\%$, normal inspection, level II

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

Find. The single sampling plan ($n$, $Ac$, $Re$) and the producer's risk of this plan.

1. Sample-size code letter. From Table 14‑4 (Sample Size Code Letters), a lot size of 501–1200 under General Inspection Level II gives code letter J.
2. Sampling plan from the master table. Table 13‑5 (Master Table for Normal Inspection, Single Sampling) gives code letter J a sample size of $n=80$; reading across to the AQL$=1.5\%$ column gives $\boxed{Ac=3,\ Re=4}$.
3. Producer's risk. The producer's risk is the probability the plan REJECTS a lot whose true fraction nonconforming equals the AQL, $p=0.015$. Using the Poisson approximation ($\lambda=np=80(0.015)=1.2$, appropriate since $n\ll N$ and $p$ is small): $$\alpha=P(D>Ac\mid p=0.015)=1-\sum_{d=0}^{3}\frac{e^{-1.2}1.2^{d}}{d!}=\boxed{0.0338\ (3.38\%)}.$$ (Exact binomial cross-check: $1-\sum_{d=0}^{3}\binom{80}{d}0.015^{d}(0.985)^{80-d}=0.0326$, in close agreement.)