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

Question 6: Acceptance Sampling 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) Purpose of acceptance sampling, traditional vs. sequential sampling, and MIL-STD-105E

Acceptance sampling exists to decide whether to accept or reject an entire lot of incoming or outgoing product based on inspecting only a sample from it, when 100% inspection is impractical (destructive testing), too costly, too slow, or itself unreliable (inspector fatigue on a very long run) — it is a lot-disposition tool, not a process-control tool, and it does not itself improve quality, only screens it at the lot boundary.

Traditional (single/double/multiple) sampling draws one fixed-size sample (or a small fixed sequence of samples) per lot, and the accept/reject decision is made once the predetermined sample(s) have been fully inspected. Sequential sampling instead inspects items one at a time (or in small increments), plotting a cumulative statistic against pre-computed accept/reject/continue boundaries after every item, and stops as soon as either boundary is crossed; on average it requires markedly fewer items inspected per lot than a fixed-size plan with comparable discriminating power, because it stops as soon as the evidence is decisive rather than always inspecting the full fixed sample.

MIL-STD-105E is a widely used, standardized system of sampling plans (single, double, and multiple) indexed by lot size, inspection level, and Acceptable Quality Level. It provides three severities of inspection — normal, tightened, and reduced — with formal, automatic switching rules between them driven by the recent run of lot acceptance/rejection history, so that a supplier with a sustained good record is rewarded with lighter (reduced) inspection while a supplier showing deterioration is automatically escalated to tightened inspection (and ultimately inspection suspended) without a separate management decision each time. Being AQL-based means each plan is designed and indexed around a stated Acceptable Quality Level — the poorest process average the standard treats as satisfactory for the purpose of routine, sustained acceptance — not around a single sharp accept/reject cutoff; a lot at exactly the AQL has a high (though not certain) probability of acceptance, and that probability falls off as the true lot/process quality worsens beyond the AQL, which is the standard's whole basis for its OC-curve behaviour and its escalation rules.

(b) Dodge-Romig vs. MIL-STD-105 plans, and AOQ/AOQL

The two systems are built around different economic objectives. MIL-STD-105 plans are indexed to a stated AQL and are designed primarily to protect the producer: given a process running at or better than the AQL, the producer's risk of having an acceptable lot rejected is held low and roughly consistent across the whole table, with consumer protection addressed indirectly through the normal/tightened/reduced switching discipline rather than a lot-by-lot numeric guarantee. Dodge-Romig plans are indexed instead to either a stated LTPD (a rejectable quality level, protecting the consumer directly) or to minimizing the AOQL, and they explicitly assume that any rejected lot is subjected to rectifying inspection (100% inspected, defectives replaced with good units) — a requirement MIL-STD-105 plans do not carry. Because Dodge-Romig plans use the process average fraction defective as an explicit input and guarantee a numeric outgoing-quality bound lot-by-lot, they generally require larger sample sizes than a comparable MIL-STD-105 AQL plan, but give a stronger, more direct consumer-protection guarantee.

AOQ (Average Outgoing Quality) is the expected fraction defective remaining in the outgoing product stream after acceptance sampling with rectification — accounting for the fact that accepted lots ship with their original (sampled but not perfect) defect rate, while rejected lots are 100%-inspected and cleaned up to (near) zero defects before shipping, so $AOQ(p)=\dfrac{p\,P_a(p)\,(N-n)}{N}$ (approximately $p\,P_a(p)$ for large lots) as a function of the incoming process fraction defective $p$. Because $P_a(p)$ falls as $p$ worsens, $AOQ(p)$ rises from zero (at $p=0$), peaks, and falls back toward zero (as $p\to1$, since almost every lot is rejected and rectified); its peak value, the AOQL (Average Outgoing Quality Limit), is the worst-case long-run average outgoing defect rate the plan can produce, regardless of how bad the incoming process quality gets — the single number a Dodge-Romig AOQL-minimizing plan is designed directly around.

(c) MIL-STD-105E single sampling plan: lot $=800$, AQL$=1.5\%$, normal inspection, general level II; risks at RQL$=10\%$

Given. Lot size $N=800$; AQL $=1.5\%$; normal inspection; general inspection level II; MIL-STD-105E Table I (sample-size code letters) and Table II-A (master table for normal inspection, single sampling).

Find. The single sampling plan ($n$, $Ac$, $Re$); the producer's risk $\alpha$ at $p=$AQL; the consumer's risk $\beta$ at $p=$RQL$=10\%$.

Approach. Look up the sample-size code letter from Table I at ($N=800$, general level II), read the $(Ac,Re)$ pair directly off Table II-A at that code letter and AQL$=1.5\%$, then compute the two risks from the plan's OC curve, $P_a(p)=P(X\le Ac\mid n,p)$ with $X\sim\text{Binomial}(n,p)$.

1. Sample-size code letter. $N=800$ falls in Table I's "501 to 1200" row; the General Inspection Level II column reads code letter $\boxed{J}$ ($n=80$).
2. Sampling plan. Table II-A, row $J$, AQL$=1.5\%$ column, gives a direct entry (no arrow-substitution needed): $\boxed{n=80,\ Ac=3,\ Re=4}$.
3. Producer's risk $\alpha$ (probability a lot at the AQL is rejected). With $X\sim\text{Binomial}(80,\,0.015)$, $$\alpha=P(X>3)=1-P(X\le3)=\boxed{0.0326}\ (3.3\%).$$ This is comfortably low, as intended — a process running right at the AQL is accepted about 97% of the time.
4. Consumer's risk $\beta$ (probability a lot at the RQL is accepted). With $X\sim\text{Binomial}(80,\,0.10)$, $$\beta=P(X\le3)=\boxed{0.0353}\ (3.5\%).$$ A lot whose true fraction defective has degraded all the way to $10\%$ (RQL, nearly $7\times$ the AQL) is still accepted only about 3.5% of the time — the plan discriminates sharply between the two quality levels.