Question 6 of 6: AQL vs. Dodge-Romig Sampling Plans, Sequential Sampling, and a MIL-STD-105E Plan

Question 6: AQL vs. Dodge-Romig Sampling Plans, Sequential Sampling, and a MIL-STD-105E 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) AQL-based vs. Dodge-Romig plans; AQL, AOQL, LTPD

MIL-STD-105E is an AQL-indexed system: for a chosen AQL, lot size, and inspection level, the standard's tables (Table 14-4 for the sample size code letter, Table 13-5 for $n$/Ac/Re) return a plan whose OC curve is anchored so that a lot genuinely at the AQL quality level has a high probability of acceptance (a fixed, low producer's risk at that one point). What happens at any other incoming quality level — in particular, how strongly the consumer is protected against a genuinely bad lot — is a by-product of whichever plan the tables happen to return, not something independently specified. Dodge-Romig plans instead anchor the OTHER end of the OC curve: LTPD plans are designed to guarantee a specified, low consumer's risk (conventionally $\beta=10\%$) at a stated Lot Tolerance Percent Defective, for the minimum sample size that achieves it (given a stated process average incoming quality); AOQL plans instead guarantee that the long-run Average Outgoing Quality — the average fraction nonconforming shipped to the customer once rejected lots are screened (100% inspected and defectives replaced/removed) — can never exceed a stated ceiling, no matter how bad the incoming quality gets, for the minimum average total inspection.

Neither family, used alone with a single index, controls both risks to independently chosen targets: MIL-STD-105E fixes the producer's-risk point and leaves consumer's risk to fall out; Dodge-Romig LTPD/AOQL plans fix a consumer-side guarantee and leave producer's risk to fall out. A plan that controls both producer's and consumer's risk to independently chosen targets requires specifying two points on the desired OC curve simultaneously — a producer's-risk point $(AQL,\alpha)$ and a consumer's-risk point $(LTPD/LQL,\beta)$ — and solving for the $(n,Ac)$ pair whose OC curve passes through both (exactly the two-point design used in part (c) below, where the same $n=200$, $Ac=7$ plan is evaluated against both a producer's-risk point at $AQL=1.5\%$ and a consumer's-risk point at $LQL=6\%$).

AQL (Acceptable Quality Level) is the maximum percent nonconforming that, for purposes of sampling inspection, is considered satisfactory as a process average — the quality level a good, well-run supplier is expected to meet, and the level at which the producer's risk of rejection is deliberately kept low. LTPD (Lot Tolerance Percent Defective, also written LQL) is the poorest quality in an individual lot that the consumer is willing to accept only a small, specified probability ($\beta$) of accepting — the "line in the sand" below which the consumer wants strong protection. AOQL (Average Outgoing Quality Limit) is the worst-case long-run average outgoing quality across many lots of possibly varying incoming quality, under a rectifying-inspection plan where rejected lots are screened; unlike LTPD, which protects against any one bad lot, AOQL is a guarantee about the long-run average the consumer actually receives.

(b) When to recommend sequential testing; multiple vs. sequential sampling

Sequential testing is recommended whenever inspection is expensive, slow, or destructive, so that minimizing the average number of items tested (the average sample number, ASN) to reach a decision is valuable — for example, life testing of an expensive component to destruction, or a slow chemical assay where each additional test unit adds real cost and delay. Example: qualifying a new lot of high-reliability relays by testing them one at a time and updating a cumulative log-likelihood-ratio statistic after every unit; the test stops and accepts as soon as the statistic crosses a lower boundary, rejects as soon as it crosses an upper boundary, and continues testing (drawing one more unit) whenever the statistic lies between the two boundaries — for most incoming quality levels this reaches a decision with far fewer units tested, on average, than a fixed-size single or multiple plan built for the same $(AQL,\alpha)$/$(LTPD,\beta)$ protection.

A multiple sampling plan is a staged plan with a fixed, pre-set number of stages (commonly 2 for double sampling, or up to 7 for multiple sampling), each with its own pre-determined sample size; after each stage's cumulative results are tallied, the lot is accepted, rejected, or another pre-sized stage is drawn, but the stage boundaries and sizes are all fixed in advance regardless of how the data come in. A sequential sampling plan removes the notion of a "stage" altogether: items (or very small groups) are tested one at a time indefinitely, with accept/reject/continue boundaries computed once (via Wald's sequential probability ratio test) and re-applied after every single new observation, so the sample size is not fixed at all — it is a random variable that depends entirely on how quickly the evidence accumulates toward one boundary or the other. Sequential sampling generally achieves the smallest ASN of any plan offering the same $(AQL,\alpha)$/$(LTPD,\beta)$ protection, at the cost of open-ended (unpredictable) inspection duration and the administrative complexity of testing and deciding after every single unit rather than in convenient batches.

(c) MIL-STD-105E single sampling plan for a lot of 5,000, AQL $=1.5\%$

Given. Lot size $N=5{,}000$; required $AQL=1.5\%$; normal inspection, general inspection level II; consumer's-side quality of interest $LQL=6\%$.

Find. The MIL-STD-105E single sampling plan ($n$, $Ac$, $Re$); the producer's risk $\alpha$ (probability of rejecting a lot at $p=AQL$); and the consumer's risk $\beta$ (probability of accepting a lot at $p=LQL$).

Approach. Table 14-4 converts lot size and inspection level to a sample size code letter; Table 13-5 converts that code letter and the required AQL to the sampling plan ($n$, Ac, Re); the plan's own OC curve (Poisson approximation, $\lambda=np$) then gives the risks at any quality level of interest.

1. Sample size code letter. $N=5{,}000$ falls in the "3,201 to 10,000" row of Table 14-4; General inspection level II gives code letter $\boxed{L}$.
2. Sampling plan. Table 13-5, code letter $L$ row, $AQL=1.5\%$ column: $\boxed{n=200,\ Ac=7,\ Re=8}$.
3. Producer's risk at AQL $=1.5\%$. $\lambda_{AQL}=np=200(0.015)=3.0$; using the Poisson approximation, $$P_a(AQL)=P(X\le7\mid\lambda=3.0)=0.9881,\qquad \alpha=1-P_a(AQL)=\boxed{0.0119\ (1.19\%)}.$$
4. Consumer's risk at LQL $=6\%$. $\lambda_{LQL}=np=200(0.06)=12.0$, $$\beta=P_a(LQL)=P(X\le7\mid\lambda=12.0)=\boxed{0.0895\ (8.95\%)}.$$