Question 6 of 6: Inspection Strategies, Dodge-Romig Plans, and a MIL-STD-105E Sampling Plan

Question 6: Inspection Strategies, Dodge-Romig Plans, and a MIL-STD-105E 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; traditional vs. sequential sampling; MIL-STD-105E

100% inspection checks every unit. In principle it catches every defective, but it is slow and costly for large lots, impossible for destructive tests, and — because monotonous, high-volume inspection causes fatigue — is in practice often only 80–90% effective, so it is not actually the perfect screen it appears to be. Acceptance sampling inspects only a random sample and infers the disposition of the whole lot: it is far cheaper and faster, is the only option for destructive tests, and can even be more reliable per-unit than a fatigued 100% inspection because fewer units are inspected more carefully; its cost is that it always carries a quantified sampling risk (it can accept a bad lot or reject a good one) and gives no information about which specific units in an accepted lot are defective, nor any direct feedback to fix the process.

Traditional (single/double/multiple) sampling draws a fixed, predetermined sample and decides accept/reject only once all of it has been inspected. Sequential sampling inspects one (or a few) units at a time and, after every unit, makes one of three decisions — accept, reject, or continue sampling — based on running the cumulative result against pre-computed accept/reject boundary lines (Wald's sequential probability ratio test). Its chief advantage is a smaller average sample number (ASN) than a fixed plan giving equivalent $\alpha,\beta$ protection at the same AQL/RQL, because clearly good or clearly bad lots are typically decided very quickly. Sequential sampling is therefore preferable when inspection or testing is expensive or destructive (so minimizing the average number inspected matters most) and when submitted quality tends to be either clearly good or clearly bad; it is less used where a predictable, fixed sample size is administratively important, since its own sample size is a random variable.

MIL-STD-105E is the most widely used attributes sampling system (not a single plan): a published set of master tables giving single, double, and multiple sampling plans indexed by lot size, inspection level (general I/II/III, or special S-1–S-4), and the desired AQL, together with switching rules between normal, tightened, and reduced inspection based on recent lot history. Calling it "AQL-based" means the plans are built around the Acceptable Quality Level — the worst tolerable long-run process-average quality — and are chosen so a submitted lot at that quality level is accepted with high probability (a low, controlled producer's risk, conventionally near 95% acceptance at the AQL); the system is optimized for producer protection at the AQL rather than for a guaranteed consumer protection point.

(b) Dodge-Romig LTPD and AOQL plans, compared with AQL-based plans

The Dodge-Romig plans assume rectifying inspection — every rejected lot is 100% screened and its defectives replaced with good units — and are indexed around consumer, not producer, protection. LTPD (lot tolerance percent defective) plans guarantee a specified consumer's risk (conventionally $\beta=0.10$) at a stated LTPD/RQL, choosing the minimum sample size that gives that guarantee — protecting against accepting any single bad lot. AOQL (average outgoing quality limit) plans instead guarantee that the long-run average outgoing quality, after rectification of rejected lots, never exceeds a stated AOQL regardless of incoming quality — protecting the consumer's average quality across many lots rather than any one lot.

Compared with AQL-based plans (MIL-STD-105E): AQL plans are optimized for producer protection and do not assume rectifying inspection of rejected lots (a rejected lot may simply be returned to the supplier); Dodge-Romig plans are optimized for consumer protection and explicitly assume rectification. For a given incoming quality that is reasonably well known and stable, Dodge-Romig plans generally require a smaller average sample size than an equivalently protective AQL system, but that efficiency depends on knowing the incoming process average reasonably accurately — AQL-based plans are more robust to an unknown or drifting incoming quality, which is one reason MIL-STD-105E remains the more widely used general-purpose system.

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

Given. Lot size $N=1000$; required $AQL=1.5\%$; general inspection level II; normal inspection; consumer's risk to be evaluated at $RQL=10\%$.

Find. The single sampling plan ($n$, $Ac$, $Re$) from the MIL-STD-105E master tables, and the producer's risk at the AQL and the consumer's risk at the RQL.

Approach. Look up the sample-size code letter from Table 13-4 (lot size $\times$ inspection level), then read $(n,Ac,Re)$ off the AQL$=1.5\%$ column of the normal-inspection master table (Table 13-5); risks follow from the binomial distribution at $p=AQL$ and $p=RQL$.

1. Sample-size code letter. A lot of 1000 falls in the "501 to 1200" row of Table 13-4; at general inspection level II this gives code letter $\boxed{J}$.
2. Plan from the master table. Code letter J corresponds to sample size $n=80$; reading the $AQL=1.5\%$ column at code J gives acceptance number $Ac=3$, rejection number $Re=4$: $\boxed{n=80,\ Ac=3,\ Re=4}$.
3. Producer's risk at $AQL=1.5\%$. The lot is rejected if more than $Ac=3$ of the 80 sampled items are nonconforming, so with $D\sim\text{Binomial}(80,\,0.015)$, $$\alpha=P(D>3)=1-P(D\le3)=\boxed{0.0326}\ (3.26\%).$$
4. Consumer's risk at $RQL=10\%$. The lot is accepted if $D\le Ac=3$, so with $D\sim\text{Binomial}(80,\,0.10)$, $$\beta=P(D\le3)=\boxed{0.0353}\ (3.53\%).$$