Question 6 of 6: Acceptance Sampling and MIL-STD-105E, Dodge-Romig LTPD/AOQL Plans, and a Single Sampling Plan for a Lot of 800

Question 6: Acceptance Sampling and MIL-STD-105E, Dodge-Romig LTPD/AOQL Plans, and a Single Sampling Plan for a Lot of 800 (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, sequential sampling, and MIL-STD-105E

The purpose of acceptance sampling is to make an accept/reject decision on an entire LOT of product based on inspecting only a sample from it, rather than either 100% inspecting every item (expensive, slow, and impossible for destructive tests) or accepting a lot with no inspection at all (no protection against a bad lot). It provides a statistically-defined level of protection to both producer (a good lot should usually be accepted) and consumer (a bad lot should usually be rejected) at a known, quantifiable risk, using far less inspection effort than 100% screening.

Traditional (single/double/multiple) acceptance sampling draws one or more FIXED-size samples up front and applies a predetermined accept/reject rule (e.g. accept if $d\le Ac$) — the sample size is fixed in advance regardless of how clearly good or bad the early results look. Sequential sampling instead inspects items ONE AT A TIME (or in small increments), plotting the cumulative number of defectives against two boundary lines (an accept line and a reject line, derived from the sequential probability ratio test, SPRT) after every observation; inspection stops and a decision is reached as soon as either boundary is crossed. Because sequential sampling can terminate very early when the lot is clearly very good or very bad, it typically achieves the SAME discriminating power (OC curve) as a fixed-size plan with a substantially smaller average sample number (ASN) — its main drawback is the added administrative complexity of item-by-item tracking and a theoretically unbounded (though rarely large in practice) maximum sample size.

MIL-STD-105E's main features are: sampling plans indexed by LOT SIZE and INSPECTION LEVEL (a sample-size code letter is looked up from a lot-size/inspection-level table) and by the desired AQL; single, double, and multiple sampling plan options at each code letter/AQL combination; and a built-in SWITCHING RULE between normal, tightened, and reduced inspection, so that if recent lots show a pattern of unusually high or low nonconformance the inspection stringency automatically adjusts (tightened inspection after evidence of poor quality; reduced inspection, with smaller samples, after a run of good lots), rather than a fixed sampling intensity applied blindly regardless of actual incoming quality history. "AQL-based" means the entire family of plans is indexed around, and DESIGNED to give a high probability of acceptance (conventionally around 0.95, the producer's-risk point) when the lot's true quality is AT the stated Acceptable Quality Level — the plans primarily protect the PRODUCER against having good-quality lots (at or better than the AQL) rejected too often, with consumer protection against poor lots relying mainly on the switching rules (tightened inspection) rather than being directly guaranteed at any single specific bad-quality level.

(b) Dodge-Romig LTPD vs. AOQL plans, and their difference from AQL-based plans

Both Dodge-Romig plan families assume RECTIFYING INSPECTION: every rejected lot is 100% inspected and all discovered nonconforming units are removed or replaced, so the "outgoing" quality after inspection is always better than the lot's incoming quality. LTPD (lot tolerance percent defective) plans are indexed to guarantee that a lot at the stated LTPD (the quality level considered "just barely unacceptable") has only a small, specified probability of acceptance — conventionally $\beta=0.10$ — i.e. they fix the CONSUMER's risk at a single specific bad-quality level and, subject to that constraint, choose the plan (from tables indexed by the process average) that minimizes the average amount of inspection (ATI). AOQL (average outgoing quality limit) plans instead guarantee that the AVERAGE OUTGOING quality, MAXIMIZED over every possible incoming lot quality level (accounting for the rectification of rejected lots), never exceeds a stated AOQL value — they protect the consumer against a poor LONG-RUN AVERAGE outgoing quality across many lots, rather than guaranteeing protection against any single bad lot the way an LTPD plan does, and are again chosen (from tables indexed by process average) to minimize ATI subject to the AOQL constraint.

Both Dodge-Romig families differ from the AQL-based MIL-STD-105E plans in orientation and in the underlying assumption: MIL-STD-105E plans are indexed around a GOOD-quality level (the AQL) and primarily protect the PRODUCER (guaranteeing high $Pa$ at the AQL), with only indirect, systemic consumer protection via the tightened/reduced switching rules across a sequence of lots; they do NOT assume rectifying inspection (rejected lots may simply be returned to the supplier, not 100% inspected). Dodge-Romig plans are indexed around a BAD-quality level (LTPD) or a long-run average bound (AOQL) and are built explicitly around the rectifying-inspection assumption, directly guaranteeing a specific CONSUMER-side protection for every individual plan selected, at the cost of requiring an estimate of the incoming process average quality to choose the correct table entry (information MIL-STD-105E's simpler AQL/lot-size/level lookup does not require).

(c) Single sampling plan for a lot of 800, AQL 1.5%, and consumer's risk at LQL 5%

Given. Lot size $N=800$; general inspection level II; normal inspection; $AQL=1.5\%$; $LQL$ (lot quality level of interest) $=5\%$.

Find. The sample-size code letter, the single sampling plan $(n,\,Ac,\,Re)$, and the consumer's risk $\beta=P(\text{accept}\mid p=0.05)$.

1. Sample-size code letter (Table 14-4, this paper). Lot size 800 falls in the "501 to 1200" row; reading the General Inspection Level II column gives code letter $\boxed{J}$.
2. Single sampling plan (Table 13-5, this paper). For code letter $J$, sample size $n=80$; reading across to the $AQL=1.5\%$ column under normal inspection: $$\boxed{n=80,\quad Ac=7,\quad Re=8}$$ (accept the lot if 7 or fewer nonconforming units are found in the 80-item sample; reject if 8 or more).
3. Consumer's risk at $LQL=5\%$. The consumer's risk is the probability the plan still ACCEPTS a lot whose true fraction nonconforming is $p=0.05$, i.e. $P(X\le Ac)$ for $X\sim\text{Binomial}(n=80,\,p=0.05)$: $$\beta=P(X\le7)=\sum_{k=0}^{7}\binom{80}{k}(0.05)^k(0.95)^{80-k}=\boxed{0.9534\ (95.3\%)}.$$
4. Cross-check with the Poisson approximation (large $n$, small $p$, $\lambda=np=80(0.05)=4.0$): $$\beta\approx P(X\le7\mid\lambda=4)=\sum_{k=0}^{7}\frac{e^{-4}4^k}{k!}=\boxed{0.9489},$$ close to the exact binomial value, confirming the plan's numbers are consistent.