Question 6 of 6: Acceptance Sampling Fundamentals, Dodge–Romig Plans, and a MIL-STD-105E Sampling Plan

Question 6: Acceptance Sampling Fundamentals, 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) Purpose of acceptance sampling; variables vs. attributes; MIL-STD-105E vs. MIL-STD-414; what "AQL-based" means

Acceptance sampling exists to decide whether to accept or reject an incoming (or outgoing) LOT of product based on inspecting only a SAMPLE from it, rather than 100% inspection, when 100% inspection is too costly, too slow, destructive, or itself unreliable (inspector fatigue on a huge lot can miss more defects than a well-designed sample would). It provides a documented, statistically-characterized balance between the cost of inspection and the risk of accepting a bad lot or rejecting a good one, and — because a supplier knows every lot may be sampled and rejected — it also functions as an ongoing incentive for suppliers to maintain quality, not merely a lot-by-lot filter.

Attributes sampling classifies each inspected unit simply as conforming or nonconforming (or counts nonconformities), which is quick and requires no special measurement equipment or normality assumption, but extracts relatively little information per unit inspected, so it needs a LARGER sample size to achieve a given level of protection. Variables sampling instead uses the actual measured value of a continuous quality characteristic (its sample mean and standard deviation, or sample range), extracting far more information per unit and therefore achieving the SAME level of protection with a substantially SMALLER sample — but it requires the characteristic to be (at least approximately) normally distributed, needs a separate plan per characteristic measured (an attributes plan can lump multiple defect types into one nonconforming/conforming count), and is more complex to administer and audit.

MIL-STD-105E is the attributes-based counterpart of this pair: it is indexed by AQL, offers a wide range of sample-size code letters and normal/tightened/reduced inspection levels with defined switching rules, and works from a simple accept/reject count against $Ac$/$Re$. MIL-STD-414 (its successor is ANSI/ASQ Z1.9) is the analogous VARIABLES-based standard: also AQL-indexed, but it computes an acceptability criterion from the sample mean and standard deviation (or sample range, in its range method) relative to the specification limit(s), using a computed quality index $Q$ compared against a tabulated acceptability constant $k$ — delivering MIL-STD-105E-equivalent protection at a smaller sample size, at the cost of requiring the underlying characteristic to be normally distributed and measured on a continuous scale.

Both standards being "AQL-based" means each sampling plan in the system is selected and indexed so that a lot submitted AT the stated Acceptable Quality Level has a HIGH probability of acceptance (by design, roughly 0.95 under normal inspection) — the AQL is a producer-protection benchmark, the maximum defect rate still considered "acceptable" as a long-run process average, not a promise that every individual lot at that quality will pass, and not a statement about consumer protection against any one bad lot (that role, addressed differently, is discussed in part (b)).

(b) Dodge–Romig LTPD and AOQL plans; comparison with AQL-based plans

The Dodge–Romig plans are RECTIFYING inspection plans: they assume any REJECTED lot is subjected to 100% screening, with all nonconforming units found replaced by good ones, so that every lot that ultimately ships (whether it passed the sample or was screened) meets a guaranteed protection level — and the plans are selected to minimize the resulting Average Total Inspection (ATI) for a stated incoming process-quality level. Two indexing philosophies are offered. LTPD plans (Lot Tolerance Percent Defective) guarantee a CONSUMER-protection point: for a stated LTPD quality level, the plan's probability of accepting a lot that is actually at that (bad) quality is fixed at a small consumer's risk, conventionally 10% — the plan protects against passing any SINGLE lot as bad as the LTPD. AOQL plans (Average Outgoing Quality Limit) instead guarantee a bound on the LONG-RUN AVERAGE fraction defective actually shipped to the customer (across many lots, accounting for the rectification of rejected lots), REGARDLESS of the incoming quality — even if incoming quality is very poor, the rectifying/screening process caps the average outgoing quality at the AOQL; the guarantee is on the long-run AVERAGE, not on any individual lot.

Compared with AQL-based plans (MIL-STD-105E/414): Dodge–Romig plans need an ASSUMED OR ESTIMATED incoming process-average quality up front to select the ATI-minimizing plan, are inherently rectifying (they presuppose 100% screening of rejects), and are indexed around consumer protection (LTPD) or long-run average outgoing quality (AOQL) for a SINGLE submission. AQL-based plans need no assumed incoming-quality distribution, are not necessarily rectifying, and are indexed around PRODUCER protection at a stated acceptable level, with a built-in SYSTEM of normal/tightened/reduced switching rules that automatically adapts the stringency of inspection to the SUBMITTED QUALITY TRACK RECORD observed over many consecutive lots — a dynamic, ongoing-relationship control mechanism that Dodge–Romig's single-lot ATI-minimization does not provide.

(c) MIL-STD-105E single sampling plan, lot 400, AQL 1%, normal inspection, general level II

Given. Lot size $N=400$; required $AQL=1\%$; Normal inspection; General Inspection Level II; consumer's-risk quality point $LQL=5\%$.

Find. The sample-size code letter, single sampling plan ($n$, $Ac$, $Re$), and the producer's risk (at $AQL$) and consumer's risk (at $LQL$) for that plan.

1. Sample-size code letter (Table 1, MIL-STD-105E). Lot size 400 falls in the "281 to 500" row; reading across to the General Inspection Level II column gives code letter $$\boxed{H}.$$
2. Single sampling plan, normal inspection (Table 13-5/Master Table, code letter H, $AQL=1.0\%$). $$\boxed{n=50,\qquad Ac=1,\qquad Re=2}$$ (accept the lot if 1 or fewer nonconforming units are found in the sample of 50; reject if 2 or more are found).
3. Producer's risk $\alpha$: $P(\text{reject}\mid p=AQL=0.01)$. Using the binomial distribution with $n=50$, $p=0.01$, $$\alpha=P(d\ge2)=1-P(d=0)-P(d=1)=1-\binom{50}{0}(0.01)^0(0.99)^{50}-\binom{50}{1}(0.01)^1(0.99)^{49}=\boxed{0.0894\ (8.9\%)}.$$
4. Consumer's risk $\beta$: $P(\text{accept}\mid p=LQL=0.05)$. Using the binomial distribution with $n=50$, $p=0.05$, $$\beta=P(d\le1)=\binom{50}{0}(0.05)^0(0.95)^{50}+\binom{50}{1}(0.05)^1(0.95)^{49}=\boxed{0.2794\ (27.9\%)}.$$ (Poisson approximation, $\lambda=np$, gives $\alpha\approx0.0902$ and $\beta\approx0.2873$ — consistent with the exact binomial values above to about one part in a hundred.)