Question 6 of 6: MIL-STD-105E vs. Dodge-Romig LTPD, Process Capability vs. Sampling, and a Single Sampling Plan

Question 6: MIL-STD-105E vs. Dodge-Romig LTPD, Process Capability vs. Sampling, and a 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) MIL-STD-105E vs. Dodge-Romig LTPD; which controls both risks; AOQL-based plans

MIL-STD-105E is an AQL-indexed attributes sampling system: plans are selected so that a lot at the Acceptable Quality Level is accepted with high probability (i.e. the producer's risk at the AQL is kept low, by convention around 5% for most plans), with no explicit control over the consumer's risk at any particular bad-quality level — the resulting consumer's risk simply falls out of whatever $(n,Ac)$ satisfies the AQL condition. Dodge-Romig LTPD plans are the opposite: they are consumer-risk-indexed, chosen so a lot at the Lot Tolerance Percent Defective is accepted only rarely (consumer's risk fixed at a stated value, conventionally 10%), and then, subject to that constraint, the plan additionally minimizes either the Average Total Inspection (ATI) or is chosen at a target Average Outgoing Quality Limit.

Neither plan by itself controls both risks simultaneously — each anchors one risk (MIL-STD-105E anchors $\alpha$ at the AQL, Dodge-Romig anchors $\beta$ at the LTPD) and lets the other risk fall out of the resulting $(n,Ac)$. A plan that explicitly fixes BOTH $\alpha$ at the AQL and $\beta$ at the RQL/LTPD simultaneously requires solving for $(n,Ac)$ from two OC-curve equations at once (a "two-point" design), which neither published table does automatically — it is a separate design procedure layered on top of either standard.

AOQL-based plan. An Average Outgoing Quality Limit plan is chosen (under 100% inspection/rectification of rejected lots) to guarantee that the long-run Average Outgoing Quality, $AOQ(p)=p\cdot P_a(p)$, never exceeds a stated ceiling regardless of how bad the incoming lot quality $p$ becomes — it is a worst-case, long-run consumer-protection guarantee rather than a single-lot risk statement, and is one of the two design objectives (with ATI) that Dodge-Romig plans can be built around.

(b) Traditional sampling vs. process capability testing; supplier's risk under capability testing

Disadvantages of traditional acceptance sampling relative to the modern process-capability approach: (1) it inspects/tests a sample from EVERY lot after the fact, adding recurring inspection cost and cycle-time delay that never goes away, whereas a demonstrated-capable process (high $C_{pk}$, verified in control) can justify skip-lot or reduced/no incoming inspection; (2) acceptance sampling only ever gives a pass/fail decision on the specific lot sampled and provides no diagnostic information about WHY a lot failed or how to improve the process, whereas capability analysis (control charts, capability indices) directly identifies whether the process itself is centred and capable, pointing at the root cause; (3) sampling plans always carry nonzero producer's and consumer's risk on every single lot by design (both types of decision error recur lot after lot), while a process shown to be in control with adequate $C_{pk}$ gives ongoing assurance without repeatedly re-incurring those sampling risks; (4) acceptance sampling can accept an out-of-control or marginally capable process indefinitely as long as each individual lot's sample happens to pass, providing no incentive or mechanism for continuous improvement, unlike control charts and capability tracking which flag drift immediately.

Can the supplier's (producer's) risk be controlled in a process capability test, and how? Yes — rather than being fixed implicitly by a sampling plan's OC curve, the producer's risk becomes the probability that a process which is genuinely capable (e.g. true $C_{pk}\ge$ some target such as 1.33) is nevertheless rejected because the ESTIMATED $\hat C_{pk}$ from a finite sample of size $n$ falls below the acceptance threshold, purely due to sampling variability in $\hat C_{pk}$. This risk is controlled directly by choosing the sample size $n$ (and, equivalently, the confidence level used to build a lower confidence bound on $C_{pk}$) large enough that the sampling distribution of $\hat C_{pk}$ has acceptably small probability of falling below the threshold when the true capability is at the target level — i.e. producer's risk in a capability test is an $n$-driven design choice, exactly analogous to how $(n,Ac)$ jointly set the producer's risk in MIL-STD-105E.

(c) MIL-STD-105E single sampling plan: lot=1,000, General II, AQL=1%, RQL=5%

Given. Normal inspection, General Inspection Level II, lot size $N=1{,}000$, $AQL=1\%$; $RQL=5\%$ for the risk estimates.

Find. Sample size code letter, sample size $n$, acceptance/rejection numbers $Ac/Re$; producer's risk $\alpha=P(\text{reject}\mid p=AQL)$ and consumer's risk $\beta=P(\text{accept}\mid p=RQL)$.

1. Sample size code letter (Table 14-4, reproduced on p.9 of the source). Lot size 1,000 falls in the row "501 to 1,200"; the General Inspection Level II column gives code letter J.
2. Sampling plan (Table 13-5, standard published MIL-STD-105E master table — not itself reproduced legibly in this paper's appendix, applied from the standard). Code letter J → sample size $n=80$. Reading the J row at $AQL=1.0\%$: $Ac=2$, $Re=3$. $$\boxed{n=80,\ Ac=2,\ Re=3}$$
3. Producer's risk at $p=AQL=0.01$. $\alpha=P(d>2\mid n=80,p=0.01)=1-\sum_{k=0}^{2}\binom{80}{k}(0.01)^k(0.99)^{80-k}$. $$\boxed{\alpha=1-0.9535=4.66\%}$$
4. Consumer's risk at $p=RQL=0.05$. $\beta=P(d\le2\mid n=80,p=0.05)=\sum_{k=0}^{2}\binom{80}{k}(0.05)^k(0.95)^{80-k}$. $$\boxed{\beta=23.06\%}$$