Acceptance Sampling: Method

OC Curves, Risks, AOQ, ATI, Plan Design and the Z1.4 Tables

What acceptance sampling decides

A lot of N units arrives and must be accepted or rejected without inspecting every unit. An attribute sampling plan draws a random sample of n units, counts the nonconforming ones, d, and accepts the lot when d is at most the acceptance number c (equivalently, rejects at the rejection number Re = c + 1). Sampling cannot sort good lots from bad with certainty: it is a decision rule whose only properties are the probabilities with which it accepts lots of each quality. Those probabilities, plotted against the lot quality p (the fraction nonconforming), make the operating characteristic (OC) curve, and everything else about a plan follows from it. Sampling is the right tool when 100 % inspection is too costly, destructive or no more reliable than sampling, and when lots come in a stream so that the long-run acceptance rates mean something; it is not a substitute for process control, which keeps p low in the first place.

The operating characteristic curve

For a process producing a fraction p nonconforming (the type-B curve, lots drawn from a continuing process) the count in a sample of n is binomial, so Pa(p) = Σ_d=0^c C(n, d) p^d (1 − p)^n−d. When the sample is small against np the Poisson form e^−np (np)^d/d! is close and is what the printed tables of MIL-STD-105E use (its Table X). For an isolated lot of N units holding exactly D nonconforming (the type-A curve) the count is hypergeometric, C(D, d) C(N − D, n − d)/C(N, n), with D = pN rounded; for n below about a tenth of N the two curves are nearly the same, and the tool draws all three so the difference can be seen. The curve of a plan with c = 0 falls steeply from Pa = 1 at p = 0, which is why such plans are hard on the producer; raising c with n in proportion makes the curve steeper around the quality of interest, which is what discriminates good lots from bad.

AQL, LTPD and the two risks

Two points on the OC curve summarise a plan. The acceptable quality level (AQL) is the poorest process average the producer regards as satisfactory; lots at the AQL should be accepted nearly always, and the producer's risk α = 1 − Pa(AQL) is the probability that one is rejected. The lot tolerance percent defective (LTPD, also called the rejectable or limiting quality) is the lot quality the consumer wants to be protected against; the consumer's risk β = Pa(LTPD) is the probability that such a lot slips through. Conventional values are α = 0.05 and β = 0.10, but a plan has whatever risks its OC curve gives at the quality levels the parties name, and the tool prints those. The plan's own quality levels at Pa = 0.95, 0.50 (indifference quality) and 0.10 are the mirror image: the lot qualities a given plan treats as acceptable, indifferent and rejectable.

Rectifying inspection: AOQ, AOQL and ATI

When a rejected lot is screened 100 % and the nonconforming units replaced by good ones, the stream leaving inspection has quality p from the accepted lots and 0 from the screened ones, so the average outgoing quality is AOQ = Pa p (N − n)/N (the sample itself has been cleared; for an infinite lot AOQ = Pa p). AOQ rises with p while most lots are accepted and falls once most are screened, and its maximum, the average outgoing quality limit (AOQL), is the worst long-run outgoing quality the plan can produce regardless of what comes in: a plan is often chosen for its AOQL alone (the Dodge–Romig tables are indexed by it). The cost of this protection is the inspection: the average total inspection per lot is ATI = n + (1 − Pa)(N − n), rising from n when every lot is accepted to N when every lot is screened. The standard's Table V-A lists the AOQL factor p Pa for every single normal plan, to be multiplied by 1 − n/N for the exact value; the tool computes the maximum directly and reproduces that table.

Double sampling

A double plan inspects n₁ units and accepts at d₁ ≤ c₁, rejects at d₁ ≥ r₁, and otherwise inspects n₂ more and accepts when d₁ + d₂ ≤ c₂. Its acceptance probability is Pa = Pa^I + Pa^II: the first-sample part P(d₁ ≤ c₁) plus, for each undecided count d₁ between c₁ and r₁, P(d₁) P(d₂ ≤ c₂ − d₁). Clearly good or clearly bad lots are settled on the first sample, so the average sample number ASN = n₁ + n₂ P(second sample) lies below the single plan of the same OC curve over most of the range, at the price of a second sample for the lots in between (the tool takes the second sample in full; curtailing it once rejection is certain lowers the ASN further). The ATI is n₁ Pa^I + (n₁ + n₂) Pa^II + N (1 − Pa). The matched double plans of MIL-STD-105E (Table III) have OC curves close to the single plans of Table II with first samples of about 63 % of the single sample.

Designing a plan from two points

Given (AQL, 1 − α) and (LTPD, β) a single plan with n and c is sought whose OC curve passes through or below and above the two points. Two constructions are given. The exact binomial search takes each c in turn, finds the smallest n whose Pa at the LTPD is at most β, and stops at the first c whose producer's risk is also met: this is the smallest plan that honours both risks exactly, and the table of candidates shows how the sample grows with c while α falls. The Poisson ratio method of Cameron (1952), the table behind the binomial nomographs of the textbooks, tabulates for each c the Poisson means np₁ at Pa = 1 − α and np₂ at Pa = β and their ratio R; the smallest c whose R does not exceed LTPD/AQL is taken, and n = np₁/AQL meets the producer's risk exactly while n = np₂/LTPD meets the consumer's, the two bounding the choice. A nomograph reading such as Montgomery's n = 89, c = 2 for AQL 1 %, LTPD 6 % falls on the c = 2 row of the candidate table, which misses α by a little (0.060 against 0.050); the exact search moves to c = 3, n = 110. Which is preferred is a matter of whose risk may be stretched.

ANSI/ASQ Z1.4 and MIL-STD-105E

The military standard (1963, revision E of 1989, carried on unchanged as ANSI/ASQ Z1.4) is an AQL system: the plans are indexed by the AQL and by a sample size code letter, so that larger lots get larger samples and a steeper OC curve while the acceptance probability at the AQL stays high. Table I gives the code letter from the lot size and the inspection level: general level II is the default, level I for less discrimination (a smaller sample), level III for more, and the special levels S-1 to S-4 for small samples where inspection is costly or destructive. Table II-A (normal), II-B (tightened) and II-C (reduced) give n, Ac and Re at the code letter and AQL. The arrows matter: a cell holding an arrow sends the reader to the first plan below (↓, a larger sample) or above (↑, a smaller one) in the same AQL column, and if the sample size reached equals or exceeds the lot size the lot is inspected 100 %. The tool transcribes the four master tables from the standard with the printed irregularities kept (code letter A has no Ac = 0 plan under tightened inspection, letters A and B of the reduced table carry their own acceptance sequences, the tightened table has an extra row S of 3,150) and follows the arrows automatically.

Tightened inspection shifts the AQL columns one step so the same sample size carries a smaller acceptance number; reduced inspection cuts the sample to about two fifths and leaves a gap between Ac and Re in which the lot is accepted but normal inspection is reinstated. The switching rules (4.7) are what make the system work as a whole: normal to tightened when 2 of 5 or fewer consecutive lots are rejected; back to normal after 5 consecutive acceptances; normal to reduced after 10 consecutive acceptances with the total nonconforming within Table VIII and steady production; reduced to normal on any rejection or gap count; discontinuation after 5 lots fail on tightened inspection. Table III-A gives the matched double plans for normal inspection (first and second samples equal, cumulative Ac and Re); the cells marked with a dot refer to the single plan. The tool prints the OC of the plan found, the three severities side by side, and the AOQL of the plan for the lot entered.

Reading the results

Check the two risks against what the parties agreed, then look at the OC curve as a whole: the AQL point should sit on the flat upper shoulder and the LTPD point low on the tail; if the curve is still high at the quality the consumer cannot tolerate, the sample is too small for that purpose whatever the AQL says. Under rectifying inspection compare the AOQL with the outgoing quality the next operation needs and the ATI with the inspection capacity. For a double plan read the ASN at the quality the process actually runs at, not at the AQL only. For a Z1.4 lookup remember that the plan is tied to the lot size through the code letter, so splitting or combining lots changes the plan, and that the protection at the LTPD comes from the switching rules as much as from the single plan.

Limits

The tool covers attribute plans for fraction nonconforming (and the standard's defects-per-hundred-units AQLs above 10 through the same tables); variables plans (ANSI/ASQ Z1.9, MIL-STD-414), multiple sampling (Table IV), sequential plans, chain and skip-lot sampling, the Dodge–Romig LTPD and AOQL tables and the tightened and reduced double tables III-B and III-C are not included. The hypergeometric curve rounds pN to a whole number of nonconforming units, so it is a step function in p. Random sampling of the lot is assumed throughout.