Acceptance sampling
4.1 Introduction to Acceptance Sampling and Single Sampling Plan
Certain important terms relevant to acceptance sampling are discussed below (see
ANSI/ASQC Standard A2-1987; Duncan (1986); Schilling (1982)):
Acceptance sampling plan: a specific plan that clearly states the rules for sampling
and the associated criteria foracceptance or otherwise. Acceptance sampling plans can
be applied for inspection of (i) end items, (ii) components, (iii) raw materials, (iv)
operations, (v) materials in process, (v) supplies in storage, (vi) maintenance
operations, (vii) data or records and
(viii) administrative procedures.
Item or Unit: an object or quantity of product or material on which observations
(attribute or variable or both) are made.
Lot or Batch: a defined quantity of product accumulated for sampling purposes. It is
expected that the quality of the product within a lot is uniform.
Attributes Method: where quality is measured by observing the presence or absence
of some characteristic or attribute in each of the units in the sample or lot under
consideration, and the number of items counted which do or do not possess the quality
attribute, or how many events occur in the unit area, etc.
Variables Method: where measurement of quality is by means of measuring and
recording the numerical magnitude of a quality characteristic for each of the items.
Nonconformity: the departure of a quality characteristic from its intended level,
causing the product or service to fail to meet the specification requirement. If the
product or service is also not meeting the usage requirements, it is called as a defect.
Usually the terms "defect" and "nonconformity" are interchangeable but the word
"defect" is more stringent.
Nonconforming (Defective) Unit: a unit containing at least one nonconformity
(defect). The terms "defective" and "nonconforming" are interchangeable but a
defective unit will fail to satisfy the normally intended usage requirements.
Proportion (Fraction) Defective or Proportion (Fraction) Nonconforming Units p:
This is the ratio of the number of nonconforming units (defectives) to the total number
of (sampled) units.
Single Sampling: the sampling inspection type in which the lot disposition is based
on the inspection of a single sample of size n.
Single Sampling Attributes Plan (n, Ac)
The operating procedure of the single sampling attributes plan is as follows:
1. From a lot of size N, draw a random sample of size n and observe the number
of nonconforming units (nonconformities) d.
2. If d is less than or equal to the acceptance number Ac, which is the maximum
allowable number of nonconforming units or nonconformities, accept the lot.
If d > Ac, do not accept the lot.
The symbol c is also used for acceptance number. The symbol Re (= Ac+1) is used to
denote the rejection number.
The acceptable quality level (AQL) is the maximum percentage or proportion of
nonconforming units in a lot that can be considered satisfactory as a process average
for the purpose of acceptance sampling. When a consumer designates some specific
value of AQL, the supplier or producer is notified that the consumer's acceptance
sampling plan will accept most of the produced lots submitted by the supplier,
provided the process average of these lots is not greater than the designated value of
AQL. Suppose that one performs an error-free 100% inspection of the lot and
observes p for each lot. Then all lots with p  AQL will be accepted and all lots with p
> AQL will not be accepted. This situation is shown graphically in Figure 4.1.
Figure 4.1 Ideal OC Curve
Due to sampling, one faces the risk of not accepting lots of AQL quality as well as the
risk of accepting lots of poorer than AQL quality. One is therefore interested in
knowing how an acceptance sampling plan will accept or not accept lots over various
lot qualities. A curve showing the probability of acceptance over various lot or
process qualities is called the operating characteristic (OC) curve and is explained
below:
Operating Characteristic (OC) Curve
The OC curve reveals the performance of the acceptance sampling plan. We consider
two types of OC curves: Type A: (For isolated or unique lots) This is a curve showing
the probability of accepting a lot as a function of the lot quality.
Type B: (For a continuous stream of lots) This is a curve showing the probability of
accepting a lot as a function of the process average. That is, the Type B OC curve will
give the proportion of lots accepted as a function of the process average p.
OC Function of a Single Sampling Plan
The OC function of the single sampling attribute plan giving the probability of
acceptance for a given lot or process quality p is:
Pa = Pa(p) = Pr (d  Ac  n, Ac, p).
For Type A situations, the hypergeometric distribution is exact for the case of
nonconforming units. Hence,
Pa(p) = Pr (d  Ac  N, n, Ac, p),
where N is the lot size, D is the number of defectives in the lot and hence p = D/N.
For Type B situations, the binomial model is exact for the case of fraction
nonconforming units and the OC function is given by
Pa(p) = Pr (d  Ac  n, Ac, p)
.
The above OC function is applicable to a continuous stream of lots and can be used as
an approximation to Type A situation when N is large compared to n (n/N < 0.10) and
p is small.
For the case of nonconformities per unit, the Poisson model is exact for both Type A
and Type B situations. The OC function in this case is
Pa(p) = Pr (d  Ac  n, Ac, p),
The above OC function is also used as an approximation to binomial when n is large
and p is small such that np < 5. A typical OC curve is shown in Figure 4.2.
Figure 4.2 OC Curve
In general, the probability of acceptance will be underestimated at good quality levels
and overestimated at poor quality levels if one approximates the hypergeometric OC
function with a binomial or Poisson OC function. The same is true when the binomial
OC function is approximated by the Poisson OC function. This is shown graphically
in Figure 4.3 assuming N = 200, n = 20 and Ac = 1.
Figure 4.3 Comparison of OC Curves
It can be observed that when n is constant and Ac is increased, Pa(p) will increase.
When Ac is constant and n is increased, then Pa(p) will decrease (see Figures 4.4 and
4.5). These two properties are useful for designing sampling plans for given quality
requirements.
Figure 4.4 Effect of Acceptance Number on OC Curve
Figure 4.5 Effect of Sample Size on OC Curve
Suppose that a company manufacturing cheese is continuously supplying its
production to two supermarkets. Supermarket A requires cheese in lots of 5000 units
and supermarket B prefers lots of size 10000. Assume that the producer uses two
sampling plans with sample size equal to the square root of the lot size. For both
plans, let the acceptance number be fixed as one. The company's true fraction of
nonconforming cheeses produced is 1%, which is considered as the acceptable quality
level by the consuming supermarkets. The effectiveness of the sampling plans (plan
for Supermarket A: n = 71 Ac = 1; plan for supermarket B: n = 100 Ac = 1) will be
revealed by the respective OC curves shown in Figure 4.6
Figure 4.6 Comparison of OC Curves for a Given AQL
The proportion of lots of AQL quality accepted by the two plans are:
Pa(AQL) = 84% for n = 71 Ac = 1 plan
Pa(AQL) = 74% for n = 100 Ac = 1 plan.
It is also evident that the plan used for supermarket B is tighter than the plan used for
supermarket A. For the fixed acceptance number, an increase in sample size means
tightening. It is always desired that the probability of acceptance at AQL be higher,
such as 95%. Both the plans do not have a high Pa at AQL. Plan n = 71 and Ac = 1 is
preferable to plan n = 100, Ac = 1 since Pa (AQL) = 84% is closer to 95%. The
manufacturer is regularly supplying cheese to both supermarkets. Under the Type B
situation of series of lots being submitted, the lots are themselves viewed as random
samples from the process producing cheese. One therefore need not sample in relation
to the lot size. If it is desired to encourage large lot sizes, then the acceptance number
should be accordingly adjusted so that the Pa at AQL is higher for large lot sizes. This
is not however the case here.
Arguments in favour of the n = 100 Ac = 1 plan can also be given. The consuming
supermarkets may need protection against bad quality lots. For example, lots having
5% nonconforming cheeses may be required to be rejected with a large probability to
protect the consumer interests. The plan n = 100 Ac = 1 is tighter and has a smaller
probability of acceptance at the rejectable quality level namely 5% nonconforming
(see Figure 4.7).
Figure 4.7 Comparison of OC Curves for a Given LQL
Thus it is seen that it will be worthwhile to prescribe an additional index for consumer
protection since the AQL does not completely describe the protection to the
consumer. Hence for consumer protection against bad quality lots, the Limiting
quality level (LQL) is defined as the percentage or proportion of nonconforming units
in a lot for which the consumer wishes the probability of acceptance to be restricted to
a specified low value. LQL is also referred to as the rejectable quality level (RQL),
the unacceptable quality level (UQL), the limiting quality (LQ), the lot tolerance
fraction defective (LTFD) or the lot tolerance percent defective (LTPD). Here LTFD
and LTPD are defined for Type A situations only.
Figure 4.8 OC Curve Showing AQL, LQL,  and 
The producer's risk () is the probability of not accepting a lot of AQL quality and the
consumer's risk () is the probability of accepting a lot of LQL quality. Generally the
parameters of the (single) sampling plan must be determined for given quality levels,
namely AQL, LQL and risks  and . Figure 4.8 shows the quality indices AQL, LQL
and the associated risks  and  respectively on the OC curve.
Consider the single sampling plans (n = 127, Ac = 5), and (n = 38, Ac = 1) and the
OC curves of these two plans (Figure 4.9).
Figure 4.9 Comparison of Discriminating Power of OC Curves
The plan (127,5) possesses a more highly discriminating OC curve than the plan
(38,1). That is, the plan (127,5) achieves a smaller producer's risk at all good quality
levels and involves a smaller consumer's risk at all poor quality levels. Consider the
OC curves of the plans (63,3) and (38,1) given in Figure 4.10. Here the plan (63,3)
achieves a smaller producer's risk compared to the plan (38,1) but maintains the
consumer's risk at poor quality levels.
Figure 4.10 Comparison of OC Curves
In general, both the sample size and acceptance numbers need to be large for reaching
the ideal shape of the OC curve. Consider a value of 1% for AQL. OC curves of plans
with different acceptance numbers giving the same producer's risk at AQL are shown
in Figure 4.11. The consumer's risks for these plans are very different, and the ideal
shape is reached only with an increasing acceptance number and sample size.
Figure 4.11 Approaching Ideal OC Curve
Average Sample Number (ASN)
The average sample number (ASN) is defined as the average number of sample units
per lot used for deciding acceptance or non-acceptance. For a single sampling plan,
one takes only a single sample of size n and hence the ASN is simply the sample size
n.
By curtailed inspection, we mean the stopping of sampling inspection, when a
decision is certain. Inspection can be curtailed when the rejection number is reached
since the rejection is certain and no further inspection is necessary in reaching that
decision. Such curtailment of inspection for rejecting a lot is known as semi-curtailed
inspection. If inspection is curtailed once acceptance or rejection is evident, then it is
known as fully curtailed inspection.
For example, consider the single sampling plan with n = 50 and Ac = 1. Let the
sample be randomly drawn and testing of units takes place unit-by-unit. One can
curtail inspection, rejecting the lot, as early as the third unit if the first two units are
nonconforming. Similarly if all the first 49 units are found to be conforming, then the
lot can be accepted without testing the last unit. The fully curtailed and semi-curtailed
ASN curves of the plan n = 50 and Ac = 1 are shown in Figure 4.12. When p = 0, the
fully curtailed plan has an ASN of 49 while the semi-curtailed plan requires all the 50
units to be tested. The policy of curtailment is effective only at poor quality levels
since more nonconforming units are likely to be sampled leading to an early rejection
decision.
Figure 4.12 Curtailed ASN Curves
Generally it is undesirable to curtail inspection in single sampling. The whole sample
is usually inspected in order to have an unbiased record of quality history.
Rectifying or Screening Inspection
Those lots not accepted by a sampling plan will usually be 100% inspected or
screened for nonconforming or defective units. After screening, nonconforming units
may be rectified or discarded or replaced by good units, usually taken from accepted
lots. Such a programme of inspection is known as a rectifying or screening inspection.
For those lots accepted by the sampling plan, no screening will be done and the
outgoing quality will be the same as that of the incoming quality p. For those lots
screened, the outgoing quality will be zero, meaning that they contain no
nonconforming items. Since the probability of accepting a lot is Pa, the outgoing lots
will contain a proportion of pPa defectives. If the nonconforming units found in the
sample of size n are replaced by good ones, the average outgoing quality (AOQ) will
be
In short, one defines the average outgoing quality as the expected quality of outgoing
product following the use of an acceptance sampling plan for a given value of the
incoming quality. Figure 4.13 gives a typical AOQ curve as a function of the
incoming quality.
Figure 4.13 AOQ Curve
If the incoming quality is good, then a large proportion of the lots will be accepted by
the sampling plan and only a smaller fraction will be screened and hence the outgoing
quality will be small (good). Similarly, when the incoming quality is not good, a large
proportion of the lots will go for screening inspection and in this case also, the
outgoing quality will be good since defective items will be either replaced or rectified.
Only for intermediate quality levels, lot acceptance will be at a moderate rate and
hence the AOQ will rise (see Figures 4.14 and 4.15). The maximum ordinate of the
AOQ curve represents the worst possible average for the outgoing quality and is
known as the average outgoing quality limit (AOQL). In other words, the AOQL is
defined as the maximum AOQ over all possible levels of the incoming quality for a
known acceptance sampling plan (see Figure 4.14).
Figure 4.14 AOQ and AOQL
Figure 4.15 Occurrence of AOQL at a Moderate Quality Level
It should be noted that both the concepts of AOQ and AOQL are valid only in Type B
situations, that is for a continuous stream of lots. For isolated lots, the AOQ and
AOQL will have no meaning and consumer protection could be achieved only
through the limiting quality. The expression for AOQ will also be affected by the
disposition policy of discarding the nonconforming unit instead of rectifying it or
replacing it with a conforming unit in the screening inspection phase.
Figure 4.16 ATI Curve When N = 1000, n = 50, Ac = 1
Similar to the measure of average sample number considered for sampling inspection,
an important measure relevant to rectifying inspection is the average total inspection
(ATI), which is defined as the average number of units inspected per lot based on the
sample for accepted lots and all inspected units in lots not accepted.
If the lot size is N and is of quality p, then the ATI for the single sampling plan is
given by
ATI(p) = n + (1-Pa) (N-n).
This expression is simple to interpret. The single sample of size n is always inspected.
If the lot is not accepted, the remainder of the lot containing (N-n) units are screened,
for which the probability is (1-Pa ). The above expression is actually obtained from the
expression
ATI(p) = nPa+ N(1-Pa),
meaning n units are only inspected for those lots accepted (the probability being Pa)
and the whole lot will be inspected on non-acceptance (the probability being (1-Pa)). A
typical ATI curve is shown in Figure 4.16. For lots of perfect quality (p=0), the ATI
will be n and for lots with all defective units (p=1), the ATI will be N.
In general, the OC and AOQ functions are known as protection measures since they
reveal the protection given by the sampling plan to the producer and consumer. The
measures ASN and ATI are known as cost measures since they give an idea of the
cost involved.
Appendix A4.10, providing MINITAB local macros, must be consulted for drawing
the OC, AOQ and ATI curves of a given single sampling plan. It also shows how to
use a macro for drawing the curtailed ASN curve.