Kleijnen, J.P.C.

Kriens, J.

Lafleur, M.

English

Research Papers in Economics

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Tii.,SUi'í~ ~SAMPLING FOR QUALITY INSPECTION AND
CORRECTION: AOQL PERFORMANCE CRITERIA
J.P.C. Kleijnen
J. Kriens
M.C.H.M. Lafleur
J.H.F. Pardoel
FEw 246
School of Business and Economics
Tilburg University
Post Box 90153
5000 LE Tilburg
Netherlands
Version 1: December 1986SAMPLING FOR QUALITY ZNSPECTION AND CORRECTION:
AOQL PERFORMANCE CRITERIA
J.P.C. KLEIJNEN
J. KRIENS
M.C.H.M. LAFLEUR
J.H.F. PARDOEL
School of Business and Economics
Tilburg University
Post Box 90153
5000 LE Tilburg
Netherlands
Keywords: Percentage defectives, quality control, Monte Carlo
simulation, auditing.
ABSTRACT
The Average Outgoing Quality Limit (AOQJ,) denotes a
sampling plan leading to inspection of the whole population, if
the sample shows a number of defective items k exceeding an
acceptance number ko. According to the literature, this con-
stant ko is chosen such that the expected value of é, the frac-
tion of defectives after inspection and possible correction,
does not exceed a prespecified constant pm; moreover several
other performance criteria are estimated, using an extensive
Monte Carlo simulation. The main conclusion is that the AOQL
scheme is useful in practice, including applications in audi-
ting, but the chance that the quality constraint is violated,
P
Lp
) pmJ , may be sizable.z
1. INTRODUCTION: AOQL
AOQL was íntroduced by Dodge and Romig around 1930; its
application to suditing was studied by Kriens and Veenstra
(1985). We can summarize this sampling scheme as follows. The
goal of AOQL as introduced by Dodge and Romig, is to guarantee
a minimum quality of the outgoing populations expressed as a
maximum for the average fraction of defectives in the popula-
tion. Kriens and Veenstra (1985) split up the yearly population
into a number of subpopulations. The quality of a yearly popu-
lation - before sampling and correction - is quantified by p,
the percentage of "defective items" in the yearly population.
The Number of items (defect plus correct) per Year is (say) NY,
for example, a company produces NY cars; in suditing, accounts
are sampled and NY is measured in dollars per years; see Kriens
and Veenstra (1985, p. 387). Consequently, after inspection and
correction the minimum quality corresponds to a maximum value
for p.
The sampling scheme has the following steps (also see
Table 1 later on).
(i) The expected yearly population is divided into a number of
subperiods S, for example, S- 52 corresponds to production per
week. These subpopulations may have different sizes, in expec-
tation and certainly in realization. We denote the realized
size of the subpopulation by N.
(ii) From each (realized) subpopulation a sample of size n is
taken (n depends on several parameters, as we shall see).
(iii) Per sample (of size n) the number of defective items k is
determined by inspection; obviously k is random (denoted by an
underscore). And its integer values k satisfy: 0 s k s n.
(iv) If and only if k exceeds a critical constant ko (which
varies with n; see step ii) then the whole subpopulation is3
inspected and, by assumption, all defective items in the subpo-
pulation are corrected. If k s ko then the defective items in
the sample are corrected. So after sampling the quality of the
subpopulation is improved, unless no defectives at all were
found (k - 0) which probably indicates that the subpopulation
had perfect quality already.
After step (iv) the fraction of defectives per subpopulation ~
should satisfy the minimum-quality requirement pm. So given a
correct selection of the sampling plan's parameters n and ko
(see next paragraph), p should satisfy the condition Er~1 s pm.
Obviously, if the original fraction of defectives (bef,o.lre sam-
pling) was very good already (say, p- 0), then E~~~ C p. If
this quality was very bad (p ~~ pm), then the sampling plan
implies that sampling is (nearly) always followed by inspection
and correction of the whole subpopulation so that (0~) p C~ pm.
s
See Figure 1 where p is the "least favorable" value of p.
Obviously k follows the hypergeometric distribution
with parameters n, p and N. The critical constant ko and n can
be computed such that, not only holds E~p~ s pm, but also the
expected costs are minimized. The original tables in Dodge and
Romig (1959). however, contain some inaccuracies. Therefore we
use the recent tables computed by Kriens and Winters (1987);
see also Veenstra and Buysse (1985) and Van Batenburg, Kriens
and Veenstra (1987). Table 1 illustrates some typical results.
In practice p, the before-sampling or prior probabili-
ty, is unknown and often only the left-most columns in tables
like Table 1 are used (low p). Even if the prior probability is
estimated wrongly, the quality constraint E~p~ 5 pm is met; the
expected costs, however, may increase. A conjecture is that,
not only does the expected value meet the quality constraint,
but also the chance of too bad a yearly quality is negligible,4
p'
Figure 1: Expected fraction of defectives after sampling E(p) versus fraction
before sampling p. -S
Table 1: Sample size n and acceptance number ko depending on
prior probability p, subpopulation size N, and quality
limit pm (pm - lz).
Subpopulation Before sampling p
size N
0-0.02 ... 0.21-0.40 ... o.81-i.oo
N n ko n ko n ko
1-25 All 0 All 0 All 0
26-50 22 0 22 0 22 0
8oi-looo 35 0 80 1 120 2
iooi-2ooo 36 0 8o i 180 3
20001-50000 85 1 255 4 990 15
5oooi-100o00 85 1 255 4 1520 22
i.e., if ~ denotes the average yearly outgoing quality then
P ~p ~ PmJ
. 0
This conjecture is investigated in our simulation. Moreover, it
hardly takes more computer time to estimate how bad p is if the
constraint p~ pm is violated, i. e., we estimate the following
conditional expectations:
Elp - pmlp ~ pmJ
(1.2)6
2. DESIGN OF MONTE CARLO EXPERIMENT
As Table 1 showed, the sample size n and acceptance
number ko are completely determined by the subpopulation size
N, the prior probability p, and the quality limit pm. In turn
the subpopulation size N depends on the yearly population size
NY and the number of subperiods S. In the simulation we study
three values for S, namely 4, 13, and 52 which correspond to
quarters, "months", and weeks. Our selection of the yearly
population size NY is based in the experience of one of the
authors with auditing applications: 10,000; 100,000 and
1,000,000. The expected subpopulation size E(N) equals NY~S. We
assume that the actual sizes N follow a uniform distribution
with expected value NY~S; its range is such that the coeffi-
cient of variation is always (roughly) 6X, an arbitrarily se-
lected value.
We selected the following six values for the quality
limit pm: O.1X, 0.5X, 1X. 2X, 5X, lOX. Selection of the prior
probability p in the simulation must be related to the quality
limit pm. There are no tables available for p) 2 pm. This is
no problem if only the left columns of the tables are used (see
section 1); obviously if p is very high, then the scheme breaks
down, that is, sampling is (nearly) always followed by inspec-
tion of the whole subpopulation; therefore we restrict our
simulation of the "practitioner" to, p S 6 pm. Obviously not
all subpopulations must have the same p, even if all subpopula-
tions have the same expected value E(~). Therefore we sample p
in the simulation. As figure 1 demonstrates, the performance
~
E(~) improves as p deviates from the least favourable value p.
In preliminary simulation experiments we sampled p from a dis-
tribution with s high variance, and indeed E(~) decreased (not
further reported in this paper). Therefore we concentrate our
simulation on worst case situations, that is, p has a small7
range. We further assume that p is uniformly distributed with a
range of only 0.2 pm. We do change the expected value E~e1; as
we explained above, we vary p between 0 and 6 pm. So we Jsample
p from the uniform distribution between 0 and 0.2 p, between m
0.2 pm and 0.4 pm, ... , between 5.8 pm and 6 pm.
Summarizing, we simulate 1620 factor combinations using
only the left-most columns of the tables ("practitioner's ap-
proach") and 540 combinations with the optimal ('n, kol combina-
tions ("theoretical approach").
L J
There is a technical issue in the simulation: how often
should each factor combination be simulated in order to obtain
reliable estimates of performance criteria such as PL
~ ~ p I?
m.l
By definition, one replicate yields a binomial variable (say) x
with q- P(x - 0) - PI~ ) pmJ. Using the normal approximation
to the binomial distl,ribution, it is straightforward to derive
the number of replications needed to estimate q with either a
relative precision of lOX or an absolute precision of 0.001;
also see Kleijnen (1987, pp. 46-51). This approach shows that
at most 16,221 replications are needed (when q-.O1) to satis-
fy either the relative precision or the absolute precision
requirement, with 90X probability. Actually we do not know q.
So we substitute the "current" estimate for q(after at least
100 replications) that is, the estimate available after (say) r
replications where r- 101, 102, ... . We found that the aver-
age number of replications is roughly 1000. We emphasize that
the simulation not only estimates the performance criterion q-
Plp ) pmJ but several more criteria. The main criterion, how-
ev`er, is q so that we concentrate on q to select the number of
replications.s
It turns out that it takes 40 hours of computer time to
simulate 1620 ; 540 factor combinations, each combination re-
plicated roughly 1000 times. To keep computer time below this
(sizable) value, we have to introduce the following technical
refinement. The number of defectives k has a hypergeometric
distribution. The binomial distribution provides s good ap-
proximation of n(t N which is often the case (but not always:
if NY is small then n~ N may occur); see Table 1. The Poisson
distribution is a good approximation to the binominal distribu-
tion, if p is small. We simulate the Poisson distribution using
the subroutine in Naylor et ai. (1966, p. 114), so that our
program runs 20 times faster on our computer (a VAX 780 running
under VMS using the NAG subroutine for the multiplicative con-
gruential random number generator).
3. MONTE CARLO RESULTS
The Monte Carlo experiment yields a mass of data. We
analyze these data through regression analysis (using SAS).
Preliminary plots look like a gamma function. Therefore we fit
such a type of non-linear regression model for p; its R2 adjus-
ted for the number of explanatory variables, is higher than
0.95 and yields Figure 2, which looks like the theoretical
Figure 1: p s p.
m
If the prior probability satisfies p 5 pm then obvious-
ly q- PI~ ) pmJ - 0. If p) pm then we again fit a function
like the g
LLL
amma function, with R2 - 0.99 for the theoretical
case and 0.74 for the practitioner's approach; see Figure 3. So
there is a sizable chance (up to 40X in Figure 3) of violating
the quality constraint p S pm. However, we repeat that our
simulation is a worst case, since p of the subpopulation is
sampled from a uniform distribution with a small range.y
~
0.000 0.005
~
0.010
~-
0.015
~-
0.020
~-
0.025 0.030
~-
0.035
~ p
0.040
Figure 2: Averaqe yearly outgoing quality P versus fraction of defectives
before sampling p(yearly population NY - 1,000,000; subperiods S- 52).10
P(P ~ Pm)
0.50
0.45
0.40
0.35
0.30
F
H 0.25
T-
0.005 0.010
(Pm)
Practice
Theory
0.015 0.020 0.025 0.030 0.035 0.040
P
Figure 3: Estimated probability of quality violation, P(p , pm) (NY - 100,000; S- 52).f:(P ' Pm I P~ Pm) ~ 1000
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0 -r
0.010
(pm)
T~
0.012 0.014
P
0.016 0.018 0.020
Figure 4: F.stimated size of quality violation, E(p - pm~ p, pm) in
theoretical approach ( NY - 1,000,000).12
0.000 0.005 0.010 0.015 0.020
(pm)
0.025
P
0.030 0.035 0.040
Figure 5: Estimated fraction of yearly population, actually inspected
(NY - 1,000,000 ; S - 52).13
If p~ pm then we wonder how bad the quality violation
is: E~~ - pm~p ~ pmJ. Figure 4 shows that smaller subperiods
(higher S) give extra protection.
Next we consider the costs of the AOQL scheme. This
scheme implies that all N units (of a subperiod) are inspected
if k~ ko. Figure 5 shows the fraction of the yearly population
actually inspected. That fraction increases drastically if p)
pm. Obviously the practitioner's approach is more expensive. We
add that the curves are hardly effected by S, the number of
subperiods (not displayed).
Our simulation shows that is it important to have a
good idea about p, the before sampling fraction of defectives.
Therefore we suggest to obtain an estimate of p, using p- k~n
if k s ko and p- K~N if k~ ko where K denotes the number of
defectives in the subpopulation (of size N). As time t goes on,
the estimators gt can be combined; for example, we may weigh
the gt with the sample sizes nt or the subpopulation sizes Nt
(if k s ko or k~ ko respectively). If pt shows serial correla-
tion or non-stationary behavior, then time series techniques
may be applied.
4. CONCLUSIONS
The AOQL sampling plan is indeed used in practice; see
Kriens and Veenstra (1985). Then it is assumed that if the
expected yearly fraction of defectives after inspection and
correction E(~) meets the quality constraint pm then the proba-
bility of exceeding the constraint, PI~ ) pmJ is negligible.
Figure 3(based on our simulation dat`a analyzed by regression)
shows that this probability is sizable, if the "before inspec-
tion" or prior probability ~ is higher than the limit pm but14
not extremely high (if p)) pm then most times sampling is
followed by inspection of the whole subpopulation). If in prac-
tice p varies much over subperiods, then PI~ ~ pmJ decreases
(we simulated worst case situations: smallll..range of p). One can
get an-estimate of p from the sampling procedure: if k s ko
then ~- k~n; else p- K~N. Increasing the number of periods S
decreases the size of the expected quality violation; see Fi-
gure 4. Underestimating p is not wise: it does not give extra
quality protection (Figure 3) and yet more inspection work is
done (Figure 5). So in practice one should build up knowledge
about p.
REFERENCFS
Dodge, H.F. and H.G. Romig, Samplíng inspectton tables. John
Wiley, New York, second editíon, 1959.
Kleijnen, J.P.C., StattsttcaZ tooLs for simulatton practttio-
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Kriens, J. and R.H. Veenstra, Statistical sampling in internal
control by using the AOQL-system. The Stattsttctan, ~,
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Kriens, J. and A.A.G.M. Winters (1987), The A.O.Q.L. tables by
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Naylor, T.H., J.L. Balintfy, D.S. Burdick and K. Chu, Computer
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applications in the administration, De Accountant, 10, no.
198, pp. 561-563. (in Dutch.)i
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Sampling for quality inspection and correction: AOQL performance criteria

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