the data set is bad in relation to the quality desires of the user.
All the previous discussion in this section is about the rela-
tion between the
AQL
and the
BaM
, but the use of
ISO
2859-2 is
somewhat different because it is indexed by means of the
LQ
.
As stated above, the
AQL
is not used directly but there is a rela-
tion between
LQ
and
AQL
. The proposed relation is to consider
LQ
= k
×
AQL
, where
k
≥
3
. In order to protect the user using this
LQ
value, the standard reduces the probability of acceptance
significantly by means of increasing the sample size, for the
same lot size, and maintaining or reducing the acceptance val-
ue. In real terms, this means a greater demand for quality and
therefore a lower acceptability for the
LQ
in order to protect the
user. For example, let us consider a Binomial distribution B(
n,
p
) where
n = 20
and
p
is the probability of success in each trial,
and two cases: (a)
p
1
= 10 % (=
AQL
)
; (b)
p
2
= 30 percent (=
LQ
=3
×
AQL
)
. Table 2 shows the results for the same count of cases (
f
= {0,1,…,5}
), and we can clearly observe that the probabilities
of case two are always fewer than the probabilities of case one.
Example of Application
This section shows two examples of the application of the
proposed method jointly with
ISO
2859-1 and
ISO
2859-2. One
of its major advantages is the possibility of working lot-by-lot
when using
ISO
2859-1, but remember that for isolated lots or
when the length of the sequence of lots is less than ten, we
must apply
ISO
2859-2. Here for both examples, the
BaM
being
used is that presented in Figure 4.
For the example of application of
ISO
2859-1, let us
consider that we wish to control a road data set supply in
Andalucía (Spain), and that the data supply is organized by
the map sheet distribution of the National Topographic Map
of Spain. In this way the mean accumulated number of road
segments (arcs between topological nodes) per map sheet is
91, with a mean accumulated length equal to 220 km. Because
the
BaM
is derived for road segments, we must consider lots of
road segments. In this case the mean lot size is 91 and for the
sake of simplicity in this example, we are going to consider
that the lot size is always 91. Now let us consider a supply
sequence of 15 lots. If we take
Tol =
16.3 m
the
BaM
(Figure 4)
gives
π
=
5 %, so we can consider that
AQL
c1
=
6.5 %
(we take
the next and greater A
LQ
value proposed by the standard).
We start the application of
ISO
2859-1 by considering general
inspection level II. In this way, entering into Table 1 of
ISO
2859-1 (shown here as Figure 7) with the mean lot size (91
items) results in a sample size code letter F. The next step
is going to Table 2.A of
ISO
2859-1, which is entitled “single
sampling plans for normal inspection (master table).” Enter-
ing into this table (here shown as Figure 8) with the sample
size code letter F and
AQL
c1
=
6.5 percent, gives a recom-
mended sample size
n =
20,an acceptance value
Ac =
3, and a
rejection value
Re =
4. As the
ISO
standard states, the sample
of control must be extracted by means of a random procedure.
This is not a problem because current
GIS
software has tools
for this purpose. But, prior to applying this sampling plan, we
must agree with the estimated performance of this sampling
plan. This can be analyzed by means of the corresponding
OC
curve, which is offered by
ISO
2859-1 in Table 10-*, where *
is the code letter, here * = F. Information of risk is offered by
ISO
2859-1 in Table 5.A for producer risks and in Table 6.A for
user risks. A summary of the principal values of these tables
of the International Standard is shown in Table 3 for
AQL
= 6.5
% and neighboring values of
AQL
.
T
able
2. E
xample
of
C
omparison
of
A
cceptance
P
robabilities
for
the
S
ame
C
ount
of
C
ases
f when
LQ = 3 × AQL
Case
Mean
f
=0
f
=1
f
=2
f
=3
f
=4
f
=5
1-B(n=20,
p1=0.1)
n×p1=2
0.1215 0.3917 0.6769 0.8670 0.9568 0.9887
2-B(n =20,
p2=0.3)
n×p2=6
0.0008 0.0076 0.0354 0.1071 0.2375 0.4164
T
able
3. U
ser
and
P
roducer
R
isks
for
S
ampling
P
lan
F O
btained
from
T
ables
of
ISO 2859-1
AQL=4% AQL=6.5% AQL=10%
α
(producer risk)
(Table 5.A of ISO 2859-1)
4.74% 4.31% 1.66%
β
(consumer risk)
(Table 6.A of ISO 2859-1)
25.5% 30.4% 41.5%
If the sampling plan is considered satisfactory we can
start its application. Now consider that control samples are
realized lot by lot. For each lot (column labeled C1) Table 4
Figure 7. Figure of “Table 1 - Sample Size Code Letters” from ISO 2859-1 (ISO 1999).
664
August 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
