and was superseded by
MIL-STD-
105
A
in 1950. The issuance of
MIL-STD-
105
A
started a sequence of revisions through to MIL-
STD-105
D until its withdrawal in 1995. This standard was di-
rectly adopted by the International Organization for Standard-
ization as
ISO
2859-1, is still in force, and the latest revision is
from 1999.
ISO
2859-1 is a well-known method and a widely
extended standard in all industrial scopes (Juran and Godfrey,
1999). When comparing
ISO
2859-1 with other Lot Acceptance
Sampling plans (e.g., Dodge-Romig and Philips), Banovac
et
al
. (2012) conclude that the D-Romig plan protects the cus-
tomer and
ISO
2859-1 protects the producer, while the Philips
plans are somewhere in between. Zero acceptance number
sampling plans (Squeglia, 2008) is another option to take into
account, but the basic ideas of the method are the same. There
is some criticism of
ISO
2859-1’s purpose and formulation and
its appropriateness to address industrial needs (von Collani,
2004). But analysis of such criticism clearly shows that it is not
centered on the statistical basics of the standard but mainly
on the unclear and confusing writing. The
ISO
2859-1 does not
include a statistical demonstration of the standard but there
are many sources (e.g., Montgomery, 2001; Stephens, 2001).
Recently, Banovac
et al
. (2012) analyzed the characteristics of
lot acceptance sampling plans by attributes and also elaborated
a mathematical base. At present, the interest in sampling plans
is related to economical aspects, and there are many recent
studies analyzing relations between quality, sample size (cost)
and risk requirements in order to determine optimal sampling
plans (Hsu and Hsu, 2012; Nikolaidis and Nenes, 2009; Stout
and Hardwick, 2005, Ferrell and Chhoker 2002).
A sampling plan is a lot-sentencing procedure in which a
decision about an entire lot of size
N
is taken from the result
of a random sample of size
n
selected from the lot. The deci-
sion is taken by comparing the accounted number of defects
that are present in the sample with an acceptance number
c
.
Thus, if the lot size is
N
a sampling plan is defined by {
n
,
c
}.
In relation to the size of the lot, larger lots are preferred over
smaller ones because statistical sampling is more efficient
when inspecting larger populations (lots) than smaller ones. It
is also important that lots be conformable to handling systems
used by the producer and the consumer.
Another important parameter for the use of
ISO
sampling
plans (e.g.,
ISO
2859 and
ISO
3951) is the
AQL
(acceptable
quality level). The
AQL
represents the worst or poorest level of
quality for the production process that the consumer would
consider to be acceptable as a process average (Montgomery,
2001). Users (buyers or consumers) are generally willing to
tolerate lots that contain small percentages of defects. This fig-
ure is known as the acceptable quality level. The
AQL
should
be set based on the criticality of the characteristic that is being
inspected (the more critical the characteristic, the smaller the
AQL
should be). For example, although there is no direct rela-
tionship an
AQL
of 5 percent could be related to class I of the
ASLSM
standard and an
AQL
of 10 percent with class II of that
same rule. Also, the
AQL
should be set so that the good incom-
ing lots are of better quality than the
AQL
. Otherwise, the sup-
plier will be overwhelmed with rejected lots and there may
not be enough accepted lots for production to continue. It is
considered that
AQL
provides a guide for the producer on the
level of quality that needs to produced so that the acceptance
criteria can be satisfied (sampling clause) most of the time for
a given desired quality. Because
AQL
is expressed in percent-
age values (1 % , 2 % ,...). the greater the value the poorer the
quality. Thus, an
AQL
of 1 % means that 1 percent of the lot
contains defects while an
AQL
of 2 % means that 2 percent of
the lot contains defects. It is obvious that a 2 percent defective
lot is worse than a 1 percent defective lot. This is confusing,
and for this reason
AQL
was renamed, from “acceptable qual-
ity level” to “acceptance quality limit.” It is the limit, and it is
not really “acceptable.” Process average refers to a sequence
of lots, and for this reason in order to apply
ISO
2859-1, we
require, at least, a sequence of ten lots. If there are not ten lots
we have to analyze lots individually and apply
ISO
2859-2,
detailed below. It has to be noted that the
AQL
is a property of
the production process that is shown in the product present-
ed as a lot, which means that the
AQL
is not a property of the
sampling plan. The
AQL
is simply a standard against which
to judge the lots, and it is hoped that the production process
will operate at a fallout level that is considerably better than
the
AQL
(Montgomery, 2001). If the latter is not true, there will
be a great rejection in the acceptance process. For this reason,
ISO
2859 and the use of
AQL
are adequate when the aim is to
maintain a quality level as a target. Based on the Technical
Report of the Geographical Survey Institute of Japan (JGSI,
2002), we can establish the following relationships:
•
AQL
= 0 % : No error is permitted (an error loses the
value as product.).
•
AQL
=5 % : It is desirable that there is no error.
•
AQL
= 10 % : Slight error is permitted.
•
AQL
= 20 % : Error is permitted to some extent.
Because random sampling cannot identify all lots that
contain more than
AQL
percentage of defectives, users recognize
that some lots that actually contain more defectives than
AQL
will be accepted. However, there is an upper limit for the per-
centage of defective items that the user is willing to tolerate in
accepted lots. This value is known as the lot tolerance percent
defective (
LTPD
). Thus, users desire a quality equal at or better
than the
AQL
, and are willing to live with poorer quality, but in
this case they do not accept any lots with a defective percentage
greater or equal to the
LTPD
. The
LTPD
should also be set based
on the criticality of the characteristic that is being inspected
(the more critical the characteristic, the smaller the
LTPD
).
The probability that a bad lot containing defectives equal to
the
LTPD
will be accepted is known as the consumer’s risk, or
beta (
β
), or the probability of making a Type II error. On the other
hand, the probability that a good lot containing defectives equal
to the
AQL
will be rejected is known as the producer’s risk, or
alpha (
α
), or the probability of making a Type I error. The latter
means that the probability of accepting a good lot with quality
better or equal to the
AQL
is 95 percent. This is because
ISO
2859
series and
ISO
3951 series are designed to have a producer’s risk
of 5 percent and a consumer’s risk of 10 percent, but in the case
of
ISO
2859 series because of the nature of counting variables,
these levels are not assured for all sampling plans.
The inspection level designates the relative amount of
inspection. There are seven inspection levels (S1, S2, S3, S4,
I, II, and III) proposed by the standard. Inspection levels I,
II, and III are for general use, and unless otherwise specified
level II shall be used.
At each inspection level three different severities of in-
spection can be applied:
• Normal inspection. The acceptance criterion has been
devised to secure the producer a high probability of
acceptance when the process average of the lot is better
than the
AQL
.
• Tightened inspection. The acceptance criterion is
tighter than that for the corresponding plan for normal
inspection and is invoked when the inspection results
of a predetermined number of consecutive lots indicate
that the process average might be poorer than the
AQL
.
• Reduced inspection. The sample size is smaller than
that for the corresponding plan for normal inspection
and with an acceptance criterion that is comparable to
that for the corresponding plan for normal inspection.
Reduced inspection is at the discretion of the respon-
sible authority.
660
August 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
