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Economic Evaluation of ISO 2859 Acceptance Sampling Plans
used with Rectifying Inspection of Rejected Lots
Yiannis Nikolaidisa,, George Nenesb
a Department of Engineering & Management of Energy Resources,
University of Western Macedonia, 50100, Kozani, Greece
b Department of Mechanical Engineering,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Abstract
To conduct Acceptance Sampling, companies often use plans that are determined by easy-to-
use standards. However, these standards do not take quality costs directly into account. Motivated
by the case of a Greek company, which uses the Greek equivalent to the ISO 2859 (1974) for the
quality control of its incoming raw materials, this paper aims at evaluating the single-sampling
plans recommended by the latest update of ISO 2859, from an economic point of view. The
evaluation shows that the use of standards rarely leads to satisfactory economic results. Therefore,
simple rules for the economical use of the standard are provided.
Key words: Statistical Quality Control; Acceptance Sampling; Sampling Plan; Economic Design;
Quality Cost
Corresponding author. Tel.: +30–24610–56700; fax: +30–24610–56601.
E-mail address: [email protected] (Yiannis Nikolaidis). Department of Engineering & Management of Energy Resources, University of Western Macedonia, 50100, Kozani, Greece
1. Introduction
Over the last decades quality issues have held the attention of both industry and research
communities because of their impact on cost and, therefore, on profit. Quality costs include any
cost that results from the fact that systems, processes, products and services are imperfect. More
specifically, it consists of prevention costs, appraisal costs, internal and external failure costs, and it
varies between 4% and 40% of the sales of a company (Montgomery, 2001). Consequently, the
analysis and estimation of quality cost are of great importance. Usually, all quality cost categories
can be expressed as functions of the actual quality of products (e.g. fraction nonconforming of a
lot), therefore, it is possible to derive the economically optimum quality of products, as well as the
optimum quality control policy, i.e., the control policy that minimizes the expected quality-related
costs.
Statistical Quality Control (SQC) aims at monitoring and improving the quality of products
produced by a process and consists of three areas/types: Design of Experiment, Statistical Process
Control and Acceptance Sampling. Modern quality assurance systems usually place less emphasis
on the latter, while they attempt to focus their efforts on the other two types of SQC. However,
there are still numerous companies all over the world where the evolution of SQC techniques is
limited and, thus, they use mainly Acceptance Sampling in order to monitor the quality of either
incoming materials or final products. For example, in most small to medium-sized Greek
companies, the typical SQC procedure is lot-by-lot Acceptance Sampling, single-sampling by
attributes in particular. In our research we focus on this type of SQC.
In general, single-sampling by attributes is carried out as follows: from a lot of N units, a
sample of size n is taken and every item of the sample is checked and characterized as conforming
or nonconforming based on an attribute-type quality characteristic, e.g., color, appearance etc. The
lot may be accepted or rejected depending on the outcome of the control and the acceptance -
rejection criterion, i.e., the maximum allowable number of nonconforming items in the sample
(acceptance number c).
In order to monitor the quality of acquired or produced lots, companies often use sampling
plans (n,c) that are recommended by specific standards, which are simple and easy to use. Two
recently released technical reports that provide guidance for Acceptance Sampling of products in
lots are ISO/TR 8550-1 (2007) and ISO/TR 8550-2 (2007). However, the most popular
international standard is ISO 2859-10 (2006). In fact, this standard is a review of the ISO 2859
series of standards and it does not include any tables of sampling plans. The latter can be found in
any of the five standards - members of the ISO 2859-10 (2006) family of standards, namely ISO
2859-1 (1999), ISO 2859-2 (1985), ISO 2859-3 (2005), ISO 2859-4 (2002) and ISO 2859-5 (2005),
which are suitable for various Acceptance Sampling cases. Although in our paper we use the
sampling plans recommended by ISO 2859-1 (1999), in the analysis of the paper hereafter, we will
use the general term ISO 2859.
In Greece, Acceptance Sampling is usually conducted by using the ELOT 398.0 (1982)
standard and its addendum ELOT 398.1 (1982). Together, ELOT 398.0 and 398.1 constitute the
Greek equivalent to the ISO 2859 (1974), namely, to an outdated version of the ISO 2859 family of
standards. Note that although the ISO 2859 standard has been updated many times since 1974, the
respective Greek standards remain the same, not following the updating procedure of the
international standard. Nevertheless, despite the updates of ISO 2859, the part of the latest version
that we examine in our research has remained unchanged through years and therefore, our results
that were based on ELOT 398.0 and 398.1 standards, hold also for the newer versions of the ISO
2859 family.
The motivation for this research was given by a project that we undertook a few years ago,
that was intended to reduce the quality-related costs incurred during the Acceptance Sampling
process, using the ELOT 398.0 and 398.1 standards, of the incoming raw materials of CRYSTAL
S.A., a commercial refrigerators Greek company. The main conclusion of that project was that the
ELOT standards were leading to unacceptably high quality costs very often. Having realized that,
we have extended our research in order to come up with a simple and easy way to use either the
ELOT 398.0 and 398.1 standards or, most importantly, the ISO 2859, without overlooking their
inevitable economic impact. Consequently, this paper focuses on determining the economic
consequences of using a quality standard without taking into account the respective quality costs.
In any case, we don’t downplay the ISO 2859 standard. We just present some rules that can lead to
its better use, considering that this particular standard per se is a tool for statistical quality control
with wide-ranging appeal. It should be mentioned that this effort has been embraced by a lot of
companies in Greece, mainly because it reveals some weaknesses of quality standards concerning
their economic efficiency, which the companies haven’t realized until now.
The implementation of single-sampling plans by attributes that are recommended by the ISO
2859, is evaluated economically for a variety of a) parameters of the control process, i.e., lot size
(N), Acceptance quality limit (AQL) etc., b) cost elements and c) quality level of lots, viz the
fraction nonconforming (p). In particular, we determine the average total quality cost
corresponding to the plans (n,c) recommended by the standard and then we compare it against the
respective cost of the optimum Acceptance Sampling plans (n*,c*). In addition, we compare the
parameters of the former plans against the optimum parameters, i.e., the plan parameters that lead
to the minimization of the average total quality cost.
The remainder of the paper is structured as follows. The following Section presents the related
literature. In Section 3 we set the notation and we formulate the average total quality cost function.
Section 4 briefly presents the case of CRYSTAL S.A. The combinations of parameters that are
examined and the extensive numerical investigation are presented in Section 5, while the final
Section of the paper summarizes the main conclusions of our research and proposes topics for
future research.
2. Related Literature
Detailed information about SQC techniques can be found in many books, such as the one by
Montgomery (2001). In Tagaras (2001) there is a section dealing with the economic aspect of
Acceptance Sampling plans. More specialized information about Acceptance Sampling and its
economic dimension can be found in Hald (1981).
Significant reviews of papers concerning Acceptance Sampling are those of Wetherrill and
Chiu (1975) and Wall and Elshennawy (1989). Both papers make extensive references to
economically optimum sampling plans. Bai and Riew (1984) develop an economic Acceptance
Sampling plan by attributes for cases where sampling is expensive or destructive. They present a
linear cost model and they consider three decision criteria. A few years later Tagaras and Lee
(1987) develop an algorithm to compute the economically optimum single-sampling plan, using
Bayes’ theorem.
Even though lately Acceptance Sampling has given place to more advanced SQC techniques,
the published research on this particular SQC area and especially on the economic design of
sampling plans is still extensive. Thus, Ferrell and Chhoker (2002) have recently proposed an
economic model for the design of Acceptance Sampling plans adopting the Taguchi approach.
González and Palomo (2003) use a Bayesian analysis in order to derive Acceptance Sampling plans
regarding the number of defects per unit of product and apply their methodology to the paper pulp
industry. The sampling plans are obtained following an economic criterion: the minimization of the
expected total cost of quality. At the same time, Cassady and Nachlas (2003) define a generic
framework for establishing three-level acceptance sampling plans, using quality value functions.
They note that there are many cases in which the quality of a product can be classified in three or
more discrete levels; for example, a food product may be classified as good, marginal or bad. Chen
et al. (2004b) present and investigate a general model of Acceptance Sampling plan for the
exponential distribution with exponentially distributed random censoring, based on Bayesian
decision theory. In order to determine the optimal sampling plan they consider a loss function,
which includes the sampling cost, the time-consuming cost and the decision loss to determine the
optimal Acceptance Sampling plan. At the same time and in a similar study Chen et al. (2004a)
develop a general Bayesian framework for designing a variable Acceptance Sampling plan with
mixed censoring. A general loss function including the three partial costs of Chen et al. (2004b) as
well as the salvage value is introduced to determine the corresponding optimal sampling plan. Stout
and Hardwick (2005) present a unified approach to the problem of response adaptive screening,
when multiple costs and constraints ought to be incorporated. In particular, they describe a cost-
and constraint-based approach, which is suitable in Acceptance Sampling where sample products
must be tested before an entire batch is accepted. Finally, Chen (2006) modifies Pulak and Al-
Sultan’s model in order to determine the optimum process mean and standard deviation under the
rectifying inspection plan with the average outgoing quality limit protection. The symmetric
quadratic quality loss function of Taguchi is adopted for evaluating the product quality.
The forerunner of the present paper is some earlier work by Nikolaidis and Nenes (2005),
regarding the project mentioned previously, which deals with the economic consequences of using
the ELOT 398.0 and 398.1 standards, for the control of some raw materials of CRYSTAL S.A.
Regarding the updates of the ISO 2859 standard, it has been updated twice since 1974. More
specifically, ISO 2859 (1974) has been revised by ISO 2859-1 (1989), which in turn has been
replaced by ISO 2859-1 (1999). The latter is currently the applied standard and, as mentioned
previously, it constitutes a part of the recently published ISO 2859-10 (2006).
3. Notation and Cost Function
For evaluating the economic results of the use of a specific sampling plan or for designing a
sampling plan using economic criteria, it is necessary to determine the economic elements of the
control process first and then to formulate an average total quality cost function. The form of this
function depends on the characteristics of the control process. The calculation or the minimization
(using either analytical or numerical methods) of that function ensues.
In the case of Acceptance Sampling by attributes, the quality cost categories mentioned
previously are modified accordingly; normally, in the average total quality cost function one can
find the sampling cost, the cost of handling nonconforming units (e.g., repair cost, returns etc.) and
the cost of nonconforming items not detected during inspection (e.g., use of nonconforming raw
materials in the production process, defamation of a company in case that nonconforming products
reach its customers etc.). The notation of the respective cost elements is the following:
ci: inspection cost per item, which is determined by reckoning the time required to inspect a unit,
the salary of the inspector(s), the cost of the control device(s) etc.,
cr: replacement cost, i.e., purchase cost, in case a nonconforming unit is immediately replaced
upon detection at a company’s expense, or repair cost per nonconforming item,
cd: cost per nonconforming item that is not detected during inspection; cd cr, where equality
holds in case a nonconforming unit is replaced or repaired with no additional cost.
The average total quality cost per lot that corresponds to the use of a single-sampling plan
(n,c) for the inspection of a lot with a fraction nonconforming p, is in general:
pcnK , [cost of accepting a lot] pPa +[cost of rejecting a lot] pPa 1 , (1)
where dnc
d
da pp
dndnpP
1!!
!0
is the probability of accepting a lot with a fraction
nonconforming p.
In the case that a lot of size N is submitted to 100% inspection if rejected (rectifying
inspection), the analytical form of (1) becomes:
pPNpcNcpPpcnNnpcncpcnK ariadri 1, . (2)
Note that although the choice to apply 100% inspection on rejected lots is not always made in real
life situations, it is very common in practice and, thus, this paper focuses only on such cases.
Remark 1: through simple modifications of (2) it is easy to verify the observation of Hald
(1981) that in case of known and constant fraction nonconforming p the optimum sampling plan is
either 100% inspection, i.e., n = N - if rdi cccp / - or acceptance without sampling, i.e., n = c
= 0 - if rdi cccp / . The ratio rdi ccc / is called break-even quality level and is denoted
hereafter by pr.
Remark 2: again, through simple modifications of (2) it can be seen that the average inspection
cost in our cost function is a product of the inspection cost ci and the average total inspection (ATI),
which Montgomery (2001) calculates by nNpPnΑΤΙ a 1 . Specifically:
pPNpcpPpcnNnpcATIcpc,nK aradri 1 . (3)
Since most of the times the fraction nonconforming p per lot is not deterministic, but is
distributed according to a probability density function (pdf) φ(p), the average total quality cost
cnK , is given by:
p
dppφpcnKcnK ,, . (4)
Remark 3: a closer look on (2) and (4) reveals that the simultaneous increase or decrease of cr
and cd by A cost units, leads to an increase or decrease respectively of c,nK by pNA cost units,
independently of the distribution of p. Note that p denotes the average fraction nonconforming of
consecutive lots, whose p is distributed according to φ(p). However, since the term pNA is
constant, i.e., it is not affected by the parameters of the selected sampling plan (n,c), it follows that
the optimum plan (n*,c*) doesn’t change, if a simultaneous change of the same magnitude on cr
and cd takes place.
4. The Case of CRYSTAL S.A.
As mentioned previously, the inspection of incoming raw materials in CRYSTAL S.A. is held
using the ELOT standards. Subsequently, an indicative raw material is presented in detail; the
palette, which is the basis where the refrigerator is placed during packaging. It is very important for
every palette to carry the necessary nogs in order for the packaged refrigerators to be properly
placed into containers. In all palettes that are found during quality control to be nonconforming
(i.e., without nogs), the company’s inspector puts the necessary nogs. If a palette is found to be
nonconforming during the packaging process, then the proper worker should lift up the refrigerator,
remove the palette, put the nogs appropriately and, finally, place again the refrigerator on the
palette. In case that a lot of palettes is rejected, it is submitted to 100% inspection at company’s
expense. According to historical data, the usual lot size is 800 palettes, while the fraction
nonconforming per lot p is distributed uniformly between 5% and 8%, i.e., p ~ U(0.05, 0.08).
In order to conduct Acceptance Sampling on the received lots of palettes, the Quality
Assurance Department of CRYSTAL S.A. has arbitrarily chosen to use general inspection level II,
normal inspection (without ever switching to tightened or reduced inspection) and AQL = 1%
(although this was not the wiser choice for AQL). According to these choices, the sampling plan
that ELOT 398.0 and 398.1 recommend is (n,c) = (80,2).
In the case of CRYSTAL S.A., due to the simple visual nature of inspections, the inspection
cost ci was determined by simply taking into account the time to inspect each unit and the salary of
the inspector. It has been estimated that ci = 0.119 € per item. The replacement cost cr (repair cost)
of a nonconforming palette was also relatively easy to be determined. Every time a palette is found
to be nonconforming during quality control, it just needs the missing nogs. Consequently, cr was
calculated taking into account the time needed for this task, the salary of the inspector who puts the
necessary nogs and the cost of nogs. It has been estimated that cr = 0.535 € per nonconforming
item. Finally, the nonconforming palettes, when detected during production, are repaired at a cost
that consists of the cost of nogs and the cost of the time required putting the nogs at the palette. It
has been estimated that cd = 2.3 € per nonconforming item. Note that cd takes a much greater value
than cr since it is more difficult and needs more time to repair a palette, when a refrigerator has
already been placed on it.
Based on the aforementioned parameters of the sampling process, the actual quality of lots of
palettes, the cost elements and using (4) and (2), the average quality cost of the sampling plan that
is recommended by the ELOT 398.0 and 398.1 standards is calculated through 2,80 cnK
09.122
1535.0800119.08003.280800535.080119.08008.0
05.0
dppφ
pPppPpp
a
a € . The minimum average
quality cost and the respective optimum sampling plan are 119.58 € and (n*,c*) = (85,7)
respectively. They are derived by computing (4) and (2) for every possible sampling plan and by
determining the optimal average quality cost and the sampling plan leading to the latter.
Consequently, the sampling plan recommended by the ELOT standards leads to an increased
quality cost of 2.1%, which for different parameters of the sampling process, actual quality of lots
and cost elements may be a lot higher. An explanation for this variation could be attributed to the
100% inspection of rejected lots, which in general is expensive. Therefore, the optimum sampling
plan (85,7) is not so tight as the (80,2) recommended by the ELOT standards, in order not to result
frequently to 100% inspection.
Note also that for the examined raw material, 067.0535.03.2/119.0/ rdir cccp ,
i.e. 6.7%, which is very close to the average value of p (6.5%), according to its distribution. In this
case, as it will be explained extensively in the following Section, the standards were expected to
provide a near-optimum sampling plan. Nevertheless, pr may be a lot different from the average
value of p and the economic loss of using the aforementioned standards is usually much higher.
Undoubtedly, the benefit from using the optimum sampling plan instead of the one
recommended by the ELOT standards is marginal for the palettes. However, considering that the
incoming raw materials of CRYSTAL S.A. are numerous made obvious to the company that the
economic benefit from using the optimum sampling plans for all its incoming materials would be
significant. In addition, it justified and crowned with success the project that was undertaken.
5. Numerical Investigation on ISO 2859
The economic evaluation of the Acceptance Sampling plans recommended by ISO 2859 is
accomplished for a variety of combinations of parameters. The set of values that are chosen, as far
as the cost elements and the pdfs are concerned, primarily aim at reflecting real cases of
Acceptance Sampling in practice. As for the values of AQL and N, they are selected with a view to
covering most sub-cases that can occur every time the standard is used in real-life applications.
More precisely:
Lot size N: 40 different values are investigated, three of each one of the 13 first classes of
the ISO 2859 standard (a small, a medium and a large N value for every class) and one of
the 14th class, covering a wide range of N values.
Acceptance quality limit AQL: 11 different values are studied, i.e., all the values between
0.065% and 6.5%, covering the majority of the possible AQL values of the standard.
Cost elements ci, cr and cd: 11 different combinations of cost elements - that lead to a
variety of values of pr - are investigated and are presented in Table 1. Note that according
to Remark 3 these combinations correspond to many more combinations of cost
parameters. For example, by setting A = 50 in combination 2, the latter corresponds to ci =
1, cr = 100, cd = 500, while for A = 650 it corresponds to ci = 1, cr = 700, cd = 1100, which
besides ci have also the same cr or cd with combination 1, respectively. This correlation of
combinations permits the conduction of further investigation in an easy way.
[Insert Table 1 about here]
Actual quality level - fraction nonconforming p: we study four pdfs for p, three uniform
ones, i.e., U(0.0005, 0.005), U(0.0015, 0.04) and U(0.01, 0.07), corresponding to lots of
very good, good - normal and bad quality, respectively, and a truncated normal pdf
between 0.15% and 4%1, which will be denoted hereafter by TN(0.0015, 0.04), for reasons
of consistency in notation.
In every case that we examine we consider the general inspection level II, since this inspection
level is the most common in practice, adding realism to our numerical investigation. In addition, it
should be mentioned that according to both ELOT standards and ISO 2859, apart from the selection
of the sampling plan, there are also rules concerning skip-lot sampling procedures or switching
procedures between normal, tightened and reduced sampling. According to what happened in the
case of CRYSTAL S.A., these rules are not taken into consideration, since they are rarely applied
in practice (at least in Greece), while they are used for adapting the sampling plans when the
quality of acquired lots changes dramatically during time, which is not the case in our study2. Note
also that in the past many practitioners and researchers have admitted that the implementation of
switching rules constitutes a very complicated - and sometimes severe - procedure to be applied in
practice (Balamurali and Kalyanasundaram, 1997, Brown and Rutemiller, 1973), and as such it is
very often ignored (Gao and Tang, 2006, Taylor, 1995, Selinger, 1995). The evaluation of such
1 In order to design this pdf, we first created a normal pdf using the same mean and variance with the
uniform pdf (0.15% - 4%) and then we truncated the areas below 0.15% and above 4%. We computed the necessary probabilities by dividing the probabilities of the initial normal pdf with the probability corresponding to the truncated area, i.e., between 0.15% and 4%.
2 Note that the quality of lots may not change dramatically during time in our study, but it is not constant either, since it is assumed to follow a specific pdf. If the fraction nonconforming p were assumed to be
cases is not in the objectives of this study. However, the present research can be extended to
account for these procedures as well.
According to our methodology, for every N and AQL values, the sampling plan (n,c)
recommended by ISO 2859 is first determined. Then, for every combination of cost elements and
pdf examined, the average cost of using this plan, cnK , , is calculated by (4) and (2). For the
same combination of cost elements and pdf, the optimum sampling plan (n*,c*), along with the
minimum expected quality cost **,cnK are found; this has been done by calculating (4) and (2)
for all possible ns and cs and, then, by finding the minimum cnK , . Finally, using
**,
**,,cnK
cnKcnKΔΚ .100%, the percentage cost penalty of using the sampling plan (n,c) of
ISO 2859 instead of the optimum (n*,c*), is calculated. Table 2 presents an indicative part of the
results that constituted the basis of our numerical investigation and comparative analysis. It should
be noted that the tool that was used for all calculations and the optimization of the cost function
cnK , was various programs developed specifically for this particular study, in Microsoft
FORTRAN PowerStation 4.
[Insert Table 2 about here]
Properties of the optimum sampling plans
Regarding the optimum sampling plans (n*,c*), we have noticed that Remark 1 stands also for
a stochastic fraction nonconforming p, distributed according to φ(p): whenever pr << p then n* =
N - or tends to N - and c* = 0 - or tends to 0 (100% or at least very tightened inspection), while
whenever pr >> p then n* = c* = 0 (acceptance without sampling). Generally, the increase of pr
gives more and more reduced optimum sampling plans, namely plans with continuously decreasing
n* and usually increasing c*.
Since in most cases ISO 2859 allows neither 100% inspection nor acceptance without
sampling, it is reasonable to think that in all these cases where the optimum sampling plan is
constant, then Acceptance Sampling would be redundant. On the other hand, if it changed dramatically during time then the use of the switching and/or skip-lot procedures would be essential.
“extreme”, the evaluation of the standard is adverse and it seems somewhat unfair. Nevertheless,
we choose to examine some combinations of cost elements that lead to such kind of comparisons,
because in practice many times a company should adopt an “extreme” control policy for a lot,
which is usually not provided by the standard.
Comparing the average cost K(n,c) of the ISO 2859 sampling plans to the optimum K(n*,c*)
The use of the sampling plans recommended by the ISO 2859 instead of the optimum ones,
frequently leads to significant cost burden, which basically varies according to the values of ci, cr,
cd and, consequently, pr. The maximum and the average percentage cost increase ΔΚmax and ΔΚ ,
for p distributed either uniformly or normally between 0.15% and 4% and for every pr investigated,
are presented numerically in Table 3 and graphically in Figure 1. Table 3 includes also the standard
deviation of the percentage cost increase sΔΚ. It should be noted that every value of ΔΚmax, ΔΚ and
sΔΚ presented in Table 3 and Figure 1 has been calculated taking into consideration all ΔKs, for
every N and AQL examined.
[Insert Table 3 about here]
[Insert Figure 1 about here]
The main conclusion is that both ΔΚmax and ΔΚ take their minimum values for a value of pr,
which seems to be close to p , namely 2.075% in the examined distributions. For any other value
of pr, both percentage cost burdens increase. In fact, for values of pr outside the range of p values,
the increase of burdens becomes progressively enormous. The specific conclusion holds for any
other distribution of p that has been examined.
Remark 4: The proper use of the following rules requires estimating as accurately as possible
ci, cr and cd, and calculating pr.
Rule 1: If pr p then trust the recommendations of ISO 2859 more than in any other
case, i.e., the economic loss of using the sampling plans of the ISO 2859 standard is minimum.
Regarding the sΔΚ values it can be noticed that for the majority of combinations presented in
Table 3, sΔΚ takes a value close to ΔΚ . This can be explained by the specific form of the frequency
distribution of ΔKs that have been considered for the determination of both statistical parameters: a
very large frequency of ΔKs (approaching 50% in some cases) appears in the class close to ΔKmax,
while another very large frequency (also approaching 50% in some cases) appears in the class
above 0%. It can be easily proved that when a random variable X usually takes two values, for
example a and b, then 2
bax and
22
2222
22
baxbxan
xbnxan
s
.
In our case α = 0 and consequently sx , i.e.. KΔsΔΚ .
The impact of an increase of the lot size N
The study of the impact of a potential increase of the lot size N (exclusively) on the percentage
cost burden ΔΚ when using the sampling plans of the standard instead of the optimum ones is
summarized in Table 4. An obvious remark regarding the lot size N is that most of the times the use
of ISO 2859 cannot be economically optimum, since it does not recommend a specific sampling
plan for any possible Ν value, but only for groups of Ν values; only 15 classes of N exist in ISO
2859.
[Insert Table 4 about here]
More specifically, when pr is much smaller than p (which means that the optimum sampling
plans are much tightened, i.e., n* N and c* 0), the increase of N (for a given AQL) leads to an
increase of the percentage economic loss of using the sampling plans recommended by the standard
(Figure 2 illustrates this behavior). Note that whenever the increase of N is combined with a change
of the inspection plan recommended by the standard, an instant reduction of the percentage
economic loss is noticed. In Table 5 an indicative series of economically optimum inspection plans
is matched against the respective series of plans recommended by ISO 2859 in order to explain this
behavior3. As long as the increase of N is not accompanied with a change of the sampling plan
recommended by the standard, the inspection recommended by the standard becomes progressively
3
p
aa dp)p()p(PP is the average probability of accepting a lot when the fraction nonconforming p is
distributed according to φ(p).
reduced compared to the optimum one and thus the percentage loss increases. When the ISO 2859
sampling plan changes, it instantly tends to the optimum one and, thus, peaks of improvement
appear periodically.
[Insert Figure 2 about here]
[Insert Table 5 about here]
On the other hand, when pr is much larger than p (and consequently the optimum inspections
are reduced, i.e., n* 0 and c* n*), the increase of N (for a given AQL) leads to a reduction of
the percentage economic burden of preferring the sampling plans of ISO 2859 instead of the
optimum ones (Figure 3 illustrates this tendency). Note again, that whenever the increase of N
leads to a change of the inspection plan recommended by the standard, an instant increase of the
percentage economic loss takes place. In order to explain this behavior we present in Table 6 -
similarly to Table 5 - a series of economically optimum inspection plans and the respective series
of plans recommended by ISO 2859. As long as the increase of N does not change the sampling
plan recommended by the standard, the concept of the inspection comes closer to the optimum one,
i.e., “reduced inspection or no inspection at all”, causing a reduction in the percentage loss. When
the plan recommended by ISO 2859 changes, there is a divergence from the optimum plan and,
thus, peaks of deterioration appear from time to time.
[Insert Figure 3 about here]
[Insert Table 6 about here]
When pr is close to p , the increase of N (for a given AQL) leads to a mixture of the two
behaviors described above. In particular, for small values of N the increase of the lot size leads to a
reduction of the percentage economic loss (Figure 4, region A), while for large values of N the
increase of the lot size leads to an increase of the percentage loss (Figure 4, region B).
[Insert Figure 4 about here]
Rule 2: Assuming that the choice of the AQL value cannot be changed, trust more the
recommendations of ISO 2859 when pr << p , for small values of N, and when pr >> p , for
large values of N.
The impact of an increase of the Acceptance quality limit AQL
The study of the impact of increasing exclusively AQL on the economic loss ΔΚ when using
the sampling plans recommended by ISO 2859 instead of the optimum plans is summarized in
Table 7. Specifically, when pr is much smaller than p , increasing AQL (for a given lot size N)
results in an increase of the percentage loss from using the standard (Figure 5). According to the
principles of ISO 2859, increasing AQL leads more and more to reduced inspections (either by
keeping n constant and increasing c, or by keeping c constant and decreasing n) and consequently
to larger deviations from the tightened optimum sampling plans. The only exception of this pattern
appears when the increase of AQL changes c from 0 to 1 (encircled points in Figure 5). In these
cases the simultaneous increase of n usually causes the reduction of aP .
[Insert Table 7 about here]
[Insert Figure 5 about here]
On the other hand, when pr is much larger than p , increasing AQL (for a given lot size N)
leads to a reduction of the percentage loss when using the sampling plans recommended by the
standard (Figure 6). In this case, it is desirable to use reduced sampling plans, through the increase
of AQL. The only exception to this behavior appears again when the increase of AQL leads to an
increase of c from 0 to 1 (encircled points in Figure 6).
[Insert Figure 6 about here]
When pr is close to p , the percentage loss is minimum for intermediate values of AQL.
It should be noted that that the aforementioned impact of AQL on the economic consequences
of preferring the ISO 2859 sampling plans stands for any pdf examined in this research. From the
numerical investigation performed, it becomes obvious that there is a correlation between pr and
the value of AQL that permits the selection of sampling plans (through the ISO 2859 standard) with
satisfactory economic results. In all cases, regardless of the size N of a lot, as pr increases, the
“optimum” AQL increases too. Rule 3 gives some simple and general directions for the selection of
AQL in tabular form (Table 8).
[Insert Table 8 about here]
Additional comments
Studying the impact of changing simultaneously N and AQL on the economic implications of
using the ISO 2859 sampling plans, we have noticed that especially for small values of N, very
small or very large values of AQL are more likely to lead to good economic results (depending on
pr) whereas for large values of N, intermediate values of AQL tend to give sampling plans with
better economic results.
Finally, the impact of a potential differentiation of each cost element on the percentage cost
increase ΔΚ becomes evident, if one keeps in mind that pr increases when ci or cr increases, or cd
decreases and vice-versa.
6. Conclusions - Future research
Going over the results of the numerical investigation presented previously, the main
conclusions are summarized below:
The use of the sampling plans recommended by the ISO 2859 standard instead of the
economically optimum ones, in order to conduct Acceptance Sampling, results in
significant increase of the quality cost met by companies.
In order to determine the economically optimum sampling plans it is important to be able
or, at least, try to estimate the values of all parameters of the Acceptance Sampling
procedure, but mainly to estimate as accurately as possible all cost elements, namely ci, cr
and cd. If this is not possible, then the so called economically optimum sampling plans will
lead to a sub-optimum average total quality cost and, consequently, not to the maximum
possible benefit.
Even in cases where the accurate estimation of the cost elements is not possible, the general
feeling about the economic issues of Acceptance Sampling can lead to the choice of
sampling plans (even through the ISO 2859 standard) that will produce near-optimum
economic results. For instance, it is preferable to choose a tightened inspection plan every
time the cost of rejecting a lot is low and cd is relatively high, while it is better to choose a
reduced sampling plan every time cr cd.
A very important finding that needs to be understood by the practitioners is that although
the sampling process is advisable most of the times, it is not always optimum. For example
it may be optimum not to inspect at all when the quality of the acquired lots is very good
and/or the ci is very high. Similarly, it may be optimum to inspect 100% when ci or cr are
very small, p is high etc. The use of the ISO standard does not allow the practitioner to
adopt either the 100% inspection or the “no inspection” policy, which are very often
economically optimum.
The application of the three simple rules presented in this paper can lead to a better use of
the ISO 2859 standard.
Bear in mind that the rules for the proper use of the ISO standard, presented in this paper, hold
when the inspection level is the general inspection level II. For other inspection levels (which,
however, are not so popular in practice) similar numerical investigations should be conducted in
order to derive accurate conclusions.
The deeper understanding of the correlation between AQL and pr will contribute to the
improvement of the process of finding (through the ISO 2859 standard) sampling plans that lead to
economically acceptable results. Consequently, the extensive study of this relation constitutes an
interesting topic of future research. Generally, an even more thorough numerical investigation can
reveal information that could help to develop more effective rules of using the ISO 2859 standard.
For example, future research can be directed to the examination of more pdfs of the fraction
nonconforming p, different inspection levels, more combinations of cost elements and/or probably
to the sensitivity analysis of the economic results.
Another interesting area of research would be the study of the complete operation of the ISO
2859 standard, i.e., taking into consideration the procedure of switching to tightened or reduced
inspection and the skip-lot sampling procedures. Moreover, the analysis of this paper has been
based on the assumption that all lots are submitted to rectifying inspection upon rejection. If this is
not the case, then the cost functions should be modified accordingly, offering an appealing area for
fruitful research.
Acknowledgment
The authors would like to thank Mr. Dimitris Hatzinikolaou for his valuable contribution to
this paper. The authors are also grateful to the two reviewers for their valuable remarks that
contributed significantly to the improvement of the paper.
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Table 1: Combinations of cost elements that were examined in the numerical investigation
Combination Cost elements pr (%)
1 ci = 1, cr = 100, cd = 1100 0.10
2 ci = 1, cr = 50, cd = 450 0.25
3 ci = 1, cr = 50, cd = 300 0.40
4 ci = 1, cr = 100, cd = 243 0.70
5 ci = 1, cr = 100, cd = 180 1.25
6 ci = 1, cr = 80, cd = 135.5 1.80
7 ci = 1, cr = 80, cd = 122.5 2.35
8 ci = 1, cr = 80, cd = 114.5 2.90
9 ci = 1, cr = 50, cd = 79 3.45
10 ci = 1, cr = 50, cd = 70 5.00
11 ci = 1, cr = 10, cd = 20 10.00
Table 2: Illustrative results for combination 7 of cost elements, selected values of AQL and p ~
U(0.0015, 0.04)
AQL Optimum values 0.40% 1.50% 6.50%
Ν n* c* K(n*,c*) n c K(n,c) ΔΚ (%) n c K(n,c) ΔΚ (%) n c K(n,c) ΔΚ (%)
3 0 0 7.63 Ν 0 7,98 4.65 Ν 0 7,98 4.65 2 0 7.86 4.65
5 0 0 12.71 Ν 0 13,30 4.65 Ν 0 13,30 4.65 2 0 12.93 4.65
7 0 0 17.79 Ν 0 18,62 4.65 Ν 0 18,62 4.65 2 0 18.00 4.65
10 0 0 25.42 Ν 0 26,60 4.65 8 0 26,33 4.65 2 0 25.61 3.57
12 0 0 30.50 Ν 0 31,92 4.65 8 0 31,37 4.65 2 0 30.68 2.86
14 0 0 35.59 Ν 0 37,24 4.65 8 0 36,42 4.65 2 0 35.76 2.34
17 0 0 43.21 Ν 0 45,22 4.65 8 0 43,99 4.65 8 1 44.12 1.80
20 0 0 50.84 Ν 0 53,20 4.65 8 0 51,56 4.65 8 1 51.74 1.42
24 0 0 61.01 Ν 0 63,84 4.65 8 0 61,65 4.65 8 1 61.89 1.07
27 0 0 68.63 Ν 0 71,82 4.65 8 0 69,23 4.65 8 1 69.51 0.87
38 0 0 96.59 32 0 100,15 4.65 8 0 96,98 3.68 8 1 97.43 0.41
49 2 0 124.53 32 0 127,70 4.66 8 0 124,74 2.55 8 1 125.35 0.17
55 3 0 139.74 32 0 142,73 4.69 8 0 139,88 2.14 13 2 141.28 0.10
70 6 0 177.70 32 0 180,30 4.78 8 0 177,73 1.46 13 2 179.39 0.02
85 8 0 215.58 32 0 217,87 4.88 8 0 215,58 1.06 13 2 217.50 0.00
100 10 0 253.41 32 0 255,45 4.97 32 1 255,60 0.81 20 3 256.50 0.87
120 12 0 303.77 32 0 305,54 5.08 32 1 305,75 0.58 20 3 307.32 0.65
140 14 0 354.08 32 0 355,64 5.17 32 1 355,89 0.44 20 3 358.14 0.51
160 16 0 404.35 32 0 405,74 5.26 32 1 406,03 0.34 32 5 410.46 0.42
215 19 0 542.45 32 0 543,50 5.24 32 1 543,92 0.19 32 5 550.26 0.27
270 52 1 680.13 32 0 681,26 4.88 32 1 681,81 0.17 32 5 690.06 0.25
300 55 1 754.65 32 0 756,41 4.82 50 2 760,28 0.23 50 7 768.46 0.75
390 61 1 977.93 32 0 981,84 4.73 50 2 986,10 0.40 50 7 997.22 0.84
480 65 1 1200.95 32 0 1207,27 4.70 50 2 1211,92 0.53 50 7 1225.98 0.91
550 68 1 1374.31 125 1 1387,72 4.69 80 3 1388,13 0.98 80 10 1407.47 1.01
850 114 2 2112.83 125 1 2132,59 4.88 80 3 2138,34 0.94 80 10 2170.03 1.21
1150 160 3 2848.51 125 1 2877,45 5.07 80 3 2888,55 1.02 80 10 2932.59 1.41
1400 164 3 3460.25 125 1 3498,17 5.21 125 5 3524,19 1.10 125 14 3573.38 1.85
2200 254 5 5410.19 125 1 5484,47 5.60 125 5 5526,82 1.37 125 14 5606.88 2.16
3000 301 6 7354.29 125 1 7470,78 5.86 125 5 7529,45 1.58 125 14 7640.37 2.38
3500 345 7 8567.16 200 2 8687,35 6.00 200 7 8733,36 1.40 200 21 8920.19 1.94
6600 521 11 16067.11 200 2 16348,43 6.50 200 7 16437,67 1.75 200 21 16800.00 2.31
9700 651 14 23547.89 200 2 24009,52 6.77 200 7 24141,99 1.96 200 21 24679.81 2.52
12500 738 16 30295.77 315 3 31044,73 6.93 315 10 30877,06 2.47 200 21 31797.05 1.92
22500 996 22 54359.44 315 3 55834,91 7.25 315 10 55529,64 2.71 200 21 57215.79 2.15
32500 1209 27 78392.95 315 3 80625,09 7.42 315 10 80182,20 2.85 200 21 82634.53 2.28
40000 1377 31 96406.91 500 5 99243,52 9.27 500 14 97561,35 2.94 200 21 101698.60 1.20
92500 2101 48 222378.40 500 5 229381,70 9.53 500 14 225463,80 3.15 200 21 235147.00 1.39
145000 2609 60 348248.30 500 5 359519,90 9.63 500 14 353366,20 3.24 200 21 368595.40 1.47
160000 2740 63 384201.20 800 7 400154,20 9.65 800 21 387015,80 4.15 200 21 406723.5 0.73
Table 3: ΔΚmax, ΔΚ and sΔΚ depending on pr, for p ~ U(0.0015, 0.04) and p ~ TN(0.0015, 0.04)
U(0.0015, 0.04) TN(0.0015, 0.04) Combination
of cost
elements
pr (%) ΔΚmax (%) ΔΚ (%) sΔΚ (%) ΔΚmax (%) ΔΚ (%) sΔΚ (%)
1 0.10 641.47 229.98 222.78 636.57 234.67 220.57
2 0.25 358.68 126.87 124.61 355.27 130.08 123.37
3 0.40 207.96 72.07 71.87 204.46 74.11 71.16
4 0.70 66.71 22.09 22.71 64.43 22.72 22.10
5 1.25 26.34 7.64 8.24 23.67 7.49 7.64
6 1.80 13.77 3.30 3.61 10.67 2.52 2.71
7 2.35 9.65 2.95 2.12 7.07 2.53 1.78
8 2.90 12.89 5.92 4.31 11.96 6.43 4.26
9 3.45 24.30 12.59 8.56 24.30 13.73 8.48
10 5.00 40.28 22.43 13.87 40.28 23.64 13.90
11 10.00 190.96 110.49 65.41 190.96 114.39 65.55
Table 4: The impact of a potential increase of N on ΔΚ
(n*,c*) The increase of N does not
lead to a change of (n,c)
The increase of N leads to a
change of (n,c)
Relevant
figure
pr << p Tightened ΔΚ ΔΚ (instantly) 2
pr >> p Reduced ΔΚ ΔΚ (instantly) 3
Small values of N (first
classes of ISO 2859)
Large values of N (last
classes of ISO 2859)
pr p Neither
tightened nor
reduced
ΔΚ ΔΚ 4
Table 5: Economically optimum inspection plans (n*,c*) for %1.0rp and respective plans
recommended by ISO 2859 (n,c) for AQL = 0.4% - p ~ U(0.0015, 0.04)
Ν n* c* aP (%) n c aP (%)
300 300 0 5.49 32 0 54.44
390 390 0 3.69 32 0 54.44
480 480 0 2.62 32 0 54.44
550 550 0 2.06 125 1 36.48
850 850 0 0.85 125 1 36.48
1150 1150 0 0.40 125 1 36.48
1400 1400 0 0.23 125 1 36.48
2200 2200 0 0.04 125 1 36.48
3000 2991 0 0.01 125 1 36.48
3500 3467 0 0.00 200 2 34.68
6600 5407 0 0.00 200 2 34.68
9700 6094 0 0.00 200 2 34.68
Table 6: Economically optimum inspection plans (n*,c*) for %5rp and respective plans
recommended by ISO 2859 (n,c) for AQL = 0.4% - p ~ U(0.0015, 0.04)
Ν n* c* aP (%) n c aP (%)
300 0 0 100.00 32 0 54.44
390 0 0 100.00 32 0 54.44
480 0 0 100.00 32 0 54.44
550 0 0 100.00 125 1 36.48
850 0 0 100.00 125 1 36.48
1150 0 0 100.00 125 1 36.48
1400 0 0 100.00 125 1 36.48
2200 0 0 100.00 125 1 36.48
3000 0 0 100.00 125 1 36.48
3500 0 0 100.00 200 2 34.68
6600 0 0 100.00 200 2 34.68
9700 0 0 100.00 200 2 34.68
Table 7: The impact of a potential increase of AQL on ΔΚ
(n*,c*) Increase of AQL The increase of AQL leads to a
change of c from 0 to 1
Relevant
figure
pr << p Tightened ΔΚ ΔΚ (instantly) 5
pr >> p Reduced ΔΚ ΔΚ (instantly) 6
Small values of AQL Large values of AQL
pr p Neither
tightened nor
reduced
ΔΚ ΔΚ -
Table 8: Rule 3
pr < pmin Don’t use the standard! Just reject the lot.
Alternatively choose the minimum value of AQL
pmin < pr < p Small values of N: Choose the
minimum value of AQL
Large values of N:
Choose AQL pr
pr p Choose AQL pr
p < pr < pmax No simple rule
App
roxi
mat
e ar
eas o
f pr
pmax < pr Don’t use the standard! Just accept the lot.
Alternatively choose the maximum value of AQL
Figure 1: ΔΚmax and ΔΚ depending on pr, for p ~ U(0.0015, 0.04) and p ~ TN(0.0015, 0.04)
0%
100%
200%
300%
400%
500%
600%
700%
0% 2% 4% 6% 8% 10%
Perc
enta
ge c
ost i
ncre
ase
ΔΚ
Maximum
Average
p r
p ~ U(0.0015, 0.04) andp ~ TN(0.0015, 0.04)
Figure 2: ΔΚ depending on N, for %1.0rp and AQL = 4%
0%
100%
200%
300%
400%
500%
600%
700%
800%3 10 17 27 55 100
160
300
550
1400
3500
1250
0
4000
0
1600
00
Lot size Ν
ΔΚ
p ~ U(0.0015, 0.04) & p ~ TN(0.0015, 0.04)
p ~ U(0.01, 0.07)
Figure 3: ΔΚ depending on N, for %5rp and AQL = 6.5%
0%
50%
100%
150%
200%
250%3 10 17 27 55 100
160
300
550
1400
3500
1250
0
4000
0
1600
00
Lot size Ν
ΔΚ-p
~ U
(0.0
005,
0.0
05)
0%
5%
10%
15%
20%
25%
30%
35%
ΔΚ-p
~ U
(0.0
015,
0.0
4) &
p ~
TN(0
.001
5, 0
.04)
p ~ U(0.0015, 0.04) & p ~ TN(0.0015, 0.04)
p ~ U(0.0005, 0.005)
Figure 4: ΔΚ depending on N, for %35.2rp , p ~ U(0.0015, 0.04) and various values of AQL
0,00%
1,00%
2,00%
3,00%
4,00%
5,00%
6,00%
3 10 17 27 55 100
160
300
550
1400
3500
1250
0
4000
0
2E+0
5
Lot size Ν
ΔΚ
BAAQL = 0.4%
AQL = 6.5%
Figure 5: ΔΚ depending on AQL, for selected values of N, %1.0rp and p ~ U(0.0015, 0.04)
0%
100%
200%
300%
400%
500%
600%
700%
0.06
5%
0.1%
0.15
%
0.25
%
0.4%
0.65
% 1%
1.5%
2.5% 4%
6.5%
AQL
ΔΚ
N = 3
N = 160,000
Figure 6: ΔΚ depending on AQL, for selected values of N, %5rp and p ~ U(0.0015, 0.04)
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
0.06
5%
0.1%
0.15
%
0.25
%
0.4%
0.65
% 1%
1.5%
2.5% 4%
6.5%
AQL
ΔΚ
N = 3
N = 160,000