Economic Evaluation of ISO 2859 Acceptance Sampling …users.uom.gr/~nikolai/5-Nikolaidis,...

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.


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 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.


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.



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.



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).


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 .



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).


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).


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.

References

1. Bai, D.S., Riew, M.C. (1984). An Economic Attributes Acceptance Sampling Plan with Three

Decision Criteria, Journal of Quality Technology, 16(3):136-143.

2. Balamurali, S., Kalyanasundaram, M. (1997). Determination of an Attribute Single Sampling

Scheme, Journal of Applied Statistics, 24(6):689-696.

3. Brown, G.G., Rutemiller, H.C. (1973). A cost analysis of sampling inspection under Military

Standard 105D, Naval Research Logistics Quarterly, 20(1):181-199.

4. Cassady, R., Nachlas, J.A. (2003). Evaluating and Implementing 3-Level Acceptance

Sampling Plans, Quality Engineering, 15(3):361-369.

5. Chen, C.-H. (2006). The Modified Pulak and Al-Sultan's Model for Determining the Optimum

Process Parameters, Communications in Statistics - Theory and Methods, 35(10):1767-1778.

6. Chen, J.W., Chou, W., Wu, H., Zhou, H. (2004a). Designing Acceptance Sampling Schemes

for Life Testing with Mixed Censoring, Naval Research Logistics, 51(4):597-612.

7. Chen, J.W., Choy, S.T.B., Li K.H. (2004b). Optimal Bayesian Sampling Acceptance Plan with

Random Censoring, European Journal of Operational Research, 155(3):683-694.

8. ELOT 398.0 (1982). Sampling Procedures and Tables for Inspection by Attributes, Hellenic

Organization for Standardization.

9. ELOT 398.1 (1982). Sampling Procedures and Tables for Inspection by Attributes -

Addendum 1, Hellenic Organization for Standardization.

10. Ferrell, W.G., Chhoker, A. (2002). Design of Economically Optimal Acceptance Sampling

Plans with Inspection Error, Computers & Operations Research, 29(10):1283-1300.

11. Gao, Y., Tang, L.C. (2006). Chain Sampling Scheme under Constant Inspection Errors,

Quality And Reliability Engineering International, 22(8):889-903.

12. González, C., Palomo, G. (2003). Bayesian Acceptance Sampling Plans Following Economic

Criteria: An Application To Paper Pulp Manufacturing, Journal of Applied Statistics,

30(3):319-333.

13. Hald, A. (1981). Statistical Theory of Sampling by Attributes; Academic Press, London.

14. ISO 2859 (1974). Sampling Procedures and Tables for Inspection by Attributes, International


15. ISO 2859-1 (1989). Sampling Procedures for Inspection by Attributes - Part 1: Sampling

Plans Indexed by Acceptable Quality Level (AQL) for Lot-by-lot Inspection, International



Schemes Indexed by Acceptance Quality Limit (AQL) for Lot-by-lot Inspection, International



Plans Indexed by Limiting Quality (QL) for Isolated Lot Inspection, International Organization

for Standardization.

18. ISO 2859-3 (2005). Sampling Procedures for Inspection by Attributes - Part 3: Skip-Lot

Sampling Procedures, International Organization for Standardization.

19. ISO 2859-4 (2002). Sampling Procedures for Inspection by Attributes - Part 4: Procedures for

Assessment of Declared Quality Levels, International Organization for Standardization.

20. ISO 2859-5 (2005). Sampling Procedures for Inspection by Attributes - Part 5: System of

Sequential Sampling Plans Indexed by Acceptance Quality Limit (AQL) for Lot-by-lot

Inspection, International Organization for Standardization.

21. ISO 2859-10 (2006). Sampling Procedures for Inspection by Attributes - Part 10: Introduction

to the ISO 2859 series of standards for sampling for inspection by attributes, International


22. ISO/TR 8550-1 (2007). Guidance on the selection and usage of acceptance sampling systems

for inspection of discrete items in lots - Part 1: Acceptance sampling, International


23. ISO/TR 8550-2 (2007). Guidance on the selection and usage of acceptance sampling systems

for inspection of discrete items in lots - Part 2: Sampling by attributes, International


24. Montgomery, D.C. (2001). Introduction to Statistical Quality Control; John Wiley, New York.

25. Nikolaidis, Y., Nenes, G. (2005). Economic Evaluation of Sampling Schemes that are

Proposed by Acceptance Sampling Standards, In Standardization in an Enlarged Europe,

Proceedings of the 10th EURAS Workshop, Tallinn, Estonia, June 2-3, Coenen, H., Gröhndal,

J., Jakobs, K., Valdlo, T., Eds.; 111-116.

26. Selinger, K.A. (1995). Inspection by variables as an acceptance criterion in bioanalysis – a

proposal, Journal of Pharmaceutical and Biomedical Analysis, 13(12):1427-1436.

27. Stout, Q.F., Hardwick, J. (2005). Optimal Screening Designs With Flexible Cost And

Constraint Structures, Journal of Statistical Planning and Inference, 132(1-2):149-162.

28. Tagaras, G. (2001). Statistical Quality Control (in Greek); Zitis, Thessaloniki.

29. Tagaras, G., Lee, H.L. (1987). Optimal Bayesian Single-Sampling Attribute Plans with

Modified Beta Prior Distribution, Naval Research Logistics, 34(6):789-801.

30. Taylor, W.A. Acceptance sampling update, Appeared in MDDI (Medical Device and

Diagnostic Industry), October 1995, Canon Communications, 17(10):92-108.

31. Wall, M.S., Elshennawy, A.K. (1989). Economically-based Acceptance Sampling

Plans, Computers & Industrial Engineering, 17(1-4):340-346.

32. Wetherrill, G.B., Chiu, W.K. (1975). A Review of Acceptance Sampling Schemes with

Emphasis on the Economic Aspect, International Statistical Review, 43(2):91-210.

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

Economic Evaluation of ISO 2859 Acceptance Sampling …users.uom.gr/~nikolai/5-Nikolaidis,...

Documents

Transcript of Economic Evaluation of ISO 2859 Acceptance Sampling …users.uom.gr/~nikolai/5-Nikolaidis,...