SIST ISO 8423:2010
Sequential sampling plans for inspection by variables for percent nonconforming (known standard deviation)
Sequential sampling plans for inspection by variables for percent nonconforming (known standard deviation)
This International Standard specifies sequential sampling plans and procedures for inspection by variables of discrete items. The plans are indexed in terms of producer's risk point and the consumer's risk point. Therefore, they are suitable not only for the purposes of acceptance sampling, but for the more general purpose of the testing of simple statistical hypotheses for proportions. The purpose of this International Standard is to provide procedures for the sequential assessment of inspection results that may be used to induce the supplier to supply lots of a quality having a high probability of acceptance. At the same time, the consumer is protected by a prescribed upper limit to the probability of accepting a lot (or process) of poor quality. This International Standard is primarily designed for use under the following conditions: a) where the inspection procedure is to be applied to a continuing series of lots of discrete products all supplied by one producer using one production process. In such a case, sampling of particular lots is equivalent to the sampling of the process. If there are different producers or production processes, this International Standard shall be applied to each one separately; b) where only a single quality characteristic x of these products is taken into consideration, which must be measurable on a continuous scale; c) where the measurement error is negligible (i.e. with a standard deviation no more than 10 % of the process standard deviation); d) where production is stable (under statistical control) and the quality characteristic x has a known standard deviation, and is distributed according to a normal distribution or a close approximation to the normal distribution; e) where a contract or standard defines an upper specification limit U, a lower specification limit L, or both; an item is qualified as conforming if and only if its measured quality characteristic, x, satisfies the appropriate one of the following inequalities: 1) x L (i.e. the lower specification limit is not violated); 3) x u U and x < L (i.e. neither the upper nor the lower specification limit is violated). Inequalities 1) and 2) are called cases with a "single specification limit", and 3) is the case with "double specification limits". In this International Standard, it is assumed that, where double specification limits apply, conformance to both specification limits is either equally important to the integrity of the product or is considered separately for both specification limits. In the first case, it is appropriate to control the combined percentage of product outside the two specification limits. This is referred to as combined control. In the second case, nonconformity beyond each of the limits is controlled separately, and this is referred to as separate control.
Plans d'échantillonnage progressif pour le contrôle par mesures des pourcentages de non-conformes (écart-type connu)
L'ISO 8423:2008 sp�cifie des plans et des r�gles d'�chantillonnage progressif pour le contr�le par mesures d'individus discrets. Les plans sont index�s en termes de point du risque fournisseur et de point du risque client. Par cons�quent, ils sont appropri�s non seulement aux fins d'�chantillonnage pour acceptation, mais �galement aux v�rifications d'ordre plus g�n�ral d'hypoth�ses statistiques simples de proportions. Le but de l'ISO 8423:2008 est de fournir des r�gles fond�es sur la d�termination progressive des r�sultats de contr�le, afin d'inciter le fournisseur � fournir des lots de qualit� ayant une forte probabilit� d'acceptation. En m�me temps, le client est prot�g� par une limite sup�rieure prescrite de la probabilit� d'accepter des lots (ou proc�d�s) de faible qualit�.
L'ISO 8423:2008 est principalement con�ue pour �tre utilis�e lorsque les conditions suivantes sont satisfaites: lorsque la r�gle de contr�le est destin�e � �tre appliqu�e � une s�rie continue de lots constitu�s d'individus discrets, tous fournis par un seul fournisseur utilisant un seul proc�d� de fabrication; lorsqu'un unique caract�re de qualit� de ces individus, qui doit �tre mesurable sur une �chelle continue, est pris en consid�ration; lorsque l'erreur de mesure est n�gligeable (c'est-�-dire avec un �cart-type non sup�rieur � 10 % de l'�cart-type du processus); lorsque la fabrication est stable (sous ma�trise statistique) et le caract�re de qualit� a un �cart-type connu, et est distribu� suivant une loi normale ou voisine d'une loi normale; lorsqu'un contrat ou une norme d�finit une limite de sp�cification sup�rieure, une limite de sp�cification inf�rieure, ou les deux. Dans le cas de limites de sp�cification doubles, il est suppos� que la conformit� aux deux limites de sp�cification s'applique conjointement � l'int�grit� du produit ou est consid�r�e s�par�ment pour les deux limites de sp�cification.
Les proc�dures de l'ISO 8423:2008 ne s'appliquent pas aux lots ayant pr�alablement fait l'objet d'une s�lection d'individus non conformes.
Sekvenčni načrti vzorčenja za kontrolo po številskih spremenljivkah za odstotkovno neskladje (znan standardni odklon)
Ta mednarodni standard opredeljuje sekvenčne načrte vzorčenja in postopke za kontrolo po številskih spremenljivkah diskretnih primerkov. Načrti so razvrščeni glede na točko tveganja proizvajalca in točko tveganja potrošnika. Zato niso primerni zgolj za namene prevzemnega vzorčenja, ampak tudi za bolj splošen namen preskušanja preprostih statističnih hipotez za proporce. Namen tega mednarodnega standarda je zagotoviti postopke za sekvenčno oceno rezultatov kontrole, ki se lahko uporabijo tako, da vzpodbudijo dobavitelja k dobavi partij takšne kakovosti, ki ima visoko verjetnost prevzema. Istočasno je potrošnik varovan s predpisano zgornjo mejo verjetnosti za prevzem partije (ali postopka) slabe kakovosti. Ta mednarodni standard je bil prvotno zasnovan za uporabo pod naslednjimi pogoji: a) kjer je potrebno kontrolni postopek uporabiti za neprekinjene serije partij diskretnih produktov, ki jih vse dobavlja en proizvajalec, z uporabo enega proizvodnega procesa. V takšnem primeru je vzorčenje določenih partij enakovredno vzorčenju procesa. Če obstajajo različni proizvajalci ali proizvodni procesi, se ta mednarodni standard uporablja za vsakega posebej; b) kjer se upošteva le posamična karakteristika x teh produktov, ki mora biti izmerljiva na nepretrgani skali; c) kjer je napaka pri merjenju zanemarljiva (tj. pri standardni deviaciji, ki ni večja od 10 odstotkov standardne deviacije procesa); kjer je proizvodnja stabilna (pod statistično kontrolo) in ima karakteristika x kvalitete poznano standardno deviacijo ter je razdeljena v skladu z normalno distribucija ali natančnim približkom normalni distribuciji; e) kjer pogodba ali standard opredeljujeta zgornjo mejo specifikacije U, spodnjo mejo specifikacijo L, ali obe; primerek je kvalificiran, da sovpada, če in samo če njegova izmerjena karakteristika kvalitete x izpolnjuje primerno izmed naslednjih neenakosti: 1) x L (tj. spodnja meja specifikacije ni kršena); 3) x u U in x < L (tj. niti zgornja niti spodnja meja specifikacije ni kršena). Neenakosti 1) in 2) se imenujejo primeri brez »enotne meje specifikacije« in 3) je primer »dvojne meje specifikacije.« V tem mednarodnem standardu se predvideva, da kjer velja dvojna meja specifikacije, je skladnost z obema mejama specifikacije bodisi enako pomembna za integriteto produkta bodisi se upošteva ločeno za obe meji specifikacije. V prvem primeru je primerno kontrolirati skupni odstotni delež produkta zunaj teh dveh mej specifikacije. To se navaja kot skupna kontrola. V drugem primeru se neskladnost prek vsake od teh meja kontrolira ločeno, kar se imenuje ločena kontrola.
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INTERNATIONAL ISO
STANDARD 8423
Second edition
2008-09-01
Sequential sampling plans for inspection
by variables for percent nonconforming
(known standard deviation)
Plans d'échantillonnage progressif pour le contrôle par mesures des
pourcentages de non-conformes (écart-type connu)
Reference number
ISO 8423:2008(E)
©
ISO 2008
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ISO 8423:2008(E)
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ISO 8423:2008(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 2
3 Terms and definitions. 2
4 Symbols . 5
5 Principles of sequential sampling plans for inspection by variables . 6
6 Selection of a sampling plan . 7
7 Operation of a sequential sampling plan . 7
8 Examples . 18
9 Tables. 23
Annex A (informative) Additional information. 28
Bibliography . 32
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ISO 8423:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 8423 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 5, Acceptance sampling.
This second edition cancels and replaces the first edition (ISO 8423:1991), which has been technically revised.
It also incorporates the Technical Corrigendum ISO 8423:1991/Cor.1:1993. Annex A of ISO 8423:1991 is
superseded by ISO 3951-5:2005.
The following improvements have been introduced:
⎯ values of the parameters h , h and g have been recalculated in order to provide plans that exactly meet
A R
stated requirements,
⎯ the average sample sizes for quality levels equal to producer’s risk quality and consumer’s risk quality
have been significantly decreased.
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ISO 8423:2008(E)
Introduction
In contemporary production processes, quality is often expected to reach such high levels that the number of
nonconforming items is reported in parts per million. Under such circumstances, popular acceptance sampling
plans by attributes, such as those presented in ISO 2859-1, require prohibitively large sample sizes. When it is
possible to apply acceptance sampling plans by variables, such as those presented in ISO 3951-1, the sample
sizes are much smaller. However, especially in the case of acceptance of a product of extremely high quality,
those sample sizes are still too large. Therefore, there is a need to apply standardized statistical procedures
that require the smallest possible sample sizes; sequential sampling plans are the only statistical procedures
that satisfy that need. It has been mathematically proved that among all possible sampling plans having
similar statistical properties the sequential sampling plan has the smallest average sample size.
The principal advantage of sequential sampling plans is the reduction in the average sample size. The
average sample size is the average of all the sample sizes that may occur under a sampling plan for a given
lot or process quality level. The use of sequential sampling plans leads to a smaller average sample size than
single sampling plans having the equivalent operating characteristic.
Other factors that should be taken into account are as follows:
a) Complexity
The rules of a sequential sampling plan are more easily misunderstood by inspectors than the simple
rules for a single sampling plan.
b) Variability in the amount of inspection
As the actual number of items inspected for a particular lot is not known in advance, the use of sequential
sampling plans brings about various organizational difficulties. For example, scheduling of inspection
operations may be difficult.
c) Difficulty of drawing sample items
If drawing sample items is rather difficult, the reduction in the average sample size by sequential sampling
plans may be cancelled out by the increased sampling cost.
d) Duration of test
If the test of a single item is of long duration and a number of items can be tested simultaneously,
sequential sampling plans are much more time-consuming than the corresponding single sampling plan.
e) Variability of quality within the lot
If the lot consists of two or more sublots from different sources and if there is likely to be any substantial
difference between the qualities of the sublots, drawing of a representative sample under a sequential
sampling plan is far more difficult than under the corresponding single sampling plan.
The balance between the advantage of a smaller average sample size of the sequential sampling plan and the
above disadvantages leads to the conclusion that sequential sampling plans are suitable only when inspection
of individual items is costly in comparison with inspection overheads.
The choice between single and sequential sampling plans should be made before the inspection of a lot is
started. During inspection of a lot, it is not permitted to switch from one type to another, because the operating
characteristic of the plan may be drastically changed if the actual inspection results influence the choice of
acceptability criteria.
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ISO 8423:2008(E)
Although a sequential sampling plan is on average much more economical than the corresponding single
sampling plan, it may occur, during inspection of a particular lot, that acceptance or non-acceptance comes at
a very late stage because the cumulative leeway (the statistic used for the determination of lot acceptability)
remains between the acceptance value and the rejection value for a long time. With the graphical method, this
corresponds to the random progress of the step-wise linear curve remaining in the indecision zone.
In order to alleviate this disadvantage, the curtailment values are set before the inspection of a lot (or a
process) is started, and inspection terminates if the cumulative sample size reaches the curtailment value, n ,
t
without determination of lot acceptability. The acceptance and non-acceptance of the lot (or the process) is
then determined using the curtailment acceptance and rejection values.
For sequential sampling plans in common use, curtailment usually represents a deviation from their intended
usage, leading to a distortion of their operating characteristics. In this International Standard, however, the
operating characteristics of the sequential sampling plans have been determined with curtailment taken into
account, so curtailment is an integral component of the provided plan.
Sequential sampling plans for inspection by variables are also provided in ISO 3951-5. However, the design
principle of those plans is fundamentally different from that of this International Standard. The sampling plans
in ISO 3951-5 are designed to supplement the ISO 3951-1 acceptance sampling system for inspection by
variables, which is a counterpart of the popular ISO 2859-1 acceptance sampling system for inspection by
attributes. Thus, they should be used for the inspection of a continuing series of lots, that is, a series long
enough to permit the switching rules of the ISO 3951 system to take effect. The application of the switching
rules is the only means of providing enhanced protection to the consumer (by means of tightened sampling
inspection criteria or discontinuation of sampling inspection) when the sequential sampling plans from
ISO 3951-5 are used. However, in certain circumstances, there is a strong need to have both producer’s and
consumer’s risks under strict control. Such circumstances occur, for example, when sampling is performed for
regulatory reasons, for the demonstration of quality of production processes or for hypothesis testing. In such
cases, individual sampling plans selected from the ISO 3951-5 sampling scheme may be inappropriate. The
sampling plans from this International Standard have been designed in order to meet these specific
requirements.
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INTERNATIONAL STANDARD ISO 8423:2008(E)
Sequential sampling plans for inspection by variables for
percent nonconforming (known standard deviation)
1 Scope
This International Standard specifies sequential sampling plans and procedures for inspection by variables of
discrete items.
The plans are indexed in terms of producer’s risk point and the consumer’s risk point. Therefore, they are
suitable not only for the purposes of acceptance sampling, but for the more general purpose of the testing of
simple statistical hypotheses for proportions.
The purpose of this International Standard is to provide procedures for the sequential assessment of
inspection results that may be used to induce the supplier to supply lots of a quality having a high probability
of acceptance. At the same time, the consumer is protected by a prescribed upper limit to the probability of
accepting a lot (or process) of poor quality.
This International Standard is primarily designed for use under the following conditions:
a) where the inspection procedure is to be applied to a continuing series of lots of discrete products all
supplied by one producer using one production process. In such a case, sampling of particular lots is
equivalent to the sampling of the process. If there are different producers or production processes, this
International Standard shall be applied to each one separately;
b) where only a single quality characteristic x of these products is taken into consideration, which must be
measurable on a continuous scale;
c) where the measurement error is negligible (i.e. with a standard deviation no more than 10 % of the
process standard deviation);
d) where production is stable (under statistical control) and the quality characteristic x has a known standard
deviation, and is distributed according to a normal distribution or a close approximation to the normal
distribution;
CAUTION — The procedures in this International Standard are not suitable for application to lots that
have been screened previously for nonconforming items.
e) where a contract or standard defines an upper specification limit U, a lower specification limit L, or both;
an item is qualified as conforming if and only if its measured quality characteristic, x, satisfies the
appropriate one of the following inequalities:
1) x u U (i.e. the upper specification limit is not violated);
2) x W L (i.e. the lower specification limit is not violated);
3) x u U and x W L (i.e. neither the upper nor the lower specification limit is violated).
Inequalities 1) and 2) are called cases with a “single specification limit”, and 3) is the case with “double
specification limits”.
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ISO 8423:2008(E)
In this International Standard, it is assumed that, where double specification limits apply, conformance to both
specification limits is either equally important to the integrity of the product or is considered separately for both
specification limits. In the first case, it is appropriate to control the combined percentage of product outside the
two specification limits. This is referred to as combined control. In the second case, nonconformity beyond
each of the limits is controlled separately, and this is referred to as separate control.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 3534-1:2006, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 3534-2:2006, Statistics — Vocabulary and symbols — Part 2: Applied statistics
ISO 3951-1:2005, Sampling procedures for inspection by variables — Part 1: Specification for single sampling
plans indexed by acceptance quality limit (AQL) for lot-by-lot inspection for a single quality characteristic and a
single AQL
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 3534-2 and ISO 3951-1
and the following apply.
3.1
inspection by variables
inspection by measuring the magnitude(s) of the characteristic(s) of an item
[ISO 3534-2:2006, definition 4.1.4]
3.2
sampling inspection
inspection of selected items in the group under consideration
[ISO 3534-2:2006, definition 4.1.6]
3.3
acceptance sampling
sampling after which decisions are made to accept or not to accept a lot, or other grouping of products,
materials or services, based on sample results
[ISO 3534-2:2006, definition 1.3.17]
3.4
acceptance sampling inspection
acceptance inspection where the acceptability is determined by means of sampling inspection
[ISO 3534-2:2006, definition 4.1.8]
3.5
acceptance sampling inspection by variables
acceptance sampling inspection in which the acceptability of a process is determined statistically from
measurements on specified quality characteristics of each item in a sample from a lot
[ISO 3534-2:2006, definition 4.2.11]
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ISO 8423:2008(E)
3.6
quality level
quality expressed as a rate of occurrence of nonconforming units
3.7
nonconformity
non-fulfillment of a requirement
[ISO 9000:2005, definition 3.6.2, and ISO 3534-2:2006, definition 3.1.11]
3.8
nonconforming unit
unit with one or more nonconformities
[ISO 3534-2:2006, definition 1.2.15]
3.9
specification limit
limiting value stated for a characteristic
[ISO 3534-2:2006, definition 3.1.3]
3.10
lower specification limit
L
specification limit that defines the lower limiting value
[ISO 3534-2:2006, definition 3.1.5]
3.11
upper specification limit
U
specification limit that defines the upper limiting value
[ISO 3534-2:2006, definition 3.1.4]
3.12
combined control
requirement when both upper and lower limits are specified for the quality characteristic and specified risks
apply to the combined percent nonconforming beyond the two limits
NOTE The use of combined control implies that nonconformities beyond either specification limit are believed to be of
equal, or at least roughly equal, importance to the lack of integrity of the product.
3.13
separate control
requirement when both upper and lower limits are specified for the quality characteristic and separate risks
are given which apply to each limit
NOTE The use of separate control implies that nonconformities beyond either specification limit are believed to be of
different importance to the lack of integrity of the product.
3.14
maximum process standard deviation
σ
max
largest process standard deviation for a given sampling plan for which it is possible to satisfy the acceptance
criteria for a combined double specification limit when the process variability is known
NOTE 1 Maximum process standard deviation σ was denoted by its acronym MPSD in older standards.
max
NOTE 2 This definition is different from the similar definition given in ISO 3534-2 in which the concept of AQL is used.
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ISO 8423:2008(E)
3.15
measurement
set of operations having the object of determining a value of a quantity
[ISO 3534-2:2006, definition 3.2.1]
3.16
leeway
quantity derived from a measured value of an item
NOTE In the case of a single lower specification limit and in the case of double specification limits, the leeway is
obtained by subtracting the numerical value of the lower specification limit from the measured value. In the case of an
upper specification limit, the leeway is obtained by subtracting the measured value from the numerical value of the upper
specification limit.
3.17
cumulative leeway
value calculated by summing the leeways obtained from the start of the inspection up to, and including, that of
the item last inspected
3.18
cumulative sample size
total number of inspected items, counting from the start of the inspection up to, and including, the item last
inspected
3.19
acceptance value for sequential sampling
value derived from the specified parameters of the sampling plan and the cumulative sample size
NOTE Whether the lot may yet be accepted is determined by comparing the cumulative leeway with the acceptance
value.
3.20
rejection value for sequential sampling
value derived from the specified parameters of the sampling plan and the cumulative sample size
NOTE Whether the lot may yet be considered unacceptable is determined by comparing the cumulative leeway with
the rejection value.
3.21
consumer's risk quality
CRQ
Q
CR
〈acceptance sampling〉 quality level of a lot or process which, in the acceptance sampling plan, corresponds to
a specified consumer's risk
NOTE The specified consumer's risk is usually 10 %.
[ISO 3534-2:2006, definition 4.6.9]
3.22
producer's risk quality
PRQ
Q
PR
〈acceptance sampling〉 quality level of a lot or process which, in the acceptance sampling plan, corresponds to
a specified producer's risk
[ISO 3534-2:2006, definition 4.6.10]
NOTE The specified producer's risk is usually 5 %.
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ISO 8423:2008(E)
3.23
average sample size
ASSI
〈acceptance sampling〉 average number of units in a sample inspected per lot in reaching decisions to accept
or not to accept when using a given acceptance sampling plan
[ISO 3534-2:2006, definition 4.7.3]
3.24
sequential acceptance sampling inspection
acceptance sampling inspection in which, after each item has been inspected, the decision to accept the lot,
not accept the lot, or to inspect another item is taken based on the cumulative sampling evidence to date
[ISO 3534-2:2006, definition 4.2.7]
3.25
sequential sampling plan
plan which states acceptance criteria in sequential acceptance sampling inspection
3.26
operating characteristic curve
curve showing the relationship between probability of acceptance of product and the incoming quality level for
a given acceptance sampling plan
[ISO 3534-2:2006, definition 4.5.1]
3.27
producer’s risk point
PRP
〈acceptance sampling〉 point on the operating characteristic curve corresponding to a predetermined high
probability of acceptance
[ISO 3534-2:2006, definition 4.6.7]
3.28
consumer’s risk point
CRP
〈acceptance sampling〉 point on the operating characteristic curve corresponding to a predetermined low
probability of acceptance
[ISO 3534-2:2006, definition 4.6.5]
4 Symbols
The symbols used are as follows.
A acceptance value for sequential sampling
A acceptance value corresponding to the curtailed value of the cumulative sample size
t
f a factor given in Tables 5 and 6, that relates the maximum process standard deviation to the
difference between U and L
g multiplier of the cumulative sample size that is used to determine the acceptance values and the
rejection values (slope of the acceptance and rejection lines)
h constant that is used to determine the acceptance values (intercept of the acceptance line)
A
h constant that is used to determine the rejection values (intercept of the rejection line)
R
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ISO 8423:2008(E)
L lower specification limit (as a suffix to a variable, denotes its value at L)
N lot size (number of items in a lot)
n sample size (number of items in a sample)
n cumulative sample size
cum
n curtailment value of the cumulative sample size
t
P probability of acceptance
a
Q consumer's risk quality
CR
Q producer's risk quality
PR
R rejection value for sequential sampling
U upper specification limit (as a suffix to a variable, denotes its value at U)
x measured value of the quality characteristic for the item of the sample
y leeway, defined as
y = U − x for a single upper specification limit
y = x − L for a single lower specification limit
y = x − L for double specification limits
Y cumulative leeway obtained by adding the leeways up to, and including, the item last inspected
α producer's risk
β consumer's risk
σ standard deviation of a process that is under statistical control
2
NOTE σ , the square of the process standard deviation, is known as the process variance.
σ maximum process standard deviation
max
5 Principles of sequential sampling plans for inspection by variables
Under a sequential sampling plan by variables, sample items are drawn at random and inspected one by one,
and the cumulative leeway (which measures a “distance” between the process level and specification limits) is
obtained. After the inspection of each item, the cumulative leeway is compared with the acceptability criteria in
order to assess whether there is sufficient information to determine lot or process acceptability at that stage of
the inspection.
If, at a given stage, the cumulative leeway is such that the risk of accepting a lot of unsatisfactory quality level
is sufficiently low, the lot is considered acceptable and the inspection is terminated.
If, on the other hand, the cumulative leeway is such that the risk of non-acceptance of a lot of satisfactory
quality level is sufficiently low, the lot is considered not acceptable and the inspection is terminated.
If the cumulative leeway does not allow either of the above decisions to be taken, then an additional item is
sampled and inspected. The process is continued until sufficient sample information has been accumulated to
warrant a decision that the lot is acceptable or not acceptable.
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ISO 8423:2008(E)
6 Selection of a sampling plan
6.1 Producer’s risk point and consumer’s risk point
The general method described in 6.1 and 6.2 is used when the requirements of the sequential sampling plan
are specified in terms of two points on the operating characteristic curve of the plan. The point corresponding
to the higher probability of acceptance shall be designated the “producer’s risk point”; the other shall be
designated the “consumer’s risk point”.
The first step when designing a sequential sampling plan is to choose these two points, if they have not
already been dictated by circumstances. For this purpose, the following combination is often used:
⎯ a producer’s risk of α u 0,05 and the corresponding producer’s risk quality (Q ), and
PR
⎯ a consumer’s risk of β u 0,10 and the corresponding consumer’s risk quality (Q ).
CR
This combination of requirements is used in this International Standard for the design of the sampling plans.
When the desired sequential sampling plan is required to have approximately the same operating
characteristic curve as an existing single sampling plan, the producer’s risk point and the consumer’s risk
point may be read off from a graph or a table of the operating characteristic of that plan. When no such plan
exists, the producer’s and the consumer’s risk points have to be determined from direct consideration of the
conditions under which the sampling plan operates.
6.2 Preferred values of Q and Q
PR CR
Table 4 gives 21 preferred values of Q (producer’s risk quality) ranging from 0,1 % to 10,0 %, and 17
PR
preferred values of Q (consumer’s risk quality) ranging from 0,8 % to 31,5 %. This International Standard is
CR
only applicable to a combination of the preferred values of Q and Q .
PR CR
6.3 Pre-operation preparations
6.3.1 Obtaining the parameters h , h and g
A R
The criteria for acceptance and non-acceptance of a lot are determined from the parameters h , h and g.
A R
Table 4 gives the values of these parameters corresponding to each combination of preferred values of Q
PR
and Q together with the producer's risk α approximately equal to 0,05 and the consumer's risk
CR
β approximately equal to 0,1.
6.3.2 Obtaining the curtailment values
The curtailment value, n , of the cumulative sample size of the sequential sampling plan is given in Table 4
t
together with the other parameters.
7 Operation of a sequential sampling plan
7.1 Specification of the plan
Before operating a sequential sampling plan, the inspector shall record on the sampling document the
specified values of the parameters, h , h , g and n .
A R t
7.2 Drawing a sample item
As a rule, the individual sample items shall be drawn at random from the lot and inspected one by one in the
order in which they were drawn. If, for convenience, successive items are drawn at the same time, the order in
which each sample item is inspected shall be at random.
© ISO 2008 – All rights reserved 7
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ISO 8423:2008(E)
7.3 Leeway and cumulative leeway
Following the inspection of each item, record the inspection result x against the current value, n , of the
cum
cumulative sample size.
Calculate the leeway y for that item as
y = x − L in the case of combined control of double specification limits or a single lower specification limit;
y = U − x in the case of a single upper specification limit.
Record the cumulative leeway Y as the sum of the leeways found so far in the sample from the lot.
7.4 Choice between numerical and graphical methods
This International Standard provides two methods of operating
...
SLOVENSKI STANDARD
SIST ISO 8423:2010
01-julij-2010
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SIST ISO 8423:1996
SIST ISO 8423:1996/C1:1996
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Sequential sampling plans for inspection by variables for percent nonconforming (known
standard deviation)
Plans d'échantillonnage progressif pour le contrôle par mesures des pourcentages de
non-conformes (écart-type connu)
Ta slovenski standard je istoveten z: ISO 8423:2008
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
SIST ISO 8423:2010 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 8423:2010
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SIST ISO 8423:2010
INTERNATIONAL ISO
STANDARD 8423
Second edition
2008-09-01
Sequential sampling plans for inspection
by variables for percent nonconforming
(known standard deviation)
Plans d'échantillonnage progressif pour le contrôle par mesures des
pourcentages de non-conformes (écart-type connu)
Reference number
ISO 8423:2008(E)
©
ISO 2008
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SIST ISO 8423:2010
ISO 8423:2008(E)
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ii © ISO 2008 – All rights reserved
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SIST ISO 8423:2010
ISO 8423:2008(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 2
3 Terms and definitions. 2
4 Symbols . 5
5 Principles of sequential sampling plans for inspection by variables . 6
6 Selection of a sampling plan . 7
7 Operation of a sequential sampling plan . 7
8 Examples . 18
9 Tables. 23
Annex A (informative) Additional information. 28
Bibliography . 32
© ISO 2008 – All rights reserved iii
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SIST ISO 8423:2010
ISO 8423:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 8423 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 5, Acceptance sampling.
This second edition cancels and replaces the first edition (ISO 8423:1991), which has been technically revised.
It also incorporates the Technical Corrigendum ISO 8423:1991/Cor.1:1993. Annex A of ISO 8423:1991 is
superseded by ISO 3951-5:2005.
The following improvements have been introduced:
⎯ values of the parameters h , h and g have been recalculated in order to provide plans that exactly meet
A R
stated requirements,
⎯ the average sample sizes for quality levels equal to producer’s risk quality and consumer’s risk quality
have been significantly decreased.
iv © ISO 2008 – All rights reserved
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SIST ISO 8423:2010
ISO 8423:2008(E)
Introduction
In contemporary production processes, quality is often expected to reach such high levels that the number of
nonconforming items is reported in parts per million. Under such circumstances, popular acceptance sampling
plans by attributes, such as those presented in ISO 2859-1, require prohibitively large sample sizes. When it is
possible to apply acceptance sampling plans by variables, such as those presented in ISO 3951-1, the sample
sizes are much smaller. However, especially in the case of acceptance of a product of extremely high quality,
those sample sizes are still too large. Therefore, there is a need to apply standardized statistical procedures
that require the smallest possible sample sizes; sequential sampling plans are the only statistical procedures
that satisfy that need. It has been mathematically proved that among all possible sampling plans having
similar statistical properties the sequential sampling plan has the smallest average sample size.
The principal advantage of sequential sampling plans is the reduction in the average sample size. The
average sample size is the average of all the sample sizes that may occur under a sampling plan for a given
lot or process quality level. The use of sequential sampling plans leads to a smaller average sample size than
single sampling plans having the equivalent operating characteristic.
Other factors that should be taken into account are as follows:
a) Complexity
The rules of a sequential sampling plan are more easily misunderstood by inspectors than the simple
rules for a single sampling plan.
b) Variability in the amount of inspection
As the actual number of items inspected for a particular lot is not known in advance, the use of sequential
sampling plans brings about various organizational difficulties. For example, scheduling of inspection
operations may be difficult.
c) Difficulty of drawing sample items
If drawing sample items is rather difficult, the reduction in the average sample size by sequential sampling
plans may be cancelled out by the increased sampling cost.
d) Duration of test
If the test of a single item is of long duration and a number of items can be tested simultaneously,
sequential sampling plans are much more time-consuming than the corresponding single sampling plan.
e) Variability of quality within the lot
If the lot consists of two or more sublots from different sources and if there is likely to be any substantial
difference between the qualities of the sublots, drawing of a representative sample under a sequential
sampling plan is far more difficult than under the corresponding single sampling plan.
The balance between the advantage of a smaller average sample size of the sequential sampling plan and the
above disadvantages leads to the conclusion that sequential sampling plans are suitable only when inspection
of individual items is costly in comparison with inspection overheads.
The choice between single and sequential sampling plans should be made before the inspection of a lot is
started. During inspection of a lot, it is not permitted to switch from one type to another, because the operating
characteristic of the plan may be drastically changed if the actual inspection results influence the choice of
acceptability criteria.
© ISO 2008 – All rights reserved v
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SIST ISO 8423:2010
ISO 8423:2008(E)
Although a sequential sampling plan is on average much more economical than the corresponding single
sampling plan, it may occur, during inspection of a particular lot, that acceptance or non-acceptance comes at
a very late stage because the cumulative leeway (the statistic used for the determination of lot acceptability)
remains between the acceptance value and the rejection value for a long time. With the graphical method, this
corresponds to the random progress of the step-wise linear curve remaining in the indecision zone.
In order to alleviate this disadvantage, the curtailment values are set before the inspection of a lot (or a
process) is started, and inspection terminates if the cumulative sample size reaches the curtailment value, n ,
t
without determination of lot acceptability. The acceptance and non-acceptance of the lot (or the process) is
then determined using the curtailment acceptance and rejection values.
For sequential sampling plans in common use, curtailment usually represents a deviation from their intended
usage, leading to a distortion of their operating characteristics. In this International Standard, however, the
operating characteristics of the sequential sampling plans have been determined with curtailment taken into
account, so curtailment is an integral component of the provided plan.
Sequential sampling plans for inspection by variables are also provided in ISO 3951-5. However, the design
principle of those plans is fundamentally different from that of this International Standard. The sampling plans
in ISO 3951-5 are designed to supplement the ISO 3951-1 acceptance sampling system for inspection by
variables, which is a counterpart of the popular ISO 2859-1 acceptance sampling system for inspection by
attributes. Thus, they should be used for the inspection of a continuing series of lots, that is, a series long
enough to permit the switching rules of the ISO 3951 system to take effect. The application of the switching
rules is the only means of providing enhanced protection to the consumer (by means of tightened sampling
inspection criteria or discontinuation of sampling inspection) when the sequential sampling plans from
ISO 3951-5 are used. However, in certain circumstances, there is a strong need to have both producer’s and
consumer’s risks under strict control. Such circumstances occur, for example, when sampling is performed for
regulatory reasons, for the demonstration of quality of production processes or for hypothesis testing. In such
cases, individual sampling plans selected from the ISO 3951-5 sampling scheme may be inappropriate. The
sampling plans from this International Standard have been designed in order to meet these specific
requirements.
vi © ISO 2008 – All rights reserved
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SIST ISO 8423:2010
INTERNATIONAL STANDARD ISO 8423:2008(E)
Sequential sampling plans for inspection by variables for
percent nonconforming (known standard deviation)
1 Scope
This International Standard specifies sequential sampling plans and procedures for inspection by variables of
discrete items.
The plans are indexed in terms of producer’s risk point and the consumer’s risk point. Therefore, they are
suitable not only for the purposes of acceptance sampling, but for the more general purpose of the testing of
simple statistical hypotheses for proportions.
The purpose of this International Standard is to provide procedures for the sequential assessment of
inspection results that may be used to induce the supplier to supply lots of a quality having a high probability
of acceptance. At the same time, the consumer is protected by a prescribed upper limit to the probability of
accepting a lot (or process) of poor quality.
This International Standard is primarily designed for use under the following conditions:
a) where the inspection procedure is to be applied to a continuing series of lots of discrete products all
supplied by one producer using one production process. In such a case, sampling of particular lots is
equivalent to the sampling of the process. If there are different producers or production processes, this
International Standard shall be applied to each one separately;
b) where only a single quality characteristic x of these products is taken into consideration, which must be
measurable on a continuous scale;
c) where the measurement error is negligible (i.e. with a standard deviation no more than 10 % of the
process standard deviation);
d) where production is stable (under statistical control) and the quality characteristic x has a known standard
deviation, and is distributed according to a normal distribution or a close approximation to the normal
distribution;
CAUTION — The procedures in this International Standard are not suitable for application to lots that
have been screened previously for nonconforming items.
e) where a contract or standard defines an upper specification limit U, a lower specification limit L, or both;
an item is qualified as conforming if and only if its measured quality characteristic, x, satisfies the
appropriate one of the following inequalities:
1) x u U (i.e. the upper specification limit is not violated);
2) x W L (i.e. the lower specification limit is not violated);
3) x u U and x W L (i.e. neither the upper nor the lower specification limit is violated).
Inequalities 1) and 2) are called cases with a “single specification limit”, and 3) is the case with “double
specification limits”.
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SIST ISO 8423:2010
ISO 8423:2008(E)
In this International Standard, it is assumed that, where double specification limits apply, conformance to both
specification limits is either equally important to the integrity of the product or is considered separately for both
specification limits. In the first case, it is appropriate to control the combined percentage of product outside the
two specification limits. This is referred to as combined control. In the second case, nonconformity beyond
each of the limits is controlled separately, and this is referred to as separate control.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 3534-1:2006, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 3534-2:2006, Statistics — Vocabulary and symbols — Part 2: Applied statistics
ISO 3951-1:2005, Sampling procedures for inspection by variables — Part 1: Specification for single sampling
plans indexed by acceptance quality limit (AQL) for lot-by-lot inspection for a single quality characteristic and a
single AQL
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 3534-2 and ISO 3951-1
and the following apply.
3.1
inspection by variables
inspection by measuring the magnitude(s) of the characteristic(s) of an item
[ISO 3534-2:2006, definition 4.1.4]
3.2
sampling inspection
inspection of selected items in the group under consideration
[ISO 3534-2:2006, definition 4.1.6]
3.3
acceptance sampling
sampling after which decisions are made to accept or not to accept a lot, or other grouping of products,
materials or services, based on sample results
[ISO 3534-2:2006, definition 1.3.17]
3.4
acceptance sampling inspection
acceptance inspection where the acceptability is determined by means of sampling inspection
[ISO 3534-2:2006, definition 4.1.8]
3.5
acceptance sampling inspection by variables
acceptance sampling inspection in which the acceptability of a process is determined statistically from
measurements on specified quality characteristics of each item in a sample from a lot
[ISO 3534-2:2006, definition 4.2.11]
2 © ISO 2008 – All rights reserved
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SIST ISO 8423:2010
ISO 8423:2008(E)
3.6
quality level
quality expressed as a rate of occurrence of nonconforming units
3.7
nonconformity
non-fulfillment of a requirement
[ISO 9000:2005, definition 3.6.2, and ISO 3534-2:2006, definition 3.1.11]
3.8
nonconforming unit
unit with one or more nonconformities
[ISO 3534-2:2006, definition 1.2.15]
3.9
specification limit
limiting value stated for a characteristic
[ISO 3534-2:2006, definition 3.1.3]
3.10
lower specification limit
L
specification limit that defines the lower limiting value
[ISO 3534-2:2006, definition 3.1.5]
3.11
upper specification limit
U
specification limit that defines the upper limiting value
[ISO 3534-2:2006, definition 3.1.4]
3.12
combined control
requirement when both upper and lower limits are specified for the quality characteristic and specified risks
apply to the combined percent nonconforming beyond the two limits
NOTE The use of combined control implies that nonconformities beyond either specification limit are believed to be of
equal, or at least roughly equal, importance to the lack of integrity of the product.
3.13
separate control
requirement when both upper and lower limits are specified for the quality characteristic and separate risks
are given which apply to each limit
NOTE The use of separate control implies that nonconformities beyond either specification limit are believed to be of
different importance to the lack of integrity of the product.
3.14
maximum process standard deviation
σ
max
largest process standard deviation for a given sampling plan for which it is possible to satisfy the acceptance
criteria for a combined double specification limit when the process variability is known
NOTE 1 Maximum process standard deviation σ was denoted by its acronym MPSD in older standards.
max
NOTE 2 This definition is different from the similar definition given in ISO 3534-2 in which the concept of AQL is used.
© ISO 2008 – All rights reserved 3
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SIST ISO 8423:2010
ISO 8423:2008(E)
3.15
measurement
set of operations having the object of determining a value of a quantity
[ISO 3534-2:2006, definition 3.2.1]
3.16
leeway
quantity derived from a measured value of an item
NOTE In the case of a single lower specification limit and in the case of double specification limits, the leeway is
obtained by subtracting the numerical value of the lower specification limit from the measured value. In the case of an
upper specification limit, the leeway is obtained by subtracting the measured value from the numerical value of the upper
specification limit.
3.17
cumulative leeway
value calculated by summing the leeways obtained from the start of the inspection up to, and including, that of
the item last inspected
3.18
cumulative sample size
total number of inspected items, counting from the start of the inspection up to, and including, the item last
inspected
3.19
acceptance value for sequential sampling
value derived from the specified parameters of the sampling plan and the cumulative sample size
NOTE Whether the lot may yet be accepted is determined by comparing the cumulative leeway with the acceptance
value.
3.20
rejection value for sequential sampling
value derived from the specified parameters of the sampling plan and the cumulative sample size
NOTE Whether the lot may yet be considered unacceptable is determined by comparing the cumulative leeway with
the rejection value.
3.21
consumer's risk quality
CRQ
Q
CR
〈acceptance sampling〉 quality level of a lot or process which, in the acceptance sampling plan, corresponds to
a specified consumer's risk
NOTE The specified consumer's risk is usually 10 %.
[ISO 3534-2:2006, definition 4.6.9]
3.22
producer's risk quality
PRQ
Q
PR
〈acceptance sampling〉 quality level of a lot or process which, in the acceptance sampling plan, corresponds to
a specified producer's risk
[ISO 3534-2:2006, definition 4.6.10]
NOTE The specified producer's risk is usually 5 %.
4 © ISO 2008 – All rights reserved
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SIST ISO 8423:2010
ISO 8423:2008(E)
3.23
average sample size
ASSI
〈acceptance sampling〉 average number of units in a sample inspected per lot in reaching decisions to accept
or not to accept when using a given acceptance sampling plan
[ISO 3534-2:2006, definition 4.7.3]
3.24
sequential acceptance sampling inspection
acceptance sampling inspection in which, after each item has been inspected, the decision to accept the lot,
not accept the lot, or to inspect another item is taken based on the cumulative sampling evidence to date
[ISO 3534-2:2006, definition 4.2.7]
3.25
sequential sampling plan
plan which states acceptance criteria in sequential acceptance sampling inspection
3.26
operating characteristic curve
curve showing the relationship between probability of acceptance of product and the incoming quality level for
a given acceptance sampling plan
[ISO 3534-2:2006, definition 4.5.1]
3.27
producer’s risk point
PRP
〈acceptance sampling〉 point on the operating characteristic curve corresponding to a predetermined high
probability of acceptance
[ISO 3534-2:2006, definition 4.6.7]
3.28
consumer’s risk point
CRP
〈acceptance sampling〉 point on the operating characteristic curve corresponding to a predetermined low
probability of acceptance
[ISO 3534-2:2006, definition 4.6.5]
4 Symbols
The symbols used are as follows.
A acceptance value for sequential sampling
A acceptance value corresponding to the curtailed value of the cumulative sample size
t
f a factor given in Tables 5 and 6, that relates the maximum process standard deviation to the
difference between U and L
g multiplier of the cumulative sample size that is used to determine the acceptance values and the
rejection values (slope of the acceptance and rejection lines)
h constant that is used to determine the acceptance values (intercept of the acceptance line)
A
h constant that is used to determine the rejection values (intercept of the rejection line)
R
© ISO 2008 – All rights reserved 5
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SIST ISO 8423:2010
ISO 8423:2008(E)
L lower specification limit (as a suffix to a variable, denotes its value at L)
N lot size (number of items in a lot)
n sample size (number of items in a sample)
n cumulative sample size
cum
n curtailment value of the cumulative sample size
t
P probability of acceptance
a
Q consumer's risk quality
CR
Q producer's risk quality
PR
R rejection value for sequential sampling
U upper specification limit (as a suffix to a variable, denotes its value at U)
x measured value of the quality characteristic for the item of the sample
y leeway, defined as
y = U − x for a single upper specification limit
y = x − L for a single lower specification limit
y = x − L for double specification limits
Y cumulative leeway obtained by adding the leeways up to, and including, the item last inspected
α producer's risk
β consumer's risk
σ standard deviation of a process that is under statistical control
2
NOTE σ , the square of the process standard deviation, is known as the process variance.
σ maximum process standard deviation
max
5 Principles of sequential sampling plans for inspection by variables
Under a sequential sampling plan by variables, sample items are drawn at random and inspected one by one,
and the cumulative leeway (which measures a “distance” between the process level and specification limits) is
obtained. After the inspection of each item, the cumulative leeway is compared with the acceptability criteria in
order to assess whether there is sufficient information to determine lot or process acceptability at that stage of
the inspection.
If, at a given stage, the cumulative leeway is such that the risk of accepting a lot of unsatisfactory quality level
is sufficiently low, the lot is considered acceptable and the inspection is terminated.
If, on the other hand, the cumulative leeway is such that the risk of non-acceptance of a lot of satisfactory
quality level is sufficiently low, the lot is considered not acceptable and the inspection is terminated.
If the cumulative leeway does not allow either of the above decisions to be taken, then an additional item is
sampled and inspected. The process is continued until sufficient sample information has been accumulated to
warrant a decision that the lot is acceptable or not acceptable.
6 © ISO 2008 – All rights reserved
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SIST ISO 8423:2010
ISO 8423:2008(E)
6 Selection of a sampling plan
6.1 Producer’s risk point and consumer’s risk point
The general method described in 6.1 and 6.2 is used when the requirements of the sequential sampling plan
are specified in terms of two points on the operating characteristic curve of the plan. The point corresponding
to the higher probability of acceptance shall be designated the “producer’s risk point”; the other shall be
designated the “consumer’s risk point”.
The first step when designing a sequential sampling plan is to choose these two points, if they have not
already been dictated by circumstances. For this purpose, the following combination is often used:
⎯ a producer’s risk of α u 0,05 and the corresponding producer’s risk quality (Q ), and
PR
⎯ a consumer’s risk of β u 0,10 and the corresponding consumer’s risk quality (Q ).
CR
This combination of requirements is used in this International Standard for the design of the sampling plans.
When the desired sequential sampling plan is required to have approximately the same operating
characteristic curve as an existing single sampling plan, the producer’s risk point and the consumer’s risk
point may be read off from a graph or a table of the operating characteristic of that plan. When no such plan
exists, the producer’s and the consumer’s risk points have to be determined from direct consideration of the
conditions under which the sampling plan operates.
6.2 Preferred values of Q and Q
PR CR
Table 4 gives 21 preferred values of Q (producer’s risk quality) ranging from 0,1 % to 10,0 %, and 17
PR
preferred values of Q (consumer’s risk quality) ranging from 0,8 % to 31,5 %. This International Standard is
CR
only applicable to a combination of the preferred values of Q and Q .
PR CR
6.3 Pre-operation preparations
6.3.1 Obtaining the parameters h , h and g
A R
The criteria for acceptance and non-acceptance of a lot are determined from the parameters h , h and g.
A R
Table 4 gives the values of these parameters corresponding to each combination of preferred values of Q
PR
and Q together with the producer's risk α approximately equal to 0,05 and the consumer's risk
CR
β approximately equal to 0,1.
6.3.2 Obtaining the curtailment values
The curtailment value, n , of the cumulative sample size of the sequential sampling plan is given in Table 4
t
together with the other parameters.
7 Operation of a sequential sampling plan
7.1 Specification of the plan
Before operating a sequential sampling plan, the inspec
...
NORME ISO
INTERNATIONALE 8423
Deuxième édition
2008-09-01
Plans d'échantillonnage progressif pour
le contrôle par mesures des
pourcentages de non-conformes (écart-
type connu)
Sequential sampling plans for inspection by variables for percent
nonconforming (known standard deviation)
Numéro de référence
ISO 8423:2008(F)
©
ISO 2008
---------------------- Page: 1 ----------------------
ISO 8423:2008(F)
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ii © ISO 2008 – Tous droits réservés
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ISO 8423:2008(F)
Sommaire Page
Avant-propos. iv
Introduction . v
1 Domaine d'application. 1
2 Références normatives . 2
3 Termes et définitions. 2
4 Symboles . 6
5 Principes des plans d'échantillonnage progressif pour le contrôle par mesures. 7
6 Choix du plan d'échantillonnage. 7
7 Mise en œuvre d'un plan d'échantillonnage progressif . 8
8 Exemples . 18
9 Tableaux . 23
Annexe A (informative) Informations supplémentaires. 28
Bibliographie . 32
© ISO 2008 – Tous droits réservés iii
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ISO 8423:2008(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée
aux comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du
comité technique créé à cet effet. Les organisations internationales, gouvernementales et non
gouvernementales, en liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec
la Commission électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI,
Partie 2.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Les projets de Normes
internationales adoptés par les comités techniques sont soumis aux comités membres pour vote. Leur
publication comme Normes internationales requiert l'approbation de 75 % au moins des comités membres
votants.
L'attention est appelée sur le fait que certains des éléments du présent document peuvent faire l'objet de
droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable de ne
pas avoir identifié de tels droits de propriété et averti de leur existence.
L'ISO 8423 a été élaborée par le comité technique ISO/TC 69, Application des méthodes statistiques,
sous-comité SC 5, Échantillonnage en vue d'acceptation.
Cette deuxième édition annule et remplace la première édition (ISO 8423:1991), qui a fait l'objet d'une
révision technique. Elle incorpore également le Correctif technique ISO 8423:1991/Cor.1:1993. L'Annexe A de
l'ISO 8423:1991 est remplacée par l'ISO 3951-5:2005.
Les améliorations suivantes ont été apportées:
⎯ les valeurs des paramètres h , h et g ont été recalculées pour fournir des plans satisfaisant pleinement
A R
aux conditions spécifiées,
⎯ les effectifs moyens d'échantillon pour les niveaux de qualité risque fournisseur et qualité risque client ont
été réduits de manière significative.
iv © ISO 2008 – Tous droits réservés
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ISO 8423:2008(F)
Introduction
De nos jours, les procédés de fabrication prévoient le plus souvent d'atteindre des niveaux de qualité
tellement élevés que le nombre d'individus non conformes est exprimé en parties par million. Dans ces
conditions, les plans d'échantillonnage pour acceptation par attributs les plus répandus, tels que ceux
présentés dans l'ISO 2859-1, nécessitent des effectifs d'échantillon excessivement important. Lorsqu'il est
possible d'appliquer des plans d'échantillonnage pour acceptation par mesures, tels que ceux présentés dans
l'ISO 3951-1, les effectifs d'échantillon sont beaucoup plus petits. Cependant, et plus particulièrement dans le
cas de l'acceptation d'un produit de très haute qualité, les effectifs d'échantillon sont encore trop importants. Il
est par conséquent nécessaire d'appliquer des méthodes statistiques normalisées qui nécessitent des
effectifs d'échantillon les plus petits possible. Les plans d'échantillonnage progressif sont les seules méthodes
statistiques permettant de satisfaire à ces conditions. Il a été mathématiquement prouvé que parmi tous les
plans d'échantillonnage possibles ayant des caractéristiques statistiques similaires, le plan d'échantillonnage
progressif présentait l'effectif moyen d'échantillon le plus petit.
Les plans d'échantillonnage progressif présentent le principal avantage d'avoir un effectif moyen d'échantillon
réduit. L'effectif moyen d'échantillon est la moyenne de tous les effectifs d'échantillon possibles auxquels peut
conduire un plan d'échantillonnage pour un lot ou un niveau de qualité du processus donné. L'utilisation de
plans d'échantillonnage progressif conduit à un effectif moyen d'échantillon plus petit que pour les plans
d'échantillonnage simple de même efficacité.
Les autres facteurs qu'il convient de prendre en compte sont les suivants:
a) Complexité
Les règles d'un plan d'échantillonnage progressif sont souvent plus difficiles à comprendre par les
contrôleurs que les règles plus simples d'un plan d'échantillonnage simple.
b) Variabilité de l'importance du contrôle
Étant donné que le nombre réel d'unités contrôlées pour un lot particulier n'est pas connu à l'avance,
l'utilisation de plans d'échantillonnage progressif présente de nombreuses difficultés organisationnelles.
Par exemple, la planification des opérations de contrôle peut présenter des difficultés.
c) Difficulté de prélèvement d'échantillons
Lorsque le prélèvement d'échantillons est relativement difficile, la réduction de l'effectif moyen
d'échantillon apportée par les plans d'échantillonnage progressif peut être annulée par l'augmentation
des coûts d'échantillonnage.
d) Durée de l'essai
Lorsque l'essai d'une unité simple est de longue durée et que plusieurs unités peuvent être soumises à
essai simultanément, les plans d'échantillonnage progressif sont beaucoup plus chronophages que le
plan d'échantillonnage simple équivalent.
e) Variabilité de la qualité au sein du lot
Si le lot comprend deux sous-lots ou plus provenant de sources différentes et qu'il est probable
d'observer des différences significatives entre les qualités des sous-lots, le prélèvement d'un échantillon
représentatif dans le cadre d'un plan d'échantillonnage progressif est bien plus difficile que dans le cadre
du plan d'échantillonnage simple équivalent.
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ISO 8423:2008(F)
Le résultat de la balance entre l'avantage d'un effectif moyen d'échantillon plus petit pour le plan
d'échantillonnage progressif et les inconvénients indiqués ci-dessus est que les plans d'échantillonnage
progressif ne sont appropriés que lorsque le contrôle des unités discrètes est coûteux par rapport aux frais
généraux de contrôle.
Le choix entre des plans d'échantillonnage progressif et simple s'effectue avant le début du contrôle d'un lot. Il
n'est pas possible pendant le contrôle d'un lot de changer un type de plan d'échantillonnage contre un autre,
l'efficacité du plan pouvant être radicalement modifiée si les résultats du contrôle réel influencent le choix du
critère d'acceptation.
Bien qu'un plan d'échantillonnage progressif soit en moyenne beaucoup plus économique que le plan
d'échantillonnage simple équivalent, il peut arriver que, pour le contrôle d'un lot particulier, l'acceptation ou la
non-acceptation arrive à un stade très tardif, car l'écart cumulé (la statistique utilisée pour déterminer
l'acceptabilité d'un lot) reste longtemps compris entre la valeur d'acceptation et la valeur de rejet. Avec la
méthode graphique, cela correspond à une progression aléatoire de la courbe à l'intérieur de la zone
d'indécision.
Afin de pallier cet inconvénient, les valeurs de troncage sont établies avant le début du contrôle d'un lot (ou
d'un procédé), et le contrôle est arrêté lorsque l'effectif cumulé d'échantillon atteint la valeur de troncage, n ,
t
sans qu'une décision ait été prise. L'acceptation ou la non-acceptation du lot (ou du procédé) est alors
déterminée en utilisant les valeurs d'acceptation et de rejet de troncage.
Pour les plans d'échantillonnage progressif d'usage courant, le troncage constitue généralement un écart par
rapport à leur usage prévu, donnant ainsi lieu à une modification de leur efficacité. Cependant, dans la
présente Norme internationale, l'efficacité des plans d'échantillonnage progressif a été déterminée en tenant
compte du troncage comme constituant une composante intégrante du plan fourni.
L'ISO 3951-5 spécifie également des plans d'échantillonnage progressif pour le contrôle par mesures.
Cependant, le principe de base de ces plans est fondamentalement différent de celui de la présente Norme
internationale. Les plans d'échantillonnage de l'ISO 3951-5 sont conçus pour compléter le système
d'échantillonnage pour acceptation pour le contrôle par mesures de l'ISO 3951-1, dont l'homologue est
représenté par le système très répandu d'échantillonnage pour acceptation pour le contrôle par attributs de
l'ISO 2859-1. Ainsi, il convient de les utiliser pour le contrôle d'une série continue de lots, à savoir une série
suffisamment longue pour pouvoir mettre en place les règles de passage du système de l'ISO 3951.
L'application des règles de passage est l'unique moyen de renforcer la protection du consommateur (au
moyen de critères renforcés de contrôle par échantillonnage ou l'interruption du contrôle par échantillonnage)
lorsqu'on utilise les plans d'échantillonnage progressifs de l'ISO 3951-5. Dans certaines circonstances
cependant, il est absolument nécessaire que les risques fournisseur et client soient parfaitement maîtrisés.
Des exemples de circonstances de cette nature peuvent comprendre l'échantillonnage réalisé pour des
raisons d'ordre réglementaire, pour démontrer la qualité de procédés de fabrication ou pour vérifier des
hypothèses. Dans ces cas, les plans d'échantillonnage individuels sélectionnés à partir du programme
d'échantillonnage de l'ISO 3951-5 peuvent ne pas convenir. Les plans d'échantillonnage de la présente
Norme internationale ont été conçus pour satisfaire à ces conditions spécifiques.
vi © ISO 2008 – Tous droits réservés
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NORME INTERNATIONALE ISO 8423:2008(F)
Plans d'échantillonnage progressif pour le contrôle par
mesures des pourcentages de non-conformes (écart-type
connu)
1 Domaine d'application
La présente Norme internationale spécifie des plans et des règles d'échantillonnage progressif pour le
contrôle par mesures d'individus discrets.
Les plans sont indexés en termes de point du risque fournisseur et de point du risque client. Par conséquent,
ils sont appropriés non seulement aux fins d'échantillonnage pour acceptation, mais également aux
vérifications d'ordre plus général d'hypothèses statistiques simples de proportions.
Le but de la présente Norme internationale est de fournir des règles fondées sur la détermination progressive
des résultats de contrôle, afin d'inciter le fournisseur à fournir des lots de qualité ayant une forte probabilité
d'acceptation. En même temps, le client est protégé par une limite supérieure prescrite de la probabilité
d'accepter des lots (ou procédés) de faible qualité.
La présente Norme internationale est principalement conçue pour être utilisée lorsque les conditions
suivantes sont satisfaites:
a) lorsque la règle de contrôle est destinée à être appliquée à une série continue de lots constitués
d'individus discrets, tous fournis par un seul fournisseur utilisant un seul procédé de fabrication. Dans ce
cas, l'échantillonnage de lots particuliers correspond à l'échantillonnage du procédé. S'il y a différents
fournisseurs ou procédés de fabrication, la présente Norme internationale doit être appliquée à chacun
d'eux séparément;
b) lorsqu'un unique caractère de qualité x de ces individus, qui doit être mesurable sur une échelle continue,
est pris en considération;
c) lorsque l'erreur de mesure est négligeable (c'est-à-dire avec un écart-type non supérieur à 10 % de
l'écart-type du processus);
d) lorsque la fabrication est stable (sous maîtrise statistique) et le caractère de qualité x a un écart-type
connu, et est distribué suivant une loi normale ou voisine d'une loi normale;
ATTENTION — Les procédures de la présente Norme internationale ne s'appliquent pas aux lots ayant
préalablement fait l'objet d'une sélection d'individus non conformes.
e) lorsqu'un contrat ou une norme définit une limite de spécification supérieure U, une limite de spécification
inférieure L, ou les deux; un produit est qualifié de non conforme si la mesure de son caractère de qualité
x satisfait l'une des inégalités suivantes:
1) x u U (c'est-à-dire que la limite de spécification supérieure est respectée);
2) x W L (c'est-à-dire que la limite de spécification inférieure est respectée);
3) x u U et x W L (c'est-à-dire que la limite de spécification supérieure et inférieure est respectée).
Les inégalités 1) et 2) répondent au cas d'une limite de spécification unique, et 3) au cas de limites de
spécification doubles.
© ISO 2008 – Tous droits réservés 1
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ISO 8423:2008(F)
La présente Norme internationale suppose que, dans le cas de limites de spécification doubles, la conformité
aux deux limites de spécification s'applique conjointement à l'intégrité du produit ou est considérée
séparément pour les deux limites de spécification. Dans le premier cas, il convient de contrôler le pourcentage
combiné de produits en dehors des deux limites de spécification; il s'agit du contrôle combiné. Dans le second
cas, la non-conformité au-delà de chacune des limites est contrôlée séparément; il s'agit du contrôle séparé.
2 Références normatives
Les documents de référence suivants sont indispensables pour l'application du présent document. Pour les
références datées, seule l'édition citée s'applique. Pour les références non datées, la dernière édition du
document de référence s'applique (y compris les éventuels amendements).
ISO 3534-1:2006, Statistique — Vocabulaire et symboles — Partie 1: Termes statistiques généraux et termes
utilisés en calcul des probabilités
ISO 3534-2:2006, Statistique — Vocabulaire et symboles — Partie 2: Statistique appliquée
ISO 3951-1:2005, Règles d'échantillonnage pour les contrôles par mesures — Partie 1: Spécifications pour
les plans d'échantillonnage simples indexés d'après la limite d'acceptation de qualité (LAQ) pour le contrôle lot
par lot pour une caractéristique de qualité unique et une LAQ unique
3 Termes et définitions
Pour les besoins du présent document, les termes et définitions donnés dans l'ISO 3534-1, l'ISO 3534-2 et
l'ISO 3951-1 ainsi que les suivants s'appliquent.
3.1
contrôle par mesures
contrôle qui consiste à mesurer la (les) valeur(s) d'une (des) caractéristique(s) d'un individu
[ISO 3534-2:2006, définition 4.1.4]
3.2
contrôle par échantillonnage
contrôle des individus sélectionnés dans le groupe considéré
[ISO 3534-2:2006, définition 4.1.6]
3.3
contrôle par échantillonnage pour acceptation
échantillonnage pour acceptation
échantillonnage où les décisions d'accepter ou de ne pas accepter un lot, ou autre groupement de produits,
de matériaux ou de services, sont prises d'après les résultats sur un échantillon
[ISO 3534-2:2006, définition 1.3.17]
3.4
contrôle par échantillonnage pour acceptation
contrôle pour acceptation où l'acceptabilité est déterminée au moyen du contrôle par échantillonnage
[ISO 3534-2:2006, définition 4.1.8]
2 © ISO 2008 – Tous droits réservés
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ISO 8423:2008(F)
3.5
contrôle par échantillonnage pour acceptation par mesures
contrôle par échantillonnage pour acceptation dans lequel l'acceptation d'un processus est déterminée
statistiquement à partir des mesures des caractéristiques qualité spécifiées de chaque individu dans un
échantillon d'un lot
[ISO 3534-2:2006, définition 4.2.11]
3.6
niveau de qualité
qualité exprimée en fréquence d'occurrence d'unités non conformes
3.7
non-conformité
non-satisfaction d'une exigence
[ISO 9000:2005, définition 3.6.2, et ISO 3534-2:2006, définition 3.1.11]
3.8
unité non conforme
unité avec une ou plusieurs non-conformités
[ISO 3534-2:2006, définition 1.2.15]
3.9
limite de spécification
valeur limite spécifiée pour une caractéristique
[ISO 3534-2:2006, définition 3.1.3]
3.10
limite de spécification inférieure
L
limite de spécification qui définit la valeur limite inférieure
[ISO 3534-2:2006, définition 3.1.5]
3.11
limite de spécification supérieure
U
limite de spécification qui définit la valeur limite supérieure
[ISO 3534-2:2006, définition 3.1.4]
3.12
contrôle combiné
exigence lorsque les limites supérieure et inférieure sont toutes deux spécifiées pour la caractéristique qualité
et que des risques spécifiés s'appliquent au pourcentage combiné de non-conformes au-delà des deux limites
NOTE L'utilisation du contrôle combiné implique que les non-conformités au-delà de l'une ou l'autre des limites de
spécification sont considérées comme d'importance égale ou quasi égale par rapport au manque d'intégrité du produit.
3.13
contrôle séparé
exigence lorsque les limites supérieure et inférieure sont toutes deux spécifiées pour la caractéristique qualité
et que des risques s'appliquent séparément à chaque limite
NOTE L'utilisation du contrôle séparé implique que les non-conformités au-delà de l'une ou de l'autre des limites de
spécification sont considérées comme d'importance différente par rapport au manque d'intégrité du produit.
© ISO 2008 – Tous droits réservés 3
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ISO 8423:2008(F)
3.14
écart-type limite du processus
σ
max
écart-type du processus le plus grand pour un plan d'échantillonnage donné pour lequel il est possible de
satisfaire aux critères d'acceptation pour une limite de spécification double combinée lorsque la variabilité du
processus est connue
NOTE 1 L'écart-type limite du processus σ était désigné dans les anciennes normes par son acronyme MPSD.
max
NOTE 2 Cette définition est différente de la définition similaire donnée dans l'ISO 3534-2 dans laquelle le concept AQL
est utilisé.
3.15
mesurage
ensemble d'opérations ayant pour but de déterminer la valeur d'une grandeur
[ISO 3534-2:2006, définition 3.2.1]
3.16
écart
grandeur obtenue à partir de la valeur mesurée sur une unité
NOTE Dans le cas d'une limite de spécification inférieure unique et dans le cas de limites de spécification doubles,
l'écart est obtenu en soustrayant la valeur numérique de la limite de spécification inférieure de la valeur mesurée. Dans le
cas d'une limite de spécification supérieure, l'écart est obtenu en soustrayant la valeur mesurée de la valeur numérique de
la limite de spécification supérieure.
3.17
écart cumulé
valeur calculée en additionnant les écarts obtenus à partir du début du contrôle jusqu'à la dernière unité
contrôlée incluse
3.18
effectif cumulé d'échantillon
total des unités contrôlées, comptées à partir du début du contrôle jusqu'à la dernière unité contrôlée incluse
3.19
valeur d'acceptation pour l'échantillonnage progressif
valeur calculée à partir des paramètres spécifiés du plan d'échantillonnage et de l'effectif cumulé d'échantillon
NOTE L'acceptabilité du lot peut ensuite être décidée en comparant l'écart cumulé avec la valeur d'acceptation.
3.20
valeur de rejet pour l'échantillonnage progressif
valeur calculée à partir des paramètres spécifiés du plan d'échantillonnage et de l'effectif cumulé d'échantillon
NOTE La non-acceptabilité du lot peut ensuite être décidée en comparant l'écart cumulé avec la valeur de rejet.
3.21
qualité du risque du client
QRC
Q
RC
〈échantillonnage pour acceptation〉 niveau de qualité d'un lot ou d'un processus qui, dans le plan
d'échantillonnage pour acceptation, correspond à un risque du client spécifié
NOTE Le risque du client spécifié est généralement de 10 %.
[ISO 3534-2:2006, définition 4.6.9]
4 © ISO 2008 – Tous droits réservés
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ISO 8423:2008(F)
3.22
qualité du risque du fournisseur
QRF
Q
RF
〈échantillonnage pour acceptation〉 niveau de qualité d'un lot ou d'un processus qui, dans le plan
d'échantillonnage pour acceptation, correspond à un risque du fournisseur spécifié
[ISO 3534-2:2006, définition 4.6.10]
NOTE Le risque du fournisseur spécifié est généralement de 5 %.
3.23
effectif moyen d'échantillon
EMC
〈échantillonnage pour acceptation〉 valeur moyenne des unités dans l'échantillon inspecté par lot pour prendre
les décisions d'accepter ou de ne pas accepter en fonction d'un plan d'échantillonnage pour acceptation
donné
[ISO 3534-2:2006, définition 4.7.3]
3.24
contrôle par échantillonnage pour acceptation progressif
contrôle par échantillonnage pour acceptation où, après contrôle de chaque individu, la décision d'accepter le
lot, de ne pas l'accepter ou de contrôler encore un autre individu est fondée sur le cumul des informations
d'échantillonnage actuelles
[ISO 3534-2:2006, définition 4.2.7]
3.25
plan d'échantillonnage progressif
plan qui spécifie les critères d'acceptation du contrôle par échantillonnage pour acceptation progressif
3.26
courbe d'efficacité
courbe montrant la relation entre la probabilité d'acceptation du produit et le niveau de qualité avant contrôle
pour un plan d'échantillonnage pour acceptation donné
[ISO 3534-2:2006, définition 4.5.1]
3.27
point du risque du fournisseur
PRF
〈échantillonnage pour acceptation〉 point sur la courbe d'efficacité correspondant à une probabilité
d'acceptation forte fixée à l'avance
[ISO 3534-2:2006, définition 4.6.7]
3.28
point du risque du client
PRC
〈échantillonnage pour acceptation〉 point sur la courbe d'efficacité correspondant à une probabilité
d'acceptation faible fixée à l'avance
[ISO 3534-2:2006, définition 4.6.5]
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ISO 8423:2008(F)
4 Symboles
Les symboles utilisés sont les suivants:
A valeur d'acceptation pour l'échantillonnage progressif
A valeur d'acceptation correspondant à la valeur de troncage de l'effectif cumulé d'échantillon
t
f un facteur donné dans les Tableaux 5 et 6, qui met en relation l'écart-type limite du processus et la
différence entre U et L
g facteur multiplicatif de l'effectif cumulé d'échantillon utilisé pour déterminer les valeurs d'acceptation
et de rejet (pente des droites d'acceptation et de rejet)
h constante utilisée pour déterminer les valeurs d'acceptation (ordonnée à l'origine de la ligne
A
d'acceptation)
h constante utilisée pour déterminer les valeurs de rejet (ordonnée à l'origine de la ligne de rejet)
R
L limite de spécification inférieure (en indice sur une mesure, désigne sa valeur pour L)
N effectif de lot (nombre d'individus dans un lot)
n effectif d'échantillon (nombre d'individus dans un échantillon)
n effectif cumulé d'échantillon
cum
n valeur de troncage de l'effectif cumulé d'échantillon
t
Q qualité du risque du client
RC
Q qualité du risque du fournisseur
RF
P probabilité d'acceptation
a
R valeur de rejet pour l'échantillonnage progressif
U limite de spécification supérieure (en indice sur une mesure, désigne sa valeur pour U)
x valeur mesurée d'un caractère pour l'individu de l'échantillon
y écart, défini par:
y = U − x pour une limite de spécification supérieure unique
y = x − L pour une limite de spécification inférieure unique
y = x − L pour des limites de spécification doubles
Y écart cumulé obtenu par l'addition des écarts, jusqu'au dernier individu contrôlé inclus
α risque fournisseur
β risque client
σ écart-type d'un processus en maîtrise statistique
2
NOTE σ , le carré de l'écart-type dans le processus est appelé variance.
σ écart-type limite du processus
max
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ISO 8423:2008(F)
5 Principes des plans d'échantillonnage progressif pour le contrôle par mesures
Dans un plan d'échantillonnage progressif par mesures, les individus sont prélevés au hasard et soumis au
contrôle un par un, et l'écart cumulé (qui mesure une «distance» entre le niveau du processus et les limites de
spécification) est obtenu. Après le contrôle de chaque individu, l'écart cumulé est comparé aux critères
d'acceptabilité pour évaluer si l'information est suffisante pour prendre une décision sur l'acceptabilité du lot
ou du procédé à ce stade du contrôle.
Si, à un stade donné, l'écart cumulé est tel que le risque d'accepter un lot d'une qualité insuffisante est
suffisamment bas, le lot est considéré acceptable et le contrôle est terminé.
Si, d'un autre côté, l'écart cumulé est tel que le risque de non-acceptation pour un lot de qualité suffisante est
suffisamment bas, le lot est considéré non acceptable et le contrôle est terminé.
Si l'écart cumulé ne permet de prendre aucune de ces décisions, alors un individu supplémentaire est
échantillonné et contrôlé. Cette procédure est reprise jusqu'à ce que l'information sur l'échantillon soit
accumulée de façon suffisante pour permettre de décider si le lot est acceptable ou non acceptable.
6 Choix du plan d'échantillonnage
6.1 Point du risque du fournisseur et point du risque du client
La méthode générale décrite en 6.1 et en 6.2 est utilisée lorsque les besoins d'un plan d'échantillonnage
progressif sont spécifiés sous la forme de deux points sur la courbe d'efficacité du plan. Le point
correspondant à la plus haute probabilité d'acceptation doit être appelé «point du risque du fournisseur»;
l'autre doit être appelé «point du risque du client».
La première étape dans la détermination d'un plan d'échantillonnage progressif est de choisir les points des
risques fournisseur et client, s'ils n'ont pas déjà été dictés par les circonstances. Dans cette perspective, la
combinaison suivante est souvent utilisée:
⎯ un risque fournisseur de α u 0,05 et la qualité du risque du fournisseur (Q ) correspondante, et
RF
⎯ un risque client de β u 0,10 et la qualité du risque du client (Q ) correspond
...
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