Acceptance Sampling-single Plan

8/12/2019 Acceptance Sampling-single Plan

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Acceptance sampling

4.1 Introduction to Acceptance Sampling and Single Sampling Plan

Certain important terms relevant to acceptance sampling are discussed below (seeANSI/ASQC Standard A2-19!" #uncan (19$%" Sc&illing (192%%'

Acceptance sampling plan' a speciic plan t&at clearl) states t&e rulesor sampling

and t&e associated criteria oracceptance or ot&erwise.Acceptance sampling plans can

be applied or inspection o (i% end items* (ii% components* (iii% raw materials* (iv%

operations* (v% materials in process* (v% supplies in storage* (vi% maintenance

operations* (vii% data or records and

(viii% administrative procedures+

Item or Unit:an ob,ect or uantit) o product or material on w&ic& observations

(attribute or variable or bot&% are made+

Lot or Batch: a deined uantit) o product accumulated or sampling purposes+ It is

e.pected t&at t&e qualityo t&e product wit&in a lot is uniorm+

Attributes Method:w&ere ualit) is measured b) observing t&e presence or absence

o some c&aracteristic or attribute in eac& o t&e units in t&e sample or lot under

consideration* and t&e number o items counted w&ic& do or do not possess t&e ualit)

attribute* or &ow man) events occur in t&e unit area* etc+

Variables Method: w&ere measurement o ualit) is b) means o measuring andrecording t&e numerical magnitude o a ualit) c&aracteristic or eac& o t&e items+

Nonconformity:t&e departure o a ualit) c&aracteristic rom its intended level*

causing t&e product or service to ail to meet t&e speciication reuirement+ I t&e

product or service is also not meeting t&e usage reuirements* it is called as a defect.

suall) t&e terms 0deect0 and 0nonconormit)0 are interc&angeable but t&e word

0deect0 is more stringent+

Nonconforming (Defectie! Unit:a unit containing at least one nonconormit)

(deect%+ &e terms 0deective0 and 0nonconorming0 are interc&angeable but a

deective unit will ail to satis) t&e normall) intended usage reuirements+

"roportion (#raction! Defectie or "roportion (#raction! Nonconforming Units p:

&is is t&e ratio o t&e number o nonconorming units (deectives% to t&e total number

o (sampled% units+

$ingle $ampling:t&e sampling inspection t)pe in w&ic& t&e lot disposition is based

on t&e inspection o asinglesample o sie n.

$ingle $ampling Attributes "lan (n% Ac!
http://www-ist.massey.ac.nz/61325/sg/#CHAPTER%204


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&e operating procedure o t&e single sampling attributes plan is as ollows'

1+ 3rom a lot o sie N* draw a random sample o sie n and observe t&e number

o nonconorming units (nonconormities% d+

2+ I d is less t&an or eual to t&e acceptance number Ac* w&ic& is t&e ma.imumallowable number o nonconorming units or nonconormities* accept t&e lot+

I d 4 Ac* do not accept t&e lot+

&e s)mbol c is also used or acceptance number+ &e s)mbol 5e (6 Ac71% is used to

denote t&e re,ection number+

&e acceptable &uality leel (A'L! is t&e maximumpercentage or proportion o

nonconorming units in a lot t&at can be considered satisactor) as a process average

or t&e purpose o acceptance sampling+ 8&en a consumer designates some speciic

value o AQ* t&e supplier or producer is notiied t&at t&e consumer:s acceptance

sampling plan will accept most o t&e produced lots submitted b) t&e supplier*provided t&e process average o t&ese lots is not greater t&an t&e designated value o

AQ+ Suppose t&at one perorms an error-ree 1;;< inspection o t&e lot and

observes p or eac& lot+ &en all lots wit& p AQ will be accepted and all lots wit& p4 AQ will not be accepted+ &is situation is s&own grap&icall) in 3igure =+1+

Figure 4.1 Ideal OC Curve

#ue to sampling* one aces t&e ris> o not accepting lots o AQ ualit) as well as t&e

ris> o accepting lots o poorer t&an AQ ualit)+ ?ne is t&ereore interested in

>nowing &ow an acceptance sampling plan will accept or not accept lots over various

lot ualities+ A curve s&owing t&e probabilit) o acceptance over various lot or process

ualities is called t&e operating c&aracteristic (?C% curve and is e.plained below'

perating )haracteristic ()! )ure

&e ?C curve reveals t&e perormance o t&e acceptance sampling plan+ 8e consider

two t)pes o ?C curves' TypeA:(3or isolated or uniue lots% &is is a curve s&owingt&e probabilit) o accepting a lotasa unction o t&e lot quality+


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Type B:(3or a continuous stream o lots% &is is a curve s&owing t&e probabilit) o

accepting a lot as a unction o t&e process average+ &at is* t&e )pe @ ?C curve will

give t&e proportion o lots accepted as a unction o t&e process average p+

) #unction of a $ingle $ampling "lan

&e ?C unction o t&e single sampling attribute plan giving t&e probabilit) o

acceptance or a given lot or process ualit) p is'

a6 a(p% 6 r (d Ac n* Ac* p%+

3or )pe A situations* t&e &)pergeometric distribution is e.act or t&e case o

nonconorming units+ Bence*

a(p% 6 r (d Ac N* n* Ac* p%*

w&ere N is t&e lot sie* # is t&e number o deectives in t&e lot and &ence p 6 #/N+

3or )pe @ situations* t&e binomial model is e.act or t&e case o raction

nonconorming units and t&e ?C unction is given b)

a(p% 6 r (d Ac n* Ac* p%

+

&e above ?C unction is applicable to a continuous stream o lots and can be used as

an appro.imation to )pe A situation w&en N is large compared to n (n/N ;+1;% and

p is small+

3or t&e case o nonconormities per unit* t&e oisson model is e.act or bot& )pe A

and )pe @ situations+ &e ?C unction in t&is case is

a(p% 6 r (d Ac n* Ac* p%*


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&e above ?C unction is also used as an appro.imation to binomial w&en n is largeand p is small suc& t&at np D+ A t)pical ?C curve is s&own in 3igure =+2+

Figure 4.2 OC Curve

In general* t&e probabilit) o acceptance will be underestimated at good ualit) levels

and overestimated at poor ualit) levels i one appro.imates t&e &)pergeometric ?C

unction wit& a binomial or oisson ?C unction+ &e same is true w&en t&e binomial?C unction is appro.imated b) t&e oisson ?C unction+ &is is s&own grap&icall)

in 3igure =+E assuming N 6 2;;* n 6 2; and Ac 6 1+

Figure 4.3 Comparison of OC Curves


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It can be observed t&at w&en n is constant and Ac is increased* a(p% will increase+

8&en Ac is constant and n is increased* t&en a(p% will decrease (see 3igures =+= and

=+D%+ &ese two properties are useul or designing sampling plans or given ualit)

reuirements+

Figure 4.4 Effect of Acceptance um!er on OC Curve

Figure 4." Effect of Sample Si#e on OC Curve

Suppose t&at a compan) manuacturing c&eese is continuouslysuppl)ing its

production to two supermar>ets+ Supermar>et A reuires c&eese in lots o D;;; units

and supermar>et @ preers lots o sie 1;;;;+ Assume t&at t&e producer uses two

sampling plans wit& sample sie eual to t&e suare root o t&e lot sie+ 3or bot&

plans* let t&e acceptance number be i.ed as one+ &e compan):s true raction o

nonconorming c&eeses produced is 1ets+ &e eectiveness o t&e sampling plans (plan

or Supermar>et A' n 6 !1 Ac 6 1" plan or supermar>et @' n 6 1;; Ac 6 1% will be

revealed b) t&e respective ?C curves s&own in 3igure =+$


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Figure 4.( Comparison of OC Curves for a %iven '&'

&us it is seen t&at it will be wort&w&ile to prescribe an additional inde. or consumer

protection since t&e AQ does not completel) describe t&e protection to t&e consumer+

Bence or consumer protection against bad ualit) lots* t&e imiting ualit) level

(Q% is deined as t&e percentage or proportion o nonconorming units in a lot or

w&ic& t&e consumer wis&es t&e probabilit) o acceptance to be restricted to a speciied

low value+ Q is also reerred to as t&e re,ectable ualit) level (5Q%* t&e

unacceptable ualit) level (Q%* t&e limiting ualit) (Q%* t&e lot tolerance raction

deective (3#% or t&e lot tolerance percent deective (#%+ Bere 3# and

# are deined or )pe A situations onl)+

Figure 4.) OC Curve S*o+ing A&', '&', and

&e producer:s ris> (% is t&e probabilit) o not accepting a lot o AQ ualit) and t&e

consumer:s ris> (% is t&e probabilit) o accepting a lot o Q ualit)+ Fenerall) t&eparameters o t&e (single% sampling plan must be determined or given ualit) levels*


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namel) AQ* Q and ris>s and + 3igure =+ s&ows t&e ualit) indices AQ* Qand t&e associated ris>s and respectivel) on t&e ?C curve+

Consider t&e single sampling plans (n 6 12!* Ac 6 D%* and (n 6 E* Ac 6 1% and t&e ?C

curves o t&ese two plans (3igure =+9%+

Figure 4.- Comparison of iscriminating Po+er of OC Curves

&e plan (12!*D% possesses a more &ig&l) discriminating ?C curve t&an t&e plan

(E*1%+ &at is* t&e plan (12!*D% ac&ieves a smaller producer:s ris> at all good ualit)

levels and involves a smaller consumer:s ris> at all poor ualit) levels+ Consider t&e

?C curves o t&e plans ($E*E% and (E*1% given in 3igure =+1;+ Bere t&e plan ($E*E%ac&ieves a smaller producer:s ris> compared to t&e plan (E*1% but maintains t&e

consumer:s ris> at poor ualit) levels+

Figure 4.1/ Comparison of OC Curves

In general* bot& t&e sample sie and acceptance numbers need to be large or reac&ing

t&e ideal s&ape o t&e ?C curve+ Consider a value o 1< or AQ+ ?C curves o planswit& dierent acceptance numbers giving t&e same producer:s ris> at AQ are s&own


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in 3igure =+11+ &e consumer:s ris>s or t&ese plans are ver) dierent* and t&e ideal

s&ape is reac&ed onl) wit& an increasing acceptance number and sample sie+

Figure 4.11 Approac*ing Ideal OC Curve

Aerage $ample Number (A$N!

&e average sample number (ASN% is deined as t&e average number o sample unitsper lot used or deciding acceptance or non-acceptance+ 3or a single sampling plan*

one ta>es onl) a single sample o sie n and &ence t&e ASN is simpl) t&e sample sie

n+

@) curtailed inspection* we mean t&e stopping o sampling inspection* w&en a

decision is certain+ Inspection can be curtailed w&en t&e re,ection number is reac&ed

since t&e re,ection is certain and no urt&er inspection is necessar) in reac&ing t&at

decision+ Suc& curtailment o inspection or re,ecting a lot is >nown as semi-curtailed

inspection+ I inspection is curtailed once acceptance or re,ection is evident* t&en it is

>nown as ull) curtailed inspection+

3or e.ample* consider t&e single sampling plan wit& n 6 D; and Ac 6 1+ et t&e

sample be randoml) drawn and testing o units ta>es place unit-b)-unit+ ?ne can

curtail inspection* re,ecting t&e lot* as earl) as t&e t&ird unit i t&e irst two units are

nonconorming+ Similarl) i all t&e irst =9 units are ound to be conorming* t&en t&e

lot can be accepted wit&out testing t&e last unit+ &e ull) curtailed and semi-curtailed

ASN curves o t&e plan n 6 D; and Ac 6 1 are s&own in 3igure =+12+ 8&en p 6 ;* t&e

ull) curtailed plan &as an ASN o =9 w&ile t&e semi-curtailed plan reuires all t&e D;

units to be tested+ &e polic) o curtailment is eective onl) at poor ualit) levels

since more nonconorming units are li>el) to be sampled leading to an earl) re,ection

decision+


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Figure 4.12 Curtailed AS Curves

Fenerall) it is undesirable to curtail inspection in single sampling+ &e w&ole sample

is usuall) inspected in order to &ave an unbiased record o ualit) &istor)+

*ectifying or $creening Inspection

&ose lots not accepted b) a sampling plan will usuall) be 1;;< inspected or

screened or nonconorming or deective units+ Ater screening* nonconorming units

ma) be rectiied or discarded or replaced b) good units* usuall) ta>en rom acceptedlots+ Suc& a programme o inspection is >nown as a recti)ing or screening inspection+

3or t&ose lots accepted b) t&e sampling plan* no screening will be done and t&e

outgoing ualit) will be t&e same as t&at o t&e incoming ualit) p+ 3or t&ose lots

screened* t&e outgoing ualit) will be ero* meaning t&at t&e) contain no

nonconorming items+ Since t&e probabilit) o accepting a lot is a* t&e outgoing lots

will contain a proportion o padeectives+ I t&e nonconorming units ound in t&e

sample o sie n are replaced b) good ones* t&e average outgoing ualit) (A?Q% will

be

In s&ort* one deines t&e average outgoing ualit) as t&e e.pected ualit) o outgoing

product ollowing t&e use o an acceptance sampling plan or a given value o t&e

incoming ualit)+ 3igure =+1E gives a t)pical A?Q curve as a unction o t&eincoming ualit)+


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Figure 4.13 AO& Curve

I t&e incoming ualit) is good* t&en a large proportion o t&e lots will be accepted b)

t&e sampling plan and onl) a smaller raction will be screened and &ence t&e outgoing

ualit) will be small (good%+ Similarl)* w&en t&e incoming ualit) is not good* a large

proportion o t&e lots will go or screening inspection and in t&is case also* t&e

outgoing ualit) will be good since deective items will be eit&er replaced or rectiied+

?nl) or intermediate ualit) levels* lot acceptance will be at a moderate rate and

&ence t&e A?Q will rise (see 3igures =+1= and =+1D%+ &e ma.imum ordinate o t&e

A?Q curve represents t&e worst possible average or t&e outgoing ualit) and is

>nown as t&e average outgoing ualit) limit (A?Q%+ In ot&er words* t&e A?Q is

deined as t&e ma.imum A?Q over all possible levels o t&e incoming ualit) or a

>nown acceptance sampling plan (see 3igure =+1=%+

Figure 4.14 AO& and AO&'


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Figure 4.1" Occurrence of AO&' at a 0oderate &ualit 'evel

It s&ould be noted t&at bot& t&e concepts o A?Q and A?Q are valid onl) in )pe @

situations* t&at is or a continuous stream o lots+ 3or isolated lots* t&e A?Q and

A?Q will &ave no meaning and consumer protection could be ac&ieved onl) t&roug&

t&e limiting ualit)+ &e e.pression or A?Q will also be aected b) t&e disposition

polic) o discarding t&e nonconorming unit instead o recti)ing it or replacing it

wit& a conorming unit in t&e screening inspection p&ase+

Figure 4.1$ AI Curve *en 1///, n "/, Ac 1

Similar to t&e measure o average sample number considered or sampling inspection*

an important measure relevant to recti)ing inspection is t&e average total inspection

(AI%* w&ic& is deined as t&e average number o units inspected per lot based on t&e

sample or accepted lots and all inspected units in lots not accepted+

I t&e lot sie is N and is o ualit) p* t&en t&e AI or t&e single sampling plan is

given b)


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AI(p% 6 n 7 (1-a% (N-n%+

&is e.pression is simple to interpret+ &e single sample o sie n is alwa)s inspected+

I t&e lot is not accepted* t&e remainder o t&e lot containing (N-n% units are screened*

or w&ic& t&e probabilit) is (1-a%+ &e above e.pression is actuall) obtained rom t&e

e.pression

AI(p% 6 na7 N(1-a%*

meaning n units are onl) inspected or t&ose lots accepted (t&e probabilit) being a%

and t&e w&ole lot will be inspected on non-acceptance (t&e probabilit) being (1- a%%+ A

t)pical AI curve is s&own in 3igure =+1$+ 3or lots o perect ualit) (p6;%* t&e AI

will be n and or lots wit& all deective units (p61%* t&e AI will be N+

In general* t&e ?C and A?Q unctions are >nown asprotectionmeasures since t&e)

reveal t&e protection given b) t&e sampling plan to t&e producer and consumer+ &e

measures ASN and AI are >nown ascostmeasures since t&e) give an idea o t&e costinvolved+

Appendi. A=+1;* providing GINIA@ local macros* must be consulted or drawing

t&e ?C* A?Q and AI curves o a given single sampling plan+ It also s&ows &ow to

use a macro or drawing t&e curtailed ASN curve+

Acceptance Sampling-single Plan

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