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Acceptance Sampling by Attributes : Multiple

Fraction – Defraction Sampling Plans

1. METHDS F DES!"#$#%& M'(T#P(E – SAMP(#%& P(A%S

If we FOLLOW the Wald approach to item – by – item sampling and the “group

sequential” sampling devired therefrom ! multiple – sampling plan will be described in

terms of acceptance and re"ection boundaries for accmulated groups of samples #uch a

description of a multiple – sampling plan will run li$e this %

Cumulative Sample Size

Acceptance Number

Rejection Number

&'

('

)'

*'

+''

+&'

+('

'

+

,

-

*

.

+'

(

-

)

*

+'

++

++

/he plan will operate as follows If at the completion of any stage the number of

defective items equals or fall below the acceptance number the lot is accepted If during

any stage the number of defective items equals or e0eceeds the re"ection number the lot is

re"ected Otherwise anoter sample ta$en /his multiple decision procedure continues untilthe seventh sample is ta$en when a decision to accept or re"ect must be made /he first

sample is usually inspected +'' percent for the sa$e of the record but inspection is often

stopped as soon as the re"ection number is reached in any stage subsequent to first

1nter and 2ama$er following the lead of 3arnard have developed an approach

to multiple sampling that is thought by some to be administratively simpler than that

following the Wald line of development /he 3arnard – 1nter – 2ama$er way of

describing a multiple – sampling plan runs as follows #tars with an initial score 1. !dd S

for each group of and items inspected and subtract the number of defective items found

!ccept the lot if at any stage the cumulated score reaches or e0ceeds 4 re"ect the lot if the

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cumulated score becomes ' or less !t the end of m stage accept the lot if the final score

is greater than the initial score 4 otherwise re"ecy the lot /he multiple – sampling plan

M ( I = 2.S = 2.A = 4.M = 8

will thus have the following scoring system #tart with a score of &5 add & for each groupinspected and subtract the number of defectives found in the group /he lot is accepted

when the total score reaches ( 4 it is re"ected when the total score becomes ' or less !t

the end of * groups if the score is greater than & accept 4 otherwise re"ect

It is claimed that inspector find the 3arnad – 1nter – 2ama$er way of

describing a plan easier to follow than one prescribing a series of acceptance and

re"ection numbers It should be noted however that this simpler method of describing a

multiple – sampling plan can readily be converted into the more conventional form /hus

the above multiple sampling plan 6 7 &5&5(5* 8 can be rewritten as follows %

Sample Size Number Rejection

Number

n...............................

2n...............................

!n...............................

4n...............................

"n...............................

#n...............................

$n...............................

8n...............................

'

&

(

)

*

+'

+&

+-

(

)

*

+'

+&

+(

+)

+)

). !MP'TAT#% F THE ! !'"*E F" A M'(T#P(E – SAMP(#%&

P(A%

/he following is an e0planations of the computation of a /ype 3 or /ype ! O9 curve

for a multiple – sampling plan described in the conventional manner /he process is

simple enough and can be done readily if an orderly procedure is followed /he step

consist in calculating the probability of accepting5 re"ecting5 and continuing to sample

at each stage in the plan We shall illustrate the plan of figure .+

Step. 1. :ote that ;< is to be calculated at p= > ''- For this value of p= write on a

strip of paper the probabilities of e0actly ' out of &' 7> ',-*-8 + out of &' 7>',??(8

& out of &' 7>'+**?8 and , out of &' 7>''-.)84 the probabilities of + or less out of

&' 7>'?,-.8 and & or less out of &' 7>'.&()8 7all evaluated at p' >''-8 /hese may

be found in tabels of the binominal probability distribution

Step. 2. #et up a wor$ sheet such as shown in figure .+ with heavy lines drawn

around the acceptance values and re"ection values 3e sure there is enough room in

each cell to write in as many numbers as the diference betwen the acceptance andre"ection numbers

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Step. +. For the first stage write in the probabilities of the result that lead to

acceptance re"ection and continuance of sampling :ote that the value written above

the re"ection limit is the probability of equaling or e0ceeding the re"ection number and

that written below the acceptance limit is the probability of equaling or falling below

the acceptance number

Step. 4. For each of the result at the first stage that lead to a second sample

compute the probability of acceptance and re"ection at the second stage and also the

probability of each of the result that will lead to third stage /o carry out these

computations it was found easiest to set the probability of a0actly + in the first sample

7',??(8 in the computing machine as a multiplicand and to multiply successively by

the probability of e0actly ' the probability of e0actly + 7giving a total of & at the

second stage8 the probability of e0actly & 7giving a total of e0actly (8 and the

probability of ( or more /hus ',??( @ ',-*- > '+,-, which is the probability of

e0actly + at the second stage and hence equal the probability of acceptance at thesecond stage !gain ',??( @ ',??( > '+(&( which is the probability of e0actly & at

the second stage 4 ',??( @ o+**? > ''?+& which is the probability of e0actly , at the

second stage 4 ',??( @ ''-.) > ''&&- which is the probability of e0actly ( at the

second stage 4 and ',??( @ ''+-. > ''')' which is the probability of - or more at

the second stage – all starting with e0actly one at the first stage :e0t we ta$e '+**?

and multiply it successively by ',-*- 5 ',??( 5 '+**? and

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