Choosing a Sampling Plan With Given OCR

8/10/2019 Choosing a Sampling Plan With Given OCR

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6.Process or Product Monitoring and Control

6.2.Test Product for Acceptability: Lot Acceptance Sampling

6.2.3.How do you Choose a Single Sampling Plan?

Sample

OC

curve

We start by looking at a typical OC curve. The OC curve for a (52 ,3)

sampling plan is shown below.

Number of

defectives is

approximately

binomial

It is instructive to show how the points on this curve are

obtained, once we have a sampling plan (n,c) - later we will

demonstrate how a sampling plan (n,c) is obtained.

We assume that the lot sizeNis very large, as compared to

the sample size n, so that removing the sample doesn't

significantly change the remainder of the lot, no matter howmany defects are in the sample. Then the distribution of the

number of defectives, d, in a random sample ofnitems is

approximately binomial with parameters nandp, wherepis

the fraction of defectives per lot.

The probability of observing exactly d defectives is given

by


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The binomial

distribution

The probability of acceptance is the probability thatd, the

number of defectives, is less than or equal to c, the accept

number. This means that

Sample table

for Pa, Pd

using the

binomial

distribution

Using this formula with n = 52 and c=3 andp= .01, .02,

...,.12 we find

Pa

Pd

.998 .01

.980 .02

.930 .03

.845 .04

.739 .05

.620 .06

.502 .07

.394 .08

.300 .09

.223 .10

.162 .11

.115 .12

Solving for (n,c)

Equations for

calculating a

sampling plan

with a given

OC curve

In order to design a sampling plan with a specified OC

curve one needs two designated points. Let us design a

sampling plan such that the probability of acceptance is 1-

for lots with fraction defectivep1and the probability of

acceptance is for lots with fraction defectivep2. Typical

choices for these points are:p1is theAQL,p2is theLTPD

and , are the Producer's Risk (Type I error)and

Consumer's Risk (Type II error), respectively.

If we are willing to assume that binomial sampling is valid,

then the sample size n, and the acceptance number care the

solution to


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These two simultaneous equations are nonlinear so there isno simple, direct solution. There are however a number of

iterative techniques available that give approximate

solutions so that composition of a computer program poses

few problems.

Average Outgoing Quality (AOQ)

Calculating

AOQ's

We can also calculate theAOQfor a (n,c) sampling plan,

provided rejected lots are 100% inspected and defectives are

replaced with good parts.

Assume all lots come in with exactly ap0proportion of

defectives. After screening a rejected lot, the final fraction

defectives will be zero for that lot. However, accepted lots

have fraction defectivep0. Therefore, the outgoing lots from

the inspection stations are a mixture of lots with fractions

defectivep0and 0. Assuming the lot size isN, we have.

For example, letN= 10000, n = 52, c= 3, andp, the quality

of incoming lots, = 0.03. Now atp= 0.03, we glean from

the OC curve table thatpa= 0.930 and

AOQ= (.930)(.03)(10000-52) / 10000 = 0.02775.

Sample table

of AOQversus p

Settingp = .01, .02, ..., .12, we can generate the following

table

AOQ p

.0010 .01

.0196 .02

.0278 .03

.0338 .04

.0369 .05


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.0372 .06

.0351 .07

.0315 .08

.0270 .09

.0223 .10

.0178 .11

.0138 .12

Sample plot

of AOQ

versus p

A plot of theAOQversuspis given below.

Interpretation

of AOQ plot

From examining this curve we observe that when the

incoming quality is very good (very small fraction of

defectives coming in), then the outgoing quality is also very

good (very small fraction of defectives going out). When

the incoming lot quality is very bad, most of the lots are

rejected and then inspected. The "duds" are eliminated or

replaced by good ones, so that the quality of the outgoing

lots, theAOQ, becomes very good. In between these

extremes, theAOQrises, reaches a maximum, and then

drops.

The maximum ordinate on theAOQcurve represents theworst possible quality that results from the rectifying

inspection program. It is called the average outgoing

quality limit,(AOQL).

From the table we see that theAOQL= 0.0372 atp= .06 for

the above example.

One final remark: ifN>> n, then theAOQ~pap .


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The Average Total Inspection (ATI)

Calculating

the Average

Total

Inspection

What is the total amount of inspection when rejected lots

are screened?

If all lots contain zero defectives, no lot will be rejected.

If all items are defective, all lots will be inspected, and the

amount to be inspected isN.

Finally, if the lot quality is 0


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Plot of ATI

versus p

A plot ofATI versusp,the Incoming Lot Quality (ILQ) is

given below.

Choosing a Sampling Plan With Given OCR

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