Process or Product Monitoring and Control_2 pptx

in the use of process quality control by: varying the quantity andseverity of acceptance inspections in direct relation to the importanceof the characteristics inspected, and in the inve

6 Process or Product Monitoring and Control

6.2 Test Product for Acceptability: Lot

Acceptance Sampling

This section describes how to make decisions on a lot-by-lot basiswhether to accept a lot as likely to meet requirements or reject the lot aslikely to have too many defective units

Contents of

section 2

This section consists of the following topics

What is Acceptance Sampling?

1

What kinds of Lot Acceptance Sampling Plans (LASPs) arethere?

2

How do you Choose a Single Sampling Plan?

Choosing a Sampling Plan: MIL Standard 105D

6 Process or Product Monitoring and Control

6.2 Test Product for Acceptability: Lot Acceptance Sampling

6.2.1 What is Acceptance Sampling?

called Lot Acceptance Sampling or just Acceptance Sampling.

acceptance plans: by attributes ("go, no-go") and by variables The

attribute case is the most common for acceptance sampling, and will

be assumed for the rest of this section

Important

point

A point to remember is that the main purpose of acceptance sampling

is to decide whether or not the lot is likely to be acceptable, not toestimate the quality of the lot

in the use of process quality control by: varying the quantity andseverity of acceptance inspections in direct relation to the importance

of the characteristics inspected, and in the inverse relation to thegoodness of the quality level as indication by those inspections."

To reiterate the difference in these two approaches: acceptancesampling plans are one-shot deals, which essentially test short-runeffects Quality control is of the long-run variety, and is part of awell-designed system for lot acceptance

According to the ISO standard on acceptance control charts (ISO

7966, 1993), an acceptance control chart combines consideration ofcontrol implications with elements of acceptance sampling It is anappropriate tool for helping to make decisions with respect to processacceptance The difference between acceptance sampling approachesand acceptance control charts is the emphasis on process acceptabilityrather than on product disposition decisions

6.2.1 What is Acceptance Sampling?

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6 Process or Product Monitoring and Control

6.2 Test Product for Acceptability: Lot Acceptance Sampling

6.2.2 What kinds of Lot Acceptance

Sampling Plans (LASPs) are there?

LASP is a

sampling

scheme and

a set of rules

A lot acceptance sampling plan (LASP) is a sampling scheme and a set

of rules for making decisions The decision, based on counting thenumber of defectives in a sample, can be to accept the lot, reject the lot,

or even, for multiple or sequential sampling schemes, to take anothersample and then repeat the decision process

Types of

acceptance

plans to

choose from

LASPs fall into the following categories:

Single sampling plans: One sample of items is selected at

random from a lot and the disposition of the lot is determinedfrom the resulting information These plans are usually denoted as

(n,c) plans for a sample size n, where the lot is rejected if there are more than c defectives These are the most common (and

easiest) plans to use although not the most efficient in terms of average number of samples needed.

If the outcome is (3), and a second sample is taken, the procedure

is to combine the results of both samples and make a finaldecision based on that information

●

Multiple sampling plans: This is an extension of the double

sampling plans where more than two samples are needed to reach

a conclusion The advantage of multiple sampling is smallersample sizes

●

Sequential sampling plans: This is the ultimate extension of

multiple sampling where items are selected from a lot one at atime and after inspection of each item a decision is made to accept

or reject the lot or select another unit

fraction of the submitted lots are inspected.

Deriving a plan, within one of the categories listed above, is discussed

in the pages that follow All derivations depend on the properties youwant the plan to have These are described using the following terms:

Acceptable Quality Level (AQL): The AQL is a percent defective

that is the base line requirement for the quality of the producer'sproduct The producer would like to design a sampling plan such

that there is a high probability of accepting a lot that has a defect

level less than or equal to the AQL

●

Lot Tolerance Percent Defective (LTPD): The LTPD is a

designated high defect level that would be unacceptable to theconsumer The consumer would like the sampling plan to have a

low probability of accepting a lot with a defect level as high as

the LTPD

●

Type I Error (Producer's Risk): This is the probability, for a

given (n,c) sampling plan, of rejecting a lot that has a defect level

equal to the AQL The producer suffers when this occurs, because

a lot with acceptable quality was rejected The symbol iscommonly used for the Type I error and typical values for range from 0.2 to 0.01

●

Type II Error (Consumer's Risk): This is the probability, for a

given (n,c) sampling plan, of accepting a lot with a defect level

equal to the LTPD The consumer suffers when this occurs,because a lot with unacceptable quality was accepted The symbol

is commonly used for the Type II error and typical values rangefrom 0.2 to 0.01

●

Operating Characteristic (OC) Curve: This curve plots the

probability of accepting the lot (Y-axis) versus the lot fraction or

percent defectives (X-axis) The OC curve is the primary tool for

displaying and investigating the properties of a LASP.

●

Average Outgoing Quality (AOQ): A common procedure, when

sampling and testing is non-destructive, is to 100% inspectrejected lots and replace all defectives with good units In thiscase, all rejected lots are made perfect and the only defects left

are those in lots that were accepted AOQ's refer to the long term

defect level for this combined LASP and 100% inspection ofrejected lots process If all lots come in with a defect level of

exactly p, and the OC curve for the chosen (n,c) LASP indicates a probability p a of accepting such a lot, over the long run the AOQ

can easily be shown to be:

●

6.2.2 What kinds of Lot Acceptance Sampling Plans (LASPs) are there?

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where N is the lot size.

Average Outgoing Quality Level (AOQL): A plot of the AOQ

(Y-axis) versus the incoming lot p (X-axis) will start at 0 for p =

0, and return to 0 for p = 1 (where every lot is 100% inspected

and rectified) In between, it will rise to a maximum This

maximum, which is the worst possible long term AOQ, is called the AOQL.

●

Average Total Inspection (ATI): When rejected lots are 100%

inspected, it is easy to calculate the ATI if lots come consistently with a defect level of p For a LASP (n,c) with a probability p a of

accepting a lot with defect level p, we have

ATI = n + (1 - p a ) (N - n)

where N is the lot size.

●

Average Sample Number (ASN): For a single sampling LASP

(n,c) we know each and every lot has a sample of size n taken and

inspected or tested For double, multiple and sequential LASP's,the amount of sampling varies depending on the the number ofdefects observed For any given double, multiple or sequential

plan, a long term ASN can be calculated assuming all lots come in with a defect level of p A plot of the ASN, versus the incoming defect level p, describes the sampling efficiency of a given LASP

knowing how much sampling and inspection will be done on aday-by-day basis

6.2.2 What kinds of Lot Acceptance Sampling Plans (LASPs) are there?

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

A single sampling plan, as previously defined, is specified by the pair of

numbers (n,c) The sample size is n, and the lot is rejected if there are more than c defectives in the sample; otherwise the lot is accepted There are two widely used ways of picking (n,c):

Use tables (such as MIL STD 105D) that focus on either the AQL

or the LTPD desired

1

Specify 2 desired points on the OC curve and solve for the (n,c)

that uniquely determines an OC curve going through these points

2

The next two pages describe these methods in detail

6.2.3 How do you Choose a Single Sampling Plan?

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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?

6.2.3.1 Choosing a Sampling Plan: MIL

acceptance at the AQL On the other side of the OC curve, the consumer

wishes to be protected from accepting poor quality from the producer

So the consumer establishes a criterion, the lot tolerance percent

defective or LTPD Here the idea is to only accept poor quality productwith a very low probability Mil Std plans have been used for over 50years to achieve these goals

The U.S Department of Defense Military Standard 105E

on attribute sampling plans

These three streams combined in 1950 into a standard called Mil Std.105A It has since been modified from time to time and issued as 105B,195C and 105D Mil Std 105D was issued by the U.S government in

1963 It was adopted in 1971 by the American National StandardsInstitute as ANSI Standard Z1.4 and in 1974 it was adopted (with minorchanges) by the International Organization for Standardization as ISOStd 2859 The latest revision is Mil Std 105E and was issued in 1989.These three similar standards are continuously being updated andrevised, but the basic tables remain the same Thus the discussion thatfollows of the germane aspects of Mil Std 105E also applies to the

6.2.3.1 Choosing a Sampling Plan: MIL Standard 105D

other two standards.

a reduced sampling plan plus rules for switching from one to the other

AQL is

foundation

of standard

The foundation of the Standard is the acceptable quality level or AQL In

the following scenario, a certain military agency, called the Consumerfrom here on, wants to purchase a particular product from a supplier,called the Producer from here on

In applying the Mil Std 105D it is expected that there is perfect

agreement between Producer and Consumer regarding what the AQL is

for a given product characteristic It is understood by both parties thatthe Producer will be submitting for inspection a number of lots whosequality level is typically as good as specified by the Consumer

Continued quality is assured by the acceptance or rejection of lotsfollowing a particular sampling plan and also by providing for a shift toanother, tighter sampling plan, when there is evidence that the

Producer's product does not meet the agreed-upon AQL.

Inspection

level

In addition to an initial decision on an AQL it is also necessary to decide

on an "inspection level" This determines the relationship between thelot size and the sample size The standard offers three general and fourspecial levels

6.2.3.1 Choosing a Sampling Plan: MIL Standard 105D

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Steps in the

standard

The steps in the use of the standard can be summarized as follows:

Decide on the AQL.

6.2.3.1 Choosing a Sampling Plan: MIL Standard 105D

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?

6.2.3.2 Choosing a Sampling Plan with a

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 size N is 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 how many defects are in the sample

Then the distribution of the number of defectives, d, in a random sample of n items is approximately binomial with parameters n and p, where p is the fraction of defectives per lot.

The probability of observing exactly d defectives is given by

the probability of acceptance is 1- for lots with fraction defective p1

and the probability of acceptance is for lots with fraction defective

p2 Typical choices for these points are: p 1 is the AQL , p 2 is the LTPD

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 c are the solution to

These two simultaneous equations are nonlinear so there is no simple,direct solution There are however a number of iterative techniquesavailable that give approximate solutions so that composition of acomputer program poses few problems

Average Outgoing Quality (AOQ)

Calculating

AOQ's

We can also calculate the AOQ for a (n,c) sampling plan, provided

rejected lots are 100% inspected and defectives are replaced with goodparts

Assume all lots come in with exactly a p0 proportion 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 defective p0 and 0 Assuming the lot size is N, we

have

For example, let N = 10000, n = 52, c = 3, and p, the quality of incoming lots, = 0.03 Now at p = 0.03, we glean from the OC curve table that p a = 0.930 and

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

6.2.3.2 Choosing a Sampling Plan with a given OC Curve

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Sample plot

of AOQ

versus p

A plot of the AOQ versus p is given below.

6.2.3.2 Choosing a Sampling Plan with a given OC Curve

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 theoutgoing quality is also very good (very small fraction of defectivesgoing out) When the incoming lot quality is very bad, most of the lotsare rejected and then inspected The "duds" are eliminated or replaced

by good ones, so that the quality of the outgoing lots, the AOQ, becomes very good In between these extremes, the AOQ rises, reaches

a maximum, and then drops

The maximum ordinate on the AOQ curve represents the worst

possible quality that results from the rectifying inspection program It

is called the average outgoing quality limit, (AOQL ).

From the table we see that the AOQL = 0.0372 at p = 06 for the above

example

One final remark: if N >> n, then the AOQ ~ p a p

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 is N.

Finally, if the lot quality is 0 < p < 1, the average amount of inspection per lot will vary between the sample size n, and the lot size N.

Let the quality of the lot be p and the probability of lot acceptance be

p a , then the ATI per lot is