Introduction to Acceptance Sampling by Moin

Introduction to Acceptance Sampling by Moin


Introduction to Acceptance Sampling

Presented by:

Chowdhury Jony Moin

ID:0409082120


• Acceptance Sampling (AS)

– Statistical quality control technique, where a random sample is taken from a lot, and upon the results of the sample taken the lot will either be rejected or accepted.

– This is also called Lot Acceptance Sampling.

– This is the “middle of the road” approach between no inspection and 100% inspection.

– This is popularized by Dodge and Romig.


The main outputs of AS• Accept Lot

– Ready for customers• Reject Lot

– Not suitable for customers• Statistical Process Control (SPC)

– Sample and determine if in acceptable limits

Note: Acceptance sampling plans do not improve quality. The most effective use of acceptance sampling is as an auditing tool to help ensure that the output of a process meets requirements.


Acceptance Sampling vs

Acceptance Quality Control by Harold Dodge in 1969

• Acceptance Sampling

Implemented in the form of specific sampling plan to indicate the conditions for reject or accept an inspected lot or batch.

• Acceptance Quality Control

Implemented in the form of Acceptance Quality Chart to compute specification limits and observation the standard deviation.


Acceptance Sampling vs

Acceptance Quality Control by Harold Dodge in 1969

• Acceptance Sampling

Example:

Acceptance sampling by Attributes,

Acceptance sampling by Variable.• Acceptance Quality Control

Example:

Control chart such as p chart, c chart etc.


When 100% inspection is not practiced

• When testing is destructive• When inspection is very costly• When many similar products are to be inspected.• When very high efforts required for testing.• When time & technology limitations are high.• When the population or lot size is very large &

probability of inspection error is high.• When the population is geographically scattered

over a large area.• Supplier’s quality history is good enough to justify

less than 100% inspection.


Types of Sampling Plan• There are two major classifications of acceptance plans: (1) Acceptance sampling by Attributes, in which the presence

or absence of a characteristic in the inspected item is only taken note of,

• Defectives-product acceptability across range• Defects-number of defects per unit• “go no-go” or similar to control chart for attributes

(2) Acceptance sampling by Variable , in which the presence or absence of a characteristic in the inspected item is measured on a predetermined scale.

• Usually measured by mean and standard deviation• Continuous or similar to control chart for variables.

Note: the attribute case is most common for acceptance sampling.


Types of Sampling Plan• There are varieties of sampling plan and the selection of

plan depends on type of product and production process.

Skip-lot sampling

Chain sampling

Acceptance sampling plans

Attribute sampling

Variable sampling

Sequential sampling

Continuous sampling

Double sampling

Multiple sampling

Single sampling

Special Attribute sampling


Factors For Designing the Plan• Acceptable Quality Level (AQL) = Max.

acceptable percentage of defectives defined by producer. Quality level of a good lot.

This is the poorest quality level of the supplier’s (or the producer’s) process that the customer to be consider to be acceptable as a process average.

So the producer would like to design a sampling plan such that there is a high probability of accepting a lot that has a defect a level less than or equal to the AQL.

Example: For apparel manufacturer AQL practiced 1.5%, 2.5% 4%, 6% and so on as per given by customer for particular style.


Factors For Designing the Plan• Lot Tolerance Percent Defective (LTPD) =

Limiting Quality Level (LQL) = Percentage of defectives that defines consumer’s rejection point.– Quality level of a bad lot.

This is the poorest quality that a customer is willing to tolerate in an individual lot.

The LTPD is a designated defect level for a lot beyond which the lot is unacceptable to the customer.

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.


Factors For Designing the Plan• Type I Error (Producer’s risk) = The probability of

rejecting a good lot. • The producer suffers when this occur because a lot

with acceptable quality was rejected.• This is the probability, for a given sampling plan, of

rejecting a lot that has a defect level less than or equal to the AQL.

• This happens mainly for sampling error.• Symbolic expression is α • Range from 0.2 to 0.01.


Factors For Designing the Plan• Type II Error (Consumer’s risk) =The probability

of accepting a bad lot.• The consumer suffers when this occur because a

lot with unacceptable quality was accepted.• This is the probability, for a given sampling plan, of

accepting a lot that has a defect level equal to or more than the LTPD.

• This happens mainly for sampling error.• Symbolic expression is β.• Range from 0.2 to 0.01.


Factors For Designing the Plan

• Type I Error (Producer’s risk) and• Type II Error (Consumer’s risk) both sampling

errors.

• Note: In statistics, sampling error is the error caused by observing a sample instead of the whole population.


Factors For Designing the Plan• Type of product to be inspected

There are two types of inspection for the product to be inspected.

1. Non destructive type of inspection

2. Destructive/expensive type of inspection• Type of inspection level

There are two types of inspection level.

1. General inspection level (Level I, II. III).

2. Special inspection level (Level S-1, S-2, S-3,S-4)

Note: Here inspection type no. 1 for inspection level no. 1 and no. 2 for 2.

*



The OC curve is the primary tool for displaying and investigating the properties of a LASP (Lot acceptance

sampling plan).

Operating Characteristic (OC) Curve: This curve plots the probability of accepting the lot (Y-axis) versus the lot fraction or percent defectives (X-axis).

**



• Operating Characteristic (OC) Curve: cont…..

The properties of a LASP (Lot acceptance sampling plan) are sample size (n) and acceptance number (c) which meet the performance requirements specified by the user.

The performance requirements are AQL, LOL or LTPD, α and β.

**


Factors For Designing the Plan• Average Outgoing Quality (AOQ):

A common procedure, when sampling and testing is non-destructive, is to 100% inspect rejected lots and replace all defectives with good units.

In this case, 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 of rejected lots process.


Factors For Designing the PlanAverage Outgoing Quality (AOQ): continue.....

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 pa of accepting such a lot, over the long run the AOQ can easily be shown to be:

Where, p = true defects level, N = lot size n = no. of product/units to be inspected,

c = acceptance number.


Factors For Designing the PlanAverage 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 number of defects 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 scheme.


Acceptance sampling by Attributes

• Attribute sampling is a statistical process.• A specific number of units from a lot or batch is

picked up for inspection.• The units are evaluated either conforming or

non-conforming.• If the number of non-conforming units is less

than a previously agreed number (known as acceptance number), then the lot or batch is accepted.

• Otherwise the lot is rejected.


Acceptance sampling by Attributes

• This process is basically statistically inferencing, where population characteristics are judged based on sample characteristics.

• The decision is largely influenced by the accuracy of the random sampling.

• Any sample error leads to business risk, either on producer or consumer.


Acceptance sampling by AttributesSingle sampling plan

• One sample of items is selected at random from a lot.

• 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 found by inspection.

• These are the most common (and easiest) plans to use although not the most efficient in terms of average number of samples needed.


Acceptance sampling by AttributesSingle sampling plan

• There are two widely used ways of picking (n,c):

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

**


Acceptance sampling by AttributesSingle sampling plan example:

Problem: A company produces a batch of 1000 product. An agreement between the producer and the customer specifics the followings:

• Order batch size, N:1000 units,• Sample size, n:30 units,• Acceptance number, c: 2

Plot the OC curve for this sampling plan.



Solution:For plotting the OC curve we required probability of

acceptance, pa values for different values of fraction nonconforming (p).

Following the binomial distribution, i is the number of nonconforming items, which can have a maximum value of c in order to accept the lot then probability of acceptance,



For fraction nonconforming value , p = 0.01 ►

= P(0 non conforming) + P(1 non conforming) + P(2 non conforming)

= [ 0.7397] + [ 0.22415] + [ 0.03283]= 0.99668



From the previous slide we get, For p = 0.01, Pa = 0.99668Similarly we get

Fraction nonconforming (p)

Probability of acceptance, Pa

0.01 0.996680.03 0.939910.05 0.812170.07 0.648730.09 0.485510.11 0.344190.13 0.232950.15 0.151390.17 0.09485

00.

020.

040.

060.

08 0.1

0.12

0.14

0.16

0.18

0

0.2

0.4

0.6

0.8

1

1.2

O C curve


Prob

abili

ty o

f acc

epta

nce,

Pa


Acceptance sampling by AttributesDouble sampling plan

• A double sampling plan provides additional cushion to the producer, as producer’s loss is more serious than that of a buyer.

• In case the lot is rejected in the first trial, another chance is given to the producer for inspection in the second trial.

• However, the sample size in the second trial may not be equal to that in the first trial.




Flowchart of the double sampling plan

Sample of size n1 is drawn

Is D1≤ c1 ? Accept the lot

Is D1> c2 ? Reject the lot

Sample of size n2 is drawn

Is (D1+D2) > c2? Reject the lot

Accept the lot

Y

Y

N

Y

N

N

Here:

n1 = Sample size in the first trial

n2 = Sample size in the second trial

c1 = Acceptance number for the first trial

c2 = Acceptance number for both trial together

D1 = Number of nonconforming units for the first trial.

D2 = Number of nonconforming units for the first trial.



Acceptance sampling by AttributesDouble sampling plan Procedure:

In the first trial, sample of size n1 is taken and inspected from the lot or batch of size N.

If the number of nonconforming units(D1) is less than or equal to acceptance number c1, then the entire lot is accepted and no second trial.

If D1 is greater than acceptance number c2, then the entire lot is rejected and no second trial.

If D1 is greater than c1 but less or equal to c2 then second sample of size n2 is taken.

If D1+D2(nonconforming units for second trial) is less than or equal c2 then the lot is accepted.




• Suppose:

• N = Lot size to be inspected =3000 units

• n1 = Sample size in the first trial = 40 units

• n2 = Sample size in the second trial = 80 units

• c1 = Acceptance number for the first trial = 1 unit

• c2 = Acceptance number for both trial = 4 units (both trials together)

• D1 = Number of nonconforming units for the first trial.

• D2 = Number of nonconforming units for the first trial.



Acceptance sampling by AttributesDouble sampling plan example

Solution:If the number of nonconforming units; D1≥ 1, then the

entire lot is accepted and no second trial.If the number of nonconforming units;D1>4, then the

entire lot is rejected and no second trial.If the number of nonconforming units; 1>D1≥4 then

second sample of size n2 is taken. Then If D1+D2(nonconforming units for second trial) is less

than or equal 4 then the lot is accepted.• If D1+D2(nonconforming units for second trial) is

greater than 4 then the lot is rejected.




• In order to plot OC curve of the previous problem, Let,

• Pa1 = Probability of acceptance in the first trial

• Pa11 =Probability of acceptance in the second trial

• P = Total probability of acceptance in the combined first & second trial = Pa

1 + Pa11




Now the lot is accepted after the first trial if D1 = 0, or 1 or 2. Thus for fraction nonconforming value, p = 0.01 ►

= P(0 non conforming) + P(1 non conforming) + P(2 non conforming)

• =0.9925 (by MATHLAB 7.8.0; binocdf(2,40,0.01))




From the previous slide we get, For p = 0.01, Pa = 0.9925Similarly we get


Prob

abili

ty o

f acc

epta

nce,

Pa1


Probability of acceptance, Pa1

0.01 0.99250.03 0.88220.05 0.676730.07 0.462520.09 0.289420.11 0.168830.13 0.09290.15 0.04860.17 0.02429

00.

020.

040.

060.

08 0.1

0.12

0.14

0.16

0.18

0

0.2

0.4

0.6

0.8

1

1.2

O C curve



• If D1 > 4, then the lot is rejected & no second trial.

• But if D1≤4, then second trial is taken and hence the probable situations for D1 & D2 for accepting the lot are

• D1=3 & D2 = 0 or,• D1=3 & D2 = 1 or ,• D1=4 & D2 = 0


• So situations for Pa11 =Probability of

acceptance in the second trial are

• For D1=3 & D2 = 0, Pa11 (D1=3,D2=0) or

• For D1=3 & D2 = 1 Pa11 (D1=3,D2=1) or

• For D1=4 & D2 = 0 Pa11 (D1=4,D2=0)




Pa11 (D1=4,D2=0)

= binopdf(4,40,0.010) x binopdf(0,80,0.010)

= 0.000285

Pa11 (D1=3,D2=1)


= 0.002387

Pa11 (D1=3,D2=0)


= 0.0068 x 0.4475 = 0.003043

Here.


• Thus for p = 0.01►• Pa

11 = 0.003043 +0.002387 +0.000285 =0.005714

• So Pa = Pa1 + Pa

11 = 0.9925 + 0.005714 =0.998214

• Similarly we can get Pa for different p .





Probability of acceptance Pa1 Probability of acceptance, Pa

0.01 0.9925 0.99830.03 0.8822 0.91060.05 0.67673 0.69410.07 0.46252 0.46790.09 0.28942 0.29060.11 0.16883 0.1690.13 0.0929 0.09290.15 0.0486 0.04860.17 0.02429 0.0243




Prob

abili

ty o

f acc

epta

nce,

Pa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.2

0.4

0.6

0.8

1

1.2

Pa1

Pa

OC curve for Double Sampling Plan


References:• Quality Control and Management

By Dr. M. Ahsan Akhter Hasin, Professor, IPE Dept., BUET.

• * M.E. Davis . The retailer’s quality requirements, Textile Institute and Industry, May 1977.

• * An Introduction to Quality Control for the Apparel Industry ,By Pradip V. Mehta

• ** Production and Operations Management, fifth edition, By Everett E. Adam Page-653-654

Documents

Introduction to Acceptance Sampling by Moin