Acceptance SamplingAcceptance Sampling

• Lot-by-lot Acceptance Sampling by AttributesLot-by-lot Acceptance Sampling by Attributes• Acceptance Sampling SystemsAcceptance Sampling Systems

LOT-BY-LOT ACCEPTANCE SAMPLING BY ATTRIBUTESLOT-BY-LOT ACCEPTANCE SAMPLING BY ATTRIBUTESAcceptance Sampling is a form of inspection applied toAcceptance Sampling is a form of inspection applied tolots or batches of items before or after a process to lots or batches of items before or after a process to

judgejudgeconformance to predetermined standards.conformance to predetermined standards.

Acceptance Sampling is very useful when:Acceptance Sampling is very useful when:1. When the test is destructive.1. When the test is destructive.2. When automated inspection is not available. 2. When automated inspection is not available. 3. When the cost of 100% inspection is high.3. When the cost of 100% inspection is high.4. Fatigue/boredom is caused by inspecting large4. Fatigue/boredom is caused by inspecting large numbers of items.numbers of items.5. When information concerning producers quality not5. When information concerning producers quality not know/available. know/available.

AdvantagesAdvantages1. Less expensive.1. Less expensive.2. Reduced damage.2. Reduced damage.3. Reduces the amount of inspection error.3. Reduces the amount of inspection error.

DisadvantagesDisadvantages1. Risk of accepting “bad” lots and rejecting “good” lots.1. Risk of accepting “bad” lots and rejecting “good” lots.2. Less information generated.2. Less information generated.3. Requires planning and documentation.3. Requires planning and documentation.

Sampling PlansSampling Plans• Sampling PlansSampling Plans specify the lot size, sample size, specify the lot size, sample size,

number ofnumber of

samples and acceptance/rejection criteria.samples and acceptance/rejection criteria.• Sampling plans involve:Sampling plans involve:

1. Single sampling1. Single sampling

2. Double sampling2. Double sampling

3. Multiple sampling3. Multiple sampling• Define by,Define by, N = lot size N = lot size

n = sample size n = sample size

c = acceptance numberc = acceptance number• If c or less non-conforming units are found in the If c or less non-conforming units are found in the

sample, the lot is accepted, else it is rejectedsample, the lot is accepted, else it is rejected..

Single Sampling PlanSingle Sampling Plan• A A representative sample of n itemsrepresentative sample of n items is drawn from a is drawn from a

lot size of N itemslot size of N items.. • Each item in the Each item in the sample is examinedsample is examined and and classified asclassified as

good/defectivegood/defective..• If the number of defective exceeds a specified If the number of defective exceeds a specified

rejection number (c) the whole lot is rejected; rejection number (c) the whole lot is rejected; otherwise the whole lot is accepted.otherwise the whole lot is accepted.

• If If Single Sampling PlanSingle Sampling Plan, , N = 9000N = 9000, , n = 300n = 300, , c = 2c = 2..

- Interpretation of Plan means, inspect 300 pieces - Interpretation of Plan means, inspect 300 pieces fromfrom

the lot of 9000 units.the lot of 9000 units.

- If nonconforming found - If nonconforming found ≤ c=2≤ c=2 then then ACCEPT LOTACCEPT LOT..

- If nonconforming found - If nonconforming found > c=2> c=2 then then REJECT LOTREJECT LOT..

Double Sampling PlanDouble Sampling Plan• A A Double Sampling PlanDouble Sampling Plan allows to take a second sample if the allows to take a second sample if the

results of the original sample are inconclusive.results of the original sample are inconclusive. • Define by, N = lot size n1 = sample size (1st sample)

c1 = acceptance number (1st sample)

r1 = non-acceptance number (1st sample)

n2 = sample size (2nd sample)

c2 = acceptance number (2nd sample)

r2 = non-acceptance number (2nd sample)

• If values are not given for r1 and r2, they are equal to c2 + 1.

• Example Double Sampling Plan, N = 9000

n1 = 60 n2 = 150

c1 = 1 c2 = 6

r1 = 5 r2 = 7

• An initial sample (n1 ) of 60 is selected from the lot (N) of 9000 and inspected. One of the following judgments is made:1. If, 1 or fewer nonconforming units (c1), the lot accepted.

2. If, 5 or more nonconforming units (r1), the lot not accepted.

3. If, 2,3 and 4 nonconforming units, no decision is made and a second sample is taken.• A second sample of 150 (n2) from the lot (N) is inspected,

and one of the following judgments is made:1. If there are 6 or fewer nonconforming units (c1. If there are 6 or fewer nonconforming units (c22) in both ) in both sample, the lot is accepted. This number is obtained by 2 in the sample, the lot is accepted. This number is obtained by 2 in the 11stst sample and 4 or fewer in the 2 sample and 4 or fewer in the 2ndnd sample, by 3 in the 1 sample, by 3 in the 1stst sample and 3 or fewer in the 2sample and 3 or fewer in the 2ndnd sample, or by 4 in the 1 sample, or by 4 in the 1stst sample and 2 or fewer in the 2sample and 2 or fewer in the 2ndnd sample. sample.

2. If there are 7 or more nonconforming units (2. If there are 7 or more nonconforming units (r2) in both ) in both sample, the lot is not accepted. This number is obtained by 2 in sample, the lot is not accepted. This number is obtained by 2 in the 1the 1stst sample and 5 or more in the 2 sample and 5 or more in the 2ndnd sample, by 3 in the 1 sample, by 3 in the 1stst sample and 4 or more in the 2sample and 4 or more in the 2ndnd sample, or by 4 in the 1 sample, or by 4 in the 1stst sample sample and 3 or more in the 2and 3 or more in the 2ndnd sample. sample.

Multiple Sampling PlanMultiple Sampling Plan• A A Multiple Sampling PlanMultiple Sampling Plan is similar to the double sampling is similar to the double sampling

plan in that successive trials are made, each of which has plan in that successive trials are made, each of which has acceptance, rejection and inconclusive options.acceptance, rejection and inconclusive options.

• Which Plan you choose depends on: Which Plan you choose depends on:

- Cost and time - Cost and time

- Number of samples needed and number of items in each- Number of samples needed and number of items in each

samplesample

Lot FormationLot FormationConsideration before inspection:Consideration before inspection:

1. Lots should be 1. Lots should be homogeneoushomogeneous (same machine, same (same machine, same operator, same input material, and so on).operator, same input material, and so on).

2. 2. Larger lotsLarger lots are more preferable than smaller lots. are more preferable than smaller lots.

3. Lots should be 3. Lots should be conformableconformable to the materials-handling to the materials-handling systems used in both the systems used in both the vendorvendor and and consumer facilitiesconsumer facilities..

Statistical AspectsStatistical AspectsThe The Operating Characteristic CurveOperating Characteristic Curve::• Measures the performance of an acceptance sampling plan.Measures the performance of an acceptance sampling plan.• Plots the probability of accepting the lot versus the lot Plots the probability of accepting the lot versus the lot

fraction defective.fraction defective.• Shows the probability that a lot submitted with a certain Shows the probability that a lot submitted with a certain

fraction defective will be either accepted or rejected.fraction defective will be either accepted or rejected.• This curve plots the probability of accepting the lot (Y-axis) This curve plots the probability of accepting the lot (Y-axis)

versus the lot fraction or percent defectives (X-axis). versus the lot fraction or percent defectives (X-axis). The OC The OC curve is the primary tool for displaying and investigating the curve is the primary tool for displaying and investigating the properties of a Lot Acceptance Sampling Planproperties of a Lot Acceptance Sampling Plan. .

• There are There are two types of OC curvestwo types of OC curves::

- - Type AType A - Gives the probability of acceptance of an - Gives the probability of acceptance of an

individual lot coming from finite productionindividual lot coming from finite production

- - Type BType B - Gives the probability of acceptance for lots coming - Gives the probability of acceptance for lots coming

from a continuous productionfrom a continuous production

OC Curve for Single Sampling PlanOC Curve for Single Sampling Plan• Example, N=3000, n = 89 and c = 2.Example, N=3000, n = 89 and c = 2.

1. Assume p1. Assume poo,,

100p100poo = 2% = 2%

ppo o = 0.02= 0.02

2. Calculate, nppo o = (89)(0.02) = (89)(0.02)

= 1.8= 1.8

3. Find Pa = P(0) + P(1) + P(2) = P(2) or less3. Find Pa = P(0) + P(1) + P(2) = P(2) or less

= 0.731 or 100Pa= 0.731 or 100Pa

= 73.1%= 73.1%

******Pa value obtained from Poisson Table page 515. Pa value obtained from Poisson Table page 515. LookLook

at npat npoo = 1.8 and c = 2 = 1.8 and c = 2

4. Plot point (100p4. Plot point (100poo, 100Pa), 100Pa)

• A table can be assist with the calculations, as shown in A table can be assist with the calculations, as shown in Table 9-2. Table 9-2.

• The curve is terminated when the Pa value is close to 0.05. Since Pa=0.055 for 100ppoo = 7%, it is not necessary to make any calculations for values greater than 7%.

• In this CASE 7 points enough to plot the curve.

• Thus, if the incoming process quality is 2.3% nonconforming, the Thus, if the incoming process quality is 2.3% nonconforming, the percent of the lots that are expected is 66%. Similarly, if 55 lots percent of the lots that are expected is 66%. Similarly, if 55 lots from a process that is 2.3% nonconforming are inspected using from a process that is 2.3% nonconforming are inspected using this sampling plan, 36 [(55)(0.66)=36] will be accepted and 19 this sampling plan, 36 [(55)(0.66)=36] will be accepted and 19 [55-36=19] will be unaccepted. [55-36=19] will be unaccepted.

• When, % nonconforming low, Pa is largeWhen, % nonconforming low, Pa is large

% nonconforming high, Pa is small% nonconforming high, Pa is small

Average Outgoing QualityAverage Outgoing Quality• A common procedure, when sampling and testing is non-A common procedure, when sampling and testing is non-

destructive, is to 100% inspect rejected lots and replace all destructive, is to 100% inspect rejected lots and replace all defectives with good units. In this case, all rejected lots are defectives with good units. In this case, all rejected lots are made perfect and the only defects left are those in lots made perfect and the only defects left are those in lots that were accepted. that were accepted.

• The Average Outgoing Quality is the average of rejected The Average Outgoing Quality is the average of rejected lots (100% inspection) and accepted lots ( a sample of lots (100% inspection) and accepted lots ( a sample of items inspected)items inspected)

ac

ac

N - nAOQ = P * p( ) where

NP = Probability of accepting a lot

p = Fraction defective

n = sample size

N = Lot size

Average Total InspectionAverage Total Inspection• When rejected lots are 100% inspected, it is easy to When rejected lots are 100% inspected, it is easy to

calculate the ATI if lots come consistently with a defect calculate the ATI if lots come consistently with a defect level of p. For a LASP (n,c) with a probability pa of level of p. For a LASP (n,c) with a probability pa of accepting a lot with defect level p, we have:accepting a lot with defect level p, we have:

ATI = n + (1 - pa) (N - n) ; where N is the lot size. ATI = n + (1 - pa) (N - n) ; where N is the lot size.

Average Sample NumberAverage Sample Number• For a single sampling (n,c) we know each and every lot has For a single sampling (n,c) we know each and every lot has

a sample of size n taken and inspected or tested. For a sample of size n taken and inspected or tested. For double, multiple and sequential plans, the amount of double, multiple and sequential plans, the amount of sampling varies depending on the number of defects sampling varies depending on the number of defects observed.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 plan scheme.

ASN = n1 + n2 (1 – P1) for a double sampling plan.

Sampling Plan DesignSampling Plan DesignSuppose α is known and the AQL is also known then :Suppose α is known and the AQL is also known then :

1. Sampling plan with stipulated producer’s risk1. Sampling plan with stipulated producer’s risk

2. Sampling plan with stipulated consumer’s risk2. Sampling plan with stipulated consumer’s risk

3. Sampling plan with stipulated producer’s and consumer’s 3. Sampling plan with stipulated producer’s and consumer’s riskrisk

can be designed.can be designed.