Acceptance 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 ATTRIBUTES Acceptance Sampling is a form of inspection applied to lots or batches of items before or after a process to judge conformance to predetermined standards.

Acceptance Sampling is very useful when: 1. When the test is destructive. 2. When automated inspection is not available. 3. When the cost of 100% inspection is high. 4. Fatigue/boredom is caused by inspecting large numbers of items. numbers of items. 5. When information concerning producers quality not know/available. know/available.Advantages 1. Less expensive. 2. Reduced damage. 3. Reduces the amount of inspection error. Disadvantages 1. Risk of accepting “bad” lots and rejecting “good” lots. 2. Less information generated. 3. Requires planning and documentation.

Sampling Plans Sampling Plans specify the lot size, sample size, number ofSampling Plans specify the lot size, sample size, number of samples and acceptance/rejection criteria. Sampling plans involve:Sampling plans involve: 1. Single sampling 2. Double sampling 3. Multiple sampling Define by,N = lot sizeDefine by,N = lot size n = sample size n = sample size c = acceptance number c = acceptance number If c or less non-conforming units are found in the sample, the lot is accepted, else it is rejected.If c or less non-conforming units are found in the sample, the lot is accepted, else it is rejected.

Single Sampling Plan A representative sample of n items is drawn from a lot size of N items.A representative sample of n items is drawn from a lot size of N items. Each item in the sample is examined and classified as good/defective.Each item in the sample is examined and classified as good/defective. If the number of defective exceeds a specified rejection number (c) the whole lot is rejected; otherwise the whole lot is accepted.If the number of defective exceeds a specified rejection number (c) the whole lot is rejected; otherwise the whole lot is accepted. If Single Sampling Plan, N = 9000, n = 300, c = 2.If Single Sampling Plan, N = 9000, n = 300, c = 2. - Interpretation of Plan means, inspect 300 pieces from the lot of 9000 units. the lot of 9000 units. - If nonconforming found ≤ c=2 then ACCEPT LOT. - If nonconforming found > c=2 then REJECT LOT.

Double Sampling Plan A Double Sampling Plan allows to take a second sample if the results of the original sample are inconclusive.A Double Sampling Plan allows to take a second sample if the results of the original sample are inconclusive. Define by, N = lot size n 1 = sample size (1 st sample) c 1 = acceptance number (1 st sample) r 1 = non-acceptance number (1 st sample) n 2 = sample size (2 nd sample) c 2 = acceptance number (2 nd sample) r 2 = non-acceptance number (2 nd sample) If values are not given for r 1 and r 2, they are equal to c Example Double Sampling Plan, N = 9000 n 1 = 60 n 2 = 150 c 1 = 1 c 2 = 6 r 1 = 5 r 2 = 7

An initial sample (n 1 ) 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 (c 1 ), the lot accepted. 2. If, 5 or more nonconforming units (r 1 ), 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 (n 2 ) from the lot (N) is inspected, and one of the following judgments is made: 1. If there are 6 or fewer nonconforming units (c 2 ) in both sample, the lot is accepted. This number is obtained by 2 in the 1 st sample and 4 or fewer in the 2 nd sample, by 3 in the 1 st sample and 3 or fewer in the 2 nd sample, or by 4 in the 1 st sample and 2 or fewer in the 2 nd sample. 2. If there are 7 or more nonconforming units () in both sample, the lot is not accepted. This number is obtained by 2 in the 1 st sample and 5 or more in the 2 nd sample, by 3 in the 1 st sample and 4 or more in the 2 nd sample, or by 4 in the 1 st sample and 3 or more in the 2 nd sample. 2. If there are 7 or more nonconforming units ( r 2 ) in both sample, the lot is not accepted. This number is obtained by 2 in the 1 st sample and 5 or more in the 2 nd sample, by 3 in the 1 st sample and 4 or more in the 2 nd sample, or by 4 in the 1 st sample and 3 or more in the 2 nd sample.

Multiple Sampling Plan A Multiple Sampling Plan is similar to the double sampling plan in that successive trials are made, each of which has acceptance, rejection and inconclusive options.A Multiple Sampling Plan is similar to the double sampling plan in that successive trials are made, each of which has acceptance, rejection and inconclusive options. Which Plan you choose depends on:Which Plan you choose depends on: - Cost and time - Number of samples needed and number of items in each sample sample Lot Formation Consideration before inspection: 1. Lots should be homogeneous (same machine, same operator, same input material, and so on). 2. Larger lots are more preferable than smaller lots. 3. Lots should be conformable to the materials-handling systems used in both the vendor and consumer facilities.

Statistical Aspects The Operating 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 fraction defective.Plots the probability of accepting the lot versus the lot fraction defective. Shows the probability that a lot submitted with a certain fraction defective will be either accepted or rejected.Shows the probability that a lot submitted with a certain fraction defective will be either accepted or rejected. 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 Lot Acceptance Sampling Plan.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 Lot Acceptance Sampling Plan. There are two types of OC curves:There are two types of OC curves: - Type A - Gives the probability of acceptance of an individual lot coming from finite production individual lot coming from finite production - Type B - Gives the probability of acceptance for lots coming from a continuous production from a continuous production

OC Curve for Single Sampling Plan Example, N=3000, n = 89 and c = 2.Example, N=3000, n = 89 and c = Assume p o, 100p o = 2% 100p o = 2% p o = 0.02 p o = 0.02 p o = (89)(0.02) 2. Calculate, np o = (89)(0.02) = 1.8 = Find Pa = P(0) + P(1) + P(2) = P(2) or less = or 100Pa = 73.1% ***Pa value obtained from Poisson Table page 515. Look ***Pa value obtained from Poisson Table page 515. Look at np o = 1.8 and c = 2 at np o = 1.8 and c = 2 4. Plot point (100p o, 100Pa)

A table can be assist with the calculations, as shown in Table 9-2.A table can be assist with the calculations, as shown in Table 9-2. p oThe curve is terminated when the Pa value is close to Since Pa=0.055 for 100p o = 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 percent of the lots that are expected is 66%. Similarly, if 55 lots from a process that is 2.3% nonconforming are inspected using this sampling plan, 36 [(55)(0.66)=36] will be accepted and 19 [55- 36=19] will be unaccepted.Thus, if the incoming process quality is 2.3% nonconforming, the percent of the lots that are expected is 66%. Similarly, if 55 lots from a process that is 2.3% nonconforming are inspected using this sampling plan, 36 [(55)(0.66)=36] will be accepted and 19 [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 Quality 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.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. The Average Outgoing Quality is the average of rejected lots (100% inspection) and accepted lots ( a sample of items inspected)The Average Outgoing Quality is the average of rejected lots (100% inspection) and accepted lots ( a sample of items inspected)

Average Total Inspection 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 pa of accepting a lot with defect level p, we have: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 pa of accepting a lot with defect level p, we have: ATI = n + (1 - pa) (N - n) ; where N is the lot size. Average Sample Number For a single sampling (n,c) we know each and every lot has a sample of size n taken and inspected or tested. For double, multiple and sequential plans, the amount of sampling varies depending on the number of defects observed.For a single sampling (n,c) we know each and every lot has a sample of size n taken and inspected or tested. For double, multiple and sequential plans, 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 plan scheme. ASN = n1 + n2 (1 – P1) for a double sampling plan.

Sampling Plan Design Suppose α is known and the AQL is also known then : 1. Sampling plan with stipulated producer’s risk 2. Sampling plan with stipulated consumer’s risk 3. Sampling plan with stipulated producer’s and consumer’s risk can be designed.