OC curve for the single sampling plan N = 3000, n=89, c= 2

Probabilities of Acceptance for the Single Sampling Plan n = 89, c = 2 Assumed Process Quality Sample size, n np0 Probability of acceptance, Pa Percent of Lots Accepted 100Pa P0 100P0 0.01 1.0 89 0.9 0.938 93.8 0.02 2.0 1.8 0.731 73.1 0.03 3.0 2.7 0.494 49.4 0.04 4.0 3.6 0.302 30.2 0.05 5.0 4.5 0.174 17.4 0.06 6.0 5.3 0.106 10.6 0.07 7.0 6.2 0.055 5.5

Double Sampling Plan Inspect a sample of 150 from lot of 2400 If 1 or less Nonconforming units accept lots and stop If 4 or more Nonconforming units the lot is not accepted and stop If 2 or 3 nonconforming units, inspect a second sample of 200 If 5 or less Nonconforming units On both samples, Accept the lot If 6 or more Nonconforming units On both samples The lot is not accepted Graphical description of the double sampling plan: N=2400,n1=150, c1=1, r1=4, n2=200, c2=5, and r2=6

OCC for Double Sampling Plan

OCC for a Multiple Sampling Plan

Type-A and Type-B OC curves: fg1506

Effect of n and c on OC curves:

Other Aspects of OC Curve Behavior

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.

Average Outgoing Quality The Average Outgoing Quality (AOQ) is the average of rejected lots (100% inspection) and accepted lots ( a sample of items inspected) Note that as the lot size N becomes large relative to the sample size n, AOQ ≈ Pacp

Average Quality of Inspected Lots Typically the term (N-n)/N is very close to 1; therefore, the equation most often used is:

Example If N = 10,000, n = 89, and c = 2, and that the incoming lots are of quality p = 0.01. What is AOQ? AOQ = Pacp(N-n)/N = ?

Average Outgoing Quality (AOQ) for the Single Sampling Plan n = 89, c = 2 Assumed Process Quality Sample size, n np0 Probability of acceptance, Pa AOQ 100Pacp P0 100P0 0.01 1.0 89 0.9 0.938 93.8 0.02 2.0 1.8 ? 0.03 3.0 2.7 0.494 1.482 0.04 4.0 3.6 0.05 5.0 4.5 0.06 6.0 5.3 0.106 0.636 0.07 7.0 6.2 0.055 0.385

AOQ and Acceptance Sampling 11 lots 2% nonconforming 15 lots 2% nonconforming Producer N=3000 n=89 c=2 Consumer 4 lots 2% nonconforming 4 lots 0% nonconforming Figure 9-15 How acceptance Sampling works

AOQ and Acceptance Sampling Total Number Number Nonconforming 11 lots- 2% Nonconforming 11(3000)=33,000 33,000(0.02)=660 4 lots- 0% Nonconforming 4(3000)(0.98)=11,760 44,760 660 Percent Nonconforming (AOQ) = 660/44,760 X 100 =1.47% Figure 9-15 cont’d.

Average Outgoing Quality curve for the sampling plan N = 3000, n = 89, and c = 2

Average Outgoing Quality Level 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 Average Outgoing Quality Level 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 pa of accepting a lot with defect level p, we have: ATI = n + (1 - pa) (N - n) where N is the lot size.

Example If a lot size N = 10,000, and sample size n = 89, number of acceptance c = 2, find ATI at p = 0.01

Average Sample Number (ASN) 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.

Average Sample Number (ASN) 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 : Sampling plan with stipulated producer’s risk Sampling plan with stipulated consumer’s risk Sampling plan with stipulated producer’s and consumer’s risk can be designed.

Sampling Plan Design Stipulated Producer’s Risk α = 0.05 AQL = 1.2% Pa=0.95 P0.95= 0.012 Assume values for C, find np0.95 for this c value, calculate n

Sampling Plan Design Stipulated Consumer’s Risk β = 0.10 LQ = 6.0% Pa=0.10 P0.10= 0.060 Assume values for C, find np0.95 for this c value, calculate n

Sampling Plan Design Stipulated Producer’s and Consumer’s risk α = 0.10 β = 0.10 AQL=0.9 LQ= 7.8 Find the ratio of P0.10/P0.95. From table 9-4 C is between 1 and 2. Find n for c =1 and n for c =2 .

Sampling Plan Design Have 4 plans. Select plan based on: Lowest sampling size Greatest sampling size Plan exactly meets consumer’s stipulation and is as close as possible to producer’s stipulation Plan exactly meets producer’s stipulation and is as close as possible to consumer’s stipulation