© 2007 Pearson Education   AQL LTPD Acceptance Sampling Plans Supplement I

© 2007 Pearson Education Acceptance Sampling  Acceptance sampling is a statistical process for determining whether to accept or reject a lot of products by testing a random sample of parts taken from the lot.  An acceptance sampling plan is specified by n and c, where,  n = the sample size, and  c = the critical number of defectives in the sample up to which the lot will be accepted.

© 2007 Pearson Education OC Curve  LetP d = Probability of defectives in the lot  P a = Probability of accepting the lot P(x< c), where x = number of defectives in the sample  OC Curve is a graph with values of P d on the x- axis and the corresponding values of P a in the y- axis.

© 2007 Pearson Education Computing P a for a given sampling plan and P d value  Compute nP d  Use Poisson Probability Table and lookup the value of P a for the value of c  Example: Given a sampling plan of n = 60 and c = 2, if P d = 1%, nP d = 60(.01) =.6 np P a =.977

© 2007 Pearson Education OC Curve 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance

© 2007 Pearson Education Constructing OC Curve The Noise King Muffler Shop, a high-volume installer of replacement exhaust muffler systems, just received a shipment of 1,000 mufflers. The sampling plan for inspecting these mufflers calls for a sample size n=60 and an acceptance number c=1. Construct the OC curve for this sampling plan.

© 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance Constructing an OC Curve Example I.1

© 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance np Constructing an OC Curve Example I.1

© 2007 Pearson Education Probability Proportionof c or less Defectivedefects (p)np(P a )Comments n = 60 c = – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance np Constructing an OC Curve Example I.1

© 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance np Constructing an OC Curve Example I.1

© 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance np Constructing an OC Curve Example I.1

© 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance Constructing an OC Curve Example I.1

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = 1 Constructing an OC Curve Example I.1

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| Proportion defective (hundredths) Probability of acceptance Constructing an OC Curve Example I.1

© 2007 Pearson Education AQL and LTPD  Acceptable Quality Level (AQL)  The poorest level of quality that is acceptable to the customer. It is specified as a percentage of defectives in the lot.  Lot Tolerance Percent Defective (LTPD)  The quality level at which the lot is considered bad. It is specified as a percentage of defectives in the lot.

© 2007 Pearson Education Risks  Producer’s risk  The probability of rejecting a good lot (i.e. P d = AQL) based on the acceptance sampling plan. This is also known as Type I error (  ).  Consumer’s risk  The probability of accepting a bad lot (i.e. P d = LTPD) based on the acceptance sampling plan. This also known as Type II error ( .

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Probability Proportionof c or less defectivedefects (p)np(P a )Comments 0.01 (AQL)  = – = (LTPD)  = n = 60 c = 1 Consumer’s and Producer’s risks - Example I.1

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – ||||||||||  = (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance  = Constructing an OC Curve Example I.1

© 2007 Pearson Education Drawing the OC Curve Application I.1

© 2007 Pearson Education Finding  (probability of rejecting AQL quality: p =.03 np =5.79 Pa =Pa =  = 1 –.965 = Drawing the OC Curve Application I.1 Cumulative Poisson Probabilities

© 2007 Pearson Education Finding  (probability of accepting LTPD quality: p =.08 np =15.44 Pa =Pa = 0.10  = P a = 0.10 Drawing the OC Curve Application I.1 Cumulative Poisson Probabilities

© 2007 Pearson Education Drawing the OC Curve Application I.1

© 2007 Pearson Education Drawing the OC Curve Application I.1

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk n (p = AQL)(p = LTPD) Understanding Changes in the OC Curve (with c = 1)

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance n = 60, c = 1 n = 80, c = 1 n = 100, c = 1 n = 120, c = 1 Operating Characteristic Curves (with c = 1)

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk c (p = AQL)(p = LTPD) Understanding Changes in the OC Curve (with n = 60)

© 2007 Pearson Education 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance n = 60, c = 1 n = 60, c = 2 n = 60, c = 3 n = 60, c = 4 Operating Characteristic Curves (with n = 60)

© 2007 Pearson Education Average Outgoing Quality AOQ =  where,  P d = probability of defectives in the lot  P a = probability of accepting the lot  N = Lot size  n = sample size

© 2007 Pearson Education Average Outgoing Quality Example I.2 Noise King example with rectified inspection for its single-sampling plan with n = 110, c = 3, N = 1000 ProportionProbability Defectiveof Acceptance (p)np(P a ) = ( )/ = ( )/ = ( )/

© 2007 Pearson Education Average Outgoing Quality Example I.2 ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ For p = 0.01, Pa = AOQ = =

© 2007 Pearson Education Average Outgoing Quality Example I.2 ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ

© 2007 Pearson Education Average Outgoing Quality Example I.2 ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ

© 2007 Pearson Education Average Outgoing Quality Example I – 1.2 – 0.8 – 0.4 – 0 – |||||||| |||||||| Defectives in lot (percent) Average outgoing quality (percent) ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ

© 2007 Pearson Education AOQL 1.6 – 1.2 – 0.8 – 0.4 – 0 – |||||||| |||||||| Defectives in lot (percent) Average outgoing quality (percent) Average Outgoing Quality Example I.2 AOQL = Average Outgoing Quality Limit

© 2007 Pearson Education AOQ Calculations Application I.2 Management has selected the following parameters:  

© 2007 Pearson Education AOQ Calculations Application I.2

© 2007 Pearson Education Solved Problem 1.0 — 0.9 — 0.8 — 0.7 — 0.6 — 0.5 — 0.4 — 0.3 — 0.2 — 0.1 — 0 — |||||||||| Proportion defective (hundredths)(p) Probability of acceptance (P a ) (AQL)(LTPD)  =  = 0.049

© 2007 Pearson Education Sequential Sampling Chart 8 8 – 7 7 – 6 6 – 5 5 – 4 4 – 3 3 – 2 2 – 1 1 – 0 0 – Reject Continue sampling Accept Cumulative sample size ||||||| Number of defectives

© 2007 Pearson Education Sequential Sampling Chart 8 8 – 7 7 – 6 6 – 5 5 – 4 4 – 3 3 – 2 2 – 1 1 – 0 0 – Reject Decision to reject Continue sampling Accept Cumulative sample size ||||||| Number of defectives