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

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Presentation on theme: "© 2007 Pearson Education   AQL LTPD Acceptance Sampling Plans Supplement I."— Presentation transcript:

1 © 2007 Pearson Education   AQL LTPD Acceptance Sampling Plans Supplement I

2 © 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.

3 © 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.

4 © 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 np012.40.670.938.992.45.638.925.989.50.607.910.986.55.577.894.982.60.549.878.977.65.522.861.972 P a =.977

5 © 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 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance

6 © 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.

7 © 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = 1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance Constructing an OC Curve Example I.1

8 © 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments n = 60 c = 1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance np012.05.951.9991.000.10.905.9951.000.15.861.990.999.20.819.982.999.25.779.974.998.30.741.963.996.35.705.951.994.40.670.938.992.45.638.925.989.50.607.910.986.55.577.894.982.60.549.878.977.65.522.861.972 Constructing an OC Curve Example I.1

9 © 2007 Pearson Education Probability Proportionof c or less Defectivedefects (p)np(P a )Comments 0.010.6 n = 60 c = 1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance np012.05.951.9991.000.10.905.9951.000.15.861.990.999.20.819.982.999.25.779.974.998.30.741.963.996.35.705.951.994.40.670.938.992.45.638.925.989.50.607.910.986.55.577.894.982.60.549.878.977.65.522.861.972 Constructing an OC Curve Example I.1

10 © 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments 0.01 0.60.878 n = 60 c = 1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance np012.05.951.9991.000.10.905.9951.000.15.861.990.999.20.819.982.999.25.779.974.998.30.741.963.996.35.705.951.994.40.670.938.992.45.638.925.989.50.607.910.986.55.577.894.982.60.549.878.977.65.522.861.972 Constructing an OC Curve Example I.1

11 © 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments 0.01 0.60.878 n = 60 c = 1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance np012.05.951.9991.000.10.905.9951.000.15.861.990.999.20.819.982.999.25.779.974.998.30.741.963.996.35.705.951.994.40.670.938.992.45.638.925.989.50.607.910.986.55.577.894.982.60.549.878.977.65.522.861.972 Constructing an OC Curve Example I.1

12 © 2007 Pearson Education Probability Proportionof c or less defectivedefects (p)np(P a )Comments 0.01 0.60.878 n = 60 c = 1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 Proportion defective (hundredths) Probability of acceptance Constructing an OC Curve Example I.1

13 © 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 – 0.663 |||||||||| 12345678910 0.308 0.199 0.048 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Probability Proportionof c or less defectivedefects (p)np(P a )Comments 0.01 0.60.878 0.021.20.663 0.031.80.463 0.042.40.308 0.053.00.199 0.06 3.60.126 0.074.20.078 0.084.80.048 0.095.40.029 0.106.00.017 n = 60 c = 1 Constructing an OC Curve Example I.1

14 © 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 – |||||||||| 12345678910 0.878 0.663 0.463 0.308 0.199 0.126 0.078 0.048 0.029 0.017 Proportion defective (hundredths) Probability of acceptance Constructing an OC Curve Example I.1

15 © 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.

16 © 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 ( .

17 © 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 – 0.663 |||||||||| 12345678910 0.308 0.199 0.048 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Probability Proportionof c or less defectivedefects (p)np(P a )Comments 0.01 (AQL)0.60.878  = 1.000 – 0.878 = 0.122 0.021.20.663 0.031.80.463 0.042.40.308 0.053.00.199 0.06 (LTPD)3.60.126  = 0.126 0.074.20.078 0.084.80.048 0.095.40.029 0.106.00.017 n = 60 c = 1 Consumer’s and Producer’s risks - Example I.1

18 © 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 – |||||||||| 12345678910 0.878 0.663 0.463 0.308 0.199 0.126 0.078 0.048 0.029 0.017  = 0.122 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance  = 0.126 Constructing an OC Curve Example I.1

19 © 2007 Pearson Education Drawing the OC Curve Application I.1

20 © 2007 Pearson Education Finding  (probability of rejecting AQL quality: p =.03 np =5.79 Pa =Pa = 0.965  = 1 –.965 = 0.035 Drawing the OC Curve Application I.1 Cumulative Poisson Probabilities

21 © 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

22 © 2007 Pearson Education Drawing the OC Curve Application I.1

23 © 2007 Pearson Education Drawing the OC Curve Application I.1

24 © 2007 Pearson Education Understanding Changes in the OC Curve (with c = 1) 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk n (p = AQL)(p = LTPD)

25 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk n (p = AQL)(p = LTPD) 600.1220.126 Understanding Changes in the OC Curve (with c = 1)

26 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk n (p = AQL)(p = LTPD) 600.1220.126 800.1910.048 Understanding Changes in the OC Curve (with c = 1)

27 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk n (p = AQL)(p = LTPD) 600.1220.126 800.1910.048 1000.2640.017 Understanding Changes in the OC Curve (with c = 1)

28 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk n (p = AQL)(p = LTPD) 600.1220.126 800.1910.048 1000.2640.017 1200.3320.006 Understanding Changes in the OC Curve (with c = 1)

29 © 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 – |||||||||| 12345678910 (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)

30 © 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 – |||||||||| 12345678910 (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)

31 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk c (p = AQL)(p = LTPD) 10.1220.126 Understanding Changes in the OC Curve (with n = 60)

32 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk c (p = AQL)(p = LTPD) 10.1220.126 20.0230.303 Understanding Changes in the OC Curve (with n = 60)

33 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk c (p = AQL)(p = LTPD) 10.1220.126 20.0230.303 30.0030.515 Understanding Changes in the OC Curve (with n = 60)

34 © 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 – |||||||||| 12345678910 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance Producer’sConsumer’sRisk c (p = AQL)(p = LTPD) 10.1220.126 20.0230.303 30.0030.515 40.0000.726 Understanding Changes in the OC Curve (with n = 60)

35 © 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 – |||||||||| 12345678910 (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)

36 © 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

37 © 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 ) 0.011.100.974 0.022.200.819 0.033.300.581= (0.603 + 0.558)/2 0.044.400.359 0.055.500.202= (0.213 + 0.191)/2 0.066.600.105 0.077.700.052= (0.055 + 0.048)/2 0.088.800.024

38 © 2007 Pearson Education Average Outgoing Quality Example I.2 ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ 0.011.100.974 0.022.200.819 0.033.300.581 0.044.400.359 0.055.500.202 0.066.600.105 0.077.700.052 0.088.800.024 For p = 0.01, Pa = 0.974 AOQ = = 0.0087

39 © 2007 Pearson Education Average Outgoing Quality Example I.2 ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ 0.011.100.9740.0087 0.022.200.819 0.033.300.581 0.044.400.359 0.055.500.202 0.066.600.105 0.077.700.052 0.088.800.024

40 © 2007 Pearson Education Average Outgoing Quality Example I.2 ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ 0.011.100.9740.0087 0.022.200.8190.0146 0.033.300.5810.0155 0.044.400.3590.0128 0.055.500.2020.0090 0.066.600.1050.0056 0.077.700.0520.0032 0.088.800.0240.0017

41 © 2007 Pearson Education Average Outgoing Quality Example I.2 1.6 – 1.2 – 0.8 – 0.4 – 0 – ||||||||12345678||||||||12345678 Defectives in lot (percent) Average outgoing quality (percent) ProportionProbability Defectiveof Acceptance (p)np(P a )AOQ 0.011.100.9740.0087 0.022.200.8190.0146 0.033.300.5810.0155 0.044.400.3590.0128 0.055.500.2020.0090 0.066.600.1050.0056 0.077.700.0520.0032 0.088.800.0240.0017

42 © 2007 Pearson Education AOQL 1.6 – 1.2 – 0.8 – 0.4 – 0 – ||||||||12345678||||||||12345678 Defectives in lot (percent) Average outgoing quality (percent) Average Outgoing Quality Example I.2 AOQL = Average Outgoing Quality Limit

43 © 2007 Pearson Education AOQ Calculations Application I.2 Management has selected the following parameters:  

44 © 2007 Pearson Education AOQ Calculations Application I.2

45 © 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 — |||||||||| 12345678910 Proportion defective (hundredths)(p) Probability of acceptance (P a ) (AQL)(LTPD) 1.000 0.996 0.951 0.810 0.587 0.363 0.194 0.092 0.039 0.015  = 0.092  = 0.049

46 © 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 ||||||| 10203040506070 Number of defectives

47 © 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 ||||||| 10203040506070 Number of defectives

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