Acceptance Sampling Plans Supplement G  AQL LTPD Acceptance Sampling Plans Supplement G Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall G- 01

What is Acceptance Sampling? An inspection procedure used to determine whether to accept or reject a specific quantity of material. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Acceptance Sampling Plan Decisions Basic procedure Take random sample Accept or reject, based on results Producer, or seller, is the origin of the material or service Consumer, or buyer, is the destination of the material or service Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Quality and Risk Decisions Acceptable quality level (AQL) The quality level desired by the consumer Producer’s risk () The risk that the sampling plan will fail to verify an acceptable lot’s quality and, thus, reject it (Type 1 Error) Most often the producer’s risk is set at 0.05, or 5 percent. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Quality and Risk Decisions Lot tolerance proportion defective (LTPD) The worst level the customer can tolerate Consumer’s risk, ( ) The probability of accepting a lot with LTPD quality (Type II Error) A common value for the consumer’s risk is 0.10, or 10 percent Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Sampling Plans Single-Sampling Plan Double-Sampling Plan Sequential Sampling Plan Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Single Sampling Plan Single Sampling Plan A decision to accept or reject a lot based on the results of one random sample from the lot. States the sample size, n, and the acceptable number of defectives, c If the quality characteristic of the sample passes the test (defects ≤ c), accept the lot. If the sample fails (defects > c) there may be complete inspection of the lot or the entire lot is rejected. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Double-Sampling Plan Double-Sampling Plan A plan in which management specifies two sample sizes, (n1 and n2), and two acceptance numbers (c1 and c2) Take a random sample of relatively small size n1, from a large lot If the sample passes the test (≤ c1), accept the lot If the sample fails (> c2), the entire lot is rejected If the sample is between c1 and c2, then take a larger second random sample, n2 If the combined number of defects ≤ c2 accept the lot, otherwise reject Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Sequential-Sampling Plan A plan in which the consumer randomly selects items from the lot and inspects them one-by-one. Results are compared to sequential-sampling chart Chart guides decision to reject, accept, or continue sampling, based on cumulative results Average number of items inspected (ANI) is generally lower with sequential sampling Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Sequential Sampling Chart 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 – 0 – Cumulative sample size | | | | | | | 10 20 30 40 50 60 70 Number of defectives Reject Continue sampling Accept Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Operating Characteristic Curve A graph that describes how well a sampling plan discriminates between good and bad Select sample size n and acceptance number c to achieve the level of performance specified by the AQL, , LTPD, and  The OC curve shows the probability of accepting a lot Pa, as a dependent function of p, the true proportion of defectives in the lot. For every possible combination of n and c, there exists a unique operating characteristic curve. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Operating Characteristic Curve 1.0 AQL LTPD Probability of acceptance Proportion defective a Ideal OC curve Typical OC curve  Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.1 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. The contract with the muffler manufacturer calls for an AQL of 1 defective muffler per 100 and an LTPD of 6 defective mufflers per 100. Calculate the OC curve for this plan, and determine the producer’s risk and the consumer’s risk for the plan. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.1 Let p = 0.01. Multiply n by p to get 60(0.01) = 0.60. Locate 0.60 in Table G.1. The probability of acceptance = 0.878. Repeat this process for a range of p values. The following table contains the remaining values for the OC curve. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.1 Values for the Operating Characteristic Curve with n = 60 and c = 1 Proportion Defective (p) np Probability of c or Less Defects (Pa) Comments 0.01 (AQL) 0.6 0.878  = 1.000 – 0.878 = 0.122 0.02 1.2 0.663 0.03 1.8 0.463 0.04 2.4 0.308 0.05 3.0 0.199 0.06 (LTPD) 3.6 0.126  = 0.126 0.07 4.2 0.078 0.08 4.8 0.048 0.09 5.4 0.029 0.10 6.0 0.017 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.1 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 Proportion defective (hundredths) Probability of acceptance 0.878 0.663 0.463 0.308 0.199 0.126 0.078 0.048 0.029 0.017  = 0.122  = 0.126 (AQL) (LTPD) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Cumulative Poisson Probabilities Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Cumulative Poisson Probabilities Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Application G.1 A sampling plan is being evaluated where c = 10 and n = 193. If AQL = 0.03 and LTPD = 0.08, what are the producer’s risk and consumer’s risk for the plan? Draw the OC curve. Finding  (probability of rejecting AQL quality) Finding  (probability of accepting LTPD quality) p = np = Pa =  = 0.03 5.79 (use 5.8 in table) 0.965 0.035 (or 1.0 – 0.965) p = np = Pa =  = 0.08 15.44 0.10 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Application G.1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Application G.1 1.0 – 0.8 – 0.6 – 0.4 – 0.2 – 0.0 – | | | | | | | | | | | 0 2 4 6 8 10 Probability of acceptance Percentage defective  = 0.035  = 0.10 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Explaining Changes in the OC Curve Sample size effect Increasing n while holding c constant increases the producer’s risk and reduces the consumer’s risk n Producer’s Risk (p = AQL) Consumer’s Risk (p = LTPD) 60 0.122 0.126 80 0.191 0.048 100 0.264 0.017 120 0.332 0.006 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Explaining Changes in the OC Curve 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance n = 60, c = 1 n = 80, c = 1 n = 100, c = 1 n = 120, c = 1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Explaining Changes in the OC Curve Acceptance level effect Increasing c while holding n constant decreases the producer’s risk and increases the consumer’s risk c Producer’s Risk (p = AQL) Consumer’s Risk (p = LTPD) 1 0.122 0.126 2 0.023 0.303 3 0.003 0.515 4 0.000 0.706 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Explaining Changes in the OC Curve 1.0 – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1 – 0.0 – | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 (AQL) (LTPD) Proportion defective (hundredths) Probability of acceptance n = 60, c = 1 n = 60, c = 2 n = 60, c = 3 n = 60, c = 4 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Acceptance Sampling Plan Data Noise King Sampling Plan: c = 3 and n = 111 are best AQL Based LTPD Based Acceptance Number Expected Defectives Sample Size 0.0509 5 2.2996 38 1 0.3552 36 3.8875 65 2 0.8112 81 5.3217 89 3 1.3675 137 6.6697 111 4 1.9680 197 7.9894 133 2.6256 263 9.2647 154 6 3.2838 328 10.5139 175 7 3.9794 398 11.7726 196 8 4.6936 469 12.9903 217 9 5.4237 542 14.2042 237 10 6.1635 616 15.4036 257 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Average Outgoing Quality AOQ The expressed proportion of defects that a plan will allow to pass. Rectified inspection The assumption that all defective items in the lot will be replaced if the lot is rejected and that any defective items in the sample will be replaced if the lot is accepted. where p = true proportion defective of the lot Pa = probability of accepting the lot N = lot size n = sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.2 Suppose that Noise King is using rectified inspection for its single-sampling plan. Calculate the average outgoing quality limit for a plan with n = 110, c = 3, and N = 1,000. Use the following steps to estimate the AOQL for this sampling plan: Step 1: Determine the probabilities of acceptance for the desired values of p. However, the values for p = 0.03, 0.05, and 0.07 had to be interpolated because the table does not have them. For example, Pa for p = 0.03 was estimated by averaging the Pa values for np = 3.2 and np = 3.4, (or 0.603 + 0.558)/2 = 0.580. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Proportion Defective (p) Probability of Acceptance (Pa) Example G.2 Proportion Defective (p) np Probability of Acceptance (Pa) 0.01 1.10 0.974 0.02 2.20 0.819 0.03 3.30 0.581 = (0.603 + 0.558)/2 0.04 4.40 0.359 0.05 5.50 0.202 = (0.213 + 0.191)/2 0.06 6.60 0.105 0.07 7.70 0.052 = (0.055 + 0.048)/2 0.08 8.80 0.024 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.2 Step 2: Calculate the AOQ for each value of p. For p = 0.01: 0.01(0.974)(1000 – 110)/1000 = 0.0087 For p = 0.02: 0.02(0.819)(1000 – 110)/1000 = 0.0146 For p = 0.03: 0.03(0.581)(1000 – 110)/1000 = 0.0155 For p = 0.04: 0.04(0.359)(1000 – 110)/1000 = 0.0128 For p = 0.05: 0.05(0.202)(1000 – 110)/1000 = 0.0090 For p = 0.06: 0.06(0.105)(1000 – 110)/1000 = 0.0056 For p = 0.07: 0.07(0.052)(1000 – 110)/1000 = 0.0032 For p = 0.08: 0.08(0.024)(1000 – 110)/1000 = 0.0017 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example G.2 1.6 – 1.2 – 0.8 – 0.4 – 0 – | | | | | | | | 1 2 3 4 5 6 7 8 Defectives in lot (percent) Average outgoing quality (percent) Step 3: Identify the largest AOQ value, which is the estimate of the AOQL. In this example, the AOQL is 0.0155 at p = 0.03. AOQL Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Application G.2 Demonstrate the model for computing AOQ Management has selected the following parameters: AQL = 0.01  = 0.05 LTPD = 0.06  = 0.10 n = 100 c = 3 What is the AOQ if p = 0.05 and N = 3000? p = np = Pa = AOQ = 0.05 100(0.05) = 5 0.265 = 0.0128 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Solved Problem In a manufacturing facility a feeder process, when operating correctly, has an AQL of 3 percent. A consuming process has a specified LTPD of 8 percent. Management wants no more than a 5 percent producer’s risk and no more than a 10 percent consumer’s risk. a. Determine the appropriate sample size, n, and the acceptable number of defective items in the sample, c. b. Calculate values and draw the OC curve for this inspection station. c. What is the probability that a lot with 5 percent defectives will be rejected? Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Solved Problem For AQL = 3 percent, LTPD = 8 percent,  = 5 percent, and  = 10 percent, use Table G.1 and trial and error to arrive at a sampling plan. If n = 180 and c = 9, 180(0.03) = 5.4  = 0.049 np = np = 180(0.08) = 14.4  = 0.092 Sampling plans that would also work are n = 200, c = 10; n = 220, c = 11; and n = 240, c = 12. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Proportion Defective (p) Probability of c or Less Defectives (Pa) Solved Problem b. The following table contains the data for the OC curve. Table G.1 was used to estimate the probability of acceptance. Proportion Defective (p) np Probability of c or Less Defectives (Pa) Comments 0.01 1.8 1.000 0.02 3.6 0.996 0.03 (AQL) 5.4 0.951  = 1 – 0.951 = 0.049 0.04 7.2 0.810 0.05 9.0 0.587 0.06 10.8 0.363 0.07 12.6 0.194 0.08 (LTPD) 14.4 0.092  = 0.092 0.09 16.2 0.039 0.10 18.0 0.015 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Solved Problem 0.996 0.951 0.810 0.587 0.363 0.194 0.092 0.039 0.015 1.000 c. According to the table, the probability of accepting a lot with 5 percent defectives is 0.587. Therefore, the probability that a lot with 5 percent defects will be rejected is 0.413, or 1 – 0.587 1.0 — 0.9 — 0.8 — 0.7 — 0.6 — 0.5 — 0.4 — 0.3 — 0.2 — 0.1 — 0 — | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 Proportion defective (hundredths)(p) Probability of acceptance (Pa) (AQL) (LTPD)  = 0.049  = 0.092 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Printed in the United States of America. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall