Zero Acceptance Number Sampling Plans [6 ed.] 1636940935, 9781636940939

Table of contents :
Cover
Contents
List of Figures and Tables
Preface
Introduction
Attribute Sampling Plans
Nonstatistical Sampling Plans
c=0 vs Effects of AQL Greater Than Zero
Estimating Potential Savings
Why Constant Sample Sizes Are Not Used
Use of the c=0 Plans Table
Physically Taking the Sample
Comments on the AOQL
Background Information
Adjustments from MIL-STD-105E/ANSI Z1.4
Sampling Plan “Switching”
Switching Guidelines
Small Lot Supplement
Operating Characteristic Curves and Values
Index

Citation preview

Zero Acceptance Number Sampling Plans Sixth Edition

We Culture

12 Skills for Growing Teams ZeroinAcceptance Number the Future of Work

Sampling Plans Sixth Edition Luciana Paulise

Nicholas L. Squeglia

Milwaukee, Wisconsin

We Culture: 12 Skills for Growing Teams in the Future of Work Luciana Paulise Zero Acceptance Number Sampling Plans, Sixth Edition American Society for Quality, Quality Press, Milwaukee 53203 Nicholas L. Squeglia All rights reserved. Published 2022 American Society for Quality, Quality Press, Milwaukee 53203  © 2022 by Luciana Paulise All rights reserved. Published 1969, 2023 ©2023 by Quality Press  No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the No part of this book may be reproduced in any form or by any means, electronic, prior written permission of the publisher. mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.  Publisher’s Cataloging-in-Publication data Publisher’s Cataloging-in-Publication Data  Names: Paulis, Nicholas Luciana,L., author. Names: Squeglia, author. Title:Zero We acceptance culture: 12number skills for growing teams in edition the future of work / Title: sampling plans , sixth / Nicholas L. Squeglia. by LucianaIncludes Paulise.index. | Milwaukee, WI: Quality Press, 2023. Description: Description: Includes bibliographical references. |(paperback) Milwaukee, Identifiers: LCCN: 2023934945 | ISBN: 9781636940939 | WI: Quality Press, (pdf) 2022.| 9781636940953 (epub) 9781636940946 Subjects: LCSH Acceptance sampling. | BISAC BUSINESS & ECONOMICS Identifiers: LCCN: 2022935206 | ISBN: 978-1-63694-017-5 (paperback) | / 978-1-63694-018-2 Quality Control | BUSINESS & ECONOMICS / Total Quality Management | (epub) MATHEMATICS / Probability & / General Subjects: LCSH Leadership. | Statistics Corporate culture. | Organizational change. Classification: LCC TS156.4 .S68 2023 | DDC 658.5/62--dc23 | Business communication. | BISAC BUSINESS & ECONOMICS / Business Communication / General | BUSINESS & ECONOMICS / ASQ advances individual, organizational, and community excellence worldwide Leadership | BUSINESS & ECONOMICS / Mentoring & Coaching through learning, quality improvement, and knowledge exchange.  Classification: LCC HD58.8 .P38 2022 | DDC 658.4/06—dc23 Bookstores, wholesalers, schools, libraries, businesses, and organizations: Quality ASQbooks advances individual, organizational, and community excellence Press are available at quantity discounts for bulk purchases for business, trade, worldwide through learning, quality improvement, and knowledge or educational uses. For more information, please contact Quality Press at 800-248exchange. 1946 or [email protected]. 

Bookstores, businesses, andvisit organizations: place orderswholesalers, or browse theschools, selectionlibraries, of all Quality Press titles, our website at: To Quality Press books are available at quantity discounts for bulk purchases http://www.asq.org/quality-press.  for business, trade, or educational uses. For more information, please contact Quality LS 7 6Press 5 4 3 at 2 1 800-248-1946 or [email protected]. 26 25 24 23 To place orders or browse the selection of all Quality Press titles, visit our website at: http://www.asq.org/quality-press Printed in the United States of America. 27 26 25 24 23 22 SP 7 6 5 4 3 2 1

This book is dedicated to my wife Joan; my children Vanessa, Nicholas, and Jacqueline; and my grandson Shane.

CONTENTS

List of Figures and Tables....................................................................... ix Preface.................................................................................................... xi Introduction............................................................................................. 1 Attribute Sampling Plans........................................................................ 3 Nonstatistical Sampling Plans................................................................. 4 c=0 vs Effects of AQL Greater Than Zero.............................................. 5 Estimating Potential Savings................................................................... 11 Why Constant Sample Sizes Are Not Used............................................. 11 Use of the c=0 Plans Table...................................................................... 12 Physically Taking the Sample................................................................. 13 Comments on the AOQL......................................................................... 13 Background Information......................................................................... 14 Adjustments from MIL-STD-105E/ANSI Z1.4....................................... 19 Sampling Plan “Switching”..................................................................... 20 Switching Guidelines.............................................................................. 21 Small Lot Supplement............................................................................. 22 Operating Characteristic Curves and Values........................................... 25 Index........................................................................................................ 41

vii

LIST OF FIGURES AND TABLES Figure 1.

Operating characteristic curve............................................. 7

Figure 2.

Effects of acceptance numbers on OC curve....................... 8

Table 1.

Original c=0 sampling plans............................................... 10

Figure 3.

AOQL curve........................................................................ 14

Table 2.

AOQL values for Table 1..................................................... 15

Figure 4.

Curves for determining AOQL values................................ 16

Table 3.

Inspection results from a large receiving inspection department over a one-month period................................... 18

Table 4.

Receiving inspection figures for one month........................ 18

Table 5.

Receiving inspection figures for one month........................ 19

Table 6.

Receiving inspection figures for one month........................ 19

Table 7.

Small lot size supplement.................................................... 23

.

Operating characteristic curves and values (sample size as indicated)........................................................................ 25

ix

PREFACE For many years, the acceptable quality level (AQL)1 concept was used largely because of the influence of MIL-STD-105 and its revisions. However, in the current business climate of immense worldwide competition and greater demands by customers, more companies are realizing that quality control does not cost—it pays. As a result, the prevailing wisdom has moved toward zero defects, and AQLs greater than zero are no longer the rule, but the exception; they are simply not compatible with today’s environment. Many companies are striving for zero defects through statistical process control, improved processes, closed loop inspection systems, and other means. The use of sampling plans with zero acceptance numbers is the norm today. The sampling plans in this book (c=0) actually represent a revision in 1963 of similar plans I developed in 1961. Because of the widespread use of MIL-STD-105C in 1961, the only way to depart from this standard was to develop a set of plans that could be compared favorably with the military standard. The c=0 plans were developed and originally implemented in a medium-sized plant that did both military and commercial work. Although the plans were not formally approved, there was no opposition to them. In 1963, MIL-STD-105D was introduced, and the c=0 plans were updated and revised. This time, the plans were proposed to a large aerospace manufacturer with a staff of resident government quality control representatives. It was necessary to deliver a formal presentation and explain the c=0 plans in great detail. The aerospace manufacturer and the government representatives agreed to accept the plans on a trial basis. While the plans were targeted essentially to the limiting quality (LQ) percentages in the military standard tables, there were departures from these targets in several instances. These special adjustments were necessary to maintain the logic of

1. ANSI Z1.4 changed “acceptable quality level” to “acceptance quality limit” and provided an explanation. The c=0 plans are not AQL plans. For comparison purposes, they are associated with particular ANSI plans. See Table 1; the numbers are index values.

xi

the c=0 plans. These adjustments were highlighted during the presentation and explained in the section “Adjustments from Mil-Std 105E/ANSI Z1.4.” The results of the trial period at this company were excellent. Not only were the savings significant, but there was a significant reduction in assembly problems as well. A check with the company in 1983 (20 years later) revealed the c=0 plans were still being used. The c=0 sampling plans were presented in a national publication in 1965 (N. L. Squeglia, “Sampling Plans for Zero Defects,” Quality Assurance 4 [August 1965]: 28). The inquiries and interest generated by the article prompted me to write the first edition of this book. Published in 1969, it described the plans in more detail and contained operating characteristic (OC) curves. The continual interest in the c=0 plans and encouragement from Professor N. L. Enrick of Kent State University resulted in the publication of the second edition in 1981. In 1983, I conducted an informal survey to get some idea of the extent of the savings realized by users of the c=0 plans who had switched from MILSTD-105D. A few said it was too early to tell, but the majority reported a range of savings from 8 percent to 30 percent, with an average of 18 percent. Of course, the extent of savings is based on the lot sizes and index value (associated AQL) used. The larger the lot and index value, the greater the savings. It was not necessary to implement the plans to determine the savings. The savings potential can be evaluated from past data as described later, in the section titled “Estimating Potential Savings.” While the hypergeometric distribution was used originally to maximize mathematical accuracy, it is my humble opinion that the most important feature of the plans is the philosophy of zero defects. The c=0 sampling plans are now in wide use throughout the United States and in other countries. In 1983, the c=0 plans became a part of the Department of Defense’s DLAM 8200.2 for use by government Defense Contract Administration Services quality assurance representatives. In 1989, MIL-STD-105E superseded 105D. This revision placed emphasis on the use of 105E as a guide in developing inspection strategies, and it recognized the limitation of the AQL concept. The sampling plans were not changed. As a

xii

result of this revision, the fourth edition of this book was published in 1994. That edition included important updates and a small-lot sampling table. In the earliest editions it was necessary to compare the c=0 plans to the MIL-STD plans in order to show the advantages of the c=0 plans. The comparisons provide important information as to their derivation that provided for a smooth transition from the MIL-STD plans to the c=0 plans, as discussed in the section titled “Relationship of c=0 Plans to ANSI Z1.4 Plans.” After years of extensive application by government contractors, commercial manufacturing, and service industries, the c=0 sampling plans presented in this book are now considered stand-alone sampling plans and are considered a standard. In 2000 the Department of Defense declared MIL-STD-105E obsolete and recommended that the c=0 plans from the fourth edition of this book be used in its place. Upon cancellation of MIL-STD-105E, the c=0 plans were authorized for use by the Defense Contract Management Agency/ Department of Defense. That correspondence in part read: “Zero Acceptance Sampling Plans By Nicholas L. Squeglia This book gives a number of zero based sampling plans and their corresponding Operating Characteristic (OC) Curves and values. It is the state of the art in zero based sampling plans.” (DCMA Guidebook) To further the understanding of the development of the c=0 plans, this edition includes the comparisons that were made to Mil Standards in previous editions. In the early 2000s, a large aerospace manufacturer was given permission by ASQ to reproduce the c=0 sampling table. They modified the table by changing several sample sizes, and that modified table found its way into the fifth edition. Although the changes were relatively minor, this resulted in some confusion among purchasers. Ironically, that large aerospace manufacturer reverted back to the original c=0 plans and changed its specification on sampling issued to all of their suppliers. The next printing of the fifth edition was revised to include the original c=0 sampling table

xiii

as well as the modified table and related AOQL tables (tables 2a and 2b). The sampling tables were labeled 1a and 1b. Table 1a was also labeled as the “original c=0 sampling plans (use this table unless otherwise specified by contract).” Table 1b was labeled as “modified c=0 plans; refer to page xii.” To eliminate any confusion, this sixth edition includes only the original c=0 table and is designated as Table 1. The modified table and its corresponding AOQL table have been eliminated. Table 1 and the related AOQLs Table 2 are now exactly the same as they were in the fourth edition and all prior editions. The c=0 plans gained wide recognition over the years. In May 2005, I was awarded ASQ’s Shainin Medal at the World Quality Congress in Milwaukee. The citation read: “Nicholas L. Squeglia is credited as one of the most significant contributors to the effort for driving Zero Defects by developing a set of zero acceptance number sampling plans (c=0) forcing preventive actions, thus saving millions of dollars in military and commercial applications. There was widespread acceptance of the plans because they proved to be practical, simple to use, and economical.” In 2018 I was inducted into the University of New Haven’s (UNH) Engineering Hall of Fame for the c=0 plans and my contribution to the safety of New York City bridges. The c=0 plans have been continually gaining in popularity for more than 55 years. Nicholas L. Squeglia

xiv

INTRODUCTION The zero acceptance number plans developed by the author were originally designed and used to provide overall equal or greater consumer protection with less inspection than the corresponding MIL-STD-105 sampling plans. In addition to the economic advantages they offer, these plans are simple to use and administer. Because of these advantages and because greater emphasis is now being placed on zero defects and product liability prevention, these plans have found their place in many commercial industries, although they were originally developed for military products. Comparisons can also be made to ANSI Z1.4 (2003) because it is a virtual copy of the cancelled MIL-STD-105E. The derivation of these plans is covered in detail. There is no specific sampling plan or procedure that can be considered best suited for all applications. It is impractical to cite all of the applications in which these c=0 plans are used. Some examples are machined, formed, cast, powered metal, plastic, and stamped parts; and electrical, electronic, and mechanical components. They have found application in receiving inspection, in-process inspection, and final inspection in many industries. Regardless of the product, wherever the potential for lot-by-lot sampling exists, the c=0 plans may be applicable. Quite often, the basic objective of sampling is overlooked. The primary objective is derived from the question, “Why sample?” Most of us are aware that sampling is employed simply to provide a degree of quality protection against accepting nonconforming material. Further, we know that what we are continually striving for is 100 percent good product. Assuming our inspection capability is 100 percent efficient in detecting nonconformances, the only way to assure 100 percent good product is to inspect everything 100 percent. This, then, is the reason for sampling: We sample because it is impractical in most cases to perform 100 percent inspection. What we are seeking, therefore, are sampling plans that economically provide us with a reasonable amount of protection to ensure 100 percent good quality. There are times when something less than 100 percent good product is

1

considered acceptable; in other words, there are times when we knowingly accept defective product. Such cases, however, should be treated on an exception basis. This book provides a set of attribute plans for lot-by-lot inspection. The acceptance number in all cases is zero. This means that for some level of protection you select a certain size sample and withhold the lot if the sample contains one or more nonconforming pieces. The phrase “withhold the lot” is significant in that it does not necessarily mean rejection. Under these plans, the inspector does not automatically reject the lot if one or more nonconformances are found. The inspector only accepts the lot if zero nonconformances are found in the sample. Withholding the lot forces a review and disposition by engineering or management personnel in regard to the extent and seriousness of the nonconformance. From this point on, we will use the terms “defective” and “defect” to describe nonconformances, regardless of whether the defective is fit for use. The term “defective” is commonly used in quality control to describe a part, component, item, or any other unit of product that contains one or more defects. The word “defect” is commonly used to describe a particular nonconforming characteristic on a unit of product. For example, a particular item contains a slot of a certain width and length, and it also contains a hole. Everything about the item conforms to specifications except the diameter of the hole. This nonconforming diameter is a defect, and the item is therefore defective.

2

ATTRIBUTE SAMPLING PLANS The following section provides a brief overview of lot-by-lot attribute plans while relating them to other plans. Two broad categories of sampling are (1) continuous and (2) lot-by-lot. Continuous Continuous sampling is often used when units of product are submitted one at a time. This can apply to assembly lines, for example. Product moving along a conveyor can also be thought of as a candidate for continuous sampling. Consider, for example, a functional test. Units are moved along a line, and assembly progresses to completion at the end of the line. Upon completion, we have a functional test. The continuous sampling plan may call for a frequency check of, say, one unit out of five. It basically starts off with 100 percent inspection until a designated consecutive number of accepted items is reached. When that number is reached, the frequency check starts and remains in effect until a defective unit is observed. When we find a defective unit on the frequency inspection, we go back to 100 percent inspection until we get the specified number of consecutive good items. We then go back to frequency inspection, and so on. MIL-STD-1235C, among others, provided continuous sampling plans. Lot-by-Lot: Attributes Lot-by-lot sampling involves units of products that are presented in a group, batch, or lot for inspection, as opposed to being presented one at a time. In these cases, a sample of a specified quantity is drawn and compared with acceptance criteria—for instance, the quantity of defectives allowed in the sample by the sampling plan. For example, a particular sampling plan may call for a sample size of 125 and an acceptance number of two. If two or fewer defectives are detected in the sample of 125 pieces, the lot is accepted.

3

If more than two defectives are detected in the 125-piece sample, the lot is rejected. Measurements of characteristics are not required. The characteristics evaluated either conform or do not conform. Go/no go type gages are often used in attribute plans. The c=0 sampling plans are lot-by-lot attribute sampling plans. Lot-by-Lot: Variables Another lot-by-lot sampling procedure involves the analysis of measured characteristics. For example, MIL-STD-414 (ANSI/ASQC Z1.9) was a variables-type sampling plan procedure. Variables sampling compared with attribute sampling essentially involves the inspection of a smaller sample size to obtain the same protection afforded by an attribute plan. The economics of the smaller sample size, however, are quite often offset by the calculating involved and the need for obtaining and recording measurements. It should be noted that variables sampling plans assume normality (normal distributions) in all cases, and that if a lot is rejected on the basis of calculation but no defectives are found in the sample, the inspector is required to use the corresponding attribute sampling plan. The c=0 plans can also be used at this point.

NONSTATISTICAL SAMPLING PLANS There are cases where we can virtually assure zero defects although the sample size cannot logically be defined in terms of statistical risks. Such sample sizes are generally exceptionally low for the more important characteristics and, therefore, knowledge of the process control features is required. In order to avoid any confusion in justifying such sample sizes on inspection plans, specific notations should be used to avoid any tie-in with statistical risks. The reason for such a selection should be noted, either directly on the plan or in the quality engineering standards. An example might be a stamping operation where just the first and last pieces from a lot are inspected dimensionally. A statistical sample may be taken for visual inspection only.

4

The higher index values in the c=0 plans are also used where favorable process control has been demonstrated and just an audit is required. Although the statistical risks seem high, the risks from a practical standpoint would be exceptionally low.

c=0 vs EFFECTS OF AQL GREATER THAN ZERO Essentially, the AQL is a specified percentage that is considered to be good quality. In any sampling plan, an Operating Characteristic (OC) curve can be generated to define the risks of accepting lots with varying degrees of percentage defective. When we use the AQL concept as such, we have a high probability of acceptance associated with this AQL percentage. Normally, this is on the order of a 0.90–0.98 probability of acceptance level. The risk of rejecting this AQL percentage is on the order of a 0.10-0.02 probability level. This rejection risk is called the producer’s risk. The assumption in employing the AQL concept is that some agreement has been reached between the producer and the consumer. And, because sampling is used, the producer must assume the risk of having a lot rejected even though the actual percentage defective in the lot is equal to or less than the specified AQL. If no prior AQL agreement exists, and sampling is to be performed simply because 100 percent inspection is impractical, then overinspection is usually the result. Also, when we sample-inspect because 100 percent inspection is impractical, we would expect to inspect a smaller number of pieces on less critical characteristics. To illustrate this point, consider the following example. Let us assume we are presently using the MIL-STD/ANSI plans. A 1.0 percent AQL is used for major characteristics, and a 4.0 percent AQL is used for minor characteristics, although no consumer/producer AQL agreement exists. Let the lot size equal 1300. From ANSI Z1.4 based on the lot size, we must take

5

a sample size of 125 pieces, with 3 defectives allowed for the 1.0 percent AQL and 10 defectives allowed for the 4.0 percent AQL: AQL

Sample size

Acceptance number

1.0%

125

3

4.0%

125

10

We see that for a less critical characteristic, the sample size remains the same. The difference is in the acceptance number. That is, for the 1.0 percent AQL, we can accept the lot if the sample contains 3 or fewer defectives. For the minor characteristic, we can accept the lot if the sample contains 10 or fewer defectives. Because the sample size is the same, there is no reduction in pieces inspected on the minor characteristics. Now, there is another part of the OC curve for a sampling plan that is often overlooked. (A sampling plan consists of a sample size and acceptance criteria.) An OC curve for a sample size of 125 and an acceptance number of 10 is shown in Figure 1. The AQL and producer’s risk, as previously described, are shown. The LQ (Limiting Quality) here is considered poor quality. Depending on the text, Lot Tolerance Percent Defective (LTPD) and Limiting Quality (LQ) have the same meaning and can be used interchangeably. Several sampling plans can have OC curves passing through the same AQL/producer’s risk point. For each of these plans, however, there will be a different LQ at some constant probability of acceptance level. This probability of acceptance level corresponding to the LQ is usually low, with 0.10 being widely accepted. This probability level is called the consumer’s risk. The user of the sampling plan, therefore, must select a plan that will provide reasonably good protection against accepting lots with a percent defective not too much greater than the AQL. With the AQL/producer’s risk point fixed, the closer the LQ gets to the AQL, the larger the sample size and acceptance number become. This is illustrated in Figure 2. 6

7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

2.0

3.0

4.0

AQL (4.0%)

6.0

7.0

Producer’s risk (0.014) a = (1 – Pa)

8.0

9.0

10.0

11.0

12.0

13.0

Consumer’s risk b = (0.10 Pa)

n = sample size = 125 c = acceptance number = 10

Incoming quality (percent defective) P´

5.0

Figure 1  Operating characteristic curve.

Probability of acceptance (Pa)

1.0

LQ (12.3%)

8 0

0.1

0.5

0.9

1.0

1

5 Δx

n = 18 c=0

Δx

n = 125 c = 10

10

Note: Where an AQL is involved, the narrower the distance between the AQL and the LQ, the higher the sample size and acceptance number are. 100 percent inspection will occur if the AQL=LQ. As the target is the LQ in the c=0 plans, we simply increase the distance as much as possible. With no AQL in the c=0 plans, this equates to zero on the x-axis. 100 percent inspection would take place if the LQ=0.

Figure 2  Effects of acceptance numbers on OC curve.

Pa

However, if we are satisfied with the specified LQ of ANSI Z1.4 and are not concerned with the AQL concept, we can get equal or greater protection with smaller sampling sizes by using corresponding c=0 plans. Figure 2 shows the effects of acceptance numbers on OC curves. With the acceptance number set to zero, we (the consumer) have greater protection with a sample size of 18, as compared with a sampling plan that has a sample size of 125 with an acceptance number of 10. Now, let us compare a set of c=0 plans from Table 1 with the previous ANSI Z1.4 example used. ANSI Z1.4 AQL

Sample size

Acceptance number

1.0%

125

3

4.0%

125

10

Associated AQL

Sample size

Acceptance number

1.0%

42

0

4.0%

18

0

c=0 plans

The c=0 plans provide essentially equal or greater LQ protection at the 0.10 consumer’s risk level. We also see that less inspection is performed on less important characteristics. The c=0 plans in Table 1 are “associated” with the AQLs of ANSI Z1.4. In all of these plans, the protection afforded to the consumer is equal to or greater than the ANSI Z1.4 plans. This method of developing the plans provided for simple conversion from the ANSI Z1.4 plans to the c=0 plans. Table 1 labels these associated AQLs as “index values” because, of course in a way, they are not AQLs as intended.

9

10

* * * * * * * * * 1250 1250 1250 1250 1250 1250

* * * * * * * * 800 800 800 800 800 800 1200

* * * * * * * * 500 500 500 500 500 750 1112

* * * * * * * 315 315 315 315 315 490 715 715

* * * * * * 200 200 200 200 200 300 476 476 556

.010 .015 .025 .040 .065

*Indicates entire lot must be inspected. Note: The acceptance number in all cases is zero. 

2–8 9–15 16–25 26–50 51–90 91–150 151–280 281–500 501–1200 1201–3200 3201–10,000 10,001–35,000 35,001–150,000 150,001–500,000 500,001 and over

Lot size

Table 1  Original c=0 sampling plans.

* 13 13 13 13 13 20 29 34 42 50 60 74 90 102

* 8 8 8 8 12 19 21 27 35 38 46 56 64 64

5 5 5 5 7 11 13 16 19 23 29 35 40 40 40

3 3 3 5 6 7 10 11 15 18 22 29 29 29 29

2 2 3 5 5 6 7 9 11 13 15 15 15 15 15

2 2 2 3 4 5 6 7 8 9 9 9 9 9 9

1.0 1.5 2.5 4.0 6.5 10.0

Zero Acceptance Number Sampling Plans, Sixth Edition by Nicholas L. Squeglia American Society for Quality

Index values (associated AQLs) .10 .15 .25 .40 .65 Sample size * * * * * * * * * * * * * * 20 * * * 32 20 * 80 50 32 20 125 80 50 32 20 125 80 50 32 20 125 80 50 48 47 125 80 75 73 47 125 120 116 73 53 192 189 116 86 68 294 189 135 108 77 294 218 170 123 96 345 270 200 156 119 435 303 244 189 143

It should be pointed out that the question of rejecting more lots under these plans arose because of the zero acceptance numbers. Aside from experience, which has shown that in fact considerable savings can be derived from using these c=0 plans, consider the following: • If your quality is very bad, acceptance numbers of greater than zero will not be much help. • When you allow acceptance numbers of greater than zero in your plan, you are in effect authorizing an inspector to accept parts that may not be usable. The zero acceptance number forces a review of any defectives by qualified personnel so that a proper disposition can take place. • If you are striving for zero defects, how can you allow defectives in your sampling plans?

ESTIMATING POTENTIAL SAVINGS An estimate of the cost reduction from switching from the ANSI Z1.4 plans to the c=0 plans can be made by examining your past records. Compare these records with the c=0 plans. For example, if ANSI Z1.4 called for a sample size of 125 and an acceptance number of 3, and the records show you inspected 125 pieces and found no defectives, then your comparison is the 125 pieces against the 42 pieces in the c=0 plans. Also, of course, if you found 4 bad pieces in the 125-piece sample, the chances are pretty high that 1 of these 4 would have been detected in the first 42 inspected. In other words, the lot would have been withheld with fewer pieces inspected. Some actual inspection results are shown and discussed in the “Background Information” section.

WHY CONSTANT SAMPLE SIZES ARE NOT USED As discussed in this book, the c=0 plans were designed essentially to be equal or greater in consumer and average outgoing quality limit (AOQL)

11

protection than the corresponding ANSI Z1.4 plans. The c=0 plans and the ANSI Z1.4 plans show that within a particular AQL column, the OC curves actually differ for the most part as the lot size increases. The reason for this common feature, in addition to satisfying the statistical relationship, is that it is generally considered more practical to obtain greater protection on larger lot sizes as well as getting a more representative sample. The use of constant sample sizes often results in a combination of overinspection and underinspection for broad lot size ranges. In summary, c=0 plans are used when: 1.  Manufactured parts are expected to completely conform to specification requirements 2. Less inspection is desired on less critical characteristics 3. Sampling is performed because 100 percent inspection in general on all characteristics of all parts is impractical 4. Inspectors, as a general rule, are not allowed to knowingly accept nonconforming products 5. Interim inspections are needed until a problem is corrected 6. Auditing is required for various reasons, such as: —Audit of stock for assurance —Audit of items for potential transit damage 7. Visual inspection in general is being done

USE OF THE C=0 PLANS TABLE Assume you are presently working with ANSI Z1.4 plans and have established AQL levels. A particular lot of 700 pieces is to be inspected. Normally you would use, for example, a 1.0 percent AQL in accordance with ANSI Z1.4. Under the c=0 plans, looking at the left-hand column in Table 1, we find that 700 lies within the lot size range of 501-1200. Reading across, and down from the index value (associated AQL) of 1.0, we find our sample size to be 34.

12

From this lot of 700 pieces we take a sample size of 34, and if one or more defectives are found in the sample, we withhold the lot. If you are not presently working with the ANSI Z1.4 AQLs, then someone must designate plans to use in terms of index value/associated AQL. This designation depends on your particular situation and requires product and process knowledge. However, once you select the desired index value, the procedure for taking the sample is identical to that previously cited. See the “Switching Guidelines” section for other ideas.

PHYSICALLY TAKING THE SAMPLE A representative sample is necessary to assure reliable results. One of the most common ways to obtain a representative sample is by random sampling. Randomness is achieved only when each piece and combination of pieces in the lot has an equal chance of being selected for the sample. Such a sample may be drawn in several ways. When material is packaged in an orderly manner, or laid out on a bench in groups or rows, each unit may be numbered from 1 to the total in the lot, and the sample selected by use of a random number table. For bulk-packed material, a bench may be marked in numbered areas on which the units are evenly spread. The units for the sample can be selected by reference to the numbered zone in which they are located using a random number table. Many variations of such approaches have been used with success. If random sampling is not appropriate, common sense should prevail in obtaining a representative sample. Stratified sampling, for example, should be considered.

COMMENTS ON THE AOQL The AOQL is the maximum average outgoing quality from an inspection station for a particular sampling plan. For example, Figure 3 shows an AOQ curve for the sampling plan n=50, c=0. The AOQL is designated. The

13

0.73

n = 50 c=0 Lot size = 10,000

AOQL

0.70

Average outgoing quality (percent)

0.60 0.50 0.40 0.30 0.20 0.10 0.0 0

1

2

3

4

5

6

7

8

Incoming quality (percent)

Figure 3  AOQL curve. AOQL values for the c=0 plans are shown in Table 2. All of the AOQLs for the c=0 are virtually the same as or less than those of the corresponding ANSI Z1.4 plans. AOQL Comparisons For informational purposes, Figure 4 on p. 16 shows a comparison of AOQL values for various sample sizes and acceptance numbers. Infinite lot sizes were used in constructing this graph. With the limitations of most graphs, the AOQL values read approximate.

BACKGROUND INFORMATION The cancelled MIL-Q-9858A permitted the use of other plans, such as the c=0 sampling plans. Reference was made to the following clause that permitted this authorization. 14

15

.25

.40

.65

1.0

1.5

2.5 2.5 2.5

0.02 0.04 0.09 0.15 0.26 0.43 0.46 0.47 0.75 1.10 1.30 1.90 2.4 0.02 0.03 0.06 0.11 0.17 0.28 0.30 0.31 0.49 0.68 0.86 1.00 1.60 2.0 0.03 0.04 0.07 0.11 0.18 0.19 0.19 0.31 0.42 0.54 0.73 0.96 1.30 1.7 0.03 0.04 0.07 0.12 0.12 0.12 0.19 0.27 0.34 0.48 0.61 0.80 1.10 1.3 0.03 0.05 0.07 0.07 0.08 0.12 0.17 0.22 0.30 0.38 0.50 0.66 0.92 1.3

501–1200

1201–3200

3201–10,000

10,001–35,000

35,001–150,000

150,001–500,000 0.03 0.05 0.05 0.05 0.08 0.11 0.14 0.18 0.24 0.31 0.41 0.57 0.92 1.3

500,001 and over 0.03 0.03 0.03 0.05 0.07 0.08 0.12 0.15 0.19 0.26 0.36 0.57 0.92 1.3

2.5

2.4

2.8

3.3

4.0

0.04 0.11 0.22 0.39 0.66 0.69 0.71 1.20 1.70 2.20 3.3

5.1

5.9

281–500

0.05 0.21 0.49 0.90 1.60 2.60 2.80 3.10 5.0

91–150

6.9

0.05 0.16 0.33 0.60 1.00 1.70 1.70 1.80 2.70 3.5

0.05 0.33 0.74 1.40 2.40 4.20 4.80 5.7

51–90

4.1

4.1

4.1

4.1

4.1

4.1

4.6

5.2

6.0

7.1

8.8

6.6 11.5

151–280

0.41 1.10 2.10 3.90 6.60 6.6

0.37 1.40 3.10 5.90 10.8 10.8 16.9

26–50

16–25

0.38 2.10 4.90 9.8 15.9 15.9

4.0 6.5 10.0

9–15

2.5

2.80 7.7 13.8 13.8

AOQL (acceptance number in all cases is zero)

.15

Index values (associated AQLs)

2–8

Lot size

.010 .015 .025 .040 .065 .10

Table 2  AOQL values for Table 1.

10 9

AOQL (percent defective)

8 7 6

15 c = = 14 c 13 c = 12 c = 11 c = 10 c= =9 c 8 c= 7 c= c=

5

6

5

c=

4

c=

4

3 c= 2 c= c=1

3 2

c=0

1 0 140 130 120 110 100

90

80

70

60

50

40

30

20

10

Sample size

Figure 4  Curves for determining AOQL values.

6.6 Statistical Quality Control and Analysis The contractor may employ sampling inspection in accordance with applicable military standards and sampling plans (e.g., from MIL-STD-105. MIL-STD-414, or Handbooks H 106, 107, and 108). If the contractor uses other sampling plans, they shall be subject to review by the cognizant Government Representative. Any sampling plan used shall provide valid confidence and quality levels. (M1L-Q-9858A)

16

0

It is interesting to note that there was no restriction or requirement to have “other” sampling plans match any specifics of MIL-STD-105 and its later revisions. The aerospace standard, AS9100 (presently revision D at this time) essentially replaced MIL-Q-9858A. It requires that sampling plans used for product acceptance be justified on the basis of recognized statistical principles. This book provides sufficient information on the statistical justification of these plans by means of the OC curves, AOQL table, and other information. The c=0 plans’ OC curves were calculated on the basis of the hypergeometric distribution. MIL-STD-105E, like the previous editions, utilized the binomial and Poisson distributions to calculate OC curves as an approximation of the hypergeometric distribution. Some Actual Data Based on the Results of a Large Receiving Inspection Department Using the c=0 Plans In the 1960s, a lengthy presentation was made to government officials on the c=0 plans. Although previous studies had shown lower appraisal costs and lower assembly failure quality costs, it was requested that one month’s current actual inspection results be taken from a large receiving inspection and the results were compared. Table 3 shows the results of 1141 lots inspected. Essentially, of the 46 lots rejected, 45 would have been rejected anyway if the MIL-STD-105D plans (whose sampling plans were the same as those in the “E” revision) had been used, because the number of allowable defects was exceeded. (We compared the corresponding sample sizes and acceptance numbers.) Table 4 shows the number of lots in which the sample size was the same as MIL-STD-105D, greater than 105D, and smaller than 105D. The reason for the “greater than” is explained later in light of the anomalies detected in 105D.

17

Table 3 Inspection results from a large receiving inspection department over a one-month period.

Number of lots inspected

Number of lots rejected

Number of lots in which number of defectives in c=0 sample exceeded acceptance number in MIL-STD-105D

1141

46

45

Number of lots in which number of defectives in c=0 sample did not exceed acceptance number in MIL-STD-105D 1*

*One defective in c=0, sample size 11. Corresponding MIL-STD-105D c=3, n=50.

Table 4  Receiving inspection figures for one month. Total number of lots inspected 1141

Number of lots in which c=0 sample size was The same as MIL-STD-105D

Greater than MIL-STD-105D

Smaller than MIL-STD-105D

430

30

681

37.7%

2.6%

59.7%

Table 5 continues the analysis in terms of pieces received. Although there were more pieces screened, there were lower total inspection costs. Table 6 shows the total number of characteristics inspected on the 1,250,762 pieces from the 1141 lots. It is here where the savings become more pronounced. Since the plans had been in use for some time prior to this study, a significant quality improvement in purchased material was realized. The c=0 plans, unlike plans with allowable defect numbers, do not allow or imply that producers may submit defective material, which eliminates or minimizes any misinterpretation about AQLs.

18

Table 5  Receiving inspection figures for one month. Total number of pieces received

Total number Inspection Inspection of pieces due to due to inspected Percent sampling screening Percent

Sample plan c=0

1,250,762 MIL-STD105D

50,032

4.0

30,672

19,360

1.5

80,436

6.4

62,332

18,104

1.4

Table 6  Receiving inspection figures for one month. Total number of characteristics on 1,250,762 pieces received

24,907,674

Sample plan

Total Number of Number of number of characteristics characteristics characteristics inspected due inspected due inspected Percent to sampling Percent to screening Percent

c=0

304,122

1.2

282,866

1.1

21,256

0.1

MIL-STD105D

1,013,489

4.1

993,179

4.0

20,310

0.1

ADJUSTMENTS FROM MIL-STD-105E/ANSI Z1.4 There were sampling plans in MIL-STD-105E/ANSI Z1.4 that contained apparent anomalies that were compensated for in the c=0 plans. These particular plans did not provide for the smooth progression required for less inspection on less critical characteristics. As such, the c=0 plans contain larger samples in those instances. Also, of course, where the acceptance number is zero in the MIL-STD-105E/ANSI Z1.4 plans, the sample size is the same in the c=0 plans. Take for example code letter “M” in the MIL-STD-105E/ANSI Z1.4 plans. Reading from more critical to less critical AQLs, we have the following sampling sizes: M=1250, 800, 500, 315, 200, 500, 315. Actually, both the 200 and following 500 are out of balance, but the 500 can be

19

adjusted mathematically to conform to the logic of the c=0 plans. Thus, for AQLs of 0.010-0.040, the sample sizes are the same in both the c=0 and MIL-STD-105E/ANSI Z1.4 sampling plans. At 0.065 AQL when M=200 occurs, the c=0 sampling plan sample size is 300, which is greater than the MIL-STD-105E/ANSI Z1.4 sample size. In 0.10 and up, the c=0 sample sizes are smaller. The other anomalies in MIL-STD-105E/ANSI Z1.4 occur at H-0.4 AQL, J-0.25 AQL, K-0.15, L-0.10, N-0.04, P-.025, and Q-.015. For example, take code letter “D” from MIL-STD-105E/ANSI Z1.4 at the 4.0% AQL. The sampling plan is N=13, C=1. An equivalent C=0 plan would require a sample size of 7. However, if 7 were used in place of 5 we would not have a smooth progression. Consideration was given to increasing the preceding sample size (i.e., the approach used for the 8 cases previously described) but because of the higher AQLs and closeness of the AOQLs, the apparent increase in risks was not considered to be of practical significance. If anyone should have a strong aversion to this, the following can be specified: N=4 (code letter C at 6.5), N=7 (code letter D at 4.0), N=11 (code letter E at 2.5), N=17 (code letter F at 1.5), and N=28 (code letter G at 1.0). If this is done and the logic of the smooth progression is to exist, then the preceding samples should be increased.

SAMPLING PLAN “SWITCHING” Considerable thought was given to tightened and reduced inspection. A table of c=0 plans for tightened inspection was calculated and prepared, but was never made a part of the c=0 plans. Actually, different OC curves are involved for tightened, normal, and reduced inspection. There were three reasons for not “switching” in the c=0 plans: Switching would have taken away from the simplicity of the plans; most companies were using the “normal” table, and perhaps correctly so; and switching assumes a continuing series of lots.

20

The original intent of going to tightened inspection is questionable in these days of quality programs. Tightened inspection was viewed as a means to force corrective action by increasing the chance of lot rejections. On reduced inspection, for all practical purposes, MIL-STD-105E/ ANSI Z1.4 plans are getting into non-statistical sampling plans. The main problem is that lots can be accepted that are much worse than the original intended plan. However, several requests were made for switching guidelines, and as a result, I have included tightened and reduced sampling inspection switching guidelines.

SWITCHING GUIDELINES In general, judgment is often used to determine tighter or reduced inspections when using the c=0 sampling plans. This could be accomplished by a periodic review of inspection results. Some items or characteristics, for example, might be considered so important that although there is no history of rejections, you may want to keep the same sample size. There may be other cases where you may experience just one rejected lot and want to go to 100 percent inspection until the cause of the problem is eliminated and a sufficient number of items, or means used, are taken to verify that the problem was corrected. In other cases, because of verifiable excellent process controls, you may just want to start off with an audit-type sample essentially equivalent to a reduced inspection. Based on your particular situation, including your process/product knowledge and other evidence of control, you can come up with your own guidelines. If, however, it is desired for any reason to use a tightened and reduced inspection sampling switching scheme, or it is required by contract or a customer representative, then a solution would be to use the next higher or lower index value of the c=0 sampling plans, and use the sampling plans in that particular column.

21

For example, for switching to tightened, if the present normal index level you are working with is 2.5, then the tightened index level would be 1.5. For switching from normal to reduced using the same index of 2.5 as an example, the reduced index value would be 4.0. The switching guidelines compatible with the MIL and ANSI plans are: 1.  Normal to tightened when two out of five consecutive lots are not acceptable 2. Tightened back to normal when five consecutive lots are accepted 3. Normal to reduced when the preceding 10 lots are accepted 4. Reduced to tightened when a lot is not acceptable

SMALL LOT SUPPLEMENT Many people have expressed a need for a special sampling plan table for small lots, when the associated AQL values they are using are 1.5 and below. Above 1.5, the main c=0 table works out well for small lot sizes. For the most part, any sampling plans developed for use with associated AQLs of less than 0.25 would not be valid. Therefore, in general, if you have broad lot size ranges and are using associated AQLs in the 0.25-1.5 range, the small lot size sampling table given in Table 7 should be of interest. Derivation The LQ for the small lot sizes was targeted to the LQ of the largest lot size range in which a constant sample size appeared. For example, looking at Table 1 we see that for an associated AQL of 1.0, 13 is used at the 51–90 lot size range; for 0.65, 20 is used at the 91–150 lot size range; and so forth. The sample sizes for the smaller lots shown in the supplement have essentially the same LQs as the targeted LQs. As in Table 1, the hypergeometric distribution is used. The hypergeometric is a discrete distribution and necessitates interpolation to arrive at a beta (consumer’s risk) of essentially 0.10.

22

Table 7  Small lot size supplement. Associated AQLS Lot size

.25

.40

.65

1.0

1.5

5–10

*

*

*

8

5

11–15

*

*

11

8

6

16–20

*

16

12

9

6

21–25

22

17

13

10

6

26–30

25

20

16

11

7

31–35

28

23

18

12

8

*Inspect 100 percent.

The small lot size supplement with lot sizes up to 35 does not give the same protection as MIL-STD-105E/ANSI Z1.4. However, it does give you the same protection as you would get on the targeted or larger lot sizes. Obviously, if we look at, for example, the 16–25 lot size in Table 1 we find that 100 percent inspection is required for the associated AQL of 0.40, and therefore our protection is 100 percent. Yet, the larger lot sizes involve samples. In Table 7, the sample size for a lot size of 21–25 with an associated AQL of 0.40 is 17. Thus, if we had a lot size of 25, we would inspect only 17 pieces—32 percent fewer—and get the same protection as if we were sampling in the lot size range of 151–280 of Table 1, or the MIL-STD-105E/ ANSI Z1.4 plans.

23

OPERATING CHARACTERISTIC CURVES AND VALUES

25

26

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

10

.10 63.7 46.7 26.0

2 3 5

30

Sample size

20

5

40

50

32.5 18.3

46.9

.25

.75

.90

18.3 10.0

8.33 3.33 5.00 2.00

27.5 12.50 5.00

.50

Probability of acceptance

Percent defective (P´)

3

.95 1.67 1.00

2.50

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.33 0.20

0.50

.99

60

2

70

Lot size 2–8

80

27

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

10

13

8

50

.75

.90

13

8.46

5.77

3.85

1.92 0.76

8.61 3.33 5.00 2.00 3.13 1.25

50.00 34.50 19.20 31.80 20.80 11.30 18.70 12.10 6.25

.50

3 5 8

.25

Probability of acceptance

Percent defective (P´)

40

66.00 48.30 28.30 12.90 5.00

.10

30

3

2

Sample size

20

5

.95

0.38

1.67 1.00 0.62

2.50

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.07

0.33 0.20 0.12

0.50

.99

60

2

70

Lot size 9–15

80

28

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

20

10

13

30

50

6.40

3.75

2.50

3.85

20

7.54

1.25 0.50

1.92 0.76

8.80 3.33 5.20 2.00 3.13 1.25

11.90

.90

13

.75

51.40 35.50 19.80 33.90 22.30 11.90 21.40 13.70 7.18

.50

67.00 49.00 28.70 13.10 5.04

.25

Probability of acceptance

Percent defective (P´)

40

3

3 5 8

.10

5

2

Sample size

20

8

.95

0.25

0.38

1.67 1.00 0.62

2.50

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.05

0.07

0.33 0.20 0.12

0.50

.99

60

2

70

Lot size 16–25

80

29

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

32

10

20

13

4.60

2.94

1.56 0.78

4.59 1.92 2.82 1.25

32

8.91 5.42

14.20 8.73

13 20

0.31

0.76 0.50

3.39 2.00 1.25

0.15

0.38 0.25

1.67 1.00 0.62

Probability of acceptance .25 .50 .75 .90 .95

50

3

Percent defective (P´)

40

52.50 36.30 20.20 8.97 35.40 23.20 12.40 5.38 23.20 14.80 7.72 3.31

.10

30

5

3 5 8

Sample size

20

8

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.03

0.07 0.05

0.33 0.20 0.12

.99

60

70

Lot size 26–50

80

30

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

80 50 32

20

3.17 1.23

50 80

0.93

1.98

3.48

0.62 0.31

1.00 0.50

1.80 0.78

4.86 2.05 3.06 1.29

5.68

9.44 5.99

32

.75

15.10 9.70

.50

13 20

.25

36.10 23.70 12.70 5.47 24.00 15.30 7.98 3.39

.10

0.12

0.20

0.31

0.76 0.49

2.04 1.26

.90

Probability of acceptance

Percent defective (P´)

24

8

5 8

Sample size

12

13

.95

0.06

0.10

0.15

0.38 0.25

1.00 0.62

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.01

0.02

0.03

0.07 0.50

0.19 0.12

.99

36

5

Lot size 51–90

48

31

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

80 32

13

.25

.50

.75

6.21 2.00

80

1.24

3.80

0.62 0.31

1.92 0.81

0.12

0.31

2.06 1.47 0.77 0.49

.90

Probability of acceptance

Percent defective (P´)

24

7

36.40 23.90 12.80 5.52 27.50 17.60 9.24 3.95 15.60 9.71 4.99 2.10 10.20 6.27 3.19 1.34

.10

32

5 7 13 20

Sample size

12

20

.95

0.06

0.15

1.01 0.71 0.38 0.24

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.01

0.03

0.19 0.14 0.07 0.05

.99

36

5

Lot size 91–150

48

32

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

125

50

32

10

6.55 4.10 1.39

50 125

6

.25

.50

.75

2.49 0.85

4.00

1.26 0.53 0.43 0.20

2.03 0.84

0.19 0.08

0.31

1.72 1.03 0.79 0.51

.90

Probability of acceptance

Percent defective (P´)

24

31.60 20.40 10.80 4.64 20.20 12.70 6.59 2.79 15.90 9.90 5.08 2.14 10.50 6.47 3.29 1.38

.10

13

32

6 10 13 20

Sample size

12

20

.95

0.09 0.04

0.15

0.84 0.50 0.38 0.24

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.02 0.00

0.03

0.16 0.09 0.07 0.05

.99

36

Lot size 151–280

48

33

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

125

50

29

24

4.28 1.59

125

7

.25

.50

0.96 0.48

2.60 1.31

0.20

0.54

4.00 2.55 1.76 0.95

.75

0.08

0.20

1.48 0.94 0.64 0.35

.90

Probability of acceptance

Percent defective (P´)

11

27.80 17.80 9.36 18.70 11.70 6.04 13.20 8.17 4.17 7.41 4.54 2.30

.10

50

7 11 16 29

Sample size

12

16

.95

0.04

0.10

0.72 0.46 0.31 0.17

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.02

0.14 0.09 0.06 0.03

.99

36

Lot size 281–500

48

34

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

125 75

47

27

.25

.50

2.93 1.73

125

1.05 0.52

1.77 0.89

0.21

0.37

2.57 1.49 1.05 0.59

.75

0.07

0.13

0.94 0.54 0.38 0.21

.90

Probability of acceptance

Percent defective (P´)

20

11

18.80 11.80 6.07 11.30 6.98 3.55 8.08 4.95 2.51 4.69 2.85 1.44

.10

75

11 19 27 47

Sample size

10

19

.95

0.03

0.06

0.46 0.26 0.18 0.10

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.01

0.09 0.05 0.03 0.02

.99

30

Lot size 501–1200

40

35

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

200 73

42

29

3.07 1.11

200

9

.25

.50

0.66 0.33

1.86 0.93

0.13

0.38

3.13 2.18 1.24 0.67

.75

0.05

0.14

1.16 0.80 0.45 0.24

.90

Probability of acceptance

Percent defective (P´)

20

22.40 14.20 7.37 16.10 10.10 5.17 9.46 5.82 2.95 5.29 3.22 1.62

.10

73

9 13 23 42

Sample size

10

13

.95

0.02

0.06

0.56 0.39 0.22 0.12

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.01

0.11 0.07 0.04 0.02

.99

30

Lot size 1201–3200

40

36

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 189 0 0 86 50 29

2.62 1.20

189

9

.25

.50

0.72 0.36

1.59 0.79

0.15

0.33

3.11 1.88 0.98 0.57

.75

0.05

0.12

1.15 0.69 0.36 0.20

.90

Probability of acceptance

Percent defective (P´)

20

22.10 14.10 7.32 14.00 8.74 4.48 7.56 4.64 2.35 4.47 2.72 1.37

.10

86

9 15 29 50

Sample size

10

15

.95

0.02

0.05

0.56 0.33 0.17 0.10

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.01

0.11 0.06 0.03 0.02

.99

30

Lot size 3201–10,000

40

37

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 189 0 0 77 46 29

2.91 1.20

189

9

.25

.50

0.72 0.36

1.77 0.89

0.15

0.37

3.02 1.85 0.97 0.61

.75

0.05

0.13

1.14 0.68 0.35 0.22

.90

Probability of acceptance

Percent defective (P´)

20

25.60 15.40 7.38 15.40 8.54 4.40 7.61 4.58 2.33 4.80 2.93 1.48

.10

77

9 15 29 46

Sample size

10

15

.95

0.02

0.06

0.55 0.33 0.17 0.11

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.01

0.10 0.06 0.03 0.02

.99

30

40

Lot size 10,001–35,000

38

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 170 0 0 96 56 29

2.40 1.35

170

9

.25

.50

0.81 0.40

1.44 0.71

0.16

0.29

3.20 1.92 0.98 0.51

.75

0.06

0.10

1.14 0.69 0.36 0.18

.90

Probability of acceptance

Percent defective (P´)

20

25.60 15.40 7.70 15.40 9.24 4.62 7.94 4.78 2.39 4.11 2.48 1.24

.10

96

9 15 29 56

Sample size

10

15

.95

0.03

0.05

0.55 0.33 0.17 0.09

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.01

0.10 0.06 0.03 0.01

.99

30

40

Lot size 35,001–150,000

39

Probability of acceptance (Pa)

100 90 80 70 60 50 40 30 20 10 0

0

156

64 29

156

9 15 29 64

Sample size

10

15

.25

.50

1.48

0.88 0.44

25.60 15.40 7.70 15.40 9.24 4.62 7.94 4.78 2.39 3.60 2.17 1.07

.10

0.18

3.20 1.92 0.99 0.44

.75

0.06

1.08 0.66 0.35 0.10

.90

Probability of acceptance

Percent defective (P´)

20

9

.95

0.03

0.52 0.32 0.17 0.07

OC curves for single sampling plans Acceptance number equal to zero (Sample size as indicated)

0.00

0.10 0.06 0.03 0.01

.99

30

40

Lot size 150,001–500,000

INDEX

Page numbers followed by f or t refer to figures or tables, respectively. A acceptable quality level (AQL), xi defined, 5 acceptance numbers, 11 ANSI Z1.4, 4, 12 adjustments from, 19 AOQL. See average outgoing quality limits (AOQLs) AQL. See acceptable quality level (AQL) attribute sampling plans, 3 average outgoing quality limits (AOQLs), 13, 14f, 15t C c=0 sampling plans, xi–xiv, 10t

c=0 sampling plans table, use of, 12 constant sample sizes, reasons for not using, 11 consumer’s risk, 6, 7f, 9, 22 continuous sampling, 3 D defect, defined, 2 defective, defined, 2 I inspection results, 18t L limiting quality (LQ) percentages, xi lot-by-lot attributes, 3 lot-by-lot variables, 4 M MIL-Q-9858A, 14 MIL-STD-105, xi–xiv adjustments from, 19 MIL-STD-414, 4 MIL-STD-1235C, 3

41

N nonstatistical sampling plans, 4–5 O operating characteristic (OC) curves, 5, 7f, 8f for single sampling plans, 26–39 lot size 2–8, 26 lot size 9–15, 27 lot size 16–25, 28 lot size 26–50, 29 lot size 51–90, 30 lot size 91–150, 31 lot size 151–280, 32 lot size 281–500, 33 lot size 501–1200, 34 lot size 1201–3200, 35 lot size 3201– 10,000, 36 lot size 10,001– 35,000, 37

lot size 35,001– 150,000, 38 lot size 150,001– 500,000, 39 P potential savings, estimating, 11 producer’s risk, 5 R risk, producer’s, 5 S samples, physically taking, 13 sample sizes, reasons for not using constant, 11 sampling continuous, 3 lot-by-lot, 3 objective of, 1 sampling plans. See also single sampling plans, OC curves for

attribute, 3 nonstatistical, 4–5 for small lots, 22 switching, 20 savings, estimating potential, 11 single sampling plans, OC curves for, 26–39. See also sampling plans lot size 2–8, 26 lot size 9–15, 27 lot size 16–25, 28 lot size 26–50, 29 lot size 51–90, 30 lot size 91–150, 31 lot size 151–280, 32 lot size 281–500, 33 lot size 501–1200, 34 lot size 1201–3200, 35 lot size 3201– 10,000, 36

42

lot size 10,001– 35,000, 37 lot size 35,001– 150,000, 38 lot size 150,001– 500,000, 39 small lots, sampling for, 22 switching guidelines, 21 sampling plans, 20 V variables-type sampling plan, 4 W “withhold the lot” phrase, significance of, 2 Z zero acceptance number plans, derivation of, 1 zero defects, xii

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