CONTROL CHARTS
CONTROL CHARTS J. MURDOCH Head of Statistics and Operational Research Unit, Cranfield Institute of Technology, Bedford M
J. Murdoch 1979 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any mean's, without permission First published 1979 by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in Delhi Dublin Hong Kong Johannesburg Lagos Melbourne New York Singapore and Tokyo Typeset by Reproduction Drawings Ltd, Sutton, Surrey British library Cataloguing in Publication Data Murdoch, John, b. 1926 (July) Control charts. 1. Production control-graphic methods I. Title 658.5'68 TS157 ISBN 978-0-333-26412-6 ISBN 978-1-349-16238-3 (ebook) DOI 10.1007/978-1-349-16238-3 This book is sold subject to the standard conditions of the Net Book Agreement The paperback edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser
Contents Preface Principal Symbols vi vii 1 Process Variation 2 Principles of Control 5 3 Statistical Theory in Control Chart Design 9 4 Process Capability 20 5 Shewhart Type Control Charts 36 6 Introduction to CuSum Control Charts 56 7 Design of CuSum Control Charts 76 8 Comparison of Shewhart and CuSum Cantrol Systems 100 9 Introduction to Sampling Inspection Schemes-Single Attribute Schemes 106 Appendix: Statistical Tables 126 Index 149
Preface This book is designed as a practical manual aimed at giving students and management personnel an understanding of the design and use of control charts. The text deals with both the conventional Shewhart type (or British Standard) Charts and the CuSum control charts. Comparisons are included of the operating efficiency ofthe~e control schemes at detecting changes. Problems for solution are given at the end of the chapters and fully worked solutions to these problems are included, thus enabling readers to test their understanding at each stage. The control systems covered in this book are general in their field of application, in that, while usually associated with controlling quality, they can in fact be used to control any management statistic, such as overheads, maintenance, cost, fuel consumption, etc. In this field of management control, the CuSum system shows clearly its superiority over the Shewhart system in that it is equally suitable for a sample size of one which is usually the requirement for general management control systems. The inclusion of Chapter 6, covering the use of the CuSum system as a diagnostic technique is special to this book and a range of practical examples are used to demonstrate the power of the CuSum as a diagnostic management tool, including the now well-known Cranfield Case Study, by my colleague J. A. Barnes, on monitoring a vehicle's fuel consumption. Again, the development of nomograms and tables has simplified the design of both variable and attribute CuSum control systems and Chapter 7 gives full details of the use of these new nomograms and tables. Since it is usual to link quality control charts to a sampling inspection scheme, an introduction to the principles of the design of sampling inspection schemes is given in the fmal chapter of the book. A knowledge of basic statistics including Binomial, Poisson and Normal distributions and significance testing is assumed in readers. However, a revision chapter (Chapter 3) is included, demonstrating the use of statistical theory in control chart design. For readers wishing to revise their basic statistical knowledge, Statistics: Problems and Solutions, published by The Macmillan Press, is recommended. The relevant statistical tables which appear in the appendix at the end of the book are reproduced from Murdoch and Barnes' Statistical Tables for Science, Engineering, Management and Business Studies (The Macmillan Press). Oanfield, 1979 J.M.
Principal Symbols Roman A and A' multipliers of a and w (respectively) to give Shewhart (British) chart limits for process average A 2 multiplier for Shewhart (American) charts to give limits for process average b vertical width of CuSum mask (in CuSum units) c acceptance number in single sampling inspection plans D and D' multipliers of a and w (respectively) to give Shewhart (British) chart limits of range D 3 and D 4 multipliers for lower and upper limits of range-american-type Charts d horizontal width CuSum mask (units= no. of samples) dn range conversion factor, a= wfdn h decision interval, CuSum control (CuSum units) I average amount of inspection/batch k reference value CuSum control L 0 Average Run Length to detection at acceptable quality level (AQL) L 1 Average Run length to detection at reject quality level (RQL) m average defects/sample m 1 average defects/sample at acceptable quality level p 1 m 2 average defects/sample at reject quality level p 2 n sample size N batch size p batch fraction defective Po process capability (fraction defective) for attribute control p1 acceptable quality level (fraction defective) P2 reject quality level (fraction defective) P(A) probability of accepting the batch P(R) probability of rejecting the batch = 1 - P(A) R ratio m2/m1 = P2/P1 s sample standard deviation r Sr CuSum value after r readings= 1::(x;- k) U standardised normal variate i= 1 w range of a sample w average range
viii PRINCIPAL SYMBOLS x variable x sample average X sample average in Shewhart control Greek a risk of rejecting batches of acceptable quality level {3 risk of accepting batches of reject quality level 8 CuSum mask angle C) a 0 process capability Shewhart schemes (variable schemes) a process capability CuSum schemes (variable schemes) f.l.o acceptable quality level (variable schemes) f.l. 1 reject quality level (variable schemes) x 2 chi-squared value Author's Note The use of an asterisk above a table number signifies that the table number is a Statistical Table in the Appendix at the end of this book. Thus, table 2* refers to Statistical Table 2 in the appendix-namely, the Normal Distribution.