Reliability and Decision Making PDF
Preview Reliability and Decision Making
RELIABILITY AND DECISION MAKING RELIABILITY AND DECISION MAKING Edited by RICHARD E. BARLOW University of California, Berkeley, California, USA CARLO A. CLAROTTI ENEA (National Committee for New Technologies, Energy and Environment, Rome, Italy) FABIO SPIZZICHINO University of Rome-La Sapienza, Rome, Italy 1m SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y. First edition 1993 © Springer Science+Business Media Dordrecht 1993 Originally published by Chapman & Hall in 1993 Softcover reprint of the hardcover 1st edition 1993 Typeset in Malta by Interprint Limited ISBN 978-0-412-53480-5 ISBN 978-1-4899-4459-7 (eBook) DOI 10.1007/978-1-4899-4459-7 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data available Preface This Volume gathers together papers given at the conference held in Siena, Italy, from October 15 through October 26, 1990. The aim of the Volume is more ambitious than just to provide a published account of the proceedings of the conference. The aim is to supply the reliability community with a coherent Bayesian predictive view on reliability theory and practice. We may ask: what is the predictive Bayesian approach to reliability? what makes the predictive Bayesian approach to reliability a preferred, if not a compulsory, choice? We shall try to make a long story short. The editors and most of the contributors to this Volume have recognised that the ultimate purpose of reliability analysis is to take decisions relative to engineering systems in the face of uncertainty. In short: any system of concern is prone to failure no matter which one of the conceivable design-solutions is implemented; system failure (success) will cause a loss (reward) to be incurred (cashed). The following decision problem is imposed upon the decision maker: which design solution should be implemented? Note that 'keeping the status quo', i.e. not building the system, is not a 'conservative' solution. Any decision-maker (a business-man, a producer or other) who system atically misses out on the chance of earning money, just because there is a remote possibility of losing money, will eventually go bankrupt. The decision maker needs a measure of his uncertainty to assess whether or not the possibility of failure is 'remote enough' with respect to the possible loss and the possible reward. He acts then in a betting frame work with the restriction that he will not consider a bet which promises certain loss. This restriction is the principle which forces us to assess uncertainty by probability and to use the calculus of probability in making decisions. No other measure of uncertainty guarantees the decision maker that his bet will never result in certain loss. v vi Preface Adopting the predictive Bayesian approach to reliability means that we must coherently apply the laws of subjective probabilities to reliability decision problems. This approach requires radical changes in current probabilistic/statistical techniques used in reliability practice. In particular: I. Only observable quantities are the 'random variables par excellence': parameters of probability distributions are just a computational aid for deriving the conditional probability of future observations given the values of the past ones. II. In any reliability decision problem, the related utility function must be a function of the random variables (e.g. the failure time of the system) whose actual values will cause the loss to be incurred or the reward to be cashed. The utility function cannot depend upon abstract variables (such as the failure rate) whose value is still unknown when the system has failed or accomplished its mission. In view of the Maximum Expected Utility (MEU) principle, taking reliability decisions via (point or interval) estimation of probability-distribution-parameters does not make then any sense. III. Stochastic dependence must be understood as synonymous with the possibility of learning and not necessarily as synonymous with physical dependence. IV. Any stochastic model must be expressed in terms of the joint probability distribution for the random variables of interest. Exchangea bility may playa central role in this analysis. V. The probabilistic concept of aging must be rethought. VI. In deriving the joint distribution of any set of life-times, one must use an approach which accounts for the dynamical character of the information. VII. The methods of optimal control theory are often needed. The use of such methods is needed whenever the statistical decision problem of concern is sequential in continuous time. Reliability decision problems are usually continuous time decision problems and, in the most general case, they are sequential. Adopting the predictive Bayesian approach to reliability is (in our view) not only compelling, but also convenient. It is compelling in that it is the only way to avoid 'Dutch Books'. It is convenient because it simplifies and unifies several major reliability problems. In a Bayesian predictive framework: there is no substantial difference between a system reliability problem and a problem of statistical analysis of failure and survival data observed in the past; Preface vii all the well known bounds on system reliability can be exploited also when probability distribution parameters are unknown; burn-in and life-testing can be regarded as particular design solutions of a more general decision problem; preferability for different decisions can be stated without condition ing on the value of some non-observable quantity; the terms 'uncertainty about the model' and 'uncertainty about the values of probabilities' can be rephrased in a rational way. This leads to the recognition that there is no difference between the 'two types of uncertainty'. The volume has been divided into four sections: 1. Fundamentals of Statistical Decision Theory; 2. Sequential Problems in Reliability: a Dynamical Approach; 3. Bayesian Models in Reliability and Quality Control; 4. Engineering Reliability. This has been done to permit selective reading. Those who are only interested in becoming acquainted with the motivation for the Bayesian predictive approach to reliability, and with its implementation in the case in which there is no forthcoming information can just look at Sections 1 and 4. In Section 1, the following issues are addressed: frequentists claim that they base their decisions only on the solid objective grounds of data; this claim is false since in orthodox statistics the likelihood principle is violated (see the papers by Clarotti and by Piccinato); avoidance of Dutch Books is attained if subjective probability is used for making decisions. This is shown in the paper by Bruce Hill where in addition he discusses: (1) how to use theorems derived for unending sequences of random variables when dealing with finite sets of random variables; and (2) the role of sensitivity analysis in the Bayesian approach; the influence-diagrams tool is the 'user-friendly' (see the article by Barlow and Pereira) version of the predictive Bayesian approach to reliability. Section 2 is rather technical but it is at the core of the Bayesian predictive approach: as stated any reliability problem is a decision problem in the face of uncertainty; viii Preface uncertainty can be mitigated by new evidence; the latter has in general a dynamical character; often one has some control over the development of new informa tion; the first decision to be considered is: what actions will result in new information which is optimal in the sense of expected cost? Optimal stopping of Markov processes is used by Spizzichino to answer this question when in particular one has to decide the sampling plan of a batch of new similar units. in solving this problem, the central role of the concept of Multi variate Conditional Failure Rate introduced by Shaked and Shanthikumar becomes evident; more generally the need of answering the question above, and the dynamical character of statistical evidence, requires that theories of stochastic control and of point processes now play an important role in the reliability theory. The reader is given an insight into these disciplines and into their reliability applica tions by the papers by Arjas, Gerardi and Koch, and Runggaldier. In Section 3 everyone (we hope) can find at least one Bayesian model specific to his/her own field of interest: Barlow and Mendel discuss a new probabilistic notion of aging based on the Bayesian approach; Polson, Singpurwalla and Verdinelli discuss designs for acceler ated life tests; they describe how an inferential problem can be tackled as a decision problem (remember anyway that a reliabi lity decision problem cannot be reconducted to an inferential problem); Barlow and Irony discuss a Bayesian approach to quality control; Singpurwalla discusses Taguchi methods from a Bayesian decision theory point of view; Muliere and Scarsini illustrate the change point problem and its role in quality control. In Section 4 the theory of coherent structures is considered from different points of view. This makes it apparent that: in a frequentist frame system optimisation can be achieved when the related probability-distribution-parameters are completely specified (Boland discusses majority voting in this context); Preface ix the approach based on the Bayesian estimation of parameters not only violates the MEV principle but also requires a great deal of mathematical skill to produce results valid under stringent hy potheses (e.g. series system formed from non-identical components, as shown in the paper by Gertzbach and Kordonski); the predictive Bayesian approach is the 'natural' companion of the fault-tree technique (Clarotti). In Section 4 furthermore Apostolakis and Wu review the debate in the safety engineering community concerning the interpretation of probability; the intriguing matter of prior knowledge in engineering applica tions is discussed by Natvig. We hope this volume succeeds in providing the information necessary to use the Bayesian predictive approach in reliability. Richard E. Barlow, Carlo A. Clarotti, Fabio Spizzichino Acknowledgements The editors efforts to circulate Bayesian predictive ideas about reliability theory and practice would have been useless had the conference 'Reliabil ity and Decision Making' not been held. The Conference was directed by C. A. Clarotti, G. Koch and F. Spizzichino and was organised under the auspices and the financial support of University of Siena and of Comitato Nazionale per Ie Scienze Matematiche of C.N.R. (National Board for Science in Italy). Particular thanks are to be expressed to Prof. Luigi Berlinguer (Rector of the University of Siena), to Prof. Carlo Ciliberto (President of Comitato Nazionale per Ie Scienze Matematiche of C.N.R.) and to Prof. Carlo Cercignani (President of the Subcommittee for Applied Mathematics of C.N.R.). Their sensibility for initiatives in the field of applied mathematics made it possible to mobilise resources for the Conference. The Editors are also much in-debted to their friend Giorgio Koch for acting Co-director of the Conference and to Prof. Vincenzo Millucci (Dept. of Mathematics, University of Siena) who strongly pleaded the cause of the Conference at University of Siena. xi
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