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Fuzzy theory in reliability analysis
Bulletin
361
2.2. Reportfrom CMCECH, Handan, China The Coal Mining and Civil Engineering College of Hebei (CMCECH) is one of the Institutes and Universities who have paid attention to the study of f u ~ / s e t s and its applications. I, myself, am primarily interested in the problems of Fuzzy AIf~ebra, such as fuzzy ideals, maximum fuzzy ideals, prime and primary fuzzy ideals, etc. I a~n also interested in fuzzy expert systems and application of fuzzy sets techniques to enginearin~ science and technology. My research in these areas has appeared in Fuzzy Sets and Sys~ems, Busefal, Prec. of NAFIPS'86, Proc. of 2rid IFSA Congress and in Chinese journals (Fuzzy Mathematics, Journal of Mathematical Practice and Recognition, Nature Journal and so on). Some of my colleagues in our Department and other Depsrtme,ts (at CMCECH) are working on the application of fuzzy set techniques to Operations research, decision theory, pattern recognition, engineering and management. I am studying for a Ph.D degree on Structural Mechanics under the direction of Prof. Wang Guangyuan at the Research Institute of Engineering Science and Technology of the Harbin Architectural and Civil Engineering Institute. Recently my interest is directed to the following problems: (1) theory of fuzzy stochastic processes, (2) fuzzy stochastic processes in dynamics, (3) fuzzy random vibrations. I am keen to establish contacts with other researchers who are also interested in the areas mentioned above. Zhang Yue Research Institute of Engineering Science and Technology Harbin Architectural and Civil Engineering Institute P.O. Box 0320 Harbin, China
3. Fuzzy Theory in Rei|ab|i|ty Ana|ysis I have studied applications of fuzzy theory to reliability analysis for several years. At the start I felt uncertain whether fuzzy theory was useful to reliability analysis since reliability analysis was based on probability theory. I also have heard from Prof. Y. Nishiwaki (Imernational Atomic Energy Agency) that many researchers and practitioners who use probabiliW theory in reliability analysis feel uncertain of the usefulness of fuzzy theory to reliahility analysis and that they say they don't know why fuzzy theory is a better tool than probability theory. In the following I would like to explain the usefulness of fuzzy theory to reliability analysis during my research. (1) It is necessary to collect data to estimate the failure probability. However, in practice, it is not likely that enough data can be collected to es." hate the failure probability. So estimation of the failure probability cannot help being dependent on engineering judgement. (2) The determination of a safety criterion is also dependent on engineering judgement, if the failure probability of the system is estimated at 10-e [1/year], can this system be assumed to be safe? If the system is assessed to be safe, does the assessment imply that the system has never a possibility to break down for lOeyears? From the above point of view the concept of a 'failure possibility' has been proposed instead of the failure probability [1]. The failure possibility represents that even if the failure probability of the system is estimated to be very small, there is a possibility that the system may break down. The failure possibili~r is c~,ived from estimation of the failure probability based on a safety criterion [2]. Much information with respect to system re,ability can be gained from the failure possibility.
362
Bulletin
Even in a highly automated system, a human being is still indispensable to the system. So it is necessary to include human reliability in the system reliability analysis. However: (3) It is not likely that enough data can be collected to estimate the error probability. Estimation of the ~rror probability is more dependent on engineering judgement than that of the failure probability. In human reliability analysis the concept of 'error possibility' has been proposed instead of the error probability [3, 4]. The error possibility implies that even if the error probability is estimated to be very small from the viewpoint of human reliability, there is a possibility that a human error may occur. The error possibility is derived from estimation of the error probability based on a safety criterion. A comparison between the results of human reliability analysis by the use of the error possibility and that by the use of the error probability has been performed and the validity of the fuzzy reliability analysis has been shown [5]. (4) Human performance is affected by many factors (called performance shaping factors), e.g., work environment, psychological stress, fatigue, competence, etc. The error probability is usually modified by experts based on their engineering judgement in order to consider the effect of performance shaping factors. Although the relation between human reliability and performance shaping factors may be expressed qualitatively, the relation is not completely certain. This relation has been analyzed [6] using fuzzy measure analysis 17]. Furthermore, human reliability analysis has been performed considering performance shaping factors [8]. In this study the relation between performance shaping factors and human reliability is expressed qualitatively and human reliability is assessed by the use of fuzzy reasoning [9|. The error possibiliW is derived from not only estimation of the error probability but also estimation of performance shaping factors. (5) It is necessary to consider both equipment reliability and human reliability in the system reliability analysis as mentioned above. However the failure probability and the error probability should not be regarded as the same measure since both are heterogeneous. It is meaningless to combine the failure and the error pobabilitics in order to analyze system reliability. The failure possibility and the error possibility are regarded as the same measure from the viewpoint of the reliability analysis. System reliability analysis has been performed using the failure possibility and the error possibility [10]. Equipment reliability is also affected by many factors. System reliability analysis has been performed considering many factors which affect equipment reliability or human reliability [11]. Sensitivity analysis in fuzzy reliability analysis has been performed to show important factors to be ameliorated in order to raise system reliability [12], Recently I have become interested in the connection of both quantitative and qualitative reliability analysis using fuzzy theory,
References (1] T. On)saws and M. Sugeno, An approach to failure analysis using fuzzy theory, Trans. of the Society of Instrument end Control Engineers20(6) (1984) 498-505 (in Japanese). [2] T. Onisawa, Reliability estimation from various subjective standpoints, Preprints of2nd IFSA Congress, Tokyo, Vol. 2 (1987) 793-796. [3] 1". On)saws, An approach to human reliability in man-machine systems using error possibility, Fuzzy Sets and Systems 27(2) (1988) 87-103. [4] T. On)saws, A representation of human reliability using fuzzy concepts, Information Sciences 46(2) (1988) 153-173. [5] 1". On)saws and Y. Nishiwaki, Fuzzy human reliability analysis on the Chernobyl accident, Fuzzy Sets and Systems •(2) (1988) 115-127. (6J 1". Onisawe, Fuzzyconcepts in human reliability, in: M.M. Gupta and T. Yamakawa,Eds., Fuzzy Logic in Knowledge-BasedSystems, Decision and Control (E~eviGr, Amsterdam, 1988) 317-33~. [7] 1", On)saws, M, Sugeno, Y, Nishiwaki, H. Kawai and Y. Her)ms, Fuzzy measure analysis of public attitude towards the use of nuclear energy, Fuzzy Sets and Systems20(3) (1986) 255-289. [8] T, On)saws, An application of fuzzy theory in human re,ability analysis considering performance shaping factors, Proc. of the Fourth Symposium on Human Interface Tokyo (1988) 227-234.
Bulletin
363
[9] T. Onisaws, Human reliability assessmentw~th fuzzy reasoning, Trans. of the Society of Instrument and Control Engineers 24(12) (1988) 1312-1319. [10] 1". Onisawa, Fuzzy theory in analysis of system reliability, Proc. of the 26th The Society of Instrument and Control Engineers Annual Conference, Hiroshima, Vol. 2 (1587) 1257-1260. [11] T. Onisaws0 Reliabi,~iW assessment of a man-machine system considering many factors, Proc. of International Wo~shop on Fuzzy System Application, lizuka (1988) 91-92. [12] T. Onisawa, Sensitivity analysis in fuzzy reliability analysis, Proc. of Korean Automatic Control Conference, Seoul, Vol. 2 (1~)88)764-769. T~kehisa Onisawe ~::~ i ~ r t m e n t o f Basic Engineering ~, rnamoto University 2-39-1, Kurokami Kumamoto, 860 Japan
4. Book Reviews
4. I. Fuzzy Mathematical Models in Engineering and Management Science Despite its title, which I found somewhat misleading, this book by A. Kaufmenn and M. Gupt8 (North-Holland, Amsterdam, 1988; ISBN O.-,H4..-70501-5) is essentially about fuzzy arithmetic with applications in Operations Research (OR)- or Management Science in the American language. There is little that could soundly be described as engineering content, although most engineers these days will be conversant with and use OR techniques. There is, for example, no mention of the recent successful applications of fuzzy sets in engineering control. The book is divided into two parts, the first on the theory of fuzzy arithmetic and the second on applications. Part I, after a brief justification of the endeavour of fuzzy modelling, gives the briefest introduction to fuzzy set theory, referring the reader to other works for details. It then covers in more detail fuzzy numbers and, in particular, triangular and trapezoidal fuzzy numbers. Chapter 4 looks at the different ways of well ordering fuzzy numbers, and the next chapter covers such issues as their entropy and degrees of fuzziness. Reflecting the importance given to triangu!ar fuzzy numbers (TFNs) in the appiicatio,s section, the next chapter shows how to approximate the results of non-linear operations on TFNs with TFNs. The remainder of the section covers the authors' own theory of 'deconvolution', t-norms, and fuzzy numbers in the unit interval. It ends with some good tutorial material on the solution of fuzzy simultaneous equations end inequalities. The second part purports to cover applications, but as the authors admit "actual problems...ars generally too complex and too long, and for this reason...illustrate the strength of fuzzy set theory by using relatively simple situations". They also point out that most of the examples are based on their own work (which is heavily referenced) so that much of the reference material is concerned with the classical (i.e. non-fuzzified) theory. From a pedagogical point of view this approach is a little wcrrying, introducing as it must a slightly one-sided view of work in areas where other fuzzy set theorists have worked in some cases but who are not acknowledged in this text. The redeeming feature of this section is that it is packed with wonderfully clear and thorough worked examples, which made me wonder whether tt.,a authors intend this book to be 8 student text or a monograph. I would have thought it more suitable as the former, if perhaps a little expensive for most students' pockets. The applications covered are all fuzzificetions of well known OR techniques, zero-based budgeting, long term (Delphic) forecasting, discounted cash flow with uncertain rates of interest, forecasting fuzzy time series (exponential smoothing), possibilistic reliability analysis, financial project evaluation, optimisation and so on. in all cases the emphasis is on tiie fuzzification of the problem anO the use of fur.zy numbers. Aiso every problem is explaia~ed clearly bv the use of simple, step-by-step numerical examples. This accords with the authors' stated aim o~ encouraging r¢:d~rs to attack their own problems by producing computer programs on the basis of the lessons 6earnt from their examples.
361
2.2. Reportfrom CMCECH, Handan, China The Coal Mining and Civil Engineering College of Hebei (CMCECH) is one of the Institutes and Universities who have paid attention to the study of f u ~ / s e t s and its applications. I, myself, am primarily interested in the problems of Fuzzy AIf~ebra, such as fuzzy ideals, maximum fuzzy ideals, prime and primary fuzzy ideals, etc. I a~n also interested in fuzzy expert systems and application of fuzzy sets techniques to enginearin~ science and technology. My research in these areas has appeared in Fuzzy Sets and Sys~ems, Busefal, Prec. of NAFIPS'86, Proc. of 2rid IFSA Congress and in Chinese journals (Fuzzy Mathematics, Journal of Mathematical Practice and Recognition, Nature Journal and so on). Some of my colleagues in our Department and other Depsrtme,ts (at CMCECH) are working on the application of fuzzy set techniques to Operations research, decision theory, pattern recognition, engineering and management. I am studying for a Ph.D degree on Structural Mechanics under the direction of Prof. Wang Guangyuan at the Research Institute of Engineering Science and Technology of the Harbin Architectural and Civil Engineering Institute. Recently my interest is directed to the following problems: (1) theory of fuzzy stochastic processes, (2) fuzzy stochastic processes in dynamics, (3) fuzzy random vibrations. I am keen to establish contacts with other researchers who are also interested in the areas mentioned above. Zhang Yue Research Institute of Engineering Science and Technology Harbin Architectural and Civil Engineering Institute P.O. Box 0320 Harbin, China
3. Fuzzy Theory in Rei|ab|i|ty Ana|ysis I have studied applications of fuzzy theory to reliability analysis for several years. At the start I felt uncertain whether fuzzy theory was useful to reliability analysis since reliability analysis was based on probability theory. I also have heard from Prof. Y. Nishiwaki (Imernational Atomic Energy Agency) that many researchers and practitioners who use probabiliW theory in reliability analysis feel uncertain of the usefulness of fuzzy theory to reliahility analysis and that they say they don't know why fuzzy theory is a better tool than probability theory. In the following I would like to explain the usefulness of fuzzy theory to reliability analysis during my research. (1) It is necessary to collect data to estimate the failure probability. However, in practice, it is not likely that enough data can be collected to es." hate the failure probability. So estimation of the failure probability cannot help being dependent on engineering judgement. (2) The determination of a safety criterion is also dependent on engineering judgement, if the failure probability of the system is estimated at 10-e [1/year], can this system be assumed to be safe? If the system is assessed to be safe, does the assessment imply that the system has never a possibility to break down for lOeyears? From the above point of view the concept of a 'failure possibility' has been proposed instead of the failure probability [1]. The failure possibility represents that even if the failure probability of the system is estimated to be very small, there is a possibility that the system may break down. The failure possibili~r is c~,ived from estimation of the failure probability based on a safety criterion [2]. Much information with respect to system re,ability can be gained from the failure possibility.
362
Bulletin
Even in a highly automated system, a human being is still indispensable to the system. So it is necessary to include human reliability in the system reliability analysis. However: (3) It is not likely that enough data can be collected to estimate the error probability. Estimation of the ~rror probability is more dependent on engineering judgement than that of the failure probability. In human reliability analysis the concept of 'error possibility' has been proposed instead of the error probability [3, 4]. The error possibility implies that even if the error probability is estimated to be very small from the viewpoint of human reliability, there is a possibility that a human error may occur. The error possibility is derived from estimation of the error probability based on a safety criterion. A comparison between the results of human reliability analysis by the use of the error possibility and that by the use of the error probability has been performed and the validity of the fuzzy reliability analysis has been shown [5]. (4) Human performance is affected by many factors (called performance shaping factors), e.g., work environment, psychological stress, fatigue, competence, etc. The error probability is usually modified by experts based on their engineering judgement in order to consider the effect of performance shaping factors. Although the relation between human reliability and performance shaping factors may be expressed qualitatively, the relation is not completely certain. This relation has been analyzed [6] using fuzzy measure analysis 17]. Furthermore, human reliability analysis has been performed considering performance shaping factors [8]. In this study the relation between performance shaping factors and human reliability is expressed qualitatively and human reliability is assessed by the use of fuzzy reasoning [9|. The error possibiliW is derived from not only estimation of the error probability but also estimation of performance shaping factors. (5) It is necessary to consider both equipment reliability and human reliability in the system reliability analysis as mentioned above. However the failure probability and the error probability should not be regarded as the same measure since both are heterogeneous. It is meaningless to combine the failure and the error pobabilitics in order to analyze system reliability. The failure possibility and the error possibility are regarded as the same measure from the viewpoint of the reliability analysis. System reliability analysis has been performed using the failure possibility and the error possibility [10]. Equipment reliability is also affected by many factors. System reliability analysis has been performed considering many factors which affect equipment reliability or human reliability [11]. Sensitivity analysis in fuzzy reliability analysis has been performed to show important factors to be ameliorated in order to raise system reliability [12], Recently I have become interested in the connection of both quantitative and qualitative reliability analysis using fuzzy theory,
References (1] T. On)saws and M. Sugeno, An approach to failure analysis using fuzzy theory, Trans. of the Society of Instrument end Control Engineers20(6) (1984) 498-505 (in Japanese). [2] T. Onisawa, Reliability estimation from various subjective standpoints, Preprints of2nd IFSA Congress, Tokyo, Vol. 2 (1987) 793-796. [3] 1". On)saws, An approach to human reliability in man-machine systems using error possibility, Fuzzy Sets and Systems 27(2) (1988) 87-103. [4] T. On)saws, A representation of human reliability using fuzzy concepts, Information Sciences 46(2) (1988) 153-173. [5] 1". On)saws and Y. Nishiwaki, Fuzzy human reliability analysis on the Chernobyl accident, Fuzzy Sets and Systems •(2) (1988) 115-127. (6J 1". Onisawe, Fuzzyconcepts in human reliability, in: M.M. Gupta and T. Yamakawa,Eds., Fuzzy Logic in Knowledge-BasedSystems, Decision and Control (E~eviGr, Amsterdam, 1988) 317-33~. [7] 1", On)saws, M, Sugeno, Y, Nishiwaki, H. Kawai and Y. Her)ms, Fuzzy measure analysis of public attitude towards the use of nuclear energy, Fuzzy Sets and Systems20(3) (1986) 255-289. [8] T, On)saws, An application of fuzzy theory in human re,ability analysis considering performance shaping factors, Proc. of the Fourth Symposium on Human Interface Tokyo (1988) 227-234.
Bulletin
363
[9] T. Onisaws, Human reliability assessmentw~th fuzzy reasoning, Trans. of the Society of Instrument and Control Engineers 24(12) (1988) 1312-1319. [10] 1". Onisawa, Fuzzy theory in analysis of system reliability, Proc. of the 26th The Society of Instrument and Control Engineers Annual Conference, Hiroshima, Vol. 2 (1587) 1257-1260. [11] T. Onisaws0 Reliabi,~iW assessment of a man-machine system considering many factors, Proc. of International Wo~shop on Fuzzy System Application, lizuka (1988) 91-92. [12] T. Onisawa, Sensitivity analysis in fuzzy reliability analysis, Proc. of Korean Automatic Control Conference, Seoul, Vol. 2 (1~)88)764-769. T~kehisa Onisawe ~::~ i ~ r t m e n t o f Basic Engineering ~, rnamoto University 2-39-1, Kurokami Kumamoto, 860 Japan
4. Book Reviews
4. I. Fuzzy Mathematical Models in Engineering and Management Science Despite its title, which I found somewhat misleading, this book by A. Kaufmenn and M. Gupt8 (North-Holland, Amsterdam, 1988; ISBN O.-,H4..-70501-5) is essentially about fuzzy arithmetic with applications in Operations Research (OR)- or Management Science in the American language. There is little that could soundly be described as engineering content, although most engineers these days will be conversant with and use OR techniques. There is, for example, no mention of the recent successful applications of fuzzy sets in engineering control. The book is divided into two parts, the first on the theory of fuzzy arithmetic and the second on applications. Part I, after a brief justification of the endeavour of fuzzy modelling, gives the briefest introduction to fuzzy set theory, referring the reader to other works for details. It then covers in more detail fuzzy numbers and, in particular, triangular and trapezoidal fuzzy numbers. Chapter 4 looks at the different ways of well ordering fuzzy numbers, and the next chapter covers such issues as their entropy and degrees of fuzziness. Reflecting the importance given to triangu!ar fuzzy numbers (TFNs) in the appiicatio,s section, the next chapter shows how to approximate the results of non-linear operations on TFNs with TFNs. The remainder of the section covers the authors' own theory of 'deconvolution', t-norms, and fuzzy numbers in the unit interval. It ends with some good tutorial material on the solution of fuzzy simultaneous equations end inequalities. The second part purports to cover applications, but as the authors admit "actual problems...ars generally too complex and too long, and for this reason...illustrate the strength of fuzzy set theory by using relatively simple situations". They also point out that most of the examples are based on their own work (which is heavily referenced) so that much of the reference material is concerned with the classical (i.e. non-fuzzified) theory. From a pedagogical point of view this approach is a little wcrrying, introducing as it must a slightly one-sided view of work in areas where other fuzzy set theorists have worked in some cases but who are not acknowledged in this text. The redeeming feature of this section is that it is packed with wonderfully clear and thorough worked examples, which made me wonder whether tt.,a authors intend this book to be 8 student text or a monograph. I would have thought it more suitable as the former, if perhaps a little expensive for most students' pockets. The applications covered are all fuzzificetions of well known OR techniques, zero-based budgeting, long term (Delphic) forecasting, discounted cash flow with uncertain rates of interest, forecasting fuzzy time series (exponential smoothing), possibilistic reliability analysis, financial project evaluation, optimisation and so on. in all cases the emphasis is on tiie fuzzification of the problem anO the use of fur.zy numbers. Aiso every problem is explaia~ed clearly bv the use of simple, step-by-step numerical examples. This accords with the authors' stated aim o~ encouraging r¢:d~rs to attack their own problems by producing computer programs on the basis of the lessons 6earnt from their examples.