Abstract
The chapter presents applications of the fuzzy set theory to the reliability engineering field. It starts by a historical perspective of reliability engineering and some questions in classical reliability. Then, it reviews the probabilistic method of the system reliability analysis from the fuzzy set theoretical point of view. Next, it makes a survey of the probability approach and the non-probabilistic measure approach. Especially, the non-probabilistic measure approach focuses on the introduction of natural language expressions and experts’ subjectivity to the system reliability analysis.
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References
Barlow, R. E. and Proschan, F.(1965). Mathematical theory of reliability, John Wiley and Sons Inc., New York.
Barlow, R. E. and Proschan, F. (1975). Statistical theory of reliability and life testing, Holt, Rinehart and Winston, New York.
Barlow, R. E. and Wu, S. A.(1978). Coherent systems with multistate components, Mathematics in Operations Research, 3, 275–281.
Barlow, R. E.(1984). Mathematical theory of reliability: A historical perspective, IEEE Transactions on Reliability 33, 16–20.
Baxter, L. A. (1984). Continuum structures I, Journal of Applied Probability, 21, 802–815.
Baxter, L. A. (1986). Continuum structures II, Mathematical Proceedings of the Cambridge Philosophical Society, 99, 331–338.
Baxter, L. A. and Kim, C. (1986). Bounding the stochastic performance of continuum structure functions I, Journal of Applied Probability, 23, 660–669.
Baxter, L. A. and Kim, C. (1987). Bounding the stochastic performance of continuum structure functions II, Journal of Applied Probability, 24, 609–618
Baxter, L. A. (1988). On the theory of cannibalization, Journal of Mathematical Analysis and Applications, 136, 290–297.
Birnbaum, Z. W., Esary, J. D. and Saunders, S. C. (1961). Multicomponent systems and structures and their reliability, Technometrics, 3, 55–77.
Birnbaum, Z. W. and Esary, J. D. (1965). Modules of coherent binary systems, SIAM Journal on Applied Mathematics, 13, 444–462.
Block, H. W. and Savits, T. H. (1982). A decomposition theorem for multistate structure functions, Journal of Applied Probability, 19, 391–402.
Bowels, J. B. and Pelaez, C. (1995). Application of fuzzy logic to reliability engineering, Proceedings of the IEEE, 83, 435–449.
Cai, K. Y., Wen, C. Y. (1990). Stree-lighting lamps replacement: A fuzzy viewpoint, Fuzzy Sets and Systems, 37, 161–172.
Cai, K. Y., Wen, C. Y. and Zhang, M. L. (1991a). Fuzzy reliability modelling of gracefully degradable computing systems, Reliability Engineering and System Safety, 33, 141–157.
Cai, K. Y., Wen, C. Y. and Zhang, M. L. (1991b). Posbist reliability behavior of typical systems with two types of failures, Fuzzy Sets and Systems, 43, 17–32.
Cai, K. Y., Wen, C. Y. and Zhang, M. L. (1991c). Fuzzy variables as a basis for theory of fuzzy reliability in the possibility context, Fuzzy Sets and Systems, 42, 145–172.
Cai, K. Y., Wen, C. Y. and Zhang, M. L. (1993). Fuzzy states as a basis for a theory of fuzzy reliability, Microelectronics and Reliability, 33, 2253–2263.
Cai, K. Y., Wen, C. Y. and Zhang, M. L. (1995). Coherent systems in profust reliability theory, in Reliability and Safety Analyses under Fuzziness, Onisawa, T. and Kacprzyk, J. eds., Physica-Verlag, A Springer-Verlag Company, Heidelberg, 81-94. Caldarola, L. (1980). Fault-tree analysis with multistate components, in Synthesis and Analysis for Safety and Reliability Studies, Apostolakis, G, Garriba, S. and Volta, G. eds., Plenum Press, New York, 199–248.
Cappelle, B. (1991). Multistate structure functions and possibility theory: An alternative approach to reliability, in Introduction to the Basic Principles of Fuzzy Set Theory and Some of its Applications, Kerre, E.E. ed., Communication and Cognition, Gent, 252–293.
Cappelle, B. and Kerre, E.E. (1994a). Possibilistic and necessitic reliability functions: Fundamental concepts and theorems to represent non-probabilistic uncertainty in reliability theory, in Uncertainty Modelling and Analysis: Theory and Applications, Ayyub, B. M. and Gupta, M. M. eds., Elsevier Science B. V., North Holland, 131–144.
Cappelle, B. and Kerre, E. E. (1994b). A general possibilistic framework for reliability theory, Proceedings of the Information Processing and management of Uncertainty in Knowledge-Based Systems Conference-IPMU 5, Bouchon-Meunier, B. and Yager, R. eds., 25–35.
Cappelle, B. (1995). On the notion state in multistate structure function theory, in Fuzzy Set Theory and Advanced Mathematical Applications, Ruan, D. ed., Kluwer Academic Publishers, Boston, 201–221.
Cappelle, B. and Kerre, E. E. (1995a). Issue in possibilistic reliability theory, in Reliability and Safety Analyses under Fuzziness, Onisawa, T. and Kacprzyk, J. eds., Physica-Verlag, A Springer-Verlag Company, Heidelberg, 61–80.
Cappelle, B. and Kerre, E. E. (1995b). Computer assisted reliability analysis: An application of possibilistic reliability theory to a subsystem of a nuclear power plant, Fuzzy Sets and Systems, 74, 103–113.
Cappelle, B. and Kerre, E. E. (1995c). An algorithm to compute possibilistic reliability, Proceedings of the Third of International Symposium on Uncertainty Modelling and Analysis and the Annual Conference of the North American Fuzzy Infromation Processing Society.
Carvallo, M. (1965). Principles et applications de l’analyse booléenne, Gauthier-Villars, Paris.
Cohen, H. (1984). Space reliability technology: A historical perspective, IEEE Transactions on Reliability, 33, 36–40.
de Cooman, G. (1993). Evaluation Sets and mappings: An order theoretic approach to vagueness and uncertainty, Doctoral Dissertation, Universiteit Gent(in Dutch).
Dhillon, B. S. (1980). A system with two kinds of 3-state elements, IEEE Transactions on Reliability, 29, 345.
Dubes, R. C. (1963). Two algorithms for computing reliability, IEEE Transactions on Reliability, 12, 55–63.
Dubois, A. and Prade, H. (1980). Fuzzy sets and systems: Theory and applications, Academic Press, New York.
El-Neweihi, E., Proschan, F. and Sethuraman, J. (1978). Multistate coherent systems, Journal of Applied Probability, 15, 675–688.
Elsayed, E. A. and Zebib, A. (1979). A repairable multistate device, IEEE, Transactions on Reliability, 28, 81–82.
Embrey, D. E. (1984). SLIM-MAUD(Success Likelihood Index Methodology Multi-Attribute Utility Decomposition): An approach to assessing human error probabilities using structured expert judgement, NUREG/CR-3518.
Esary, J. D. and Proschan, F. (1962). The reliability of coherent systems, in Redundancy Techniques for Computing Systems, Wilcox, R. H. and Mann, C.W. eds., Spartan Books, Washington D. C., 47–61.
French, S. (1990). On being enumerate, The University of Leeds Review, 33, 119–134.
Fussell, J. B. (1973). Fault-tree analysis: Concepts and techniques, Proceedings of the NATO Conference on Reliability, NATO Advanced Study Institute on Generic Techniques of System Reliability, Liverpool.
Fussell, J.B. (1984). Nuclear power system reliability: A historical perspective, IEEE Transactions on Reliability, 33, 41–47.
Gazdík, I. (1986). RELSHELL, An expert system shell for fault diagnosis, in Fault Diagnosis, Reliability and related Knowledge-Based Approaches, Tzafestas, Singh, Schmidt, eds., D Reidel.
Gazdik, I. (1996). Zadeh’s extension principle in design reliability, Fuzzy Sets and Systems, 83, 169–178.
Gnedenko, B., Beliaev, Y. and Soloviev, A. (1972). Méthodes mathématiques en théorie de la fiabilité, Mir, Moscow.
Griffith, W. S. (1980). Multistate reliability models, Journal of Applied Probability, 17, 735–744.
Griffith, W. S. (1982). A multistate availability model: System performance and component importance, IEEE Transactions on Reliability, 31, 97–98.
IAEA and NEA (OECD) (1992). INES: The international nuclear event scale user’s manual, revised and extended edition.
Kaufmann, A. Cyouokko, D. and Cryon, R. (1977). Mathematical models for the study of the reliability of systems, Academic Press, New York.
Kenarangui, R. (1991). Event-tree analysis by fuzzy probability, IEEE Trans. on Reliability, 40, 120–124.
Misra, K. B. and Sharma, A. (1981). Performance index to quantify reliability using fuzzy subset theory, Microelectronics and Reliability, 21, 543–549.
Misra, K. B. and Weber, G. G. (1990). Use of fuzzy set theory for level-1 studies in probabilistic risk assessment, Fuzzy Sets and Systems, 37, 139–160.
Mizumoto, M. (1989). Pictorial representations of fuzzy connectives, Part I: Cases of t-norms, t-conorms, and averaging operators, Fuzzy Sets and Systems, 31, 217–242.
Montero, J. Tejada, J. and Yáñez, J. (1988). General structure functions, Proceedings of Workshop on Knowledge-Based Systems and Models of Logical Reasoning, Egypt.
Montero, J., Cappelle, B. and Kerre, E. E. (1995). The usefulness of complete lattices in reliability theory, in Reliability and Safety Analyses under Fuzziness, Onisawa, T. and Kacprzyk, J. eds., Physica-Verlag, A Springer-Verlag Company, Heidelberg, 95–110.
Moore, E. F. and Shannon, C. E. (1956a). Reliable circuits using less reliable relays I, Journal of the Franklin Institute, 262, 191–208.
Moore, E. F. and Shannon, C. E. (1956b). Reliable circuits using less reliable relays II, Journal of the Franklin Institute, 262, 281–297.
Murchland, J. D. (1975). Fundamental concepts and relations for reliability analysis of multi state systems. in Reliability and Fault Tree Analysis, Barlow, R. E, Fussell, J. B. and Singpurwalla, N. D. eds., SIAM, Philadelphia, 581–618.
Natvig, B. (1982). Two suggestions of how to define a multistate coherent system, Advances in Applied Probability, 14, 434–455.
Noma, K. Tanaka, H. and Asai, K. (1981). On fault tree analysis with fuzzy probability, The Japanese Journal of Ergonomics, 17, 291–297(in Japanese).
Onisawa, T. (1986). Performance shaping factors modelling using the error possibility and fuzzy integrals, The Japanese Journal of Ergonomics, 22, 81–89(in Japanese).
Onisawa, T. (1988). An approach to human reliability in man-machine systems using error possibility, Fuzzy Sets and Systems, 27. 87–103.
Onisawa, T. (1990). An application of fuzzy concepts to modelling of reliability analysis, Fuzzy Sets and Systems, 37, 269–286.
Onisawa, T. (1991). Fuzzy reliability assessment considering the influence of many factors on reliability, International Journal of Approximate Reasoning, 5, 265–280.
Onisawa, T. (1993). On fuzzy sets operations in system reliability analysis with natural language, Journal of Japan Society for Fuzzy Theory and Systems, 5, 43–54(in Japanese).
Onisawa, T. and Misra, K. B. (1993). Use of fuzzy sets theory(Part-II: Applications), in New Trends in System Reliability Evaluation, Misra, K. B. ed., Elsevier, The Netherlands, 551–586.
Onisawa, T. (1994). A few remarks on membership function in fuzzy reliability analysis, Proc. of 3rd International Conference on Fuzzy Logic, Neural Nets and Soft Computing, Iizuka, Japan, 635–636.
Onisawa, T. and Kacprzyk, J. eds. (1995) Reliability and Safety Analyses under Fuzziness, Physica-Verlag, A Springer-Verlag Company, Heidelberg.
Onisawa, T. (1995). Subjective system reliability analysis and mutual agreements of its results, in Fuzzy Logic and Its Applications to Engineering, Information Sciences, and Intelligent Systems, Bien, Z. and Min, K. C. eds., Kluwer Academic Publishers, The Netherlands, 265–274.
Onisawa, T. (1996). Subjective analysis of system reliability and its analyzer, Fuzzy Sets and Systems, 83, 249–269.
Premo, A. F. (1963). The use of Boolean algebra and a truth table in the formulation of a mathematical model of success, IEEE Transactions on Reliability, 12, 45–49.
Sadatoku, H., Nagamachi, M., Matsubara, Y. and Onisawa, T. (1993). An analysis of human-error factors using fuzzy integral-measure model and natural language, The Japanese Journal of Ergonomics, 29, 289–297(in Japanese).
Schmucker, K. J. (1984). Fuzzy sets, natural language computations, and risk analysis, Computer Press, Rockville, M. D.
Shooman, M. L. (1984). Software reliability: A historical perspective, IEEE Transactions on Reliability, 33, 48–55.
Singer, D. (1990). A fuzzy set approach to fault and reliability analysis, Fuzzy Sets and Systems, 34, 145–155.
Svenson, O. (1989). On expert judgement in safety analyses in the process industries, Reliability Engineering and System Safety, 25, 219–256.
Swain, A. D. (1963). A method for performing a human factors reliability analysis, Report SCR-685, Sandia Corporation.
Swain, A. D. and Guttmann, H. E. (1983). Handbook of human reliability analysis with emphasis on nuclear power plant applications, NUREG/CR-1278.
Tanaka, H., Fan, L. T., Lui, F. S. and Toguchi, K. (1983). Fault tree analysis by fuzzy probability, IEEE Trans. on Reliability, 32, 453–457.
U. S. Atomic Energy Commission (1974). Reactor safety study: An assessment of accident risks in U.S. commercial nuclear power plants, WASH-1400.
Washio, T., Takahashi, H. and Kitamura, M. (1992). A method for supporting decision-making on plant operation based on human reliability analysis with fuzzy integral, Proc. of the 2nd Int. Conf. on Fuzzy Logic and Neural Networks, Iizuka, 2, 841–845.
Zadeh, L. A. (1965). Fuzzy sets, Information and Control, 8, 338–353.
Zadeh, L. A. (1968). Probability measures of fuzzy events, J. Math. Analysis and Appl., 23, 421–427.
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Kerre, E., Onisawa, T., Cappelle, B., Gazdik, I. (1998). Reliability. In: Słowiński, R. (eds) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. The Handbooks of Fuzzy Sets Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5645-9_12
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DOI: https://doi.org/10.1007/978-1-4615-5645-9_12
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