This article deals with an implementation of probist reliability problems in evidential networks to propagate imprecise probabilities expressed as fuzzy numbers. First, the problem of imprecise knowledge in reliability problems is described concerning system and data representation. Then, the basics of the evidence theory and its use in a directed acyclic graph approach are given. The imprecise probist reliability of a complex system by modelling the component failure probabilities as real, interval, or fuzzy numbers is pointed out. Two numerical studies of systems are carried out. The results are discussed and some comparisons with a Monte-Carlo simulation and a fuzzy fault tree approach are made.
UtkinL.CoolenFLevitinG (Eds.) New metaheuristics, neural, and fuzzy techniques in reliability. Computational intelligence in reliability engineering2007, Vol. 2, ch. 10, (Springer) 2007: pp. 261–306
2.
RaoK. DKushwahaHVermaASrividyaA‘Quantification of epistemic and aleatory uncertainties in level-1 probabilistic safety assessment studies’Reliability Engng System Saf.92 (7) (2007): 947–956
3.
ZioE.PedroniN‘Building confidence in the reliability assessment of thermal-hydraulic passive systems’Reliability Engng System Safety94 (2) (2009): 268–281
4.
BaudritC.DuboisD‘Practical representations of incomplete probabilistic knowledge’Comp. Statistics Data Analysis51 (2006): 86–106
5.
FersonS.GinzburgL‘Different methods are needed to propagate ignorance and variability’Reliability Engng System Saf.54 (1996): 133–144
6.
HeltonJ. COberkampfW. L‘Alternative representations of epistemic uncertainty’Reliability Engng System Saf.85 (2004): 1–10
7.
DuboisD.PradeHPossibility theory. An approach to computerized processing of uncertainty (New York: Plenum Press1988)
8.
FersonS.KreinovichV.GinzburgL.MyersD.SentzK., Constructing probability boxes and Dempster–Shafer structures. Sandia National Laboratory Technical Report, 2002.
9.
ShaferG.A mathematical theory of evidence (New York: Princeton University Press1976)
10.
WalleyP.Statistical reasoning with imprecise probabilities (London: Chapman & Hall1991)
NeumaierA.‘Clouds, fuzzy sets and probability intervals’Reliability Computing10 (2004): 249–272
13.
LimbourgP.GermannD‘Reliability assessment and optimization under uncertainty in the Dempster–Shafer framework’ (2004)
14.
SmetsP.KornreichFBlockPBernardHVainselRVan BemelJ. HWillemsJ. L (Eds.) ‘Fuzzy diagnostic, degrees of belief and utility’Trends in computer processed ECG (1977): pp. 257–259
15.
SmetsP.AndersonJ (Ed.) ‘Theory of evidence and medical diagnostic’Medical informatics Europe 78 (Berlin: Springer Verlag1978): pp. 285–291
16.
MassonM.-H.DenoeuxT‘Clustering interval-valued proximity data using belief functions’Pattern Recognition Lett.25 (2) (2004): 163–171
17.
TononF.‘Using random set theory to propagate epistemic uncertainty through a mechanical system’Reliability Engng System Saf.85 (2004): 169–181
18.
SimonC.WeberPEvsukoffA‘Bayesian networks inference algorithm to implement Dempster–Shafer theory in reliability analysis’Reliability Engng System Saf.93 (7) (2008): 950–963
19.
RakowskyU.GochtU‘Reasoning in reliability centred maintenance based on a Dempster–Shafer approach’J. Risk and Reliability222 (O4) (2008): 605–612
20.
HalpernJ. YFaginR‘Two views of belief: belief as generalized probability and belief as evidence’Artif. Intell.54 (3) (1992): 275–317
21.
UtkinL. V‘Extensions of belief functions and possibility distributions by using the imprecise Dirichlet model’Fuzzy Sets and Systems154 (3) (2005): 413–431
22.
DempsterA. P‘Upper and lower probabilities induced by a multivalued mapping’Ann. Mathl Statists38 (1967): 325–339
23.
CaiK.WenCZhangM‘Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context’Fuzzy Sets and Systems42 (1991): 145–172
24.
CaiK.-Y.WenC.-YZhangM.-L‘Fuzzy states as a basis for a theory of fuzzy reliability’Microelectronics and Reliability33 (15) (1993): 2253–2263
25.
CaiK.-Y.‘System failure engineering and fuzzy methodology: An introductory overview’Fuzzy Sets and Systems83 (2) (1996): 113–133
26.
BuckleyJ.Fuzzy probabilities (New York: Springer Verlag2005)
27.
ZadehL. A‘Fuzzy sets’Inf. Control (1965): 338–353
28.
KholasJ.MonneyPA mathematical theory of hints: an approach to Dempster–Shafer theory of evidence (New York: Springer Verlag1995)
SentzK.FersonS., Combination of evidence in Dempster–Shafer theory. Sandia National Laboratory Technical Report, 2002.
31.
OuzounovaE.Incertitude et mesure de performance: Une nouvelle approche à travers la théorie mathématique des indications. PhD dissertation, Faculté des Sciences Economiques et Sociales, Université de Freiburg, Switzerland 2005.
32.
DuongM. HModéle de précaution en économie: introduction aux probabilities imprecises. PhD Dissertation, Université de Paris I- Panthon Sorbonne 2005.
33.
KlirG. JFolgerT. AFuzzy sets, uncertainty, and information (Upper Saddle River, NJ: Prentice-Hall1987)
34.
ChokrB. AKreinovichV. Y‘How far are we from the complete knowledge? Complexity of knowledge acquisition in the Dempster–Shafer approach’Advances in the Dempster–Shafer theory of evidence (New York: John Wiley1994): pp. 555–576
35.
BoudraaA.-O.BentabetASalzensteiFGuillonL‘Dempster–Shafer's basic probability assignment based on fuzzy membership functions’Electronic Lett. Computer Vision and Image Analysis4 (1) (2004): 1–9
36.
GilboaI.SchmeidlerD‘Maximin expected utility with non-unique prior’J. Mathl Econ.18 (1989): 141–153
37.
KlirG. JWiermanM‘Uncertainty-based information; elements of generalized information theory’Studies in fuzziness and soft-computing (Heidelberg: Physica-Verlag1998): p. 168.
38.
SmetsPHeckermanDPooleD. R.de MantarasL. (Eds.) ‘Belief induced by the knowledge of some probabilities’Uncertainty in AI'94 (San Mateo, California: Morgan Kaufmann1994): pp. 523–530
39.
KlirG.YuanBFuzzy sets and fuzzy logic. Theory and applications (Upper Saddle River, NJ: Prentice Hall1995)
40.
GuthM.‘A probability foundation for vagueness and imprecision in fault tree analysis’IEEE Trans Reliability40 (5) (1991): 563–570
41.
ChengY.‘Uncertainty in fault tree analysis’Tamkang J. Sci. Engng3 (1) (2000): 23–29
42.
HallJ.LawryJ‘Imprecise probabilities of engineering system failure from random and fuzzy set reliability analysis’. In Proceedings of ISIPTA'01, Ithaca, New York, 26–29 June 2001.
43.
RakowskyU. K‘Fundamentals of the Dempster–Shafer theory and its applications to system safety and reliability modelling’. In Proceedings of the ESRA summer safety and reliability seminars — SSARS 2007, Sopot, Poland, July 2007.
44.
SmetsP.‘The application of the matrix calculus to belief functions’Int. J. Approximate Reasoning31 (2002): 1–30
45.
SmetsP.‘Probability of provability and belief functions’Logique et Analyse133 (1991): 177–195
46.
SimonC.SallakMAubryJ.-F‘SIL allocation of SIS by aggregation of experts’ opinions’ (2007)
47.
JensenF.An introduction to Bayesian networks (London: University Press1996)
48.
BobbioA.PortinaleLMinichinoMCiancamerlaE‘Improving the analysis of dependable systems by mapping fault trees into Bayesian networks’Reliability Engng System Saf.71 (3) (2001): 249–260
49.
BoudaliH.DuganJ‘A discrete-time Bayesian network reliability modeling and analysis framework’Reliability Engng System Saf.87 (2005): 337–349
50.
VillemeurA.Reliability, availability, maintainability safety assessment: methods and techniques (New York: John Wiley1992) Translated from French edition by CartierA.LartisienM.C.
51.
Torres-ToledanoJSucarLCoelhoH (Eds.) ‘Bayesian networks for reliability of complex systems’Progress in artificial intelligence, IBERAMIA'98 (New York: Springer-Verlag1998): pp. 195–206
