Bounds on System Reliability by Linear Programming
This article has a reply.
VIEW THE REPLYThis article has a reply.
VIEW THE REPLYThis article has a reply.
VIEW THE REPLYPublication: Journal of Engineering Mechanics
Volume 129, Issue 6
Abstract
Bounds on system probability in terms of marginal or joint component probabilities are of interest when exact solutions cannot be obtained. Currently, bounding formulas employing unicomponent probabilities are available for series and parallel systems, and formulas employing bi- and higher-order component probabilities are available for series systems. No theoretical formulas exist for general systems. It is shown in this paper that linear programming (LP) can be used to compute bounds for any system for any level of information available on the component probabilities. Unlike the theoretical bicomponent and higher-order bounds, the LP bounds are independent of the ordering of the components and are guaranteed to produce the narrowest possible bounds for the given information. Furthermore, the LP bounds can incorporate any type of information, including an incomplete set of component probabilities or inequality constraints on component probabilities. Numerical examples involving series, parallel and general structural systems are used to demonstrate the methodology.
Get full access to this article
View all available purchase options and get full access to this article.
References
Bertsimas, D., and Tsitsiklis, J. N. (1997). Introduction to linear optimization, Athena Scientific, Belmont, Mass.
Boole, G. (1854). Laws of thought, American Reprint of 1854 ed., Dover, New York.
Corotis, R. B., and Nafday, A. M.(1989). “Structural system reliability using linear programming and simulation.” J. Struct. Eng., 115(10), 2435–2447.
Daniels, H. E.(1945). “The statistical theory of the strength of bundles of threads.” Proc. R. Soc. London, Ser. A, 183(995), 405–435.
Dantzig, G. B. (1951). “Application of the simplex method to a transportation problem.” Activity analysis of production and allocation, T. C. Koopmans, ed., Wiley, New York, 359–373.
David, H. A. (1970). Order statistics, Wiley, New York.
Ditlevsen, O.(1979). “Narrow reliability bounds for structural systems.” J. Struct. Mech., 7(4), 453–472.
Dunnett, C. W., and Sobel, M.(1955). “Approximations to the probability integral and certain percentage points of a multivariate analogue of Student’s -distribution.” Biometrika, 42, 258–260.
Fréchet, M.(1935). “Généralizations du théorème des probabilités to-tales.” Fundam. Math., 25, 379–387.
Hailperin, T.(1965). “Best possible inequalities for the probability of a logical function of events.” Am. Math. Monthly, 72(4), 343–359.
Hunter, D.(1976). “An upper bound for the probability of a union.” J. Appl. Probab., 13, 597–603.
Kounias, E. G.(1968). “Bounds for the probability of a union, with applications.” Ann. Math. Stat., 39(6), 2154–2158.
Kounias, S., and Marin, J.(1976). “Best linear Bonferroni Bounds.” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 30(2), 307–323.
Nafday, A. M., Corotis, R. B., and Cohen, J. L.(1987). “Failure mode identification for structural frames.” J. Struct. Eng., 113(7), 1415–1432.
Prékopa, A.(1988). “Boole-Bonferroni inequalities and linear programming.” Oper. Res., 36(1), 145–162.
Zhang, Y. C.(1993). “High-order reliability bounds for series systems and application to structural systems.” Comput. Struct., 46(2), 381–386.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 129 • Issue 6 • June 2003
Pages: 627 - 636
Copyright
Copyright © 2003 American Society of Civil Engineers.
History
Received: May 1, 2002
Accepted: Oct 14, 2002
Published online: May 15, 2003
Published in print: Jun 2003
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.