Bounds on System Reliability by Linear Programming

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 demon- strate the methodology.

[1]  M. Fréchet.  Généralisation du théorème des probabilités totales , 1935 .

[2]  H. Daniels.  The statistical theory of the strength of bundles of threads. I , 1945, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  O. H. Brownlee,et al.  ACTIVITY ANALYSIS OF PRODUCTION AND ALLOCATION , 1952 .

[4]  C. Dunnett,et al.  Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution* , 1955 .

[5]  T. Hailperin,et al.  Best Possible Inequalities for the Probability of a Logical Function of Events , 1965 .

[6]  E. Kounias.  Bounds for the Probability of a Union, with Applications , 1968 .

[7]  Stratis Kounias,et al.  Best Linear Bonferroni Bounds , 1976 .

[8]  David J. Hunter.  An upper bound for the probability of a union , 1976, Journal of Applied Probability.

[9]  O. Ditlevsen.  Narrow Reliability Bounds for Structural Systems , 1979 .

[10]  Kanagasabai Ramachandran.  System bounds: a critical study , 1984 .

[11]  Ross B. Corotis,et al.  Failure Mode Identification for Structural Frames , 1987 .

[12]  András Prékopa,et al.  Boole-Bonferroni Inequalities and Linear Programming , 1988, Oper. Res..

[13]  Ross B. Corotis,et al.  STRUCTURAL SYSTEM RELIABILITY USING LINEAR PROGRAMMING AND SIMULATION , 1989 .

[14]  Kanagasabai Ramachandran.  Best ordering for system bounds , 1990 .

[15]  Pierre Hansen,et al.  Column Generation Methods for Probabilistic Logic , 1989, INFORMS J. Comput..

[16]  Glenn L. Greig.  An assessment of high-order bounds for structural reliability☆ , 1992 .

[17]  Yong Cang Zhang.  High-order reliability bounds for series systems and application to structural systems , 1993 .

[18]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[19]  Armen Der Kiureghian,et al.  Closure to “Bounds on System Reliability by Linear Programming” by Junho Song and Armen Der Kiureghian , 2005 .