Gradient Computations and Geometrical Meaning of Importance Measures

Abstract Reliability importance is a scalar which is a partial derivative of system reliability function with respect to component reliability variable. Gradient is a vector consisting of partial derivatives of one continuous function with respect to all variables and gradient direction points in the direction of the greatest rate of the function increase. In order to analyze the direction that the system performance rises most quickly, this paper studies the gradient computations and geometrical meaning of importance measures. The expressions and physical meaning of importance measures are described at first. Then, the representations of importance measures in gradient are introduced, and relationships between importance measures and gradient are analyzed. Thirdly, the characteristics and geometrical meaning of gradient representations of importance measures are discussed. Finally, numerical examples in typical systems are demonstrated to verify the representation methods of importance measures in gradient.

[1]  Wei Li,et al.  The Hierarchical Weighted Multi-State $k$-out-of- $n$ System Model and Its Application for Infrastructure Management , 2010, IEEE Transactions on Reliability.

[2]  F. Proschan,et al.  Importance of System Components and Fault Tree Events. , 1975 .

[3]  Ke Jyh-Bin,et al.  Reliability Measures of a Repairable System with Standby Switching Failures and Reboot Delay , 2011 .

[4]  J. B. Fussell,et al.  How to Hand-Calculate System Reliability and Safety Characteristics , 1975, IEEE Transactions on Reliability.

[5]  David W. Coit,et al.  Composite importance measures for multi-state systems with multi-state components , 2005, IEEE Transactions on Reliability.

[6]  Lirong Cui,et al.  Reliability evaluation of generalised multi-state k-out-of-n systems based on FMCI approach , 2010, Int. J. Syst. Sci..

[7]  David W. Coit,et al.  Multi-state component criticality analysis for reliability improvement in multi-state systems , 2007, Reliab. Eng. Syst. Saf..

[8]  Z W Birnbaum,et al.  ON THE IMPORTANCE OF DIFFERENT COMPONENTS IN A MULTICOMPONENT SYSTEM , 1968 .

[9]  B. Natvig.  NEW LIGHT ON MEASURES OF IMPORTANCE OF SYSTEM COMPONENTS , 1984 .

[10]  Shudong Sun,et al.  The Integrated Importance Measure of Multi-State Coherent Systems for Maintenance Processes , 2012, IEEE Transactions on Reliability.

[11]  Shaomin Wu,et al.  Performance utility-analysis of multi-state systems , 2003, IEEE Trans. Reliab..

[12]  Lirong Cui,et al.  Analysis for joint importance of components in a coherent system , 2007, Eur. J. Oper. Res..

[13]  W. E. Vesley.  A Time Dependent Methodology for Fault Tree Evaluation , 1970 .

[14]  Michael J. Armstrong.  Joint reliability-importance of components , 1995 .

[15]  Anatoly Lisnianski,et al.  The Markov Reward Model for a Multi-State System Reliability Assessment with Variable Demand , 2007 .

[16]  Lirong Cui,et al.  Analytical method for reliability and MTTF assessment of coherent systems with dependent components , 2007, Reliab. Eng. Syst. Saf..

[17]  Kailash C. Kapur,et al.  Customer-Centered Multi-State System Reliability and Repair Models , 2010 .

[18]  Shenggui Zhang,et al.  Integrated Importance Measure of Component States Based on Loss of System Performance , 2012, IEEE Transactions on Reliability.

[19]  Bent Natvig,et al.  Simulation based analysis and an application to an offshore oil and gas production system of the Natvig measures of component importance in repairable systems , 2009, Reliab. Eng. Syst. Saf..