Optimal Reliability of Components of Complex Systems Using Hierarchical System Models

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1 Special Workshop on Risk Acceptance and Risk Communication, Stanford University 1/21 Optimal Reliability of Components of Complex Systems Using Hierarchical System Models K. Nishijima, ETH Zurich M. Maes Civil Engineering Department, Schulich School of Engineering, University of Calgary J. Goyet Marine Division, Research Department, Bureau Veritas M.H. Faber, ETH Zurich

2 2/21 Background Engineered systems such as: electricity/water distribution systems structural systems are complex in the sense that: components are functionally/geographically interrelated different levels of analyses provided by different experts are required.

3 3/21 Ship hull structure is an example: components/(sub-)structures are interrelated: joints plates tanks different levels of analyses are required such as: yielding, fracture and corrosion of materials structural analysis consequence analysis.

4 4/21 Objective To develop a framework for optimization of engineered systems. What are effectively designed are components. Acceptance criteria are often given for system performance and system performance is our direct concern.

5 5/21 Objective The bridge between component and system performance needs to be developed.

6 6/21 Clues The proposed framework relies on the typical characteristics that individual components are standardized components are hierarchically interrelated and takes basis in the fact that system performance depends on way of interrelation between components reliability of components.

7 7/21 Approach Modelling of complex systems by Bayesian probabilistic Networks Consideration of a-priori given acceptance criteria for system performance Setting up an optimization problem

8 8/21 Bayesian probabilistic network: is a graphical representation of probabilistic structure of variables by nodes and arrows is quantitatively characterized by conditional probability tables can provide e.g. expected value, conditional probability Object-oriented Bayesian probabilistic network: is useful when a structure has many identical (sub-) structures/components.

9 9/21 Hierarchical modeling by use of Bayesian probabilistic network Component level X 1 X 2 X 3 X 4 Y Z

10 10/21 Hierarchical modeling by use of Bayesian probabilistic network Sub-structure level A 1 A 2 A n B C

11 11/21 Hierarchical modeling by use of Bayesian probabilistic network Structure level D 1 D 2 D m E F U G

.., x N ) where x i is design variable for components, e.g. component reliability.

12 12/21 On Bayesian probabilistic networks: expected total cost is written as: u = f( x1, x2,..., x N ) where x i is design variable for components, e.g. component reliability. acceptance criteria for system performance are written as: g ( x, x,..., x ) c j 1 2 N j

13 13/21 Optimization of component reliability can be reduced to be a standard constrained optimization problem: Minimize u = f( x1, x2,..., x N ) such that g ( x, x,..., x ) c,( j = 1, 2,..., M ) j 1 2 N j solving the optimization problem with commonly available techniques, e.g. Microsoft Excel and Hugin.

14 14/21 Optimization of reliability of welded joints in ship hull structure Acceptance criterion: probability of failure of ship hull < 10-3 /yr Objective function : expected total cost

15 15/21 Hierarchical structure of the ship hull:

16 16/21 Corresponding BPN s:

17 17/21 Conditional probability table

18 18/21 Objective function: u = f( x1, x2,..., x10) [x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 ]

19 19/21 Constraints: gx ( 1, x2,..., x10) c [x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 ]

20 Eidgenossische Technische Hochschule Zürich Problem Setting Framework Excel platform Example Conclusion 20/21 ActiveX Solver Add-in iplan (Straub and Faber 2006)

21 21/21 Use of BPN Engineered system can be hierarchically modelled by use of BPN and especially by object-oriented BPN. Standard constrained optimization - Objective function: expected total cost - Constraints: acceptance criteria for system performance - Variables: component reliability Use of commonly available techniques/algorithms For example, Hugin and Microsoft Excel.

K. Nishijima. Definition and use of Bayesian probabilistic networks 1/32

K. Nishijima. Definition and use of Bayesian probabilistic networks 1/32 The Probabilistic Analysis of Systems in Engineering 1/32 Bayesian probabilistic bili networks Definition and use of Bayesian probabilistic networks K. Nishijima nishijima@ibk.baug.ethz.ch 2/32 Today s