Heterogeneous redundancy optimization for multi-state

series-parallel systems subject to common cause failures

Chun-yang Li, Xun Chen, Xiao-shan Yi, Jun-youg Tao

Reliability Engineering and System SafetyVolumes 95 (2010) p.202-207

Advisor: Yung-Sung, LinPresented by : Hui-Yu, Chung

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Agenda Introduction Problem Formulation

A multi-state series-parallel system with CCFs Mathematics Model

Reliability Estimation of the System The UGF of a Component The UGF of Subsystems without CCFs The UGF of the Subsystem with CCFs The Reliability of the System

Genetic Algorithms Solution Encoding and Initial Population Creation Individual Evaluation by Fitness Function Selection, Crossover & Mutation New Population Formation and Termination

Numerical Example Conclusion

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Introduction Common Cause Failures (CCFs)

The simultaneous failure of multiple components due to a common cause (CC).

Exists in many systems composed of redundant components

CC Events Environmental loads Errors in maintenance System design flaws.

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Introduction Multi state system (MSS)

Compared with Binary Systems Work in different performance levels

Universal generating function (UGF) A technique first stated from Levitin Gregory’s boo

k ”The universal generating function in reliability analysis and optimization”

Used to estimate the system reliability Genetic Algorithm (GA)

Used to optimize the system structure

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Introduction Problem Formulation

A multi-state series-parallel system with CCFs Mathematics Model

Reliability Estimation of the System The UGF of a Component The UGF of Subsystems without CCFs The UGF of the Subsystem with CCFs The Reliability of the System

Genetic Algorithms Solution Encoding and Initial Population Creation Individual Evaluation by Fitness Function Selection, Crossover & Mutation New Population Formation and Termination

Numerical Example Conclusion

Agenda

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Problem Formulation Four General Assumptions:

The components are not repaired The system and components are multi-state Mixing of components of different types in

the same subsystem is allowed When any load is beyond the limit of

components, all components of this type will fail

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Problem Formulation

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A multi-state series-parallel system with CCFS

N subsystems connected in seriesSubsystem I consists of different types of components in parallelih

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Mathematics model

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Introduction Problem Formulation

A multi-state series-parallel system with CCFs Mathematics Model

Reliability Estimation of the System The UGF of a Component The UGF of Subsystems without CCFs The UGF of the Subsystem with CCFs The Reliability of the System

Genetic Algorithms Solution Encoding and Initial Population Creation Individual Evaluation by Fitness Function Selection, Crossover & Mutation New Population Formation and Termination

Numerical Example Conclusion

Agenda

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Reliability Estimation of the System 4 approaches to estimate MSS reliability

UGF technique Structure function Stochastic process Monte Carlo simulation

Here, we use UGF approach System structure, performance & reliability

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A UGF Component Random performance G

Represented by two sets gij & qij

Definition:

1 2

1 2

{ , ,..., }

{ , ,..., }ij ij ij ijM

ij ij ij ijM

g g g g

q q q q

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The UGF of subsystems without CCFs Operations:

(SUM)

(MAX)

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The UGF of the subsystem with CCFs Subsystem f composed of identical components of type j:

Subsystem f composed of different types of components:

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The Reliability of the System Usually, the performance of the system is equal to

the minimum of performance of subsystems Operation:

UGF of the System:

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The Reliability of the System Define operation :

The reliability of the system is:

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Introduction Problem Formulation

A multi-state series-parallel system with CCFs Mathematics Model

Reliability Estimation of the System The UGF of a Component The UGF of Subsystems without CCFs The UGF of the Subsystem with CCFs The Reliability of the System

Genetic Algorithms Solution Encoding and Initial Population Creation Individual Evaluation by Fitness Function Selection, Crossover & Mutation New Population Formation and Termination

Numerical Example Conclusion

Agenda

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Genetic Algorithms Genetic Algorithm

An optimization technique based on concepts from Robert Darwin’s “evolution theory” InitializationSelectionReproductionTermination

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Solution encoding & Initial population creation

Representation of chromosomes: Integer strings

Population size: pop_size

1 211 12 1 21 22 2 1 2( , ,..., , , ,..., ,..., , ,..., )

Nk H H N N NHv n n n n n n n n n

, 0ij ijn Z n

max1 in n

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Individuals evaluation by fitness function

Fitness function The sum of the objective and a penalty functi

on determined by the relative degree of infeasibility

Rk: system reliability; R0: acceptable reliability

0max(0, )k k kf C K R R

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Selection, crossover & mutation Selection

Use roulette wheel selection method to select individuals for reproduction

Single-point crossover & mutation Crossover probability: pc Mutation probability: pm

max

max min

kk

f fff f

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New population formation and termination

Termination: A pre-defined maximum generation Nmax_gen is

reached The best feasible solution has not changed for

consecutive Nstall_gen generations.

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Introduction Problem Formulation

A multi-state series-parallel system with CCFs Mathematics Model

Reliability Estimation of the System The UGF of a Component The UGF of Subsystems without CCFs The UGF of the Subsystem with CCFs The Reliability of the System

Genetic Algorithms Solution Encoding and Initial Population Creation Individual Evaluation by Fitness Function Selection, Crossover & Mutation New Population Formation and Termination

Numerical Example Conclusion

Agenda

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Numerical example Multi-state series-parallel system 4 subsystems Formulation:

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H1=4

H2=5

H3=6

H4=4

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According to Eq.12, the UGF of subsystem 1 is:

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Using Genetic Algorithm to solve K=100 =0.001 Nmax_gen=1000 Nstall_gen=500 {20,50,100} {0.5,0.8,1.0} {0.01,0.1,0.2}

_pop sizecpmp

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Using GA & Hybrid GA Hybrid GA:

Fuzzy logic controller regulates GA parameters automatically

Exploitation around the near optimum solution

Longer, but the parameter tuning time is not taken into account.

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Results of reliability optimization with/without CCFs Optimal result is different Mixing of components of different types enhance

s the system reliability and controls the cost The effect of CCFs cannot be ignored

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Introduction Problem Formulation

A multi-state series-parallel system with CCFs Mathematics Model

Reliability Estimation of the System The UGF of a Component The UGF of Subsystems without CCFs The UGF of the Subsystem with CCFs The Reliability of the System

Genetic Algorithms Solution Encoding and Initial Population Creation Individual Evaluation by Fitness Function Selection, Crossover & Mutation New Population Formation and Termination

Numerical Example Conclusion

Agenda

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Conclusion CCFs reduce the effect of components redun

dancy CCFs make the redundancy allocation strate

gy different Mixing of components of different types is ver

y useful to improve the reliability of MSSs subject to CCFs

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~The End~★~Thanks for Your Attention~★