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 Safety Volumes 95 (2010) p.202-207 Advisor: Yung-Sung, Lin Presented by : Hui-Yu, Chung

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

Introduction Common Cause Failures (CCFs) CC Events 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.

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 book ”The universal generating function in reliability analysis and optimization” Used to estimate the system reliability Genetic Algorithm (GA) Used to optimize the system structure

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

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

Problem Formulation

A multi-state series-parallel system with CCFS N subsystems connected in series Subsystem I consists of different types of components in parallel

Mathematics model

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

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

A UGF Component Random performance G Represented by two sets gij & qij Definition:

The UGF of subsystems without CCFs Operations: (SUM) (MAX)

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

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:

The Reliability of the System Define operation : The reliability of the system is:

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

Genetic Algorithms Genetic Algorithm An optimization technique based on concepts from Robert Darwin’s “evolution theory” Initialization Selection Reproduction Termination

Solution encoding & Initial population creation Representation of chromosomes: Integer strings Population size: pop_size

Individuals evaluation by fitness function The sum of the objective and a penalty function determined by the relative degree of infeasibility Rk: system reliability; R0: acceptable reliability

Selection, crossover & mutation Use roulette wheel selection method to select individuals for reproduction Single-point crossover & mutation Crossover probability: pc Mutation probability: pm

New population formation and termination A pre-defined maximum generation Nmax_gen is reached The best feasible solution has not changed for consecutive Nstall_gen generations.

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

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

H1=4 H2=5 H3=6 H4=4

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

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}

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.

Results of reliability optimization with/without CCFs Optimal result is different Mixing of components of different types enhances the system reliability and controls the cost The effect of CCFs cannot be ignored

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

Conclusion CCFs reduce the effect of components redundancy CCFs make the redundancy allocation strategy different Mixing of components of different types is very useful to improve the reliability of MSSs subject to CCFs

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