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



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


A multi-state series-parallel system with CCFs
Mathematics Model
Solution Encoding and Initial Population Creation
Individual Evaluation by Fitness Function
Selection, Crossover & Mutation
New Population Formation and Termination
Numerical Example
Conclusion
2
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.
3
Introduction

Multi state system (MSS)



Universal generating function (UGF)



Compared with Binary Systems
Work in different performance levels
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
4
Agenda


Introduction
Problem Formulation



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


A multi-state series-parallel system with CCFs
Mathematics Model
Solution Encoding and Initial Population Creation
Individual Evaluation by Fitness Function
Selection, Crossover & Mutation
New Population Formation and Termination
Numerical Example
Conclusion
5
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
6
Problem Formulation
7
A multi-state series-parallel
system with CCFS
N subsystems connected in series
Subsystem I consists of hi different types of components in parallel
8
Mathematics model
9
Agenda


Introduction
Problem Formulation



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


A multi-state series-parallel system with CCFs
Mathematics Model
Solution Encoding and Initial Population Creation
Individual Evaluation by Fitness Function
Selection, Crossover & Mutation
New Population Formation and Termination
Numerical Example
Conclusion
10
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
11
A UGF Component

Random performance G

Represented by two sets gij & qij
g ij  {g ij 1, g ij 2 ,..., g ijM }
qij  {qij 1, qij 2 ,..., qijM }

Definition:
12
The UGF of subsystems without
CCFs

Operations:
(SUM)
(MAX)
13
The UGF of the subsystem with
CCFs

Subsystem f composed of identical components of type j:

Subsystem f composed of different types of components:
14
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:

15
The Reliability of the System

Define operation  :

The reliability of the system is:
16
Agenda


Introduction
Problem Formulation



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


A multi-state series-parallel system with CCFs
Mathematics Model
Solution Encoding and Initial Population Creation
Individual Evaluation by Fitness Function
Selection, Crossover & Mutation
New Population Formation and Termination
Numerical Example
Conclusion
17
Genetic Algorithms

Genetic Algorithm

An optimization technique based on concepts
from Robert Darwin’s “evolution theory”
 Initialization
 Selection
 Reproduction
 Termination
18
Solution encoding & Initial
population creation

Representation of chromosomes:

Integer strings
vk  (n11, n12,..., n1H1 , n21, n22,..., n2H2 ,..., nN1, nN 2,..., nNHN )

Population size: pop_size
nij  Z, nij  0
1  ni  nmax
19
Individuals evaluation by fitness
function

Fitness function


The sum of the objective and a penalty
function determined by the relative degree of
infeasibility
Rk: system reliability; R0: acceptable
reliability
fk  Ck  K  max(0,R0  Rk )
20
Selection, crossover &
mutation

Selection


fmax  fk  
fk 
fmax  fmin  
Use roulette wheel selection method to select
individuals for reproduction
Single-point crossover & mutation


Crossover probability: pc
Mutation probability: pm
21
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.
22
Agenda


Introduction
Problem Formulation



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


A multi-state series-parallel system with CCFs
Mathematics Model
Solution Encoding and Initial Population Creation
Individual Evaluation by Fitness Function
Selection, Crossover & Mutation
New Population Formation and Termination
Numerical Example
Conclusion
23
Numerical example



Multi-state series-parallel system
4 subsystems
Formulation:
24
H1=4
H2=5
H3=6
H4=4
25
According to Eq.12, the UGF of subsystem 1 is:
26
Using Genetic Algorithm to solve
K=100
  =0.001
 Nmax_gen=1000
 Nstall_gen=500
 pop _ size  {20,50,100}
 p  {0.5,0.8,1.0}
c
 p  {0.01,0.1,0.2}

m
27
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.
28
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
29
Agenda


Introduction
Problem Formulation



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


A multi-state series-parallel system with CCFs
Mathematics Model
Solution Encoding and Initial Population Creation
Individual Evaluation by Fitness Function
Selection, Crossover & Mutation
New Population Formation and Termination
Numerical Example
Conclusion
30
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
31
~The End~
★~Thanks for Your Attention~★
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