Reliability-Redundancy Allocation for Multi-State Series-Parallel Systems

Zhigang Tian, Ming J. Zuo, and Hongzhong HuangIEEE Transactions on Reliability, vol. 57, No. 2, June

2008

Presented by: Hui-Yu, ChungAdvisor: Frank Yeong-Sung, Lin

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AgendaIntroductionProblem Formulation

◦ Design Variables◦ System Utility Evaluation◦ Formulation of System Cost◦ Characteristics of the Optimization Problem◦ Physical Programming-Based Optimization Problem

Formulation◦ Optimization Solution Method – Genetic Algorithm

An Example◦ The Joint Reliability-Redundancy Optimization Results◦ The Redundancy Optimization Results◦ Sensitivity Analysis for System Cost and System

UtilityConclusions

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IntroductionComponent – An “Entity”

◦Can be connected in a certain configuration to form a subsystem, or system.

Multi-State System◦Many systems can perform their intended

functions at more than two different levels From perfectly working to completely failed Provide more flexibility for modeling

Performance Measure – System Utility

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Introduction

State Distribution◦Used to describe the reliability of a

MMSTwo ways to improve the utility

of a multi-state series-parallel system:◦To provide redundancy at each stage◦To improve the component state

distribution Make a component in states w.r.t. high

utilities and probabilities

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IntroductionPrevious studies on optimization

of MMSs focused on only redundancies◦Only partial optimization

The option of selecting different versions of components provides more flexibility

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Notation & Acronym

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AssumptionsThe states of the components in

a subsystem is independent identically distributed (i.i.d.)

The components, and the system may be in M + 1 possible states, namely, 0, 1, 2, …, M

The multi-state series parallel systems under consideration are coherent systems

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AgendaIntroductionProblem Formulation

◦ Design Variables◦ System Utility Evaluation◦ Formulation of System Cost◦ Characteristics of the Optimization Problem◦ Physical Programming-Based Optimization Problem

Formulation◦ Optimization Solution Method – Genetic Algorithm

An Example◦ The Joint Reliability-Redundancy Optimization Results◦ The Redundancy Optimization Results◦ Sensitivity Analysis for System Cost and System

UtilityConclusions

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Problem FormulationThe structure of a multi-state

series-parallel system:

N subsystems connected in series, each subsystem i has independent identically distributed components connected in parallel

The prob. of component i in state j is

in

ijp

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Design Variables

State distributions◦i = 1, 2, …, N; j = 1, 2, …, M

Redundancies◦i = 1, 2, …, N

Reliability means the prob. of working

ijp

in

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Reliability of a componentConsider a three-state system

◦Three States: { 0,1,2 }◦State Distributions:

Statements:◦1)The prob. that a component is in state 1

or 2 is the reliability of this component that its state is greater or equal to 1(“working” ) Reliability:

◦2) The prob. of component in state 2 is the reliability of it that its state is greater than or equal to 2 Reliability:

0 1 2{ , , }p p p

1 2p p

2p

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System Utility EvaluationSystem utility: The expected

utilityThe prob. that the system is in

state s or above: (s = 0, 1, …, M)

The System Utility U:

: Utility when the system is in state ssu

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Formulation of System CostThe cost of subsystem i with

parallel components:

◦ : cost-reliability relationship function for a component in subsystem i

◦ : cost of the components in subsystem i

◦ : interconnecting cost in parallel subsystem

◦ , : characteristic constants

i i

( )i ic r

( )i i ic r n( )exp( )

4i

i i

nc r

in

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In a (M + 1) state MMS:◦Reliability of component i under treatment

k:

Assumption:◦There are M treatments that can influence

the component’s state distribution, and treatment k will increase the prob. of the component in state k, but will not influence the prob. of the component in the states above k

Formulation in System Cost

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Formulation of System CostThe cost of the component:

The system cost:

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Characteristics of the Optimization ProblemObjective to be optimized:

◦System Utility,◦System Cost

Determine and to maximize system utility and minimize cost

Mixed integer optimization problem◦Continuous variables: state distributions◦Integer variables: redundancies

ijp in

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Characteristics of the Optimization ProblemFormulated as a single-objective

optimization problem:◦Either cost or utility can be a design

objective, while the other can be a constraint

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Characteristics of the Optimization ProblemFormulated as a multi-objective

optimization problem:◦Three approaches:

The surrogate worth trade-off method The fuzzy optimization method Physical programming method

In this case, physical programming approach is used

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Physical Programming-Based Optimization Problem Formulation

Physical Programming Optimization◦The Decision Maker’s preference is

considered in the optimization process◦Use of class functions

Class Functions:◦The value reflects the preference of the

designer on objective function value◦Four types of of “soft” class function:

Smaller is better, Larger is better, Value is better, and Range is better

Here, we use “Smaller is better”

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Physical Programming-Based Optimization Problem Formulation

Class-1S Class Function (for Cost)◦Monotonously increasing function◦Used to represent the objectives to

be minimized

Class-2S Class Function (for Utility)◦Monotonously decreasing function◦Used to represent the objectives to

be maximized

Design Objective

Corresponding class function value

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Physical Programming-Based Optimization Problem Formulation

Transforming a physical programming problem to a single-objective optimization problem:

f: aggregate objective function

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Genetic Algorithm as the Optimization Solution Method

Genetic Algorithm:◦Most effective algorithm to solve

mixed integer optimization problems◦Chromosome: one solution in GA◦Population: a group of chromosome

in each iterationFour stages in GA:

Initialization, selection, reproduction, termination

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The procedure of GAInitialization

◦Specify the GA operators◦Specify the GA parameters

Evaluation◦Using fitness value to get P(k) and B(k)

Construct new population◦Chromosome is replaced by the best

fitness value.Terminate

◦When reaching a maximal iteration

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AgendaIntroductionProblem Formulation

◦ Design Variables◦ System Utility Evaluation◦ Formulation of System Cost◦ Characteristics of the Optimization Problem◦ Physical Programming-Based Optimization Problem

Formulation◦ Optimization Solution Method – Genetic Algorithm

An Example◦ The Joint Reliability-Redundancy Optimization Results◦ The Redundancy Optimization Results◦ Sensitivity Analysis for System Cost and System

UtilityConclusions

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An Example

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The Joint Reliability-Redundancy Optimization ResultsIn Physical programming

framework◦System utility: Class-2S objective

function◦System cost: Class-1S objective

functionMixed integer programming

problem9 design variables:

GA parameters (run 30 times)

Population Size

Chromosome Length

Selection Scheme

Crossover rate

Mutation Rate

Maximum epoch

100 15 Roulette-wheel

0.25 (One-point)

0.1 1000

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The Joint Reliability-Redundancy Optimization Results

The result of the optimization:

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The Redundancy Optimization ResultsConsider different versions of components

◦ Otherwise, the results may not be optimalInteger programming problem

The other conditions remain the same

Component version for stage i

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The Redundancy Optimization Results

The Result of the optimization:

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Sensitivity Analysis for System Cost and System Utility

Sensitivity analysis of system cost:◦9 design variables◦Model parameters ,

Since they are affect the system costs

Using the partial derivative to analyze◦While keeping all the others the

same

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Sensitivity Analysis for System Cost

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Sensitivity Analysis for System Cost

For any , is always positive◦System cost increases with the

increase in The sensitivity of system cost

decreases a bit with the increase of ,( < 0.05 )

When > 0.05, the sensitivity always increases with the increase of

System cost is more sensitive to stage 3 ◦Since in stage 3 is larger

ijp

ijp

ijp

ij

Cp

ijp

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Sensitivity Analysis for System CostSensitivity w.r.t. the parameter,

and

◦Positive Constant Value Cost increases with the increase of the

parameter

◦Positive and more sensitive

ij ij

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Sensitivity Analysis for System Utility

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Sensitivity Analysis for System Utility

For any , is always positive◦The system utility increases with the

increases of System utility becomes less

sensitive to with the increase of it.

The utility is more sensitive to the distribution variables associated with state 2

ijp ij

Cp

ijp

ijp

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AgendaIntroductionProblem Formulation

◦ Design Variables◦ System Utility Evaluation◦ Formulation of System Cost◦ Characteristics of the Optimization Problem◦ Physical Programming-Based Optimization Problem

Formulation◦ Optimization Solution Method – Genetic Algorithm

An Example◦ The Joint Reliability-Redundancy Optimization Results◦ The Redundancy Optimization Results◦ Sensitivity Analysis for System Cost and System

UtilityConclusions

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ConclusionsTwo options too improve the system

utility of a multi-state series-parallel system:◦Provide redundancy at each stage◦Improve the component state

distributionsPhysical programming-based

optimization is introduced and used in this problem

Sensitivity Analysis◦Which can reflect the facts on the model

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~The End~Thanks for Your Attention!!!