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

Zhigang Tian, Ming J. Zuo, and Hongzhong Huang

IEEE Transactions on Reliability, vol. 57, No. 2, June 2008 Presented by: Hui-Yu, Chung Advisor: Frank Yeong-Sung, Lin

Agenda

   

Introduction

◦ ◦ ◦ ◦ Problem 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 Utility Conclusions 2

Introduction

   Component – 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 3

Introduction

  State Distribution ◦ Used to describe the reliability of a MMS ◦ ◦ Two 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 4

Introduction

 Previous studies on optimization of MMSs focused on only redundancies ◦ Only partial optimization  The option of selecting different versions of components provides more flexibility 5

Notation & Acronym

6

Assumptions

   The 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 7

Agenda

    Introduction ◦ ◦ ◦ ◦

Problem 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 Utility Conclusions 8

Problem Formulation

 The structure of a multi-state series parallel system: identically distributed components connected in parallel

n

The prob. of component i in state j is

p ij

9

Design Variables

 State distributions ◦

p ij

i = 1, 2, …, N; j = 1, 2, …, M  Redundancies ◦ i = 1, 2, …, N

n i

 Reliability means the prob. of working 10

Reliability of a component

  Consider a three-state system ◦ Three States: { 0,1,2 } ◦ State Distributions: 0 1 2 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:

p

1 

p

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

p

2 11

System Utility Evaluation

  System utility: The expected utility The prob. that the system is in state s or above: (s = 0, 1, …, M)  The System Utility U:

u s

: Utility when the system is in state s 12

Formulation of System Cost

 components:

n

◦ ◦ ◦ ◦

i

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

i c r i i

: cost of the components in subsystem i subsystem

n

( )exp( ) 4 : interconnecting cost in parallel 

i



i

13

Formulation in System Cost

 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 14

Formulation of System Cost

 The cost of the component:  The system cost: 15

Characteristics of the Optimization Problem  Objective to be optimized: ◦ System Utility, ◦ System Cost 

p n

system utility and minimize cost 

Mixed integer optimization problem

◦ ◦ Continuous variables: state distributions Integer variables: redundancies 16

Characteristics of the Optimization Problem  Formulated as a single-objective optimization problem: ◦ Either cost or utility can be a design objective, while the other can be a constraint 17

Characteristics of the Optimization Problem  Formulated 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 18

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” 19

Physical Programming-Based Optimization Problem Formulation  Class-1S Class Function (for Cost) ◦ Monotonously increasing function ◦ Used to represent the objectives to be minimized Corresponding class function value  Class-2S Class Function (for Utility) ◦ ◦ Monotonously decreasing function Used to represent the objectives to be maximized 20

Physical Programming-Based Optimization Problem Formulation  Transforming a physical programming problem to a single-objective optimization problem: f: aggregate objective function 21

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 iteration Four stages in GA:  Initialization, selection, reproduction, termination 22

The procedure of GA

    Initialization ◦ 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 23

Agenda

    Introduction ◦ ◦ ◦ ◦ Problem 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 Utility Conclusions 24

An Example

25

The Joint Reliability-Redundancy Optimization Results  In Physical programming framework ◦ System utility: Class-2S objective function ◦ System cost: Class-1S objective function  Mixed integer programming problem  9 design variables:  GA parameters (run 30 times)

Population Size

100

Chromosome Length

15

Selection Scheme

Roulette wheel

Crossover rate

0.25 (One-point)

Mutation Rate

0.1

Maximum epoch

1000 26

The Joint Reliability-Redundancy Optimization Results  The result of the optimization: 27

The Redundancy Optimization Results  Consider different versions of components ◦ Otherwise, the results may not be optimal  Integer programming problem Component version for stage i  The other conditions remain the same 28

The Redundancy Optimization Results  The Result of the optimization: 29

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 30

Sensitivity Analysis for System Cost 31

Sensitivity Analysis for System Cost  

p



C



p

◦ The sensitivity of system cost decreases a

p

 

p ij p ij

◦ System cost is more sensitive to stage 3  32

Sensitivity Analysis for System Cost  

ij



ij

◦  Positive Constant Value Cost increases with the increase of the parameter ◦ Positive and more sensitive 33

Sensitivity Analysis for System Utility 34

Sensitivity Analysis for System Utility    For any , is always positive ◦ of

p ij



C ij

The system utility increases with the increases System utility becomes less sensitive to with the increase of it.

p ij

The utility is more sensitive to the distribution variables associated with state 2 35

Agenda

    Introduction ◦ ◦ ◦ ◦ Problem 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 Utility

Conclusions

36

Conclusions

   Two options too improve the system utility of a multi-state series-parallel system: ◦ Provide redundancy at each stage ◦ Improve the component state distributions Physical programming-based optimization is introduced and used in this problem Sensitivity Analysis ◦ Which can reflect the facts on the model 37

~The End~

Thanks for Your Attention!!!

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