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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Quality and Reliability Engineering
- Sprache: Englisch
This book offers a comprehensive overview of recently developed methods for assessing and optimizing system reliability. It consists of two main parts, for treating assessment methods and optimization methods, respectively. The first part covers methods of multi-state system reliability modelling and evaluation, Markov processes, Monte Carlo simulation and uncertainty analysis. The methods considered range from piecewise-deterministic Markov processes to belief function analysis. The second part covers optimization methods of mathematical programming and evolutionary algorithms, and problems of multi-objective optimization and optimization under uncertainty. The methods of this part range from non-dominated sorting genetic algorithm to robust optimization. The book also includes the application of the assessment and optimization methods considered on real case studies, particularly with respect to the reliability assessment and optimization of renewable energy systems, and bridges the gap between theoretical method development and engineering practice.
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Wiley Series in Quality & Reliability Engineering
Dr. Andre V. Kleyner
Series Editor
The Wiley Series in Quality & Reliability Engineering aims to provide a solid educational foundation for both practitioners and researchers in the Q&R field and to expand the reader’s knowledge base to include the latest
developments in this field. The series will provide a lasting and positive contribution to the teaching and practice of engineering.
The series coverage will contain, but is not exclusive to,
Statistical methods
Physics of failure
Reliability modeling
Functional safety
Six-sigma methods
Lead-free electronics
Warranty analysis/management
Risk and safety analysis
Wiley Series in Quality & Reliability Engineering
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Reliability Culture: How Leaders can Create Organizations that Create Reliable Productsby Adam P. BahretFebruary 2021
Lead-free Soldering Process Development and Reliabilityby Jasbir Bath (Editor) August 2020
Automotive System Safety: Critical Considerations for Engineering and Effective ManagementJoseph D. MillerFebruary 2020
Prognostics and Health Management: A Practical Approach to Improving SystemReliability Using Condition-Based Databy Douglas Goodman, James P. Hofmeister, Ferenc SzidarovszkyApril 2019
Improving Product Reliability and Software Quality: Strategies, Tools, Processand Implementation, 2nd EditionMark A. Levin, Ted T. Kalal, Jonathan RodinApril 2019
Practical Applications of Bayesian ReliabilityYan Liu, Athula I. AbeyratneApril 2019
Dynamic System Reliability: Modeling and Analysis of Dynamic and Dependent BehaviorsLiudong Xing, Gregory Levitin, Chaonan WangMarch 2019
Reliability Engineering and ServicesTongdan JinMarch 2019
Design for Safetyby Louis J. Gullo, Jack Dixon February 2018
Thermodynamic Degradation Science: Physics of Failure, Accelerated Testing,Fatigue and Reliability by Alec Feinberg October 2016
Next Generation HALT and HASS: Robust Design of Electronics and Systemsby Kirk A. Gray, John J. Paschkewitz May 2016
Reliability and Risk Models: Setting Reliability Requirements, 2nd Editionby Michael Todinov November 2015
System Reliability Assessment and Optimization
Methods and Applications
Yan-Fu Li
Department of Industrial Engineering, Tsinghua University, China
Enrico Zio
MINES ParisTech/PSL Université, Italy
This edition first published 2022
©2022 John Wiley & Sons Ltd
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The right of Enrico Zio and Yan-Fu Li to be identified as the authors of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication Data
Names: Li, Yan-Fu, author. | Zio, Enrico, author. Title: System reliability assessment and optimization: methods and applications / Yan-Fu Li, Enrico Zio. Description: Hoboken, NJ : John Wiley & Sons, 2022. | Series: Wiley series in quality & reliability engineering | Includes bibliographical references and index. Identifiers: LCCN 2022009590 (print) | LCCN 2022009591 (ebook) | ISBN 9781119265870 (hardback) | ISBN 9781119265924 (pdf) | ISBN 9781119265863 (epub) | ISBN 9781119265856 (ebook) Subjects: LCSH: Reliability (Engineering) | Industrial safety. Classification: LCC TA169 .Z57 2022 (print) | LCC TA169 (ebook) | DDC 620/.00452--dc23/eng/20220316 LC record available at https://lccn.loc.gov/2022009590LC ebook record available at https://lccn.loc.gov/2022009591
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To all the students, collaborators and colleagues, to whom we are forever indebted for all the enriching experience and knowledge that they have shared with us and that have made us grow professionally and personally.
Contents
Cover
Serious page
Title page
Copyright
Dedication
Series Editor’s Foreword
Preface
Acknowledgments
List of Abbreviations
Notations
Part I The Fundamentals
1 Reliability Assessment
1.1 Definitions of Reliability
1.1.1 Probability of Survival
1.2 Component Reliability Modeling
1.2.1 Discrete Probability Distributions
1.2.2 Continuous Probability Distributions
1.2.3 Physics-of-Failure Equations
1.3 System Reliability Modeling
1.3.1 Series System
1.3.2 Parallel System
1.3.3 Series-parallel System
1.3.4 K-out-of-n System
1.3.5 Network System
1.4 System Reliability Assessment Methods
1.4.1 Path-set and Cut-set Method
1.4.2 Decomposition and Factorization
1.4.3 Binary Decision Diagram
1.5 Exercises
References
2 Optimization
2.1 Optimization Problems
2.1.1 Component Reliability Enhancement
2.1.2 Redundancy Allocation
2.1.3 Component Assignment
2.1.4 Maintenance and Testing
2.2 Optimization Methods
2.2.1 Mathematical Programming
2.2.2 Meta-heuristics
2.3 Exercises
References
Part II Reliability Techniques
3 Multi-State Systems (MSSs)
3.1 Classical Multi-state Models
3.2 Generalized Multi-state Models
3.3 Time-dependent Multi-State Models
3.4 Methods to Evaluate Multi-state System Reliability
3.4.1 Methods Based on MPVs or MCVs
3.4.2 Methods Derived from Binary State Reliability Assessment
3.4.3 Universal Generating Function Approach
3.4.4 Monte Carlo Simulation
3.5 Exercises
References
4 Markov Processes
4.1 Continuous Time Markov Chain (CMTC)
4.2 In homogeneous Continuous Time Markov Chain
4.3 Semi-Markov Process (SMP)
4.4 Piecewise Deterministic Markov Process (PDMP)
4.5 Exercises
References
5 Monte Carlo Simulation (MCS) for Reliability and Availability Assessment
5.1 Introduction
5.2 Random Variable Generation
5.2.1 Random Number Generation
5.2.2 Random Variable Generation
5.3 Random Process Generation
5.3.1 Markov Chains
5.3.2 Markov Jump Processes
5.4 Markov Chain Monte Carlo (MCMC)
5.4.1 Metropolis-Hastings (M-H) Algorithm
5.4.2 Gibbs Sampler
5.4.3 Multiple-try Metropolis-Hastings (M-H) Method
5.5 Rare-Event Simulation
5.5.1 Importance Sampling
5.5.2 Repetitive Simulation Trials after Reaching Thresholds (RESTART)
5.6 Exercises
Appendix
References
6 Uncertainty Treatment under Imprecise or Incomplete Knowledge
6.1 Interval Number and Interval of Confidence
6.1.1 Definition and Basic Arithmetic Operations
6.1.2 Algebraic Properties
6.1.3 Order Relations
6.1.4 Interval Functions
6.1.5 Interval of Confidence
6.2 Fuzzy Number
6.3 Possibility Theory
6.3.1 Possibility Propagation
6.4 Evidence Theory
6.4.1 Data Fusion
6.5 Random-fuzzy Numbers (RFNs)
6.5.1 Universal Generating Function (UGF) Representation of Random-fuzzy Numbers
6.5.2 Hybrid UGF (HUGF) Composition Operator
6.6 Exercises
References
7 Applications
7.1 Distributed Power Generation System Reliability Assessment
7.1.1 Reliability of Power Distributed Generation (DG) System
7.1.2 Energy Source Models and Uncertainties
7.1.3 Algorithm for the Joint Propagation of Probabilistic and Possibilistic Uncertainties
7.1.4 Case Study
7.2 Nuclear Power Plant Components Degradation
7.2.1 Dissimilar Metal Weld Degradation
7.2.2 MCS Method
7.2.3 Numerical Results
References
Part III Optimization Methods and Applications
8 Mathematical Programming
8.1 Linear Programming (LP)
8.1.1 Standard Form and Duality
8.2 Integer Programming (IP)
8.3 Exercises
References
9 Evolutionary Algorithms (EAs)
9.1 Evolutionary Search
9.2 Genetic Algorithm (GA)
9.2.1 Encoding and Initialization
9.2.2 Evaluation
9.2.3 Selection
9.2.4 Mutation
9.2.5 Crossover
9.2.6 Elitism
9.2.7 Termination Condition and Convergence
9.3 Other Popular EAs
9.4 Exercises
References
10 Multi-Objective Optimization (MOO)
10.1 Multi-objective Problem Formulation
10.2 MOO-to-SOO Problem Conversion Methods
10.2.1 Weighted-sum Approach
10.2.2 ε-constraint Approach
10.3 Multi-objective Evolutionary Algorithms
10.3.1 Fast Non-dominated Sorting Genetic Algorithm (NSGA-II)
10.3.2 Improved Strength Pareto Evolutionary Algorithm (SPEA 2)
10.4 Performance Measures
10.5 Selection of Preferred Solutions
10.5.1 “Min-max” Method
10.5.2 Compromise Programming Approach
10.6 Guidelines for Solving RAMS+C Optimization Problems
10.7 Exercises
References
11 Optimization under Uncertainty
11.1 Stochastic Programming (SP)
11.1.1 Two-stage Stochastic Linear Programs with Fixed Recourse
11.1.2 Multi-stage Stochastic Programs with Recourse
11.2 Chance-Constrained Programming
11.2.1 Model and Properties
11.2.2 Example
11.3 Robust Optimization (RO)
11.3.1 Uncertain Linear Optimization (LO) and its Robust Counterparts
11.3.2 Tractability of Robust Counterparts
11.3.3 Robust Optimization (RO) with Cardinality Constrained Uncertainty Set
11.3.4 Example
11.4 Exercises
References
12 Applications
12.1 Multi-objective Optimization (MOO) Framework for the Integration of Distributed Renewable Generation and Storage
12.1.1 Description of Distributed Generation (DG) System
12.1.2 Optimal Power Flow (OPF)
12.1.3 Performance Indicators
12.1.4 MOO Problem Formulation
12.1.5 Solution Approach and Case Study Results
12.2 Redundancy Allocation for Binary-State Series-Parallel Systems (BSSPSs) under Epistemic Uncertainty
12.2.1 Problem Description
12.2.2 Robust Model
12.2.3 Experiment
References
Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1 The pmf of the binomial...
Figure 1.2 The pmf of the Poisson...
Figure 1.3 The pdf of the exponential...
Figure 1.4 The pdf of the Weibull...
Figure 1.5 The pdf of the gamma...
Figure 1.6 The pdf of the lognormal...
Figure 1.7 Illustration of Paris...
Figure 1.8 Reliability block diagram...
Figure 1.9 Reliability block diagram...
Figure 1.10 Reliability block diagram...
Figure 1.11 Bridge system.
Figure 1.12 Block decision diagram...
Figure 1.13 Electrical generating system.
Figure 1.14 Reliability block diagram...
Figure 1.15 Diagram of the power grid...
Figure 1.16 Reliability block diagram...
Chapter 2
Figure 2.1 Example for the search tree...
Figure 2.2 Shortest...
Chapter 4
Figure 4.1 The Markov Diagram of the...
Figure 4.2 The Markov Diagram of the...
Figure 4.3 Degradation Process of...
Figure 4.4 Sample Path of...
Figure 4.5 A Three-state semi-Markov...
Figure 4.6 An illustrative example of a system...
Figure 4.7 The Evolution of Degradation Processes...
Figure 4.8 The Diagram of the Series-parallel...
Figure 4.9 State Transition Diagram for...
Chapter 5
Figure 5.1 comparison between inverse sampling...
Figure 5.2 Distribution obtained by...
Figure 5.3 State distribution...
Figure 5.4 Markov jump process...
Figure 5.5 M-H sampling
Figure 5.6 Sampling with Gibbs...
Figure 5.7 Sampling with Multiple-try...
Chapter 6
Figure 6.1 (a). A non-normal convex fuzzy...
Figure 6.2 Possibility distribution...
Figure 6.3 Three-dimensional and two-dimensional...
Chapter 7
Figure 7.1 Conceptual diagram of the representative...
Figure 7.2 Comparison of joint propagation...
Figure 7.4 Comparison of joint propagation...
Figure 7.3 Comparison of joint propagation...
Figure 7.5 Transition diagram of the multi-state...
Figure 7.6 Convergence plots of state...
Chapter 8
Figure 8.1 Graphical solution to the...
Figure 8.2 Search tree.
Figure 8.3 The search tree...
Figure 8.4 The search tree...
Chapter 9
Figure 9.1 Example of binary...
Figure 9.2 Example of tree-based...
Figure 9.3 Swap operator for...
Figure 9.4 One example of tree-based...
Figure 9.5 Single-point crossover
Figure 9.6 Two-point crossover....
Figure 9.7 Uniform crossover.
Figure 9.8 Example of Order 1 crossover.
Figure 9.9 Example of crossover for...
Chapter 10
Figure 10.1 Pareto dominance and...
Figure 10.2 Fast non-dominance sorting...
Figure 10.3 An example of crowding distance.
Figure 10.4 An example of generational...
Figure 10.5 An example of computing spacing.
Figure 10.6 An example of illustration...
Figure 10.7 Best-compromise solution...
Figure 10.8 Illustration of the compromise Min-Max...
Chapter 12
Figure 12.1 Example of distribution generation...
Figure 12.2 Graphical representation...
Figure 12.3 Flow chart of NSGA-II...
Figure 12.4 Pareto fronts for different...
Figure 12.5 EENS v/s ECg [4].
Figure 12.6 Violation probabilities...
Figure 12.7 Change percentage in...
List of Tables
Chapter 4
Table 4.1 Results for the degradation mode...
Chapter 7
Table 7.1 Different uncertainties of the energy...
Table 7.2 Case study parameter definitions and...
Table 7.3 Comparison of the simulation results...
Chapter 10
Table 10.1 Payoff Table of the MOO problem...
Table 10.2 Steps of fast non-dominance sorting...
Table 10.3 Example of removal iterations.
Chapter 11
Table 11.1 The cost ($) of different maintenance...
Table 11.2 Tractable RC representations
Guide
Cover
Serious page
Title page
Copyright
Dedication
Table of Contents
Series Editor’s Foreword
Preface
Acknowledgments
List of Abbreviations
Notations
Begin Reading
Index
End User License Agreement
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Series Editor’s Foreword
Dr. Andre V. Kleyner
The Wiley Series in Quality & Reliability Engineering aims to provide a solid educational foundation for researchers and practitioners in the field of quality and reliability engineering and to expand the knowledge base by including the latest developments in these disciplines.
The importance of quality and reliability to a system can hardly be disputed. Product failures in the field inevitably lead to losses in the form of repair cost, warranty claims, customer dissatisfaction, product recalls, loss of sale, and in extreme cases, loss of life.
Engineering systems are becoming increasingly complex with added functions and capabilities; however, the reliability requirements remain the same or even growing more stringent. Modeling and simulation methods, such as Monte Carlo simulation, uncertainty analysis, system optimization, Markov analysis and others, have always been important instruments in the toolbox of design, reliability and quality engineers. However, the growing complexity of the engineering systems, with the increasing integration of hardware and software, is making these tools indispensable in today’s product development process.
The recent acceleration of the development of new technologies including digitalization, forces the reliability professionals to look for more efficient ways to deliver the products to market quicker while meeting or exceeding the customer expectations of high product reliability. It is important to comprehensively measure the ability of a product to survive in the field. Therefore, modeling and simulation is vital to the assessment of product reliability, including the effect of variance on the expected product life, even before the hardware is built. Variance is present in the design parameters, material properties, use conditions, system interconnects, manufacturing conditions, lot-to-lot variation and many other product inputs, making it difficult to assess. Thus, modeling and simulation may be the only tools to fully evaluate the effect of variance in the early product development phases and to eventually optimize the design.
The book you are about to read has been written by leading experts in the field of reliability modeling, analysis, simulation and optimization. The book covers important topics, such as system reliability assessment, modeling and simulation, multi-state systems, optimization methods and their applications, which are highly critical to meeting the high demands for quality and reliability. Achieving the optimal feasible performance of the system is eventually the final objective in modern product design and manufacturing, and this book rightfully puts a lot of emphasis on the process of optimization.
Paradoxically, despite its evident importance, quality and reliability disciplines are somewhat lacking in today’s engineering educational curricula. Only few engineering schools offer degree programs, or even a sufficient set of courses, in quality and reliability methods. The topics of reliability analysis, accelerated testing, reliability modeling and simulation, warranty data analysis, reliability growth programs, reliability design optimization and other aspects of reliability engineering receive very little coverage in today’s engineering students curricula. As a result, the majority of the quality and reliability practitioners receive their professional training from colleagues, professional seminars and professional publications. In this respect, this book is intended to contribute to closing this gap and provide additional educational material as a learning opportunity for a wide range of readers from graduate level students to seasoned reliability professionals.
We are confident that this book, as well as this entire book series, will continue Wiley’s tradition of excellence in technical publishing and provide a lasting and positive contribution to the teaching and practice of reliability and quality engineering.
Preface
Engineering systems, like process and energy systems, transportation systems, structures like bridges, pipelines, etc., are designed to ensure successful operation throughout the anticipated service lifetime in compliance with given all-around sustainability requirements. This calls for design, operation, and maintenance solutions to achieve the sustainability targets with maximum benefit from system operation. Reliability, availability, maintainability and Safety criteria (RAMS) are among the indicators for measuring system functionality with respect to these intended targets.
Today, modern engineering systems are becoming increasingly complex to meet the high expectations by the public for high functionality, performance, and reliability, and with this, RAMS properties have become further key issues in design, maintenance, and successful commercialization.
With high levels of RAMS being demanded on increasingly complex systems, the reliability assessment and optimization methods and techniques need to be continuously improved and advanced. As a result, many efforts are being made to address various challenges in complex engineering system lifecycle management under the global trend of systems integration. Mathematically and computationally, the reliability assessment and optimization are challenged by various issues related to the uncertain, dynamic, multi-state, non-linear interdependent characteristics of the modern engineering systems and the problem of finding optimal solutions in irregular search spaces characterized by non-linearity, non-convexity, time-dependence and uncertainty.
In the evolving and challenging RAMS engineering context depicted above, this book provides a precise technical view on system reliability methods and their application to engineering systems. The methods are described in detail with respect to their mathematical formulation and their application is illustrated through numerical examples and is discussed with respect to advantages and limitations. Applications to real world cases are given as a contribution to bridging the gap between theory and practice.
The book can serve as a solid theoretical and practical basis for solving reliability assessment and optimization problems regarding systems of different engineering disciplines and for further developing and advancing the methods to address the newly arising challenges as technology evolves.
Reliability engineering is founded on scientific principles and deployed by mathematical tools for analyzing components and systems to guarantee they provide their functions as intended by design.
On the other hand, technological advances continuously bring changes of perspectives, in response to the needs, interests, and priorities of the practical engineering world. As technology advances at a fast pace, the complexity of modern engineered systems increases and so do, at the same time, the requirements for performance, efficiency, and reliability. This brings new challenges that demand continuous developments and advancements in complex system reliability assessment and optimization.
Therefore, system reliability assessment and optimization is inevitably a living field, with solution methodologies continuously evolving through the advancements of mathematics and simulation to follow up the development of new engineering technology and the changes in management perspectives. For this, advancements in the fields of operations research, reliability, and optimization theory and computation continuously improve the methods and techniques for system reliability assessment and optimization and for their application to very large and increasingly complex systems made of a large number of heterogeneous components with many interdependencies under various physical and economic constraints.
Within the ongoing efforts of development and advancement, this book presents an overview of methods for assessing and optimizing system reliability. We address different types of system reliability assessment and optimization problems and the different approaches for their solutions. We consider the development and advancement in the fields of operations research, reliability, and optimization theory to tackle the reliability assessment and optimization of complex systems in different technological domains.
The book is directed to graduate students, researchers and practitioners in the areas of system reliability, availability, maintainability and Safety (RAMS), and it is intended to provide an overview of the state of knowledge of and tools for reliability assessment and system optimization. It is organized in three parts to introduce fundamentals, and illustrate methods and applications.
The first part reviews the concepts, definitions and metrics of reliability assessment and the formulations of different types of reliability optimization problems depending on the nature of the decision variables and considering redundancy allocation and maintenance and testing policies. Plenty of numerical examples are provided to accompany the understanding of the theoretical concepts and methods.
The second part covers multi-state system (MSS) modeling and reliability evaluation, Markov processes, Monte Carlo simulation (MCS), and uncertainty treatment under poor knowledge. The reviewed methods range from piecewise-deterministic Markov processes (PDMPs) to belief functions.
The third part of the book is devoted to system reliability optimization. In general terms, system reliability optimization involves defining the decision variables, the constraints and the single or multiple objective functions that describe the system reliability performance and involves searching for the combination of values of the decision variables that realize the target values the objective functions. Different formulations and methods are described with precise mathematical details and illustrative numerical examples, covering mathematical programming, evolutionary algorithms, multi-objective optimization (MOO) and optimization under uncertainty, including robust optimization (RO).
Applications of the assessment and optimization methods to real-world cases are also given, concerning for example the reliability of renewable energy systems. From this point of view, the book bridges the gap between theoretical development and engineering practice.
Acknowledgments
Live long and prosper, RAMS and system reliability! The authors would express the deepest appreciations to the great scholars along the line of honors and achievements for their inspirations and role modeling.
Many thanks to the postgraduate students in Tsinghua: Tianli Men, Hanxiao Zhang, Ruochong Liu, Chen Zhang and Chuanzhou Jia. Thanks for their priceless efforts in editing, depicting, and proofreading in various chapters.
The authors would like to specially thank the Wiley colleagues for their continuous and kindhearted monitoring and encouragement throughout the years.
At last, this work is supported in part by the National Natural Science Foundation of China under a key project grant No. 71731008 and the Beijing Natural Science Foundation grant No. L191022.
List of Abbreviations
ABC
artificial bee colony algorithm
ACO
ant colony optimization
AGAN
as-good-as-new
B&B
branch-and-bound
BBA
basic belief assignment
BDD
binary decision diagram
BFS
basic feasible solution
BSS
binary state system
BSSPS
binary-state series-parallel system
cdf
cumulative distribution function
CG
column generation
CLT
Central Limit Theorem
CTMC
continuous time Markov chain
CVaR
conditional value-at-risk
DC
direct current
DE
deterministic equivalent
DE
differential evolution
DG
distributed generation
DM
decision maker
DP
dynamic programming
DTMC
discrete time Markov chain
EA
evolutionary algorithm
ENS
energy not supplied
EENS
expected energy not supplied
EV
electrical vehicles
FV
finite-volume
GA
genetic algorithm
GD
generational distance
HCTMC
homogeneous CTMC
HPIS
high-pressure injection system
HUGF
hybrid UGF
HV
hyper-volume
ICTMC
inhomogeneous CTMC
ILP
integer linear programming
IP
integer programming
LO
Linear optimization
LP
linear programming
LPM
LP master problem
MCMC
Markov Chain Monte Carlo
MCS
Monte Carlo simulation
MCS-OPF
Monte Carlo simulation – optimal power flow
MCV
minimal cut vector
MDD
multi-valued decision diagram
MH
Metropolis-Hastings
MIP
mixed integer programming
MOO
multi-objective optimization
MP
mathematical programming
MPV
minimal path vector
MRC
Markov renewal chain
MS
Main supply power spot
MSCS
multi-state coherent system
MSM
multi-state model
MSMS
multi-state monotone system
MSS
multi-state system
MTBF
mean time between failures
MTBR
mean time between repairs
MTTF
mean time to failure
NLP
non-linear programming
NPGA
niched Pareto GA
NPP
nuclear power plant
NSGA-II
fast non-dominated sorting genetic algorithm
OPF
optimal power flow
probability density function
PDMP
piecewise-deterministic Markov process
pmf
probability mass function
P-o-F
Physics-of-Failure
PSO
particle swarm optimization
PV
solar photovoltaic
RAM
reliability, availability, and maintainability
RAMS
RAM and Safety criteria
RAMS+C
RAMS and Cost
RAP
redundancy allocation problem
RC
robust counterpart
RESTART
Repetitive Simulation Trials After Reaching Thresholds
RFN
random-fuzzy number
RLPM
restricted LPM
RO
robust optimization
SMP
semi-Markov process
SODE
single-object DE
SOEA
single-objective EA
SOGA
single-objective GA
SOO
single-objective optimization
SOPSO
single-objective PSO
SP
stochastic programming
SPEA
strength Pareto evolutionary algorithm
SPEA 2
improved strength Pareto evolutionary algorithm
SSO
social spider optimization
ST
storage device
TDMSM
time-dependent MSM
TIMSM
time-independent MSM
TS
Tabu search
UGF
universal generating function
VEGA
vector-evaluated GA
W
wind turbine
Notations
Notations: Part I
t
time point
n
f
(
t
)
number of failed items
n
s
(
t
)
number of the survived items
n
0
sample size
T
random variable of the failure time
F
(
t
)
cdf of failure time
f
(
t
)
pdf of failure time
R
(
t
)
reliability at time
t
h
(
t
)
hazard function at time
t
H
(
t
)
cumulative hazard function at time
t
Q
^
(
t
)
estimate of the unreliability
R
^
(
t
)
estimate of the reliability
D
(
t
)
component or system demand at time
t
G
(
t
)
performance function at time
t
M
T
T
F
mean time to failure
X
random variable
a
crack length
N
load cycle
Q
total volume of wear debris produced
R
s
(
t
)
reliability of the system at time
t
(
⋅
)
unreliability function of the system
C
cost
x
decision variable
g
(
x
)
inequality constraints
h
(
x
)
equality constraints
f
(
x
)
criterion function
D
(
V
,
A
)
directed graph
d
(
⋅
)
length of the shortest path
Notations: Part II
t
time point
S
state set
M
perfect state
x
(
x
1
,
…
,
x
n
)
component state vector
X
(
X
1
,
…
,
X
n
)
state of all components
ϕ
(
⋅
)
structure function of the system
g
i
performance level of component
i
λ
k
j
i
transition rate of component
i
from state
k
to state
j
Q
k
j
i
(
t
)
kernel of the SMP analogous to
λ
k
j
i
of the CTMC
T
n
i
time of the
n
-th transition of component
i
G
n
i
performance of component
i
at the
n
-th transition
θ
j
k
i
(
t
)
probability that the process of component
i
starts from state
j
at time
t
A
φ
W
(
t
)
availability with a minimum on performance of total
φ
at time
t
u
i
(
z
)
universal generating function of component
i
p
i
j
Pr
(
X
i
j
)
probability of component
i
being at state
j
p
(
t
)
state probability vector
λ
i
j
(
t
)
transition rate from state
i
to state
j
at time
t
in Markov process
Λ
transition rate matrix
Π
(
⋅
)
possibility function
N
(
⋅
)
necessity function
B
e
l
(
⋅
)
belief function
P
l
(
⋅
)
plausibility function
F
−
1
(
⋅
)
inverse function
E
(
⋅
)
expectation equation
⊗
UGF composition operator
Π
(
⋅
)
possibility function
N
(
⋅
)
necessity function
B
e
l
(
⋅
)
belief function
P
l
(
⋅
)
plausibility function
F
−
1
(
⋅
)
inverse function
E
(
⋅
)
expectation equation
S
(
⋅
)
system safety function
R
i
s
k
(
⋅
)
system risk function
C
(
⋅
)
cost function
Notation: Part III
r
i
reliability of subsystem
i
x
(
x
1
,
…
,
x
n
)
T
decision variable vector
c
(
c
1
,
…
,
c
n
)
T
coefficients of the objective function
b
(
b
1
,
…
,
b
m
)
T
right-hand side values of the inequality constraints
z
(
z
1
,
z
2
,
…
,
z
M
)
objective vector
x
l
*
,
l
1
,
2
,
…
,
L
set of optimal solutions
w
(
w
1
,
w
2
,
…
,
w
M
)
weighting vector
x
*
global optimal solution
R
(
⋅
)
system reliability function
A
(
⋅
)
system availability function
M
(
⋅
)
system maintainability function
S
(
⋅
)
system safety function
C
(
⋅
)
cost function
R
i
s
k
(
⋅
)
system risk function
R
N
N
-dimensional solution space
f
i
i
-
th objective functions
g
j
j
-th equality constraints
h
k
k-th inequality constraints
ω
random event
ξ
(
q
(
ω
)
T
,
h
(
ω
)
T
,
T
(
ω
)
T
)
second-stage problem parameters
W
recourse matrix
y
(
ω
)
second-stage or corrective actions
Q
(
x
)
expected recourse function
U
uncertainty set
u
uncertainty parameters
ζ
perturbation vector
Z
perturbation set
x
u
*
optimal solution under the uncertainty parameter
u
PART I The Fundamentals
1 Reliability Assessment
Reliability is a critical attribute for the modern technological components and systems. Uncertainty exists on the failure occurrence of a component or system, and proper mathematical methods are developed and applied to quantify such uncertainty. The ultimate goal of reliability engineering is to quantitatively assess the probability of failure of the target component or system [1]. In general, reliability assessment can be carried out by both parametric or nonparametric techniques. This chapter offers a basic introduction to the related definitions, models and computation methods for reliability assessments.
1.1 Definitions of Reliability
According to the standard ISO 8402, reliability is the ability of an item to perform a required function, under given environmental and operational conditions and for a stated period of time without failure. The term “item” refers to either a component or a system. Under different circumstances, the definition of reliability can be interpreted in two different ways:
1.1.1 Probability of Survival
Reliability of an item can be defined as the complement to its probability of failure, which can be estimated statistically on the basis of the number of failed items in a sample. Suppose that the sample size of the item being tested or monitored is n0. All items in the sample are identical, and subjected to the same environmental and operational conditions. The number of failed items is nf and the number of the survived ones is ns, which satisfies
The percentage of the failed items in the tested sample is taken as an estimate of the unreliability, ,
Complementarily, the estimate of the reliability, R^(t), of the item is given by the percentage of survived components in the sample:
Example 1.1
A valve fabrication plant has an average output of 2,000 parts per day. Five hundred valves are tested during a reliability test. The reliability test is held monthly. During the past three years, 3,000 valves have failed during the reliability test. What is the reliability of the valve produced in this plant according to the test conducted?
Solution
The total number of valves tested in the past three years is
The number of failed components is
According to Equation 1.3, an estimate of the valve reliability is
1.1.2 Probability of Time to Failure
Let random variable T denote the time to failure. Then, the reliability function at time t can be expressed as the probability that the component does not fail at time t, that is,
Denote the cumulative distribution function (cdf) of T as F(t). The relationship between the cdf and the reliability is
Further, denote the probability density function (pdf) of failure time T as f(t). Then, equation (1.5) can be rewritten as
In all generality, the expected value or mean of the time to failure T is called the mean time to failure (MTTF), which is defined as
It is equivalent to
Another related concept is the mean time between failures (MTBF). MTBF is the average working time between two consecutive failures. The difference between MTBF and MTTF is that the former is used only in reference to a repairable item, while the latter is used for non-repairable items. However, MTBF is commonly used for both repairable and non-repairable items in practice.
The failure rate function or hazard rate function, denoted by h(t), is defined as the conditional probability of failure in the time interval [t, t+Δt] given that it has been working properly up to time t, which is given by
Furthermore, the cumulative failure rate function, or cumulative hazard function, denoted by H(t), is given by
Example 1.2 The failure time of a valve follows the exponential distribution with parameter h(t) (in arbitrary units of time-1). The value is new and functioning at time h(t). Calculate the reliability of the valve at time h(t) (in arbitrary units of time).
Solution
The pdf of the failure time of the valve is
The reliability function of the valve is given by
At time, the value of the reliability is
1.2 Component Reliability Modeling
As mentioned in the previous section, in reliability engineering, the time to failure of an item is a random variable. In this section, we briefly introduce several commonly used discrete and continuous distributions for component reliability modeling.
1.2.1 Discrete Probability Distributions
If random variable X can take only a finite number k of different values x1,x2,…,xk or an infinite sequence of different values x1,x2,…, the random variable X has a discrete probability distribution. The probability mass function (pmf) of X is defined as the function f such that for every real number x,
If x is not one of the possible values of X, then f(x)0. If the sequence x1,x2,… includes all the possible values of X, then ∑if(xi)1. The cdf is given by
1.2.1.1 Binomial Distribution
Consider a machine that produces a defective item with probability p (0<p<1) and produces a non-defective item with probability 1−p. Assume the events of defects in different items are mutually independent. Suppose the experiment consists of examining a sample of n of these items. Let X denote the number of defective items in the sample. Then, the random variable X follows a binomial distribution with parameters n and p and has the discrete distribution represented by the pmf in (1.14), shown in Figure 1.1. The random variable with this distribution is said to be a binomial random variable, with parameters n and p,
Figure 1.1 The pmf of the binomial distribution with n5, p0.4.
The pmf of the binomial distribution is
For a binomial distribution, the mean, μ, is given by
and the variance, σ2, is given by
1.2.1.2 Poisson Distribution
Poisson distribution is widely used in quality and reliability engineering. A random variable X has the Poisson distribution with parameter λ, λ>0, the pmf (shown in Figure 1.2) of X is as follows:
Figure 1.2 The pmf of the Poisson distribution with λ0.6.
The mean and variance of the Poisson distribution are
1.2.2 Continuous Probability Distributions
We say that a random variable X has a continuous distribution or that X is a continuous random variable if there exists a nonnegative function f, defined on the real line, such that for every interval of real numbers (bounded or unbounded), the probability that X takes a value in an interval [a, b] is the integral of f over that interval, that is,
If X has a continuous distribution, the function f will be the probability density function (pdf) of X. The pdf must satisfy the following requirements:
The cdf of a continuous distribution is given by
The mean, μ, and variance, σ2, of the continuous random variable are calculated by
1.2.2.1 Exponential Distribution
A random variable T follows the exponential distribution if and only if the pdf (shown in Figure 1.3) of T is
Figure 1.3 The pdf of the exponential distribution with λ1.
where λ>0 is the parameter of the distribution. The cdf of the exponential distribution is
If T denotes the failure time of an item with exponential distribution, the reliability function will be
The hazard rate function is
The mean, μ, and variance, σ2 are
1.2.2.2 Weibull Distribution
A random variable T follows the Weibull distribution if and only if the pdf (shown in Figure 1.4) of T is
Figure 1.4 The pdf of the Weibull distribution with β1.79, η1.
where β>0 is the shape parameter and η>0 is the scale parameter of the distribution. The cdf of the Weibull distribution is
If T denotes the time to failure of an item with Weibull distribution, the reliability function will be
The hazard rate function is
The mean, μ, and variance, σ2, are
1.2.2.3 Gamma Distribution
A random variable T follows the gamma distribution if and only if the pdf (shown in Figure 1.5) of T is
Figure 1.5 The pdf of the gamma distribution with β1.99,λ1.
where β>0 is the shape parameter and η>0 is the scale parameter of the distribution. The cdf of the gamma distribution is
If T denotes the failure time of an item with gamma distribution, the reliability function will be
The hazard rate function is
The mean, μ, and variance, σ2, are
1.2.2.4 Lognormal Distribution
A random variable T follows the lognormal distribution if and only if the pdf (shown in Figure 1.6) of T is
Figure 1.6 The pdf of the lognormal distribution with μ0, σ0.954.
where σ>0 is the shape parameter and μ>0 is the scale parameter of the distribution. Note that the lognormal variable is developed from the normal distribution. The random variable XlnT is a normal random variable with parameters μ and σ. The cdf of the lognormal distribution is
where Φ(x) is the cdf of a standard normal random variable. If T denotes the failure time of an item with lognormal distribution, the reliability function of T will be
The hazard rate function is
The mean, μ, and variance, σ2, are
Example 1.3
The random variable of the time to failure of an item, T, follows the following pdf:
where t is in days and t≥0.
What is the probability of failure of the item in the first 100 days?
Find the MTTF of the item.
Solution
The cdf of the random variable is
The probability of failure in the first 100 days is
