Post on 19-Aug-2018
STUDIES ON
RELIABILITY OPTIMIZATION PROBLEMS
BY GENETIC ALGORITHM
Thesis submitted to
THE UNIVERSITY OF BURDWAN
For the Award of Degree of
DOCTOR OF PHILOSOPHY IN MATHEMATICS
By
LAXMINARAYAN SAHOOLAXMINARAYAN SAHOOLAXMINARAYAN SAHOOLAXMINARAYAN SAHOO
Under the supervision of
Dr Asoke Kumar BhuniaDr Asoke Kumar BhuniaDr Asoke Kumar BhuniaDr Asoke Kumar Bhunia
Associate Professor, Department of Mathematics
&&&&
DrDrDrDr Dilip RoyDilip RoyDilip RoyDilip Roy
Professor, Centre for Management Studies
THE UNIVERSITY OF BURDWANTHE UNIVERSITY OF BURDWANTHE UNIVERSITY OF BURDWANTHE UNIVERSITY OF BURDWAN
BURDWANBURDWANBURDWANBURDWAN----713104713104713104713104
WEST BENGAL, INDIAWEST BENGAL, INDIAWEST BENGAL, INDIAWEST BENGAL, INDIA
2012201220122012
STUDIES ON
RELIABILITY OPTIMIZATION PROBLEMS
BY GENETIC ALGORITHM
LAXMINARAYAN SAHOO
M.Sc (Applied Mathematics)
A THESIS SUBMITTED FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS
THE UNIVERSITY OF BURDWAN
2012
Dedicated
to
my beloved and respected teachers
Dr Asoke Kumar Bhunia and Dr Dilip Roy
DECLARATION
I hereby declare that the thesis entitled “Studies on Reliability Optimization
Problems by Genetic Algorithm” submitted for the degree of Doctor of Philosophy
in Mathematics is my original work carried out under the supervision of Dr Asoke
Kumar Bhunia and Professor Dilip Roy. I further declare that the work embodied in
this thesis has not been submitted previously, in whole or in part, to any University
or Institution for any academic award.
Place: The University of Burdwan (Laxminarayan Sahoo)
Date:
THE UNIVERSITY OF BURDWAN
GOLAPBAG, BURDWAN – 713 104
WEST BENGAL, INDIA
Professor Dilip Roy
E-mail: dr.diliproy@gmail.com
Dr Asoke Kumar Bhunia
Associate Professor
E-mail: math_akbhunia@buruniv.ac.in
CERTIFICATE
This is to certify that the thesis entitled “S“S“S“Studies on Reliability Optimization tudies on Reliability Optimization tudies on Reliability Optimization tudies on Reliability Optimization
Problems by Genetic AlgorithmProblems by Genetic AlgorithmProblems by Genetic AlgorithmProblems by Genetic Algorithm” ” ” ” being submitted by Sri LLLLaxminarayan Sahooaxminarayan Sahooaxminarayan Sahooaxminarayan Sahoo
for the award of the degree of “D“D“D“Doctor of Philosophyoctor of Philosophyoctor of Philosophyoctor of Philosophy”””” to the University of
Burdwan is a record of bonafide research work carried out by him under our
guidance and supervision. Sri SSSSahooahooahooahoo has done this research work in the
DDDDepartment of Mathematicsepartment of Mathematicsepartment of Mathematicsepartment of Mathematics, T, T, T, Thehehehe UUUUniversity of Burdwanniversity of Burdwanniversity of Burdwanniversity of Burdwan, according to the
regulations of this University.
In our opinion, this thesis is of the standard required for the award of the
degree of ““““DDDDoctor of octor of octor of octor of PPPPhilosophyhilosophyhilosophyhilosophy”.”.”.”.
The research works, embodied in this thesis, have not been submitted to
any University or Institution for the award of any degree or diploma.
--------------------------------------
(Dr Asoke Kumar Bhunia)
--------------------------------------
(Professor Dilip Roy) Department of Mathematics Centre for Management Studies
The University of Burdwan The University of Burdwan
Burdwan-713104
India
Burdwan-713104
India
Acknowledgements
I feel myself not eligible enough unable to adequately thank to my supervisors Dr
Asoke Kumar Bhunia, Associate Professor, Department of Mathematics, The
University of Burdwan and Professor Dilip Roy, Centre for Management Studies, The
University of Burdwan, for their valuable guidance, constant help and
encouragement throughout my research work. I would like to thank them heartily
not only for their scholarly guidance and encouragement, but also for their endless
love and support.
I take this opportunity to thank the authorities of The University of Burdwan
and Raniganj Girls’ College for providing me the opportunity to carry out this
research work and for extending whole hearted cooperation and support.
I sincerely acknowledge all the help, cooperation and constructive suggestions
from the Head of the Department and all other faculty members and staff of the
Department of Mathematics, The University of Burdwan, during the course of my
research work. I would like to extend my thanks especially to Professor Gora Chand
Layek, Dr Absos Ali Shaikh and Dr Mantu Saha.
I feel privileged to thank Dr Krishna Bardhan (Ghosh), Principal, Raniganj
Girls’ College, for her cooperation and valuable suggestions.
I would like to express my heart-felt thanks to all of my colleagues and staff of
Raniganj Girls’ College for their encouragement, support and cooperation. In
particular, I thank Dr Sucheta Mukherjee, Associate Professor of English, Raniganj
Girls’ College for helping me by reading the proof of my manuscript.
I am grateful to Professor P. K. Kapur, University of Delhi for his valuable
suggestions and cooperation.
Acknowledgements vi
My sincere thanks also go to Professor Manoranjan Maiti, Vidyasagar
University, for his very sincere encouragement and helpful suggestions.
I would like to express my gratitude to Dr Monimohan Mandal, Midnapore
College, Dr Ranjan Kumar Gupta, West Bengal State University, Dr Jayanta Majumdar,
Hooghly Mohsin College and my co-researchers, Mr Pintu Pal, Mr Samiran Karmakar,
Mr Sanat Mahato, Mr Avijit Duary, Mr Debkumar Pal, Mr Amiya Biswas, Mr Akbar Ali
Shaikh and Mr Subhra Sankha Samanta for their constant help and suggestions. I also
thank Mr Biswajit Ta and Mr Tarun Das for helping me a lot during my research
work.
I wish to thank all of my friends at Golapbag, The University of Burdwan for
making me enjoy every moment of my research work there and making it
memorable.
I also acknowledge the generosity of the University Grants Commission (UGC),
India, for providing financial support to me through Minor Research Project.
My deepest appreciation goes to my parents and other family members
including my two little lovely and sweet nieces, Senjuti and Sangbrita. I could not
have finished this research work without their unending love.
Most specially, I thank my wife, Anima, for her constant inspiration and
encouragement and also for her patience and continuous support.
Finally, I want to acknowledge the help, support and love of my elder brother,
Sri Durlav Sahoo, who laid the foundations of my career.
In fine, I am solely responsible for any errors and omissions in this thesis.
Place: The University of Burdwan (Laxminarayan Sahoo)
Date:
Publications
The thesis includes the following published works and a few works communicated
for publication:
Published/Accepted papers
1. Genetic algorithm based multi-objective reliability optimization in interval
environment, Computers & Industrial Engineering, 62, 152-160, 2012.
2. Reliability stochastic optimization for a series system with interval component
reliability via genetic algorithm, Applied Mathematics and Computation, 216,
929-939, 2010.
3. A genetic algorithm based reliability redundancy optimization for interval valued
reliabilities of components, Journal of Applied Quantitative Methods, 5(2),
270-287, 2010.
4. An application of genetic algorithm in solving reliability optimization problem
under interval component Weibull parameters, Mexican Journal of Operations
Research, 1(1), 2-19, 2012.
5. Genetic Algorithm Based Mixed-integer Nonlinear Programming in Reliability
optimization Problems, Quality, Reliability and Infocom Technology: Trends
and Future Directions, ISBN: 978-81-8487-172-2, 25-43, Narosa, 2012.
6. Genetic algorithm based reliability optimization in interval environment,
Innovative Computing Methods and Their Applications to Engineering
Problems, SCI 357, 13-36, Springer-Verlog Berlin Heidelberg, 2011.
7. Reliability optimization in imprecise environment via genetic algorithm,
Proceedings of IIT Roorke, India, ISBN: 81-86224-71-2, 372-379, AMOC 2011.
8. Optimization of Constrained multi-objective reliability problems with interval
valued reliability of components via genetic algorithm, Indian Journal of
Industrial and Applied Mathematics, 2011 (Accepted).
Communicated papers
1. Reliability optimization in Stochastic Domain via Genetic Algorithm,
International Journal of Quality & Reliability Management, Emerald.
2. Reliability optimization under high and low level redundancies via genetic
algorithm for imprecise parametric values, Computers & Structures, Elsevier.
Contents
Table of contents Page No.
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v
Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii
Notations used in the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xii
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xv
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvi
Acronyms used in the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
1.2 Basic Concepts and Terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
1.3 Historical Review of Reliability Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . .9
1.4 Objectives and Motivation of the Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
1.5 Organization of the Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Chapter 2: Solution Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1 Interval Approach in Reliability Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
2.2 Mathematical Backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
2.2.1 Finite Interval Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Interval Order Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
2.2.3 Metric Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Solution Methodologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Contents ix
2.3.1 Genetic Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36
2.3.2 GA-Based Constrained Handling Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
Chapter 3: Reliability Redundancy Allocation Problems in Interval
Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Constrained Redundancy Optimization Problem for Different System. . . . . . . . . 56
3.2.1 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
3.2.2 Series System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
3.2.3 Hierarchical Series-Parallel System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
3.2.4 Complex/Complicated System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
3.2.5 K-out-of-N System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
3.2.6 Reliability Network System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
3.3 Solution Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
3.4 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
3.6 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Chapter 4: Reliability Optimization under High and Low-level
Redundancies for Imprecise Parametric Values. . . . . . . . . . . . . . . . . . 77
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
4.3 Low-level and High-level Redundancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Formulation of Reliability-Redundancy Optimization Problems. . . . . . . . . . . . . . . 80
4.5 Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82
4.6 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
Contents x
4.8 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter 5: Reliability Optimization under Weibull Distribution
. with Interval Valued Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96
5.3 Weibull Distribution with Interval Valued Parameters. . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
5.6 Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter 6: Stochastic Optimization of System Reliability for.
Series System with Interval Component Reliabilities. . . . . . . . . . . 103
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
6.3 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
6.5 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.6 Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.7 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter 7: Reliability Optimization with Interval Parametric Values
in the Stochastic Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
7.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
7.3 Normal Distribution with Interval Valued Parameters. . . . . . . . . . . . . . . . . . . . . . . 118
7.4 Stochastic Mixed Integer Programming: A Complicated System with
Chance Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.5 Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
Contents xi
7.6 Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.7 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Chapter 8: Multi-objective Reliability Optimization in Interval
Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
8.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134
8.3 Multi-objective Optimization in Interval Environment. . . . . . . . . . . . . . . . . . . . . . . 135
8.4 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.4.1 Global Criteria Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
8.4.2 Tchebycheff Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.4.3 Weighted Tchebycheff Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.4.4 Lexicographic Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.4.5 Lexicographic Weighted Tchebycheff Problem. . . . . . . . . . . . . . . . . . . . . . . . 143
8.5 Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.6 Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.7 Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.8 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Chapter 9: General Conclusion and Scope of Future Research. . . . . . . . . . . . . . 150
9.1 General Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2 Scope of Future Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
Notations used in the Thesis
jx Number of redundant components of j -th subsystem
jl , ju Lower and upper bounds of jx
[ , ]j jL jRr r r= Interval valued reliability of component at stage j
n Number of decision variables or number of redundant
components
m Number of resource constraints
h Number of redundant subsystems, arranged in parallel in
case of high-level redundancy.
ib Total amount of i th− resource available
( ) 1 (1 ) jx
j j jR x r= − − The reliability of j -th subsystem
x 1 2( , ,..., )nx x x
( )jLR x , ( )jRR x Lower and upper bounds of ( )jR x
iR The reliability of i-th subsystem, 1, 2, ,i q q n= + + ⋅⋅⋅
iL ,iU Lower and upper bounds of
iR , 1, 2, ,i q q n= + + ⋅⋅⋅
SR =[ , ]SL SRR R System reliability which is interval valued
1j jQ R= − Unreliability of j-th subsystem
( , )x R 1 2 1( , ,..., , ,..., )q q nx x x R R+
( )SR x System reliability depending on x
( )SLR x , ( )SRR x Lower and upper bounds of ( )SR x
( , )SR x R System reliability depending on ( , )x R
Notations… xiii
( , )SLR x R , ( , )SRR x R Lower and upper bounds of ( , )SR x R
( )SR h System reliability depending on h
( )SLR x , ( )SRR h Lower and upper bounds of ( )SR h
( , )SR x t System reliability depending on ( , )x t
( , )SLR x t , ( , )SRR x t Lower and upper bounds of ( , )SR x t
( , )jC x R Consumption of j-th resource ( 1,2,..., )j m=
( , )wC x R Weighted cost
* * *[ , ]R R R= Minimum prescribed reliability in case of cost minimization
problem
( )ig x i-th resource constraint
t Mean time-to-failure
[ , ]i iL iRα α α= Interval valued Weibull scale parameter for i-th subsystem
[ , ]i iL iRβ β β= Interval valued Weibull shape parameter for i-th subsystem
( ) [ ( ), ( )]i iL iRr t r t r t= [ , ][ , ]
, 1, 2, ,iL iR
iL iR te i n
β βα α− = ⋅⋅⋅
[ ( , ), ( , )]iL i iR iR x t R x t 1 (1 [ ( ), ( )]) ix
iL iRr t r t− − , the reliability of i-th parallel
subsystem
[ , ] 1 [ , ]iL iR iL iRq q r r= − Unreliability of i-th component
[ , ]j jL jRc c c= Interval valued cost coefficients for the j-th component
[ , ]j jL jRw w w= Interval valued weight coefficients for the j-th component
[ , ]T TL TRR R R= Interval valued target system reliability
iγ Level of significant
( , )i iU ξ η Uniform distribution between iξ and iη
2( , )
i ib bN µ σ Normal distribution with mean ibµ and variance 2
ibσ
Notations… xiv
2( , )
i ib bLN µ σ Log normal distribution with mean ibµ and variance 2
ibσ
( , )i iC g x R′� �
Total consumption of i-th resource ( 1,2,..., )i m=
( ) [ ( ), ( )]i iL iRA x f x f x= Interval valued objective function
* * *[ , ]i iL iRz z z= i-th component of interval valued ideal objective vector
** ** **[ , ]i iL iRz z z= i-th component of interval valued utopian objective vector
[ ( ), ( )]S SL SRR R x R x= Interval valued system reliability
[ ( ), ( )]S SL SRC C x C x= Interval valued system cost
[ , ]iL iRc c Interval valued cost coefficients
iP Constant associated with volume
iW Constant associated with weight
S Feasible region
n� n-dimensional Euclidian space
* *[ , ]SL SRR R Optimal value of [ ( ), ( )]SL SRR x R x
* *[ , ]SL SRC C Optimal value of [ ( ), ( )]SL SRC x C x
** **[ , ]SL SRR R Infeasible solution of [ ( ), ( )]SL SRR x R x
** **[ , ]SL SRC C Infeasible solution of [ ( ), ( ) ]SL SRC x C x
[ , ]i iL iLε ε ε= Small positive and computationally significant interval
number
� Set of all positive integers
� Set of all real numbers
Tables
List of tables Page No.
Table 3.1: Parameters used in Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Table 3.2: Parameters used in Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Table 3.3: Computational results for Examples 1-4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Table 3.4: Computational results for Examples 5-6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Table 3.5: Computational results for Examples 7-8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Table 3.6: Computational results for Examples 9-10. . . . . . . . . . . . . . . . . . . . . . . . . . . .73
Table 3.7: Computational results for Examples 11-12. . . . . . . . . . . . . . . . . . . . . . . . . . 73
Table 4.1: Values of the parameters related to Examples 1-4. . . . . . . . . . . . . . . . . . . .87
Table 4.2: Computational results for Examples 1-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
Table 5.1: Values of iα and ( 1,2,3,4,5)i iβ = for four different cases. . . . . . . . . . . .101
Table 5.2: Computational results of Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102
Table 6.1: Numerical data of Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
Table 6.2: Numerical data of Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Table 6.3: Computational results of Examples 1-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Table 7.1: Optimum solution sets of Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Table 7.2: Comparative results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Table 8.1: Shows the data for the Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Table 8.2: Computational results of Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147
Figures
List of figures Page No.
Figure 1.1: Organization of research work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Figure 2.1: Type-1 intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 2.2: Type-2 intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 2.3: Type-3 intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 3.1: Series system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 3.2: Parallel-series system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
Figure 3.3: Hierarchical series-parallel system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 3.4: Complex/Complicated system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
Figure 3.5: 2-out-of-3 system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Figure 3.6: P_size vs. centre of the objective function for Example 1. . . . . . . . . . . . . .74
Figure 3.7: P_cross vs. centre of the objective function for Example 1. . . . . . . . . . . . 74
Figure 3.8: P_mute vs. centre of the objective function for Example 1. . . . . . . . . . . . 75
Figure 4.1: Low-level redundancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 4.2: High-level redundancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 4.3: Low-level redundancy of five-link bridge system. . . . . . . . . . . . . . . . . . . . .86
Figure 4.4: High-level redundancy of five-link bridge system. . . . . . . . . . . . . . . . . . . . 88
Figure 4.5: P_size vs. interval valued system reliability for Example 1. . . . . . . . . . . .90
Figure 4.6: Max_gen vs. interval valued system reliability for Example 1. . . . . . . . . 90
Figure 4.7: P_cross vs. interval valued system reliability for Example 1. . . . . . . . . . .91
Figure 4.8: P_mute vs. interval valued system reliability for Example 1. . . . . . . . . . .91
Figure 5.1: Five-link bridge network system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 6.1: A n-stage series system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figures xvii
Figure 6.2: P_size vs. interval valued system reliability for series system. . . . . . . .113
Figure 6.3: P_cross vs. interval valued system reliability for series system . . . . . .114
Figure 6.4: P_mute vs. interval valued system reliability for series system . . . . . .114
Figure 6.5: Max_gen vs. interval valued system reliability for series system. . . . .114
Figure 7.1: Bridge network system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Figure 8.1: A n-stage series system for MOOP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
Figure 8.2: P_size vs. interval valued system reliability for MOOP. . . . . . . . . . . . . . .148
Figure 8.3: P_cross vs. interval valued system reliability for MOOP. . . . . . . . . . . . . 148
Figure 8.4: P_mute vs. interval valued system reliability for MOOP. . . . . . . . . . . . . 148
Acronyms used in the Thesis
CPU time Execution time
GA Genetic Algorithm
GAs Genetic Algorithms
HSP Hierarchical Series-Parallel
INLPP Integer Non-linear Programming Problem
IVNLIP Interval Valued Non-linear Integer Programming Problem
IVNLP Interval Valued Non-linear Programming Problem
max_gen Maximum Number of Generation
MINLPP Mixed-integer Non-linear Programming Problem
MOEA Multi-objective Evolutionary Algorithm
MOOP Multi-objective Optimization Problem
p_cross Crossover Probability
p_mute Mutation Probability
p_size Population Size
PC Personal Computer
PFP Parameter Free Penalty
RAP Redundancy Allocation Problem
VLSI Very Large Scale Integration
CHAPTER 1
Introduction
• General Introduction
• Basic Concepts and Terminologies
• Historical Review of Reliability Optimization Problems
• Objectives and Motivation of the Thesis
• Organization of the Thesis
Studies on Reliability Optimization Problems by Genetic Algorithm 2
1.1 General Introduction
The subject “Reliability Optimization” appeared in the literature in due late 1940s
and was first applied to communication and transportation systems. Most of the
earlier works were confined to an analysis of certain performance aspects of an
operating system. One of the goals of the reliability engineer is to find the best way to
increase the system reliability. The reliability of a system can be defined as the
probability that the system will be operating successfully at least up to a specified
point of time (i.e., mission time) under stated conditions. As systems are becoming
more complex, the consequences of their unreliable behavior have become severe in
terms of cost, effort and so on. The interests in accessing the system reliability and
the need to improve the reliability of products and system have become more and
more important.
The primary objective of reliability optimization is to find the best way to
increase the system reliability. This can be done by different ways. Some of these are
as follows:
(i) Increasing the reliability of each component in the system.
(ii) Using redundancy for the less reliable components.
(iii) Using standby redundancy which is switched to active components when
failure occurs.
(iv) Using repair maintenance where failed components are replaced.
(v) Using preventive maintenance such that components are replaced by new ones
whenever they fail or at some fixed interval, whichever is earlier.
(vi) Using better arrangement for exchangeable components.
Introduction 3
To improve the system reliability, implementation of the above steps will normally
result in the consumption of resources. Hence, a balance between the system
reliability of a system and resource consumption is an important task.
When, redundancy is used to improve the system reliability, the
corresponding problem is known as redundancy allocation problem. The objective of
this problem is to find the number of redundant components that maximizes the
system reliability under several resource constraints. This problem is one of the most
popular ones in reliability optimization since 1950s because of its potentiality for
broad applications. When it is difficult to improve the reliability of unreliable
components, system reliability can easily be enhanced by adding redundancies on
those components. However, for design engineers improving of component
reliability have been generally preferred over by adding redundancy, because, in
many cases, redundancy is difficult to add to real systems due to technical limitations
and relatively large quantities of resources, such as weight, volume and cost that are
required.
Network reliability design problems have attracted many researchers, such as
network designers, network analysts, and network administrators, in order to share
expansive hardware and software resources and provide the access of main systems
from different locations. These problems have many applications in the areas of
telecommunications and computer networking and related domains in the electrical,
gas sewer networks. During the designing of network systems, one of the important
steps is to find the best layout of components to optimize some performance criteria,
such as cost, transmissions delay or reliability. The corresponding optimal design
problem can be formulated as a combinatorial problem.
Studies on Reliability Optimization Problems by Genetic Algorithm 4
However, recently developed advanced technologies such as semiconductor,
integrated circuits and nano technology, however, have revived the importance of
the redundancy strategy. The current downscaling trend in the semiconductor
manufacturing has caused many inevitable defects and subsequent faults in
integrated circuits. It is widely accepted that there are certain limitations on
enhancing reliability or yield in semiconductor manufacturing by developing
relevant physical technologies. Hence, various fault-tolerant and self-repairable
techniques have been studied. These approaches are mainly based on adding
redundancies on components and controlling the usage of redundancies. In fact, most
memory integrated circuits and VLSI, which includes internal memory blocks,
currently use a hierarchical redundancy scheme to increase the yield and reliability
of the chip.
To efficiently constitute the fault-tolerant systems with redundancy, the
number of redundancies should be optimized. However, for improving the system
reliability the addition of redundant components to the system is a formidable task
due to several resource constraints arising out of the size, cost and quantities of
resources coupled with technical constraints. Thus, the redundancy allocation
problem is a practical problem of determining the appropriate number of redundant
components that maximize the system reliability under different resource
constraints. Equivalently, the problem is a non-linear constrained optimization
problem. To solve this type of problem, several researchers have proposed different
approaches. In their works, the reliabilities of the system components are assumed to
be known at a fixed positive level, which lies between zero and one. However in real-
life situations, the reliabilities of these individual components may fluctuate due to
different reasons. It is not always possible for a technology to produce different
Introduction 5
components with exactly identical reliabilities. Moreover the human factor, improper
storage facilities and other environmental factors may affect the reliabilities of the
individual components. Hence, it is sensible to treat the component reliabilities as
positive imprecise numbers between zero and one instead of fixed real numbers. To
define the problem associated with such imprecise numbers, generally different
approaches like stochastic, fuzzy and fuzzy-stochastic approaches are used. In
stochastic approach, the parameters are assumed to be random variables with
known probability distribution whereas in fuzzy approach, the parameters,
constraints and goals are considered as fuzzy sets with known membership
functions. On the other hand, in fuzzy-stochastic approach, some parameters are
viewed as fuzzy sets and others as random variables. However, to select the
appropriate membership function for fuzzy approach, probability distribution for
stochastic approach and both for fuzzy-stochastic approach is a very complicated
task for a decision-maker and it arises a controversary situation as to solve a
decision-making problem, other decisions are to be taken intermediately. Therefore,
to overcome the difficulties arisen in the selection of those, the imprecise numbers
may be represented by interval numbers. As a result, the objective function of
reliability optimization problem will be interval valued, which is to be optimized.
These types of optimization problems with interval objective can be solved by
a well known powerful computerized heuristic search and optimization method, viz.
genetic algorithm (GA), which is based on the mechanics of natural selection
(depending on the evolution principle “Survival of the fittest”) and natural genetics. It
is executed iteratively on the set of real/binary coded solutions called population. In
each iteration (which is called generation), three basic genetic operations, viz.
selection/reproduction, crossover and mutation are performed.
Studies on Reliability Optimization Problems by Genetic Algorithm 6
1.2 Basic Concepts and Terminologies
1.2.1 Reliability Definition
According to the Aeronautical Radio Inc. (1994), the definition of reliability is as
follows:
“Reliability is the probability that a system will perform satisfactorily for at least a
given period of time when used under stated conditions”.
So, the reliability is defined as the probability of a device performing its
intended purpose adequately for the period of time intended under the operating
conditions encountered. The reliability is the probability with which the devices will
not fail to perform a required operation for certain duration of time. Such problem is
known as the problem of survival. This definition brings into the focus of four
important factors, viz.
(i) The reliability of a device is expressed as a probability.
(ii) The device is required to give adequate performance.
(iii) The duration of adequate performance is specified.
(iv) The environmental or operating conditions are specified.
However, in practice, even the best design manufacturing and maintenance efforts do
not completely eliminate the occurrence of failure.
1.2.2 System Reliability
According to Kuo, Prasad, Tillman and Hwang (2001), “System reliability is a
measure of how well a system meets its design objective and it is usually expressed
in terms of the reliabilities of the subsystems of components”.
Generally, to determine the reliability factor of a system, the system is blown
up into down to sub systems and elements whose individual reliability factors can be
Introduction 7
estimated or determined. Depending on the manner in which these subsystems and
elements are connected to constitute the given system. The combinatorial rules are
applied to obtain the system reliability.
1.2.3 Fundamental System Configurations
A system in many cases is not made of a single component. We always want to
evaluate the reliability of a simple as well as complex/complicated system. Let us
consider a reliability system consisting of a number component. These components
can be hardware or human or even software. If some of the components are software
products, then the modeling requires special attentions.
Now, we shall discuss several important reliability configurations.
1.2.4 Series Configuration
The series configuration is the simplest and perhaps one of the most common
structures. In this configuration, all the components must be operating to ensure the
system operation. In other words, the system fails when any one of the components
fails.
1.2.5 Parallel Configuration
A parallel system is a system that is not considered to have failed unless all
components have failed. This is sometimes called a redundant configuration. The
word “redundant” is used only when the system configuration is deliberately
changed to produce additional parallel paths in order to improve the system
reliability. In a parallel configuration consisting of a number of components, the
system works if any one of those components is working.
Studies on Reliability Optimization Problems by Genetic Algorithm 8
1.2.6 Series-Parallel Configuration
Let us consider a system which consists of k subsystems connected in parallel, with i-
th subsystem consisting of in components in series for 1, 2, ,i k= ⋅⋅⋅ . Such a system is
called a series-parallel system.
1.2.7 Parallel-Series Configuration
Let us consider a system consisting of k subsystems in series and subsystem i,
1 i k≤ ≤ , in turn in components in parallel. Such a system is called a parallel-series
system.
1.2.8 Hierarchical Series-Parallel Systems
A system is called a hierarchical series-parallel system (HSP) if the system can be
viewed as a set of subsystems arranged in a series-parallel; each subsystem has a
similar configuration; subsystems of each subsystem have a similar configuration
and so on. This system has a non-linear and non-separable structure and consists of
nested parallel and series system.
1.2.9 Complex/Complicated System
Sometimes a system cannot be reduced to series and parallel configurations, because
there exist combinations of components which are connected neither in a series nor
in parallel that system is called complex/complicated or non-parallel series systems.
1.2.10 K-out-of-N System
A k out of n− − − system is an n -component system which functions when at least
k components out of n components function satisfactorily. This redundant system is
sometimes used in the place of a pure parallel system. It is also referred to as
Introduction 9
:k out of n G− − − system. An n -component series system is a :n out of n G− − −
system whereas a parallel system with n -components is a 1 :out of n G− − − system.
1.2.11 Coherent System
In non-series systems, it is not necessary that all components operate to make the
system operational. In such systems, we can also find subsets of components such
that the failure of all components in the subset leads to the system failure
irrespective of the states of the other components. The theory of coherent systems
deals with the deterministic functional relationship between the system and its
components. Such a relationship is useful for finding the reliability of large and
complex/complicated systems.
1.3 Historical Review of Reliability Optimization Problems
Now-a-days, our society is mostly dependent on modern technological systems and
there is no doubt that these technological systems have improved the productivity,
health and affluence of our society. However, this increasing dependence on modern
technological systems requires dealing with the complicated operations and
sophisticated management. For each of the complex/complicated systems, the
system reliability plays an important role. The reliability of any system is very
important to manufacturers, designers and also to the users. During the design phase
of a product, reliability engineers/designers are called upon to measure the
reliability of that product. They desire the larger reliability of their products which
raise the production cost of the items. In such a case, there arises a question as to
how to meet the goal for the system reliability. As a result, the increase in the
production cost has negative effects on the user’s budget. Therefore, the design
reliability optimization problem is phrased as reliability improvement at a minimum
Studies on Reliability Optimization Problems by Genetic Algorithm 10
cost. In this connection, a widely known method for improving the system reliability
of a system is to introduce several redundant components. For better designing a
system using components with known cost, reliability, weight and other attributes,
the corresponding problem can be formulated as a combinatorial optimization
problem, where either system reliability is maximized or system cost is minimized.
Therefore both the formulations generally involve constraints on allowable weight,
cost and/or minimum targeted system reliability level. The corresponding problem is
known as the reliability redundancy allocation problem. The primary objective of the
reliability redundancy allocation problem is to select the best combination of
components and levels of redundancy either to maximize the system reliability
and/or to minimize the system cost subject to several constraints.
In the existing literature, reliability optimization problems are classified into
three categories according to the types of decision variables. These are reliability
allocation, redundancy allocation and reliability redundancy allocation. If the
component reliabilities are the only variables, then the problem is called reliability
allocation. If the number of redundant components is the only variable, then the
problem is called redundancy allocation problem (RAP). On the other hand, if both
the component reliabilities and redundancies are variables of the problem then the
problem is called reliability redundancy allocation problem. For reliability allocation
problems, one may refer to the works of Allella, Chiodo and Lauria (2005), Yalaoui,
Chatelet and Chu (2005) and Salzar, Rocco and Galvan (2006). Researchers like Kim
and Yum (1993), Coit and Smith (1996, 1998), Prasad and Kuo (2000), Liang and
Smith (2004), Ramirez-Marquez and Coit (2004), Yun and Kim (2004), Nourelfath
and Nash (2005), You and Chen (2005), Agarwal and Gupta (2006), Coit and Konak
(2006), Ha and Kuo (2006b), Tian and Zuo (2006), Liang and Chen (2007), Nash,
Introduction 11
Nourelfath and Ait-Kadi (2007), Onishi, Kimura, James and Nakagawa (2007), Zhao,
Liu and Dao (2007) and others have solved redundancy allocation problem. Also,
Federowicz and Mazumdar (1968), Tillman, Hwang and Kuo (1977b), Misra and
Sharma (1991), Dhingra (1992), Painton and Campbell (1995), Ha and Kuo (2005,
2006a), Chen (2006), Kim, Bae and Park (2006) and others have solved the reliability
redundancy allocation problem. Several researchers have considered standby
redundancy [Gordon (1957), Messinger and Shooman (1970), Misra (1975), Sakawa
(1978a, 1981a), Zhao and Liu (2003), Yu, Yalaoui, Chatelet and Chu (2007)], multi-
state system reliability [Boland and EL-Neweihi (1995), Prasad and Kuo (2000),
Ramirez-Marquez and Coit (2004), Meziane, Massim, Zeblah, Ghoraf and Rahil
(2005), Tian, Levitin and Zuo (2009) and Li, Chen, Yi and Tao (2010)] and
modular/multi-level redundancy [Yun and Kim (2004) and Yun, Song and
Kim(2007)].
To solve these problems, several researchers have developed different
optimization methods which include exact methods, approximate methods,
heuristics, meta-heuristics, hybrid heuristics and multi-objective optimization
techniques etc. Dynamic programming, branch and bound, cutting plane technique,
implicit enumeration search technique are exact methods which provide exact
solution to reliability optimization problems. The variational method, least square
formulation and geometric programming and Lagrange multiplier give an
approximate solution. A detailed review of the different optimization approaches to
determine the optimal solutions is presented in Tillman, Hwang and Kuo (1977a,
1980), Sakawa (1978b, 1981b), Kuo, Prasad, Tillman and Hwang (2001) and Kuo and
Wan (2007a, 2007b). On the other hand, heuristic, meta- heuristic and hybrid
heuristic have been used to solve complicated reliability optimization problems.
Studies on Reliability Optimization Problems by Genetic Algorithm 12
They can provide optimal or near optimal solution in reasonable computational time.
Genetic algorithm, simulated annealing, tabu search, ant colony optimization and
particle swarm optimization are some of the approaches in those categories. For
detailed discussion, one may refer to the works of Kuo, Hwang and Tillman (1978)
Coit and Smith (1996), Hansen and Lih (1996), Ravi, Murty and Reddy (1997), Zhao
and Song (2003), Liang and Smith (2004), Coelho (2009a) and others.
Bellman (1957) and Bellman and Dreyfus (1958, 1962) used dynamic
programming to maximize the reliability of a system with single cost constraint. In
their works, the problem was to identify the optimal levels of redundancy for only
one component in each subsystem.
In the year 1968, Fyffe, Hines and Lee (1968) considered a system having 14
subsystems with both cost and weight constraints and solved the corresponding
reliability optimization problem by dynamic programming approach. In their work,
for each subsystem there are three or four different choices of components each with
different reliability, weight and cost. They used Lagrange’s multiplier technique to
accommodate the multiple constraints.
Nakagawa and Miyazaki (1981) used a surrogate constraints approach, by
showing the inefficiency of the use of a Lagrange multiplier with dynamic
programming. Their algorithm was tested for 33 different Fyffe’s problems of which
feasible solutions were obtained only for 30 problems.
Redundancy allocation problem can be solved by another important approach
i.e., integer programming approach. Ghare and Taylor (1969) first used the branch
and bound method to maximize the system reliability under given non-linear but
separable constraints. Bulfin and Liu (1985) formulated the problem as a knapsack
problem using surrogate constraints (approximated by Lagrangian multipliers found
Introduction 13
by subgradient optimization (Fisher (1981)) and used integer programming to solve
it. Nakagawa and Miyazaki (1981) investigated the Fyffe problem and formulated 33
variances of the problem as integer programming problem. Bulfin and Liu (1985)
solved the same problem.
Misra and Sharma (1991) presented a fast algorithm to solve integer
programming problems like those of Ghare and Taylor (1969). The problem was
formulated as a multi-objective decision-making problem with distinct goals for
reliability, cost and weight and also solved by integer programming by Gen, Ida,
Tsujimura and Kim (1993).
To solve this type of problem several other methodologies have been
proposed by researchers, like, Kuo, Lin, Xu and Zhang (1987), Hikita, Nakagawa and
Narihisa (1992), Sung and Cho (1999), Mettas (2000), Coit and Smith (1996, 2002),
Sun and Li (2002), Ha and Kuo (2006b), Liang and Chen (2007), Ramirez-Marquez
and Coit (2007b), Coelho (2009a, 2009b) and others.
Among these methodologies, applications of GA in reliability optimization
problems have been received warm reception among the researchers. In this
connection one may refer to the work of Painton and Campbell (1994, 1995). They
used GA in solving an optimization model that identifies the types of component
improvements and the level of effort spent on those improvements to maximize one
or more performance measures (e.g., system reliability or availability) subject to the
constraints (e.g., cost) in the presence of uncertainty. In the year 1994, a redundancy
allocation problem with several failure modes was solved by Ida, Gen and Yokota
(1994) with the help of GA. Coit and Smith (1996) have solved a redundancy
optimization problem by applying GA to a series-parallel system with mix of
components in which each subsystem is a k-out-of-n: G system. In the year 1998, Coit
Studies on Reliability Optimization Problems by Genetic Algorithm 14
and Smith (1998) used GA-based approach to solve the redundancy allocation
problem for series-parallel system, where the objective is to maximize a lower
percentile of the system time to failure distribution. Yun and Kim (2004) and Yun,
Song and Kim (2007) solved multi-level redundancy allocation in series-parallel
system using genetic algorithm.
From the earlier-mentioned discussion, it may be observed that all the
problems solved by several researchers are of single objective. However, in most of
the real-world design or decision-making problems involving reliability optimization,
there occurs the simultaneous optimization of more than one objective function.
When designing a reliable system, as formulated by multi-objective optimization
problem, it is always desirable to simultaneously optimize several objectives such as
system reliability, system cost, volume and weight. For this reason multi-objective
optimization problem attracts a lot of attention from the researchers. The objective
of this problem is to maximize the system reliability and minimize the system cost,
volume and weight. A Pareto optimal set, which includes all of the best possible
solutions between the given objectives than a single objective, is usually identified
for multi-objective optimization problems. Dhingra (1992), Rao and Dhingra (1992)
used goal programming formulation and the goal attainment method to generate
Pareto optimal solutions. Ravi, Reddy and Zimmermann (2000) presented fuzzy
multi-objective optimization problem using linear membership functions for all of
the fuzzy goals. Busacca, Marseguerra and Zio (2001) developed a multi-objective GA
to obtain an optimal system configuration and inspection policy by considering every
target as a separate objective. Sasaki and Gen (2003a, 2003b) solved multi-objective
reliability-redundancy allocation problems using linear membership function for
both objectives and constraints. Elegbede and Adjallah (2003) solved multi-objective
Introduction 15
optimization problem by transforming it into single objective optimization problem.
Tian and Zuo (2006) and Salzar, Rocco and Galvan (2006) have used a genetic
algorithm to solve the non-linear multi-objective reliability optimization problems.
Limbourg and Kochs (2008) solved multi-objective optimization of generalized
reliability design problems using feature models. In the year 2009, Okasha and
Frangopal (2009) solved lifetime-oriented multi-objective optimization of structural
maintenance considering system reliability, redundancy and life-cycle cost model
using genetic algorithm. Li, Liao and Coit (2009) have used multiple objective
evolutionary algorithm (MOEA) to solve multi-objective reliability optimization
problem.
1.4 Objectives and Motivation of the Thesis
In reliability engineering, the reliability optimization is an important problem. As
mentioned earlier this problem came into the existence in the late 1940s and was
first applied to communication and transport system. After that a lot of works has
been done by several researchers incorporating different factors. To solve those
problems, a number of methods/techniques has been proposed. In most of these
works, the reliability of a component was considered as precise value i.e., fixed lying
between zero and one. However, due to some factors mentioned in Section 1.1, it may
not be fixed though it may vary between zero and one. So, to represent the same,
some of the researchers have used either stochastic or fuzzy or fuzzy-stochastic
approaches. On the other hand, it may be represented by an interval which is
significant. To the best of our knowledge, very few works have been done
considering interval valued reliabilities. Even today, there is a lot of scope to work in
this area considering interval valued reliabilities of components. The detailed scheme
Studies on Reliability Optimization Problems by Genetic Algorithm 16
of works along with the works presented in this thesis and also the further scope of
research has been shown in Figure 1.1.
Figure 1.1: Organization of research work
Introduction 17
It may be noted from Figure 1.1 that the objectives of the thesis is
(i) to formulate the different types of redundancy allocation problems involving
reliability maximization and cost minimization considering component
reliabilities as interval valued numbers.
(ii) to formulate chance constraints reliability stochastic optimization problem,
network reliability design problem and multi-objective reliability
optimization with fixed and interval valued values of reliability of
components.
(iii) to solve the problems mentioned in (i) and (ii) by real coded genetic
algorithm, interval mathematics and order relations proposed in the thesis.
1.5 Organization of the Thesis
In this thesis, some reliability optimization problems have been formulated and
solved in interval environment with the help of interval mathematics, our proposed
interval order relations and real coded genetic algorithm. The entire thesis has been
divided into nine chapters as follows:
Chapter 1 Introduction
Chapter 2 Solution Methodologies
Chapter 3 Reliability Redundancy Allocation Problems in Interval
Environment
Chapter 4 Reliability Optimization under High and Low-level Redundancies
for Imprecise Parametric Values
Chapter 5 Reliability Optimization under Weibull Distribution with Interval
Valued Parameters
Studies on Reliability Optimization Problems by Genetic Algorithm 18
Chapter 6 Stochastic Optimization of System Reliability for Series System with
Interval Component Reliabilities
Chapter 7 Reliability Optimization with Interval Parametric Values in the
Stochastic Domain
Chapter 8 Multi-objective Reliability Optimization in Interval Environment
Chapter 9 General Conclusion and Scope of Future Research
Chapter 2 deals with an overview of existing finite interval mathematics,
interval order relations and real coded genetic algorithm. In this chapter, we have
also proposed new definition of interval power of an interval and new order
relations of intervals irrespective of decision-makers’ value system.
The objective of Chapter 3 is to develop and solve the reliability redundancy
allocation problems of series-parallel, parallel-series and complex/complicated
systems considering the reliability of each component as interval valued number. For
optimization of system reliability and system cost separately under resource
constraints, the corresponding problems have been formulated as constrained
integer/mixed-integer programming problems with interval objectives with the help
of interval arithmetic and interval order relations. Then the problems have been
converted into unconstrained optimization problems by two different penalty
function techniques. To solve these problems, two different real coded genetic
algorithms (GAs) for interval valued fitness function with tournament selection,
whole arithmetical crossover and boundary mutation for floating point variables,
intermediate crossover and uniform mutation for integer variables and elitism with
size one have been developed. To illustrate the models, some numerical examples
have been solved and the results have been compared. As a special case, taking lower
Introduction 19
and upper bounds of the interval valued reliabilities of component as same, the
corresponding problems have been solved and the results have been compared with
the results available in the existing literature. Finally, to study the stability of the
proposed GAs with respect to the different GA parameters (like, population size,
crossover and mutation rates), sensitivity analyses have been shown graphically.
Chapter 4 deals with redundancy allocation problem in interval environment
that maximizes the overall system reliability subject to the given resource
constraints and also minimizes the overall system cost subject to the given resources
including an additional constraint on system reliability where reliability of each
component is interval valued and the cost coefficients as well as the amount of
resources are imprecise and interval valued. These types of problems have been
formulated as an interval valued non-linear integer programming problem (IVNLIP).
In this work, we have formulated two types of redundancy, viz. component level
redundancy known as low-level redundancy and the system level redundancy known
as high-level redundancy. These problems have been transformed as an
unconstrained problem using penalty function technique and solved using genetic
algorithm. Finally, two numerical examples (one for low-level redundancy and
another for high-level redundancy) have been presented and solved and the
computational results have been compared.
Chapter 5 presents the reliability optimization problem of a
complex/complicated system where time-to-failure of each component follows the
Weibull distribution with imprecise parameters. In the earlier work, either both the
scale and shape parameters of Weibull distribution or the scale parameter as a
random variable with known distribution are considered as fixed. However, in
Studies on Reliability Optimization Problems by Genetic Algorithm 20
reality, both the parameters may vary due to some factors and it is sensible to treat
them as imprecise numbers. Here, this imprecise number is represented by an
interval number. In this chapter, we have formulated the reliability optimization
problem with Weibull distributed time-to-failure for each component. The
corresponding problem has been formulated as an unconstrained mixed-integer
programming problem with interval coefficients using penalty function technique
and solved by genetic algorithm. Finally, a numerical example has been solved for
different types of scale and shape parameters of Weibull distribution.
Chapter 6 deals with chance constraints based reliability stochastic
optimization problem in the series system. This problem can be formulated as a non-
linear integer programming problem of maximizing the overall system reliability
under chance constraints due to resources. In this chapter, we have formulated the
reliability optimization problem as a chance constraints based reliability stochastic
optimization problem with interval valued reliabilities of components. Then, the
chance constraints of the problem are converted to the equivalent deterministic
form. The transformed problem has been formulated as an unconstrained integer
programming problem with interval coefficients by Big-M penalty technique. Then to
solve this problem, we have developed a real coded genetic algorithm (GA) for
integer variables with tournament selection, intermediate crossover and one
neighborhood mutation. To illustrate the model, two numerical examples have been
considered and solved by our developed GA. Finally to study the stability of our
developed GA with respect to the different GA parameters, sensitivity analyses have
been carried out and presented graphically.
Introduction 21
In Chapter 7, the problem of reliability optimization has been examined in
the stochastic domain with respect of resource constraints and the concept of
interval valued parameters has been integrated with the stochastic setup so as to
increase the applicability of the resultant solutions. In particular, the five-link bridge
network system has been studied under a normal setup with Genetic Algorithm as
the optimization tool. Deterministic solution and non-interval valued parametric
solutions follow from the general optimization results.
In Chapter 8, we have solved the constrained multi-objective reliability
optimization problem of a system with interval valued reliability of each component
by maximizing the system reliability and minimizing the system cost under several
constraints. For this purpose, five different multi-objective optimization problems
have been formulated in interval environment with the help of interval mathematics
and our newly proposed order relations of interval valued numbers. Then these
optimization problems have been solved by advanced genetic algorithm and the
concept of Pareto optimality. Finally, for the purpose of illustration and comparison,
a numerical example has been solved.
In Chapter 9, general concluding remarks drawn from our studies and further
scope of research have been presented.
CHAPTER 2
Solution Methodologies
• Interval Approach in Reliability Optimization
• Finite Interval Mathematics
• Interval Order Relations
• Metric Space
• Genetic Algorithm
• GA-Based Constrained Handling Technique
Solution methodologies 23
2.1 Interval Approach in Reliability Optimization
During the last few decades, several researchers formulated and solved either single
objective or multi-objective reliability optimization problems as integer non-linear
programming problems (INLPP) and/or mixed-integer non-linear programming
problems (MINLPP) with single or several resource constraints. To solve those
problems, they proposed different techniques. In this connection, one may refer to
the works of Tillman, Hwang and Kuo (1977a, 1977b and 1980), Nakagawa,
Nakashima and Hattori (1978), Misra and Sharma (1991), Chern (1992), Ohtagaki,
Nakagawa, Iwasaki, and Narihisa (1995), Kuo, Prasad, Tillman and Hwang (2001),
Sun and Li (2002), Gen and Yun (2006), Ha and Kuo (2006b), Coelho (2009a, 2009b)
among others. In their works, the design parameters involved in reliability
optimization have been taken to be precise values. This means that every probability
involved is perfectly determinable. In this case, it is usually assumed that there exist
some complete probabilistic information about the system and the component
behavior. However, in real-life situations, there are not sufficient statistical data
available in most of the cases where the system is either new or it exists only as a
project. It is not always possible to observe the stability from the statistical point of
view. This means that only some partial information about the system components
are known. In these cases, parameters are said to be imprecise. To tackle the problem
with such imprecise parameters, generally stochastic, fuzzy and fuzzy-stochastic
approaches are applied and the corresponding problems are converted into
deterministic problems for solving them. In the stochastic approach, the parameters
are assumed to be random variables with known probability distributions. In the
fuzzy approach, the parameters, constraints and goals are considered as fuzzy sets
with known membership functions or fuzzy numbers. On the other hand, in the
Studies on Reliability Optimization Problems by Genetic Algorithm 24
fuzzy-stochastic approach, some parameters are viewed as fuzzy sets/fuzzy numbers
and others as random variables. However, it is a formidable task for a decision-
maker to specify the appropriate membership function for fuzzy approach and
probability distribution for stochastic approach and both for fuzzy-stochastic
approach. So, to avoid these difficulties for handling the imprecise parameters by
different approaches, one may use an interval number to represent an imprecise
number, as this representation is the most significant representation among others.
2.2 Mathematical Backgrounds
2.2.1 Finite Interval Mathematics
An interval number A is a closed interval connected subset of � denoted
by [ , ]L RA a a= and is defined by [ , ] { : , }L R L RA a a x a x a x= = ≤ ≤ ∈� , where La and
Ra are the left and right limits respectively and � is the set of all real numbers.
An interval A can also be expressed in terms of centre and radius
as ,c wA a a= ={ : , }c w c wx a a x a a x− ≤ ≤ + ∈� , where ca and wa be the centre and
radius of the interval A respectively i.e., ( ) 2c L Ra a a= + and ( ) 2w R La a a= − .
Actually, every real number can be treated as an interval, such as for all x∈� , x can
be written as an interval [ , ]x x having zero width.
Here, we shall give the concise definitions of arithmetical operations like
addition, subtraction, multiplication, and division of interval numbers.
Let [ , ] ,L R c wA a a a a= = ⟨ ⟩ and [ , ] ,L R c wB b b b b= = ⟨ ⟩ be two intervals.
Then the addition of two intervals A and B is given by
[ , ]L L R RA B a b a b+ = + + , ,c c w wA B a b a b+ = ⟨ + + ⟩
The subtraction of two intervals A and B is given by
Solution methodologies 25
[ , ]L R R LA B a b a b− = − −
or, , , , , ,c w c w c w c w c c w wA B A B a a b b a a b b a b a b− = + = ⟨ ⟩ − ⟨ ⟩ = ⟨ ⟩ + ⟨− ⟩ = ⟨ − + ⟩ .
The multiplication of an interval A by a real number λ is defined by
[ , ] for 0,
[ , ] for 0,
L R
R L
a aA
a a
λ λ λλ
λ λ λ
≥=
<
or, , ,c w c wA a a a aλ λ λ λ= ⟨ ⟩ = ⟨ ⟩ .
The mid-point of an interval A is denoted by ( )m A and is defined by
( )2
L Ra am A
+=
The product of two different intervals A and B is defined by
[min( , , , ),max( , , , )]L L L R R L R R L L L R R L R RA B a b a b a b a b a b a b a b a b× = .
The division of the interval B by the interval A is defined as
1 1 1[ , ] [ , ], provided 0 [ , ]L R L R
R L
BB b b a a
A A a a= × = × ∉ .
The above definitions are given in the books written by Moore (1979) and Hansen
and Walster (2004).
2.2.1.1 Integral Power of an Interval
Let [ , ]L RA a a= be an interval and n be any non-negative integer number then
according to Hansen and Walster (2004) the definition of integer power of an
interval is as follows:
[1,1] if 0
[ , ] if 0 or if isodd
[ , ] if 0 and iseven
[0,max( , )] if 0 and 0 is even
n n
L R Ln
n n
R L R
n n
L R L R
n
a a a nA
a a a n
a a a a n
=
≥=
≤
≤ ≤ >
Studies on Reliability Optimization Problems by Genetic Algorithm 26
2.2.1.2 n-th Root of an Interval
According to Karmakar, Mahato and Bhunia (2009), the -thn root of an
interval [ , ]L RA a a= is defined as
1 1[ , ] if 0or if is odd
( ) [ , ] [ , ] [0, ] if 0, 0 and iseven
if 0 and is even
n nL R L
n n n nL R L R R L R
R
a a a n
A a a a a a a a n
a nϕ
≥
= = = ≤ ≥ <
where φ is the empty interval.
2.2.1.3 Rational Power of an Interval
Again applying the definitions of power and different roots of an interval, the rational
power of an interval [ , ]L RA a a= is defined as follows:
1
( ) ( )
p
pq qA A= or equivalently, ( ) exp log
p
q pA A
q
=
, provided 0La > .
2.2.1.4 Complex Interval
Let [ , ]L RA a a= and [ , ]L RB b b= . A complex interval z is identified with the interval
vector [ , ] [ , ]L R L Rz A iB a a i b b= + = + . The basic arithmetical operations of complex
intervals like, addition, subtraction, division and multiplication be the same as the
real interval arithmetic [Kearfott (1996)].
2.2.1.5 Functions of Finite Interval
Here we shall define different types of functions of interval arguments.
For a monotonically increasing function ( )f x in the interval [ , ]L RA a a= , where x ∈�
( ) ([ , ]) [ ( ), ( )]L R L Rf A f a a f a f a= = .
Solution methodologies 27
Similarly, if ( )f x is a monotonically decreasing function in the interval [ , ]L RA a a= ,
where x ∈� , then ( ) ([ , ]) [ ( ), ( )]L R R Lf A f a a f a f a= = .
Using the above definitions the exponential and logarithmic function can be
expressed for interval arguments as they are strictly monotonic function.
(i) exp( ) exp([ , ]) [exp( ),exp( )]L R L RA a a a a= =
(ii) log( ) log([ , ]) [log( ), log( )]L R L RA a a a a= = , provided 0La > .
For non-monotonic functions, functions of interval arguments are very much
complicated.
For bounded periodic functions
(iii) sin([ , ]) [ , ]L R L Ra a b b=
where 1 if : 2 [ , ]
2
min{sin( ),sin( )} otherwise
L R
L
L R
k k a ab
a a
ππ
− ∃ ∈ − ∈
=
�
and 1 if : 2 [ , ]
2
max{sin( ),sin( )} otherwise
L R
R
L R
k k a ab
a a
ππ
∃ ∈ + ∈
=
�
(iv) cos([ , ]) [ , ]L R L Ra a b b=
where 1 if : (2 1) [ , ]
min{cos( ),cos( )} otherwise
L R
L
L R
k k a ab
a a
π− ∃ ∈ + ∈=
�
and 1 if : 2 [ , ]
max{cos( ),cos( )} otherwise
L R
R
L R
k k a ab
a a
π∃ ∈ ∈=
�
2.2.1.6 Integration of an Interval Function
According to Moore (1979), the integration of an interval function is defined by
( ) [ ( ) , ( ) ]
b b b
L R
a a a
f y dy f y dy f y dy=∫ ∫ ∫ for any y ∈� .
Studies on Reliability Optimization Problems by Genetic Algorithm 28
Here ( ) [ ( ), ( )]L Rf y f y f y= and both ( )Lf y and ( )Rf y are continuous real valued
functions.
2.2.1.7 Interval Power of an Interval
Till now, none has developed the interval power of an interval number. In this thesis,
in Chapter 5 and Chapter 8, the interval power of an interval numbers occurs in the
formulation of the optimization problems. For this purpose, we have introduced the
formula of interval power of an interval as follows:
Let [ , ]L RA a a= and [ , ]L RB b b= be two intervals, then
(i) [ , ]( ) [ , ] L Rb bB
L RA a a=
[exp( ),exp( )] if 0
a complex interval if 0
L
L
u v a
a
≥=
<
where min{ log , log , log , log }L L L R R L R Ru b a b a b a b a=
and max{ log , log , log , log }L L L R R L R Rv b a b a b a b a= .
(ii) [ , ][ , ] L Rb b
L Ra a− −
[ , ][ , ] cos[(2 1) , (2 1) ] sin[(2 1) , (2 1) ]
if , 0, 0,1, 2,3,
L Rb b
R L L R L R
L R
a a k b k b i k b k b
a a k
π π π π= + + + + + +
≥ = ⋅⋅⋅
2.2.1.8 Mean, Variance and Standard Deviation of Interval Numbers
The mean, variance and standard deviation of n interval numbers are defined as
follows:
Let [ , ]i iL iRx x x= , 1, 2,...,i n= be the -thi observation which is an interval
number. Then mean ( )x , variance [Var( )]x and standard deviation ( )xσ of these
numbers are given by
Solution methodologies 29
1 1
1 1[ , ] ,
n n
L R iL iR
i i
x x x x xn n= =
= =
∑ ∑ ,
2
2 2
1 1 1
1 1 1Var( ) [ , ] [ , ]
n n n
L R iL iR iR iL
i i i
x x x x xn n n
σ σ= = =
= = − −
∑ ∑ ∑ ,
and [ , ] Var(x)x L Rσ σ σ= = .
2.2.2 Interval Order Relations
For obtaining the optimum solution in solving the optimization problems with
interval valued objectives we need to define the order relations of interval numbers.
Let [ , ]L RA a a= and [ , ]L RB b b= be two unequal intervals. Then these two intervals
may be one of the following types:
Type-1: Two intervals are disjoint [see Figure 2.1].
Type-2: Two intervals are partially overlapping [see Figure 2.2].
Type-3: One of the intervals contains the other one [see Figure 2.3].
Figure 2.1: Type-1 intervals
Figure 2.2: Type-2 intervals
RbLb
B
RaLa
A
RaLa
A
RbLb
B
A
RbLb
B
RaLa
A
Rb
B
Lb RaLa
Studies on Reliability Optimization Problems by Genetic Algorithm 30
Figure 2.3: Type-3 intervals
It is to be noted that both the intervals [ , ]L RA a a= and [ , ]L RB b b= will be equal in
case of fully overlapping intervals, i.e., A B= iff L La b= and
R Ra b= .
In this area, very few researchers defined the order relations of interval
valued numbers. Moore (1979) first proposed two order relations of interval
numbers.
For any two intervals [ , ]L RA a a= and [ ],L RB b b= , Moore (1979) first gave the
two order relations which are as follows:
(i) transitive order relation ‘ < ’ as iff R LA B a b< <
(ii) transitive order relation set inclusion ‘ ⊆ ’ as iff andL L R RA B b a a b⊆ ≤ ≤ .
However, these two order relations cannot order two partially or fully overlapping
intervals. Then Ishibuchi and Tanaka (1990) defined the order relations for
minimization problems of two closed intervals [ , ] ,L R c wA a a a a= =
and [ , ] ,L R c wB b b b b= = which are as follows:
(i) iff andLR L L R RA B a b a b≤ ≤ ≤
iff andLR LRA B A B A B< ≤ ≠
(ii) iff andcw c c w wA B a b a b≤ ≤ ≤
iff andcw cwA B A B A B< ≤ ≠
RaLb
B
RbLa
A
RaLa
B
RbLb
A
Solution methodologies 31
These order relations are reflexive, transitive and anti-symmetric i.e., these are
partial order. From these definitions it is clear that, for minimization problem, a
decision-maker will prefer the interval A . Generalizing the definitions of Ishibuchi
and Tanaka (1990), Chanas and Kuchta (1996) proposed the concept of 0 1t t − cut of
an interval for the ranking of interval numbers.
Let [ , ]L RA a a= be any interval and 0t and 1t be any two fixed numbers such
that 0 10 1t t≤ ≤ ≤ then the 0 1t t − cut of the interval is given by
0 1[ , ] 0 1/ [ ( ), ( )]t t L R L L R LA a t a a a t a a= + − + − .
According to Chanas and Kuchta (1996), the order relations for the intervals
[ , ]L RA a a= and [ , ]L RB b b= are as follows:
(i) 0 1 0 1 0 1[ , ] [ , ] [ , ]/ iff / /LR t t t t LR t tA B A B≤ ≤
0 1 0 1 0 1[ , ] [ , ] [ , ]/ iff / /LR t t t t LR t tA B A B< <
(ii) 0 1 0 1 0 1[ , ] [ , ] [ , ]/ iff / /cw t t t t cw t tA B A B≤ ≤
0 1 0 1 0 1[ , ] [ , ] [ , ]/ iff / /cw t t t t cw t tA B A B< <
After Chanas and Kuchta (1996), Kundu (1997) first noticed that the interval ranking
methods discussed earlier could not find the measure ‘How much larger the interval
A is, if it is greater than the other?’ He attempted to answer this question by
introducing the ‘fuzzy leftness relation’. For the intervals A and B , let a A∈ and
b B∈ are uniformly and independently distributed in A and B respectively.
Then A is left to B if Left( , ) max{0, ( ) ( )}A B P a b P a b= < − > 0> and A is right to B
if Right( , ) max{0, ( ) ( )} 0A B P a b P a b= > − < > , where ( )P a b< denotes the probability
that a b< . This is a probabilistic approach.
In the year 2000, two other approaches of ranking of two intervals were given
by Sengupta and Pal (2000). In the first approach, they defined order relations with
Studies on Reliability Optimization Problems by Genetic Algorithm 32
respect to the decision-makers’ point of view using the acceptability function
: [0, )I Iχ × → ∞ for the intervals A and B as ( , ) c c
w w
b aA B
b aχ
−=
+, where 0w wb a+ ≠ .
( , )A Bχ may be considered as a grade of acceptability of the ‘first interval to be
inferior to the second’. If ( , ) 0A Bχ = then for a minimization problem, the interval
A cannot be accepted as smaller. If 0< ( , ) 1A Bχ < , A can be accepted with the grade of
acceptability c c
w w
b a
b a
−
+. The interval A is accepted fully if ( , ) 1A Bχ = .
According to them, the acceptability index is only a value based ranking index
and it can be applied partially to select the best alternative from the pessimistic point
of view of the decision-maker. So, only the optimistic decision-maker can use it
completely. In another approach, Sengupta and Pal (2000) introduced the fuzzy
preference ordering for the ranking of a pair of interval numbers on the real line with
respect to a pessimistic decision-makers’ point of view. The fuzzy preference method
was described for maximizing the profit interval. However, this method is equally
applicable to minimize the cost/time intervals also. In this definition, they assumed
that two intervals A and B are profit intervals and the problem is to find the
maximum profit interval from among them. In this approach, they considered the
fuzzy set “Rejection of an interval A in comparison to the interval B ” or “Acceptance
of B in comparison to A ”.
The membership function of this fuzzy set is given by
1 if
( , ) max 0, if
0 otherwise
c c
c L wL w c c
c L w
b a
b a bB A a b b a
a a bµ
=
− −= + ≤ ≤
− −
This non-linear membership function lies in the interval [0, 1]. When the values of
Solution methodologies 33
this membership function lies within the interval [0.333, 0.666], this definitions fails
to find the order relations.
According to the optimistic and pessimistic decision-makers’ point of view,
Mahato and Bhunia (2006) proposed the revised definitions of order relations
between interval costs/times for minimization problems and interval profits for
maximization problems. Let the two intervals [ , ] ,L R c wA a a a a= = and
[ , ] ,L R c wB b b b b= = be the uncertain interval costs/time or profits.
Now, we explain their proposed definitions which depend on the decision-
makers’ risk taking attitude. In this case, they considered two types of decision-
making, viz. (i) Optimistic decision-making and (ii) Pessimistic decision-making.
2.2.2.1 Optimistic Decision-Making
In this decision-making, decision-maker expects the lowest value for minimization
problems and highest value for maximization problems ignoring the uncertainty.
Definition: For minimization problems, the order relation omin≤ between the
intervals [ , ]L RA a a= and [ ],L RB b b= is
omin L LA B a b≤ ⇔ ≤
and omin ominA B A B A B< ⇔ ≤ ∧ ≠ .
This implies that A is superior to (i.e., better than) B and A is accepted but B is not
inferior to A. This order relation is obviously not symmetric.
Definition: For maximization problems, the order relation omax≥ between the
intervals [ , ]L RA a a= and [ ],L RB b b= is
omax R RA B a b≥ ⇔ ≥
and omax omaxA B A B A B> ⇔ ≥ ∧ ≠ .
Studies on Reliability Optimization Problems by Genetic Algorithm 34
This implies that A is superior to B and the optimistic decision-maker accepts the
profit interval A. Here also, the order relation omax≥ is not symmetric.
2.2.2.2 Pessimistic Decision-Making
In this decision-making, the decision-maker expects the lowest/highest value with
less uncertainty for minimization/maximization problems according to the principle
“Less uncertainty is better than more uncertainty”.
Definition: For minimization problems, the order relation pmin< between the
intervals [ , ] ,L R c wA a a a a= = and [ , ] ,L R c wB b b b b= = is
(i) pmin c cA B a b< ⇔ < for Type-1 and Type-2 intervals and
(ii) ( ) ( )pmin c c w wA B a b a b< ⇔ ≤ ∧ < for Type-3 intervals
However, for Type-3 intervals with ( ) ( )c c w wa b a b< ∧ > , a pessimistic decision
cannot be taken. Here, the optimistic decision is considered.
Definition: For maximization problems the order relation pmax> between the
intervals [ , ] ,L R c wA a a a a= = and [ , ] ,L R c wB b b b b= = is
(i) pmax c cA B a b> ⇔ > , for Type -1 and Type-2 intervals and
(ii) pmax ( ) ( )c c w wA B a b a b> ⇔ ≥ ∧ < , for type-3 intervals.
Again, for Type-3 intervals with ( ) ( )c c w wa b a b> ∧ > , a pessimistic decision cannot be
taken. In this situation, the optimistic decision may be taken.
2.2.2.3 Proposed Definition of Interval Order Relations
In the definitions of Mahato and Bhunia (2006) of pessimistic decision-making of
Type-3 intervals, it is observed that sometimes optimistic decisions are to be taken.
To overcome this situation, we have proposed two new definitions of order relations
Solution methodologies 35
irrespective of optimistic as well as pessimistic decision-makers’ point of view for
maximization and minimization problems separately.
Definition: The order relation max> between the intervals [ , ] ,L R c wA a a a a= =
and [ , ] ,L R c wB b b b b= = , then for maximization problems
(i) max c cA B a b> ⇔ > for Type-1 and Type-2 intervals and
(ii) maxA B> ⇔ either c c w wa b a b≥ ∧ < or for Type-3 intervals.c c R Ra b a b≥ ∧ >
According to this definition, the interval A is accepted for maximization case. Clearly,
this order relation max> is reflexive and transitive but not symmetric.
Definition: The order relation min< between the
intervals [ , ] ,L R c wA a a a a= = and [ , ] ,L R c wB b b b b= = , then for minimization
problems
(i) min for Type-1and Type-2intervals andc cA B a b< ⇔ <
(ii) minA B< ⇔ either c c w wa b a b≤ ∧ < or for Type-3 intervals.c c L La b a b≤ ∧ <
According to this definition, the interval A is accepted for minimization case. Clearly,
the order relation min< is reflexive and transitive but not symmetric.
It is to be noted that the definitions given by Mahato and Bhunia (2006) and our
proposed definitions give the same results. So these two definitions are equivalent.
2.2.3 Metric Space
The term metric is derived from the term metor (measure). The concept of a metric
space is essentially due to a French Mathematician M. Frechet, though the definition
presently in use is that given by the German Mathematician F. Housdorff in 1914.
Definition: Let X be an arbitrary non-empty set. A mapping :d X X× → � is said to
be metric on X if it satisfies the following properties:
Studies on Reliability Optimization Problems by Genetic Algorithm 36
(i) ( , ) 0, ,d x y x y X≥ ∀ ∈
(ii) ( , ) 0 , ,d x y x y x y X= ⇔ = ∀ ∈
(iii) ( , ) ( , ), ,d x y d y x x y X= ∀ ∈
(iv) ( , ) ( , ) ( , ), , ,d x y d x z d z y x y z X≤ + ∀ ∈
If X is a non-empty set and d is a metric on X , then ( , )X d is called a metric space. A
metric d is also called a distance function. The real number ( , )d x y is called the
distance between x and y .
Here, we shall give some metric spaces which are used in this thesis.
(i) Let nX = � and suppose 1 2( , ,..., )nx ξ ξ ξ= and 1 2( , ,..., )ny η η η= be any two points in
n� . Define the mapping , :pd d X X∞ × →� as follows:
{ }1
1 1 2 2( , ) , 1pp p p
p n nd x y pξ η ξ η ξ η= − + − + ⋅⋅⋅+ − ≤ < ∞
{ } { }1 1 2 21
( , ) max , , , maxn n i ii n
d x y ξ η ξ η ξ η ξ η∞≤ ≤
= − − ⋅⋅⋅ − = −
Then, pd for each 1 p≤ < ∞ and d∞ are metrics on the same underlying set nX = � .
(iii)Let ,1pX l p= ≤ < ∞ , be the set of all sequences { }ix ξ= of real scalars such
that1
p
i
i
ξ∞
=
< ∞∑ . Define the mapping :d X X× → � by
1
1
( , )n p
p
i i
i
d x y ξ η=
= − ∑ ,
where { }ix ξ= and { }iy η= are in pl .
2.3 Solution Methodologies
2.3.1 Genetic Algorithm
Genetic algorithm (GA) is a well-known stochastic search iterative method based on
the evolutionary theory of Charles Darwin “survival of the fittest” and natural
Solution methodologies 37
genetics. GA has successfully been applied to optimization problems in different
fields, like engineering design, reliability optimization, optimal control,
transportation and assignment problems, job scheduling, inventory control and other
real-life decision-making problems. The most fundamental idea of Genetic Algorithm
is to imitate the natural evolution process artificially in which populations undergo
continuous changes through genetic operators, like crossover, mutation and
selection. The concept of GA was first introduced by Prof. John Holland of the
University of Michigan, Ann Arbor. He is considered to be the father of GA. His idea of
genetic algorithm was first used to solve optimization problem by De-Jang (1975).
Thereafter, a researcher has contributed much to the major development of this field.
Most of the initial research work in this field can be found in several International
Conference Proceedings. The detailed discussion of genetic algorithms, including
extensions along with related topics, can be found in the books on GA [Holland
(1975), Goldberg (1989), Davis (1991), Michalewicz (1996), Mitchell (1996), Gen
and Cheng (1997) and Vose (1999)].
Genetic algorithm can easily be implemented with the help of computer
programming. In particular, it is very useful for solving complicated optimization
problems which cannot be solved easily by direct or gradient based mathematical
techniques. It is very effective to handle large-scale, real-life, discrete and continuous
optimization problems without making unrealistic assumptions and approximations.
Keeping the imitation of natural evolution as the foundation, genetic algorithm can
be designed appropriately and modified to exploit special features of the problem to
solve. This algorithm starts with an initial population of probable solutions, called
individuals, to a given problem where each individual is represented using different
form of coding as a chromosome. These chromosomes are evaluated for their fitness.
Studies on Reliability Optimization Problems by Genetic Algorithm 38
Based on their fitness, chromosomes in the population are to be selected for
reproduction and selected individuals are manipulated by two known genetic
operations, like crossover and mutation. The crossover operation is applied to create
offspring from a pair of selected chromosomes. The mutation operation is used for a
slight modification/change to reproduce offspring. The repeated applications of
genetic operators to the relatively fit chromosomes result in an increase in the
average fitness of the population over generation and identification of improved
solutions to the problem under investigation. This process is applied iteratively until
the termination criterion is satisfied.
2.3.1.1 GA Terminology
It is important to first understand the terminology that is used with respect to
genetic algorithm. Some of the commonly used terms are as follows:
Population : A collection of several alternate solutions to the given
problem
Population size : The population size determines the amount of
information stored by the GA.
Chromosome : Each individual in the population is called a
chromosome.
Genes : Often these individuals are coded as binary/real
strings and the individual character or symbol in the
string is named as genes.
Solution methodologies 39
Fitness function : It is an evolution function, which is used to determine
the fitness of each chromosome. The fitness function is
usually user defined and problem specific.
Solution space : The range of possible solutions is referred to as the
solution space and the cost and the fitness of each
point is referred to as the altitude in the landscape of
the problem.
Generation gap : It is the fraction of the individuals in the population
that are replaced from one generation to the next and
is equal to one for simple GA.
Termination criterion
: The termination criterion is a condition for which the
algorithm/process is going to stop. For this purpose
any one of the following three conditions is
considered as the termination criterion.
(i) The best individual does not improve over
specified generations.
(ii) The total improvement of the last certain number
of best solutions is less than a pre-assigned small
positive number.
(iii) The number of generations reaches a prescribed
finite number of generation (called maximum
number of generations).
Studies on Reliability Optimization Problems by Genetic Algorithm 40
The procedural algorithm of the working principle of GA is as follows:
2.3.1.2 Algorithm
Step-1 : Set population size (p_size), crossover probability (p_cross), mutation
probability (p_mute), maximum generation (max_gen) and bounds of
the variables.
Step-2 : 0t = [ t represents the number of current generation].
Step-3 : Initialize the chromosome of the population ( )P t [ ( )P t represents the
population at -t th generation].
Step-4 : Evaluate the fitness function of each chromosome of ( )P t considering
the objective function as the fitness function.
Step-5 : Find the best chromosome from the population ( )P t .
Step-6 : t is increased by unity.
Step-7 : If the termination criterion is satisfied go to Step-14, otherwise, go to
next step.
Step-8 : Select the population ( )P t from the population ( 1)P t − of earlier
generation by tournament selection process.
Step-9 : Alter the population ( )P t by crossover, mutation and elitism operators.
Step-10 : Evaluate the fitness function value of each chromosome of ( )P t .
Step-11 : Find the best chromosome from ( )P t .
Step-12 : Compare the best chromosome of ( )P t and ( 1)P t − and store better one.
Step-13 : Go to step-6.
Step-14 : Print the best chromosome (which is the solution of the optimization
problem).
Step-15 : End.
Solution methodologies 41
To implement the GA, the following basic components are to be considered:
(i) GA parameters (population size, maximum number of generation,
crossover rate and mutation rate)
(ii) Chromosome representation
(iii) Initialization of population
(iv) Evaluation of fitness function
(v) Selection process
(vi) Genetic operators (crossover, mutation and elitism)
2.3.1.3 GA Parameters
There are several GA parameters, viz. population size (p_size), maximum number of
generation (max_gen), crossover rate i.e.,the probability of crossover (p_cross) and
mutation rate i.e., the probability of mutation (p_mute). There is no hard and fast rule
for selecting the population size for GA, how large it should be. The population size is
problem dependent and will need to increase with the dimensions of the problem. If
the population size is very large, storing of data in intermediate steps of GA may arise
some difficulties at the time of execution. When the population size is very small,
some genetic operators do not work properly. However, population size is restricted
by both time complexity and space complexity. Regarding the maximum number of
generations, there is no clear indication for considering this value. It varies from
problem to problem and depends upon the number of genes (variables) of a
chromosome and prescribed as stopping/termination criteria to make sure that the
solution has converged. From natural genetics, it is obvious that the rate of crossover
is always greater than that of rate of mutation. Generally, the crossover rate varies
from 0.60 to 0.95 whereas the mutation rate varies from 0.05 to 0.20. Sometimes
Studies on Reliability Optimization Problems by Genetic Algorithm 42
mutation rate is considered as1 n where n is the number of genes (variables) of the
chromosome.
2.3.1.4 Representation of Chromosomes
To represent an appropriate chromosome is an important issue in the application of
GA for solving the optimization problem and users of GA face a hard situation how to
represent the appropriate chromosome (individual). There are different types of
representations, like, binary, real, octal, hexadecimal coding, available in the existing
literature. Among different representations, mainly binary coding and real coding
representations are very popular. In the initial implementation of GAs, the
chromosomes were represented by the strings of binary numbers. In this
representation, binary sub strings of each variable with the desired precision are
concatenated to represent an individual. As a result, the string length of an individual
will be large. In this case, genetic algorithm would perform poorly. To overcome
these difficulties, real numbers are used to represent the chromosomes in GAs. In
this case, a chromosome is coded in the form of vector/matrix of integer/ floating
point or combination of the both numbers and every component of that
chromosome represents a decision variable of the problem. In this representation,
each chromosome is encoded as a vector of integer/ floating or combination of the
both numbers, with the same component as the vector of decision variables of the
problem. This type of representation is accurate and more efficient as it is closed to
the real design space and moreover, the string length of each chromosome is the
number of design variables. In this representation, for a given problem with
n decision variables, a n-component vector 1 2( , , , )nx x x x= ⋅⋅⋅ is used as a
chromosome to represent a solution to the problem. A chromosome denoted as kv
( 1, 2,..., _ )k p size= is an ordered list of n genes as 1 2{ , ,..., ,..., }k k k ki knv v v v v= .
Solution methodologies 43
2.3.1.5 Initialization
After representation of chromosome, the next step is to initialize the chromosome
that will take part in the artificial genetics. To initialize the population, first of all we
have to find the independent variables and their bounds for the given problem. Then
the initialization process produces population size number of chromosomes in which
every component for each chromosome is randomly generated within the bounds of
the corresponding decision variable. There are several procedures for selecting a
random number of either integer type or float point type. In our whole work, we have
used the following algorithm for selecting of an integer random number.
An integer random number between a and b can be generated as
either x a g= + or, x b g= −
where g is a random integer between 1 and a b− .
On the other hand, to generate a floating point random number, we have used
uniform distribution as follows:
Uniform Distribution
Using this distribution, a random number on an interval [ , ]a b can be generated
as ( )x a r b a= + − where r is another random number between 0 and 1.
The initialization procedure produces p_size chromosomes where p_size denotes the
population size, by the following algorithm.
Step-1: Generate a random number between 0 and 1.
Step-2: Assign ( )i i i ix l r u l= + − ), where il and iu are lower and upper bounds of ix .
Step-3: Repeat the steps 1 and 2 for n times and produce a vector ( 1 2, ,..., nx x x ).
Step-4: Repeat the steps 1, 2 and 3 for p_size times and produce p_size initial feasible
solutions.
Studies on Reliability Optimization Problems by Genetic Algorithm 44
Step-5: Stop.
It may be noted that a random value can be selected alternatively from the discrete
set of values {0, 0.1, 0.2, 0.3,…, 0.9}. Generally, this process is used to find the low
precision solutions of a problem. However, if a solution has a high-precision value, a
random number r is selected from either discrete set of values or Step-1 of earlier
algorithm. Again, for getting the boundary points in a chromosome by uniform
distribution, random value selection from discrete values is very efficient.
2.3.1.6 Evaluation/ Fitness Value Computation
Evaluation/fitness function plays an important role in GA. This role is same for
natural evolution process in the biological and physical environments. After
initialization of chromosomes of potential solutions, we need to see how relatively
good they are. Therefore, we have to calculate the fitness value for each chromosome.
In our work, the value of objective function of the reduced unconstrained
optimization problems corresponding to the chromosome is considered as the fitness
value of that chromosome.
2.3.1.7 Selection
The selection operator which is the first operator in artificial genetics plays an
interesting role in GA. This selection process is based on the Darwin’s principle on
natural evolution “survival of the fittest”. The primary objective of this process is to
select the above average individuals/chromosomes from the population according to
the fitness value of each chromosome and eliminate the rest of the
individuals/chromosomes. There are several methods for implementing the selection
process.
Some of the well known selection operators are as follows:
Solution methodologies 45
(a) Ranking Selection
(b) Roulette wheel selection
(c) Tournament selection
(d) Stochastic Universal Sampling selection
(e) Steady state selection
In our whole work, we have solved different types of constrained optimization
problems. As a result, for solving those problems we have used only the tournament
selection with size two. In this selection, two individuals in the population are
selected based on their fitness. The following assumptions for this selection
procedure are to be considered:
(i) when both the individuals/chromosomes are feasible then the one with better
fitness value is selected.
(ii) when one individual/chromosome is feasible and another is infeasible then the
feasible one is selected.
(iii) when both the individuals/chromosomes are infeasible with unequal constraint
violation, then the chromosome with less constraint violation is selected.
(iv) when both the individuals/chromosomes are infeasible with equal constraint
violation, then any one individual/chromosome is selected.
2.3.1.8 Crossover
The exploration and exploitation of the solution space can be made possible by
exchanging genetic information of the current chromosomes. After the selection
process, other genetic operators, like crossover and mutation are applied to the
resulting chromosomes those which have survived. Crossover is an operator that
creates new individuals/chromosomes (offspring) by combining the features of both
parent solutions. It operates on two or more parent solutions at a time and produces
Studies on Reliability Optimization Problems by Genetic Algorithm 46
offspring for next generation. In this operation, expected [ ]_ * _p cross p size number
of chromosomes will take part (* and [ ] denote the product and the integral value
respectively).
In this thesis, the following crossover operators have been used:
(a) Whole arithmetical crossover ( for floating point variables)
(b) Intermediate crossover ( for integer variables)
The different steps of general crossover operation are as follows:
Step-1: Find the integral value of [ ]_ * _p cross p size and store it in N .
Step-2: Select two parent chromosomes kv and iv randomly from the population.
Step-3: Compute the components kjv and ijv ( 1,2,..., )j n= of two offspring from the
parent chromosomes kv and
iv .
Step-4: Repeat Step-2 and Step-3 for 2
Ntimes.
In case of whole arithmetic crossover, components kjv and ( 1, 2,..., )ijv j n= of two
offspring will be created by
(1 )kj kj ijv cv c v= + −
(1 )ij kj ijv c v cv= − +
where c is a random number between 0 and 1.
In case of intermediate crossover, components kjv and ( 1,2,..., )ijv j n= of two
offspring will be created by
kj kjv v g= − and ij ijv v g= + if kj ijv v>
or, kj kjv v g= + and ij ijv v g= − ,
where g is a random integer number between 0 and kj ijv v− , 1, 2,...,j n= .
Solution methodologies 47
The aim of mutation operator is to introduce the random variations into the
population and is used to prevent the search process from converging to the local
optima. This operator helps to regain the information lost in earlier generations and
is responsible for fine tuning capabilities of the system and is applied to a single
individual only. Usually, its rate is very low; because otherwise it would defeat the
order building being generated through the selection and crossover operations.
The different steps of mutations operations are as follows:
Step-1: Find the integral value of [ ]_ * _p mute p size and store it in N .
Step-2: Select a chromosome iv randomly from the population.
Step-3: Select a particular gene ( 1, 2,..., )ikv k n= on chromosome iv for mutation and
domain of ikv is [ , ]ik ikl u .
Step-4: Create new gene ikv′ ( 1, 2,..., )k n= corresponding to the selected gene ikv by
mutation process.
Step-5: Repeat Step-2 to Step-4 for N times.
In this thesis, the following mutation operators have been used:
(a) Uniform mutation ( for integer variables)
(b) One-neighborhood mutation ( for integer variables)
(c) Boundary mutation ( floating point variables)
Among these mutation operators, one-neighborhood mutation is new. For the first
time, we have proposed this operator. Basically, this operator is used in GA to mutate
the gene corresponding to integer variables. Other two operators are well known
mutation operators available in the existing literature.
After mutation process, let '( 1, 2,..., )ikv k n= be the mutated gene corresponding to the
selected gene ikv .
Studies on Reliability Optimization Problems by Genetic Algorithm 48
In case of one-neighborhood mutation,
'
1 if
1 if
1 if a random digit is 0
1 if a random digit is 1
ik ik ik
ik ik ik
ik
ik
ik
v v l
v v uv
v
v
+ =
− ==
+ −
In case of boundary mutation,
'if a random digit is 0
if a random digit is 1
ik
ik
ik
lv
u
=
In case of uniform mutation
( ) if a random digit is 0
( ) if a random digit is 1
ik ik ik
ik
ik ik ik
v u vv
v v l
+ ∆ −′ =
− ∆ −
where ( )y∆ returns a value in the range [0, ]y .
2.3.1.9 Elitism
In any generation of GA, sometimes there arises a situation when the best
chromosome may get lost from the population when a new population is created by
crossover and mutation operations. To overcome this situation the worst
individual/chromosome of the current generation is replaced by the best
individual/chromosome of previous generation. Instead of single chromosome one
or more chromosomes may take part in this operation. This process is named as
elitism.
2.3.1.10 Advantages and Disadvantages of Genetic Algorithm
The main advantages of Genetic algorithm are as follows:
It
(i) can easily be implemented.
Solution methodologies 49
(ii) optimizes the objective function with continuous, discrete, permutation and
mixed variables.
(iii) does not normally require derivative information.
(iv) deals with large number of decision/design variables.
(v) produces global optimum, does not stuck to local optimum.
(vi) is problem, as well as variable independent.
(vii) gives alternative solution (near optimum).
(viii) works not only with the analytical functions, but also works with experimental
data.
(ix) works with a set of solutions instead of single solution in each iteration/
generation.
(x) is able to solve problem with non-convex solution space, where classical
methods usually fail.
(xi) It also performs well with problems where the objective function is interval
valued and highly non-linear, discontinuous or has many local optima.
All these advantages make the GAs superior from the classical optimization
techniques in real world applications, mainly for very complicated engineering/
scientific problems. Though there are several advantages of GA in solving different
types of optimization problems, there are few disadvantages also. These are as
follows:
(i) It is often seen that a genetic algorithm caught in a local optimum, and that all or
most of the population concentrates on a small part of the search space located
around the local optimum. This is usually known as premature convergence.
(ii) The most difficult and time consuming issue in the successful application of GAs
is to determine the approximate settings of GA parameters. The parameters of
Studies on Reliability Optimization Problems by Genetic Algorithm 50
genetic algorithm need to be tuned for efficiency. However, Michalewicz (1996)
mentioned that the determination of proper values of these genetic parameters
is an art and the quality of this tuning gently depends on the user-experience as
well as their knowledge of the problem.
(iii) Computational efficiency can be lower than that of other methods.
2.3.2 GA-Based Constrained Handling Technique
In applications of GA for solving optimization problem with interval objective, there
arises an important question for handling the constraints relating to the problem.
During the past, several methods have been proposed to handle the constraints in
evolutionary algorithms for solving the same problem with fixed objective. These
methods can be classified into several types, viz. penalty function techniques,
methods that preserve the feasibility of solutions, methods that clearly distinguish
between feasible and infeasible solutions. Among these methods, penalty function
technique is very well known and widely applicable. In this technique, the amount of
constraint violations is added /subtracted to the objective function in different ways.
When the objective function is increased/ decreased with a penalty term multiplied
by so called penalty coefficient there arises a difficulty to select the initial value and
upgrading strategy for the penalty coefficient. To overcome this difficulty, Deb
(2000) proposed a GA-based Parameter Free Penalty (PFP) technique. In this
technique, the worst fitness value of GA for feasible solutions is considered as the
fitness value of infeasible solution without multiplying the penalty coefficient i.e., the
fitness function values of infeasible solutions are independent of the objective
function value for the same solution.
Let us consider a constrained optimization problem with interval objective
Solution methodologies 51
Maximize [ ( ), ( )]L Rf x f x (2.1)
subject to the constraints
( ) , 1, 2,...,j jg x b j m≤ = ,
Therefore, according to the PFP technique, the converted problem of problem (2.1) is
as follows:
1 1
ˆ ˆMaximize[ ( ), ( )]
[ ( ), ( )] [ max(0, ( ) ), max(0, ( ) )] ( )
L R
m m
L R j j j j
j j
f x f x
f x f x g x b g x b xθ= =
= − − − +∑ ∑ (2.2)
where [0,0] if
( )[ ( ), ( )] min[ ( ), ( )] ifL R L R
x Sx
f x f x f x f x x Sθ
∈=
− + ∉
and { : ( ) , 1, 2,..., }j jS x g x b j m= ≤ =
Here the parameter min[ ( ), ( )]L Rf x f x is the value of interval valued objective
function of the worst feasible solution in the population. Alternatively, the problem
may be solved with another fitness function by penalizing a large positive number
(say M which can be written in the interval form as[ , ]M M ) [Gupta, Bhunia and Roy
(2009)]. We denote this penalty as Big-M penalty and its form is as follows:
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )L R L Rf x f x f x f x xθ= + (2.3)
where [0,0] if
( )[ ( ), ( )] [ , ] ifL R
x Sx
f x f x M M x Sθ
∈=
− + − − ∉
and { : ( ) , 1,2,..., and }j jS x g x c j m l x u= ≤ = ≤ ≤
The above problems (2.2) and (2.3) are non-linear unconstrained optimization
problem with interval valued objective.
In our work, we have used Big-M penalty technique with the value of M as 99999.
CHAPTER 3
Reliability Redundancy Allocation
Problems in Interval
Environment
• Introduction
• Constrained Redundancy Optimization Problem for Different Systems
• Solution Procedures
• Numerical Examples
• Sensitivity Analysis
• Concluding Remarks
Reliability Redundancy Allocation Problems in Interval Environment 53
3.1 Introduction
While advanced technologies have raised the world to an unprecedented level of
productivity, our modern society has become more delicate and vulnerable due to
the increasing dependence on modern technological systems that often require
complicated operations and highly sophisticated management. From any respect, the
system reliability is a crucial measure to be considered in systems operation and risk
management. At the time of designing a highly reliable system, there arises an
important question how to obtain a balance between reliability and other resources
e.g., cost, volume and weight. In the last few decades, several researchers considered
reliability optimization problems, like redundancy allocation and cost minimization
problems as integer non-linear programming problems (INLPP) and/or mixed-
integer non-linear programming problems (MINLPP) with single or several resource
constraints [Tillman, Hwang and Kuo (1977a, 1977b), Nakagawa, Nakashima and
Hattori (1978), Misra and Sharma (1991), Chern (1992), Ohtagaki, Nakagawa,
Iwasaki and Narihisa (1995), Kuo, Prasad, Tillman and Hwang (2001), Sun and Li
(2002), Gen and Yun (2006), Ha and Kuo (2006b) and Coelho (2009a, 2009b)]. To
solve those problems, different techniques have been proposed by the several
researchers. In their works, the reliability of each component is a known fixed
positive number which lies between zero and one.
However, in real-life situations, the reliability of an individual component
may not be fixed. It may vary due to several reasons. There is no technology by which
different components can be produced with exactly identical reliabilities. So, the
reliability of each component is sensible and it may be treated as a positive imprecise
number instead of a fixed real number. Studies of the system reliability where the
component reliabilities are imprecise and/or interval valued have already been
Studies on Reliability Optimization Problems by Genetic Algorithm 54
initiated by some authors [Coolen and Newby (1994), Utkin and Gurov (1999, 2001)
and Gupta, Bhunia and Roy (2009)]. To tackle the problem with such imprecise
numbers, generally stochastic, fuzzy and fuzzy-stochastic approaches are applied and
the corresponding problems are converted into deterministic problems for solving
them. In the stochastic approach, the parameters are assumed to be random
variables with known probability distributions. In the fuzzy approach, the
parameters, constraints and goals are considered as fuzzy sets with known
membership functions or fuzzy numbers. On the other hand, in the fuzzy-stochastic
approach, some parameters are viewed as fuzzy sets/fuzzy numbers and others as
random variables. However, it is a formidable task for a decision-maker to specify the
appropriate membership function for fuzzy approach and probability distribution for
stochastic approach and both for fuzzy -stochastic approach. So, to avoid these
difficulties for handling the imprecise numbers by different approaches, one may use
an interval number to represent an imprecise number, as this representation is the
most significant representation among others. Due to this representation, the system
reliability would be interval valued. In this chapter, we have considered GA-based
approaches for solving reliability optimization problems with the interval objective.
As the objective function of the reliability optimization is interval valued, to solve this
type of problem by GA method, order relations of interval numbers are essential for
selection operation as well as for finding the best chromosome in each generation.
In this chapter, we have considered the problem of constrained redundancy
allocation in the series system, the hierarchical series-parallel system, the
complex/complicated or non-parallel-series system and the network reliability
system with interval valued reliability components (redundancy allocation and
network cost minimization). These problems can be formulated as non-linear
Reliability Redundancy Allocation Problems in Interval Environment 55
constrained integer programming problems and/or mixed-integer programming
problems with interval coefficients for maximizing the overall system reliability and
system cost under some resource/budget constraints. During the last few years,
several techniques were proposed for solving the constrained optimization problem
with fixed coefficients with the help of GAs [Goldberg (1989), Michalawich (1996)
and Gen and Cheng (2000)]. Among these methods, penalty function techniques are
very popular in solving the same by GAs [Deb (2000), Miettinen, Makela and
Toivanen (2003) and Aggarwal and Gupta (2005)]. This method transforms the
constrained optimization problem to an unconstrained optimization problem by
penalizing the objective function corresponding to the infeasible solution. Hence, to
solve the constrained optimization problem, the problem is converted into
unconstrained one by two different types of penalty function techniques and the
resulting objective function would be interval valued. So, to solve this problem we
have developed two different GAs for integer variables with the same GA operators
like tournament selection, intermediate crossover for integer variables and whole
arithmetical crossover for floating point variables, uniform mutation for integer
variables and boundary mutation for floating point variables and elitism of size one
but different fitness function depending on different penalty approaches. These
methods have been illustrated with some numerical examples and to test the
performance of these methods, the results have also been compared. As a special
case considering the lower and upper bounds of interval valued reliabilities of
components as same, the resulting problem becomes identical with the existing
problem available in the literature. Finally, to study the stability of the proposed GAs
with respect to the different GA parameters (like, population size, crossover and
mutation rates), sensitivity analyses have been carried out graphically.
Studies on Reliability Optimization Problems by Genetic Algorithm 56
3.2 Constrained Redundancy Optimization Problem for Different Systems
3.2.1 Assumptions
(i) The component reliabilities are imprecise and interval valued.
(ii) The chance of failure of any component is independent of those of other
components.
(iii) Each redundancy is active redundancy without repair.
3.2.2 Series System
It is well known that a series system (see Figure 3.1) with n independent
components must be operating only if all the components are functioning. In order to
improve the overall reliability of the system; one can use more reliable components.
However, the expenditure and more often the technological limits may prohibit an
adoption of this strategy. An alternative technique is to add some redundant
components as shown in Figure 3.2. The goal of this problem is to determine an
optimal redundancy allocation so as to maximize the overall system reliability under
limited resource constraints. These constraints may arise out of the size, cost and
quantities of the resources. Mathematically, the constrained redundancy
optimization problem for such a system for interval valued values of reliability can
be formulated as follows:
Problem 1 Maximize 1
[ , ] [{1 (1 ) },{1 (1 ) }]j j
qx x
SL SR jL jR
j
R R r r
=
= − − − −∏
subject to ( ) , 1, 2,...,i ig x b i m≤ =
and j j jl x u≤ ≤ , jx being integer for 1, 2,...,j q=
where [ , ]j jL jRr r r=
Reliability Redundancy Allocation Problems in Interval Environment 57
This is a constrained non-linear integer programming problem with interval valued
objective.
Figure 3.1: Series system
Figure 3.2: Parallel-series system
3.2.3 Hierarchical Series-Parallel System
A system is called a hierarchical series-parallel system (HSP) if the system can be
viewed as a set of subsystems arranged in a series-parallel; each subsystem has a
similar configuration; subsystems of each subsystem have a similar configuration
and so on. For example let us consider a HSP system ( 10)n = shown in the Figure 3.3.
Figure 3.3: Hierarchical series-parallel system
This system has a non-linear and non-separable structure and consists of nested
parallel and series systems. The system reliability of HSP is given by
1 2
3
5 6
4 7
8
9
10
1
2
x1
1
2
x2
1
2
xq
1 2 3 n
Studies on Reliability Optimization Problems by Genetic Algorithm 58
3 1 2 4 5 6 7 8 9 10{1 1 [1 (1 )] (1 )}(1 )SR Q R R R R R Q Q Q R= − ⟨ − − − ⟩ − − .
The corresponding constrained redundancy optimization problem for this
system for interval valued reliability can be formulated as follows:
Problem 2
3 3 1 1 2 2 4 4 5 5Maximize[ , ] {1 1 (1 [ , ](1 [ , ][ , ]))[ , ] (1 [ , ]SL SR L R L R L R L R L RR R Q Q R R R R R R R R= − ⟨ − − − ⟩ −
6 6 7 7 8 8 9 9 10 10[ , ])}(1 [ , ][ , ][ , ])[ , ]L R L R L R L R L RR R Q Q Q Q Q Q R R−
subject to ( ) , 1, 2,...,i ig x b i m≤ =
and j j jl x u≤ ≤ , jx being integer for 1,2,...,j q=
This is an INLPP with interval valued objective.
3.2.4 Complex/Complicated System
When a system can be reduced to series and parallel configurations, there exist
combinations of components which are connected neither in a series nor in parallel.
Such systems are called complex/complicated or non-parallel series systems. This
system is also called the bridge system. For example, let us consider a bridge
system ( 5)n = shown in Figure 3.4. This system consists of five subsystems and three
non-linear and non-separable constraints. The overall system reliability SR is given
by the expression as follows:
5 1 3 2 4 5 1 2 3 4(1 )(1 ) [1 (1 )(1 )]SR R Q Q Q Q Q R R R R= − − + − − − ,
where ( )i i iR R x= and 1i iQ R= − for all 1,2,3,4,5i = .
The corresponding constrained redundancy optimization problem for such a
complex/complicated system for interval valued reliability can be formulated as
follows:
Reliability Redundancy Allocation Problems in Interval Environment 59
Problem 3
5 5 1 1 3 3 2 2 4 4Maximize[ , ] [ , ](1 [ , ][ , ])(1 [ , ][ , ])SL SR L R L R L R L R L RR R R R Q Q Q Q Q Q Q Q= − −
5 5 1 1 2 2 3 3 4 4[ , ]{1 (1 [ , ][ , ])(1 [ , ][ , ])}L R L R L R L R L RQ Q R R R R R R R R+ − − −
subject to ( ) , 1,2,...,i ig x b i m≤ =
and j j jl x u≤ ≤ , jx being integer for 1,2,...,j q=
Figure 3.4: Complex/Complicated system
Figure 3.5: 2- out- of -3 system
3.2.5 K-out-of-N System
A k out of n− − − system is a n -component system which functions when at least k of
its n components function. This redundant system is sometimes used in place of a
1 2
2 3
1 3
1 2
5
3 4
Studies on Reliability Optimization Problems by Genetic Algorithm 60
pure parallel system. It is also referred to as :k out of n G− − − system. An n -
component series system is a :n out of n G− − − system whereas a parallel system
with n -components is a 1 :out of n G− − − system. When all of the components are
independent and identical, the reliability of k out of n− − − system can be written
as (1 )n
j n j
S
j k
nR r r
j
−
=
= −
∑ , where r is the component reliability.
The corresponding constrained redundancy optimization problem for this
system for interval valued reliability can be formulated as follows:
Problem 4 Maximize 1
[ , ] [ , ] ([1,1] [ , ])j
j
xqx lj l
SL SR L R L R
l kj
xR R r r r r
l
−
==
= −
∑∏
subject to ( ) , 1, 2,...,i ig x b i m≤ =
and j j jl x u≤ ≤ , jx being integer for 1,2,...,j q=
3.2.6 Reliability Network System
Let us consider a network with n subsystems. The goal of the redundancy allocation
problem is to determine the number of redundant components in each of q parallel
subsystems and reliability levels of ( )n q− general subsystems so as to maximize the
overall system reliability subject to the given resource constraints and also to
minimize the overall system cost subject to the given constraint on system reliability.
The corresponding problems are mixed-integer non-linear programming problems
as follows:
Problem 5 Maximize 1 1 2 2 1( , ) ( ( ), ( ),..., ( ), ,..., )S q q q nR x R f R x R x R x R R+=
subject to ( , ) , 1,...., ,i iC x R b i m≤ =
and 1 , being integer for 1,..., ,j j j jl x u x j q≤ ≤ ≤ =
Reliability Redundancy Allocation Problems in Interval Environment 61
10 1, 1,..., ,j j jL R U j n q+< ≤ ≤ < = −
where ( ) [ ( ), ( )] 1 (1 [ , ]) ix
i i iL i iR i iL iRR x R x R x r r= = − −
Problem 6 Minimize ( , )wC x R
subject to *( , )SR x R R≥
where 1 1 2 2 1( , ) ( ( ), ( ),..., ( ), ,..., )S q q q nR x R f R x R x R x R R+=
and ( ) [ ( ), ( )] 1 (1 [ , ]) ix
i i iL i iR i iL iRR x R x R x r r= = − −
j j jl x u≤ ≤ , jx being integer, 1, 2,...,j q= ,
10 1, 1,...,j j jL R U j n q+< ≤ ≤ < = −
3.3 Solution Procedures
In this section we shall discuss the solution procedure for all the problems
mentioned in earlier section i.e., problems (1)-(6). These problems are non-linear
optimization problems (all integer/mixed-integer) with interval valued objective
function. Using PFP and Big-M penalty techniques and real coded genetic algorithm
with advanced operators these problems are converted into unconstrained
optimization problems.
The converted problems of (1)-(4) are as follows:
Maximize ˆ ˆ[ ( ), ( )]SL SRR x R x
1 1
[ ( ), ( )] [ max(0, ( ) ), max(0, ( ) )] ( )m m
SL SR i i i i
i i
R x R x g x b g x b xθ= =
= − − − +∑ ∑ (3.1)
where [0,0] if
( )[ ( ), ( )] min[ ( ), ( )] ifSL SR SL SR
x Sx
R x R x R x R x x Sθ
∈=
− + ∉
and { : ( ) , 1, 2,..., and }i iS x g x b i m l x u= ≤ = ≤ ≤
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= and 1 2( , ,..., )qx x x x= .
Studies on Reliability Optimization Problems by Genetic Algorithm 62
Here the parameter min[ ( ), ( )]SL SRR x R x is the value of interval valued objective
function of the worst feasible solution in the population. Alternatively, the problem
may be solved with another fitness function by penalizing a large positive number
(say M which can be written in the interval form as[ , ]M M ) [Gupta, Bhunia and Roy
(2009)]. We denote this penalty as Big-M penalty and its form is as follows:
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )SL SR SL SRR x R x R x R x xθ= + (3.2)
where [0,0] if
( )[ ( ), ( )] [ , ] ifSL SR
x Sx
R x R x M M x Sθ
∈=
− + − − ∉
and { : ( ) , 1, 2,..., and }i iS x g x b i m l x u= ≤ = ≤ ≤
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= and 1 2( , ,..., )qx x x x= .
The above problems (3.1) and (3.2) are non-linear unconstrained integer
programming problems with interval coefficients.
Also, according to the PFP technique, the converted problem of Problem 5 is as
follows:
Maximize [ ]1
ˆ ( , ) ( , ) max(0, ( , ) ),max(0, ( , ) ) ( , )m
S S i i i i
i
R x R R x R C x R b C x R b x Rθ=
= − − − +∑
(3.3)
where [ ]
[0,0] if ( , )( , )
( , ) min ( , ), ( , ) if ( , )S SL SR
x R Sx R
R x R R x R R x R x R Sθ
∈=
− + ∉
and { }( , ) : ( , ) , 1,..., and ,i iS x R C x R b i m l x u L R U= ≤ = ≤ ≤ ≤ ≤
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= , 1 2( , ,..., )qx x x x= , 1 2( , ,..., )q q nL L L L+ += ,
1 2( , ,..., )q q nU U U U+ += and 1 2( , ,..., )q q nR R R R+ += .
Here [ ]min ( , ), ( , )SL SRR x R R x R is the value of the interval valued objective function of
the worst feasible solution in the population.
Reliability Redundancy Allocation Problems in Interval Environment 63
Alternatively, the problem may also be solved with another fitness function by
penalizing a large positive number. The converted form is as follows:
Maximize ˆ ( , ) ( , ) ( , )S SR x R R x R x Rθ= + (3.4)
where [ ]
[0,0] if ( , )( , )
( , ) , if ( , )S
x R Sx R
R x R M M x R Sθ
∈=
− + − − ∉
and { }( , ) : ( , ) , 1,..., and ,i iS x R C x R b i m l x u L R U= ≤ = ≤ ≤ ≤ ≤
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= , 1 2( , ,..., )qx x x x= , 1 2( , ,..., )q q nL L L L+ += ,
1 2( , ,..., )q q nU U U U+ += and 1 2( , ,..., )q q nR R R R+ += .
For Problem 6, the fitness function is of the following form:
Minimize *
1
ˆ ( , ) ( , ) max(0, ( , ) ) ( , )m
w w SL
j
C x R C x R R x R R x Rθ=
= + − + + ∑ (3.5)
where { }
[0,0] if ( , )( , )
( , ) max ( , ) if ( , )w w
x R Sx R
C x R C x R x R Sθ
∈=
− + ∉
and { }*( , ) : ( , ) 0, 1,2,..., and ,SLS x R R x R R i m l x u L R U= − + ≤ = ≤ ≤ ≤ ≤
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= , 1 2( , ,..., )qx x x x= , 1 2( , ,..., )q q nL L L L+ += ,
1 2( , ,..., )q q nU U U U+ += and 1 2( , ,..., )q q nR R R R+ += .
Here { }max ( , )wC x R is the value of the interval valued objective function of the worst
feasible solution in the population.
Alternatively, the problem may also be solved with another fitness function by
penalizing a large positive number. The converted problem is as follows:
Minimize ˆ ( , ) ( , ) ( , )w wC x R C x R x Rθ= + (3.6)
where [0,0] if ( , )
( , )( , ) if ( , )w
x R Sx R
C x R M x R Sθ
∈=
− + ∉
Studies on Reliability Optimization Problems by Genetic Algorithm 64
and { }*( , ) : ( , ) 0, 1,2..., and ,SLS x R R x R R i m l x u L R U= − + ≤ = ≤ ≤ ≤ ≤
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= , 1 2( , ,..., )qx x x x= , 1 2( , ,..., )q q nL L L L+ += ,
1 2( , ,..., )q q nU U U U+ += and 1 2( , ,..., )q q nR R R R+ += .
The above problems (3.3)-(3.6) are non-linear unconstrained mixed-integer
programming problems with interval coefficients.
3.4 Numerical Examples
The proposed GAs (viz. PFP-GA and Big-M-GA) have been illustrated for solving
earlier mentioned optimization problems with interval valued reliabilities of
components. In these GAs we have applied intermediate crossover and uniform
mutation corresponding to the integer variables and whole arithmetic crossover and
boundary mutation corresponding to the floating point variables. For illustration
purpose we have solved twelve numerical examples. It may be noted that for solving
the said problem with fixed valued reliabilities of components, the reliability of each
component is taken as interval with the same lower and upper bounds of interval.
For each example, 20 independent runs have been performed by both the GAs, of
which the following measurements have been collected to compare the
performances of PFP-GA and Big-M-GA.
(i) Best found system reliability
(ii) Average generations
(iii) Average CPU times
The proposed Genetic Algorithms are coded in C programming language and
run in LINUX environment. The computational work has been done on the PC which
has Intel Core-2 duo processor with 2.5 GHz. In this computation, different
population size has been taken for different problems. However, the crossover and
Reliability Redundancy Allocation Problems in Interval Environment 65
mutation rates are taken as 0.95 and 0.15 respectively. The computational results of
different examples have been shown in Tables 3.3, 3.4, 3.5 and 3.6.
Example 1 (relating to Problem 1)
Maximize 5
1
[ , ] [{1 (1 ) },{1 (1 ) }]j jx x
SL SR jL jR
j
R R r r
=
= − − − −∏
subject to
52
1
0j j
j
p x P
=
− ≤∑
5
1
[ exp( 4)] 0j j j
j
c x x C
=
+ − ≤∑
5
1
exp( 4) 0j j j
j
w x x W
=
− ≤∑
The values of different parameters along with the interval valued reliabilities of
Example 1 are given in Table 3.1.
Example 2 (relating to Problem 2)
3 3 1 1 2 2 4 4Maximize [ , ] {1 1 (1 [ , ](1 [ , ][ , ]))[ , ]SL SR L R L R L R L RR R Q Q R R R R R R= − ⟨ − − − ⟩
5 5 6 6 7 7 8 8 9 9 10 10(1 [ , ][ , ])}(1 [ , ][ , ][ , ])[ , ]L R L R L R L R L R L RR R R R Q Q Q Q Q Q R R× − −
subject to
21 1 2 2 3 3 4 4 5 5 5 6 7 6 8exp( 2) exp( 2) [ exp( 4)]c x x c x c x c x x c x x c x+ + + + + +
37 9 10exp( 2) 120 0c x x+ − ≤
2 31 1 2 2 3 4 3 5 6 4 7 8 5 9 9exp( 2) exp( 4) [ exp( 2)]w x x w x x w x x w x x w x x+ + + + +
6 2 10exp( 4) 130 0w x x+ − ≤
The values of different parameters along with the interval valued reliabilities of
Example 2 are given in Table 3.2.
Studies on Reliability Optimization Problems by Genetic Algorithm 66
Table 3.1: Parameters used in Example 1
j jr jp
P jc C jw W
1 [0.76, 0.83] 1
110
7
175
7
200
2 [0.82, 0.87] 2 7 8
3 [0.88, 0.93] 3 5 8
4 [0.61, 0.67] 4 9 6
5 [0.70, 0.80] 2 4 9
Table 3.2: Parameters used in Example 2
j jr jc jw jl ju
1 [0.80, 0.84] 8 16 1 4
2 [0.87, 0.90] 4 6 1 5
3 [0.89, 0.93] 2 7 1 6
4 [0.84, 0.86] 2 12 1 7
5 [0.88, 0.90] 1 7 1 5
6 [0.90, 0.95] 6 1 1 5
7 [0.80, 0.85] 2 9 1 3
8 [0.91, 0.95] 8 - 1 3
9 [0.80, 0.83] - - 1 4
10 [0.88, 0.92] - - 1 6
Example 3 (relating to Problem 3)
5 5 1 1 3 3 2 2 4 4Maximize[ , ] [ , ](1 [ , ][ , ])(1 [ , ][ , ])SL SR L R L R L R L R L RR R R R Q Q Q Q Q Q Q Q= − −
5 5 1 1 2 2 3 3 4 4[ , ]{1 (1 [ , ][ , ])(1 [ , ][ , ])}L R L R L R L R L RQ Q R R R R R R R R+ − − −
subject to
21 2 3 4 510exp( 2) 20 3 8 200 0x x x x x+ + + − ≤
3 21 2 3 4 4 510exp( 2) 4exp( ) 2 6[ exp( 4)] 7exp( 4) 310 0x x x x x x+ + + + + − ≤
Reliability Redundancy Allocation Problems in Interval Environment 67
2 2 32 2 3 3 1 4 512[ exp( )] 5 exp( 4) 3 2 520 0x x x x x x x+ + + + − ≤
1 2 3 4 5(1,1,1,1,1) ( , , , , ) (6,3,5,6,6)x x x x x≤ ≤
where
1 1( ) {[0.78,0.82],[0.83,0.88],[0.89,0.91],[0.915,0.935],[0.94,0.96],[0.965,0.985]} R x =
22 2( ) 1 (1 [0.73,0.77])
xR x = − −
3
3
13 1
3 3
2
1( ) ([0.87,0.89]) ([0.11,0.13])
xx kk
k
xR x
k
++ −
=
+ =
∑
44 4( ) 1 (1 [0.68,0.72])
xR x = − −
55 5( ) 1 (1 [0.83,0.86])
xR x = − −
Example 4 (relating to problem 3)
5 5 1 1 3 3 2 2 4 4Maximize[ , ] [ , ](1 [ , ][ , ])(1 [ , ][ , ])SL SR L R L R L R L R L RR R R R Q Q Q Q Q Q Q Q= − −
5 5 1 1 2 2 3 3 4 4[ , ]{1 (1 [ , ][ , ])(1 [ , ][ , ])}L R L R L R L R L RQ Q R R R R R R R R+ − − −
subject to
21 2 3 4 510exp( 2) 20 3 8 200 0x x x x x+ + + − ≤
3 21 2 3 4 4 510exp( 2) 4exp( ) 2 6[ exp( 4)] 7exp( 4) 310 0x x x x x x+ + + + + − ≤
2 2 32 2 3 3 1 4 512[ exp( )] 5 exp( 4) 3 2 520 0x x x x x x x+ + + + − ≤
1 2 3 4 5(1,1,1,1,1) ( , , , , ) (6,3,5,6,6)x x x x x≤ ≤
where
1 1( ) {[.8,.8],[.85,.85],[.9,.9],[.925,.925],[.95,.95],[.975,.975]}R x =
22 2( ) 1 (1 [0.75,0.75])
xR x = − −
3
3
13 1
3 3
2
1( ) ([0.88,0.88]) ([0.12,0.12])
xx kk
k
xR x
k
++ −
=
+ =
∑
44 4( ) 1 (1 [0.7,0.7])
xR x = − −
Studies on Reliability Optimization Problems by Genetic Algorithm 68
55 5( ) 1 (1 [0.85,0.85])
xR x = − −
Example 5 (relating to the Problem 5)
1 1 2 2 2 2 3 3 4 4Maximize[ ( , ), ( , )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R R x R R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RR R Q Q Q Q R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R Q Q R R+
subject to
1 1 2 2 3 2 4
5
0.01( ) 2.2 1.5 2exp 28
1C x x x x x x x
R
= + + + ≤
−
2 1 2 3 4
5
0.01( ) 0.1 2 5exp 25
1C x x x x x
R
= + + + + ≤
−
2 33 1 2 3 4
5
0.01( ) ( 2) 1.5 0.6exp 21
1C x x x x x
R
= + − + + + ≤
−
where 51 6, and are integers, 1,2,3,4, 0.50 0.99ix i R≤ ≤ = ≤ ≤
and ( ) 1 (1 ) , 1,2,3, 4, 1 , 1,...,5ix
i i i i i iR R x r i Q R i= = − − = = − =
1 [0.69,0.72]r = , 2 [0.83,0.86]r = , 3 [0.73,0.76]r = , 4 [0.79,0.81]r = .
Example 6 (relating to the Problem 5)
1 1 2 2 2 2 3 3 4 4Maximize[ ( , ), ( , )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R R x R R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RR R Q Q Q Q R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R Q Q R R+
subject to
1 1 2 2 3 2 4
5
0.01( ) 2.2 1.5 2exp 28
1C x x x x x x x
R
= + + + ≤
−
Reliability Redundancy Allocation Problems in Interval Environment 69
2 1 2 3 4
5
0.01( ) 0.1 2 5exp 25
1C x x x x x
R
= + + + + ≤
−
2 33 1 2 3 4
5
0.01( ) ( 2) 1.5 0.6exp 21
1C x x x x x
R
= + − + + + ≤
−
where 51 6 and are integers, 1,2,3,4, 0.50 0.99ix i R≤ ≤ = ≤ ≤
and ( ) 1 (1 ) , 1,2,3,4, 1 , 1,...,5ix
i i i i i iR R x r i Q R i= = − − = = − =
1 [0.70,0.70]r = , 2 [0.85,0.85]r = , 3 [0.75,0.75]r = , 4 [0.8,0.8]r = .
Table 3.3: Computational results for Examples 1-4
Me
tho
d
Example Population
Size
x Best found system
reliability sR
Average
CPU time
(sec.)
PF
P-G
A
1 50 (3,2,2,3,3) [0.860808, 0.930985] 0.0001
2 100 (1,2,2,5,4,4,2,2,1,5) [0.999909, 0.999987] 0.0105
3 200 (5,1,2,4,4) [0.991225, 0.999872] 0.0200
4 100 (3,2,4,4,2) [0.999382, 0.999382] 0.0100
Big
-M-G
A
1 50 (3,2,2,3,3) [0.860808, 0.930985] 0.0001
2 100 (1,2,2,5,4,4,2,2,1,5) [0.999909, 0.999987] 0.0110
3 200 (5,1,2,4,4) [0.991225, 0.999872] 0.0200
4 100 (3,2,4,4,2) [0.999382, 0.999382] 0.0100
Example 7 (relating to the Problem 6)
Minimize 1 1 2 2 3 3( , ) 0.3 ( ) 0.5 ( ) 0.2 ( )wC x R C x C x C x= + +
subject to
( , ) [0.999,0.999]SR x R ≥
where 51 6 and are integers, 1,2,3,4, 0.50 0.99ix i R≤ ≤ = ≤ ≤
Studies on Reliability Optimization Problems by Genetic Algorithm 70
and ( , )SR x R , ( 1,2,3)iC i = are defined in Example 5
Example 8 (relating to the Problem 6)
Minimize 1 1 2 2 3 3( , ) 0.3 ( ) 0.5 ( ) 0.2 ( )wC x R C x C x C x= + +
subject to
( , ) [0.999,0.999]SR x R ≥
where 51 6and are integers, 1, 2,3,4, 0.50 0.99ix i R≤ ≤ = ≤ ≤
and ( , )SR x R , ( 1,2,3)iC i = are defined in Example 6
Example 9 (relating to problem 5)
Maximize [ ( , ), ( , )]SL SRR x R R x R
6 6 7 7 1 1 2 2 3 3[ , ][ , ] [ , ][ , ][ , ]([ , ]L R L R L R L RR R R R R R R R R R Q Q= +
6 6 7 7 1 1 4 4 7 7 6 6[ , ][ , ]) [ , ][ , ][ , ][ , ]L R L RR R Q Q R R R R R R Q Q+ +
2 2 2 2 3 3 3 3 5 5 6 6([ , ] [ , ][ , ]) [ , ][ , ][ , ]L R L R L R L R L RQ Q R R Q Q R R R R R R× + +
7 7 1 1 1 1 2 2 1 1 2 2[ , ]([ , ] [ , ][ , ]) [ , ][ , ]L R L R L R L R L RQ Q Q Q R R Q Q R R R R× + +
5 5 7 7 3 3 4 4 6 6 2 2 3 3[ , ][ , ][ , ][ , ][ , ] [ , ][ , ]L R L R L R L R L RR R R R Q Q Q Q Q Q R R R R× +
4 4 6 6 1 1 5 5 7 7 1 1 3 3[ , ][ , ][ , ][ , ][ , ] [ , ][ , ]L R L R L R L R L RR R R R Q Q Q Q Q Q R R R R× +
4 4 5 5 2 2 6 6 7 7[ , ][ , ][ , ][ , ][ , ]L R L R L RR R R R Q Q Q Q Q Q×
subject to
1 1 2 1 3 4 5
6 7
0.01 0.01( ) 0.5 log(1 ) 2 0.3exp 0.3exp 27
1 1C x x x x x x x
R R
= + + + + + + ≤
− −
2 1 2 3 1 2 3 4 5
6 7
0.02 0.01( ) ( 2 1.2 ) log(1 2 ) 0.4 0.2 exp 0.5exp 29
1 1C x x x x x x x x x
R R
= + + + + + + + + ≤
− −
where 1 4 and are integers, 1,...,5, 0.50 0.99, 6,7i ix i R i≤ ≤ = ≤ ≤ =
and ( ) 1 (1 ) , 1,...,5, 1 , 1,...,7ix
i i i i i iR R x r i Q R i= = − − = = − =
Reliability Redundancy Allocation Problems in Interval Environment 71
1 [0.69,0.71]r = , 2 [0.88,0.92]r = , 3 [0.78,0.81]r = , 4 [0.63,0.66]r = , 5 [0.68,0.71]r = .
Table 3.4: Computational results for Examples 5-6
Example 10 (relating to problem 5)
Maximize [ ( , ), ( , )]SL SRR x R R x R
6 6 7 7 1 1 2 2 3 3[ , ][ , ] [ , ][ , ][ , ]([ , ]L R L R L R L RR R R R R R R R R R Q Q= +
6 6 7 7 1 1 4 4 7 7 6 6[ , ][ , ]) [ , ][ , ][ , ][ , ]L R L RR R Q Q R R R R R R Q Q+ +
2 2 2 2 3 3 3 3 5 5 6 6([ , ] [ , ][ , ]) [ , ][ , ][ , ]L R L R L R L R L RQ Q R R Q Q R R R R R R× + +
7 7 1 1 1 1 2 2 1 1 2 2[ , ]([ , ] [ , ][ , ]) [ , ][ , ]L R L R L R L R L RQ Q Q Q R R Q Q R R R R× + +
5 5 7 7 3 3 4 4 6 6 2 2 3 3[ , ][ , ][ , ][ , ][ , ] [ , ][ , ]L R L R L R L R L RR R R R Q Q Q Q Q Q R R R R× +
4 4 6 6 1 1 5 5 7 7 1 1 3 3[ , ][ , ][ , ][ , ][ , ] [ , ][ , ]L R L R L R L R L RR R R R Q Q Q Q Q Q R R R R× +
4 4 5 5 2 2 6 6 7 7[ , ][ , ][ , ][ , ][ , ]L R L R L RR R R R Q Q Q Q Q Q×
subject to
1 1 2 1 3 4 5
6 7
0.01 0.01( ) 0.5 log(1 ) 2 0.3exp 0.3exp 27
1 1C x x x x x x x
R R
= + + + + + + ≤
− −
Method Example Population
Size
(x, R) Best found system
reliability SR
Average
CPU
seconds
PF
P-G
A 5 150 (2,3,1,2,0.9900) [0.958412, 0.997223] 0.2705
6 150 (2,1,6,5,0.9396) [0.999927, 0.999927] 0.2655
Big
-M-G
A 5 150 (2,3,1,2,0.9900) [0.958412, 0.997223] 0.2700
6 150 (2,1,6,5,0.9396) [0.999927, 0.999927] 0.2590
Studies on Reliability Optimization Problems by Genetic Algorithm 72
2 1 2 3 1 2 3 4 5
6 7
0.02 0.01( ) ( 2 1.2 ) log(1 2 ) 0.4 0.2 exp 0.5exp 29
1 1C x x x x x x x x x
R R
= + + + + + + + + ≤
− −
where 51 4 and are integers, 1,...,5, 0.50 0.99, 6,7ix i R i≤ ≤ = ≤ ≤ =
are ( ) 1 (1 ) , 1,...,5, 1 , 1,...,7ix
i i i i i iR R x r i Q R i= = − − = = − =
1 [0.70,0.70]r = , 2 [0.90,0.90]r = , 3 [0.80,0.80]r = , 4 [0.65,0.65]r = , 5 [0.70,0.70]r = .
Table 3.5: Computational results for Examples 7-8
Method Example Population
size ( , )x R Best found
system cost
wC
Best found system
reliability SR
Average
CPU
seconds
PF
P-G
A
7 150 (6,4,2,1,0.8601) 33.03866 [0.997290, 0.999885] 0.3675
8 150 (2,1,4,4,0.5) 17.97505 [0.999081, 0.999081] 0.3525
Big
-M-G
A
7 150 (6,4,2,1,0.8601) 33.03866 [0.997290, 0.999885] 0.3010
8 150 (2,1,4,4,0.5) 17.97505 [0.999081, 0.999081] 0.2815
Example 11(relating to problem 6)
Minimize 1 1 2 2( , ) 0.4 ( ) 0.6 ( )wC x R C x C x= +
subject to
( , ) [0.999,0.999]SR x R ≥
where1 4 and are integers, 1,...,5, 0.50 0.99, 6,7i ix i R i≤ ≤ = ≤ ≤ =
and ( , )SR x R , ( 1,2)iC i = are defined in Example 9.
Example 12 (relating to problem 6)
Minimize 1 1 2 2( , ) 0.4 ( ) 0.6 ( )wC x R C x C x= +
subject to
( , ) [0.999,0.999]SR x R ≥
Reliability Redundancy Allocation Problems in Interval Environment 73
where1 4 and are integers, 1,...,5, 0.50 0.99, 6,7i ix i R i≤ ≤ = ≤ ≤ =
and ( , )SR x R , ( 1,2)iC i = are defined in Example 10.
Table 3.6: Computational results for Examples 9-10
Method Example Population
Size ( , )x R Best found system
reliability SR
Average
CPU
seconds
PF
P-G
A
9 200 (4,2,2,3,2,0.984528,0.99000) [0.998951,0.999921] 0.6075
10 200 (4,1,3,4,3,0.984589,0.989918) [0.999745,0.999745] 0.5280
Big
-M-G
A
9 200 (4,2,2,3,2,0.984528,0.99000) [0.998951,0.999921] 0.6070
10 200 (4,1,3,4,3,0.984589,0.989918) [0.999745,0.999745] 0.5340
Table 3.7: Computational results for Examples 11-12
Method Example Population
size ( , )x R Best
found
system
cost wC
Best found
system reliability
SR
Average
CPU
seconds
PF
P-G
A
11 150 (2,1,4,4,0.5) 17.97505 [0.999081,0.999081] 0.3525
12 200 (3,1,2,2,2,0.9870,0.9900) 17.31398 [0.999001,0.999001] 0.6265
Big
-M-G
A
11 150 (2,1,4,4,0.5) 17.97505 [0.999081,0.999081] 0.2815
12 200 (3,1,2,2,2,0.9870,0.9900) 17.31398 [0.999001,0.999001] 0.7080
3.5 Sensitivity Analysis
To study the performance of our proposed GAs like PFP-GA and Big-M-GA based on
two different types of penalty techniques, sensitivity analyses (for Example 1) have
been carried out graphically on the centre of the interval valued system reliability
with respect to GA parameters like, population size, crossover and mutation rates
separately keeping the other parameters at their original values. These are shown in
Studies on Reliability Optimization Problems by Genetic Algorithm 74
Figures 3.6, 3.7 and 3.8. From Figure 3.6, it is evident that in case of PFP-GA, a smaller
population size gives the better result.
However, both the GAs are stable when population size exceeds the number
30. From Figure 3.7, it is observed that the system reliability is stable if we consider
the crossover rate between the interval (0.65, 0.95) in case of PFP-GA. In both GAs, it
is stable when crossover rate is greater than 0.8. In Figure 3.8, sensitivity analyses
have been done with respect to the mutation rate. In both GAs, the values of system
reliability are the same.
Figure 3.6: P_size vs. centre of the objective function for Example 1
Figure 3.7: P_cross vs. centre of the objective function for Example 1
0
0.25
0.5
0.75
1
5 10 15 20 25 30 35 40 45 50 55
Population size
Cen
tre o
f th
e inte
rval valu
ed
ob
jecti
ve f
un
cti
on
BIG-M-GA
PFP-GA
0.884
0.888
0.892
0.896
0.9
0.6 0.8 1
Crossover rate
Centr
e o
f th
e inte
rval valu
ed o
bje
ctive
function Big-M-GA
PFP-GA
Reliability Redundancy Allocation Problems in Interval Environment 75
Figure 3.8: P_mute vs. centre of the objective function for Example 1
3.6 Concluding Remarks
In this chapter, the problems of redundancy allocation problems of series system,
hierarchical series-parallel system, complex/complicated system and reliability
network system with some resource constraints have been solved. In those systems,
reliability of each component has been considered as an imprecise number and this
imprecise number has been represented by an interval number which is more
appropriate representation among other representations like, random variable
representation with known probability distribution, fuzzy set with known fuzzy
membership function or fuzzy number. For handling the resource constraints, the
corresponding problem has been converted into unconstrained optimization
problem with the help of two different parameter free penalty techniques. Therefore,
the transformed problem is of unconstrained interval valued optimization problem
with integer and/or mixed-integer variables. To solve the transformed problems, two
different real coded GA-based on different fitness functions have been developed for
integer and mixed-integer variables with interval valued fitness function,
tournament selection, crossover (intermediate crossover for integer variables and
whole arithmetical crossover for floating point variables), mutation (uniform
0.5
0.75
1
0 0.05 0.1 0.15 0.2 0.25
Mutation rateC
entr
e o
f th
e inte
rval valu
ed o
bje
ctive
function
Big-M-GA
PFP-GA
Studies on Reliability Optimization Problems by Genetic Algorithm 76
mutation for integer variables and boundary mutation for floating point variables)
and elitism of size one. In the existing penalty function technique, tuning of penalty
parameter is a formidable task. However, here tuning of parameters is not required
as these are penalty parameter free techniques. From the performance of GAs, it is
observed that the GAs with both fitness functions due to different penalty techniques
take less CPU times with very small generations to solve the problems. It is clear
from the expression of the system reliability that the system reliability is a
monotonically increasing function with respect to the individual reliabilities of the
components. Therefore, there is one optimum setup irrespective of the choice of the
upper bound or lower bound of the component reliabilities. As a result, the optimum
setup obtained from the upper bound/lower bound will provide both the upper and
the lower bounds of the optimum system reliability.
CHAPTER 4
Reliability Optimization under High
and Low-level Redundancies for
Imprecise Parametric Values
• Introduction
• Assumptions
• Low-Level and High-level Redundancy
• Formulation of Reliability-Redundancy Optimization Problems
• Solution Procedure
• Numerical Examples
• Sensitivity Analysis
• Concluding Remarks
Studies on Reliability Optimization Problems by Genetic Algorithm 78
4.1 Introduction
Generally, the designing of a modern technological system design depends on the
selection of components and their configurations to satisfy the functional as well as
performance specifications. For a system with known cost, reliability, weight, volume
and other system parameters, the corresponding design problem becomes a
combinatorial optimization problem. The best known reliability design problem of
this type is known as the redundancy allocation problem. The basic objective of the
redundancy allocation problem is to determine the number of redundant
components that either maximizes the system reliability or minimize the system cost
under several resource constraints. Redundancy allocation problem is nothing but a
non-linear integer programming problem. Most of these problems cannot be solved
by direct/indirect or mixed search methods due to discrete search space. According
to Chern (1992), redundancy allocation problem with multiple constraints is quite
often hard to find feasible solutions. This redundancy allocation problem is NP-hard
and it has been well discussed in Tillman, Hwang and Kuo (1977a) and Kuo and
Prasad (2000). In this chapter, we have formulated two types of redundancy, viz.
component level redundancy known as low-level redundancy and the system level
redundancy known as high-level redundancy for a five-link bridge system where the
objective function as well as constraints functions are considered as interval valued.
To the best of our knowledge, studies of the system reliability with component
reliability as interval valued have already been initiated by Gupta, Bhunia and Roy
(2009). Also, a number of researchers has presented different situations and
solutions methodologies on redundancy allocation problem in different
environments [Park (1987), Mahapatra and Roy (2006) and Liu (2010)].
Reliability Optimization under High and Low-level Redundancies… 79
In this chapter, we have proposed GA-based approaches for solving IVNLP
type redundancy allocation problem. To find the optimal solution of this type of
problem by GA, order relations of interval numbers assume an important role for GA
operators. Here, we have used our proposed interval order relations discussed in
Chapter 2. Using these we have developed a real coded elitist GA with tournament
selection, intermediate crossover and one-neighborhood mutation for solving the
proposed problems. Finally, to illustrate the proposed models, for high-level as well
as low-level redundancy, two numerical examples have been presented.
4.2 Assumptions
(i) Reliability of each component is imprecise and interval valued.
(ii) Chances of failures of components are mutually statistically independent.
(iii) The system will not be damaged or failed due to failed components.
(iv) All redundancy is active and there is no provision for repair.
(v) The components as well as the system have two different states, viz. operating
state and failure state.
(vi) The cost coefficients as well as the amount of resources are imprecise and
interval valued.
4.3 Low-level and High-level Redundancy
Let us consider a n component system. Now, we can either provide redundant
components, which give a system design diagram as shown in Figure 4.1, or provide
a total redundant system as shown in Figure 4.2. The component level redundancy is
known as low-level redundancy and the system level redundancy is known as high-
level redundancy. Here 1 2min( , ,..., )nh x x x= .
Studies on Reliability Optimization Problems by Genetic Algorithm 80
4.4 Formulation of Reliability-Redundancy Optimization Problems
Let us consider a network with n subsystems. The goal of the redundancy allocation
problem is to determine the number of redundant components in each of n parallel
subsystems so as to maximize the overall system reliability subject to the given
resource constraints and also to minimize the overall system cost subject to the given
constraint on system reliability.
The general form of the redundancy allocation problem is as follows:
Problem 1 Maximize ( )SR x
subject to ( )i ig x b≤ 1,2, ,i m= ⋅⋅⋅
1 , integer, 1,...,j j j jl x u x j n≤ ≤ ≤ =
The goal of the Problem 1 is to determine the number of redundant components so
as to maximize the overall system reliability. This problem belongs to the category of
constrained integer non-linear programming problems (INLPP).
The general form of the cost minimization problem is as follows:
Problem 2 Minimize ( )SC x
subject to ( )S TR x R≥
This formulation is designed to achieve a minimum total system cost, subject to TR , a
target limit on the system reliability.
For low-level redundant system (see. Figure 4.1), the corresponding reliability-
redundancy optimization problems are as follows:
Problem 3 Maximize 1 1 2 2( ) ( ( ), ( ),..., ( ),..., ( ))S q q n nR x f R x R x R x R x=
subject to ( )i ig x b≤ 1, 2, ,i m= ⋅⋅⋅
1 , integer, 1,...,j j j jl x u x j n≤ ≤ ≤ =
Reliability Optimization under High and Low-level Redundancies… 81
where ( ) [ ( ), ( )] 1 (1 [ , ]) , 1,2, ,jx
j j jL j jR j jL jRR x R x R x r r j n= = − − = ⋅⋅⋅
and [ , ] (0,1)j jL jRr r r= ∈
Problem 4 Minimize ( )SC x
subject to 1 1 2 2( ) ( ( ), ( ),..., ( ),..., ( ))S q q n n TR x f R x R x R x R x R= ≥
where ( ) [ ( ), ( )] 1 (1 [ , ]) , 1,2, ,jx
j j jL j jR j jL jRR x R x R x r r j n= = − − = ⋅⋅⋅
For high-level redundant system (see Figure 4.2), the corresponding reliability-
redundancy optimization problems are as follows:
Figure 4.1: Low-level redundancy
Figure 4.2: High-level redundancy
2
1x 2x nx
2 2
1 1 1
1 1 1
2 2 2
h h h
Studies on Reliability Optimization Problems by Genetic Algorithm 82
Problem 5
Maximize ( )SR h = 1 1[ ( ), ( )] 1 (1 ([ , ],...,[ , ],...,[ , ]))R
h
SL SR L R qL q nL nRR h R h f r r r r r r= − −
subject to ( ) ,i ig h b≤ 1,2, ,i m= ⋅⋅⋅
, integerl h u h≤ ≤
[ , ] (0,1)i iL iRr r r= ∈ , 1,2, ,i n= ⋅⋅⋅
Problem 6
Minimize ( )SC h
subject to ( )S TR h R≥
where 1 1( ) [ ( ), ( )] 1 (1 ([ , ],...,[ , ],...,[ , ]))R
h
S SL SR L R qL q nL nRR h R h R h f r r r r r r= = − −
4.5 Solution Procedure
In this section we shall discuss the solution procedure for all the problems
mentioned in earlier section i.e., Problems (3)-(6). These problems are non-linear
integer programming optimization problems, each with interval valued objective
function. Using Big- M penalty technique and real coded genetic algorithm with
advanced operators, these problems are converted into unconstrained optimization
problems.
The converted problems of Problems (3)-(6) are as follows:
For the constrained optimization Problem 3
Maximize ( ) [ ( ), ( )]S SL SR
R x R x R x=
subject to ( ) ,i i
g x b≤ 1, 2, ,i m= ⋅⋅⋅
1 , integer, 1,...,j j j jl x u x j n≤ ≤ ≤ =
The form of Big-M penalty is as follows:
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )SL SR SL SRR x R x R x R x xθ= + (4.1)
Reliability Optimization under High and Low-level Redundancies… 83
where [0,0] if
( )[ ( ), ( )] [ , ] ifSL SR
x Sx
R x R x M M x Sθ
∈=
− + − − ∉
and { : ( ) , 1, 2, , and 1 , integer, 1,..., }i i j j j j
S x g x b i m l x u x j n= ≤ = ⋅⋅⋅ ≤ ≤ ≤ =
For the constrained optimization Problem 4
Minimize ( ) [ ( ), ( )]S SL SRC x C x C x=
subject to ( )S TR x R≥
The form of Big-M penalty is as follows:
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )SL sR SL SRC x C x C x C x xθ= − + (4.2)
where [0,0] if
( )[ , ] [ , ] ifSL SR
x Sx
C C M M x Sθ
∈=
+ − − ∉
and { }: ( ) , and 1 , integer, 1,...,S T j j j jS x R x R l x u x j n= ≥ ≤ ≤ ≤ =
For the constrained optimization Problem 5
Maximize ( ) [ ( ), ( )]S SL SRR h R h R h=
subject to ( ) ,i ig h b≤ 1,2, ,i m= ⋅⋅⋅
, integerl h u h≤ ≤
The form of Big-M penalty is as follows:
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )SL SR SL SRR h R h R h R h hθ= + (4.3)
where [0,0] if
( )[ ( ), ( )] [ , ] ifSL SR
h Sh
R h R h M M h Sθ
∈=
− + − − ∉
and { : ( ) , 1, 2, , and 1 , integer}i iS h g h b i m l h u h= ≤ = ⋅⋅⋅ ≤ ≤ ≤
For the constrained optimization Problem 6
Minimize ( ) [ ( ), ( )]S SL SRC h C h C h=
subject to ( )S TR h R≥
The form of Big-M penalty is as follows:
Studies on Reliability Optimization Problems by Genetic Algorithm 84
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )SL SR SL SRC h C h C h C h hθ= − + (4.4)
where [0,0] if
( )[ , ] [ , ] ifSL SR
h Sh
C C M M h Sθ
∈=
+ − − ∉
and { }: ( ) , and 1 , integerS TS h R h R l h u h= ≥ ≤ ≤ ≤
Here ˆ ˆ[ ( ), ( )]SL SRR x R x , ˆ ˆ[ ( ), ( )]sL sRC x C x , ˆ ˆ[ ( ), ( )]SL SRR h R h and ˆ ˆ[ ( ), ( )]sL sRC h C h are the
interval valued auxiliary objective functions. Problems (4.1) and (4.2) are integer
non-linear unconstrained optimization problems with interval objective of n integer
variables 1 2, ,..., nx x x whereas problems (4.3) and (4.4) are integer non-linear
unconstrained optimization problems with interval objective of integer variable h .
These problems (4.1)-(4.4) are non-linear unconstrained integer programming
problems with interval coefficients.
4.6 Numerical Examples
In this section, we have considered the redundancy allocation problem for low-level
redundancy (see Figure 4.3) and for high-level redundancy of five-link bridge system
(see Figure 4.4) for numerical experiments. Bridge system is of use in system
network with subsystem-5 representing a hub and rest of the systems representing
servers/client with processors arranged in parallel. This five-link bridge network
system [Kuo, Prasad, Tillman and Hwang (2001)] works successfully as long as one
of the paths, (subsystem-1-subsystem-2) or (subsystem-3- subsystem-4), is active
independently of subsystem-5. However, if the pair of subsystems (1, 4) or (2, 3)
fails, then subsystem-5 plays an important role in the system operation. In each
subsystem- i , 1,2,3, 4,5,i = which is imprecise in nature, there is a parallel
Reliability Optimization under High and Low-level Redundancies… 85
configuration consisting of ix identical components having reliability ir . If iR be the
system reliability of subsystem- i , 1, 2,3,4,5i = then 1 (1 ) ,ix
i iR r= − − 1, 2,3,4,5i = .
The system reliability of the low-level five-link bridge network system is given by the
expression as follows:
1 2 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5( )SR x R R Q R R Q R R R R Q Q R R Q R R Q R= + + + + ,
where 1i iR Q= − , 1, 2,3,4,5i =
The system reliability of the high-level five-link bridge network system is given by
the expression as follows:
1 2 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5( ) 1 (1 ( )) ,h
SR h r r q r r q r r r r q q r r q r r q r= − − + + + +
where 1i ir q= − , 1,2,3,4,5i = and h be the number of redundant subsystems,
arranged in parallel. For low-level redundancy, the corresponding system reliability
maximization and cost minimization problems are of the following forms:
Example 1
1 1 2 2 2 2 3 3 4 4Maximize[ ( ), ( )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R x R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L R L RR R Q Q Q Q R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L R L RQ Q R R R R Q Q R R+
subject to
5
1 1
1
( ) [ exp( 4) 0j j j
j
g x c x x b
=
= + − ≤∑
5
2 2
1
( ) exp( 4) 0j j j
j
g x w x x b
=
= − ≤∑
Studies on Reliability Optimization Problems by Genetic Algorithm 86
Figure 4.3: Low-level redundancy of five-link bridge system
Example 2
Minimize 5
1
( ) exp( )4
j
S j j
j
xC x c x
=
= +
∑
subject to ( )S TR x R≥
where
1 1 2 2 2 2 3 3 4 4[ ( ), ( )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R x R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L R L RR R Q Q Q Q R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L R L RQ Q R R R R Q Q R R+
For high–level redundancy, the corresponding system reliability maximization and
cost minimization problems are of the form that follows:
Example 3
Maximize 1 2 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5( ) 1 (1 ( ))h
SR h r r q r r q r r r r q q r r q r r q r= − − + + + +
subject to
Reliability Optimization under High and Low-level Redundancies… 87
[ ]5
1 1
1
( ) exp( 4) 0j
j
g h h h c b
=
= + − ≤∑
[ ]5
2 2
1
( ) exp( 4) 0j
j
g h h h w b
=
= − ≤∑
where [ , ], 1, 2,3,4,5i iL iRr r r i= = and [ , ] 1 [ , ]i iL iR iL iRq q q r r= = −
Example 4
Minimize [ ]5
1
( ) exp( 4)S j
j
C h h h c
=
= + ∑
subject to ( )S TR h R≥
where 1 2 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5( ) 1 (1 ( ))h
SR h r r q r r q r r r r q q r r q r r q r= − − + + + +
and [ , ], 1, 2,3,4,5i iL iRr r r i= = and [ , ] 1 [ , ]i iL iR iL iRq q q r r= = −
All the values of the parameters related to above Examples are given in Table 4.1:
Table 4.1: Values of the parameters related to Examples 1-4
j jr jc 1b jw 2b TR
1 [0.64,0.66] [3,5]
[105, 115]
[1.5,1.6]
[30,35]
[0.99,0.999]
2 [0.73,0.76] [4.5,5] [2,2.5]
3 [0.75,0.77] [5.5,7.5] [2,2.25]
4 [0.83,0.86] [5,7] [1.5,1.75]
5 [0.88,0.90] [2,2.5] [1.75,2]
The proposed method has been coded in C programming language. The
computational work has been done on a PC with Intel Core-2 duo processor in LINUX
environment. For each example 20 independent runs have been performed to
calculate the best found system reliability and best found system cost which are
nothing but the optimal values of system reliability and system cost. Also we have to
Studies on Reliability Optimization Problems by Genetic Algorithm 88
calculated the mean and variance of system reliability as well as system cost. In this
computation, the values of genetic parameters like p_size, max_gen, p_mute and
p_cross have been taken as 100, 100, 0.15 and 0.85 respectively. The computational
results have been shown in Table 4.2.
Figure 4.4: High-level redundancy of five-link bridge system
It has been observed from the computational results that the mean system
reliability/mean system cost coincides with the best found system reliability/system
cost. This strict coincidence is due to the fact that each trial run provides us the
optimum solution. Also, the lower ends of the standard deviations, measured in
interval form, assume zero value. This speaks of high precision in our optimization
process. It may also be noted that the average CPU time required for implementing
the genetic algorithm, is very less.
Reliability Optimization under High and Low-level Redundancies… 89
Table 4.2: Computational results for Examples 1-4
Example 1 3 2 4
x ’s/ h (2,2,1,2,2) (1) (1,1,2,1,1) (3)
Best found
system
reliability
[0.939545,0.999027] [0.819842,0.928286] - -
Mean value
of system
reliability
[0.939545,0.999027] [0.819842,0.928286] - -
Best found
system cost - - [53.186,71.904] [102.34,138.159]
Mean value
of system
cost
- - [53.186,71.904] [102.34,138.159]
Standard
deviation of
system
reliability
[0, 0.059482] [0,0.108444] - -
Standard
deviation of
system cost
- - [0,18.718] [0,35.819]
CPU time in
seconds 0.04000 0.07000 0.03000 0.18000
4.7 Sensitivity Analysis
To investigate the overall performance of the proposed GA-based penalty technique
for solving low-level redundancy as well as high level redundancy, sensitivity
analyses have been carried out graphically on the interval valued system reliability
with respect to different GA parameters separately taking other parameters at their
original values. These have been shown in Figure 4.5-Figure 4.8. From Figure 4.5 it
may be observed that both the bounds of the interval valued system reliability are
same for all the values of population size greater than or equal to 30. This implies
that our proposed GA is stable when population size exceeds 30. Similarly, from
Studies on Reliability Optimization Problems by Genetic Algorithm 90
Figure 4.6 it is clear that our proposed GA is stable when maximum number of
generation is greater than or equal to 10. In Figure 4.7 and Figure 4.8, the values of
interval valued system reliability have been computed with respect to the probability
of crossover within the range 0.45 to 0.95 and the probability of mutation within the
range 0.05 to 0.30 respectively. From these figures, it is clear that the proposed GA is
stable with respect to probability of crossover as well as the probability of mutation.
0.9
0.92
0.94
0.96
0.98
1
10 20 30 40 50 60
Population size
Inte
rval valu
ed
syste
m
reliab
ilit
y
Low er bound of system
reliability
Upper bound of system
reliability
Figure 4.5: P_size vs. interval valued system reliability for Example 1
0.9
0.92
0.94
0.96
0.98
1
10 20 30 40 50 60
Max_gen
Inte
rval valu
ed
syste
m
reliab
ilit
y
Low er bound of interval
valued system reliability
Upper bound of interval
valued system reliability
Figure 4.6: Max_gen vs. interval valued system reliability for Example 1
Reliability Optimization under High and Low-level Redundancies… 91
0.92
0.935
0.95
0.965
0.98
0.995
0.45 0.55 0.65 0.75 0.85 0.95
Probability of crossover
inte
rval
valu
ed
syste
m
reliab
ilit
y
Low er bound of
interval valued
system reliabilityUpper bound of
interval valued
system reliability
Figure 4.7: P_cross vs. interval valued system reliability for Example 1
0.92
0.935
0.95
0.965
0.98
0.995
0 0.05 0.1 0.15 0.2 0.25 0.3
Probability of mutation
Inte
rval valu
ed
syste
m
reliab
ilit
y
Low er bound of
interval valued system
reliability
Upper bound of
interval valued system
reliability
Figure 4.8: P_mute vs. interval valued system reliability for Example 1
4.8 Concluding Remarks
In this chapter, we have investigated two different redundancies known as low-level
redundancy and high-level redundancy and proposed four problems where each
problem belongs to the category of interval valued non-linear integer programming
problems. Then we have solved these problems as constrained single objective
interval valued reliability optimization problem. The reduced problem has been
converted into unconstrained interval valued integer programming problem using
Studies on Reliability Optimization Problems by Genetic Algorithm 92
Big-M penalty technique and solved by genetic algorithm. To solve the problem, we
have developed a real coded GA for integer variables with interval valued fitness
function, tournament selection, intermediate crossover and one-neighborhood
mutation and elitism of size one. It is well known that the penalty coefficient plays a
crucial role in solving constrained optimization problem by penalty function
technique. Therefore, the selection of this parameter is a formidable task. To avoid
this difficulty, we have used Big-M penalty technique which does not require any
penalty coefficient. This entire approach opens up scope for reliability optimization
when reliability values and other design parameters are interval/imprecise valued.
Thus, it can be claimed that the generalization attempted in this chapter can handle
the real-life problem of imprecise reliability optimization and cost minimization.
CHAPTER 5
Reliability Optimization under Weibull
Distribution with Interval
Valued Parameters
• Introduction
• Assumptions
• Weibull Distribution with Interval valued Parameters
• Problem Formulation
• Solution Procedure
• Numerical Example
• Concluding Remarks
Studies on Reliability Optimization Problems by Genetic Algorithm 94
5.1 Introduction
Over the last few decades, attention is being paid to reliability redundancy allocation
problems which have started drawing the attention of the reliability designers for
arriving at reliability optimization designs [Chern (1992), Coit and Smith (1996), Kuo
and Prasad (2000) and Sun and Li (2002)]. The basic objective of a redundancy
allocation problem (RAP) is to increase the reliability of subsystems so as to arrive at
a prefixed reliability goal for the system as a whole, subject to several operating
constraints on the system/subsystem. RAP is basically a non-linear integer/mixed-
integer programming problem. According to Chern (1992) RAP is NP-hard and it has
been well studied as summarized in Tillman, Hwang and Kuo (1977a) and Kuo and
Prasad (2000). In the earlier stage of development, several deterministic methods,
like heuristic methods [Nakagawa and Nakashima (1977), Kim and Yum (1993) and
Aggarwal and Gupta (2005)], mixed-integer non-linear programming [Tillman,
Hwang and Kuo (1977b)], reduced gradient method [Hwang, Tillman and Kuo
(1979)], integer programming [Misra and Sharma (1991)], linear programming
approach [Kim and Yum (1993)], dynamic programming method [Kuo, Prasad,
Tillman and Hwang (2001)], branch and bound method [Sun and Li (2002)] were
used for solving such RAP. However, these methods have both advantages and
disadvantages. Dynamic programming is not useful for reliability optimization of a
general system as it can be used only for a few particular structures of the objective
function and constraints that are decomposable. In branch and bound method, the
effectiveness depends on sharpness of the bounds and the required memory
increases exponentially with the problem size.
With the advent of genetic algorithm (GA) [Goldberg (1989) and Deb (2000)]
and other evolutionary algorithms, researchers have started paying more attention
Reliability Optimization under Weibull Distribution… 95
to RAP using numerical methods [Coit and Smith (1996, 2002) and Coelho (2009a,
2009b)] as these methods provide more flexibility and require less assumptions on
the objective as well as the constraints and are also effective irrespective of whether
the search space is discrete or not. These have enabled the reliability
planners/designers to undertake and reasonably compromise with several goals. In
the literature in almost all the studies referred above, the design parameters
involved in RAP have usually been taken to be precise values. This means that every
probability involved is perfectly determinable. In this case, it is usually assumed that
there exist some complete probabilistic information about the system and the
component behavior. However, in real-life situations, there are not sufficient
statistical data available in most of the cases where either the system is new or if
exists only as a project. It is not always possible to observe the stability from the
statistical point of view. This means that only some partial information about the
system components is known. So the reliability of a component of a system will be an
imprecise number which can be represented by an interval number and is calculated
by imprecise probabilities [Coolen and Newby (1994) and Utkin and Gurov (1999,
2001)]. Further, distributional parameters may not be of precise value. They may be
allowed to vary over an interval to take care of the effects of several factors. Keeping
these considerations in mind, the reliability optimization problem can be described
as a problem with distributional parameters assuming interval/imprecise values. In
this chapter, we have considered the RAP under imprecise reliability and component
reliability following the Weibull distribution with interval valued distributional
parameters. The problem is formulated as a non-linear constrained mixed-integer
programming problem with interval coefficients for maximizing the overall system
reliability under resource constraints. In this chapter, to solve the constrained
Studies on Reliability Optimization Problems by Genetic Algorithm 96
optimization problem, we have converted it into an unconstrained one by using the
penalty function technique discussed in Chapter 2 and the resulting objective
function would be interval valued. For solving such optimization problem by GA, we
have developed a real coded elitist GA with tournament selection, intermediate
crossover and one-neighborhood mutation for integer variables and whole
arithmetical crossover and boundary mutation for floating point variables. Finally, to
illustrate the proposed model, a numerical example has been solved for different
cases of scale and shape parameters of the Weibull distribution.
5.2 Assumptions
In formulation of the problem, the following assumptions have been considered.
(i) The chance of failure of any component is independent.
(ii) All the redundancy is active redundancy without repair.
(iii) Failure of each component follows the Weibull distribution.
(iv) Both the Weibull scale and shape parameters are imprecise and interval valued.
According to the assumptions, the system reliability would be interval valued. So to
optimize this system reliability under certain constraints, the following topics play
important role in solving the problem by genetic algorithm.
5.3 Weibull Distribution with Interval Valued Parameters
The probability density function for a Weibull distributed t is given by
1( )
( ) exp , 0( )
t tf t t
ββ
β
β δ δδ
θ δθ δ
− − − = − ≥ ≥
−− .
where β is known as the shape parameter and ( )θ δ− , known as the scale
parameter. Both the parameters are always positive.
Reliability Optimization under Weibull Distribution… 97
If 0δ = and βθ α− = then 1( ) exp , 0f t t t t
β βαβ α− = − ≥
Now if [ , ]L Rα α α= and [ , ]L Rβ β β=
then ( )f t can be written as an interval [ ( ), ( )]L Rf t f t
where 1( ) expL L
L L L Lf t t tβ βα β α− = − and 1
( ) expR RR R R Rf t t t
β βα β α− = − , 0t ≥ .
We can easily ensure from interval mathematics that for such a distribution, the
following properties hold.
Property-1: max[ ( ), ( )] [0,0] for 0L Rf t f t t> ≥
Property-2: [ ( ), ( )] [1,1]L Rf t f t dt
∞
−∞
=∫
So, it can easily be proved that [ ( ), ( )]L Rf t f t is interval valued probability density
function. The interval valued probability distribution function for a Weibull
distributed t is given by
( ) [ ( ), ( )] 1 exp( ) ,1 exp( )L RL R L RF t F t F t t t
β βα α = = − − − −
As ( ) 1 ( )r t F t= − , therefore the interval valued reliability function of interval valued
Weibull distribution is given by ( ) [ ( ), ( )] exp( ) ,exp( )R LL R R Lr t r t r t t t
β βα α = = − −
Therefore ( ) exp( )RL Rr t t
βα= − and ( ) exp( )LR Lr t t
βα= −
5.4 Problem Formulation
Our objective is to formulate the redundancy allocation problem of a
complex/complicated system with n subsystems. The goal of the redundancy
allocation problem is to determine the number of redundant components in each of
n parallel subsystems and mission time for overall system so as to maximize the
overall system reliability subject to the given constraints mostly arriving in linear
Studies on Reliability Optimization Problems by Genetic Algorithm 98
form. The time-to-failure for each available component is distributed according to a
two-parameter Weibull distribution with imprecise scale and shape parameters.
Then, the corresponding problem becomes a mixed-integer non-linear programming
problem with m constraints, which can be formulated as follows:
Maximize 1 1 2 2( , ) ( ( , ), ( , ),..., ( , ),..., ( , ))S q q n nR x t f R x t R x t R x t R x t= (5.1)
subject to Ax b≤
where
11 12 1
21 22 2
1 2
...
...
. . ... .
. . ... .
...
n
n
m m mn
c c c
c c c
A
c c c
=
and [ ]1 2 ...T
mb b b b=
1 , being an integer, 1,...,i i i il x u x i n≤ ≤ ≤ = ,
where ( , ) [ ( , ), ( , )] 1 (1 [ ( ), ( )]) , 1,2,ix
i i iL i iR i iL iRR x t R x t R x t r t r t i n= = − − = ⋅⋅⋅ ,
and [ ],[ , ]
( ) [ ( ), ( )] , 1, 2, ,iL iR
iL iR t
i i ir t r t r t e i nβ β
α α−= = = ⋅⋅⋅ , under Weibull setup with
scale parameter iα and shape parameter iβ ,
, 0and real valuedl ut t t t≤ ≤ > .
5.5 Solution Procedure
In this section we shall discuss the solution procedure for the problem mentioned in
the section 5.4. This problem is a non-linear mixed-integer optimization problem
with interval valued objective function. Using Big-M penalty technique this problem
is converted into unconstrained optimization problem.
The form of Big-M penalty is as follows:
Maximize ˆ ˆ[ ( , ), ( , )] [ ( , ), ( , )] ( , )SL SR SL SRR x t R x t R x t R x t x tθ= +
Reliability Optimization under Weibull Distribution… 99
where [0,0] if ( , )
( , )[ ( , ), ( , )] [ , ] if ( , )SL SR
x t Sx t
R x t R x t M M x t Sθ
∈=
− − ∉
and { }( , ) : ,1 andi i i l uS x t Ax b l x u t t t= ≤ ≤ ≤ ≤ ≤ ≤ be the feasible space.
This is a mixed-integer non-linear unconstrained optimization problem with interval
objective of n integer variables 1 2, ,..., nx x x and a single floating point variable t.
5.6 Numerical Example
To study the performance of the Genetic Algorithm for solving reliability
optimization problem for a complex/complicated system, a numerical example of
five-link bridge network system has been considered (see Figure 5.1).The proposed
method is coded in C programming language and run in the LINUX operating system.
The computational procedure has been implemented on PC with Intel Core-2 duo
processor with 2.5 GHz. For each case, 50 independent runs have been performed to
calculate the best found system reliability which is nothing but the optimal value of
system reliability, mean and standard deviation of system reliability in interval forms
and average CPU time. In this computational work, the values of different genetic
parameters like, population size (p_size), mutation rate (p_mute), crossover rate
(p_cross) and maximum generation (max_gen) have been taken as 200, 0.15, 0.85 and
150 respectively.
Example 1
1 1 2 2 2 2 3 3 4 4Maximize[ ( , ), ( , )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R R x R R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L R L RR R Q Q Q Q R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L R L RQ Q R R R R Q Q R R+
Studies on Reliability Optimization Problems by Genetic Algorithm 100
subject to
1 2 3 4 5( ) 0.1 2.0 25c x x x x x x= + + + + ≤
1 2 3 4 5( ) 2 0.1 21w x x x x x x= + + + + ≤
1 2 3 4 5( ) 28v x x x x x x= + + + + ≤
where
[ , ] 1 (1 [ ( ), ( )]) , 1,2,3,4,5ix
i iL iR iL iRR R R r t r t i= = − − =
[ , ] 1 [ , ], 1, 2,3,4,5i iL iR iL iRQ Q Q R R i= = − =
and [ ],[ , ]
( ) [ ( ), ( )] , 1, 2,3, 4,5iL iR
iL iR t
i iL iRr t r t r t e iβ β
α α−= = =
1 6, being integer, 1, 2,3,4,5i ix x i≤ ≤ =
1 5, being real valuedt t≤ ≤ .
Figure 5.1: Five-link bridge network system
To study the variation of the parameters, we consider the following four cases
as follows:
Case-I When both the parameters iα and iβ are interval valued numbers
Reliability Optimization under Weibull Distribution… 101
Case-II When iα ’s are interval valued and iβ ’s are fixed valued numbers
Case-III When iα ’s are fixed valued and iβ ’s are interval valued numbers
Case-IV When both the parameters iα and iβ are fixed valued numbers
The values of iα and iβ ( 1, 2,3,4,5i = ) are given in Table 5.1.
Table 5.1: Values of iα and
iβ ( 1,2,3,4,5i = ) for four different cases
Case-I Case-II Case-III Case-IV
1 1 1[ , ]L Rα α α= [0.257,0.258] [0.257,0.258] [0.257,0.258] [0.257,0.258]
2 2 2[ , ]L Rα α α= [0.118,0.119] [0.118,0.119] [0.118,0.119] [0.118,0.118]
3 3 3[ , ]L Rα α α= [0.214,0.215] [0.214,0.215] [0.214,0.214] [0.214,0.214]
4 4 4[ , ]L Rα α α= [0.165,0.166] [0.165,0.166] [0.165,0.165] [0.165,0.165]
5 5 5[ , ]L Rα α α= [0.210,0.211] [0.210,0.211] [0.210,0.210] [0.210,0.210]
1 1 1[ , ]L Rβ β β= [1.99,2.1] [1.99,1.99] [1.99,2.1] [1.99,1.99]
2 2 2[ , ]L Rβ β β= [1.99,2.1] [1.99,1.99] [1.99,2.1] [1.99,1.99]
3 3 3[ , ]L Rβ β β= [1.99,2.1] [1.99,1.99] [1.99,2.1] [1.99,1.99]
4 4 4[ , ]L Rβ β β= [1.99,2.1] [1.99,1.99] [1.99,2.1] [1.99,1.99]
5 5 5[ , ]L Rβ β β= [1.99,2.1] [1.99,1.99] [1.99,2.1] [1.99,1.99]
The solutions of five-link bridge network system for different cases have been
displayed in Table 5.2. From Table 5.2 it is seen that the best found values of system
reliability in all cases are the same with mean values of the same. Again in Case-III
and Case-IV the standard deviations of system reliability of the system are zero
whereas in Case-I and Case-II, these are interval valued with lower bounds and
significantly small widths. Also average CPU time required for implementing the
genetic algorithm is also on the lower side.
Studies on Reliability Optimization Problems by Genetic Algorithm 102
Table 5.2 Computational results of Example 1
Case x Best found value of
system
reliability R
Mean of R Standard
deviation of
R
Mean
time-
to-
failure t
CPU
(in
sec)
I (2,5,5,5,5) [0.999966,0.999999] [0.999966,0.999999] [0,0.000010] 1.0 0.158
II (3,5,5,5,5) [0.999966,0.999999] [0.999966,0.999999] [0,0.000010] 1.0 0.154
III (3,4,6,5,4) [0.999997,0.999997] [0.999997,0.999997] [0,0] 1.0 0.154
IV (2,6,6,5,5) [0.999998,0.999998] [0.999998,0.999998] [0,0] 1.0 0.146
5.7 Concluding Remarks
In this chapter, for the first time, the reliability optimization problem with Weibull
distributed (with interval valued parameters) time-to-failure of each component of a
complex/complicated system with some resource constraints have been solved.
Now, for handling the problem with resource constraints, the corresponding problem
has been converted into unconstrained optimization problem with the help of
penalty technique (called Big-M penalty technique [Gupta, Bhunia and Roy (2009)]).
To solve the problem, we have developed a real coded GA for mixed-integer variables
with interval valued fitness function, tournament selection, intermediate crossover
and one-neighborhood mutation for the genes corresponding to integer variables,
whole arithmetical crossover and boundary mutation for the gene corresponding to
the floating point variable and elitism of size one.
CHAPTER 6
Stochastic Optimization of System
Reliability for series System
with Interval Component
Reliabilities
• Introduction
• Assumptions
• Problem Formulation
• Solution Procedure
• Numerical Examples
• Sensitivity Analysis
• Concluding Remarks
Studies on Reliability Optimization Problems by Genetic Algorithm 104
6.1 Introduction
In most of the probabilistic methods of the reliability engineering system, it is
assumed that all the probabilities are precise. This means that every probability
involved is perfectly determinable. In this case it is usually assumed that there exist
some complete probabilistic information about the components and system
behavior. For the completeness of probabilistic information, the following two
conditions must be satisfied.
(i) All the probabilities or probability distributions are known or perfectly
determinable.
(ii) The system components are independent, i.e., all the random variables,
describing the component reliability behavior, are independent.
During the past, the assumption of uncertainty in most of the methods in reliability is
based on precise probabilities and the reliabilities of the system components are to
be known at a fixed positive number which lies in the open interval (0,1) . The precise
system reliability can be computed theoretically if both the above two conditions are
satisfied (it is assumed that the system structure is defined precisely and there exists
a function linking the system time to failure as well as the times to failure of the
components). If at least one condition is violated, then only the interval measure of
reliability [Barlow and Proschan (1965), Coolen and Newby (1994) and Lindqvist
and Langseth (1998)] can be obtained. However, in real-life situations, there are not
sufficient statistical data in most of the cases where either the system is new or if
exists only as a project. It is not always possible to observe the stability from the
statistical point of view if such data exists. This means that only some partial
information about the system components is known. So the reliability of a
component of a system will be an imprecise number which can be represented by an
Stochastic Optimization of System Reliability for Series System… 105
interval number [Gupta, Bhunia and Roy (2009)] and is calculated by imprecise
probabilities [Gurov and Utkin (1999) and Utkin and Gurov (1999, 2001)] define
reliability as the probability of survival that a system will perform satisfactorily at
least up to a given period of time under stated conditions. For designing a highly
reliable system, there arises a question as to how to get a balance between the
reliability of a system and resources such as cost, volume and weight. As a result,
inclusion of redundant components or the increase of the components’ reliability
leads to increase the system reliability.
In the last two decades, a number of techniques have been proposed for
solving reliability optimization problems [Chern (1992), Gen and Yun (2006) and Ha
and Kuo (2006b)]. These techniques can be classified as dynamic programming
method, branch and bound method, Lagrange multiplier method, etc [Kuo, Prasad,
Tillman and Hwang (2001), Sun, Mckinnon and Li (2001) and Sun and Li (2002)]. In
the year 2003, Zhao and Liu (2003) developed stochastic programming technique for
redundancy allocation problems. Stochastic reliability optimization problem is either
an extension or reformulation of reliability optimization problem with random
variations of parameters. Moreover, the resource elements vary and it is reasonable
to regard them as stochastic variables. It is also known that a stochastic
programming problem is harder than all other combinatorial optimization problems.
In this chapter, we have solved chance constrained reliability optimization
problem with interval valued component reliabilities. Here, various types of
randomness have been discussed with known probability distributions, viz. uniform,
normal and log-normal distributions, when the resource variables are random. The
corresponding chance constrained redundancy allocation problem for the series
system has been solved with the help of GA.
Studies on Reliability Optimization Problems by Genetic Algorithm 106
6.2 Assumptions
(i) Reliability of each component is imprecise and interval valued.
(ii) If a component of any subsystem fails to function, the entire system will not be
damaged or failed.
(iii) All redundancy is active redundancy without repair.
(iv) The state of components and system has only two states like operating state
or failure state.
(v) The resource constraints are chance constraints with resource vector as
stochastic in nature.
(vi) Life distributions of components are statistically independent.
6.3 Problem Formulation
Let us consider a system consisting of n subsystem in series in which the j-th
(1 )j n≤ ≤ subsystem consists of jx components in parallel. Such a system is called
parallel-series system or n -stage series system (see Figure 6.1). Assuming all the
components in the j -th subsystem as identical, the system reliability SR is given by
1
( ) [ ( ), ( )] ( ), ( )n
S SL SR jL jR
j
R x R x R x R x R x
=
= = ∏
where ( ) 1 (1 ) jx
jL jLR x r = − −
and ( ) 1 (1 ) jx
jR jRR x r = − −
The chance constrained optimization problem for a parallel-series system with
m chance constraints can be formulated as
Maximize1
[ ( ), ( )] ( ), ( )n
SL SR jL jR
j
R x R x R x R x
=
= ∏ (6.1)
subject to
Stochastic Optimization of System Reliability for Series System… 107
Prob[ ( ) ] 1i i ig x b γ≤ ≥ − , 1,2,...i m=
and j j jl x u≤ ≤ , 1,2,...j n= .
Figure 6.1: A n-stage series system
Definition: A random variable X is said to have a normal distribution with
parameters µ (mean) and 2σ (variance) if its probability density function is given by
21 1
( ; , ) exp ; , , 022
xf x x
µµ σ µ σ
σσ π
− = − − ∞ < < ∞ >
When a random variable X is normally distributed with mean µ and standard
deviationσ , we shall express it as 2( , )X N µ σ∼ .
Definition: A random variable X is said to have a uniform distribution if its
probability density function is given by
1 1 1 1
2 2 2 2
3 3 3 3
1x
2
x
j
x n
x
Stage 1 2 j n
Studies on Reliability Optimization Problems by Genetic Algorithm 108
1;
( )
0 ;otherwise
a x bf x b a
< <
= −
When a random variable X is uniformly distributed, we shall express it
as ( , )X U a b∼ .
Definition: A positive random variable X is said to have a log-normal distribution
with parameters µ (mean) and 2σ (variance) if its probability density function is
given by
21 1 log
exp , 0( ; , ) 22
0, 0
xx
f x x
x
µ
µ σ σσ π
− − > =
<
When a random variable X is log-normally distributed with mean µ and standard
deviationσ , we shall express it as 2( , )X LN µ σ∼ .
Case-I When ib is uniformly distributed
In this case, ( , )i i ib U ξ η∼ . Then the constraint Prob[ ( ) ] 1i i ig x b γ≤ ≥ − can be written
in the equivalent deterministic constraint as ( )i ig x δ≤ ,
where 1
1
i
i
i
i i
dx
η
δ
γη ξ
= −−∫
or, i i i i iδ ξ ζ γ η= + where 1i iζ γ= − (6.2)
Hence the problem (6.1) reduces to
Maximize1
[ ( ), ( )] ( ), ( )n
SL SR jL jR
j
R x R x R x R x
=
= ∏ (6.3)
subject to
( )i i i i ig x ξ ζ γ η≤ + , 1,2,...i m= and j j jl x u≤ ≤ , 1,2,...j n=
Case-II When ib is normally distributed
Stochastic Optimization of System Reliability for Series System… 109
In this case, 2( , )
i ii b bb N µ σ∼ where ibµ = mean of ib = ( )iE b and 2
ibσ = variance of
ib = Var( )ib
Then Prob[ ( ) ] 1i i ig x b γ≤ ≥ − can be written as ( )i ii b i bg x eµ σ≤ +
where ie is the value of the standard normal variate for which ( )i ie γΦ =
Again 2
1( ) exp( )
22
zx
z dxπ −∞
Φ = −∫
Hence the problem (6.1) is equivalent to
Maximize 1
[ ( ), ( )] ( ), ( )n
SL SR jL jR
j
R x R x R x R x
=
= ∏ (6.4)
subject to
( )i ii b i bg x eµ σ≤ + , 1,2,...i m= and j j jl x u≤ ≤ , 1,2,...j n=
Case-III When ib is log-normally distributed
In this case, 2( , )
i ii b bb LN µ σ∼ where ibµ = mean of log( )ib and 2
ibσ = variance of log( )ib
Then Prob[ ( ) ] 1i i ig x b γ≤ ≥ − can be written as ( ) exp( )i ii b i bg x eµ σ≤ +
where ie is the value of the standard normal variate for which ( )i ie γΦ =
Hence the problem (6.1) reduces to
Maximize 1
[ ( ), ( )] ( ), ( )n
SL SR jL jR
j
R x R x R x R x
=
= ∏ (6.5)
subject to
( ) exp( )i ii b i bg x eµ σ≤ + , 1,2,...i m= and j j jl x u≤ ≤ , 1,2,...j n=
Now we shall solve the deterministic problems (6.3), (6.4) and (6.5) by GA-based
constraint handling techniques.
Studies on Reliability Optimization Problems by Genetic Algorithm 110
6.4 Solution Procedure
In this section, we shall discuss the solution procedure for all the problems [(6.3),
(6.4) and (6.5)] mentioned in earlier section. These problems are non-linear integer
optimization problems with interval valued objective function. Using Big-M penalty
techniques and real coded genetic algorithm with advanced operators these
problems are converted into unconstrained optimization problems.
The converted problems of problems (6.3)-(6.5) are as follows:
Using Big-M penalty technique, the transformed problem is as follows:
Maximize ˆ ˆ[ ( ), ( )] [ ( ), ( )] ( )SL SR SL SRR x R x R x R x xθ= +
where [0,0] if
( )[ ( ), ( )] [ , ] ifSL SR
x Sx
R x R x M M x Sθ
∈=
− + − − ∉
and { }: Prob[ ( ) ] 1 , 1, 2,... and , 1,2,...i i i j j jS x g x b i m l x u j nγ= ≤ ≥ − = ≤ ≤ = be
the feasible space.
This is a non-linear unconstrained optimization problem with interval valued
objective.
6.5 Numerical Examples
To illustrate our proposed GA-based on Big-M penalty technique for solving the
reliability stochastic optimization problem with interval valued as well as fixed
valued reliabilities of components, we have considered two numerical examples.
Each example has been formulated using Case-I. In the first example, the reliability of
components are interval valued whereas the second one taken from Yadavalli,
Malada and Charles (2007), the reliabilities of components are fixed. The proposed
GA is coded in C programming language and run in the LINUX environment. The
computation work has been done on the PC which has Intel Core-2 duo processor
Stochastic Optimization of System Reliability for Series System… 111
with 2.5 GHz. For each example, 20 independent runs have been performed to
calculate the best found system reliability, mean and standard deviation and average
CPU time(s). In this computation, we have taken population size, mutation rate,
crossover rate and maximum generation as 100, 0.15, 0.85 and 50 respectively. The
simulation results have been displayed in Table 6.3. Example 2 has been solved by
our proposed technique expressing the reliability of each component as interval with
the same upper and lower bounds.
Example 1
A four stage system with chance constraints is formulated as a pure stochastic
integer programming problem using the data given in the Table -6.1.
Maximize1
[ ( ), ( )] ( ), ( )n
SL SR jL jR
j
R x R x R x R x
=
= ∏
subject to
4
1
Prob[ ] 1ij j i i
j
a x b γ=
≤ ≥ −∑ , 1,2i =
Table 6.1: Numerical data of Example 1
Stage j 1 2 3 4 Available resource
jr [0.74,0.76] [0.78,0.81] [0.73,0.78] [0.83,0.86] iξ iη iγ
1 ja 1.5 3.3 3.2 4.4 1b 50 60 0.10
2 ja 4.0 5.0 7.0 9.0 2b 110 140 0.15
Studies on Reliability Optimization Problems by Genetic Algorithm 112
Example 2
A four stage system with chance constraints is formulated as a pure stochastic
integer programming problem using the data given in the Table 6.2.
Maximize1
[ ( ), ( )] ( ), ( )n
SL SR jL jR
j
R x R x R x R x
=
= ∏
subject to
4
1
Prob[ ] 1ij j i i
j
a x b γ=
≤ ≥ −∑ , 1,2i =
Table 6.2: Numerical data of Example 2
Stage j 1 2 3 4 Available resource
jr [0.75,0.75] [0.80,0.80] [0.75,0.75] [0.85,0.85] iξ iη iγ
1 ja 1.5 3.3 3.2 4.4 1c 50 60 0.10
2 ja 4.0 5.0 7.0 9.0 2c 110 140 0.15
Table 6.3: Computational results of Examples 1-2
Example
No. jx ’s Best found
System
Reliability *
R
Mean of R Standard
deviation
of R
Averag
e CPU
time
1 (5,4,5,3) [0.990154,0.994650] [0.990154,0.994650] [0.00000,0.001421] 0.001
2 (5,4,5,3) [0.993088,0.993088] [0.993088,0.993088] [0,0] 0.001
6.6 Sensitivity Analysis
To study the performance of our developed GA-based on Big-M penalty technique,
sensitivity analyses have been done graphically on the interval valued system
Stochastic Optimization of System Reliability for Series System… 113
reliability with respect to GA parameters separately keeping the other parameters at
their original values. These are shown in Figure 6.2-Figure 6.5. The graphs (Ref.
Figure 6.2-Figure 6.5) have been drawn for lower and upper bounds of the system
reliability in the same graph. In Figure 6.2, the effect of population size (p_size) on the
system reliability has been studied from the range 10 to 90 of population size
(p_size). In this study, it is observed that both the bounds of the interval be the same
for all values of p_size after 30. This means that our proposed GA is stable when
population size exceeds the number 30. In Figures 6.3-6.5, the values of system
reliability have examined with respect to the probability of crossover (p_cross)
within the range from 0.35 to 0.95, probability of mutation (p_mute) within the range
0.05 to 0.3 and maximum number of generation (max_gen) respectively. From these
figures, it is evident that the proposed GA is stable with respect to probability of
crossover (p_cross), probability of mutation (p_mute) and maximum number of
generation (max_gen).
0.8
0.85
0.9
0.95
1
1.05
10 20 30 40 50 60 70 80 90
P_size
Sy
ste
m R
elia
bilit
y
Lower bound of theinterval valued system
reliability
Upper bound of theinterval valued system
reliability
Figure 6.2: P_size vs. interval valued system reliability for series system
Studies on Reliability Optimization Problems by Genetic Algorithm 114
0.8
0.85
0.9
0.95
1
1.05
0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
P_cross
Sy
ste
m R
eliab
ility Lower bound of the
interval valued
system reliability
Upper bound of theinterval valued
system reliability
Figure 6.3: P_cross vs. interval valued system reliability for series system
0.8
0.85
0.9
0.95
1
1.05
0 0.05 0.1 0.15 0.2 0.25 0.3
P_mute
Syste
m R
elia
bili
ty
Lower bound ofthe interval valuedsystem reliability
Upper bound ofthe interval valuedsystem reliability
Figure 6.4: P_mute vs. interval valued system reliability for series system
0.8
0.85
0.9
0.95
1
1.05
10 20 30 40 50 60 70
Max_gen
Sy
ste
m R
elia
bilit
y
Lower bound of theinterval valued system
reliability
Upper bound of the
interval valued system
reliability
Figure 6.5: Max_gen vs. interval valued system reliability for series system
Stochastic Optimization of System Reliability for Series System… 115
6.7 Concluding Remarks
In this chapter, chance constrained reliability optimization problem in the series
system with some resource constraints have been formulated and solved considering
the reliability of each component as an interval number. The interval number
representation is more appropriate among other representations like, random
variable representation with known probability distribution, fuzzy set with known
fuzzy membership function or fuzzy number. For handling the resource constraints,
the corresponding problem has been converted into unconstrained optimization
problem with the help of our developed penalty technique (called Big-M penalty
technique). Therefore, the transformed problem is of unconstrained interval valued
optimization problem with integer variables. To solve the transformed problem, we
have developed a real coded GA for integer variables with interval valued fitness
function, tournament selection, intermediate crossover, one neighborhood mutation
and elitism of size one. In tournament selection process and elitism operation we
have used the definitions of interval order relations. It is well known that the penalty
coefficient plays a crucial role in solving constrained optimization problem.
Therefore, the selection of these parameters is a formidable task. To avoid this
difficulty, we have also proposed Big-M penalty technique which does not require
any penalty coefficient. Here, we have also proposed a new mutation scheme (called
one-neighborhood mutation) in which the selected gene will take either its next
number or its previous number (if exists). In the study of statistical analysis for
interval valued numbers, it has been observed that for the set of same interval valued
numbers, the standard deviation will be an interval with negligible width. However,
in case of the set of fixed real numbers, it will be zero.
CHAPTER 7
Reliability Optimization with Interval
Parametric Values in the
Stochastic Domain
• Introduction
• Assumptions
• Normal Distribution with Interval Valued Parameters
• Stochastic Mixed-integer Programming: A Complicated System with
Chance Constraints
• Solution Procedure
• Numerical Example
• Concluding Remarks
Reliability Optimization with Interval Parametric Values… 117
7.1 Introduction
In almost all the studies referred in earlier Chapters 3-6, the design parameters
involved in the optimization problem have usually been taken to be constants. In that
case, the problem of optimization becomes a deterministic optimization problem. But
mostly, the design parameters are not deterministic in nature. While in-house
determination of parameters can be nearly deterministic, parameters determined
from the factor market, especially in respect of cost can hardly be deterministic.
These parameters can be viewed as estimated values, which in turn follow certain
stochastic laws. Unfortunately constraints involving these estimated values of the
optimization problems are usually solved in the deterministic domain and need to be
solved in the stochastic domain. For results in the stochastic domain, one may refer
to the works of Coit, Jin and Wattanapongsakorn (2004), Zafiropoulos and Dialynas
(2004) and Marseguerra, Zio, Podofillini and Coit (2005). Further, the distributional
parameters may not be of single value. They may be allowed to vary over an interval
to take care of the sensitivity of the factor market.
Keeping these considerations in the backdrop, the reliability optimization
problem can be described as a problem of chance constraints with distributional
parameters assuming interval values. The resultant solutions will have a greater
appeal, so far as their applications are concerned, because interval values will make
the optimum solution less sensitive and more robust.
Study of the system reliability where the component reliabilities are
imprecise and/or interval valued have already been initiated by some authors
[Coolen and Newby (1994), Utkin and Gurov (1999, 2001) and Gupta, Bhunia and
Roy (2009)]. Here we extended the domain of reliability optimization by considering
the optimum hot redundancy allocation problem under imprecise reliability with
Studies on Reliability Optimization Problems by Genetic Algorithm 118
constraints, expressed in terms of coefficient matrix and availability vectors, as
chance constraints. Our major contribution to this chapter is the introduction of
random variables with interval valued parameters. In fact, even for the random
coefficient matrix and availability vector, we have considered normal distribution
with interval valued means and variances so that the optimization problem can be
dealt with under a generalized setup. Specific solutions obtained in earlier studies
can be arrived at by collapsing an interval valued parameter into a point.
7.2 Assumptions
(i) The reliability of some of the components are imprecise and interval valued.
(ii) The chance of failure of any component is statistically independent with respect
to those of other components.
(iii) All the redundancy is active redundancy without repair.
(iv) All the resource coefficients and available resource amounts are random in
nature.
(v) Mean and variance of each random variable are interval valued.
(vi) Each random variable follows the normal distribution.
7.3 Normal Distribution with Interval Valued Parameters
It is well known that a random variable X follows 2( , )N µ σ if its probability density
function is given by
21 1
( ; , ) exp ; , , 022
xf x x
µµ σ µ σ
σσ π
− = − − ∞ < < ∞ − ∞ < < ∞ >
If 2( , )X N µ σ∼ , then
XZ
µ
σ
−= , is called a standard normal variate with mean zero
and variance one.
Reliability Optimization with Interval Parametric Values… 119
The corresponding distribution function is denoted by ( )zΦ .
Some important properties of the distribution function (.)Φ of standard normal
variate are as follows:
( ) 1 ( ), 0z z zΦ − = − Φ >
Prob( ) ( ) ( )b a
a X bµ µ
σ σ
− −≤ ≤ = Φ − Φ , where 2
( , )X N µ σ∼
But mostly these parameters are estimated from a given set of observations. In case
of confidence intervals proposed for the parametric measures one has to examine the
distribution with interval valued parameter. Keeping this practical requirement in
mind we have introduced interval valued normal distribution with both the mean
and variance parameters as interval valued. We can symbolically denote the
underlying distribution as 2 2([ , ],[ , ])L U L UN µ µ σ σ where [ , ]L Uµ µ is the interval
valued mean parameter and 2 2[ , ]L Uσ σ , the interval valued variance parameter. We
can easily ensure, from interval arithmetic, that for such a distribution, the following
property holds.
Property-1: Prob( ) ( ) ( )UL
L U
aba X b
µµ
σ σ
−−≤ ≤ = Φ − Φ
Proof: By definition, Prob( ) Prob( ) Prob( )a X b X b X a≤ ≤ = ≤ − ≤
By treating the respective definite integrals as limiting cases of summation under
interval analysis [Moore (1979)] and using interval arithmetic we get
Prob( )a X b≤ ≤ =[ , ] [ , ]
( ) ( )[ , ] [ , ]
L U L U
L U L U
b aµ µ µ µ
σ σ σ σ
− −Φ − Φ
([ , ]) ([ , ])U UL L
U L U L
b ab aµ µµ µ
σ σ σ σ
− −− −= Φ − Φ
Now, from the monotonically increasing nature of (.)Φ function and the standard
interval arithmetic operations, introduced earlier, we can write
Studies on Reliability Optimization Problems by Genetic Algorithm 120
Prob( ) ( ) ( )UL
L U
aba X b
µµ
σ σ
−−≤ ≤ = Φ − Φ
This property of interval valued normal distribution will be of use for deterministic
reduction of chance constraints.
7.4 Stochastic Mixed-integer Programming: A Complicated System with
Chance Constraints
Let us introduce the problem of reliability optimization under chance constraints. Let
us consider a complicated system with n subsystems. The goal of the redundancy
allocation problem is to determine the number of hot redundant components in each
of q parallel subsystems involving imprecise/interval valued reliabilities and
reliability levels of the rest ( )n q− general subsystems which are of precise reliability
values so as to maximize the overall system reliability subject to the given chance
constraints mostly arriving out of cost consideration. Then the corresponding
problem becomes a mixed-integer non-linear programming problem with m chance
constraints, which can be formulated as follows:
Problem 1 Maximize 1 1 2 2 1( , ) ( ( ), ( ),..., ( ), ,..., )S q q q nR x R f R x R x R x R R+=
subject to { }Prob ( , ) 1 , 1,2,...,j j j jC g x R b j mγ′ ≤ ≥ − =� �
(7.1)
and 1 , integer, 1,..., ,i i i il x u x i q≤ ≤ ≤ =
0 1, 1, 2,...,v v vL R U v q q n< ≤ ≤ < = + +
where
1
2
.
.
.
j
j
j
j
jn
c
c
C
c
=
�,
1
2
( , )
( , )
.( , )
.
.
( , )j
j
j
j
jn
g x R
g x R
g x R
g x R
=
�
Reliability Optimization with Interval Parametric Values… 121
and ( ) [ ( ), ( )] 1 (1 [ , ]) , 1,2,ix
i i iL i iR i iL iRR x R x R x r r i q= = − − = ⋅⋅⋅ .
and (1 jγ− )’s are specified probabilities with 0 1jγ< < .
It is assumed that all ' ( 1, 2,..., , 1, 2,..., )jk jc s j m k n= = and ( 1,2,..., )jb j m= are
random variables following normal distributions with interval valued parameters.
Since each (.)jg
�
is a technical relationship no such distribution has been assumed for
the same.
Let us denote1
( , )jn
j jk jk j
k
h c g x R b
=
= −∑ , 1, 2,...,j m= .
Then (7.1) can be expressed as
{ }Prob 0 1 , 1, 2,...,j jh j mγ≤ ≥ − = (7.2)
Since each ( 1,2,..., )jh j m= is a linear combination of the normally distributed
random variables jkc ’s and jb then each ( 1,2,..., )jh j m= will also follows normal
distribution.
Now, the mean of each ( 1,2,..., )jh j m= is given by
1
( ) ( ) ( ) , 1, 2,...,
jn
j jk jk j
k
E h E c g E b j m
=
= − =∑
Also, under independent of jkc ’s and jb ’s the variance of each ( 1,2,..., )jh j m= is
given by
2
1
Var( ) var( ) var( )jn
j jk jk j
k
h g c b
=
= +∑ , 1,2,...,j m= (7.3)
Then the constraint { }Prob 0 1j jh γ≤ ≥ − can be written as
( )( ) ( )
var( )
j
j
j
E he
h
−Φ ≥ Φ (7.4)
where je is the upper 100% point of the standard normal distribution.
Studies on Reliability Optimization Problems by Genetic Algorithm 122
Then (7.4) can be written as
( ) var( ) 0j j jE h e h+ ≤ , 1,2,...,j m= .
or, 2
1 1
( ) var( ) var( ) ( )j jn n
jk jk j jl jp jp j
k p
E c g e g c b E b′= =
+ + ≤∑ ∑ , 1,2,...,j m= (7.5)
which is the deterministic form of the given chance constraints.
Let Var( ) [ , ]jp jpL jpRc c c= , Var( ) [ , ]i jL jRb b b= , ( ) [ , ]jk jkL jkRE c c c= and ( ) [ , ]j jL jRE b b b= .
Then (7.5) reduces to
2
1 1
[ , ] [ , ] [ , ] [ , ]j jn n
jkL jkR jk j jp jpL jpR jL jR jL jR
k p
c c g e g c c b b b b
= =
+ + ≤∑ ∑
or, 2 2
1 1 1 1
, [ , ]
j j j jn n n n
jkL jk j jp jpL jL jkR jk j jp jpR jR jL jR
k p k p
c g e g c b c g e g c b b b
= = = =
+ + + + ≤ ∑ ∑ ∑ ∑ ,
1,2,...,j m= (7.6)
Thus the chance constrained optimization problem becomes equivalent to the
following deterministic constrained optimization problem.
Maximize 1 1 2 2 1( , ) ( ( ), ( ),..., ( ), ,..., )S q q q nR x R f R x R x R x R R+=
subject to
2 2
1 1 1 1
, [ , ]
j j j jn n n n
jkL jk j jp jpL jL jkR jk j jp jpR jR jL jR
k p k p
c g e g c b c g e g c b b b
= = = =
+ + + + ≤ ∑ ∑ ∑ ∑ ,
1, 2,...,j m= .
1 , integer, 1,..., ,i i i il x u x i q≤ ≤ ≤ =
0 1, 1, 2,...,v v vL R U v q q n< ≤ ≤ < = + +
and ( ) [ ( ), ( )] 1 (1 [ , ]) , 1,2,ix
i i iL i iR i iL iRR x R x R x r r i q= = − − = ⋅⋅⋅ .
This optimization problem, being analytically intractable, can be solved numerically
via genetic algorithm.
Reliability Optimization with Interval Parametric Values… 123
In particular, we would like to examine the reliability optimization problem of a
bridge network system as given in Figure 7.1. Bridge system is of use in system
network with subsystem-5 representing a hub and rest of the systems representing
servers/ clients with processors arranged in parallel.
This five-link bridge network system [Kuo, Prasad, Tillman and Hwang (2001)]
works successfully as long as one of the paths, (subsystem-1, 2) or (subsystem-3, 4),
is active independently of subsystem-5. However, if the pair of subsystems (1, 4) or
(2, 3) fails, then subsystem-5 plays an important role in the system operation. As a
result, estimation of its parameters is to be made with the highest precision and this
can be treated as a component with precise reliability. In each subsystem- i ,
1,2,3,4i = which is imprecise in nature, there is a parallel configuration consisting of
ix identical components having reliability ir . If ( )i iR x be the system reliability of
subsystem- i , 1,2,3,4i = then ( ) 1 (1 ) ix
i i iR x r= − − , 1,2,3,4i = . Let 5R be the reliability
of subsystem-5.
The system reliability of the bridge network system is given by the expression as
follows:
1 2 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5( , )SR x R R R Q R R Q R R R R Q Q R R Q R R Q R= + + + + ,
where 1i iR Q= − , 1,2,3,4i =
7.5 Solution Procedure
In this section we shall discuss the solution procedure for the problem mentioned in
earlier section. This problem is a non-linear mixed-integer optimization problem
with interval valued objective. Using Big-M penalty technique this problem is
converted into unconstrained optimization problem.
Studies on Reliability Optimization Problems by Genetic Algorithm 124
The form of Big-M penalty is as follows:
Maximize ˆ ˆ[ ( , ), ( , )] [ ( , ), ( , )] ( , )SL SR SL SRR x R R x R R x R R x R x Rθ= +
where [0,0] if ( , )
( , )[ ( , ), ( , )] [ , ] if ( , )SL SR
x R Sx R
R x R R x R M M x R Sθ
∈=
− + − − ∉
and { }{ }( , ) : Prob ( , ) 1 , 1, 2,..., and ,j j j jS x R C g x R b j m l x u L R Uγ′= ≤ ≥ − = ≤ ≤ ≤ ≤� �
where 1 2( , ,..., )ql l l l= , 1 2( , ,..., )qu u u u= , 1 2( , ,..., )qx x x x= , 1 2( , ,..., ),q q nL L L L+ +=
1 2( , ,..., )q q nU U U U+ += and 1 2( , ,..., )q q nR R R R+ += .
This problem can easily be solved by real coded genetic algorithm and interval order
relations.
7.6 Numerical Example
To study the performance of GA for solving stochastic reliability optimization
problem with interval valued reliabilities of components, a numerical example of the
five-link bridge network system under chance constraints has been considered for
different choices of jkc (consumption of j-th resource components) and
jb (availability of j-th resource). The proposed method is coded in C programming
language and run in a LINUX environment. The computational work has been done
on a PC with Intel Core-2 duo processor and 2.5 GHz. For each case, 20 independent
runs have been performed to calculate the best found system reliability, mean,
standard deviation of system reliability and average CPU time for different sets of
random numbers. In this computation, the values of genetic parameters like,
population size, mutation rate, crossover rate and maximum number of generations
have been taken as 200, 0.15, 0.85 and 150 respectively.
Reliability Optimization with Interval Parametric Values… 125
Figure 7.1: Bridge network system
Example 1 (Example on Bridge Network System)
The optimization problem considered here can be expressed as
1 1 2 2 2 2 3 3 4 4Maximize[ ( , ), ( , )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R R x R R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RR R Q Q Q Q R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R Q Q R R+
subject to
11 1 2 12 2 3 13 2 4 1 1
5
0.01Prob( 2exp ) 1
1c x x c x x c x x b
Rγ
+ + + ≤ ≥ −
−
21 1 22 2 23 3 24 4 2 2
5
0.01Prob( 5exp ) 1
1c x c x c x c x b
Rγ
+ + + + ≤ ≥ −
−
Studies on Reliability Optimization Problems by Genetic Algorithm 126
2 331 1 32 2 33 3 34 4 3 3
5
0.01Prob( ( 2) 0.6exp ) 1
1c x c x c x c x b
Rγ
+ − + + + ≤ ≥ −
−
51 6, being integer, 1,2,3, 4 and 0.50 0.99i ix x i R≤ ≤ = ≤ ≤
where [ ( ), ( )] 1 (1 [ , ]) , 1,2,3, 4 and 1 , 1,2,3,4,5.ix
i iL i iR i iL iR i iR R x R x r r i Q R i= = − − = = − =
1 [0.69,0.72]r = , 2 [0.83,0.86]r = , 3 [0.73,0.76]r = , 4 [0.79,0.81]r = .
To describe the stochastic variations in the resource coefficient we consider
(in sequences) the following six possibilities.
(i) All the random variables ( 1, 2,3; 1, 2,3, 4)jkc j k= = and ( 1, 2,3)jb j = have fixed
means and zero variances as follows:
11 ([1,1],[0,0])c N∼ , 12 ([2.2, 2.2],[0,0])c N∼ , 13 ([1.5,1.5],[0,0])c N∼ ,
1 ([28, 28],[0,0])b N∼ , 21 ([1,1],[0,0])c N∼ , 22 ([0.1,0.1],[0,0])c N∼ ,
23 ([2, 2],[0,0])c N∼ , 24 ([1,1],[0,0])c N∼ , 2 ([25, 25],[0,0])b N∼ ,
31 ([1,1],[0,0])c N∼ , 32 ([1,1],[0,0])c N∼ , 33 ([1.5,1.5],[0,0])c N∼ ,
34 ([1,1],[0,0])c N∼ , 3 ([21, 21],[0,0])b N∼ .
(ii) All the random variables ( 1, 2,3; 1, 2,3, 4)jkc j k= = and ( 1, 2,3)jb j = have interval
means and zero variances as follows:
11 ([0.9,1.1],[0,0])c N∼ , 12 ([2.1, 2.4],[0,0])c N∼ , 13 ([1.4,1.6],[0,0])c N∼ ,
1 ([26, 29],[0,0])b N∼ , 21 ([0.8,1.1],[0,0])c N∼ , 22 ([0.09,0.11],[0,0])c N∼ ,
23 ([1.9,2.1],[0,0])c N∼ , 24 ([0.8,1.2],[0,0])c N∼ , 2 ([23, 27],[0,0])b N∼ ,
31 ([0.85,1.15],[0,0])c N∼ , 32 ([0.9,1.1],[0,0])c N∼ , 33 ([1.4,1.6],[0,0])c N∼ ,
34 ([0.9,1.1],[0,0])c N∼ , 3 ([20,22],[0,0])b N∼ .
Reliability Optimization with Interval Parametric Values… 127
(iii) All the random variables ( 1, 2,3; 1, 2,3, 4)jkc j k= = and ( 1, 2,3)jb j = have interval
means and fixed variances as follows:
11 ([0.9,1.1],[0.1,0.1])c N∼ , 12 ([2.1,2.4],[0.1,0.1])c N∼ , 13 ([1.4,1.6],[0.1,0.1])c N∼ ,
1 ([26,29],[1,1])b N∼ , 1 0.05γ =
21 ([0.8,1.1],[0.1,0.1])c N∼ , 22 ([0.09,0.11],[0.01,0.01])c N∼ ,
23 ([1.9,2.1],[0.1,0.1])c N∼ , 24 ([0.8,1.2],[0.1,0.1])c N∼ , 2 ([23, 27],[1,1])b N∼ ,
2 0.05γ =
31 ([0.85,1.15],[0.1,0.1])c N∼ , 32 ([0.9,1.1],[0.1,0.1])c N∼ ,
33 ([1.4,1.6],[0.1,0.1])c N∼ , 34 ([0.9,1.1],[0.1,0.1])c N∼ , 3 ([20, 22],[1,1])b N∼ ,
3 0.05γ =
(iv) All the random variables ( 1, 2,3; 1,2,3,4)jkc j k= = and ( 1, 2,3)jb i = have fixed
means and interval variances as follows:
11 ([1,1],[0.1,0.2])c N∼ , 12 ([2.2,2.2],[0.1,0.15])c N∼ , 13 ([1.5,1.5],[0.1,0.3])c N∼ ,
1 ([28, 28],[1,2])b N∼ , 1 0.05γ =
21 ([1,1],[0.1,0.2])c N∼ , 22 ([0.1,0.1],[0.01,0.02])c N∼ , 23 ([2,2],[0.1,0.3])c N∼ ,
24 ([1,1],[0.1,0.25])c N∼ , 2 ([25,25],[1,1.9])b N∼ , 2 0.05γ =
31 ([1,1],[0.1,0.2])c N∼ , 32 ([1,1],[0.1,0.3])c N∼ , 33 ([1.5,1.5],[0.1,0.2])c N∼ ,
34 ([1,1],[0.1,0.15])c N∼ , 3 ([21,21],[1,2])b N∼ , 3 0.05γ =
(v) All the random variables ( 1, 2,3; 1,2,3,4)jkc j k= = and ( 1,2,3)jb j = have interval
means and interval variances as follows:
11 ([0.9,1.1],[0.1,0.2])c N∼ , 12 ([2.1, 2.4],[0.1,0.15])c N∼ ,
13 ([1.4,1.6],[0.1,0.3])c N∼ , 1 ([26,29],[1,2])b N∼ , 1 0.05γ =
Studies on Reliability Optimization Problems by Genetic Algorithm 128
21 ([0.8,1.1],[0.1,0.2])c N∼ ,
22 ([0.09,0.11],[0.01,0.02])c N∼ , 23 ([1.9, 2.1],[0.1,0.3])c N∼ ,
24 ([0.8,1.2],[0.1,0.25])c N∼ , 2 ([23,27],[1,1.9])b N∼ , 2 0.05γ =
31 ([0.85,1.15],[0.1,0.2])c N∼ , 32 ([0.9,1.1],[0.1,0.3])c N∼ ,
33 ([1.4,1.6],[0.1,0.2])c N∼ , 34 ([0.9,1.1],[0.1,0.15])c N∼ , 3 ([20,22],[1, 2])b N∼ ,
3 0.05γ =
(vi) All the random variables ( 1, 2,3; 1, 2,3, 4)jkc j k= = and ( 1, 2,3)jb j = have fixed
means and fixed variances as follows:
11 ([1,1],[.1,.1])c N∼ , 12 ([2.2, 2.2],[.1,.1])c N∼ , 13 ([1.5,1.5],[.1,.1])c N∼ ,
1 ([28, 28],[1,1])b N∼ , 1 0.05γ =
21 ([1,1],[.1,.1])c N∼ , 22 ([0.1,0.1],[.01,.01])c N∼ , 23 ([2, 2],[.1,.1])c N∼ ,
24 ([1,1],[.1,.1])c N∼ , 2 ([25, 25],[1,1])b N∼ , 2 0.05γ =
31 ([1,1],[.1,.1])c N∼ , 32 ([1,1],[.1,.1])c N∼ , 33 ([1.5,1.5],[.1,.1])c N∼ ,
34 ([1,1],[.1,.1])c N∼ , 3 ([21, 21],[1,1])b N∼ , 3 0.05γ =
Optimum solution sets, as obtained via genetic algorithm for the six possible
alternatives, are given in Table 7.1.
To perform a comparative study with earlier results we consider the following
optimization problem keeping the same parametric values of Sun, Mckinnon and Li
(2001). Similar study under the proposed setup is carried out in Table-7.2.
Using case-(i) the earlier mentioned optimization problem reduces as follows:
1 1 2 2 2 2 3 3 4 4Maximize[ ( , ), ( , )] [ , ][ , ] [ , ][ , ][ , ]SL SR L R L R L R L R L RR x R R x R R R R R Q Q R R R R= +
1 1 2 2 3 3 4 4[ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R R R+
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RR R Q Q Q Q R R R R+
Reliability Optimization with Interval Parametric Values… 129
1 1 2 2 3 3 4 4 5 5[ , ][ , ][ , ][ , ][ , ]L R L R L R L RQ Q R R R R Q Q R R+
subject to
1 2 2 3 2 4
5
0.012.2 1.5 2exp 28
1x x x x x x
R
+ + + ≤
−
1 2 3 4
5
0.010.1 2 5exp 25
1x x x x
R
+ + + + ≤
−
2 31 2 3 4
5
0.01( 2) 1.5 0.6exp 21
1x x x x
R
+ − + + + ≤
−
where 51 6 and are integers, 1,..., 4, 0.50 0.99ix i R≤ ≤ = ≤ ≤
and [ ( ), ( )] 1 (1 [ , ]) , 1, 2,3,4 and 1 , 1,2,3,4,5.ix
i iL i iR i iL iR i iR R x R x r r i Q R i= = − − = = − =
If we take 1 [0.70,0.70]r = , 2 [0.85,0.85]r = , 3 [0.75,0.75]r = , 4 [0.8,0.8]r = then
the problem is the same as presented in Sun, Mckinnon and Li (2001).
Table 7.1: Optimum solution sets of Example 1
Example Best found
system reliability 5( , )x R Mean system
reliability
Standard
deviation of
system
reliability
Average
CPU
time in
second
(i) [0.958412,0.997223] (2,3,1,2,0.9900) [0.958412,0.997223] [0.0,0.008678] 0.2220
(ii) [0.958412,0.997223] (2,3,1,2,0.9900) [0.958412,0.997223] [0.0,0.008678] 0.2210
(iii) [0.957211,0.997215] (2,3,1,1,0.9900) [0.957211,0.997215] [0.0,0.008945] 0.2370
(iv) [0.957211,0.997215] (2,3,1,1,0.9900) [0.957211,0.997215] [0.0,0.008945] 0.2420
(v) [0.946767,0.999800] (2,2,1,1,0.9900) [0.946767,0.999980] [0.0,0.011899] 0.2270
(vi) [0.946767,0.999800] (2,2,1,1,0.9900) [0.946767,0.999980] [0.0,0.011899] 0.2260
Table-7.2 presents comparative results obtained from the proposed method and
those reported in the earlier studies [Sun, Mckinnon and Li (2001)].
Studies on Reliability Optimization Problems by Genetic Algorithm 130
Table 7.2: Comparative results
5( , )x R SR CPU seconds
Sun, Mckinnon and Li (2001) (2,1,6,5,0.9396) 0.99992653 9.84
This works using case-(i) (2,1,6,5,0.9396) [0.999927,0.999927] 0.2590
From the above table, it is observed that the system reliability has been increased
and at the same time, the processing time has also been decreased.
7.7 Concluding Remarks
For the first time, we have examined the reliability optimization problem in the
stochastic domain with respect to available resources and in the interval domain
with respect to the component reliabilities and employed the genetic algorithm to
arrive at an optimum solution set. It has been observed from the computational
results, under taken herein for a five-link bridge network, that the mean system
reliability coincides with the best found system reliability. This strict coincidence is
due to the fact that each trial run provides us optimum solution in our study. Also,
the lower ends of the standard deviations, measured in interval form, assume zero
value. It may also be noted that the average CPU time required for implementing the
genetic algorithm, is also on the lower side. Further, optimization under stochastic
setup converges to optimization under deterministic setup, as expected.
CHAPTER 8
Multi-objective Reliability
Optimization in Interval
Environment
• Introduction
• Assumptions
• Multi-Objective Optimization In Interval Environment
• Global Criterion Method
• Tchebycheff Problem
• Weighted Tchebycheff Problem
• Lexicographic Ordering
• Lexicographic Problem
• Lexicographic Weighted Tchebycheff Problem
• Problem Formulation
• Solution Procedure
• Numerical Example
• Sensitivity Analysis
• Concluding Remarks
Studies on Reliability Optimization Problems by Genetic Algorithm 132
8.1 Introduction
Most of the real-world design or decision-making problems involving reliability
optimization require the simultaneous optimization of more than one objective
function. Mostly the researchers have framed the reliability optimization problem as
a single objective optimization problem. An early discussion in this field of multiple
objectives was reported by Sakawa (2002). He considered a multi-objective
formulation to maximize the system reliability and minimize the cost for reliability
allocation by using a surrogate worth trade-off method. To the best of our
knowledge, Inagaki, Inoue and Akashi (1978) first solved a multi-objective
optimization problem by maximizing the system reliability and minimizing the
system cost and weight by using an interactive optimization method. To have an
overview of the trend of research in this area, one may refer to the works of Park
(1987), Dhingra (1992), Rao and Dhingra (1992), Srinivas and Deb (1994), Ravi,
Reddy and Zimmermann (2000), Huang, Tian and Zuo (2005), Coit and Konak (2006)
and others. In the recent years, Taboada and Coit (2007) proposed a new method
based on the sequential combination of multi-objective evolutionary algorithms and
data clustering on the prospective solutions. Taboada, Baheranwala, Coit and
Wattanapongsakorn (2007) proposed two approaches to reduce the size of the
Pareto optimal set for multi-objective reliability optimization design problems. In the
first approach, they developed pseudo-ranking scheme to select the solutions by the
decision-maker according to his objective function priorities. In second approach,
they used data mining clustering techniques to group the data by using k-means
algorithm to find the clusters of similar solutions. Ramirez-Marquez and Coit (2007a)
proposed multi-state component critical analysis for the improvement of reliability
in multi-state systems. In the same area, Taboada, Espiritu and Coit (2008a)
Multi-objective Reliability Optimization in Interval Environment 133
presented an extension and application of earlier developed multi-objective
evolutionary algorithm for solving the design allocation problems of multi-state
series-parallel system for power system. Taboada, Espiritu and Coit (2008b) solved
multiple objective multi-state reliability optimization design problems by
maximizing the system reliability and minimizing both the system cost and weight.
In the year 2009, Li, Liao and Coit (2009) proposed a two-stage approach for multi-
objective decision-making with applications to system reliability optimization.
Ramirez-Marquez and Rocco (2010) developed a new evolutionary optimization
technique for multi-state two-terminal reliability allocation in multi-objective
problems. For identifying the combination of component failures that provide
maximum reduction of network performance, Rocco, Ramirez-Marquez, Salazar and
Hernandez (2010) presented the vulnerability analysis of a complex network. All
these multi-objective reliability optimization problems are based on the assumption
of fixed/constant reliabilities of components which lie between zero and one.
In the single objective optimization one attempts to obtain the best design or
decision, which is usually a global minimum or the global maximum depending on
whether the optimization problem is of minimization or maximization. On the other
hand for the multiple objectives, there may not exist one solution which is best
(global minimum or maximum) with respect to all the objectives. In multi-objective
optimization, there exists a set of solutions which are superior to the rest of the
solutions in the search space when all the objectives are considered, but are inferior
to other solutions in the space in one or more objectives. These solutions are known
as Pareto optimal solutions or non-dominated solutions [Srinivas and Deb (1994)]
and the rest of the solutions are known as dominated solutions. Since none of the
solutions in the non-dominated set is absolutely better than any other, any one of
Studies on Reliability Optimization Problems by Genetic Algorithm 134
them is an acceptable solution. As reliability of each component is interval valued the
system reliability would be interval valued. In this chapter, we have proposed GA-
based approaches for solving the multi-objective reliability optimization problem
with interval objectives. The objectives considered are the maximization of the
system reliability and minimization of the system cost. Here also, we have considered
the interval valued cost coefficients. For this purpose we have formulated several
problems for solving multi-objective reliability optimization problems with interval
valued objectives. In this connection, we have also proposed the definition of Pareto
optimality in interval environment. To obtain the optimal solution of multi-objective
optimization problem we have converted the same into a single objective
constrained optimization problem. Then, we have converted the reduced
optimization problem into unconstrained optimization problem by using penalty
function technique. For solving such typical problems, we have developed a real
coded elitist GA with tournament selection, intermediate crossover and one-
neighborhood mutation. Finally, to illustrate the different approaches based on
different multi-objective optimization techniques, a numerical example has been
solved and to investigate the overall performance of the proposed GA-based penalty
technique for solving multi-objective optimization problems, sensitivity analyses
have been carried out graphically.
8.2 Assumptions
The following assumptions have been used in the entire chapter.
(i) Reliability of each component is imprecise and interval valued.
(ii) Chances of failures of components are statistically independent.
(iii) The system will not be damaged or failed due to failed components.
(iv) Each redundancy is active and there is no provision for repair.
Multi-objective Reliability Optimization in Interval Environment 135
(v) The components as well as the system have two different states, viz. operating
state and failure state.
(vi) The cost coefficients are imprecise and interval valued.
8.3 Multi-Objective Optimization in Interval Environment
According to the existing literature, there are several methods developed for solving
the multi-objective optimization problem with non-interval valued objectives. Till
now, none has developed the techniques/ methods for solving multi-objective
optimization problems with interval valued objectives. In this section, we shall
discuss the solution methodologies/techniques for solving multi-objective
optimization problem with interval valued objectives for several decision variables.
These types of multi-objective optimization problems can be written as
Minimize 1 2{ ( ), ( ),..., ( )}kA x A x A x
subject to x S∈
where ( ) [ ( ), ( )], 1,2, ,i iL iRA x f x f x i k= = ⋅⋅⋅
and { : ( ) 0, 1, 2, , }jS x g x j m= ≤ = ⋅⋅⋅
Before we discuss about the solution methodologies of the optimization
problem, we propose to define the Pareto optimality (with respect to general
decision-makers’ point of view) and ideal objectives and different types of ideal
objective vectors for the above problem.
Definition: A decision vector *x S∈ is Pareto optimal if there does not exist another
decision vector x S∈ such that *min( ) ( )i iA x A x< for at least one index i .
i.e., for Type-1 and Type-2 intervals * *( ) ( ) ( ) ( )iL iR iL iRf x f x f x f x+ < +
and for Type-3 intervals,
Studies on Reliability Optimization Problems by Genetic Algorithm 136
either * * * *[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]iL iR iL iR iR iL iR iLf x f x f x f x f x f x f x f x+ ≤ + ∧ − < −
or* * *
[ ( ) ( ) ( ) ( )] [ ( ) ( )]iL iR iL iR iL iLf x f x f x f x f x f x+ ≤ + ∧ < .
Definition: Let X be a metric space. The (open) ball of radius 0δ > centered at a
point *x in X is defined as * *
( , ) { : ( , ) }B x x X d x xδ δ= ∈ < where d is the distance
function or metric. If the less than symbol ( )< is replaced by a less than or equal
to ( )≤ , the above definition becomes the same of a closed
ball: * *( , ) { : ( , ) }B x x X d x xδ δ= ∈ ≤ .
Definition: A decision vector *
x S∈ is locally Pareto optimal if there exists 0δ > such
that *
x is Pareto optimal in *
( , )S B x δ∩ where *
( , )B x δ is an open ball of radius 0δ >
centered at a point*
x .
Definition: A decision vector *x S∈ is weakly Pareto optimal if there does not exist
another decision vector x S∈ such that *min( ) ( ) for all 1,2, ,i iA x A x i k< = ⋅⋅⋅ .
Definition: An objective vector minimizing each of the objective functions is called
an ideal (or perfect) objective vector.
Definition: A utopian objective vector ** kz ∈� is an infeasible objective vector
whose components are formed by ** *i i iz z ε= − for all 1,2, ,i k= ⋅⋅⋅ , where *
iz is the
component of the ideal objective vector and 0iε > is a relatively small but
computationally significant scalar.
Definition: Let nX = � and suppose that 1 2{ , , , }nξ ξ ξ ξ= ⋅⋅⋅ and 1 2{ , , , }nη η η η= ⋅⋅⋅ be
any two points in n� . Define the mapping :
n
pd X X× → � and :n
d X X∞ × →� as
follows:
Multi-objective Reliability Optimization in Interval Environment 137
1
1
( , )
p pn
p i i
i
d ξ η ξ η=
= − ∑ where 1 p≤ < ∞
and { }1
( , ) max i i
i n
d ξ η ξ η∞≤ ≤
= −
Then ,pd d∞ are metrics on the same set nX = � .
Definition: Let pX l= , 1 p≤ < ∞ , be the set of all sequences { }iξ ξ= of real scalars
such that 1
p
i
i
ξ∞
=
< ∞∑ . Define the mapping :d X X× → � by
1
1
( , )
p pn
i i
i
d ξ η ξ η=
= − ∑
where { }iξ ξ= and { }iη η= are in pl .
It has been noted in the literature that pl is a metric space.
According to the existing literature there are several techniques for solving the multi-
objective optimization problems with non-interval valued objectives. In these
techniques, the multi-objective optimization problems have been formulated as
different types of problems. Some of these problems are as follows:
(i) Global criteria method
(ii) Tchebycheff problem
(iii) Weighted Tchebycheff problem
(iv) Lexicographic problem
(v) Lexicographic weighted Tchebycheff problem
Now, we shall formulate all these problems with interval valued objectives.
Global Criterion Method
In this method, the different steps are as follows:
Step 1: Solve the problem:
Maximize ( )iA x
Studies on Reliability Optimization Problems by Genetic Algorithm 138
subject to x S∈
and obtain the optimal value ( say,) * * *[ , ]i iL iRz z z= for 1, 2, ,i k= ⋅⋅⋅ .
Step 2: Using the above reference point and pd -metric used for measuring, we get
the following auxiliary problem:
Minimize
1
* *
1
[ ( ) , ( ) ]k pp
iL iR iR iL
i
f x z f x z
=
− −
∑ (8.1)
subject to x S∈
The exponent 1
p may be dropped. Problems with or without the exponent
1
p are
equivalent for1 p≤ < ∞ , since problem (8.1) is an increasing function of the
corresponding problem without exponent.
Tchebycheff Problem
When p → ∞ , the pd metric reduces to a Tchebycheff metric. The corresponding
d∞ problem (which is called Tchebycheff problem) with interval objective is of the
form:
Minimize ( )*
1,2, ,Max ( )i i
i k
A x z= ⋅⋅⋅
− (8.2)
subject to x S∈
Weighted Tchebycheff Problem
When p → ∞ and 0iw ≥ , the pd metric is called a Tchebycheff metric and the
corresponding problem (called Weighted Tchebycheff problem) with interval
objectives is of the form
Multi-objective Reliability Optimization in Interval Environment 139
Minimize ( )*
1,2, ,Max ( )i i i
i k
w A x z= ⋅⋅⋅
− (8.3)
subject to 1
1k
i
i
w
=
=∑
and x S∈
Lexicographic Ordering
In lexicographic ordering the decision-maker sorts the objective functions according
to their absolute importance. This means that the more important objective is
infinitely more important. After ordering, the most important objective function is
optimized subject to the given constraints. If this problem has a unique solution, it is
the solution of the whole multi-objective optimization problem. Otherwise, the
second most important objective function is to be optimized. If this problem has a
unique solution, it is the solution of the original problem and so on.
Lexicographic Problem
Let the objective functions be sorted according to the lexicographic order from the
most important to the less important. In this technique, the given multi-objective
optimization problem reduces to
lex minimize ( ( )iA x ) (8.4)
subject to x S∈
Lexicographic Weighted Tchebycheff Problem
If p → ∞ and 0iw ≥ , the pd metric is called a Tchebycheff metric and the
corresponding lexicographic weighted Tchebycheff problem is as follows:
Studies on Reliability Optimization Problems by Genetic Algorithm 140
lex Minimize ( ) ( )* **
1,2, ,1
Max ( ) , ( )k
i i i i ii k
i
w A x z A x z= ⋅⋅⋅
=
− −
∑ (8.5)
subject to 1
1k
i
i
w
=
=∑
and x S∈
where ** *i i iz z ε= − is the i-th component of utopian objective vector which is an
infeasible objective vector and iε , 1,2, ,i k= ⋅⋅⋅ be relatively small positive interval
numbers and computationally significant.
8.4 Problem Formulation
Let us consider a system consisting of n subsystems in series in which the i-th
(1 )i n≤ ≤ subsystem consists of ix components in parallel (see Figure 8.1) and the
reliability of each component as well as the cost of resources are interval valued.
Such a system is called series-parallel system or n -stage series system. In this
system, the system reliability and also the system cost would be interval valued.
Assuming all the components in i-th subsystem as identical, the system reliabilitySR
is given by
[ ] [ ]1
( ) ( ), ( ) ( ), ( )n
S SL SR iL iR
i
R x R x R x R x R x
=
= = ∏
where ( ) 1 (1 ) ix
iL iLR x r = − − and ( ) 1 (1 ) ix
iR iRR x r = − −
Hence our problem is to determine the number of redundant components ix ,
1,2,...,i n= by maximizing the system reliability [ ( ), ( )]SL SRR x R x and minimizing the
system cost[ ( ), ( )]SL SRC x C x , subject to the given constraints. Hence the problem can
be written as
Multi-objective Reliability Optimization in Interval Environment 141
Maximize [ ] [ ]1
( ), ( ) ( ), ( )n
SL SR iL iR
i
R x R x R x R x
=
= ∏ (8.6)
Minimize [ ( ), ( )]SL SRC x C x
subject to the constraints ( ) 0, 1, 2,...,jg x j m≤ =
Figure 8.1: A n-stage series system for MOOP
The above problem is a multi-objective optimization problem with interval valued
objectives.
8.4.1 Global Criteria Method
As the objective functions of (8.6) are interval valued, we have to modify the existing
methods for solving the said problem. In global criteria method for solving MOOPs
with fixed (non-interval) objective, the ideal objective vector is used as a reference
point. This vector is obtained by minimizing each of the objective functions
individually subject to the constraints. In problem (8.6), the objective functions are
interval valued. Hence, in this case, each component of ideal objective vector would
be interval valued. Hence, different steps of modified global criteria method are as
follows:
1 1 1 1
2 2 2 2
3 3 3 3
1x
2
x
i
x
n
x
Stage 1 2 i n
Studies on Reliability Optimization Problems by Genetic Algorithm 142
Step 1: Solve the problem:
Minimize [ ( ), ( )]SR SLR x R x− −
subject to ( ) 0, 1,2,...,kg x k m≤ =
and obtain the optimal value ( say,) * *[ , ]SL SRR R .
Step 2:
Minimize [ ( ), ( )]SL SRC x C x
subject to ( ) 0, 1,2,...,kg x k m≤ =
and find the optimal value ( say,) * *[ , ]SL SRC C .
Step 3: Form the ideal objective vector * * * *([ , ],[ , ])SL SR SL SRR R C C with the interval valued
components.
Step 4: Using the above reference point and pd -metric used for measuring.
The problem is then to solve the following auxiliary problem:
Minimize
1
* * * *[ ( ) , ( ) ] [ ( ) , ( ) ]
p p p
SR SR SL SL SL SR SR SLR x R R x R C x C C x C − − − − + − −
(8.7)
subject to ( ) 0, 1,2,...,kg x k m≤ =
The exponent 1
p may be dropped. Problems with or without the exponent
1
p are
equivalent for1 p≤ < ∞ .
8.4.2 Tchebycheff Problem
The Tchebycheff problem with interval objective corresponding to (8.6) is of the
form:
Minimize ( )* * * *Max [ ( ) , ( ) ] , [ ( ) , ( ) ]SR SR SL SL SL SR SR SLR x R R x R C x C C x C− − − − − − (8.8)
Multi-objective Reliability Optimization in Interval Environment 143
subject to ( ) 0, 1, 2,...,jg x j m≤ =
8.4.3 Weighted Tchebycheff Problem
The Weighted Tchebycheff problem with interval objectives is of the form:
Minimize
( )* * * *Max [ ( ) , ( ) ] , (1 ) [ ( ) , ( ) ]SR SR SL SL SL SR SR SLw R x R R x R w C x C C x C− − − − − − − (8.9)
subject to ( ) 0, 1, 2,...,jg x j m≤ =
8.4.4 Lexicographic Problem
Let the objective functions be arranged according to the lexicographic order from the
most important [ ( ), ( )]SR SLR x R x− − to the less important[ ( ), ( )]SL SRC x C x . In this
technique, the multi-objective optimization problem (8.6) reduces to
lex Minimize ([ ( ), ( )]SR SLR x R x− − ,[ ( ), ( )]SL SRC x C x ) (8.10)
subject to ( ) 0, 1,2,...,jg x j m≤ =
8.4.5 Lexicographic Weighted Tchebycheff Problem
The Lexicographic Weighted Tchebycheff problem is of the form:
( )* * * *lex Minimize Max [ ( ) , ( ) ] , (1 ) [ ( ) , ( ) ] ,SR SR SL SL SL SR SR SLw R x R R x R w C x C C x C− − − − − − −
( )** ** ** **[ ( ) , ( ) ] [ ( ) , ( ) ]SR SR SL SL SL SR SR SLR x R R x R C x C C x C− − − − + − −
subject to ( ) 0, 1,2,...,jg x j m≤ = . (8.11)
where ** ** ** **([ , ],[ , ])SL SR SL SRR R C C is the utopian objective vector which is an infeasible
objective vector. Hence this vector is equivalent to
* * * *1 1 2 2([ , ], [ , ])SL R SR L SL R SR LR R C Cε ε ε ε− − − − , where 1 1[ , ],L Rε ε 2 2[ , ]L Rε ε are
relatively small positive interval numbers but computationally significant scalars.
Studies on Reliability Optimization Problems by Genetic Algorithm 144
8.5 Solution Procedure
In this section we shall discuss the solution procedure for all the problems
mentioned in earlier section. These problems are non-linear integer optimization
problem with interval valued objectives. Using Big-M penalty technique these
problems are converted into unconstrained optimization problems. In this technique
any given constrained optimization problem with an interval valued fitness function
can be converted into an interval valued unconstrained optimization problem by
penalizing a large positive number say, M which can be written in the interval form
as [ , ]M M and called this penalty as Big-M penalty.
Let us consider a constrained optimization problem of the following form:
Maximize ( )[ , ]auxL auxRf f−
subject to the constraints
( ) 0, 1, 2,...,jg x j m≤ =
The form of Big-M penalty is as follows:
maximize ˆ ˆ[ , ]auxL auxRf f = [ , ] ( )auxL auxRf f xθ− + (8.12)
where [0,0] if
( )[ , ] [ , ] ifauxL auxR
x Sx
f f M M x Sθ
∈=
− ∉
and { }: ( ) 0, 1,2,...,jS x g x j m= ≤ = be the feasible space.
Here ( )[ , ]auxL auxRf f− is the interval valued auxiliary objective function. Problem
(8.12) is an integer non-linear unconstrained optimization problem with interval
objective of n integer variables 1 2, ,..., nx x x .
Multi-objective Reliability Optimization in Interval Environment 145
8.6 Numerical Example
To illustrate the different techniques for solving constrained multi-objective
optimization problem with interval valued reliabilities of components by genetic
algorithm, the following numerical example has been considered.
Example 1
Maximize 5
1
[ ( ), ( )] 1 [1 ,1 ] ix
SL SR iR iL
i
R x R x r r
=
= − − − ∏
Minimize [ ]5
1
[ ( ), ( )] [ , ] exp( 4)SL SR iL iR i i
i
C x C x C C x x
=
= +∑
subject to the constraints
52
1 1
1
( ) 0i i
i
g x P x b
=
= − ≤∑
5
2 2
1
( ) [ exp( 4)] 0i i i
i
g x W x x b
=
= − ≤∑
and ix being a non-negative integer for 1,2,3,4,5i = ; where the values of iP , iW ,
1b and 2b are given in Table 8.1. The proposed method/technique has been coded in
C programming language. The computational work has been done on a PC with Intel
Core-2-duo 2.5 GHz processor in LINUX environment. For each case 20 independent
runs have been performed to calculate the best found system reliability which is
nothing but the optimal value of the system reliability. In this computation, the
values of genetic parameters like, population size (p_size), mutation rate (p_mute),
crossover rate (p_cross) and maximum number of generations (max_gen) have been
taken as 100, 0.15, 0.85 and 150 respectively. The computational results have been
shown in Table 8.2.
Studies on Reliability Optimization Problems by Genetic Algorithm 146
Table 8.1: Shows the data for the Example 1
i 1 2 3 4 5
ir [0.78, 0.82] [0.84, 0.85] [0.87, 0.91] [0.63, 0.66] [0.74, 0.76]
iC [6, 8] [5, 8] [3, 6] [6, 9] [3, 6]
iP 1 2 3 4 2
iW 7 8 8 6 9
1 110b = , 2 200b =
From Table 8.2, the following observations can be made:
(i) the results obtained by Weighted Tchebycheff and Lexicographic Tchebycheff
problems be the same.
(ii) the best found value, mean value of system reliability *R and the corresponding
system cost *C in Lexicographic problem are higher than the same in other
problems.
(iii) all the results in Tchebycheff problems are greater than weighted Tchebycheff
and lexicographic weighted Tchebycheff problems.
Hence from the above observation, it can be concluded that the solution of
Lexicographic problem is the best solution. In this case, the best found value as well
as mean value of system reliability *R are far away from the same results obtained
from other problems. If a decision-maker is interested for a system with minimum
system cost, in that case, he/she may take the solution of either weighted
Tchebycheff or Lexicographic weighted Tchebycheff problem as these provide the
same solution.
Multi-objective Reliability Optimization in Interval Environment 147
Table 8.2: Computational results of Example 1
Problem 'x s Best *R Mean value of
*R
Best *C
Global Criterion (2,2,2,3,1) [0.6404, 0.6850] [0.6404, 0.6850] [88.6362, 140.0290]
Tchebycheff (2,1,2,1,3) [0.4864, 0.5310] [0.4864, 0.5310] [73.3138, 120.6125]
Weighted Tchebycheff (1,2,2,1,3) [0.4625, 0.5175] [0.4625, 0.5175] [71.9491, 120.6125]
Lexicographic (3,2,2,3,3) [0.8839, 0.9132] [0.8839, 0.9132] [105.9448, 168.7731]
Lexicographic
Weighted Tchebycheff
(1,2,2,1,3) [0.4625, 0.5175] [0.4625, 0.5175] [71.9491, 120.6125]
8.7 Sensitivity Analysis
To investigate the overall performance of the proposed GA-based penalty technique
for solving Lexicographic problem corresponding to multi-objective optimization
problems, sensitivity analyses have been carried out graphically on the system
reliability with respect to different GA parameters separately taking other
parameters at their original values. These have been shown in Figures 8.2- 8.4. From
Figure 8.2, it is observed that both the bounds of the system reliability be the same
for all the values of population size (p_size) greater than or equal to 60. This means
that our proposed GA is stable when population size exceeds 60. In Figures 8.3 and
8.4, the values of system reliability have been computed with respect to the
probability of crossover (p_cross) within the range from 0.55 to 0.95 and the
probability of mutation (p_mute) within the range 0.05 to 0.25 respectively. From
these figures, it is evident that the proposed GA is stable with respect to probability
of crossover as well as the probability of mutation.
Studies on Reliability Optimization Problems by Genetic Algorithm 148
0.8
0.84
0.88
0.92
0.96
1
50 60 70 80 90 100 110 120
Popsize
Sys
tem
Re
lia
bil
ity
Lower limit of interval
valued system reliability
Upper limit of interval
valued system reliability
Figure 8.2: P_size vs. interval valued system reliability for MOOP
0.6
0.7
0.8
0.9
1
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
Probability of Crossover
Sy
ste
m r
eli
ab
ilit
y
Lower limit ofinterval valued
system reliabilityUpper limit of
interval valuedsystem reliability
Figure 8.3: P_cross vs. interval valued system reliability for MOOP
0.5
0.6
0.7
0.8
0.9
1
0.05 0.10 0.15 0.20 0.25
Probability of Mutation
Sy
ste
m R
eli
ab
ilit
y
Lower limit of intervalvalued system reliability
Upper limit of intervalvalued system reliability
Figure 8.4: P_mute vs. interval valued system reliability for MOOP
Multi-objective Reliability Optimization in Interval Environment 149
8.8 Concluding Remarks
In this chapter, we have extended the idea of multi-objective optimization in interval
environment. For this purpose, we have proposed the definition of Pareto optimality
and formulated different problems (viz. Global criteria method, Tchebycheff
problem, Weighted Tchebycheff problem, lexicographic problem and lexicographic
Weighted Tchebycheff problem) in interval environment for solving multi-objective
optimization problem. These methodologies have been applied to solve constrained
multi-objective optimization problem by maximizing system reliability and
minimizing system cost under the assumption that the reliability of each component
as well as the cost coefficients are interval valued. Then the problem has been
reduced to single objective constrained optimization problem in different forms. The
reduced problem has been converted into single objective optimization problem
using Big-M penalty technique and solved by genetic algorithm and our newly
proposed interval order relations.
CHAPTER 9
General Conclusion and Scope
of Future Research
• General Conclusion
• Scope of Future Research
General Conclusions and Scope of Future Research 151
9.1 General Conclusion
In this thesis, for the first time, we have investigated different types of reliability
optimization problems in interval environment under the assumption that either the
reliability of each component is interval valued or distributional parameters and/or
resource parameters are interval valued. In Chapter 3, the problems of redundancy
allocation problems of series system, hierarchical series-parallel system, complicated
system and reliability network system with some resource constraints have been
solved. We have also formulated and solved two different redundancies known as
low-level redundancy and high-level redundancy addressed in Chapter 4. In Chapter
5, the reliability optimization problem with Weibull distributed (with interval valued
parameters) time-to-failure of each component of a complicated system with some
resource constraints have been solved. On the other hand chance constrained
reliability optimization problem of series system with some resource constraints
have been formulated and solved in Chapter 6. We have also examined the reliability
optimization problem in stochastic domain with respect to available resources. This
is addressed in Chapter 7. In Chapter 8, we have formulated and solved the
constrained multi-objective optimization problems with interval valued objectives.
In Chapter 5 and Chapter 8, the interval power of an interval number occurs
in the formulation of optimization problems. For this purpose, we have developed
the formula of interval power of an interval. In Chapter 8, we have developed the
definition of Pareto optimality for multi-objective optimization problems with
interval objective functions. In the whole work of the thesis, the reliability of each
component is considered as interval valued number. As a result, the objective
function is converted into interval valued objective. To solve those problems, the
interval order relations are very essential. For this purpose, we have proposed a new
Studies on Reliability Optimization Problems by Genetic Algorithm 152
definition of interval order relations discussed in Chapter 2 by rectifying the
drawbacks of the existing definitions proposed by Ishibuchi and Tanaka (1990),
Chanas and Kuchata (1996), Kundu (1997), Sengupta and Pal (2000) and Mahato and
Bhunia (2006).
In this thesis, we have introduced the definition of Weibull distribution and
Normal distribution with interval valued parameters to formulate the optimization
problems in Chapter 5 and 7 respectively.
9.2 Scope of Future Research
For further researches a lot of scope may arise from this thesis.
(i) The proposed techniques may be applied for solving real-life decision-making
problems in the form of interval valued constrained optimization problems,
interval valued multi-objective optimization problems, chance constrained
optimization problems arising in different fields of engineering, management,
manufacturing firms, etc.
(ii) In this thesis, we have solved all the optimization problems with the help of
genetic algorithm with interval valued fitness function. These problems can be
solved by other evolutionary algorithms/ hybrid algorithms.
(iii) In this thesis, we have used Big-M penalty and PFP penalty techniques to solve
the constrained optimization problems. Alternatively, by using the other
penalty techniques, one may solve the same problem.
(iv) The proposed approach of Chapter 5 opens up the scope for reliability
optimization when component reliabilities and the Weibull distribution with
interval valued parameters, estimated from sample observations, vary over
interval sets.
General Conclusions and Scope of Future Research 153
(v) In Chapter 8, we have formulated and solved multi-objective optimization
problems considering only two objectives, viz. system reliability and cost. One
may extend the problem for higher objectives, viz. system reliability, cost,
volume and weight. The same methodologies can be applied to solve the multi-
objective problem in the areas of manufacturing, scheduling, marketing,
assignment, transportation, inventory, etc.
(vi) For solving the problem in Chapter 8, we have formulated Tchebycheff,
Weighted Tchebycheff, Lexicographic, Lexicographic Weighted Tchebycheff
problems in interval environment and developed genetic algorithm with
different types of fitness function. In this connection, one may develop other
methods in interval environment to solve the same problems.
(vii) In interval as well as stochastic environments, there is a lot of scopes to work in
the area of multi-objective optimization problems.
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