studies on reliability optimization problems by genetic...

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


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:


GOLAPBAG, BURDWAN – 713 104

WEST BENGAL, INDIA

Professor Dilip Roy

E-mail: [email protected]

Dr Asoke Kumar Bhunia

Associate Professor

E-mail: [email protected]

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


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


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


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 3.6: 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 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.


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.


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.


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


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.


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


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


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


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


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


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


[ , ]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

=

≥=

≤

≤ ≤ >


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= = .


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 ∈� .


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


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


RbLb

B

RaLa

A

RaLa

A

RbLb

B

A

RbLb

B

RaLa

A

Rb

B

Lb RaLa



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


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


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


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> ⇔ ≥ ∧ ≠ .


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


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:


(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


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.


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.


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).


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.


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


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= .


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.


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:


(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


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= .


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 .


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.


(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


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


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


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


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.


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=


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


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−


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:


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≤ =


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


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

−

==

= −

∑∏



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≤ ≤ ≤ =


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= .


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.


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= ≤ = ≤ ≤ ≤ ≤



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= − + ≤ = ≤ ≤ ≤ ≤



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

∈=

− + ∉


and { }*( , ) : ( , ) 0, 1,2..., and ,SLS x R R x R R i m l x u L R U= − + ≤ = ≤ ≤ ≤ ≤



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


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.


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.


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




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( ) {[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)



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 = − −


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 = .






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 [0.70,0.70]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


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≤ ≤ = ≤ ≤


and ( , )SR x R , ( 1,2,3)iC i = are defined in Example 5


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


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 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= = − − = = − =


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 = .



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× + +



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


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 = .



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.


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 10.



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



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


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


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


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





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= .


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= ⋅⋅⋅



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


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


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= ⋅⋅⋅


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)


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= ≤ = ⋅⋅⋅ ≤ ≤ ≤ =


Minimize ( ) [ ( ), ( )]S SL SRC x C x C x=



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= ≥ ≤ ≤ ≤ =


Maximize ( ) [ ( ), ( )]S SL SRR h R h R h=

subject to ( ) ,i ig h b≤ 1,2, ,i m= ⋅⋅⋅

, integerl h u h≤ ≤


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= ≤ = ⋅⋅⋅ ≤ ≤ ≤


Minimize ( ) [ ( ), ( )]S SL SRC h C h C h=




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.


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


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 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

=

= − ≤∑


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

=

= +

∑


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= +




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


subject to


[ ]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

=

= + ∑


where 1 2 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5( ) 1 (1 ( ))h


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


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.



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


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


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


Figure 4.6: Max_gen vs. interval valued system reliability for Example 1


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


reliability

Figure 4.8: P_mute vs. interval valued system reliability for Example 1


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


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


• Numerical Example



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


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.


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


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≤ ≤ > .


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.


Maximize ˆ ˆ[ ( , ), ( , )] [ ( , ), ( , )] ( , )SL SR SL SRR x t R x t R x t R x t x tθ= +


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






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


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.


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


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







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.


6.2 Assumptions


(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.


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


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


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


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.



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.


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


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


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


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


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


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


reliability

Figure 6.5: Max_gen vs. interval valued system reliability for series system



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




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


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.


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


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

=

�


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.


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.


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 =


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.



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+ +=


This problem can easily be solved by real coded genetic algorithm and interval order

relations.


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.


Figure 7.1: Bridge network system

Example 1 (Example on Bridge Network System)

The optimization problem considered here can be expressed as





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

+ + + + ≤ ≥ −

−


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∼ .


(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 =


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:






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)].


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.


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







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


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.


(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.


(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,


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:


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


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


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:


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.


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


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


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)


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.




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−


( ) 0, 1, 2,...,jg x j m≤ =


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 .



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

=

= +∑


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.


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.


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]


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.


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


Upper limit of interval


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



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


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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