Concise Guide to Optimization Models and Methods: A Problem-Based Test Prep for Students

132 Pages·2021·16.103 MB·English

by  Xian Wen Ng

Xian Wen Ng
Concise Guide
to Optimization
Models and
Methods
A Problem-Based Test Prep for Students
Concise Guide to Optimization Models and Methods
Xian Wen Ng
Concise Guide
to Optimization Models
and Methods
A Problem-Based Test Prep for Students
XianWenNg
Singapore,Singapore
ISBN978-3-030-84416-5 ISBN978-3-030-84417-2 (eBook)
https://doi.org/10.1007/978-3-030-84417-2
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Preface
Optimizationinvolvesfindingthebestsolutiontoasystemorprocessthatissubject
to constraints. It is highly valued in numerous fields and professions, from the
scientist and engineer to the market analyst and regular business owner, since
most real-life scenarios invariably require one to make the best decisions that
would achieve desired outcomes, while keeping within any imposed limitations
andconstraints.
Withanincreasinglyunpredictablegloballandscapewithever-evolvingenviron-
mentalconditionsandotherconstraints,theabilitytodevelopandapplysystematic
methodsthatefficientlyoptimizecomplexproblemshasbecomeevermorerelevant
inordertorespondandadaptwelltochangingcircumstances.
Inlightofmodern-dayissuessuchasescalatingenergycostsandtighterresource
availabilities,increasinglystringentenvironmentalregulations,andstiffeningcom-
petitioninproductqualityandpricing,masteringoptimizationtechniqueshaseasily
becomeone ofthe most importantskillsetsofthefuture.The optimizationprocess
quickly turns challenging with a large number of variables and dimensions to deal
with,andsosystematicmethodstohelponeformulatemodelsthatwouldefficiently
and accurately lead to optimized solutions hold the key to breaking down and
solvingsomeoftheseeminglymostonerousandconvolutedproblems.
This book is targeted at beginning students taking an introductory course in
optimization, or entry-level practising engineers looking to hone their problem-
solving skills and practise building optimization models. While this book focuses
largely on problems with static constraints, it is worth noting that the subject of
optimalcontrolextendswidelyintooptimizationsofperformanceindicessubjectto
a range of dynamic constraints, which are typical of real-world physical systems
withdynamiccontrolproblems.
Written in problem-solution format, this book will be useful as supplementary
reference to mainstream textbooks for students, as it serves as a guide in handling
challenging problems commonly encounteredin test and examinations. With com-
prehensive worked solutions and detailed explanations provided for each problem,
studentswillbeabletofollowthethoughtprocessofproblem-solvingfromstartto
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vi Preface
finish, thereby hone their skills in applying abstract theoretical concepts to solving
practical problems, a critical step to acing examinations. The mix of academic and
real-world problems presented in this book will also help students tackling term
projectsorgraduate-leveloptimizationcourses.
The balance of academic and practical examples in this book will help students
develop skills in building optimization models and formulating solutions to linear,
non-linearandconvexprogrammingproblems.Studentswillbecomeproficient,not
only in tackling tests and examinations, but also in relating the significance of
desktopproblemstoalargerreal-worldcontext.
Singapore,Singapore XianWenNg
Acknowledgements
MyheartfeltgratitudegoestotheteamatSpringerfortheirunrelentingsupportand
professionalismthroughoutthepublicationprocess.SpecialthankstoMichaelLuby
andBrianHalmforyourconstanteffortandattentiontowardsmakingthispublica-
tionpossible.Iamalsodeeplyappreciativeofthereviewersformymanuscriptwho
had provided excellent feedback and numerous enlightening suggestions to help
improvethebook’scontents.
Finally,Iwishtothankmylovedoneswhohave,asalways,offeredonlypatience
andunderstandingthroughouttheprocessofmakingthisbookareality.
vii
Contents
BasicConcepts,LagrangianMethodsandLinearProgramming
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Problem1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Solution1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Problem2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Solution2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Problem3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Solution3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Problem4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Solution4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Problem5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Solution5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Problem6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Solution6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Non-linearProgrammingProblemswithConstraintsandEuler’s
Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Problem7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Solution7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Problem8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Solution8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Problem9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Solution9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Problem10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Solution10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
ComplexOptimizationProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Problem11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Solution11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Problem12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Solution12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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Problem13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Solution13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Problem14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Solution14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Problem15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Solution15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Problem16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Solution16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
About the Author
X. W. Ng graduated with First-Class Honours from the University of Cambridge,
UK, with a master’s degree in chemical engineering and Bachelor of Arts in 2011
andwassubsequentlyconferredaMasterofArtsin2014.Shewasrankedsecondin
her graduating class and was the recipient of a series of college scholarships,
including the Samuel Taylor Marshall Memorial Scholarship, Thomas Ireland
Scholarship and British Petroleum Prize in Chemical Engineering, for top perfor-
manceinconsecutiveyearsofacademicexaminations.Shewasalsooneofthetwo
students from Cambridge University selected for the Cambridge-Massachusetts
Institute of Technology (MIT) exchange programme in chemical engineering,
which she completed with honours. During her time at MIT, she was also a part-
time tutor for junior classesin engineering and pursued other disciplines including
economics, real estate development and finance at MIT and the John F. Kennedy
School of Government, Harvard University. Upon graduation, she was elected by
hercollegefellowshiptothetitleofscholar,asamarkofheracademicdistinction.
Since graduation, she has been keenly involved in teaching across various
academic levels. Her area of specialization includes mathematics, science and
engineering topics. Some of her recent works include Engineering Problems for
UndergraduateStudentsandPocketGuidetoRheology,bothofwhichwerewritten
inasimilarproblem-basedformat,specificallyaimedatstudentstakingengineering
and related STEM courses at higher education and university levels. These books
aimtosharpenstudents’problem-solvingskillsandputthemingoodsteadfortests
andexaminations.
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