Concise Guide to Optimization Models and Methods: A Problem-Based Test Prep for Students PDF
Preview Concise Guide to Optimization Models and Methods: A Problem-Based Test Prep for Students
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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similarordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 v 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 ix x Contents 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. xi
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