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Instructor’sManualChapter891ManualtoaccompanyData,Models&Decisions:TheFundamentalsofManagementSciencebyBertsimasandFreund.Copyright2000,South-WesternCollegePublishing.PreparedbyManuelNunez,ChapmanUniversity.Chapter8IChapterOutline8.1.FormulatingaManagementProblemasaNonlinearOptimizationModelDiscussionofanoptimalportfoliomanagementproblemDiscussionofanoptimalproductionandpricingproblemDiscussionofafacilitylocationproblem8.2.GraphicalAnalysisofNonlinearOptimizationModelsinTwoVariablesFeasibleregionofanonlinearoptimizationproblemIsoquantlinesofanonlinearproblemLocalandglobaloptimalsolutions8.3.ComputerSolutionofNonlinearOptimizationProblems8.4.ShadowPricesInformationinNonlinearOptimizationProblemsLagrangemultipliersandtheirinterpretationasshadowprices8.5.ACloserLookatPortfolioOptimizationExtensionsofthebasicmodelEfficientfrontier8.6.TaxonomyoftheSolvabilityofNonlinearOptimizationProblemsConvexminimizationproblemsConcavemaximizationproblemsGeneralnonlinearproblemsIITeachingTips1.IfyouuseCase2asanassignment,youmaywanttoaddafewhintsconcerningtheformulationoftheconstraintsinthisnonlinearmodel.Manystudentsfindthiscaseparticularlyhardwithouthints.2.YoucancomplementthediscussionontheportfolioproblembyexplaininghowtoobtainthecorrelationmatrixorthecovariancematrixfromExcel,basedonobservationsofthebehaviorofdifferentstockswithrespecttotime.3.Aswithlinearoptimizationmodels,youmaywanttostresstheimportanceoflinearoptimizationmodelsbypresentingapracticalapplicationinindustry(otherthanthecasesinthischapter).Therearemanymanagementsciencejournalswithsuchexamplesandthestudentsalwaysgetapositivereinforcementinthissubjectafterhearingofasuccessstoryinlinearoptimization.Instructor’sManualChapter892ManualtoaccompanyData,Models&Decisions:TheFundamentalsofManagementSciencebyBertsimasandFreund.Copyright2000,South-WesternCollegePublishing.PreparedbyManuelNunez,ChapmanUniversity.IIIAnswerstoChapterExercises8.1(a)Anonlinearoptimizationmodelissimilartoalinearoptimizationmodelinthefollowing:BothhaveafeasibleregiondeterminedbyconstraintsBothhaveanobjectivefunctionthatshouldbemaximizedorminimizedBothcanbeformulatedasspreadsheetcomputermodels(b)Thechiefdifferencesbetweenanonlinearoptimizationproblemandalinearoptimizationproblemare:Thefeasibleregionofanonlinearoptimizationproblemisnotnecessarilydeterminedbylinearconstraints,andsoitisnotnecessarilyapolyhedronasinthelinearoptimizationcaseTheobjectivefunctionofanonlinearoptimizationmodelisnotnecessarilylinearGenerally,nonlinearoptimizationproblemsaremoredifficulttosolveonthecomputerThedecisionvariablesinanonlinearoptimizationproblemarenotnecessarilycontinuousInanonlinearoptimizationproblemtheremightbeseverallocalsolutionsinadditiontoglobalsolutions.Inalinearoptimizationproblemeverylocalsolutionisaglobalsolution(c)Thesedifferencesmakemoredifficulttosolveanonlinearoptimizationproblem.Forinstance,sinceanoptimalsolutionofanonlinearoptimizationproblemisnotnecessarilyatcornerpointsofthefeasibleregion,itismoredifficultforthecomputertosearchforthissolution(itiseasiertonarrowthesearchtocornerpoints).8.2Theplotoftheobjectivefunctionisshowninthefollowingchart(thescaleonthexaxisis1:1000):Thelocalminimaareatx=0.9455,x=6.1385,andx=10.Thevalueoftheglobalminimumisx=0.9455.Thelocalmaximaareatx=0,x=3.5795,andx=9.5125.Thevalueoftheglobalmaximumisx=9.5125.-200204060801000200040006000800010000xf(x)Instructor’sManualChapter893ManualtoaccompanyData,Models&Decisions:TheFundamentalsofManagementSciencebyBertsimasandFreund.Copyright2000,South-WesternCollegePublishing.PreparedbyManuelNunez,ChapmanUniversity.8.3Wesolvethefollowingnonlinearoptimizationproblem:Maximize4(1–0.75t)+1/(1+t).Subjectto:Non-negativity:t=0,tinteger.Thegraphoftheobjectivefunctionisshownbelow.UsingSolver,wefoundthattheoptimalsolutionisattainedatt=22yearswithanoptimalobjectivevalueof$4.03millions.8.4Theplotoftheobjectivefunctionisshowninthefollowingchart.Thelocalminimaareatx=0,x=4.255,andx=8.545.Thevalueoftheglobalminimumisx=0.Thelocalmaximaareatx=1.215,x=7.605,andx=10.Thevalueoftheglobalmaximumisx=10.Revenue+SalvageValue00.511.522.533.544.50102030Time$millions0510152025300246810xf(x)Instructor’sManualChapter894ManualtoaccompanyData,Models&Decisions:TheFundamentalsofManagementSciencebyBertsimasandFreund.Copyright2000,South-WesternCollegePublishing.PreparedbyManuelNunez,ChapmanUniversity.8.5(a)IfPF=270,thenDF=490–270=220.IfPC=230,thenDC=640–2(230)=180.Therefore,thedemandsagreewiththelinearmodelfromthepreviouschapter.(b)Theoptimalstrategyistoproduce151.8full-sizemicrowaveovensand165.5compact-sizemicrowaveovens,settingtheunitpricesat$338.18and$237.27,respectively.Theoptimalcontributiontoearningsis$51,300,whichisgreaterthantheoptimalcontributionforthelinearmodel($40,500).IwouldrecommendMagnetrontosetthepricesatthislevelandtostudytheresultstodeterminehowaccurateisthemodelorifthemodelneedstoberevised.8.6(a)LetPandMbethedecisionvariablesdefinedinthiscase.Weobtainthefollowingnonlinearoptimizationmodel:Maximize(506–0.75P)P+(653.5–M)M.Subjectto