数据、模型与决策（第12版）全套课件 第1-16章管理科学 -库存管理

ManagementScienceChapter1ChapterTopicsTheManagementScienceApproachtoProblemSolvingManagementScienceandBusinessAnalyticsModelBuilding:Break-EvenAnalysisComputerSolutionManagementScienceModelingTechniquesBusinessUsageofManagementScienceTechniquesManagementScienceModelsinDecisionSupportSystemsTheManagementScienceApproachManagementscienceisascientificapproachtosolvingmanagementproblems.Itisusedinavarietyoforganizationstosolvemanydifferenttypesofproblems.Itencompassesalogicalmathematicalapproachtoproblemsolving.Managementscience,alsoknownasoperationsresearch,quantitativemethods,businessanalytics,etc.,involvesaphilosophyofproblemsolvinginalogicalmanner.TheManagementScienceProcessFigure1.1ThemanagementscienceprocessStepsintheManagementScienceProcessObservation-Identificationofaproblemthatexists(ormayoccursoon)inasystemororganization.ProblemDefinition-Theproblemmustbeclearlyandconsistentlydefined,showingitsboundariesandinteractionswiththeobjectivesoftheorganization.ModelConstruction-Developmentofthefunctionalmathematicalrelationshipsthatdescribethedecisionvariables,objectivefunctionandconstraintsoftheproblem.ModelSolution-Modelssolvedusingmanagementsciencetechniques.ModelImplementation-Actualuseofthemodeloritssolution.InformationandData:BusinessfirmmakesandsellsasteelproductProductcosts$5toproduceProductsellsfor$20Productrequires4poundsofsteeltomakeFirmhas100poundsofsteelBusinessProblem:Determinethenumberofunitstoproducetomakethemostprofit,giventhelimitedamountofsteelavailable.ExampleofModelConstruction(1of3)Variables: x=#unitstoproduce(decisionvariable) Z=totalprofit(in$)Model: Z=$20x-$5x(objectivefunction) 4x=100lbofsteel(resourceconstraint)Parameters: $20,$5,4lbs,100lbs(knownvalues)FormalSpecificationofModel:

maximizeZ=$20x-$5x subjectto4x=100ExampleofModelConstruction(2of3)ExampleofModelConstruction(3of3)Solvetheconstraintequation: 4x=100 (4x)/4=(100)/4

x=25unitsSubstitutethisvalueintotheprofitfunction: Z=$20x-$5x=(20)(25)–(5)(25) =$375(Produce25units,toyieldaprofitof$375)ModelSolution:ManagementScience

andBusinessAnalyticsBusinessanalyticsuseslargeamountsofdatawithmanagementsciencetechniquestohelpmanagersmakedecisionsBringstogetherinformationtechnology,statistics,managementscience,andmathematicalmodelingBigdataModelBuilding:Break-EvenAnalysis(1of9)Usedtodeterminethenumberofunitsofaproducttosellorproducethatwillequatetotalrevenuewithtotalcost.Thevolumeatwhichtotalrevenueequalstotalcostiscalledthebreak-evenpoint.Profitatbreak-evenpointiszero.ModelComponentsFixedCost(cf)-coststhatremainconstantregardlessofnumberofunitsproduced.VariableCost(cv)-unitproductioncostofproduct.Volume(v)–thenumberofunitsproducedorsoldTotalvariablecost(vcv)-functionofvolume(v)andunitvariablecost.ModelBuilding:

Break-EvenAnalysis(2of9)ModelComponentsTotalCost(TC)-totalfixedcostplustotalvariablecost.Profit(Z)-differencebetweentotalrevenuevp(p=unitprice)andtotalcost,i.e.ModelBuilding:

Break-EvenAnalysis(3of9)ModelBuilding:

Break-EvenAnalysis(4of9)ComputingtheBreak-EvenPoint Thebreak-evenpointisthatvolumeatwhichtotalrevenueequalstotalcostandprofitiszero:

Thebreak-evenpointModelBuilding:Break-EvenAnalysis(5of9)Example:

WesternClothingCompany

FixedCosts:cf=$10000 VariableCosts:cv=$8perpair Price:p=$23perpair

TheBreak-EvenPointis: v=(10,000)/(23-8) =666.7pairsofjeansModelBuilding:Break-EvenAnalysis(6of9)Figure1.2Break-evenmodelModelBuilding:Break-EvenAnalysis(7of9)Figure1.3Break-evenmodelwithanincreaseinpriceModelBuilding:Break-EvenAnalysis(8of9)Figure1.4Break-evenmodelwithanincreaseinvariablecostModelBuilding:Break-EvenAnalysis(9of9)Figure1.5Break-evenmodelwithachangeinfixedcostBreak-EvenAnalysis:ExcelSolution(1of4)Exhibit1.1Formulaforv,break-evenpoint,=D4/(D8-D6)Break-EvenAnalysis:ExcelQMSolution(2of4)

EntermodelparametersincellsB10:B13Clickon“ExcelQM,”thenonAlphabetical”listofmodelsandselect“BreakevenAnalysis”Exhibit1.2Exhibit1.3WesternClothingCompanyinQMBreak-EvenAnalysis:ExcelQMSolution(3of4)

Break-EvenAnalysis:QMSolution(4of4)Exhibit1.4QMbreak-evengraphforWesternClothingCompanyFigure1.6Classificationofmanagementsciencetechniques

ClassificationofManagementScienceTechniquesLinearMathematicalProgramming-

clearobjective;restrictionsonresourcesandrequirements;parametersknownwithcertainty.(Chap2-6,9)ProbabilisticTechniques-

resultscontainuncertainty.(Chap11-13)NetworkTechniques-modeloftenformulatedasdiagram;deterministicorprobabilistic.(Chap7-8)OtherTechniques-varietyofdeterministicandprobabilisticmethodsforspecifictypesofproblemsincludingforecasting,inventory,simulation,multicriteria,AHP(analytichierarchyprocess),etc.(Chap9,14-16)CharacteristicsofModelingTechniquesSomeapplicationareas: -ProjectPlanning -CapitalBudgeting -InventoryAnalysis -ProductionPlanning -SchedulingInterfaces-ApplicationsjournalpublishedbyInstituteforOperationsResearchandManagementSciences(INFORMS)BusinessUsageofManagementScience Adecisionsupportsystemisacomputer-basedsystemthathelpsdecisionmakersaddresscomplexproblemsthatcutacrossdifferentpartsofanorganizationandoperations.FeaturesofDecisionSupportSystemsInteractiveUsesdatabases&managementsciencemodelsAddress“whatif”questionsPerformsensitivityanalysisExamplesinclude:ERP–EnterpriseResourcePlanningOLAP–OnlineAnalyticalProcessingManagementScienceModelsinDecisionSupportSystems(DSS)Figure1.7Adecisionsupportsystem

ManagementScienceModelsDecisionSupportSystems(2of2)Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher.PrintedintheUnitedStatesofAmerica.LinearProgramming:ModelFormulationandGraphicalSolutionChapter2ChapterTopicsModelFormulationAMaximizationModelExampleGraphicalSolutionsofLinearProgrammingModelsAMinimizationModelExampleIrregularTypesofLinearProgrammingModelsCharacteristicsofLinearProgrammingProblemsObjectivesofbusinessdecisionsfrequentlyinvolvemaximizingprofitorminimizingcosts.Linearprogramminguseslinearalgebraicrelationshipstorepresentafirm’sdecisions,givenabusinessobjective,andresourceconstraints.Stepsinapplication:Identifyproblemassolvablebylinearprogramming.Formulateamathematicalmodeloftheunstructuredproblem.Solvethemodel.ImplementationLinearProgramming:AnOverviewDecisionvariables-mathematicalsymbolsrepresentinglevelsofactivitybythefirm.Objectivefunction-alinearmathematicalrelationshipdescribinganobjectiveofthefirm,intermsofdecisionvariables-thisfunctionistobemaximizedorminimized.Constraints–requirementsorrestrictionsplacedonthefirmbytheoperatingenvironment,statedinlinearrelationshipsofthedecisionvariables.Parameters-numericalcoefficientsandconstantsusedintheobjectivefunctionandconstraints.ModelComponentsSummaryofModelFormulationStepsStep1:DefinethedecisionvariablesStep2:Definetheobjectivefunction

Step3:Definetheconstraints

LPModelFormulationAMaximizationExample(1of3)Productmixproblem-BeaverCreekPotteryCompanyHowmanybowlsandmugsshouldbeproducedtomaximizeprofitsgivenlaborandmaterialsconstraints?Productresourcerequirementsandunitprofit:ResourceRequirementsProductLabor(Hr./Unit)Clay(Lb./Unit)Profit($/Unit)Bowl1440Mug2350Figure2.6BeaverCreekPotteryCompanyLPModelFormulationAMaximizationExample(2of3)Resource 40hrsoflaborperdayAvailability: 120lbsofclayDecision x1=numberofbowlstoproduceperdayVariables: x2=numberofmugstoproduceperdayObjective MaximizeZ=$40x1+$50x2Function: WhereZ=profitperdayResource 1x1+2x2

40hoursoflaborConstraints: 4x1+3x2

120poundsofclayNon-Negativity x10;x20Constraints: LPModelFormulationAMaximizationExample(3of3)CompleteLinearProgrammingModel:Maximize Z=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0Afeasiblesolutiondoesnotviolateanyoftheconstraints:

Example: x1=5bowls x2=10mugs Z=$40x1+$50x2=$700

Laborconstraintcheck: 1(5)+2(10)=25≤40hoursClayconstraintcheck: 4(5)+3(10)=70≤120poundsFeasibleSolutions Aninfeasiblesolutionviolatesatleastoneoftheconstraints:

Example: x1=10bowls x2=20mugs

Z=$40x1+$50x2=$1400Laborconstraintcheck: 1(10)+2(20)=50>40hoursInfeasibleSolutionsGraphicalsolutionislimitedtolinearprogrammingmodelscontainingonlytwodecisionvariables(canbeusedwiththreevariablesbutonlywithgreatdifficulty).Graphicalmethodsprovidevisualizationofhowasolutionforalinearprogrammingproblemisobtained.GraphicalSolutionofLPModelsCoordinateAxesGraphicalSolutionofMaximizationModel(1of12)Figure2.2CoordinatesforgraphicalanalysisMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0X1isbowlsX2ismugsLaborConstraintGraphicalSolutionofMaximizationModel(2of12)Figure2.3GraphoflaborconstraintMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0LaborConstraintAreaGraphicalSolutionofMaximizationModel(3of12)Figure2.4LaborconstraintareaMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0ClayConstraintAreaGraphicalSolutionofMaximizationModel(4of12)Figure2.5The

constraintareaforclayMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0BothConstraintsGraphicalSolutionofMaximizationModel(5of12)Figure2.6GraphofbothmodelconstraintsMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0FeasibleSolutionAreaGraphicalSolutionofMaximizationModel(6of12)Figure2.7ThefeasiblesolutionareaconstraintsMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0ObjectiveFunctionSolution=$800GraphicalSolutionofMaximizationModel(7of12)Figure2.8ObjectivefunctionlineforZ=$800MaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0AlternativeObjectiveFunctionSolutionLinesGraphicalSolutionofMaximizationModel(8of12)Figure2.9Alternativeobjectivefunctionlinesforprofits,Z,of$800,$1,200,and$1,600MaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0OptimalSolutionGraphicalSolutionofMaximizationModel(9of12)Figure2.10IdentificationofoptimalsolutionpointMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0OptimalSolutionCoordinatesGraphicalSolutionofMaximizationModel(10of12)Figure2.11OptimalsolutioncoordinatesMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0Extreme(Corner)PointSolutionsGraphicalSolutionofMaximizationModel(11of12)Figure2.12SolutionsatallcornerpointsMaximizeZ=$40x1+$50x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0OptimalSolutionforNewObjectiveFunctionGraphicalSolutionofMaximizationModel(12of12)MaximizeZ=$70x1+$20x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0Figure2.13OptimalsolutionwithZ=70x1+20x2Standardformrequiresthatallconstraintsbeintheformofequations(equalities).Aslackvariableisaddedtoa

constraint(weakinequality)toconvertittoanequation(=).Aslackvariabletypicallyrepresentsanunusedresource.Aslackvariablecontributesnothingtotheobjectivefunctionvalue.SlackVariablesLinearProgrammingModel:StandardFormMaxZ=40x1+50x2+s1+s2subjectto:1x1+2x2+s1=404x2+3x2+s2=120x1,x2,s1,s2

0Where:x1=numberofbowlsx2=numberofmugss1,s2areslackvariablesFigure2.14SolutionsatpointsA,B,andCwithslackLPModelFormulation–Minimization(1of7)Twobrandsoffertilizeravailable-Super-gro,Crop-quick.Fieldrequiresatleast16poundsofnitrogenand24poundsofphosphate.Super-grocosts$6perbag,Crop-quick$3perbag.Problem:Howmuchofeachbrandtopurchasetominimizetotalcostoffertilizergivenfollowingdata?Figure2.15Fertilizingfarmer’sfieldDecisionVariables: x1=bagsofSuper-gro x2=bagsofCrop-quickTheObjectiveFunction: MinimizeZ=$6x1+3x2 Where: $6x1=costofbagsofSuper-Gro $3x2=costofbagsofCrop-QuickModelConstraints: 2x1+4x2

16lb(nitrogenconstraint) 4x1+3x2

24lb(phosphateconstraint) x1,x2

0(non-negativityconstraint)LPModelFormulation–Minimization(2of7)MinimizeZ=$6x1+$3x2subjectto: 2x1+4x2

16 4x2+3x2

24 x1,x2

0Figure2.16ConstraintlinesforfertilizermodelConstraintGraph–Minimization(3of7)Figure2.17FeasiblesolutionareaFeasibleRegion–Minimization(4of7)MinimizeZ=$6x1+$3x2subjectto: 2x1+4x2

16 4x2+3x2

24 x1,x2

0Figure2.18TheoptimalsolutionpointOptimalSolutionPoint–Minimization(5of7)MinimizeZ=$6x1+$3x2subjectto: 2x1+4x2

16 4x2+3x2

24 x1,x2

0Theoptimalsolutionofaminimizationproblemisattheextremepointclosesttotheorigin.Asurplusvariableissubtractedfroma

constrainttoconvertittoanequation(=).Asurplusvariablerepresentsanexcessaboveaconstraintrequirementlevel.Asurplusvariablecontributesnothingtothecalculatedvalueoftheobjectivefunction.Subtractingsurplusvariablesinthefarmerproblemconstraints:2x1+4x2-s1=16(nitrogen) 4x1+3x2-s2=24(phosphate)SurplusVariables–Minimization(6of7)Figure2.19GraphofthefertilizerexampleGraphicalSolutions–Minimization(7of7)MinimizeZ=$6x1+$3x2+0s1+0s2subjectto: 2x1+4x2–s1

=16 4x2+3x2–s2=24 x1,x2,s1,s2

0 Forsomelinearprogrammingmodels,thegeneralrulesdonotapply.Specialtypesofproblemsincludethosewith:MultipleoptimalsolutionsInfeasiblesolutionsUnboundedsolutionsIrregularTypesofLinearProgrammingProblemsFigure2.20GraphofBeaverCreekPotteryexamplewithmultipleoptimalsolutionsMultipleOptimalSolutionsBeaverCreekPotteryTheobjectivefunctionisparalleltoaconstraintline.MaximizeZ=$40x1+30x2subjectto: 1x1+2x2

40 4x2+3x2

120 x1,x2

0Where:x1=numberofbowlsx2=numberofmugsAnInfeasibleProblemFigure2.21GraphofaninfeasibleproblemEverypossiblesolutionviolatesatleastoneconstraint:MaximizeZ=5x1+3x2subjectto: 4x1+2x2

8 x1

4 x2

6 x1,x2

0AnUnboundedProblemFigure2.22GraphofanunboundedproblemValueoftheobjectivefunctionincreasesindefinitely:MaximizeZ=4x1+2x2subjectto:x1

4x2

2x1,x2

0CharacteristicsofLinearProgrammingProblemsAdecisionamongalternativecoursesofactionisrequired.Thedecisionisrepresentedinthemodelbydecisionvariables.Theproblemencompassesagoal,expressedasanobjectivefunction,thatthedecisionmakerwantstoachieve.Restrictions(representedbyconstraints)existthatlimittheextentofachievementoftheobjective.Theobjectiveandconstraintsmustbedefinablebylinearmathematicalfunctionalrelationships.Proportionality

-Therateofchange(slope)oftheobjectivefunctionandconstraintequationsisconstant.Additivity

-Termsintheobjectivefunctionandconstraintequationsmustbeadditive.Divisibility

-Decisionvariablescantakeonanyfractionalvalueandarethereforecontinuousasopposedtointegerinnature.Certainty-Valuesofallthemodelparametersareassumedtobeknownwithcertainty(non-probabilistic).PropertiesofLinearProgrammingModelsProblemStatementExampleProblemNo.1(1of3)Hotdogmixturein1000-poundbatches.Twoingredients,chicken($3/lb)andbeef($5/lb).Reciperequirements:atleast500poundsof“chicken”atleast200poundsof“beef”Ratioofchickentobeefmustbeatleast2to1.Determineoptimalmixtureofingredientsthatwillminimizecosts.Step1:Identifydecisionvariables.x1=lbofchickeninmixturex2=lbofbeefinmixtureStep2:Formulatetheobjectivefunction. MinimizeZ=$3x1+$5x2 whereZ=costper1,000-lbbatch$3x1=costofchicken$5x2=costofbeefSolutionExampleProblemNo.1(2of3)Step3:EstablishModelConstraintsx1+x2=1,000lbx1

500lbofchickenx2

200lbofbeefx1/x2

2/1orx1-2x2

0x1,x2

0TheModel:MinimizeZ=$3x1+5x2subjectto:x1+x2=1,000lbx1

50x2

200x1-2x2

0x1,x2

0SolutionExampleProblemNo.1(3of3)Solvethefollowingmodelgraphically:MaximizeZ=4x1+5x2subjectto:x1+2x2

106x1+6x2

36x1

4x1,x2

0Step1: Plottheconstraints asequationsExampleProblemNo.2(1of3)Figure2.23ConstraintequationsExampleProblemNo.2(2of3)MaximizeZ=4x1+5x2subjectto:x1+2x2

106x1+6x2

36x1

4x1,x2

0Step2:DeterminethefeasiblesolutionspaceFigure2.24FeasiblesolutionspaceandextremepointsExampleProblemNo.2(3of3)MaximizeZ=4x1+5x2subjectto:x1+2x2

106x1+6x2

36x1

4x1,x2

0Step3and4:DeterminethesolutionpointsandoptimalsolutionFigure2.25OptimalsolutionpointAllrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher.PrintedintheUnitedStatesofAmerica.LinearProgramming:ComputerSolutionandSensitivityAnalysisChapter3Copyright©2016PearsonEducation,Inc.ChapterTopicsComputerSolutionSensitivityAnalysisCopyright©2016PearsonEducation,Inc.EarlylinearprogrammingusedlengthymanualmathematicalsolutionprocedurecalledtheSimplexMethod(SeewebsiteModuleA).StepsoftheSimplexMethodhavebeenprogrammedinsoftwarepackagesdesignedforlinearprogrammingproblems.Manysuchpackagesavailablecurrently.Usedextensivelyinbusinessandgovernment.TextfocusesonExcelSpreadsheetsandQMforWindows.ComputerSolutionCopyright©2016PearsonEducation,Inc.BeaverCreekPotteryExampleExcelSpreadsheet–DataScreen(1of5)Exhibit3.1Clickon“Data”tabtoinvoke“Solver.”=C6*B10+D6*B11=G6-E6=G7-E7=C7*B10+D7*B11Decisionvariables—bowls(x1)=B10;mugs(x2)=B11Objectivefunction=C4*B10+D4*B11Copyright©2016PearsonEducation,Inc.BeaverCreekPotteryExample“Solver”ParameterScreen

(2of5)Exhibit3.2SolverparametersCopyright©2016PearsonEducation,Inc.Exhibit3.3LaborconstraintBeaverCreekPotteryExampleAddingModelConstraints(3of5)=C6*B10+D6*B11=40Copyright©2016PearsonEducation,Inc.BeaverCreekPotteryExample“Solver”Settings(4of5)Exhibit3.4SolutionscreenSlack-S1=0andS2=0Copyright©2016PearsonEducation,Inc.Exhibit3.5AnswerreportBeaverCreekPotteryExampleSolutionScreen(5of5)Copyright©2016PearsonEducation,Inc.LinearProgrammingProblem:StandardFormStandardformrequiresallvariablesintheconstraintequationstoappearontheleftoftheinequality(orequality)andallnumericvaluestobeontheright-handside.Examples:x3

x1+x2mustbeconvertedtox3-x1-x2

0x1/(x2+x3)

2becomesx1

2(x2+x3) andthenx1-2x2-2x3

0

Copyright©2016PearsonEducation,Inc.BeaverCreekPotteryExampleQMforWindows(1of4)Exhibit3.6DataentryscreenSetnumberofconstraintsanddecisionvariables.Clickherewhenfinished.Copyright©2016PearsonEducation,Inc.BeaverCreekPotteryExampleQMforWindows(2of4)Exhibit3.7DatatableCopyright©2016PearsonEducation,Inc.BeaverCreekPotteryExampleQMforWindows(3of4)Exhibit3.8ModelsolutionThemarginalvalueisthedollaramountonewouldbewillingtopayforoneadditionalresourceunit.Copyright©2016PearsonEducation,Inc.BeaverCreekPotteryExampleQMforWindows(4of4)Exhibit3.9GraphicalsolutionCopyright©2016PearsonEducation,Inc.Sensitivityanalysisdeterminestheeffectontheoptimalsolutionofchangesinparametervaluesoftheobjectivefunctionandconstraintequations.Changesmaybereactionstoanticipateduncertaintiesintheparametersortoneworchangedinformationconcerningthemodel.BeaverCreekPotteryExampleSensitivityAnalysis

(1of4)Copyright©2016PearsonEducation,Inc.MaximizeZ=$40x1+$50x2subjectto:x1+2x2

404x1+3x2

120 x1,x2

0Figure3.1OptimalsolutionpointBeaverCreekPotteryExampleSensitivityAnalysis

(2of4)Copyright©2016PearsonEducation,Inc.MaximizeZ=$100x1

+$50x2subjectto:x1+2x2

404x1+3x2

120 x1,x2

0Figure3.2Changingthex1objectivefunctioncoefficientBeaverCreekPotteryExampleChangex1ObjectiveFunctionCoefficient(3of4)Copyright©2016PearsonEducation,Inc.MaximizeZ=$40x1+$100x2subjectto:x1+2x2

404x1+3x2

120 x1,x2

0Figure3.3Changingthex2objectivefunctioncoefficientBeaverCreekPotteryExampleChangex2ObjectiveFunctionCoefficient(4of4)Copyright©2016PearsonEducation,Inc.Thesensitivityrangeforanobjectivefunctioncoefficientistherangeofvaluesoverwhichthecurrentoptimalsolutionpointwillremainoptimal.Thesensitivityrangeforthexicoefficientisdesignatedasci.ObjectiveFunctionCoefficientSensitivityRange(1of3)objectivefunctionZ=$40x1+$50x2

Theslopeoftheobjectivefunctionis-4/5givenby:50X2=Z-40X1X2=Z/50–4/5X1Iftheslopeoftheobjectivefunctionchangesto-4/3,thelineisparalleltotheconstraintline(nextslide).Copyright©2016PearsonEducation,Inc.objectivefunctionZ=$40x1+$50x2sensitivityrangefor: x1:25

c1

66.67 x2:30

c2

80Figure3.4Determiningthesensitivityrangeforc1ObjectiveFunctionCoefficientSensitivityRangeforc1andc2(2of3)Copyright©2016PearsonEducation,Inc.MinimizeZ=$6x1+$3x2subjectto: 2x1+4x2

16 4x1+3x2

24 x1,x2

0sensitivityranges: 4

c1

0

c2

4.5ObjectiveFunctionCoefficientFertilizerCostMinimizationExample(3of3)Figure3.5Fertilizerexample:sensitivityrangeforc1Copyright©2016PearsonEducation,Inc.Exhibit3.10SolverresultsscreenObjectiveFunctionCoefficientRangesExcel“Solver”ResultsScreenCopyright©2016PearsonEducation,Inc.Exhibit3.11ObjectiveFunctionCoefficientRangesBeaverCreekExampleSensitivityReportSensitivityrangesforobjectivefunctioncoefficientsCopyright©2016PearsonEducation,Inc.Exhibit3.11BeaverCreekPotteryCompanySensitivityReportObjectiveFunctionCoefficientRangesBeaverCreekExampleSensitivityReportCopyright©2016PearsonEducation,Inc.Exhibit3.12SensitivityrangesforobjectivefunctioncoefficientsObjectiveFunctionCoefficientRangesQMforWindowsSensitivityRangeScreenSensitivityrangesforobjectivefunctioncoefficientsCopyright©2016PearsonEducation,Inc.ChangesinConstraintQuantityValuesSe