管理数学PPT课件讲义

管理数学

Chapter2:

SystemofLinearEquationsAgendaLinearSystemsasMathematicalModelsLinearSystemsHavingOneorNoSolutionsLinearSystemsHavingManySolutionsLinear<--Linea1x+a2y=b ora1x1+a2x2+....+anxn=b2x+3y=4x2+y2=1x1

-

x2+x3

-

x4=6z=5-3x+y/2sinx+ey

=1xy=27x1

+3x2+9/x3

+

2x4=1x1

+

2x2+3x3

+....+

nxn=1Whichoneislinear?Example1AfirmproducesbargainanddeluxeTVsetsbybuyingthecomponents,assemblingthem,andtestingthesetsbeforeshipping.ResourcesThebargainsetrequires3hourstoassembleand1hourtotest.Thedeluxesetrequires4hourstoassembleand2hourstotest.Thefirmhas390hoursforassemblyand170hoursfortestingeachweek.QuestionUseasystemoflinearequationstomodelthenumberofeachtypeofTVsetthatthecompanycanproduceeachweekwhileusingallofitsavailablelabor.ProblemFormulationDefinedecisionvariables(unitofscale)Definethelinearrelationbetweenvariables(writethelinearequations)Example2Adietitianistocombineatotalof5servingsofcreamofmushroomsoup,tuna,andgreenbeans,amongotheringredients,inmakingacasserole.IngredientNutritionsEachservingofsouphas15caloriesand1gramofprotein,eachservingoftunahas160caloriesand12gramofprotein,andeachservingofgreenbeanshas20caloriesand1gramofprotein.QuestionIfthesethreefoodsaretofurnish380caloriesand27gramsofproteinincasserole,howmanyservingsofeachshouldbeused?Example3AretailerhaswarehousesinLimaandCanton,fromwhichtwostores—oneinTiffinandoneinDanville—placeordersforbicycles.Tiffinorders38andDanvilleorders46.LimitationsandQuestionEachwarehousehasenoughtosupplyallordersbuttwiceasmanyaretobeshippedfromLimatoDanvilleasfromCantontoTiffin.Writethelinearequations.AgendaLinearSystemsasMathematicalModelsLinearSystemsHavingOneorNoSolutionsLinearSystemsHavingManySolutionsSolvingaSystemofLinearEquationsProblemformulation:variabledefinitionandequationsAlgorithmsorformulaInterpretationofsolutionsx+3y=9-2x+y=-4{(1)-2x+y=3-4x+2y=2{(2)4x-2y=66x-3y=9{(3)Systemof2LinearEquationsx1+x2+x3=2 --------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6--------(3)(1)(2)(3)Systemof3LinearEquationsASystemofLinearEquations

(ALinearSystem)Afinitecollectionoflinearequationsa11x1+a12x2+....+a1nxn=b1a21x1+a22x2+....+a2nxn=b2

....am1x1+am2x2+....+amnxn=bmASolutionToaequation:a1x1+a2x2+....+anxn=b(t1,t2,....,tn)Toalinearsystem:Asolutiontoeachoflinearequationsimultaneouslyps.SolutionSetElementaryTransformations1.Interchangethepositionoftwoequations.2.Multiply

bothsidesofanequationbyanonzeroconstant.3.Addamultipleofoneequationtoanotherequation.x1+x2+x3=2 --------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6--------(3)Interchange(1)and(3).Multiply(2)by1/2.Adda-1multiple(2)to(1).ContinuousOperationsx1+x2+x3=2-------------(1)x1+1.5x2+0.5x3=3-----(2)x1-x2-2x3=-6------------(3)Ax1-x2-2x3=-6-----------(3)

x1+1.5x2+0.5x3=1.5--(2)-0.5x2+0.5x3=0.5---(1)BResultofoperationsx1+x2+x3=2--------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6--------(3)Ax1=?------(1)

x2=?------(2)x3=?------(3)DTheObjectiveSolvetheproblembyelementarytransformationx1+x2+x3=2------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6------(3)Adda-2multiple(1)to(2).Adda-1multiple(1)to(3).Iteration1x1+x2+x3=2 ------(1)x2-x3=-1------(2)-2x2-3x3=-8------(3)Adda-1multiple(2)to(1).Adda2multiple(2)to(3).Iteration2x1+2x3=3 ------(1)x2-x3=-1------(2)-5x3=-10------(3)Multiply(2)by-1/5.Adda-2multiple(3)to(1).Adda1multiple(3)to(2).FinalIterationx1=-1 ------(1)x2=1------(2)x3=2------(3)Finalanswer:(x1,x2,x3)=(-1,1,2)DifficultyItisveryhardtocarryvariables,xi’s,throughthecalculationprocesswhenapplyingelementarytransformation2 3-47 5-17 10 5-8 33 5 60 -2 58 9 128 9 123 50 -28 95-293 50 -2MatrixTransfertobe:AX=BMatrixNotationa11x1+a12x2+....+a1nxn=b1a21x1+a22x2+....+a2nxn=b2am1x1+am2x2+....+amnxn=bm................x1+x2+x3=2 ------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6------(3)Example1Transfer(I)tomatrixformat.(I)x1x2

x3

X=13 -2B=1 -1 -22 -3 -5-1 3 5A=MatrixRepresentationAX=B3x1+2x2-5x3=7 x1-8x2+4x3=92x1+6x2-7x3=-2Transfer(I)tomatrixformat.(I)3 2 -51 -8 42 6 -7A=x1x2

x3

X=79 -2B=Example2MatrixofCoefficients1 1 12 3 11 -1 -2x1+x2+x3=22x1+3x2+x3=3x1-x2-2x3=-6CoefficientsofthesystemorMatrixAAugmentedMatrixx1+x2+x3=22x1+3x2+x3=3x1-x2-2x3=-61 1 1 22 3 1 31 -1 -2 -6CoefficientsandRHS,or[A|B].ReducedEchelonForm1.Anyrowswithallzerosareatthebottom.2.Leading1.3.Leading1totheright.4.Allotherelementsinaleading1

columnarezeros.1 0 80 1 20 0 01 2 0 40 0 0 00 0 1 31 0 0 20 0 1 40 1 0 31 2 3 00 0 0 10 0 0 0ExamplesI1 2 0 3 00 0 3 4 00 0 0 0 11 7 0 80 1 0 30 0 1 20 0 0 0ExamplesIIElementaryRowOperations1.Interchange

tworows2.Multiplytheelementsofarowbyanonzeroconstant3.Addamultipleoftheelementsofonerowtothecorrespondingelementsofanotherrow.x1+x2+x3=2 --------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6--------(3)111223131-1-2-6ContinuousOperations111223131-1-2-6Interchanger.1andr.3.Multiplyr.2by1/2.Adda-1multipler.2tor.3.1-1-2-60-Result111223131-1-2-6ThroughElementaryRowoperationsTheObjectiveReducedechelonForm???EquivalentSystemsSupposethatAandBarebothsystemsoflinearequations.AandBareequivalentiftheyarerelatedthroughelementarytransformations.AandBhasthesamesolutioniftheyareequivalent.SolvingasystemoflinearequationsGauss-JordanEliminationGaussEliminationGauss-JordanElimination1.Writetheaugmentedmatrix.2.Derivethereducedechelonformoftheaugmentedmatrix3.Writethesystemofequationscorrespondingtothereducedechelonform.x1+x2+x3=2 --------(1)2x1+3x2+x3=3------(2)x1-x2-2x3=-6--------(3)111223131-1-2-6PerformJ-GElimination111223131-1-2-6111201-1-10-2-3-8+*(-2)+*(-1)PivotingStep11