Download - EPEF007 Decision Methods Topic1
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DecisionMethod(EPEF007)LowBengYewEN22767805670
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Topic1:LinearProgrammingModelsLearningObjectives
After completing this chapter, students will be able to:
1. Understandthebasicassumptionsandpropertiesoflinearprogramming(LP).
2. GraphicallysolveanyLPproblemthathasonlytwovariablesbycornerpointmethod.
3. UnderstandspecialissuesinLPsuchasinfeasibility,unboundedness,redundancy,andalternativeoptimalsolutions.
4. UseExcelspreadsheetstosolveLPproblems.
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Topic1:LinearProgrammingModelsTopicOutline(TextbookChapter7)
1.1 Introduction
1.2 RequirementsofaLinearProgrammingProblem
1.3 FormulatingLPProblems
1.4 GraphicalSolutiontoanLPProblem
1.5 SolvingFlairFurnituresLPProblemusingExcel
1.6 SolvingMinimizationProblems
1.7 FourSpecialCasesinLP
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Topic1:LinearProgrammingModels1.1Introduction
Manymanagementdecisionsinvolvetryingtomakethemosteffectiveuseoflimitedresources.
Linearprogramming (LP)isawidelyusedmathematicalmodelingtechniquedesignedtohelpmanagersinplanninganddecisionmakingrelativetoresourceallocation.
Thisbelongstothebroaderfieldofmathematicalprogramming.
Inthissense,programming referstomodelingandsolvingaproblemmathematically.
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Topic1:LinearProgrammingModels1.2RequirementsofaLinearProgrammingProblem
AllLPproblemshave4propertiesincommon:1. Allproblemsseektomaximize orminimize somequantity
(theobjectivefunction).2. Restrictionsorconstraints thatlimitthedegreetowhichwe
canpursueourobjectivearepresent.3. Theremustbealternativecoursesofactionfromwhichto
choose.4. Theobjectiveandconstraintsinproblemsmustbeexpressed
intermsoflinear equationsorinequalities.
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Topic1:LinearProgrammingModels1.2RequirementsofLP:BasicAssumptions
Weassumeconditionsofcertainty existandnumbersintheobjectiveandconstraintsareknownwithcertaintyanddonotchangeduringtheperiodbeingstudied.
Weassumeproportionality existsintheobjectiveandconstraints. Weassumeadditivity inthatthetotalofallactivitiesequalsthe
sumoftheindividualactivities. Weassumedivisibility inthatsolutionsneednotbewhole
numbers. Allanswersorvariablesarenonnegative.
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Topic1:LinearProgrammingModels1.2RequirementsofLP:LPProperties
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Topic1:LinearProgrammingModels1.3FormulatingLPProblems
Formulatingalinearprograminvolvesdevelopingamathematicalmodeltorepresentthemanagerialproblem.
Thestepsinformulatingalinearprogramare:1. Completelyunderstandthemanagerialproblembeingfaced.2. Identifytheobjectiveandtheconstraints.3. Definethedecisionvariables.4. Usethedecisionvariablestowritemathematicalexpressions
fortheobjectivefunctionandtheconstraints.
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Topic1:LinearProgrammingModels1.3FormulatingLPProblems
OneofthemostcommonLPapplicationsistheproductmixproblem.
Twoormoreproductsareproducedusinglimitedresourcessuchaspersonnel,machines,andrawmaterials.
Theprofitthatthefirmseekstomaximizeisbasedontheprofitcontributionperunitofeachproduct.
Thecompanywouldliketodeterminehowmanyunitsofeachproductitshouldproducesoastomaximizeoverallprofitgivenitslimitedresources.
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Topic1:LinearProgrammingModelsProblemExample:FlairFurnitureCompany
TheFlairFurnitureCompanyproducesinexpensivetablesandchairs.
Processesaresimilarinthatbothrequireacertainamountofhoursofcarpentryworkandinthepaintingandvarnishingdepartment.
Eachtabletakes4hoursofcarpentryand2hoursofpaintingandvarnishing.
Eachchairrequires3hoursofcarpentryand1hourofpaintingandvarnishing.
Thereare240hoursofcarpentrytimeavailableand100hoursofpaintingandvarnishing.
Eachtableyieldsaprofitof$70andeachchairaprofitof$50.
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Topic1:LinearProgrammingModelsFlairFurnitureCompanyData
Thecompanywantstodeterminethebestcombinationoftablesandchairstoproducetoreachthemaximumprofit.
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Topic1:LinearProgrammingModelsFlairFurnitureCompanyData
Theobjectiveisto:Maximizeprofit Theconstraintsare:
1. Thehoursofcarpentrytimeusedcannotexceed240hoursperweek.
2. Thehoursofpaintingandvarnishingtimeusedcannotexceed100hoursperweek.
Thedecisionvariablesrepresentingtheactualdecisionswewillmakeare:
T =numberoftablestobeproducedperweek.C =numberofchairstobeproducedperweek.
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Topic1:LinearProgrammingModelsFlairFurnitureCompany WecreatetheLPobjectivefunctionintermsofT andC:
Maximizeprofit=$70T +$50C
Developmathematicalrelationshipsforthetwoconstraints: Forcarpentry,totaltimeusedis:
(4hourspertable)(Numberoftablesproduced)+(3hoursperchair)(Numberofchairsproduced).
Weknowthat:CarpentrytimeusedCarpentrytimeavailable.
4T +3C 240(hoursofcarpentrytime)
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Topic1:LinearProgrammingModelsFlairFurnitureCompany
Similarly,Paintingandvarnishingtimeused
Paintingandvarnishingtimeavailable.2T +1C 100(hoursofpaintingandvarnishingtime)
Thismeansthateachtableproducedrequirestwohoursofpaintingandvarnishingtime.
Bothoftheseconstraintsrestrictproductioncapacityandaffecttotalprofit.
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Topic1:LinearProgrammingModelsFlairFurnitureCompany
ThevaluesforT andCmustbenonnegative.T 0(numberoftablesproducedisgreater
thanorequalto0)C 0(numberofchairsproducedisgreater
thanorequalto0)Thecompleteproblemstatedmathematically:
Maximizeprofit=$70T +$50C
subjectto4T +3C 240 (carpentryconstraint)2T +1C 100 (paintingandvarnishingconstraint)T,C 0 (nonnegativity constraint)
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Topic1:LinearProgrammingModels1.4GraphicalSolutiontoanLPProblem
The easiest way to solve a small LP problems is by graphical method.The graphical method only works when there are just two decision variables. When there are more than two variables, a more complex approach is needed as it is not possible to plot the solution on a two-dimensional graph.The graphical method provides valuable insight into how other approaches work.
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
QuadrantContainingAllPositiveValues
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This Axis Represents the Constraint T 0
This Axis Represents the Constraint C 0
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
Thefirststepinsolvingtheproblemistoidentifyasetorregionoffeasiblesolutions.
Todothisweploteachconstraintequationonagraph. Westartbygraphingtheequalityportionofthe
constraintequations:4T +3C =240
Wesolvefortheaxisinterceptsanddrawtheline.
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
WhenFlairproducesnotables,thecarpentryconstraintis:
4(0)+3C =2403C =240C =80
Similarlyfornochairs:4T +3(0)=240
4T =240T =60
Thislineisshownonthefollowinggraph:
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
Graphofcarpentryconstraintequation
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
RegionthatSatisfiestheCarpentryConstraint
Anypointonorbelowtheconstraintplotwillnotviolatetherestriction.
Anypointabovetheplotwillviolatetherestriction.
(30, 40)
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
Thepoint(30,40)liesontheplotandexactlysatisfiestheconstraint
4(30)+3(40)=240.
Thepoint(30,20)liesbelowtheplotandsatisfiestheconstraint
4(30)+3(20)=180.
Thepoint(70,40)liesabovetheplotanddoesnotsatisfytheconstraint
4(70)+3(40)=400.
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
RegionthatSatisfiesthePaintingandVarnishingConstraint
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(T = 0, C = 100)
(T = 50, C = 0)
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
Toproducetablesandchairs,bothdepartmentsmustbeused.
Weneedtofindasolutionthatsatisfiesbothconstraintssimultaneously.
Anewgraphshowsbothconstraintplots.
Thefeasibleregion (orareaoffeasiblesolutions)iswhereallconstraintsaresatisfied.
Anypointinsidethisregionisafeasible solution.
Anypointoutsidetheregionisaninfeasible solution.
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
FeasibleSolutionRegionfortheFlairFurnitureCompanyProblem
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Painting/Varnishing Constraint
Carpentry Constraint
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
Forthepoint(30,20)
Carpentry constraint
4T + 3C 240 hours available(4)(30) + (3)(20) = 180 hours used
Painting constraint
2T + 1C 100 hours available(2)(30) + (1)(20) = 80 hours used
Forthepoint(70,40)Carpentry constraint
4T + 3C 240 hours available(4)(70) + (3)(40) = 400 hours used
Painting constraint
2T + 1C 100 hours available(2)(70) + (1)(40) = 180 hours used
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Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint
Forthepoint(50,5)
Carpentry constraint
4T + 3C 240 hours available(4)(50) + (3)(5) = 215 hours used
Painting constraint
2T + 1C 100 hours available(2)(50) + (1)(5) = 105 hours used
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Topic1:LinearProgrammingModelsCornerPointSolutionMethod
Oncethefeasibleregionhasbeengraphed,weneedtofindtheoptimalsolutionfromthemanypossiblesolutions.
Itinvolveslookingattheprofitateverycornerpointofthefeasibleregion.Thisisknownascornerpointmethod.
ThemathematicaltheorybehindLPisthattheoptimalsolutionmustlieatoneofthecornerpoints,orextremepoint,inthefeasibleregion.
ForFlairFurniture,thefeasibleregionisafoursidedpolygonwithfourcornerpointslabeled1,2,3,and4onthegraph.
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Topic1:LinearProgrammingModelsCornerPointSolutionMethod
FourCornerPointsoftheFeasibleRegion
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Topic1:LinearProgrammingModelsCornerPointSolutionMethod
TofindthecoordinatesforPointaccuratelywehavetosolvefortheintersectionofthetwoconstraintlines.
Usingthesimultaneousequationsmethod,wemultiplythepaintingequationby2andaddittothecarpentryequation
4T +3C = 240 (carpentryline) 4T 2C = 200 (paintingline)
C = 40 Substituting40forC ineitheroftheoriginalequations
allowsustodeterminethevalueofT.4T +(3)(40)= 240 (carpentryline)
4T +120= 240T = 30
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Topic1:LinearProgrammingModelsCornerPointSolutionMethod
Point:(T =0,C =0) Profit=$70(0)+$50(0)=$0
Point:(T =0,C =80) Profit=$70(0)+$50(80)=$4,000
Point:(T =50,C =0) Profit=$70(50)+$50(0)=$3,500
Point:(T =30,C =40) Profit=$70(30)+$50(40)=$4,1003
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BecausePointreturnsthehighestprofit,thisistheoptimalsolution.
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Topic1:LinearProgrammingModelsSlackandSurplus
Slackistheamountofaresourcethatisnotused.Foralessthanorequalconstraint:
Slack=Amountofresourceavailable amountofresourceused.
Surplusisusedwithagreaterthanorequalconstrainttoindicatetheamountbywhichtherighthandsideoftheconstraintisexceeded.
Surplus=Actualamount minimumamount.
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Topic1:LinearProgrammingModelsSummaryofGraphicalSolutionUsingCornerPointMethod
1. Graphallconstraintsandfindthefeasibleregion.
2. Findthecornerpointsofthefeasibleregion.
3. Computetheprofit(orcost)ateachofthefeasiblecornerpoints.
4. SelectthecornerpointwiththebestvalueoftheobjectivefunctionfoundinStep3.Thisistheoptimalsolution.
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Topic1:LinearProgrammingModels1.5SolvingFlairFurnituresLPProblemUsingExcel
MostorganizationshaveaccesstosoftwaretosolvebigLPproblems.
Whiletherearedifferencesbetweensoftwareimplementations,theapproacheachtakestowardshandlingLPisbasicallythesame.
OnceyouareexperiencedindealingwithcomputerizedLPalgorithms,youcaneasilyadjusttominorchanges.
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Topic1:LinearProgrammingModels1.5SolvingFlairFurnituresLPProblemUsingExcel
TheSolvertoolinExcelcanbeusedtofindsolutionsto: LPproblems. Integerprogrammingproblems. Nonintegerprogrammingproblems.
Solverislimitedto200variablesand100constraints.
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
RecallthemodelforFlairFurnitureis:
Maximizeprofit= $70T + $50CSubjectto 4T + 3C 240
2T + 1C 100
TouseSolver,itisnecessarytoenterformulasbasedontheinitialmodel.
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
1. Enterthevariablenames,thecoefficientsfortheobjectivefunctionandconstraints,andtherighthandsidevaluesforeachoftheconstraints.
2. Designatespecificcellsforthevaluesofthedecisionvariables.
3. Writeaformulatocalculatethevalueoftheobjectivefunction.
4. Writeaformulatocomputethelefthandsidesofeachoftheconstraints.
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
ExcelDataInputfortheFlairFurnitureExample
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
FormulasfortheFlairFurnitureExample
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
Excel Spreadsheet for the Flair Furniture Example
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
Oncethemodelhasbeenentered,thefollowingstepscanbeusedtosolvetheproblem.
InExcel2010,selectData Solver.IfSolverdoesnotappearintheindicatedplace,seeAppendixFforinstructionsonhowtoactivatethisaddin.
1. IntheSetObjectivebox,enterthecelladdressforthetotalprofit.
2. IntheByChangingCellsbox,enterthecelladdressesforthevariablevalues.
3. ClickMax foramaximizationproblemandMin foraminimizationproblem.
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
4. ChecktheboxforMakeUnconstrainedVariablesNonnegative.
5. ClicktheSelectSolvingMethodbuttonandselectSimplexLP
fromthemenuthatappears.
6. ClickAdd toaddtheconstraints.
7. Inthedialogboxthatappears,enterthecellreferencesforthelefthandsidevalues,thetypeofequation,andtherighthandsidevalues.
8. ClickSolve.
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
StartingSolver
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
SolverParametersDialogBox
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
SolverAddConstraintDialogBox
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
SolverResultsDialogBox
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Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem
SolutionFoundbySolver
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Topic1:LinearProgrammingModels1.6SolvingMinimizationProblems
ManyLPproblemsinvolveminimizinganobjectivesuchascostinsteadofmaximizingaprofitfunction.
Minimizationproblemscanbesolvedgraphicallybyfirstsettingupthefeasiblesolutionregionandthenusingthecornerpointmethodtofindthevaluesofthedecisionvariables(e.g.,X1 andX2)thatyieldtheminimumcost.
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanchTheHolidayMealTurkeyRanchisconsideringbuyingtwodifferentbrandsofturkeyfeedandblendingthemtoprovideagood,lowcostdietforitsturkeys
X1 =numberofpoundsofbrand1feedpurchasedX2 =numberofpoundsofbrand2feedpurchased
Let
Minimizecost(incents)=2X1 +3X2subjectto:
5X1+10X2 90ounces (ingredientconstraintA)4X1 +3X2 48ounces (ingredientconstraintB)
0.5X1 1.5ounces (ingredientconstraintC)X1 0 (nonnegativityconstraint)
X2 0 (nonnegativityconstraint)
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch
HolidayMealTurkeyRanchData
INGREDIENT
COMPOSITION OF EACH POUND OF FEED (OZ.) MINIMUM MONTHLY
REQUIREMENT PER TURKEY (OZ.)BRAND 1 FEED BRAND 2 FEED
A 5 10 90
B 4 3 48
C 0.5 0 1.5Cost per pound 2 cents 3 cents
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch
Usethecornerpointmethod.
Firstconstructthefeasiblesolutionregion.
Theoptimalsolutionwilllieatoneofthecornersasitwouldinamaximizationproblem.
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch FeasibleRegion
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Feasible Region
DECISIONANALYSIS(ESZ2001)
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch
Solveforthevaluesofthethreecornerpoints.
Pointa istheintersectionofingredientconstraintsCandB.
4X1 +3X2 =48
X1 =3
Substituting3inthefirstequation,wefindX2 =12.
SolvingforpointbwithbasicalgebrawefindX1 =8.4andX2=4.8.
SolvingforpointcwefindX1 =18andX2 =0.
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch
Substitutingthesevaluebackintotheobjectivefunctionwefind
Cost =2X1 +3X2Costatpointa =2(3)+3(12)=42Costatpointb =2(8.4)+3(4.8)=31.2Costatpointc =2(18)+3(0)=36
Thelowestcostsolutionistopurchase8.4poundsofbrand1feedand4.8poundsofbrand2feedforatotalcostof31.2centsperturkey.
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch ExcelSolver
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Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch ExcelSolver
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
FourspecialcasesanddifficultiesariseattimeswhenusingthegraphicalapproachtosolvingLPproblems.
Nofeasiblesolution
Unboundedness
Redundancy
AlternateOptimalSolutions
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
Nofeasiblesolution
Thisexistswhenthereisnosolutiontotheproblemthatsatisfiesalltheconstraintequations.
Nofeasiblesolutionregionexists.
Thisisacommonoccurrenceintherealworld.
Generallyoneormoreconstraintsarerelaxeduntilasolutionisfound.
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
Aproblemwithnofeasiblesolution
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Region Satisfying Third Constraint
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
Unboundedness
Sometimesalinearprogramwillnothaveafinitesolution.
Inamaximizationproblem,oneormoresolutionvariables,andtheprofit,canbemadeinfinitelylargewithoutviolatinganyconstraints.
Inagraphicalsolution,thefeasibleregionwillbeopenended.
Thisusuallymeanstheproblemhasbeenformulatedimproperly.
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Topic1:LinearProgrammingModelsFourSpecialCasesinLP
AFeasibleRegionThatisUnboundedtotheRight
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
Redundancy
Aredundantconstraintisonethatdoesnotaffectthefeasiblesolutionregion.
Oneormoreconstraintsmaybebinding.
Thisisaverycommonoccurrenceintherealworld.
Itcausesnoparticularproblems,buteliminatingredundantconstraintssimplifiesthemodel.
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
ProblemwithaRedundantConstraint
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
AlternateOptimalSolutions
Occasionallytwoormoreoptimalsolutionsmayexist.
Graphicallythisoccurswhentheobjectivefunctionsisoprofitorisocostlinerunsperfectlyparalleltooneoftheconstraints.
Thisactuallyallowsmanagementgreatflexibilityindecidingwhichcombinationtoselectastheprofitisthesameateachalternatesolution.
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Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP
ExampleofAlternateOptimalSolutions
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Isoprofit Line for $8
Optimal Solution Consists of All Combinations of X1 and X2 Along the AB Segment
Isoprofit Line for $12 Overlays Line Segment AB
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Topic1:LinearProgrammingModels
EndofTopic1