EPEF007 Decision Methods Topic1

DecisionMethod(EPEF007)LowBengYewEN22767805670

Topic1:LinearProgrammingModelsLearningObjectives

After completing this chapter, students will be able to:

1. Understandthebasicassumptionsandpropertiesoflinearprogramming(LP).

2. GraphicallysolveanyLPproblemthathasonlytwovariablesbycornerpointmethod.

3. UnderstandspecialissuesinLPsuchasinfeasibility,unboundedness,redundancy,andalternativeoptimalsolutions.

4. UseExcelspreadsheetstosolveLPproblems.

Topic1:LinearProgrammingModelsTopicOutline(TextbookChapter7)

1.1 Introduction

1.2 RequirementsofaLinearProgrammingProblem

1.3 FormulatingLPProblems

1.4 GraphicalSolutiontoanLPProblem

1.5 SolvingFlairFurnituresLPProblemusingExcel

1.6 SolvingMinimizationProblems

1.7 FourSpecialCasesinLP

Topic1:LinearProgrammingModels1.1Introduction

Manymanagementdecisionsinvolvetryingtomakethemosteffectiveuseoflimitedresources.

Linearprogramming (LP)isawidelyusedmathematicalmodelingtechniquedesignedtohelpmanagersinplanninganddecisionmakingrelativetoresourceallocation.

Thisbelongstothebroaderfieldofmathematicalprogramming.

Inthissense,programming referstomodelingandsolvingaproblemmathematically.

Topic1:LinearProgrammingModels1.2RequirementsofaLinearProgrammingProblem

AllLPproblemshave4propertiesincommon:1. Allproblemsseektomaximize orminimize somequantity

(theobjectivefunction).2. Restrictionsorconstraints thatlimitthedegreetowhichwe

canpursueourobjectivearepresent.3. Theremustbealternativecoursesofactionfromwhichto

choose.4. Theobjectiveandconstraintsinproblemsmustbeexpressed

intermsoflinear equationsorinequalities.

Topic1:LinearProgrammingModels1.2RequirementsofLP:BasicAssumptions

Weassumeconditionsofcertainty existandnumbersintheobjectiveandconstraintsareknownwithcertaintyanddonotchangeduringtheperiodbeingstudied.

Weassumeproportionality existsintheobjectiveandconstraints. Weassumeadditivity inthatthetotalofallactivitiesequalsthe

sumoftheindividualactivities. Weassumedivisibility inthatsolutionsneednotbewhole

numbers. Allanswersorvariablesarenonnegative.

Topic1:LinearProgrammingModels1.2RequirementsofLP:LPProperties

Topic1:LinearProgrammingModels1.3FormulatingLPProblems

Formulatingalinearprograminvolvesdevelopingamathematicalmodeltorepresentthemanagerialproblem.

Thestepsinformulatingalinearprogramare:1. Completelyunderstandthemanagerialproblembeingfaced.2. Identifytheobjectiveandtheconstraints.3. Definethedecisionvariables.4. Usethedecisionvariablestowritemathematicalexpressions

fortheobjectivefunctionandtheconstraints.

Topic1:LinearProgrammingModels1.3FormulatingLPProblems

OneofthemostcommonLPapplicationsistheproductmixproblem.

Twoormoreproductsareproducedusinglimitedresourcessuchaspersonnel,machines,andrawmaterials.

Theprofitthatthefirmseekstomaximizeisbasedontheprofitcontributionperunitofeachproduct.

Thecompanywouldliketodeterminehowmanyunitsofeachproductitshouldproducesoastomaximizeoverallprofitgivenitslimitedresources.

Topic1:LinearProgrammingModelsProblemExample:FlairFurnitureCompany

TheFlairFurnitureCompanyproducesinexpensivetablesandchairs.

Processesaresimilarinthatbothrequireacertainamountofhoursofcarpentryworkandinthepaintingandvarnishingdepartment.

Eachtabletakes4hoursofcarpentryand2hoursofpaintingandvarnishing.

Eachchairrequires3hoursofcarpentryand1hourofpaintingandvarnishing.

Thereare240hoursofcarpentrytimeavailableand100hoursofpaintingandvarnishing.

Eachtableyieldsaprofitof$70andeachchairaprofitof$50.

Topic1:LinearProgrammingModelsFlairFurnitureCompanyData

Thecompanywantstodeterminethebestcombinationoftablesandchairstoproducetoreachthemaximumprofit.

Topic1:LinearProgrammingModelsFlairFurnitureCompanyData

Theobjectiveisto:Maximizeprofit Theconstraintsare:

1. Thehoursofcarpentrytimeusedcannotexceed240hoursperweek.

2. Thehoursofpaintingandvarnishingtimeusedcannotexceed100hoursperweek.

Thedecisionvariablesrepresentingtheactualdecisionswewillmakeare:

T =numberoftablestobeproducedperweek.C =numberofchairstobeproducedperweek.

Topic1:LinearProgrammingModelsFlairFurnitureCompany WecreatetheLPobjectivefunctionintermsofT andC:

Maximizeprofit=$70T +$50C

Developmathematicalrelationshipsforthetwoconstraints: Forcarpentry,totaltimeusedis:

(4hourspertable)(Numberoftablesproduced)+(3hoursperchair)(Numberofchairsproduced).

Weknowthat:CarpentrytimeusedCarpentrytimeavailable.

4T +3C 240(hoursofcarpentrytime)

Topic1:LinearProgrammingModelsFlairFurnitureCompany

Similarly,Paintingandvarnishingtimeused

Paintingandvarnishingtimeavailable.2T +1C 100(hoursofpaintingandvarnishingtime)

Thismeansthateachtableproducedrequirestwohoursofpaintingandvarnishingtime.

Bothoftheseconstraintsrestrictproductioncapacityandaffecttotalprofit.

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)

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.

Topic1:LinearProgrammingModels1.4GraphicalRepresentationofaConstraint

QuadrantContainingAllPositiveValues

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This Axis Represents the Constraint T 0

This Axis Represents the Constraint C 0


Thefirststepinsolvingtheproblemistoidentifyasetorregionoffeasiblesolutions.

Todothisweploteachconstraintequationonagraph. Westartbygraphingtheequalityportionofthe

constraintequations:4T +3C =240

Wesolvefortheaxisinterceptsanddrawtheline.


WhenFlairproducesnotables,thecarpentryconstraintis:

4(0)+3C =2403C =240C =80

Similarlyfornochairs:4T +3(0)=240

4T =240T =60

Thislineisshownonthefollowinggraph:


Graphofcarpentryconstraintequation

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(T = 60, C = 0)

(T = 0, C = 80)

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(70, 40)

(30, 20)


RegionthatSatisfiestheCarpentryConstraint

Anypointonorbelowtheconstraintplotwillnotviolatetherestriction.

Anypointabovetheplotwillviolatetherestriction.

(30, 40)


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.


RegionthatSatisfiesthePaintingandVarnishingConstraint

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(T = 0, C = 100)

(T = 50, C = 0)


Toproducetablesandchairs,bothdepartmentsmustbeused.

Weneedtofindasolutionthatsatisfiesbothconstraintssimultaneously.

Anewgraphshowsbothconstraintplots.

Thefeasibleregion (orareaoffeasiblesolutions)iswhereallconstraintsaresatisfied.

Anypointinsidethisregionisafeasible solution.

Anypointoutsidetheregionisaninfeasible solution.


FeasibleSolutionRegionfortheFlairFurnitureCompanyProblem

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Painting/Varnishing Constraint

Carpentry Constraint


Forthepoint(30,20)

Carpentry constraint

4T + 3C 240 hours available(4)(30) + (3)(20) = 180 hours used

Painting constraint


Forthepoint(70,40)Carpentry constraint


Painting constraint



Forthepoint(50,5)

Carpentry constraint


Painting constraint


Topic1:LinearProgrammingModelsCornerPointSolutionMethod

Oncethefeasibleregionhasbeengraphed,weneedtofindtheoptimalsolutionfromthemanypossiblesolutions.

Itinvolveslookingattheprofitateverycornerpointofthefeasibleregion.Thisisknownascornerpointmethod.

ThemathematicaltheorybehindLPisthattheoptimalsolutionmustlieatoneofthecornerpoints,orextremepoint,inthefeasibleregion.

ForFlairFurniture,thefeasibleregionisafoursidedpolygonwithfourcornerpointslabeled1,2,3,and4onthegraph.


FourCornerPointsoftheFeasibleRegion

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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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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.

Topic1:LinearProgrammingModelsSummaryofGraphicalSolutionUsingCornerPointMethod

1. Graphallconstraintsandfindthefeasibleregion.

2. Findthecornerpointsofthefeasibleregion.

3. Computetheprofit(orcost)ateachofthefeasiblecornerpoints.

4. SelectthecornerpointwiththebestvalueoftheobjectivefunctionfoundinStep3.Thisistheoptimalsolution.

Topic1:LinearProgrammingModels1.5SolvingFlairFurnituresLPProblemUsingExcel

MostorganizationshaveaccesstosoftwaretosolvebigLPproblems.

Whiletherearedifferencesbetweensoftwareimplementations,theapproacheachtakestowardshandlingLPisbasicallythesame.

OnceyouareexperiencedindealingwithcomputerizedLPalgorithms,youcaneasilyadjusttominorchanges.

Topic1:LinearProgrammingModels1.5SolvingFlairFurnituresLPProblemUsingExcel

TheSolvertoolinExcelcanbeusedtofindsolutionsto: LPproblems. Integerprogrammingproblems. Nonintegerprogrammingproblems.

Solverislimitedto200variablesand100constraints.

Topic1:LinearProgrammingModels1.5UsingSolvertoSolvetheFlairFurnitureProblem

RecallthemodelforFlairFurnitureis:

Maximizeprofit= $70T + $50CSubjectto 4T + 3C 240

2T + 1C 100

TouseSolver,itisnecessarytoenterformulasbasedontheinitialmodel.


1. Enterthevariablenames,thecoefficientsfortheobjectivefunctionandconstraints,andtherighthandsidevaluesforeachoftheconstraints.

2. Designatespecificcellsforthevaluesofthedecisionvariables.

3. Writeaformulatocalculatethevalueoftheobjectivefunction.

4. Writeaformulatocomputethelefthandsidesofeachoftheconstraints.


ExcelDataInputfortheFlairFurnitureExample


FormulasfortheFlairFurnitureExample


Excel Spreadsheet for the Flair Furniture Example


Oncethemodelhasbeenentered,thefollowingstepscanbeusedtosolvetheproblem.

InExcel2010,selectData Solver.IfSolverdoesnotappearintheindicatedplace,seeAppendixFforinstructionsonhowtoactivatethisaddin.

1. IntheSetObjectivebox,enterthecelladdressforthetotalprofit.

2. IntheByChangingCellsbox,enterthecelladdressesforthevariablevalues.

3. ClickMax foramaximizationproblemandMin foraminimizationproblem.


4. ChecktheboxforMakeUnconstrainedVariablesNonnegative.

5. ClicktheSelectSolvingMethodbuttonandselectSimplexLP

fromthemenuthatappears.

6. ClickAdd toaddtheconstraints.

7. Inthedialogboxthatappears,enterthecellreferencesforthelefthandsidevalues,thetypeofequation,andtherighthandsidevalues.

8. ClickSolve.


StartingSolver


SolverParametersDialogBox


SolverAddConstraintDialogBox


SolverResultsDialogBox


SolutionFoundbySolver

Topic1:LinearProgrammingModels1.6SolvingMinimizationProblems

ManyLPproblemsinvolveminimizinganobjectivesuchascostinsteadofmaximizingaprofitfunction.

Minimizationproblemscanbesolvedgraphicallybyfirstsettingupthefeasiblesolutionregionandthenusingthecornerpointmethodtofindthevaluesofthedecisionvariables(e.g.,X1 andX2)thatyieldtheminimumcost.

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)

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


Usethecornerpointmethod.

Firstconstructthefeasiblesolutionregion.

Theoptimalsolutionwilllieatoneofthecornersasitwouldinamaximizationproblem.

Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch FeasibleRegion

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Ingredient C Constraint

Ingredient B Constraint

Ingredient A Constraint

Feasible Region

DECISIONANALYSIS(ESZ2001)


Solveforthevaluesofthethreecornerpoints.

Pointa istheintersectionofingredientconstraintsCandB.

4X1 +3X2 =48

X1 =3

Substituting3inthefirstequation,wefindX2 =12.

SolvingforpointbwithbasicalgebrawefindX1 =8.4andX2=4.8.

SolvingforpointcwefindX1 =18andX2 =0.


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.

Topic1:LinearProgrammingModelsHolidayMealTurkeyRanch ExcelSolver

Topic1:LinearProgrammingModels1.7FourSpecialCasesinLP

FourspecialcasesanddifficultiesariseattimeswhenusingthegraphicalapproachtosolvingLPproblems.

Nofeasiblesolution

Unboundedness

Redundancy

AlternateOptimalSolutions


Nofeasiblesolution

Thisexistswhenthereisnosolutiontotheproblemthatsatisfiesalltheconstraintequations.

Nofeasiblesolutionregionexists.

Thisisacommonoccurrenceintherealworld.

Generallyoneormoreconstraintsarerelaxeduntilasolutionisfound.


Aproblemwithnofeasiblesolution

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RegionSatisfyingFirstTwoConstraints

Region Satisfying Third Constraint


Unboundedness

Sometimesalinearprogramwillnothaveafinitesolution.

Inamaximizationproblem,oneormoresolutionvariables,andtheprofit,canbemadeinfinitelylargewithoutviolatinganyconstraints.

Inagraphicalsolution,thefeasibleregionwillbeopenended.

Thisusuallymeanstheproblemhasbeenformulatedimproperly.

Topic1:LinearProgrammingModelsFourSpecialCasesinLP

AFeasibleRegionThatisUnboundedtotheRight

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Feasible Region

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Redundancy

Aredundantconstraintisonethatdoesnotaffectthefeasiblesolutionregion.

Oneormoreconstraintsmaybebinding.

Thisisaverycommonoccurrenceintherealworld.

Itcausesnoparticularproblems,buteliminatingredundantconstraintssimplifiesthemodel.


ProblemwithaRedundantConstraint

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AlternateOptimalSolutions

Occasionallytwoormoreoptimalsolutionsmayexist.

Graphicallythisoccurswhentheobjectivefunctionsisoprofitorisocostlinerunsperfectlyparalleltooneoftheconstraints.

Thisactuallyallowsmanagementgreatflexibilityindecidingwhichcombinationtoselectastheprofitisthesameateachalternatesolution.


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

Topic1:LinearProgrammingModels

EndofTopic1

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EPEF007 Decision Methods Topic1