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21CHAPTER2LINEARPROGRAMMINGBASICCONCEPTSREVIEWQUESTIONS211PONDEROSAINDUSTRIALUSESLINEARPROGRAMMINGMONTHLYTOGUIDETHEPRODUCTMIXDECISION212OVERALLPROFITABILITYHASINCREASEDBY20BETTERUTILIZATIONOFRAWMATERIAL,CAPITALEQUIPMENT,ANDPERSONNELALSORESULTED213THEGOALWASTOIMPROVETHEUTILIZATIONOFRESERVATIONPERSONNELBYMATCHINGWORKSCHEDULESTOCUSTOMERNEEDS214UNITEDAIRLINESSAVEDMORETHAN6MILLIONANNUALLYINDIRECTSALARYANDBENEFITCOSTSCUSTOMERSERVICEALSOIMPROVEDANDWORKLOADSWEREREDUCEDFORSUPPORTSTAFF215THESDMSYSTEMISUSEDTOCOORDINATETHESUPPLY,DISTRIBUTIONANDMARKETINGOFEACHOFCITGOSMAJORPRODUCTSTHROUGHOUTTHEUNITEDSTATES216CITGOSAVEDABOUT14MILLIONANNUALLYININTERESTEXPENSESIMPROVEMENTSINCOORDINATION,PRICING,ANDPURCHASINGDECISIONSADDEDATLEAST25MILLIONMORETOANNUALPROFITS221THEYPROVIDETHEHIGHESTQUALITYAVAILABLEINTHEINDUSTRYFORTHEMOSTDISCRIMINATINGBUYERS2221SHOULDTHECOMPANYLAUNCHTHETWONEWPRODUCTS2WHATSHOULDBETHEPRODUCTMIXFORTHETWONEWPRODUCTS223THEGROUPWASASKEDTOANALYZEPRODUCTMIX224WHICHCOMBINATIONOFPRODUCTIONRATESFORTHETWONEWPRODUCTSWOULDMAXIMIZETHETOTALPROFITFROMBOTHOFTHEM2251AVAILABLEPRODUCTIONCAPACITYINEACHOFTHEPLANTS2HOWMUCHOFTHEPRODUCTIONCAPACITYINEACHPLANTWOULDBENEEDEDBYEACHPRODUCT3PROFITABILITYOFEACHPRODUCT2311WHATARETHEDECISIONSTOBEMADE2WHATARETHECONSTRAINTSONTHESEDECISIONS3WHATISTHEOVERALLMEASUREOFPERFORMANCEFORTHESEDECISIONS232WHENFORMULATINGALINEARPROGRAMMINGMODELONASPREADSHEET,THECELLSSHOWINGTHEDATAFORTHEPROBLEMARECALLEDTHEDATACELLSTHECHANGINGCELLSARETHECELLSTHATCONTAINTHEDECISIONSTOBEMADETHEOUTPUTCELLSARETHECELLSTHATPROVIDEOUTPUTTHATDEPENDSONTHECHANGINGCELLSTHETARGETCELLISASPECIALKINDOFOUTPUTCELLTHATSHOWSTHEOVERALLMEASUREOFPERFORMANCEOFTHEDECISIONTOBEMADE22233THEEXCELEQUATIONFOREACHOUTPUTCELLCANBEEXPRESSEDASASUMPRODUCTFUNCTION,WHEREEACHTERMINTHESUMISTHEPRODUCTOFADATACELLANDACHANGINGCELL2411GATHERTHERELEVANTDATA2IDENTIFYTHEDECISIONSTOBEMADE3IDENTIFYTHECONSTRAINTSONTHESEDECISIONS4IDENTIFYTHEOVERALLMEASUREOFPERFORMANCEFORTHESEDECISIONS5CONVERTTHEVERBALDESCRIPTIONOFTHECONSTRAINTSANDMEASUREOFPERFORMANCEINTOQUANTITATIVEEXPRESSIONSINTERMSOFTHEDATAANDDECISIONS242ALGEBRAICSYMBOLSNEEDTOBEINTRODUCEDTOREPRESENTSTHEMEASUREOFPERFORMANCEANDTHEDECISIONS243ADECISIONVARIABLEISANALGEBRAICVARIABLETHATREPRESENTSADECISIONREGARDINGTHELEVELOFAPARTICULARACTIVITYTHEOBJECTIVEFUNCTIONISTHEPARTOFALINEARPROGRAMMINGMODELTHATEXPRESSESWHATNEEDSTOBEEITHERMAXIMIZEDORMINIMIZED,DEPENDINGONTHEOBJECTIVEFORTHEPROBLEMANONNEGATIVITYCONSTRAINTISACONSTRAINTTHATEXPRESSTHERESTRICTIONTHATAPARTICULARDECISIONVARIABLEMUSTBEGREATERTHANOREQUALTOZEROALLCONSTRAINTSTHATARENOTNONNEGATIVITYCONSTRAINTSAREREFERREDTOASFUNCTIONALCONSTRAINTS244AFEASIBLESOLUTIONISONETHATSATISFIESALLTHECONSTRAINTSOFTHEPROBLEMTHEBESTFEASIBLESOLUTIONISCALLEDTHEOPTIMALSOLUTION251TWO252THEAXESREPRESENTPRODUCTIONRATESFORPRODUCT1ANDPRODUCT2253THELINEFORMINGTHEBOUNDARYOFWHATISPERMITTEDBYACONSTRAINTISCALLEDACONSTRAINTBOUNDARYLINEITSEQUATIONISCALLEDACONSTRAINTBOUNDARYEQUATION254THECOEFFICIENTOFX1GIVESTHESLOPEOFTHECONSTRAINTBOUNDARYLINETHECONSTANTTERMGIVESTHEVALUEWHERETHELINEINTERCEPTSTHEX2AXIS255THEEASIESTWAYTODETERMINEWHICHSIDEOFTHELINEISPERMITTEDISTOCHECKWHETHERTHEORIGIN0,0SATISFIESTHECONSTRAINTIFITDOES,THENTHEPERMISSIBLEREGIONLIESONTHESIDEOFTHECONSTRAINTWHERETHEORIGINISOTHERWISEITLIESONTHEOTHERSIDE261THESOLVERDIALOGUEBOX262THEADDCONSTRAINTDIALOGUEBOX263THEASSUMELINEARMODELOPTIONANDTHEASSUMENONNEGATIVEOPTION271CLEANINGPRODUCTSFORHOMEUSE272TELEVISIONANDPRINTMEDIA273DETERMINEHOWMUCHTOADVERTISEINEACHMEDIUMTOMEETTHEMARKETSHAREGOALSATAMINIMUMTOTALCOST23274THECHANGINGCELLSAREINTHECOLUMNFORTHECORRESPONDINGADVERTISINGMEDIUM275THEOBJECTIVEISTOMINIMIZETOTALCOSTRATHERTHANMAXIMIZEPROFITTHEFUNCTIONALCONSTRAINTSCONTAINRATHERTHAN276NO277CLOSERTOTHEORIGIN281NO282THEGRAPHICALMETHODHELPSAMANAGERDEVELOPAGOODINTUITIVEFEELINGFORTHELINEARPROGRAMMINGIS2831WHERELINEARPROGRAMMINGISAPPLICABLE2WHEREITSHOULDNOTBEAPPLIED3DISTINGUISHBETWEENCOMPETENTANDSHODDYSTUDIESUSINGLINEARPROGRAMMING4HOWTOINTERPRETTHERESULTSOFALINEARPROGRAMMINGSTUDYPROBLEMS21ATHETWOFACTORSTHATOFTENHINDERTHEUSEOFOPTIMIZATIONMODELSBYMANAGERSARECULTURALDIFFERENCESANDRESPONSETIMECULTURALDIFFERENCESCAUSEMANAGERSANDMODELDEVELOPERSTOOFTENHAVEAHARDTIMEUNDERSTANDINGEACHOTHERRESPONSETIMEISOFTENSLOWDUETOTHETIMETOTRANSLATE,FORMULATE,ANDSOLVETHEMANGERSPROBLEMUSINGOPTIMIZATIONSYSTEMSBTHECOMPANYSHIFTEDFROMANEMPHASISONTHEMANUFACTUREOFTHICKERPLYWOODSTOTHINNERPLYWOODSCPONDEROSAPLANSTOUSEOPTIMIZATIONINTHEUSEOFTIMBERFOROTHERPRODUCTSALSOINADDITION,OPTIMIZATIONMAYBEUSEDFORRAWMATERIALANDINVENTORYMANAGEMENTANDFORFINANCIALPLANNING22ATHESHIFTSCHEDULESATAIRPORTSANDRESERVATIONOFFICESWEREDONEBYHANDPRIORTOTHISSTUDYBTHEPROJECTREQUIREMENTSWEREITODETERMINETHENEEDSFORINCREASEDMANPOWER,IITOIDENTIFYEXCESSMANPOWERFORREALLOCATION,IIITOREDUCETHETIMEREQUIREDFORPREPARINGSCHEDULES,IVTOMAKEMANPOWERALLOCATIONMOREDAYANDTIMESENSITIVE,ANDVTOQUANTIFYTHECOSTASSOCIATEDWITHSCHEDULINGCFLEXIBILITY,SUCHASTHENUMBEROFSTARTTIMES,THEPREFERREDSHIFTLENGTHS,THELENGTHOFBREAKS,THEPREFERREDDAYSOFFCOMBINATIONS,ETCWERECONSIDEREDTHISVERSAT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