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51CHAPTER5WHATIFANALYSISFORLINEARPROGRAMMINGREVIEWQUESTIONS511THEPARAMETERSOFALINEARPROGRAMMINGMODELARETHECONSTANTSCOEFFICIENTSORRIGHTHANDSIDESINTHEFUNCTIONALCONSTRAINTSANDTHEOBJECTIVEFUNCTION512MANYOFTHEPARAMETERSOFALINEARPROGRAMMINGMODELAREONLYESTIMATESOFQUANTITIESTHATCANNOTBEDETERMINEDPRECISELYANDTHUSRESULTININACCURACIES513WHATIFANALYSISREVEALSHOWCLOSEEACHOFTHESEESTIMATESNEEDSTOBETOAVOIDOBTAININGANERRONEOUSOPTIMALSOLUTION,ANDTHEREFOREPINPOINTSTHESENSITIVEPARAMETERSWHEREEXTRACAREISNEEDEDTOREFINETHEIRESTIMATES514NO,IFTHEOPTIMALSOLUTIONWILLREMAINTHESAMEOVERAWIDERANGEOFVALUESFORAPARTICULARCOEFFICIENT,THENITMAYBEAPPROPRIATETOMAKEONLYAFAIRLYROUGHESTIMATEFORAPARAMETEROFAMODEL515CONDITIONSTHATIMPACTTHEPARAMETERSOFAMODEL,SUCHASUNITPROFIT,MAYCHANGEOVERTIMEANDRENDERTHEMINACCURATE516IFCONDITIONSCHANGE,WHATIFANALYSISLEAVESSIGNPOSTSTHATINDICATEWHETHERARESULTINGCHANGEINAPARAMETEROFTHEMODELCHANGESTHEOPTIMALSOLUTION517SENSITIVITYANALYSISISSTUDYINGHOWCHANGESINTHEPARAMETERSOFALINEARPROGRAMMINGMODELAFFECTTHEOPTIMALSOLUTION518WHATIFANALYSISPROVIDESGUIDANCEABOUTWHATTHEIMPACTWOULDBEOFALTERINGPOLICYDECISIONSTHATAREREPRESENTEDBYPARAMETERSOFAMODEL521THEESTIMATESOFTHEUNITPROFITSFORTHETWOPRODUCTSAREMOSTQUESTIONABLE522THENUMBEROFHOURSOFPRODUCTIONTIMETHATISBEINGMADEAVAILABLEPERWEEKINTHETHREEPLANTSMIGHTCHANGEAFTERANALYSIS531THEALLOWABLERANGEFORACOEFFICIENTINTHEOBJECTIVEFUNCTIONISTHERANGEOFVALUESOVERWHICHTHEOPTIMALSOLUTIONFORTHEORIGINALMODELREMAINSOPTIMAL532IFTHETRUEVALUEFORACOEFFICIENTINTHEOBJECTIVEFUNCTIONLIESOUTSIDEITSALLOWABLERANGETHENTHEOPTIMALSOLUTIONWOULDCHANGEANDTHEPROBLEMWOULDNEEDTOBERESOLVED533THEOBJECTIVECOEFFICIENTCOLUMNGIVESTHECURRENTVALUEOFEACHCOEFFICIENTTHEALLOWABLEINCREASECOLUMNANDTHEALLOWABLEDECREASECOLUMNGIVETHEAMOUNTTHATEACHCOEFFICIENTMAYDIFFERFROMTHESEVALUESTOREMAINWITHINTHEALLOWABLERANGEFORWHICHTHEOPTIMALSOLUTIONFORTHEORIGINALMODELREMAINSOPTIMAL52541THE100RULECONSIDERSTHEPERCENTAGEOFTHEALLOWABLECHANGEINCREASEORDECREASEFOREACHCOEFFICIENTINTHEOBJECTIVEFUNCTION542IFTHESUMOFTHEPERCENTAGECHANGESDONOTEXCEED100THENTHEORIGINALOPTIMALSOLUTIONDEFINITELYWILLSTILLBEOPTIMAL543NO,EXCEEDING100MAYORMAYNOTCHANGETHEOPTIMALSOLUTIONDEPENDINGONTHEDIRECTIONSOFTHECHANGESINTHECOEFFICIENTS551THEPARAMETERSINTHECONSTRAINTSMAYONLYBEESTIMATES,OR,ESPECIALLYFORTHERIGHTHANDSIDES,MAYWELLREPRESENTMANAGERIALPOLICYDECISIONS552THERIGHTHANDSIDESOFTHEFUNCTIONALCONSTRAINTSMAYWELLREPRESENTMANAGERIALPOLICYDECISIONSRATHERTHANQUANTITIESTHATARELARGELYOUTSIDETHECONTROLOFMANAGEMENT553THESHADOWPRICEFORAFUNCTIONALCONSTRAINTISTHERATEATWHICHTHEVALUEOFTHEOBJECTIVEFUNCTIONCANBEINCREASEDBYINCREASINGTHERIGHTHANDSIDEOFTHECONSTRAINTBYASMALLAMOUNT554THESHADOWPRICECANBEFOUNDWITHTHESPREADSHEETBYINCREASINGTHERIGHTHANDSIDEBYONE,ANDTHENRESOLVINGTODETERMINETHEINCREASEINTHEOBJECTIVEFUNCTIONVALUEITCANBEFOUNDSIMILARLYWITHASOLVERTABLEBYCREATINGATABLETHATSHOWSTHEINCREASEINPROFITFORAUNITINCREASEINTHERIGHTHANDSIDETHESHADOWPRICEISGIVENDIRECTLYINTHESENSITIVITYREPORT555THESHADOWPRICEFORAFUNCTIONALCONSTRAINTINFORMSMANAGEMENTABOUTHOWMUCHTHETOTALPROFITWILLINCREASEFOREACHEXTRAUNITOFARESOURCERIGHTHANDSIDEOFACONSTRAINT556YESTHESHADOWPRICEALSOINDICATESHOWMUCHTHEVALUEOFTHEOBJECTIVEFUNCTIONWILLDECREASEIFTHERIGHTHANDSIDEWERETOBEDECREASEDBY1557ASHADOWPRICEOF0TELLSAMANAGERTHATASMALLCHANGEINTHERIGHTHANDSIDEOFTHECONSTRAINTWILLNOTCHANGETHEOBJECTIVEFUNCTIONVALUEATALL558THEALLOWABLERANGEFORTHERIGHTHANDSIDEOFAFUNCTIONALCONSTRAINTISFOUNDINTHESOLVERSSENSITIVITYREPORTBYUSINGTHECOLUMNSLABELEDCONSTRAINTRHSIDE,ALLOWABLEINCREASE,ANDALLOWABLEDECREASE559THEALLOWABLERANGESFORTHERIGHTHANDSIDESAREOFINTERESTTOMANAGERSBECAUSETHEYTELLTHEMHOWLARGECHANGESINTHERIGHTHANDSIDESCANBEBEFORETHESHADOWPRICESARENOLONGERAPPLICABLE561THEREMAYBEUNCERTAINTYABOUTTHEESTIMATESFORANUMBEROFTHEPARAMETERSINTHEFUNCTIONALCONSTRAINTSALSO,THERIGHTHANDSIDESOFTHECONSTRAINTSOFTENREPRESENTMANAGERIALPOLICYDECISIONSTHESEDECISIONSAREFREQUENTLYINTERRELATEDANDSONEEDTOBECONSIDEREDSIMULTANEOUSLY562THESPREADSHEETCANBEUSEDTODIRECTLYDETERMINETHEIMPACTOFSEVERALSIMULTANEOUSCHANGESSIMPLYCHANGETHEPAREMETERSANDRESOLVE563USINGSOLVERTABLE,TRIALVALUESCANBEENUMERATEDSIMULTANEOUSLYFORONEORTWODATACELLS,WITHTHEPOSSIBILITYOFENTERINGFORMULASFORADDITIONALDATACELLSINTERMSOFTHESEONEORTWODATACELLS53564THERIGHTHANDSIDESOFTHECONSTRAINTSOFTENREPRESENTMANAGERIALPOLICYDECISIONSTHESEDECISIONSAREFREQUENTLYINTERRELATEDANDSONEEDTOBECONSIDEREDSIMULTANEOUSLY565THE100PERCENTRULEBASICALLYSAYSTHATWECANSAFELYUSETHESHADOWPRICESTOPREDICTTHEEFFECTOFSIMULTANEOUSCHANGESINTHERIGHTHANDSIDESIFTHESUMOFTHEPERCENTAGESOFTHECHANGESDOESNOTEXCEED100PERCENT566THEDATANEEDEDTOAPPLYTHE100RULEFORSIMULTANEOUSCHANGESINRIGHTHANDSIDESAREGIVENBYTHESENSITIVITYREPORTCONSTRAINTRHSIDE,ALLOWABLEINCREASE,ANDALLOWABLEDECREASE567IFTHESUMOFTHEPERCENTAGECHANGESDOESNOTEXCEED100,THESHADOWPRICESDEFINITELYWILLSTILLBEVALID568IFTHESUMOFTHEPERCENTAGESOFALLOWABLECHANGESINTHERIGHTHANDSIDESDOESEXCEED100,THENWECANNOTBESUREIFTHESHADOWPRICESWILLSTILLBEVALIDPROBLEMS51A123456789ABCDEFTOYSSUBASSEMBLIESUNITPROFIT300250USEDAVAILABLESUBASSEMBLYA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