Page 195 Chapter 5 Simulation Modeling: Concepts and Practice CONTENTS 5.1 A Simple Problem: Operations at Conley Fisheries 5.2 Preliminary Analysis of Conley Fisheries 5.3 A Simulation Model of the Conley Fisheries Problem 5.4 Random Number Generators 5.5 Creating Numbers That Obey a Discrete Probability Distribution 5.6 Creating Numbers That Obey a Continuous Probability Distribution 5.7 Completing the Simulation Model of Conley Fisheries 5.8 Using the Sample Data for Analysis 5.9 Summary of Simulation Modeling, and Guidelines on the Use of Simulation 5.10 Computer Software for Simulation Modeling 5.11 Typical Uses of Simulation Models 5.12 Case Modules The Gentle Lentil Restaurant To Hedge or Not to Hedge? Ontario Gateway Casterbridge Bank This chapter presents the basic concepts and demonstrates the managerial use of a simulation model, which is a computer representation of a problem that involves random variables. The chief advantage of a simulation model of a problem is that the simulation model can forecast the consequences of various management decisions before such decisions must be made. Simulation models are used in a very wide variety of management settings, including modeling of manufacturing operations, modeling of service operations where queues form (such as in banking, passenger air travel, food services, etc.), modeling of investment alternatives, and analyzing and pricing of sophisticated financial instruments. A simulation model is an extremely useful tool to help a manager make difficult decisions in an environment of uncertainty.
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Chapter5SimulationModeling:ConceptsandPractice
CONTENTS
5.1ASimpleProblem:OperationsatConleyFisheries
5.2PreliminaryAnalysisofConleyFisheries
5.3ASimulationModeloftheConleyFisheriesProblem
5.4RandomNumberGenerators
5.5CreatingNumbersThatObeyaDiscreteProbabilityDistribution
5.6CreatingNumbersThatObeyaContinuousProbabilityDistribution
5.7CompletingtheSimulationModelofConleyFisheries
5.8UsingtheSampleDataforAnalysis
5.9SummaryofSimulationModeling,andGuidelinesontheUseofSimulation
5.10ComputerSoftwareforSimulationModeling
5.11TypicalUsesofSimulationModels
5.12CaseModules
TheGentleLentilRestaurant
ToHedgeorNottoHedge?
OntarioGateway
CasterbridgeBank
Thischapterpresentsthebasicconceptsanddemonstratesthemanagerialuseofasimulationmodel,whichisacomputerrepresentationofaproblemthatinvolvesrandomvariables.Thechiefadvantageofasimulationmodelofaproblemisthatthesimulationmodelcanforecasttheconsequencesofvariousmanagementdecisionsbeforesuchdecisionsmustbemade.Simulationmodelsareusedinaverywidevarietyofmanagementsettings,includingmodelingofmanufacturingoperations,modelingofserviceoperationswherequeuesform(suchasinbanking,passengerairtravel,foodservices,etc.),modelingofinvestmentalternatives,andanalyzingandpricingofsophisticatedfinancialinstruments.Asimulationmodelisanextremelyusefultooltohelpamanagermakedifficultdecisionsinanenvironmentofuncertainty.
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5.1ASimpleProblem:OperationsatConleyFisheries
Thecentralideasofasimulationmodelarebestunderstoodwhenpresentedinthecontextofamanagementproblem.Toinitiatetheseideas,considerthefollowingpracticalproblemfacedbyConleyFisheries,Inc.
OperationsatConleyFisheries,Inc.
ClintConley,presidentofConleyFisheries,Inc.,operatesafleetoffiftycodfishingboatsoutofNewburyport,Massachusetts.Clint'sfatherstartedthecompanyfortyyearsagobuthasrecentlyturnedthebusinessovertoClint,whohasbeenworkingforthefamilybusinesssinceearninghisMBAtenyearsago.Everyweekdayoftheyear,eachboatleavesearlyinthemorning,fishesformostoftheday,andcompletesitscatchofcodfish(3,500lbs.ofcodfish)bymid-afternoon.Theboatthenhasanumberofportswhereitcansellitsdailycatch.Thepriceofcodfishatsomeportsisveryuncertainandcanchangequiteabitevenonadailybasis.Also,thepriceofcodfishtendstobedifferentatdifferentports.Furthermore,someportshaveonlylimiteddemandforcodfish,andsoifaboatarrivesrelativelylaterthanotherfishingboatsatthatport,thecatchoffishcannotbesoldandsomustbedisposedofinoceanwaters.
TokeepConleyFisheries'problemsimpleenoughtoanalyzewithease,assumethatConleyFisheriesonlyoperatesoneboat,andthatthedailyoperatingexpensesoftheboatare$10,000perday.Alsoassumethattheboatisalwaysabletocatchallofthefishthatitcanhold,whichis3,500lb.ofcodfish.
AssumethattheConleyFisheries'boatcanbringitscatchtoeithertheportinGloucesterortheportinRockport,Massachusetts.Gloucesterisamajorportforcodfishwithawell-establishedmarket.ThepriceofcodfishinGloucesteris$3.25/lb.,andthispricehasbeenstableforquitesometime.ThepriceofcodfishinRockporttendstobeabithigherthaninGloucesterbuthasalotofvariability.ClinthasestimatedthatthedailypriceofcodfishinRockportisNormallydistributedwithameanofµ=$3.65/lb.andwithastandarddeviationofs=$0.20/lb.
TheportinGloucesterhasaverylargemarketforcodfish,andsoConleyFisheriesneverhasaproblemsellingtheircodfishinGloucester.Incontrast,theportinRockportismuchsmaller,andsometimestheboatisunabletosellpartorallofitsdailycatchinRockport.Basedonpasthistory,ClinthasestimatedthatthedemandforcodfishinRockportthathefaceswhenhisboatarrivesattheportinRockportobeysthediscreteprobabilitydistributiondepictedinTable5.1.
TABLE5.1DailydemandinRockportfacedbyConleyFisheries.Demand(lbs.ofcodfish) Probability
0 0.021,000 0.032,000 0.053,000 0.084,000 0.335,000 0.296,000 0.20
ItisassumedthatthepriceofcodfishinRockportandthedemandforcodfishinRockportfacedbyConleyFisheriesareindependentofoneanother.Therefore,thereisnocorrelationbetweenthedailypriceofcodfishandthedailydemandinRockportfacedbyConleyFisheries.
Atthestartofanygivenday,thedecisionClintConleyfacesiswhichporttouseforsellinghisdailycatch.ThepriceofcodfishthatthecatchmightcommandinRockportisonlyknownifandwhentheboatdocksattheportandnegotiateswithbuyers.Aftertheboatdocksatoneofthetwoports,itmustsellitscatchatthatportornotatall,sinceittakestoomuchtimetopilottheboatoutofoneportandpoweritallthewaytotheotherport.
ClintConleyisjustasanxiousasanyotherbusinesspersontoearnaprofit.Forthisreason,hewondersifthesmartstrategymightbetosellhisdailycatchinRockport.Afterall,theexpectedpriceofcodfishishigherinRockport,andalthoughthe
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standarddeviationofthepriceishigh,andhencethereisgreaterriskwiththisstrategy,heisnotaversetotakingchanceswhentheymakegoodsense.However,italsomightbetruethatthesmartstrategycouldbetosellthecodfishinGloucester,sinceinGloucesterthereisampledemandforhisdailycatch,whereasinRockportthereisthepossibilitythathemightnotsellallofhiscatch(andsopotentiallylosevaluablerevenue).Itisnotcleartohimwhichstrategyisbest.
OnecanstarttoanalyzethisproblembycomputingthedailyearningsifClintchoosestosellhisdailycatchofcodfishinGloucester.TheearningsfromusingGloucester,denotedbyG,issimply:
G=($3.25)(3,500)$10,000=$1,375,
whichistherevenueof$3.25perpoundtimesthenumberofpoundsofcodfish(3,500lbs.)minusthedailyoperatingcostsof$10,000.
ThecomputationofdailyearningsifClintchoosesRockportisnotsostraight-forward,becausethepriceandthedemandareeachuncertain.ThereforethedailyearningsfromchoosingRockportisanuncertainquantity,i.e.,arandomvariable.Inordertomakeaninformeddecisionastowhichporttouse,itwouldbehelpfultoanswersuchquestionsas:
(a)WhatistheshapeoftheprobabilitydistributionofdailyearningsfromusingRockport?
(b)Onanygivenday,whatistheprobabilitythatConleyFisherieswouldearnmoremoneyfromusingRockportinsteadofGloucester?
(c)Onanygivenday,whatistheprobabilitythatConleyFisherieswilllosemoneyiftheyuseRockport?
(d)WhatistheexpecteddailyearningsfromusingRockport?
(e)WhatisthestandarddeviationofthedailyearningsfromusingRockport?
Theanswerstothesefivequestionsare,inalllikelihood,allthatisneededforClintConleytochoosetheportstrategythatwillbestservetheinterestsofConleyFisheries.
5.2PreliminaryAnalysisofConleyFisheries
OnecanbegintoanalyzethedecisionproblematConleyFisheriesbylookingattheproblemintermsofrandomvariables.Wefirstdefinethefollowingtworandomvariables:
Page198PR=priceofcodfishattheportinRockportin$/lb.
D=demandfacedbyConleyFisheriesattheportinRockportinlbs.
Accordingtothestatementoftheproblem,theassumptionsaboutthedistributionsofthesetworandomvariablesareasshowninTable5.2.
TABLE5.2Summaryofrandomvariablesandtheirdistributions.RandomVariable Distribution
PR Normal,µ=3.65,s=0.20D Discretedistribution,asgiveninTable5.1.
InordertoanalyzethedecisionproblematConleyFisheries,wenextdefineonenewrandomvariableFtobethedailyearnings(indollars)iftheboatdocksattheportinRockporttosellitscatchofcodfish.NotethatFisindeedarandomvariable.ThequantityFisuncertain,andinfact,FisafunctionofthetwoquantitiesPRandD,whicharethemselvesrandomvariables.Infact,itiseasytoexpressFasafunctionoftherandomvariablesPRandD.TheformulaforFisasfollows:
i.e.,Fissimplythepricetimesthequantityofcodfishthatcanbesold(totalsalesrevenue)minusthecostofdailyoperations.However,inthiscase,thequantityofcodfishthatcanbesoldistheminimumofthequantityofthecatch(3,500lbs.)andthedemandforcodfishfacedbyConleyFisheriesatthedock(D).Infact,theaboveexpressioncanalternativelybewrittenas:
F=PR×min(3,500,D)10,000,
wheretheexpressionmin(a,b)standsfortheminimumofthetwoquantitiesaandb.
Thisformulaisaconcisewayofstatingtheproblemintermsoftheunderlyingrandomvariables.WiththeterminologyjustintroducedthequestionsattheendofSection5.1canberestatedas:
(a)WhatistheshapeoftheprobabilitydensityfunctionofF?
(b)WhatisP(F>$1,375)?
(c)WhatisP(F<$0)?
(d)WhatistheexpectedvalueofF?
(e)WhatisthestandarddeviationofF?
Nowthatthesefivequestionshavebeenrestatedconciselyintermsofprobabilitydistributions,onecouldattempttoanswerthefivequestionsusingthetoolsofChapters2and3.NoticethateachofthesefivequestionspertainstotherandomvariableF.Furthermore,fromtheformulaaboveforF,weseethatthatFisarelativelysimplefunctionofthetworandomvariablesPRandD.However,itturnsoutthatwhenrandomvariablesarecombined,eitherbyaddition,multiplication,orsomemorecomplicatedoperation,thenewrandomvariablerarelyhasaconvenientformforwhichthereareconvenientformulas.(AnexceptiontothisdictumisthecaseofthesumofjointlyNormallydistributedrandomvariables.)Almostallotherinstanceswhererandomvariablesarecombinedareverycomplextoanalyze;andthereareseldomformulasfor
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themean,thestandarddeviation,fortheprobabilitydensityfunction,orforthecumulativedistributionfunctionoftheresultingrandomvariable.
Inlightoftheprecedingremarks,therearenoformulasortablesthatwillallowustoanswerthefivequestionsposedaboveabouttherandomvariableF.However,asweshallsoonsee,acomputersimulationmodelcanbeusedtoeffectivelygainalloftheinformationweneedaboutthedistributionoftherandomvariableF,andsoenableustomakeaninformedandoptimalstrategydecisionaboutwhichporttousetosellthedailycatchofcodfish.
5.3ASimulationModeloftheConleyFisheriesProblem
SupposeforthemomentthatClintConleyisaverywealthyindividual,whodoesnotneedtoearnanymoney,andwhoissimplycurioustoknowtheanswerstothefivequestionsposedaboveforthepureintellectualpleasureitwouldgivehim!Withallofhistimeandmoney,Clintcouldaffordtoperformthefollowingexperiment.Foreachofthenext200weekdays,ClintcouldsendtheboatouttofishforitscatchandthenbringtheboattotheportinRockportattheendofthedaytosellthecatch.Hecouldrecordthedailyearningsfromthisstrategyeachdayand,insodoing,wouldobtainsampledataof200observedvaluesoftherandomvariableF.ClintcouldthenusethemethodologyofChapter4toanswerquestionsabouttheprobabilitydistributionofFbasedonsampledatatoobtainapproximateanswerstothefivequestionsposedearlier.
Ofcourse,ClintConleyisnotaverywealthyindividual,andhedoesneedtoearnmoney,andsohecannotaffordtospendthetimeandmoneytocollectadatasetof200valuesofdailyearningsfromtheRockportstrategy.Wenowwillshowhowtoconstructanelementarysimulationmodelonacomputer,whichcanbeusedtoanswerthefivequestionsposedbyClint.
Thecentralnotionbehindmostcomputermodelsistosomehowre-createtheeventsonthecomputerwhichoneisinterestedinstudying.Forexample,ineconomicmodeling,onebuildsacomputermodelofthenationaleconomytoseehowvariouspricesandquantitieswillmoveovertime.Inmilitarymodeling,oneconstructsa''wargame"modeltostudytheeffectivenessofnewweaponsormilitarytactics,withouthavingtogotowartotesttheseweaponsand/ortactics.Inweatherorclimateforecasting,oneconstructsanatmosphericmodeltoseehowstormsandfrontalsystemswillmoveovertimeinordertopredicttheweatherwithgreateraccuracy.Ofcourse,thesethreeexamplesareobviouslyverycomplexmodelsinvolvingsophisticatedeconomicprinciples,orsophisticatedmilitaryinteractions,orsophisticatedconceptsaboutthephysicsofweathersystems.
IntheConleyFisheriesproblem,onecanalsobuildacomputermodelthatwillcreatetheeventsonthecomputerwhichoneisinterestedinstudying.Forthisparticularproblem,theeventsofinterestarethepriceofcodfishandthedemandforcodfishthatConleyFisherieswouldfaceinRockportovera200dayperiod.Forthesakeofdiscussion,letusreducethelengthoftheperiodfrom200daysto20days,asthiswillsufficeforpedagogicalpurposes.
Asitturnsout,theConleyFisheriesproblemisafairlyelementaryproblemtomodel.OnecanstartbybuildingtheblanktableshowninTable5.3.Thistablehasalistofdays(numbered1through20)inthefirstcolumn,followedbyblanksintheremainingcolumns.Ourfirsttaskwillbetofillinthesecondcolumnofthetableby
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modelingthedemandforcodfishfacedbyConleyFisheries.RecallthatthisdemandobeysadiscreteprobabilitydistributionandisgiveninTable5.1.Thus,wewouldliketofillinalloftheentriesofthecolumn''DemandinRockport"withnumbersthatobeythediscretedistributionfordemandofTable5.1.Putaslightlydifferentway,wewouldliketofillinalloftheentriesofthecolumn"DemandinRockport"withnumbersdrawnfromthediscreteprobabilitydistributionofTable5.1.
TABLE5.3ComputerworksheetforasimulationofConleyFisheries.
DayDemandinRockport
(lbs.)QuantityofCodfishSold
(lbs.)PriceofCodfishinRockport
($/lb.)DailyEarningsinRockport
($)123456789
1011121314151617181920
Oncewehavefilledintheentriesofthedemandcolumn,itistheneasytofillintheentriesofthenextcolumnlabeled"QuantityofCodfishSold."BecausetheConleyFisheriesboatalwayshasadailycatchof3,500lbs.ofcodfish,thequantitysoldwillbeeither3,500orthe"DemandinRockport"quantity,whicheverofthetwoissmaller.(Forexample,ifthedemandinRockportis5,000,thenthequantitysoldwillbe3,500.IfthedemandisRockportis2,000,thenthequantitysoldwillbe2,000.)
Thefourthcolumnofthetableislabeled"PriceofCodfishinRockport."WewouldliketofillinthiscolumnofthetablebymodelingthepriceofcodfishinRockport.RecallthatthepriceofcodfishobeysaNormaldistributionwithmeanµ=$3.65andstandarddeviations=$0.20.Thus,wewouldliketofillinthealloftheentriesofthecolumn"PriceofCodfishinRockport"withnumbersthatobeyaNormaldistributionwithmeanµ=$3.65andstandarddeviations=$0.20.Putaslightlydifferentway,wewouldliketofillinalloftheentriesofthefourthcolumnwithnumbersdrawnfromaNormaldistributionwithmeanµ=3.65andstandarddeviations=0.20.
ThefifthcolumnofTable5.3willcontainthedailyearningsinRockportforeachofthenumbereddays.Thisquantityiselementarytocomputegiventheentriesintheothercolumns.Itis:
DailyEarnings=(QuantityofCodfishSold)×(PriceofCod)$10,000.
Thatis,thedailyearningsforeachofthedaysissimplythepricetimesthequantity,minusthedailyoperatingcostof$10,000.Putinamoreconvenientlight,foreachrowofthetable,theentryinthefifthcolumniscomputedbymultiplyingthecorrespondingentriesinthethirdandfourthcolumnsandthensubtracting$10,000.
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Summarizingsofar,wewouldliketofillintheentriesofTable5.3foreachofthe20rowsofdays.Ifwecanaccomplishthis,thenthelastcolumnofTable5.3willcontainasampleofcomputer-generated,i.e.,simulated,valuesofthedailyearningsinRockport.Thesampleof20simulateddailyearningsvaluescanthenbeusedtoanswerthefivequestionsposedabouttherandomvariableF,usingthemethodsofstatisticalsamplingthatweredevelopedinChapter4.
Inordertosimulatetheeventsof20daysofsellingthecodfishinRockport,wewillneedtofillinalloftheentriesofthecolumn''DemandinRockport"withnumbersdrawnfromthediscreteprobabilitydistributionofTable5.1.Wewillalsoneedtofillinalloftheentriesofthecolumn"PriceofCodfishinRockport"withnumbersdrawnfromaNormaldistributionwithmeanµ=3.65andstandarddeviations=0.20.Onceweareabletodothis,thecomputationofalloftheothernumbersinthetableisextremelysimple.
TherearetwocriticalstepsinfillingintheentriesofTable5.3thatareasyetunclear.ThefirstistosomehowgenerateasequenceofnumbersthataredrawnfromandhenceobeythediscretedistributionofTable5.1.Thesenumberswillbeusedtofillinthe"DemandinRockport"columnofthetable.ThesecondstepistosomehowgenerateasequenceofnumbersthataredrawnfromandhenceobeyaNormaldistributionwithmeanµ=3.65andstandarddeviations=0.20.Thesenumberswillbeusedtofillinthe"PriceofCodfishinRockport"columnofthetable.Oncewehavefilledinalloftheentriesofthesetwocolumns,thecomputationsofallothernumbersinthetablecanbeaccomplishedwithease.
Therefore,thecriticalissueincreatingthesimulationmodelistobeabletogenerateasequenceofnumbersthataredrawnfromagivenprobabilitydistribution.Tounderstandhowtodothis,oneneedsacomputerthatcangeneraterandomnumbers.Thisisdiscussedinthenextsection.
5.4RandomNumberGenerators
Arandomnumbergeneratorisanymeansofautomaticallygeneratingasequenceofdifferentnumberseachofwhichisindependentoftheother,andeachofwhichobeystheuniformdistributionontheintervalfrom0.0to1.0.
Mostcomputersoftwarepackagesthatdoanykindofscientificcomputationhaveamathematicalfunctioncorrespondingtoarandomnumbergenerator.Infact,mosthand-heldscientificcalculatorsalsohavearandomnumbergeneratorfunction.Everytimetheuserpressesthebuttonfortherandomnumbergeneratoronahand-heldscientificcalculator,thecalculatorcreatesadifferentnumberanddisplaysthisnumberonthescreen;andeachofthesenumbersisdrawnaccordingtoauniformdistributionontheintervalfrom0.0to1.0.
TheExcel®spreadsheetsoftwarealsohasarandomnumbergenerator.Thisrandomnumbergeneratorcanbeusedtocreatearandomnumberbetween0.0and1.0inanycellbyentering"=RAND()"inthedesiredcell.EverytimethefunctionRAND()iscalled,thesoftwarewillreturnadifferentnumber(whosevalueisalwaysbetween0.0and1.0);andthisnumberwillbedrawnfromauniformdistributionbetween0.0and1.0.Forexample,supposeoneweretotype"=RAND()"intothefirstrowofcolumn"A"ofanExcel®spreadsheet.ThenthecorrespondingspreadsheetmightlooklikethatshowninFigure5.1.
FIGURE5.1Spreadsheetillustrationofarandomnumbergenerator.
LetXdenotetherandomvariablethatisthevaluethatwillbereturnedbyacalltoarandomnumbergenerator.ThenXwillobeyauniformdistributionontheintervalfrom0.0to1.0.Herearesomeconsequencesofthisfact:
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P(X£0.5)=0.5,
P(X³0.5)=0.5,
P(0.2£X£0.9)=0.7.
Infact,foranytwonumbersaandbforwhich0£a£b£1,then:
P(a£X£b)=ba.
StatedinplainEnglish,thissaysthatforanyintervalofnumbersbetweenzeroandone,theprobabilitythatXwilllieinthisintervalisequaltothewidthoftheinterval.Thisisaveryimportantpropertyoftheuniformdistributionontheintervalfrom0.0to1.0,whichwewilluseshortlytogreatadvantage.
(Onemightask,''Howdoesacomputertypicallygeneratearandomnumberthatobeysauniformdistributionbetween0.0and1.0?"Theanswertothisquestionisrathertechnical,butusuallyrandomnumbersaregeneratedbymeansofexaminingthedigitsthatarecutoffwhentwoverylargenumbersaremultipliedtogetherandplacingthesedigitstotherightofthedecimalplace.Theextradigitsthatthecomputercannotstoreareinsomesenserandomandareusedtoformthesequenceofrandomnumbers.)
(Actually,onanevenmoretechnicalnote,therandomnumbergeneratorsthatareprogrammedintocomputersoftwarearemorecorrectlycalled"pseudo-randomnumbergenerators."Thisisbecauseacomputerscientistorothermathematicallytrainedprofessionalcouldinfactpredictthesequenceofnumbersproducedbythesoftwareifhe/shehadasophisticatedknowledgeofthewaythenumbergenerator'ssoftwareprogramisdesignedtooperate.Thisisaminortechnicalpointthatisofnoconsequencewhenusingsuchnumbergenerators,andsoitsufficestothinkofsuchnumbergeneratorsasactualrandomnumbergenerators.)
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Wenextaddresshowwecanusearandomnumbergeneratortocreatenumbersthatobeyadiscreteprobabilitydistribution.
5.5CreatingNumbersThatObeyaDiscreteProbabilityDistribution
ReturningtotheConleyFisheriesproblem,recallthatournexttaskistogenerateasequenceofdemandvaluesforeachofthe20daysbeingsimulated.RecallthatthedemandvaluesmustobeythediscreteprobabilitydistributionofTable5.1.
AccordingtoTable5.1,theprobabilitythatthedemandwillbe0lbs.is0.02.Also,theprobabilitythatthedemandis1,000lbs.is0.03,etc.Nowconsiderthefollowingruleforcreatingasequenceofdemandsthatwillobeythisdistribution.First,makeacalltoarandomnumbergenerator.Theoutputofthiscallwillbesomevaluex,andrecallthatxwillbeanobservedvaluefromauniformdistributionontheintervalfrom0.0to1.0.Supposethatwecreateademandvalueofd=0wheneverxliesbetween0.0and0.02.Thenbecausexhasbeendrawnfromauniformdistributionontheintervalfrom0.0to1.0,thelikelihoodthatxwillliebetween0.0and0.02willbeprecisely0.02;andsothelikelihoodthatdisequalto0willbeprecisely0.02.Thisisexactlywhatwewant.Wecandevelopasimilarruleinordertodecidewhentocreateademandvalueofd=1,000asfollows:Createademandvalueofd=1,000wheneverxliesbetween0.02and0.05.Notethatbecausexhasbeendrawnfromauniformdistributionontheintervalfrom0.0to1.0,thelikelihoodthatxwillliebetween0.02and0.05willbeprecisely0.03(whichisthewidthoftheinterval,i.e.,0.03=0.050.02),whichisexactlywhatwewant.Similarly,wecandeveloparuleinordertodecidewhentocreateademandvalueofd=2,000asfollows:Createademandvalueofd=2,000wheneverxliesbetween0.05and0.10.Noteonceagainthatbecausexhasbeendrawnfromauniformdistributionontheintervalfrom0.0to1.0,thelikelihoodthatxwillliebetween0.05and0.10willbeprecisely0.05(whichisthewidthoftheinterval,i.e.,0.05=0.100.05),whichisalsoexactlywhatwewant.Ifwecontinuethisprocessforallofthesevenpossibledemandvaluesoftheprobabilitydistributionofdemand,wecansummarizethemethodinTable5.4.
TABLE5.4IntervalruleforcreatingdailydemandinRockportfacedbyConleyFisheries.
Interval DemandValueCreated(lbs.ofcodfish)0.000.02 00.020.05 1,0000.050.10 2,0000.100.18 3,0000.180.51 4,0000.510.80 5,0000.801.00 6,000
Table5.4summarizestheintervalruleforcreatingasequenceofdemandvaluesthatobeytheprobabilitydistributionofdemand.Foreachofthe20daysunderconsideration,calltherandomnumbergeneratoronceinordertoobtainavaluexthatisdrawnfromauniformdistributionontheintervalfrom0.0to1.0.Next,findwhichofthesevenintervalsofTable5.4containsthevaluex.Lastofall,usethatintervaltocreatethedemandvaluedinthesecondcolumnofthetable.
Forexample,supposetherandomnumbergeneratorhasbeenusedtogenerate20randomvalues,andthesevalueshavebeenenteredintothesecondcolumn
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ofTable5.5.ConsiderthefirstrandomnumberinTable5.5,whichis0.3352.Asthisnumberliesbetween0.18and0.51,wecreateademandvalueonday1of4,000lbs.usingtheintervalruleofTable5.4.Next,considerthesecondrandomnumberinTable5.5,whichis0.4015.Asthisnumberliesbetween0.18and0.51,wealsocreateademandvalueonday2of4,000lbs.usingtheintervalruleofTable5.4.Next,considerthethirdrandomnumberinTable5.5,whichis0.1446.Asthisnumberliesbetween0.10and0.18,wecreateademandvalueonday3of3,000lbs.usingtheintervalruleofTable5.4.Ifwecontinuethisprocessforallofthe20daysportrayedinTable5.5,wewillcreatethedemandvaluesasshowninthethirdcolumnofTable5.5.
TABLE5.5WorksheetforgeneratingdemandforcodfishinRockport.Day RandomNumber DemandinRockport(lbs.ofcodfish)1 0.3352 4,0002 0.4015 4,0003 0.1446 3,0004 0.4323 4,0005 0.0358 1,0006 0.4999 4,0007 0.8808 6,0008 0.9013 6,0009 0.4602 4,00010 0.3489 4,00011 0.4212 4,00012 0.7267 5,00013 0.9421 6,00014 0.7059 5,00015 0.1024 3,00016 0.2478 4,00017 0.5940 5,00018 0.4459 4,00019 0.0511 2,00020 0.6618 5,000
NoticethatifwegeneratethedemandvaluesaccordingtotheintervalruleofTable5.4,thenthedemandvalueswillindeedobeytheprobabilitydistributionfordemandasspecifiedinTable5.1.Toseewhythisisso,consideranyparticularvalueofdemand,suchasdemandequalto5,000lbs.Ademandof5,000lbs.willbegeneratedforaparticulardaywhenevertherandomnumbergeneratedforthatdayliesintheintervalbetween0.51and0.80.Becausethisintervalhasawidthof0.29(0.800.51=0.29),andbecausexwasdrawnfromauniformdistributionbetween0.0and1.0,thelikelihoodthatanygivenxwilllieinthisparticularintervalisprecisely0.29.Thisargumentholdsforalloftheothersixvaluesofdemand,becausethewidthoftheassignmentintervalsofTable5.4areexactlythesameastheprobabilityvaluesofTable5.1.Infact,thatiswhytheywereconstructedthisparticularway.Forexample,accordingtoTable5.1,theprobabilitythatdemandforcodfishwillbe4,000lbs.is0.33.InTable5.4,thewidthoftherandomnumberassignmentintervalis0.510.18=0.33.Therefore,thelikelihoodinthesimulationmodelthatthemodelwillcreateademandof4,000lbs.is0.33.
NoticehowtheentriesinTable5.4arecomputed.Onesimplydividesupthenumbersbetween0.0and1.0intonon-overlappingintervalswhosewidthscorrespondtotheprobabilitydistributionfunctionofinterest.Thegeneralmethodforgeneratingasequenceofnumbersthatobeyagivendiscreteprobabilitydistributionissummarizedasfollows:
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GeneralMethodforCreatingSampleDatadrawnfromaDiscreteProbabilityDistribution
1.Divideupthenumberlinebetween0.0and1.0intonon-overlappingintervals,oneintervalforeachofthepossiblevaluesofthediscretedistribution,andsuchthatthewidthofeachintervalcorrespondstotheprobabilityforeachvalue.
2.Usearandomnumbergeneratortogenerateasequenceofrandomnumbersthatobeytheuniformdistributionbetween0.0and1.0.
3.Foreachrandomnumbergeneratedinthesequence,assignthevaluecorrespondingtotheintervalthattherandomnumberliesin.
ThereisoneminortechnicalpointthatneedstobementionedabouttheendpointsoftheassignmentintervalsofTable5.4.AccordingtoTable5.4,itisnotclearwhatdemandvalueshouldbecreatedifthevalueoftherandomnumberxisexactly0.51(eitherd=4,000ord=5,000),orifxisexactlyanyoftheotherendpointvalues(0.02,0.05,0.10,0.18,0.51,or0.80).Becausethenumberxwasdrawnfromacontinuousuniformdistribution,theprobabilitythatitwillhaveavalueofprecisely0.51iszero,andsothisisextremelyunlikelytooccur.Butjustincase,wecanbemoreexactintheruleofTable5.4tospecifythatwhenthenumberisexactlyoneoftheendpointvalues,thenchoosethesmallerofthepossibledemandvalues.Thus,forexample,ifx=0.51,onewouldchoosed=4,000.
Thefinalcommentofthissectionconcernscomputersoftwareandcomputerimplementationofthemethod.Itshouldbeobviousthatitispossiblebyhandtoimplementthismethodforanydiscreteprobabilitydistribution,ashasbeendone,forexample,tocreatethedemandvaluesinTable5.5fortheConleyFisheriesproblem.Itshouldalsobeobviousthatitisfairlystraightforwardtoprogramaspreadsheettoautomaticallycreateasequenceofnumbersthatobeyagivendiscreteprobabilitydistribution,usingcommandssuchasRAND(),etc.Inaddition,therearespecialsimulationsoftwareprogramsthatwilldoallofthisautomatically,wheretheuseronlyneedstospecifythediscreteprobabilitydistribution,andthesoftwareprogramdoesalloftheotherwork.ThiswillbediscussedfurtherinSection5.10.
Wenextaddresshowwecanusearandomnumbergeneratortocreatenumbersthatobeyacontinuousprobabilitydistribution.
5.6CreatingNumbersThatObeyaContinuousProbabilityDistribution
AgainreturningtotheproblemofConleyFisheries,thenexttaskistocreateasequenceofdailypricesofcodfishattheportinRockport,onepriceforeachofthe20daysunderconsideration,insuchawaythatthepricesgeneratedareobservedvaluesthathavebeendrawnfromtheprobabilitydistributionofthedailypriceofcodfishinRockport.RecallthatthedailypriceofcodfishinRockportobeysaNormaldistributionwithmeanµ=$3.65/lb.andwithastandarddeviations=$0.20/lb.Thusthepriceofcodfishobeysacontinuousprobabilitydistribution,andsounfortunatelythemethodoftheprevioussection,whichwasdevelopedfordiscreterandomvariables,cannotbedirectlyapplied.Forthecasewhenitisnecessarytocreate
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numbersthataredrawnfromacontinuousprobabilitydistribution,thereisaverysimplemethodthatisdescribedasfollowsingenerality.
SupposethatYisanyrandomvariablethatobeysacontinuousprobabilitydistributionwhosecumulativedistributionfunction(cdf)isF(y),whererecallthat
F(y)=P(Y£y),
andthatitisnecessaryaspartofasimulationmodeltocreateasequenceofnumbersthatobeysthisparticularcontinuousprobabilitydistribution.Thereisagraphicalprocedureforcreatingsuchasequenceofnumbers,whichisquitesimpleandisstatedasfollows:
GraphicalMethodforCreatingSampleDatadrawnfromaContinuousProbabilityDistribution
1.Usearandomnumbergeneratortogenerateasequenceofrandomnumbersthatobeytheuniformdistributionbetween0.0and1.0.
2.ForeachrandomnumberxgeneratedinthesequenceofStep1,placethatnumberontheverticalaxisofthegraphofthecumulativedistributionfunction(cdf)F(y).ThenfindthepointyonthehorizontalaxiswhosecdfvalueF(y)equalsx.
Wenowillustratethisgraphicalmethod.SupposeYisacontinuousrandomvariablewhoseprobabilitydensityfunction(pdf)f(y)isgiveninFigure5.2,andwhosecumulativedistributionfunction(cdf)F(y)isgiveninFigure5.3.Supposethatweareinterestedincreatingasampleoftenvaluesdrawnfromthisdistribution.First,inStep1,wegenerateasequenceoftenrandomnumbersusingarandomnumbergenerator,whosevaluesmightlooklikethoseinthefirstcolumnofTable5.6.
FIGURE5.2Probabilitydensityfunctionf(y)oftherandomvariableY.
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FIGURE5.3CumulativedistributionfunctionF(y)oftherandomvariableY.
TABLE5.6Asequenceoftenrandomnumbers.RandomNumber(x) Valueofy
0.80540.64230.88490.69700.24850.07930.70020.14910.40670.1658
Inordertocreatethesamplevalueforthefirstrandomnumber,whichisx=0.8054,weplacethatvalueontheverticalaxisofthecumulativedistributionfunctionF(y)ofFigure5.3anddeterminethecorrespondingyvalueonthehorizontalaxisfromthegraph.ThisisillustratedinFigure5.4.Forx=0.8054,thecorrespondingyvalueis(approximately)y=6.75.Proceedinginthismannerforallofthetendifferentrandomnumbervaluesx,oneobtainsthetenvaluesofydepictedinTable5.7.ThesetenvaluesofythenconstituteasampleofobservedvaluesdrawnfromtheprobabilitydistributionofY.
FIGURE5.4IllustrationofthegraphicalmethodforcreatingobservedvaluesofacontinuousrandomvariablewithcumulativedistributionfunctionF(y).
TABLE5.7Valuesofxandy.RandomNumber(x) Valueofy
0.8054 6.750.6423 6.040.8849 7.160.6970 6.240.2485 4.510.0793 3.580.7002 6.280.1491 4.020.4067 5.150.1658 4.12
Asjustshown,itisquiteeasytoperformthenecessarystepsofthegraphicalmethod.However,itmaynotbeobviouswhythismethodcreatesasequenceofvaluesofythatareinfactdrawnfromtheprobabilitydistributionofY.Wenowgivesomeintuitionastowhythemethodaccomplishesthis.
Considertheprobabilitydensityfunction(pdf)f(y)ofYshowninFigure5.2anditsassociatedcumulativedistributionfunction(cdf)F(y)showninFigure5.3.RememberthatthecdfF(y)istheareaunderthecurveofthepdff(y).Therefore,theslopeofF(y)willbesteeperforthosevaluesofywherethepdff(y)ishigher,and
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theslopeofF(y)willbeflatterforthosevaluesofywherethepdff(y)islower.Forexample,aty=5.5,f(y)isquitelarge,andtheslopeofF(y)isquitesteep.Also,forexample,aty=3.0,f(y)isquitesmall,andtheslopeofF(y)isquiteflat.
Nowconsidertheapplicationofthegraphicalmethodforcreatingvaluesofy.Notethatbywayoftheconstruction,themethodwillproduceagreaterconcentrationofvaluesofywheretheslopeofthecdfF(y)issteeper.Likewise,themethodwillproducealesserconcentrationofvaluesofywheretheslopeofthecdfF(y)isflatter.Therefore,usingtheobservationintheparagraphabove,themethodwillproduceagreaterconcentrationofvaluesofywherethepdff(y)ishigherandwillproducealesserconcentrationofvaluesofywherethepdff(y)islower.Asitturnsout,thiscorrespondenceisprecise,andsoindeedthemethodwillproducevaluesofythatareexactlyinaccordwiththeunderlyingprobabilitydistributionofY.
AlthoughthemethodisbestillustratedbyconsideringagraphofthecdfF(y),itisnotnecessarytohavealiteralgraphofthecumulativedistributionfunctioninordertousethemethod.Infact,anotherwaytostatethemethodwithouttheuseofgraphsisasfollows:
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GeneralMethodforCreatingSampleDataDrawnfromaContinuousProbabilityDistribution
1.Usearandomnumbergeneratortogenerateasequenceofrandomnumbersthatobeytheuniformdistributionbetween0.0and1.0.
2.ForeachrandomnumberxgeneratedinthesequenceofStep1,computethatvalueofywhosecumulativedistributionfunctionvalueisequaltox.Thatis,givenx,solvetheequation:
F(y)=P(Y£y)=x
toobtainthevaluey.Thenassignyasthevaluecreated.
ThismoregeneralformofthemethodisnowillustratedontheConleyFisheriesproblem,wherethetaskathandistocreateasequenceofpricesforcodfishattheportinRockport,onepriceforeachofthe20daysunderconsideration,insuchawaythatthecreatedpricesaresamplesdrawnfromtheprobabilitydistributionofthedailypriceofcodfishinRockport.RecallthatthedailypriceofcodfishinRockportobeysaNormaldistributionwithmeanµ=$3.65/lb.andwithastandarddeviations=$0.20/lb.
Thefirststepistousearandomnumbergeneratortogenerate20randomnumbers.Supposethishasbeendone,andthe20randomnumbersareasshowninthesecondcolumnofTable5.8.
TABLE5.8WorksheetforgeneratingthepriceofcodfishinRockport.Day RandomNumber PriceofCodfishinRockport($/lb.)1 0.3236 3.55852 0.1355 3.42993 0.5192 3.65964 0.9726 4.03425 0.0565 3.33306 0.2070 3.48667 0.2481 3.51398 0.8017 3.81969 0.2644 3.524010 0.2851 3.536511 0.7192 3.766112 0.7246 3.769313 0.9921 4.133014 0.5227 3.661415 0.0553 3.330916 0.5915 3.696317 0.0893 3.381018 0.3136 3.552919 0.0701 3.355020 0.8309 3.8416
Thesecondstepisthentocomputethatvalueofyforwhichtheequation
F(y)=P(Y£y)=x
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issolved,foreachofthe20differentrandomnumbervaluesxinTable5.8,andwhereF(y)isthecdfforaNormaldistributionwithmeanµ=$3.65/lb.andwithstandarddeviations=$0.20/lb.Considerthefirstrandomnumbervalue,whichisx=0.3236.Thenthecorrespondingvalueofyisthatvalueofywhichsolves:
P(Y£y)=0.3236,
andwhereYobeysaNormaldistributionwithmeanµ=3.65andwithastandarddeviations=0.20.Butthen,ifwedefine
andrecallthatZisastandardNormalrandomvariable,then
P(Y£y)=0.3236
isequivalentto
SolvingthisequationviaastandardNormaltable,oneobtainsthat
andsubstitutingthevaluesofµ=$3.65/lb.ands=$0.20/lb.inthisexpressionandsolvingforyyields
y=µ0.4575s=3.650.4575×0.20=$3.5585/lb.
Therefore,withy=$3.5585,itistruethat
F(y)=P(Y£y)=0.3236,
andsowecreateapriceof$3.5585/lb.inthefirstrowofthethirdcolumnofTable5.8.Continuinginpreciselythismannerforalloftheother19randomnumbervalues,wecreatetheremainingentriesinthetable.
Thefinalcommentofthissectionconcernscomputersoftwareandcomputerimplementationofthegeneralmethod.Itshouldbeobviousthatitispossiblebyhandtoimplementeitherthegraphicalmethodorthemoregeneralmethodforanycontinuousprobabilitydistribution,solongasitispossibletoworkwiththecumulativedistributionfunctionofthedistributionviaeitheragraphoratable.FortheConleyFisheriesproblem,forexample,themethodisimplementedbyusingthetablefortheNormaldistribution.Whenthecontinuousdistributionisofaveryspecialform,suchasaNormaldistributionorauniformdistribution,itisquiteeasytocreateaspreadsheetthatwilldoallofthenecessarycomputationsautomatically.WhenthecontinuousdistributionisotherthantheNormalorauniformdistribution,therearespecialsimulationsoftwareprogramsthatwilldoallofthecomputationautomatically,wheretheuseronlyneedstospecifythegeneralparametersofthecontinuousdistribution,andthesoftwareprogramdoesalloftheotherwork.ThiswillbediscussedfurtherinSection5.10
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5.7CompletingtheSimulationModelofConleyFisheries
WecannowcompletethesimulationoftheConleyFisheriesproblembycombiningthedemandvaluesofTable5.5andthepricevaluesofTable5.8tocompletetheanalysis.ThisisshowninTable5.9,whichisnowdiscussedindetail.Table5.9portraystheresultsofthesimulationeffortsonthesimulationofthefirst20daysofthe200-dayperiod.ThesecondandthirdcolumnsofTable5.9aresimplythesimulatedvaluesofdemandinRockport(column3)basedontherandomnumberinColumn2,ascreatedinTable5.5andsimplycopiedovertoTable5.9.ThefourthcolumnofTable5.9isthequantityofcodfishsold,which,asonemayrecall,istheminimumofthedemand(Column3)andthedailycatchof3,500lbs.ThefifthandsixthcolumnsofTable5.9aresimplythesimulatedvaluesofthepriceofcodfishinRockport(Column6)basedontherandomnumberinColumn5,ascreatedinTable5.8andsimplycopiedovertoTable5.9.Lastofall,Column7ofTable5.9isthedailyearningsinRockport,whichiscomputedbytheformula:
DailyEarnings=(QuantityofCodfishSold)(PriceofCod)$10,000.
Thus,forexample,forday1,thedailyearningsis:
DailyEarnings=(3,500lbs.)($3.5585/lb.)$10,000=$2,455.
TABLE5.9CompletedworksheetfortheConleyFisheriesproblem.Day
NumberRandomNumber
DemandinRockport(lbs.)
QuantityofCodfishSold(lbs.)
RandomNumber
PriceofCodfishinRockport($/lb.)
DailyEarningsinRockport($)
1 0.3352 4,000 3,500 0.3236 3.5585 $2,4552 0.4015 4,000 3,500 0.1355 3.4299 $2,0053 0.1446 3,000 3,000 0.5192 3.6596 $9794 0.4323 4,000 3,500 0.9726 4.0342 $4,1205 0.0358 1,000 1,000 0.0565 3.3330 ($6,667)6 0.4999 4,000 3,500 0.2070 3.4866 $2,2037 0.8808 6,000 3,500 0.2481 3.5139 $2,2998 0.9013 6,000 3,500 0.8017 3.8196 $3,3689 0.4602 4,000 3,500 0.2644 3.5240 $2,33410 0.3489 4,000 3,500 0.2851 3.5365 $2,37811 0.4212 4,000 3,500 0.7192 3.7661 $3,18112 0.7267 5,000 3,500 0.7246 3.7693 $3,19313 0.9421 6,000 3,500 0.9921 4.1330 $4,46514 0.7059 5,000 3,500 0.5227 3.6614 $2,81515 0.1024 3,000 3,000 0.0553 3.3309 ($7)16 0.2478 4,000 3,500 0.5915 3.6963 $2,93717 0.5940 5,000 3,500 0.0893 3.3810 $1,83418 0.4459 4,000 3,500 0.3136 3.5529 $2,43519 0.0511 2,000 2,000 0.0701 3.3550 ($3,290)20 0.6618 5,000 3,500 0.8309 3.8416 $3,445
Thecomputationsarerepeatedinasimilarfashionforallofthe20daysinthetable.
TheimportantnumbersinTable5.9arethedailyearningsnumbersinthelastcolumnofthetable.Bygeneratingthedailyearningsnumbersforalargenumberofdays,itshouldbepossibletogetafairlyaccuratedescriptionoftheprobabilitydistributionofthedailyearningsinRockport,i.e.,theprobabilitydistributionoftherandomvariableF.
Althoughthenumbern=20hasbeenusedtoshowsomeoftheintermediarydetailedworkinthesimulationoftheConleyFisheriesproblem,thereisnoreasonwhythemodelcannotbeextendedtorunforn=200simulateddays,andhencesimulate
Document
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n=200daysofdailyearnings.Indeed,Table5.10shows200differentdailyearningsnumbersgeneratedbyusingthesimulationmethodologyfor200differentdays.Notethatthefirst20entriesinTable5.10aresimplythedailyearningsinRockportofthefirst20daysofthesimulation,i.e.,theyarethenumbersfromthefinalcolumnofTable5.9.
TABLE5.10Simulationoutputforn=200simulationtrialsoftheConleyFisheriesmodel.DayNumber DailyEarningsinRockport($)
1 $2,4552 $2,0053 $9794 $4,1205 ($6,667)6 $2,2037 $2,2998 $3,3689 $2,33410 $2,37811 $3,18112 $3,19313 $4,46514 $2,81515 ($7)16 $2,93717 $1,83418 $2,43519 ($3,290)20 $3,44521 $2,07622 $3,58323 $2,16924 $3,06425 ($6,284)26 $3,60227 $4,40628 $2,91129 $2,38930 $2,75231 $2,16332 $3,55333 $3,31534 $1,93635 $3,01336 $40537 $2,44338 $2,82539 $1,81840 ($1,808)41 $3,10442 $2,80243 $55644 $2,55445 $2,79246 $3,09947 $2,46548 $2,90949 $2,38650 $2,50551 $2,87052 $2,53053 $4,28954 $1,96855 ($2,382)56 $3,27157 $2,457
58 $2,24059 $2,65860 $1,44361 ($6,491)62 $1,95463 ($6,284)64 $2,49465 $3,64966 $3,25867 ($2,034)68 $2,79169 $2,85670 $2,02671 $2,67772 $3,364
72 $3,36473 $3,47274 $1,87375 $2,10476 $2,58677 $2,20178 $1,82579 $2,95580 $1,46981 $1,84382 $3,93683 $2,57284 ($10,000)85 $1,60186 $4,23887 $2,42388 $1,07289 $2,65190 $1,82391 $2,78292 ($5,963)93 $2,90494 $3,97295 $2,53996 $1,53097 $1,62998 $2,61099 $2,821100 $2,067101 $3,188102 $2,907103 $4,192104 $2,792105 $2,727106 $1,930107 $2,569108 $2,858109 $3,783110 ($2,523)111 ($2,290)112 $4,229113 $3,317114 ($1,769)115 $2,581116 $2,361117 ($10,000)118 $919119 $2,493120 $3,973121 $3,189122 ($3,654)123 $2,492124 $2,843
125 ($3,020)126 $2,725
127 $2,194128 $1,883129 $3,329130 $2,372131 $1,010132 $3,161133 $2,769134 $3,184135 $2,786136 $3,233137 $2,230138 $3,338139 $2,670140 ($6,362)141 $2,500142 ($3,068)143 $2,036144 $4,030145 $3,826146 $3,527147 $3,196148 $3,573149 $4,020150 $3,012151 ($3,039)152 $1,822153 $2,217
154 $3,068155 ($224)156 $3,662157 $3,829158 $1,628159 ($10,000)160 $2,254161 $3,406162 $441163 ($3,159)164 $3,243165 $1,351166 $3,649167 $3,156168 $2,104169 $2,573170 $2,011171 $3,706172 $2,017173 ($2,860)174 $2,247175 $2,165176 $4,134177 $3,031178 $2,345179 $1,416180 $3,025181 $156182 $2,737183 $3,025184 ($3,328185 $2,163186 ($2,383)187 $1,641188 $2,310189 $2,980190 $3,109191 $3,246192 $2,567193 $3,340
194 $2,244195 $3,219196 $2,496197 $2,011198 $2,731199 ($6,464)200 $3,614
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5.8UsingtheSampleDataforAnalysis
Table5.10containsasampleofn=200observedvaluesoftheearningsfromusingRockport.Fixingsomenotation,letxidenotethedailyearningsfromusingRockportfordaynumberi,forallvaluesofi=1,...,200.Thatis,x1=$2,455,x2=$2,005,...,x200=$3,614.TheneachxiistheobservedvalueoftherandomvariableFandhasbeendrawnfromthe(unknown)probabilitydistributionoftherandomvariableF.Infact,thearrayofvaluesx1,x2,...,x200constitutesafairlylargesampleofn=200differentobservedvaluesoftherandomvariableF.Therefore,wecanusethissampleofobservedvaluestoestimatetheanswerstothefivequestionsposedinSection5.1andSection5.2,whichwerephrasebelowinthelanguageofstatisticalsampling:
(a)WhatistheshapeoftheprobabilitydensityfunctionofF?
(b)WhatisP(F>$1,375)?
(c)WhatisP(F<$0)?
(d)WhatistheexpectedvalueofF?
(e)WhatisthestandarddeviationofF?
Wenowproceedtoanswerthesefivequestions.
Question(a):
WhatisanestimateoftheshapetheprobabilitydensityfunctionofF?
RecallfromChapter4thatinordertogainintuitionabouttheshapeofthedistributionofF,itisusefultocreateafrequencytableandahistogramoftheobservedsamplevaluesx1,x2,...,x200ofTable5.10.SuchafrequencytableispresentedinTable5.11.
TABLE5.11Frequencytableofthe200observedvaluesofearningsinRockport.
IntervalFrom IntervalTo NumberintheSample($10,500) ($10,000) 3($10,000) ($9,500) 0($9,500) ($9,000) 0($9,000) ($8,500) 0($8,500) ($8,000) 0($8,000) ($7,500) 0($7,500) ($7,000) 0($7,000) ($6,500) 1($6,500) ($6,000) 5($6,000) ($5,500) 1($5,500) ($5,000) 0($5,000) ($4,500) 0($4,500) ($4,000) 0($4,000) ($3,500) 1($3,500) ($3,000) 5($3,000) ($2,500) 2($2,500) ($2,000) 4($2,000) ($1,500) 2($1,500) ($1,000) 0($1,000) ($500) 0($500) $0 2$0 $500 3$500 $1,000 3$1,000 $1,500 6$1,500 $2,000 17$2,000 $2,500 43$2,500 $3,000 41$3,000 $3,500 35$3,500 $4,000 16$4,000 $4,500 10$4,500 $5,000 0
AhistogramofthevaluesofTable5.11isshowninFigure5.5.Thishistogramgivesanicepictorialviewofthedistributionoftheobservedvaluesofthesample,anditalsoisanapproximationoftheshapeoftheprobabilitydensityfunction(pdf)ofF.WeseefromthehistograminFigure5.5thatthevaluesofFaremostlyclusteredintherange$0through$4,500,inaroughlybellshapeorNormalshape,butthattherearealsoasignificantnumberofothervaluesscatteredbelow$0,whosevaluescanbeaslowas$10,000.ThishistogramindicatesthatwhilemostobservedvaluesofearningsfromRockportarequitehigh,thereissomedefiniteriskofsubstantiallossesfromusingRockport.
FIGURE5.5HistogramofearningsfromRockport.
Question(b):
WhatisanestimateofP(F>$1,375)?
ThisquestioncanbeansweredbyusingthecountingmethoddevelopedinChapter4.RecallfromChapter4thatthefractionofvaluesofx1,x2,...,x200inTable5.10thatarelargerthan$1,375isanestimateoftheprobabilitypforwhich
p=P(F>$1,375).
Ifwecountthenumberofvaluesofx1,x2,...,x200inTable5.10thatarelargerthan$1,375,weobtainthat165ofthese200valuesarelargerthan$1,375.Therefore,anestimateofp=P(F>$1,375)is:
Wethereforeestimatethatthereisan83%likelihoodthattheearningsinRockportonanygivendaywouldexceedtheearningsfromGloucester.ThissupportsthestrategyoptionofchoosingRockportoverGloucester.
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Question(c):
WhatisanestimateofP(F<0)?
ThisquestioncanalsobeansweredbyusingthecountingmethodofChapter4.RecallfromChapter4thatthefractionofvaluesofx1,x2,...,x200inTable5.10thatarelessthan$0isanestimateoftheprobabilitypforwhichp=P(F<$0).Ifwecountthenumberofvaluesofx1,x2,...,x200inTable5.10thatarelessthan$0,weobtainthat26ofthese200valuesarelessthan$0.Therefore,anestimateofp=P(F<$0)is:
Wethereforeestimatethatthereisa13%likelihoodthatConleyFisherieswouldlosemoneyonanygivenday,iftheychosetoselltheircatchinRockport.ThisshowsthattheriskofchoosingRockportisnottoolarge,butitisnotinsubstantialeither.
Question(d):
WhatisanestimateoftheexpectedvalueofF?
Weknowthattheobservedsamplemean ofthissampleof200observedvaluesisagoodestimateoftheactualexpectedvalueµoftheunderlyingdistributionofF,especiallywhenthesamplesizeislarge(andhere,thesamplesizeisn=200,whichisquitelarge).Therefore,theobservedsamplemeanofthe200valuesx1,x2,...,x200inTable5.10shouldbeaverygoodestimateoftheexpectedvalueµoftherandomvariableF.Itisstraightforwardtoobtainthesamplemean forthesamplegiveninTable5.10.Itsvalueis
ThereforeourestimateofthemeanoftherandomvariableFis$1,768.38.Noticethatthisvalueislargerthan$1,375,whichistheearningsthatConleyFisheriescanobtain(withcertainty)bysellingitsfishinGloucester.Thus,anestimateoftheexpectedincreaseinrevenuesfromsellinginRockportis
$393.38/day=$1,768.38/day$1,375.00/day.
Question(e):
WhatisanestimateofthestandarddeviationofF?
RecallfromthestatisticalsamplingmethodologyofChapter4thattheobservedsamplestandarddeviationsisagoodestimateoftheactualstandarddeviationsoftherandomvariableF,especiallywhenthesamplesizeislarge.ItisstraightforwardtoobtaintheobservedsamplestandarddeviationforthesamplegiveninTable5.10,asfollows:
Therefore
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andourestimateofthestandarddeviationofFiss=$2,672.59.Thisstandarddeviationisratherlarge,whichconfirmsClintConley'sintuitionthatthereissubstantialriskinvolvedinusingRockportastheportatwhichtosellhisdailycatch.
Anadditionalquestionthatonemightwanttoansweris:Whatisa95%confidenceintervalforthemeanofthedistributionofdailyearningsfromsellingthecatchofcodfishinRockport?TheanswertothisquestionisprovidedbytheformulainChapter4fora95%confidenceintervalforthemeanofadistributionwhennislarge.Thisformulais:
Substitutingthevaluesof =$1,768.38,s=$2,672.59,andn=200,wecomputethe95%confidenceintervalforthetruemeanofthedistributionofdailyearningstobe
[$1,397.98,$2,138.78].
Noticethatthisintervaldoesnotcontainthevalue$1,375.Therefore,atthe95%confidencelevel,wecanconcludethattheexpecteddailyearningsfromsellingthecatchofcodfishinRockportishigherthanfromsellingthecatchofcodfishinGloucester.
Finally,supposeClintConleyweretoconstructandusethesimulationmodelthathasbeendevelopedherein.Withtheanswerstothequestionsposedearlier,hewouldbeinagoodpositiontomakeaninformeddecision.Hereisalistofthekeydatagarneredfromtheanalysisofthesimulationmodel:
WeestimatethattheshapeofthedistributionofdailyearningsfromRockportwillbeasshowninFigure5.5.Onmostdaystheearningswillbebetween$0and$4,500perday.However,onsomedaysthisnumbercouldbeaslowas$10,000.
Weestimatethattheprobabilityis0.83thatthedailyearningsinRockportwouldbegreaterthaninGloucesteronanygivenday.
Weestimatethattheprobabilityis0.13thatthedailyearningsinRockportwillbenegativeonanygivenday.
WeestimatethattheexpecteddailyearningsfromRockportis$1,768.38.ThisishigherthantheearningsinGloucesterwouldbe,by$393.38/day.
WeestimatethatthestandarddeviationofthedailyearningsinRockportis$2,672.59.
The95%confidenceintervalfortheactualexpecteddailyearningsfromusingRockportexcludes$1,375.Therefore,weare95%confidentthattheexpecteddailyearningsfromRockportishigherthanfromGloucester.
Basedonthisinformation,ClintConleywouldprobablyoptimallychoosetosellhiscatchinRockport,inspiteoftherisk.Notethattheriskisnottrivial,asthereisalwaysthepossibilitythathecouldlosemoneyonanygivenday(withprobability0.13),andinfacthecouldhavecash-flowproblemsifhewereextremelyunlucky.Butunlessheisextremelyaversetotakingrisks(inwhichcasehemightnotwanttobeinthefishingindustrytobeginwith),hewouldprobablyfurtherthelong-terminterestsofConleyFisheriesbychoosingtosellhisfishinRockport.
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5.9SummaryofSimulationModeling,andGuidelinesontheUseofSimulation
Asimulationmodelattemptstomeasureaspectsofuncertaintythatsimpleformulascannot.AswesawintheConleyFisheriesproblem,eventhissimpleproblemhasnoconvenientanalysisviaformulasrelatedtoprobabilityandrandomvariables.Ofcourse,whenformulasandtablescanbeusedinsteadofasimulationmodel,thenthemanager'staskisthatmucheasier.Butalltoooften,therearesituationsthatmustbeanalyzedwhereasimulationmodelistheonlyappropriatemethodologicaltool.
Thesuccessfulapplicationofasimulationmodeldependsontheabilitytocreatesamplevaluesofrandomvariablesthatobeyavarietyofdiscreteandcontinuousprobabilitydistributions.Thisisthekeytoconstructingandusingasimulationmodel.Throughtheuseofarandomnumbergenerator,itispossibletocreatesampledatavaluesthatobeyanydiscretedistributionoranycontinuousdistributionbyapplyingthegeneralmethodsoutlinedinSection5.5(fordiscreterandomvariables)andSection5.6(forcontinuousrandomvariables).
Unlikedecisiontreesandcertainothermodelingtools,asimulationhasnointernaloptimaldecision-makingcapability.NotethatthesimulationmodelconstructedforanalyzingtheConleyFisheriesproblemonlyproduceddataastheoutput,andthemodelitselfdidnotchoosethebestportstrategy.SupposethatConleyFisheriesfacedthedecisionofchoosingamongfivedifferentports.Asimulationmodelwouldbeabletoanalyzetheimplicationsofusingeachport,butunlikeadecisiontreemodel,thesimulationmodelwouldnotchoosetheoptimalportstrategy.Inordertouseasimulationmodel,themanagermustenumerateallpossiblestrategyoptionsandthendirectthesimulationmodeltoanalyzeeachandeveryoption.
Theresultsthatonecanobtainfromusingasimulationmodelarenotpreciseduetotheinherentrandomnessinasimulation.Thetypicalconclusionsthatonecandrawfromasimulationmodelareestimatesoftheshapesofdistributionsofparticularquantitiesofinterest,estimatesofprobabilitiesofeventsofinterest,andmeansandstandarddeviationsoftheprobabilitydistributionsofinterest.Onecanalsoconstructconfidenceintervalsandotherinferencesofstatisticalsampling.
Thequestionofthenumberoftrialsorrunstoperforminasimulationmodelismathematicallycomplex.Fortunately,withtoday'scomputingpower,thisisnotaparamountissueformostproblems,becauseitispossibletorunevenverylargeandcomplexsimulationmodelsformanyhundredsorthousandsoftrials,andsoobtainaverylargesetofsampledatavaluestoworkwith.
Finally,oneshouldrecognizethatgainingmanagerialconfidenceinasimulationmodelwilldependonatleastthreefactors:
agoodunderstandingoftheunderlyingmanagementproblem,
one'sabilitytousetheconceptsofprobabilityandstatisticscorrectly,and
one'sabilitytocommunicatetheseconceptseffectively.
5.10ComputerSoftwareforSimulationModeling
TheConleyFisheriesexampleillustrateshoweasyitistoconstructasimulationmodelusingstandardspreadsheetsoftware.The''input"totheConleyFisheriesexampleconsistedofthefollowinginformation:
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thedescriptionofprobabilitydistributionoftherandomvariableD,thedailydemandinRockport.ThisdistributionwasshowninTable5.1.
thedescriptionoftheprobabilitydistributionoftherandomvariablePR,thedailypriceofcodfishinRockport.ThisisaNormaldistributionwithmeanµ=$3.65andstandarddeviations=$0.20.
theformulafortheearningsfromsellingthecatchinRockport,namely
F=PR×min(3,500,D)10,000.
Basedontheseinputs,weconstructedasimulationmodelpredicatedontheabilitytogeneraterandomvariablesthatobeycertaindistributions,namelyadiscretedistribution(fordailydemandinRockport)andtheNormaldistribution(fordailypricesinRockport).TheoutputofthemodelwasthesampleofobservedvaluesofearningsinRockportshowninTable5.10.BasedontheinformationinTable5.10,weconstructedahistogramofthesampleobservations(showninFigure5.5),andweperformedavarietyofothercomputations,suchascountingthenumberofobservationsinagivenrange,thecomputationofthesamplemeanandthesamplestandarddeviation,etc.
Becausesimulationmodelingissuchausefultool,thereareavarietyofsimulationmodelingsoftwareproductsthatfacilitatetheconstructionofasimulationmodel.Atypicalsimulationsoftwarepackageisusuallydesignedtobeusedasanadd-ontotheExcel®spreadsheetsoftwareandhaspull-downmenusthatallowstheusertochoosefromavarietyofprobabilitydistributionsforthegenerationofrandomvariables(suchastheuniformdistribution,theNormaldistribution,thebinomialdistribution,adiscretedistribution,etc.)Thesoftwareisdesignedtoautomaticallygeneraterandomnumbersthatobeythesedistributions.Furthermore,thetypicalsimulationsoftwarepackagewillautomaticallyperformtheroutinetasksinvolvedinanalyzingtheoutputofasimulationmodel,suchascreatinghistograms,estimatingprobabilities,andestimatingmeansandstandarddeviations.Allofthiscapabilityisdesignedtofreethemanagertofocusonmanagerialanalysisofthesimulationmodel,asopposedtogeneratingrandomnumbersandcreatingchartoutput.AnexampleofsuchasoftwarepackagethatisusedinsomeofthecasesinSection5.12iscalledCrystalBall®andisdescribedattheendoftheOntarioGatewaycase.
Therearealsoalargenumberof''specialtysimulation"softwarepackagesthataredesignedforspecificusesinspecificapplicationsdomains.Forexample,therearesomesimulationmodelingsoftwarepackagesdesignedtomodelmanufacturingoperations,andthatofferspecialgraphicsandotherfeaturesuniquetoamanufacturingenvironment.Thereareothersimulationmodelingsoftwarepackagesdesignedforotherspecialapplicationdomains,suchasserviceapplications,militaryapplications,andfinancialmodeling.
5.11TypicalUsesofSimulationModels
Perhapsthemostfrequentuseofsimulationmodelsisintheanalysisofacompany'sproductionoperations.Manycompaniesusesimulationtomodeltheeventsthatoccurintheirfactoryproductionprocesses,wherethetimesthatvariousjobstakeisuncertainandwheretherearecomplexinteractionsintheschedulingoftasks.Thesemodelsareusedtoevaluatenewoperationsstrategies,totesttheimplicationsofusingnewprocesses,andtoevaluatevariousinvestmentpossibilitiesthatareintendedtoimproveproductionandoperationalefficiency.
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Anotherfrequentuseofsimulationmodelsisintheanalysisofoperationswheretherearelikelytobequeues(thatis,waitinglines).Forexample,thebestmanagedfast-foodchainsusesimulationmodelstoanalyzetheeffectsofdifferentstaffingstrategiesonhowlongcustomerswillwaitforservice,andontheimplicationsofofferingnewproducts,etc.BanksusesimulationmodelstoassesshowmanytellersorhowmanyATMs(automatedtellermachines)toplanforatagivenlocation.Airlinesusesimulationmodelingtoanalyzethroughputofpassengersatticketcounters,atgates,andinbaggagehandling.Withtheincreasinguseoftoll-freenumbersforservingcustomersinavarietyofbusinesses(fromcatalogshoppingtotoll-freesoftwaresupportforsoftwareproducts),manytelecommunicationscompaniesnowoffersimulationmodelingaspartoftheirbasicservicetoalloftheir''800"businesscustomerstohelpthemassignstaffinglevelsfortheirtoll-freeservices.
Anotheruseofsimulationmodelingisincapitalbudgetingandthestrategicanalysisofinvestmentalternatives.Simulationmodelsareusedtoanalyzetheimplicationsofvariousassumptionsconcerningthedistributionofcostsofaninvestment,possiblemarketpenetrationscenarios,andthedistributionofcash-flows,bothinanygivenyearaswellasoverthelifeoftheinvestment.
Asmentionedearlier,simulationmodelsareusedquiteabundantlyintheanalysisofmilitaryprocurementandintheanalysisofmilitarystrategy.Hereinparticular,simulationmodelingisusedtogreatadvantage,asthealternativeoftestingnewhardwareortacticsinthefieldisnotparticularlyattractive.
Simulationisalsousedinfinancialengineeringtoassignpricesandanalyzeotherquantitiesofinterestforcomplexfinancialinstruments,suchasderivativesecurities,options,andfuturescontracts.
Thisisonlyasmalllistofthewaysthatsimulationmodelsarecurrentlyusedbymanagers.Withtherapidadvancesinbothcomputerhardwareandtheeaseofuseoftoday'ssimulationmodelingsoftware,thereisenormouspotentialforsimulationmodelstoaddevenmorevaluetotheeducatedandcreativemanagerwhoknowshowtowiselyusethetoolsofsimulation.
5.12CaseModules
TheGentleLentilRestaurant
AnExcellentJobOffer
SanjayThomas,asecond-yearMBAstudentattheM.I.T.SloanSchoolofManagement,isinaveryenviableposition:Hehasjustreceivedanexcellentjobofferwithatop-flightmanagementconsultingfirm.Furthermore,thefirmwassoimpressedwithSanjay'sperformanceduringtheprevioussummerthattheywouldlikeSanjaytostartupthefirm'spracticeinitsnewBombayoffice.ThesynergyofSanjay'spreviousconsultingexperiences(bothpriortoSloanandduringtheprevioussummer),hisSloanMBAeducation,andhisfluencyinHindiofferanextremelyhighlikelihoodthatSanjaywouldbeverysuccessful,ifheweretoacceptthefirm'sofferanddeveloptheBombayoffice'spracticeinIndiaandsouthernAsia.
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