©CopyrightJASSS
MertEdaliandHakanYasarcan(2014)
AMathematicalModeloftheBeerGame
JournalofArtificialSocietiesandSocialSimulation 17(4)2
<http://jasss.soc.surrey.ac.uk/17/4/2.html>
Received:29-Nov-2013Accepted:28-Mar-2014Published:31-Oct-2014
Abstract
Thebeerproduction-distributiongame,inshort"TheBeerGame",isamultiplayerboardgame,whereeachindividualplayeractsasanindependentagent.Thegameiswidelyusedinmanagementeducation
aimingtogiveanexperiencetotheparticipantsaboutthepotentialdynamicproblemsthatcanbeencounteredinsupplychainmanagement,suchasoscillationsandamplificationofoscillationsasonemoves
fromdownstreamtowardsupstreamechelons.Thegameisalsousedinnumerousscientificstudies.Inthispaper,weconstructamathematicalmodelthatisanexactone-to-onereplicaoftheoriginalboard
versionofTheBeerGame.Weapplymodelreplicationprinciplesanddiscussthedifficultieswefacedintheprocessofconstructingthemathematicalmodel.Accordingly,themodelispresentedinfullprecision
includingnecessaryassumptions,explanations,andunitsforallparametersandvariables.Inaddition,theadjustableparametersarestated,theequationsgoverningtheartificialagents'decisionmaking
processesarementioned,andanRcodeofthemodelisprovided.WealsoshortlydiscusshowtheRcodecanbeusedinexperimentationandhowitcanalsobeusedtocreateasingle-playerormulti-player
beergameonacomputer.OurcodecanproducetheexactsamebenchmarkcostvaluesreportedbySterman(1989)verifyingthatitiscorrectlyimplemented.ThemathematicalmodelandtheRcodepresented
inthispaperaimstofacilitatepotentialfuturestudiesbasedonTheBeerGame.
Keywords:
AcquisitionLag,ArtificialAgents,BeerGame,MathematicalModel,Replication,SystemDynamics
Introduction
1.1
Thebeerproduction-distributiongame,inshort"TheBeerGame",isamultiplayerboardgame,whereeachindividualplayeractsasanindependentagent.ItwasfirstintroducedbyJayForrester'sSystem
Dynamics(SD)researchgroupoftheSloanSchoolofManagementattheMassachusettsInstituteofTechnologyinthe1960s.TheBeerGameisanapplicationofSDmodelingandsimulationmethodology,which
iswidelyusedinmanagementeducationandaimstogiveanexperiencetotheparticipantsaboutthepotentialdynamicproblemsthatcanbeencounteredinsupplychainmanagement,suchasoscillationsand
amplificationofoscillationsasonemovesfromdownstreamtowardsupstreamechelons(Akkermans&Vos2003;Barlas2002;Chen&Samroengraja2000;Forrester1961,1971;Größleretal.2008;Sterman
2000).Adetaileddescriptionoftheoriginalbeergame,whichiswidelyplayedbynumerouspeoplewithdifferenteducationalbackgroundsandisalsousedinscientificstudies,isgiveninSterman(1989)and
CrosonandDonohue(2006).WelistsomeoftheworkonTheBeerGametogiveanideaabouttherangeofstudiesbasedonthegame.Jacobs(2000)introducedtheinternet-basedversionofTheBeerGameand
reportedthatthisversionofthegamesignificantlyreducedthetimerequiredtoplaythegame.Accordingtohim,themainreasonforthisdifferenceisthatintheboardversionofthegame,participantsmanually
keeptherecordsofinventoriesandbacklogsandcalculatethetotalcost,butintheinternet-basedversionofthegame,thegamesoftwaretakescareofthesecalculationsanddoessoinafasterandmoreaccurate
mannercomparedtohumanagentsparticipatinginthegame.DayandKumar(2010)usedmobilephonestorunthegameandreportedimprovedaccuracyandspeedduetotheautomatedcalculations.
IndependentfromJacobs'work(2000),Samuretal.(2004,2005)developedamulti-playercomputerizedversionofTheBeerGame.Theyfirstpresentverificationrunstodemonstratethattheirmodelcorrectly
representstheboardgame.Thentheyconcludethatparticipantswhoplayedtheboardgameweremoresuccessfulthanthosewhoplayedthecomputerizedversionofthegame.Thepotentialcausesforthisresult
includedtheslowerpaceofprogressoftheboardversion,whichgivesmoretimetothinkabouttheorderquantity,andtherelativelymorerealisticenvironmentoftheboardversion.Steckeletal.(2004)examined
theeffectofreducedcycletimesandtheeffectofsharedpoint-of-sale(POS)informationamongthesupplychainmembersinTheBeerGame.Theyreportedthatreducedcycletimesleadtoreducedcosts,which
isanexpectedresult.TheyfurtherreportedaninterestingresultthatthebenefitofPOSinformationsharingdependsonthecustomerdemandpattern.Chaharsooghietal.(2008)proposedareinforcementlearning
(RL)modelfororderingpoliciesinsupplychainsandusedTheBeerGamemodelasanexperimentalplatform.Kim(2009)extendedTheBeerGameintoasupplynetworkandconductedagent-basedsimulations
inhisstudy.MosekildeandLaugesen(2007)conductedanextensivebifurcationanalysisandshowedthatTheBeerGamecanproducecomplexdynamics.Thomsenetal.(1991)alsoshowedthatitispossibleto
obtaincomplexdynamicsfromTheBeerGame,includinghyperchaos.
1.2
TheBeerGameisafourechelonsupplychainconsistingofaretailer,wholesaler,distributor,andfactory,whereeachoneoftheseechelonsismanagedbyanindependentagent.Inthismulti-agentgame,thereis
aninventorycontrolproblemforeachoneoftheseechelons.Duringthegame,everyhumanagentinateamoffourisresponsibleforoneofthefourechelonsandmanagestheassociatedinventorybyplacing
orders.Theordersflowfromdownstreamechelonstowardsupstreamechelonsandcasesofbeerflowintheoppositedirection.Theaimofthegameistominimizetheaccumulatedtotalcostobtainedbythe
participantsofateammanagingeachechelon.Theaccumulatedcostgeneratedbyeachindividualecheloniscalculatedattheendofthegamebyaddingupallinventoryholdingandbacklogcostsobtainedatthe
endofeachsimulatedweek(Sterman1989).
1.3
Atthebeginningofanongoingresearchstudy,amathematicalmodelthatisanexactone-to-onereplicaoftheoriginalboardversionofTheBeerGame,wasneeded.Moreover,wedecidedtouseamodelthathad
equationsorganizedandexecutedinexactlythesameorderasthe'fivesteps'oftheboardgame.Webelievedthatsuchamodelwouldfacilitatetheverificationofourresultsandalsopotentiallycontributetothe
analysisandunderstandingoftheboardgame.Wewerenotabletofindacomputermodelintheliteraturethatprovidedsuchanexactreplicaoftheboardgame.Therefore,wefirstconstructedamathematical
modelthatisindependentfromprogramminglanguagesbasedonthedescriptionsofTheBeerGameprovidedinSterman(1989).Later,wecodedthismodelinR [1].Themathematicalmodelisgiveninfulldetail
inthesectionnamed"Mathematicalmodelofthebeergame".
1.4
Axelrod(1997,pp.20–21)notedthat:"Replicationisoneofthehallmarksofcumulativescience.Itisneededtoconfirmwhethertheclaimedresultsofagivensimulationarereliableinthesensethattheycanbe
reproducedbysomeonestartingfromscratch."Therearemanyotherresearcherswhoalsoemphasizetheimportanceofreplicationasavalidationapproach(seeforexample,Edmonds&Hales2003,2005;
Miodowniketal.2010;Wilensky&Rand2007).Modelreplicationrequiresasignificanteffortbecausecomplexdynamicmodelsarehighlysensitivetotheimplementationdetails(Merloneetal.2008).Inhispaper
entitled"AdvancingtheArtofSimulationintheSocialSciences",Axelrod(1997)reportsproblemsthatwereencounteredinreplicatingsimulationmodelsdescribedinotherpublishedwork.AccordingtoAxelrod,
someofthereplicationproblemsarecausedbyambiguities,gaps,anderrorsinthemodeldescriptions.DespiteallthedetailsprovidedbySterman(1989),asignificanteffortwasrequiredtoobtainaone-to-one
mathematicalreplicaoftheboardversionofthegame,andweexperienceddifficultiessimilartotheonesexperiencedbyAxelrod(1997):(1)Stermanprovidedequationsforthegeneralstockmanagementtask,
whichcanformabasisinobtainingTheBeerGameequations.However,theexactequationsforTheBeerGamearenotpresentinSterman'spaper,exceptfortheorderingequation.(2)Thereisanambiguityin
thetie-breakingruleusedinroundingthevaluesoftheorders.Hence,weareforcedtoassumeatie-breakingruleinroundingthevalues.(3)Expectationformationisassumedtobeperformedinformallybya
humanparticipantinhismindand,therefore,isnotlistedamongthefivestepsofTheBeerGame.However,inthemathematicalmodel,thedecisionmakingprocessisalsocapturedasapartofthemodeland,
therefore,wehavetodetermineitsplaceamongthestepsofthegame.(4)Thereisanerrorregardingtheconceptualizationofthedelaydurations.Inthesectionnamed"Adiscussiononacquisitionlags",we
explainthemisconceptualizationofthedurationofacquisitionlagsthatarepresentinthegameandtheerrorthatcanpotentiallybecausedbyit.NotethatNorthandMacal(2002)alsoreporteddifficultiesthatthey
facedintheirBeerGamedockingprocess.
1.5
AlthoughTheBeerGameisanapplicationofSDmethodology,aone-to-oneSDmodelofthegamecannotbedirectlyobtainedbecausetheorderofcalculationsfollowedinthegameandtheorderofcalculations
followedinSDmethodologywillnotmatchunlesstheorderofcalculationsinthecorrespondingSDmodeliscarefullyalteredbyintroducingadditionalvariablestothemodel.Thismismatchalsocontributestothe
difficultyinobtainingacompletemathematicalmodelofthegame.NorthandMacal(2002)implementedTheBeerGameusingthreedifferentplatformsandtheyreportedsubtledifferencesintheoutputsofthose
threedifferentimplementations,whichwethinkthosedifferencescouldonlybeeliminatedbysubstantialamountofadditionaleffort.
1.6
Inthesectionnamed"Rcodeofthemathematicalmodelasanexperimentalplatform",weprovideanRcode[1](RCoreTeam2013)ofthemathematicalmodelpresentedinthispapertoeasethesimulation
replicationsofourmodel.Inthesamesection,wealsoshortlydiscusshowthecodecanbeusedinexperimentationandhowitcanbeusedtocreateasingle-playerormulti-playerbeergameonacomputer.
1.7
InSterman(1989),theanchor-and-adjustheuristicformulationthatissuggestedtobeusedindecisionmakingandtheanchor-and-adjustheuristicformulationthatisusedinmodelingtheparticipantbehaviorare
slightlydifferent.Inthesectionnamed"Verificationofthemathematicalmodel",weexplainthedifferencesbetweenthetwoformulations,provideadifferentsetofequationsfortheanchor-and-adjustheuristic
formulationthatisusedinmodelingtheparticipantbehavior,executethecorrespondingRcode[2]withtheoptimumbenchmarkparametervaluesgivenbySterman,andobtaintheexactsamebenchmarkcost
valuesalsopresentedbyhim.ThisprocessverifiesthatourRcodeiscorrectlyimplementedanditalsovalidatesthatourmodelisacorrectandexactrepresentationoftheboardversionofTheBeerGame.
Mathematicalmodelofthebeergame
2.1
Toconductthisresearch,wefirstconstructedamathematicalmodelofTheBeerGamebasedonafigureoftheboardgame(seeFigure1inthispaper),equations,thefivestepsofthegame,anddescriptions
giveninSterman(1989).
http://jasss.soc.surrey.ac.uk/17/4/2.html
1
19/10/2015
Figure1.TheBoardofTheBeerDistributionGame(Figure2inSterman1989,p.327)
Parametersandinitialvaluesofthemathematicalmodel
2.2
sat i=1[week]fori=R,W,D,F
(1)
mdti=1[week]fori=R,W,D
(2)
st i=2[week]fori=W,D,F
(3)
plt =2[week]
(4)
Wheresat (1/αSinSterman1989)standsforthestockadjustmenttime ,mdtstandsforthemailingdelaytime,st standsfortheshipmenttime ,andplt standsfortheproductionleadtime.R,W,D,andF stand,
respectively,fortheretailer,wholesaler,distributor,andfactoryechelons(Figure1).Notethatsat ,wsl,θ,andI*arethedecisionparameters(equations1,5,6,and10).Thedifferentsetsofvaluesofthese
parametersrepresentthedifferentinstancesoftheanchor-and-adjustorderingpolicyand,togetherwiththedecisionmakingvariables(equations8,9,11–14,47–50,54–57,and63–68),theydefinehowtheartificial
agentsmakedecisions.Notethatthedifferentsetsofvaluesfortheseparameterswillnotmakethemathematicalmodeldivertfrombeinganexactone-to-onereplicaoftheoriginalboardversionofTheBeer
Game.
wsli=1[dimensionless]fori=R,W,D,F
(5)
θi=0.2[1/week]fori=R,W,D,F
(6)
wsl(βinSterman1989)standsfortheweightofsupplylineandθ(alsoθinSterman1989)standsforthesmoothingfactorusedbyeachecheloninthegameinformingexpectationsusingthesimpleexponential
smoothingforecastingmethod.Theoretically, θcantakeavaluebetween0and1.
(7)
Intheequationabove,ENDCDstandsfortheend-customerdemand.Tosavespace,theunitcaseisusedtorepresentacaseofbeer.
EECD 0=4[case/week]
(8)
EOi,0 =4[case/week]fori=R,W,D
(9)
EECD standsfortheexpectedend-customerdemand ,whichisassumedtobecalculatedbytheretailer.EOrepresentsexpectedorderscalculatedbythewholesaler,distributor,andfactoryechelonsbasedonthe
orderstheyreceivefromtheirrespectivecustomers(i.e.,theretailer'sordersreceivedbythewholesaler,thewholesaler'sordersreceivedbythedistributor,andthedistributor'sordersreceivedbythefactory).
Ii* =0[case]fori=R,W,D,F
(10)
(11)
(12)
(13)
(14)
I*representsthedesiredinventory,andSL*standsforthedesiredsupplyline.
2.3
Bi,0 =0[case]fori=R,W,D,F
(15)
Ii,0 =12[case]fori=R,W,D,F
(16)
ITI1i,0 =4[case]fori=R,W,D
(17)
WIPI10=4[case]
(18)
ITI2i,0 =4[case]fori=R,W,D
(19)
WIPI20=4[case]
(20)
Theequations15through20representinitialbacklogs,initialinventories,andinitialin-transitinventories(i.e.,thevaluesofthestatevariablesatweekzero).ITI2(in-transitinventory2)representstheshippingdelay
boxjustbeforetheinventorybox,andITI1(in-transitinventory1)representstheshippingdelayboxbeforethat(seeFigure1).ThevalueofITI1belongingtoanechelonisshiftedtoITI2ofthesameechelonafter
onesimulatedweek.ITI2isaddedtotheinventory( I)orsubtractedfromthebacklog(B)afteraweek.Likewise,WIPI1andWIPI2standforwork-in-processinventories.WIPI1isthework-in-processinventoryofthe
factorythatwillbeshiftedto WIPI2afteraweekandthatwilleventuallyreachtothefactory'sinventory.WIPI2isthework-in-processinventorythatwillbeaddedtothefactory'sinventory(IF)orsubtractedfromthe
backlog( BF)afteraweek.
O i,1 =4[case/week]fori=R,W,D
(21)
PSR 1=4[case/week]
(22)
IO i,1 =4[case/week]fori=W,D,F
(23)
O istandsforordersthatareplacedbyecheloni(retailer,wholesaler,anddistributor).PSR standsfortheproductionstartrate,whichistheproductionordergivenbythefactoryitself.IO istandsfortheincoming
orders(theboxrighttotheboxofordersplacedinFigure1)thatarereceivedbyecheloni.Thepurchaseorders(O )placedbytheretailer,wholesaler,anddistributorandtheproductionorders(PSR )givenby
factoryatweektareplacedforweek( t+1).Orders placedatweek(t+1)bytheretailer,wholesaler,anddistributorbecometheincomingorders(IO ),respectively,forthewholesaler,distributor,andfactoryat
week(t+2).Therefore,theend-customerdemand(ENDCD),whichisgivenbyEquation7,orders(O ),andproductionstartrate(PSR )havenovalueatweekzero,buttheyareassignedavalueforthefirsttimeat
week1.
http://jasss.soc.surrey.ac.uk/17/4/2.html
2
19/10/2015
TC i,0 =0[$]fori=R,W,D,F
(24)
uihc=0.5[$/(week·case)]
(25)
ubc=1[$/(week·case)]
(26)
TC ,uihc,andubcstandforthetotalcostgeneratedbyanechelon,theunitinventoryholdingcost,andtheunitbacklogcost,respectively.
2.4
TheremainingmodelequationsaregiveninanorderbasedonthestepsofthegamepresentedinSterman(1989).Thissequenceshouldstrictlybefollowedwhileperformingcalculationstoensureanaccurate
representationoftheboardversionofTheBeerGame(Figure1).
2.5
Afterinitializingtheboard,thegamefacilitatordeclaresthetimeasweek1,andthegamestartsfromStep1.Afterthecompletionofallsteps(attheendofStep5),thefacilitatoraddsoneweektothecurrenttime,
declaresit,andthegamecontinuesrepeatingthesameprocess.
Step1.Receiveinventoryandadvanceshippingdelays
2.6
2.7
InTheBeerGame,casesofbeerflowfromrighttoleft(i.e.,fromtheupperechelontothelower)andordersflowfromlefttoright(i.e.,fromlowerechelontotheupper).Inthefirststepofthegame,thein-transit
inventory(work-in-processinventoryforthefactory)thatisimmediatelytotherightofaninventoryisaddedtotheinventorybytheparticipants.Afterthat,thecontentsoftherightmostin-transitinventoryboxesare
shiftedtothenear-rightin-transitinventoryboxes,andthecontentsoftherightmostwork-in-processinventoryboxareshiftedtothenear-rightwork-in-processinventorybox.Asaresult,therightmostin-transit
inventoryboxesandWIPI1becomeempty.
Ii, t =Ii, t -1+ITI2i, t -1[case]fori=R,W,D
(27)
IF, t =IF, t -1+WIPI2t -1[case]
(28)
ITI2i, t =ITI1i, t -1[case]fori=R,W,D
(29)
WIPI2t =WIPI1t -1[case]
(30)
ITI1i, t =0[case]fori=R,W,D
(31)
WIPI1t =0[case]
(32)
Equations31and32areredundantinthesensethattheexclusionoftheseequationswillnotpreventthecorrectsimulationofthemathematicalmodel,butwepresenttheminordertofollowthesameexact
processasintheboardversionofthegame.
Step2.Fillorders
2.8
Inthisstep,eachecheloncalculatestheamountofbeertobeshippedtoitscustomer(i.e.,shipmentsfromretailertoendcustomer,fromwholesalertoretailer,fromdistributortowholesaler,andfromfactoryto
distributor)byconsideringincomingordersfromthecustomer,thebacklogoforders,andtheinventoryofthatechelon.Aftercalculatingshipments(S),eachparticipant,exceptfortheretailer,puts shipmentstothe
boxesontheirnearleft(i.e.,ITI1R,ITI1W,ITI1D,andWIPI1areupdated).
(33)
(34)
(35)
(36)
2.9
Theshipmentvariablesareflowvariablesinessence,andtheunitofthesevariablesis[case/week].However,weusestockvariablesthathave[case]astheirunitincalculatingtheshipmentvariables.Therefore,
correctionsintheunitsarenecessary.Accordingly,(1/week)and(1week)areusedintheshipmentequations33–36.Thesecorrectionshavenoeffectonthenumericalvalues,buttheycorrecttheunits.Toeasily
comprehendtheissue,consideracarthattraveledforonehourandcovered50milesinthisjourney.Insuchacase,theaveragespeedofthecarduringthatonehourwouldbe50milesperhour.Althoughthe
unitsofthedistancecoveredbythecaranditsaveragespeedaredifferent,theirnumericalvaluesarenot.Notethat,forsimilarreasons,wealsousethesametypeofcorrectioninmanyoftheremaining
equations.Asafurtherexplanation;inadiscretetimemodel,thesimulationtimestepbecomesequaltotheunittimeofthatsimulationmodel.Insuchamodel,whenanumericalvaluewithaunitof[items/time]
accumulateforoneunittime,theresultingnumericalvaluewillnotchange,butitsunitwillbe[items].
ITI1i, t =Si, t ·(1week)[case]fori=R,W,D
(37)
WIPI1t =PSR t ·(1week)[case]
(38)
Step3.Recordinventoryorbacklogontherecordsheet
2.10 Afterfillingorders,participantseithercounttheirinventoriesiftheyhavechipsrepresentingthecasesofbeerintheirinventoryboxesorcalculatethebacklogsiftheyfailtosatisfythetotalityofthepastbacklogs
andthecurrentordersreceivedfromtheircustomers.Theyrecordtheinventoryorbacklogontheirrecordsheets.Thisisrepresentedbytheequationsbelow:
BR, t =BR, t -1+(ENDCDt −SENDC, t )·(1week)[case]
(39)
BW, t =BW, t -1+(IO W, t −SR, t )·(1week)[case]
(40)
BD, t =BD, t -1+(IO D, t −SW, t )·(1week)[case]
(41)
BF, t =BF, t -1+(IO F, t −SD, t )·(1week)[case]
(42)
IR, t =IR, t −SENDC, t ·(1week)[case]
(43)
IW, t =IW, t −SR, t ·(1week)[case]
(44)
ID, t =ID, t −SW, t ·(1week)[case]
(45)
IF, t =IF, t −SD, t ·(1week)[case]
(46)
Inthisstudy,theequalitysignisusedinassigningvaluestoparametersandvariables,anditdoesnotimplyamathematicalequality.Therefore,thesamevariablecanappearonleftandrightsidesofthesame
equation(see,forexample,equations43–46).
Expectationformation
2.11 Expectationformationisassumedtobeperformedinformallybyaparticipantinhismindand,therefore,isnotlistedamongthefivestepsofTheBeerGame.Sterman(1989)modeledtheexpectationformation
processusingthesimpleexponentialsmoothingmethod(seeEquation9inSterman1989).Thisprocessisreflectedbytheequationsgivenbelow:
EECD t =EECD t -1+θR·(ENDCDt −EECD t -1)·(1week)[case/week]
(47)
EOR, t =EOR, t -1+θW·(IO W, t −EOR, t -1)·(1week)[case/week]
(48)
EOW, t =EOW, t -1+θD·(IO D, t −EOW, t -1)·(1week)[case/week]
(49)
http://jasss.soc.surrey.ac.uk/17/4/2.html
3
19/10/2015
EOD, t =EOD, t -1+θF·(IO F, t −EOD, t -1)·(1week)[case/week]
(50)
Step4.Advancetheorderslips
2.12 Orders (O )placedbytheretailer,wholesaler,anddistributorbecomeincomingorders(IO ),respectively,forthewholesaler,distributor,andfactoryafteraweek.
IO W, t +1=O R, t [case/week]
(51)
IO D, t +1=O W, t [case/week]
(52)
IO F, t +1=O D, t [case/week]
(53)
Step5.Placeorders
2.13 IntheboardversionofTheBeerGame,participantsplaceordersandproductionrequestsinthislaststep.AccordingtoSterman(1989),thedecisionmakingprocessofparticipantscanberepresentedbyusingthe
anchor-and-adjustdecisionheuristic.Theequationsbelow(54–68)areallpartofthisheuristic,whichfinallyresultsinordersandproductionrequests.Theretailer,wholesaler,anddistributorplaceorders(equations
65–67)andthefactorydecidesontheproductionrequests(Equation68).SL*standsforthedesiredsupplyline,EIstandsfortheeffectiveinventory ,SLstandsforthesupplyline,SLAstandsforthesupplyline
adjustment,IAstandsfortheinventoryadjustment ,andPSR standsfortheproductionstartrate.Theanchoroftheanchor-and-adjustheuristicistheexpectedlossfromthestock,whichisEECD t inEquation65,
EOR,tinEquation66,EOW,tinEquation67,andEOD,tinEquation68.
(54)
(55)
(56)
(57)
EIi, t =Ii, t −Bi, t [case]fori=R,W,D,F
(58)
SLR, t =IO W, t +1·(1week)+BW, t +ITI1R, t +ITI2R, t [case]
(59)
SLW, t =IO D, t +1·(1week)+BD, t +ITI1W, t +ITI2W, t [case]
(60)
SLD, t =IO F, t +1·(1week)+BF, t +ITI1D, t +ITI2D, t [case]
(61)
SLF, t =WIPI1t +WIPI2t [case]
(62)
(63)
IAi, t =(Ii* −EIi, t )/sat i[case/week]fori=R,W,D,F
(64)
(65)
(66)
(67)
(68)
2.14 Intheboardversionofthegame,humanagentsgiveorders.However,inourmodel,artificialagentsgiveordersasdescribedbyequations65–68.Allequationsbelongingtothefivestepsofthegame(equations
27–69)arecalculatedforthecurrentsimulatedweek,exceptfortheequationsforincomingorders(51–53),orders(65–67),andproductionstartrate(68).Theequationsfor incomingorders,orders,andproduction
startratearecalculatedforthenextsimulatedweek.Intheboardversionofthegame,orderscanonlybeintegers.Theroundingfunction(||||)usedinequations65–68(andalsoequations72–75)reflectsthis
aspectofthegame.Inroundingthevalues,weassumethatthe"roundhalfawayfromzero"tie-breakingruleisused.
2.15 Intheoriginalgame,theaccumulatedtotalcostofeachecheloniscalculatedattheendofthegamefromtherecordsheetkeptbytheparticipantmanagingthatechelon.Inthemathematicalmodel,the
totalcostis
calculatedattheendofeachsimulatedweekusingtheequationgivenbelow:
TC i, t =TC i, t -1+(uihc·Ii, t +ubc·Bi, t )·(1week)[$]fori=R,W,D,F
(69)
2.16 Afterthisfinalstep,thesimulatedtimeisincreasedbyoneweekandannouncedtotheparticipants.ThegamecontinuesbyrepeatingthewholeprocessstartingfromStep1untilStep5ofweek36iscompleted.
Afterthesimulationends,themainperformancemeasure,whichisteamtotalcost(TTC),iscalculated.
TTC=TC R,36 +TC W,36 +TC D,36 +TC F,36 [$]
(70)
Rcodeofthemathematicalmodelasanexperimentalplatform
3.1
TheRcode[1](RCoreTeam2013)ofthemathematicalmodelisreadytobeexecuted;oneneedstosimplycopythecodefromthetextfileandpasteittoRconsoleinordertoexecuteit.Asmentionedbefore,
equations1,5,6,and10arethedecisionparameters(sat ,wsl,θ,andI*)thatdescribethedecisionmakingprocessesoftheartificialagents,andtheirdifferentvaluesrepresentdifferentinstancesoftheanchorand-adjustorderingpolicy.Hence,onecanchangethevaluesoftheseparametersinthecodewithoutmakingthemathematicalmodeldivertfrombeinganexactone-to-onereplicaoftheoriginalboardversionof
TheBeerGame.Afterthechange,onecanre-executethecodetoobtainthecorrespondingcostvalues.TheweeklyvaluesofanyvariablecaneasilybedisplayedbysimplytypingthenameofthatvariableinR
console.Inthisway,thecodeservesasanexperimentalplatform.Onecanalsogenerategraphicaloutputsusingthecode [3].Wesetsat =2,wsl=0.8,θ=0.4,andI*=0forallechelonstoobtaintheexample
dynamicspresentedinfigures2–4.
http://jasss.soc.surrey.ac.uk/17/4/2.html
4
19/10/2015
Figure2.AnexampleofInventoryDynamics
Figure3.AnexampleofBacklogDynamics
http://jasss.soc.surrey.ac.uk/17/4/2.html
5
19/10/2015
Figure4.AnexampleofDynamicsofOrders
3.2
Tocreatedifferentsettingsforasimulationexperimentwithoutalteringthemodelstructure,onecanchangetheendcustomerdemandpatterngivenbyEquation7andplaywiththeinitialvaluesofthestock
variables,butthosechangeswillimplyadiversionfromtheoriginalsettingofTheBeerGame.Themodelstructurecanalsobealtered.Forexample,onecanchangethedelaytimes mdt(mailingdelaytime),st
( shipmenttime ),or plt (productionleadtime).However,suchachangewouldimplyanincreaseordecreaseinthenumberoftheassociatedvariables.Forinstance,ifonewantstoincreasetheshipmenttimeof
anechelonfromtwoweekstothreeweeks,hemustintroduceonemorein-transitinventoryvariable(i.e.,ITI3).AnotherexamplecouldbetheadditionofproductioncapacitytoEquation68ortheadditionof
shipmentcapacitiestoequations33–36.
3.3
Simulationexperimentscanalsobeconductedinanotherprogrammingenvironmentbyre-writingthecodeusingthatprogramminglanguage.Onecanalsodevelopamulti-playerbeergameusingaprogramming
environmentthatsupportsuserinterfacecreation.Inthatcase,therespectiveorderequationsdescribingtheartificialagents'decisionmakingprocessesshouldberemovedfromthecodeandfourhumanagents
wouldbeaskedtoinsertvaluesforordersofthefourechelons.Alternatively,asingle-playerversionofthegamecanbeconstructed.Inthiscase,ahumanagentwouldgiveordersforoneofthefourechelonsand
threeartificialagentswouldcontroltheordersoftheremainingechelons.
Adiscussiononacquisitionlags
4.1
InTheBeerGame,theacquisitionlagisthesummationofthemailingdelaytimeandshipmenttime fortheretailer,wholesaler,anddistributor,anditisdirectlyequaltotheproductionleadtimeforthefactory.In
thegame,ordersplacedbyanechelon(i.e.,theretailer,wholesaler,ordistributor)atweek twillreachtheinventoryofthatechelonatweek(t+4)giventhatthesupplierofthatechelonhassufficientinventoryto
fulfilltheorder.Forthefactory,ordersgivenatweektwillbereceivedatweek(t+3).Therefore,Sterman(1989)statesmanytimesinhispaperthattheacquisitionlagfortheretailer,wholesaler,anddistributoris
atleast4weeks,anditisalways3weeksforthefactory.However,weclaimthatordersplacedatStep5ofweektareforweek( t+1)(seeequations21,22,65-68,and72-75).Therefore,slightlychangingthe
game,byplacingordersatthebeginningofStep1ofweek(t+1)insteadofplacingthemattheendofStep5ofweekt,willmakenodifference.Accordingly,inourmathematicalmodel,theacquisitionlagsusedin
calculatingthevaluesofthe desiredsupplylineare3,3,3,and2weeksfortheretailer,wholesaler,distributor,andfactory,respectively(seeequations2–4,11–14,and54–57).Inthedesiredsupplylineequations
(11–14and54–57),whichcorrespondtoEquation7inSterman(1989),usingacquisitionlagsof4weeks(fortheretailer,wholesaler,anddistributor)and3weeks(forthefactory)insteadof3weeks(fortheretailer,
wholesaler,anddistributor)and2weeks(forthefactory)willcreateasteady-stateerrorinthedynamics.
Verificationofthemathematicalmodel
5.1
Afterconstructingaone-to-onemodelofTheBeerGame,weenteredtheoptimaldecisionparametervaluessuggestedbySterman(1989)intoourmodel.Theseparametervaluesare0,1,and1for θ(smoothing
factor;alsoθinSterman1989),sat (stockadjustmenttime ;1/αSinSterman1989),andwsl(weightofsupplyline;βinSterman1989),respectively,forallechelons.TheotherdecisionparametergivenbySterman
isS',whichisdefinedasI*(desiredinventory ;S* inSterman1989)plus wsltimesSL*(desiredsupplyline;SL*inSterman1989);seeEquation71andtheunnumberedS'equationinSterman(1989,p.334).
StermangivestheoptimalvaluesofS'as28,28,28,and20fortheretailer,wholesaler,distributor,andfactoryechelons,respectively.
Si'=Ii* +wsli·SLi* [case]fori=R,W,D,F
5.2
(71)
Inourmathematicalmodel,theSL*valuesaredynamicallyupdatedastheexpectedordersfromthecustomerschange.Thus,S'shouldalsobeavariable.However,StermanusesconstantSL*andS'values.If
theS'valueisusedinsteadofseparateI*andSL*valuesandifitisaconstant,theorderequations65–68becomeasfollows:
(72)
(73)
(74)
(75)
5.3
Aftersimulatingourmodelwiththeoptimumθ,sat ,wsl,andS'valuesusingtheorderequations72–75,weobtainedtheexactsamebenchmarkcostvaluesreportedbySterman,whichsupportsourclaimthatour
modelisanexactrepresentationofTheBeerGame.NotethattheconceptualerrorregardingtheacquisitionlagshasnoeffectonthemodelusedbyStermaninoptimizingtheparametersbecauseS'isaconstant;
itisnotdynamicallycalculatedduringtheoptimizationruns.TheRcodeofthemathematicalmodelthatismodifiedwithandfortheorderingequations72–75isalsoprovided[2].InthisversionoftheRcode,sat ,
wsl,θ,andS'arethedecisionparameters(equations1,5,6,and71).Thedifferentsetsofvaluesoftheseparametersrepresentthedifferentinstancesoftheanchor-and-adjustorderingpolicyand,togetherwiththe
decisionmakingvariables(equations8,9,47–50,and72–75),theydefinehowtheartificialagentsmakedecisions.
5.4
Asafinalstepinverificationofthemathematicalmodel,wefirstremovetheorderingequationsfromtheRcode[1], [2]and,instead,insertpseudo-randomordersintothecode.Wealsoplaytheboardversionofthe
gamewiththesamepseudo-randomorders.ThedynamicsobtainedfromtheRcodeandtheboardversionofthegameexactlymatch.
Conclusions
6.1
Inthisstudy,weconstructedadetailedmathematicalmodelthatrepresentsandreplicatestheexactexecutionorderofthestepsoftheoriginalboardversionofTheBeerGame.Asapartoftheintroductionsection,
wealsostatethedifficultiesthatwefacedintheconstructionprocessofsuchanexactone-to-onereplica.Oneofthemaindifficultiesisanerrorregardingtheconceptualizationofthedelaydurationsasexplainedin
detailinthesectionnamed"Adiscussiononacquisitionlags".Wepresenttheconstructedmodelinfullprecisionincludingnecessaryassumptions,explanations,andunitsforallparametersandvariablesinthe
sectionnamed"Mathematicalmodelofthebeergame".
6.2
AccordingtoAxelrod(1997),internalvalidity(i.e.,verification),usability,andextendibilityarethethreegoalsoftheprogrammingofasimulationmodel.Toincreasetheusabilityofthemodelpresentedinthispaper,
westatedtheadjustableparameters,mentionedtheequationsgoverningtheartificialagents'decisionmakingprocesses,andwroteanRcode[1]ofthemodel.Forextendibility,inthesectionnamed"Rcodeofthe
http://jasss.soc.surrey.ac.uk/17/4/2.html
6
19/10/2015
mathematicalmodelasanexperimentalplatform",weshortlydiscussedhowthecodecanbeusedinexperimentationandhowitcanbeusedtocreateasingle-playerormulti-playerbeergameonacomputer.For
verification,wemodifiedsomeoftheequationsbelongingtothedecisionmakingformulationsoftheartificialagentsandpresentedtheminthesectionnamed"Verificationofthemathematicalmodel".Wealso
modifiedtheRcodeaccordingly.Finally,weexecutedthemodifiedRcode [2]withtheoptimumbenchmarkparametervaluesgivenbySterman(1989)andobtainedtheexactsamebenchmarkcostvaluesalso
presentedbyhimverifyingthatourRcodeiscorrectlyimplemented.ThisalsovalidatesthatourmodelisacorrectandexactrepresentationoftheboardversionofTheBeerGame.
Acknowledgements
TheauthorsthankDr.YamanBarlas,Dr.NigelGilbert,Dr.GönençYücel,andthesecondrefereeofthisarticlefortheirvaluablesuggestions.
ThisresearchissupportedbyaMarieCurieInternationalReintegrationGrantwithinthe7thEuropeanCommunityFrameworkProgramme(grantagreementnumber:PIRG07-GA-2010-268272)andalsoby
BogaziciUniversityResearchFund(grantno:6924-13A03P1).
Notes
1TheRcodeofthemathematicalmodelcanbefoundathttp://www.openabm.org/model/4161/version/1/view
2TheRcodeofthemodifiedmathematicalmodelcanbefoundathttp://www.openabm.org/model/4163/version/1/view
3TheRcodeofthemathematicalmodel(codeforgraphicaloutputadded)canbefoundathttp://www.openabm.org/model/4166/version/1/view
References
AKKERMANS,H.&Vos,B.(2003).Amplificationinservicesupplychains:Anexploratorycasestudyfromthetelecomindustry.ProductionandOperationsManagement,12(2) ,204–223.[doi:10.1111/j.19375956.2003.tb00501.x]
AXELROD,R.(1997).Advancingtheartofsimulationinthesocialsciences.Complexity,3(2) ,16–22.[doi:10.1002/(SICI)1099-0526(199711/12)3:2<16::AID-CPLX4>3.0.CO;2-K]
BARLAS,Y.(2002).SystemDynamics:SystemicFeedbackModelingforPolicyAnalysis.InKnowledgeforSustainableDevelopment:AnInsightintotheEncyclopediaofLifeSupportSystems(pp.1131–1175).
Paris,FranceandOxford,UK:UNESCO/EOLSS.
CHAHARSOOGHI,S.K.,Heydari,J.&Zegordi,S.H.(2008).Areinforcementlearningmodelforsupplychainorderingmanagement:Anapplicationtothebeergame.DecisionSupportSystems,45(4) ,949–959.
[doi:10.1016/j.dss.2008.03.007]
CHEN,F.&Samroengraja,R.(2000).Thestationarybeergame. ProductionandOperationsManagement,9(1) ,19–30.[doi:10.1111/j.1937-5956.2000.tb00320.x]
CROSON,R.&Donohue,K.(2006).Behavioralcausesofthebullwhipeffectandtheobservedvalueofinventoryinformation.ManagementScience,52(3) ,323–336.[doi:10.1287/mnsc.1050.0436]
DAY,J.M.&Kumar,M.(2010).UsingSMStextmessagingtocreateindividualizedandinteractiveexperiencesinlargeclasses:Abeergameexample.DecisionSciencesJournalofInnovativeEducation,8(1) ,
129–136.[doi:10.1111/j.1540-4609.2009.00247.x]
EDMONDS,B.&Hales,D.(2003).Replication,ReplicationandReplication:SomeHardLessonsfromModelAlignment.JournalofArtificialSocietiesandSocialSimulation,6(4) ,11
<http://jasss.soc.surrey.ac.uk/6/4/11.html>
EDMONDS,B.&Hales,D.(2005).ComputationalSimulationasTheoreticalExperiment.TheJournalofMathematicalSociology,29(3) ,209–232.[doi:10.1080/00222500590921283]
FORRESTER,J.W.(1961).IndustrialDynamics.Waltham,MA:PegasusCommunications.
FORRESTER,J.W.(1971).PrinciplesofSystems.Waltham,MA:PegasusCommunications.
GRÖßLER,A.,Thun,J.H.&Milling,P.M.(2008).Systemdynamicsasastructuraltheoryinoperationsmanagement.ProductionandOperationsManagement,17(3) ,373–384.[doi:10.3401/poms.1080.0023]
JACOBS,F.R.(2000).Playingthebeerdistributiongameovertheinternet.ProductionandOperationsManagement,9(1) ,31–39.[doi:10.1111/j.1937-5956.2000.tb00321.x]
KIM,W.-S.(2009).EffectsofaTrustMechanismonComplexAdaptiveSupplyNetworks:AnAgent-BasedSocialSimulationStudy.JournalofArtificialSocietiesandSocialSimulation,12(3) ,4
<http://jasss.soc.surrey.ac.uk/12/3/4.html>
MERLONE,U.,Sonnessa,M.&Terna,P.(2008).HorizontalandVerticalMultipleImplementationsinaModelofIndustrialDistricts.JournalofArtificialSocietiesandSocialSimulation,11(2) ,5
<http://jasss.soc.surrey.ac.uk/11/2/5.html>
MIODOWNIK,D.,Cartrite,B.&Bhavnani,R.(2010).BetweenReplicationandDocking:"AdaptiveAgents,PoliticalInstitutions,andCivicTraditions"Revisited.JournalofArtificialSocietiesandSocialSimulation,
13(3) ,1<http://jasss.soc.surrey.ac.uk/13/3/1.html>.
MOSEKILDE,E.&Laugesen,J.L.(2007).Nonlineardynamicphenomenainthebeermodel.SystemDynamicsReview,23(2–3),229–252.[doi:10.1002/sdr.378]
NORTH,M.&Macal,C.(2002).TheBeerDock:ThreeandaHalfImplementationsoftheBeerDistributionGame.<http://backspaces.net/sun/SCSim/BeerDock.pdf>.Archivedat:
<http://www.webcitation.org/6LUrLLIVQ>.
RCORETEAM.(2013).R:Alanguageandenvironmentforstatisticalcomputing.RFoundationforStatisticalComputing.Vienna,Austria.<http://www.R-project.org/>
SAMUR,M.G.,Eren,E.C.,Hancer,O.&Tunakan,B.B.(2004).TestingSimultaneousMultipleDecisionMakerCharacteristicsinaDynamicFeedbackEnvironment.B.Sc.Project,BogaziciUniversity.
SAMUR,M.G.,Eren,E.C.,Hancer,O.&Tunakan,B.B.(2005).Acomputerizedbeergameanddecision-makingexperiments.InProceedingsofthe23rdInternationalSystemDynamicsConference.Boston,MA.
STECKEL,J.H.,Gupta,S.&Banerji,A.(2004).Supplychaindecisionmaking:Willshortercycletimesandsharedpoint-of-saleinformationnecessarilyhelp?ManagementScience,50(4) ,458–464.
[doi:10.1287/mnsc.1030.0169]
STERMAN,J.D.(1989).Modelingmanagerialbehavior:misperceptionsoffeedbackinadynamicdecisionmakingexperiment.ManagementScience,35(3) ,321–339.[doi:10.1287/mnsc.35.3.321]
STERMAN,J.D.(2000). BusinessDynamics:SystemsThinkingandModelingforaComplexWorld.Boston,MA:Irwin/McGraw-Hill.
THOMSEN,J.S,Mosekilde,E.&Sterman,J.D.(1991).Hyperchaoticphenomenaindynamicdecisionmaking.InE.MosekildeandL.Mosekilde(Eds.),Complexity,Chaos,andBiologicalEvolution(pp.397–420).
NewYork:PlenumPress.[doi:10.1007/978-1-4684-7847-1_30]
WILENSKY,U.&Rand,W.(2007).MakingModelsMatch:ReplicatinganAgent-BasedModel.JournalofArtificialSocietiesandSocialSimulation,10(4) ,2<http://jasss.soc.surrey.ac.uk/10/4/2.html>
http://jasss.soc.surrey.ac.uk/17/4/2.html
7
19/10/2015