The Beer Dock: Three and a Half Implementations of the Beer Distribution Game

Transcription

1 The Beer Dock :55: The Beer Dock: Three and a Half Implemenaions of he Beer Disribuion Game Michael J. Norh[1] and Charles M. Macal Argonne Naional Laboraory, Argonne, Illinois Absrac The Beer Disribuion Game is a classic supply chain problem widely used in graduae business programs o each he conceps of supply chain managemen (Moseklide, Larsen, and Serman, 1991). I is well suied for his purpose since i is simple enough o be easily undersandable bu complex enough o be ineresing. In paricular, he Beer Disribuion Game is noable for is abiliy o confound ypical human players (Serman, 1987, 1989). Many people who play he game find i difficul, if no impossible, o avoid he chaoic operaing regimes ha are he game s hallmark. According o Buron, docking is a compelling meaphor from space exploraion ha offers much promise o give simulaion modeling greaer validiy (1998). Docking is used o deermine wheher wo models can produce he same resuls, which in urn is he basis for criical experimens and for ess of wheher one model can subsume anoher (Axell, Axelrod, Epsein, and Cohen, 1996). The original Beer Disribuion Game was implemened as a sysems dynamics model. The auhors have implemened he game using Mahemaica funcional programming, RePas agen-based modeling and simulaion (ABMS), and Swarm ABMS. The auhors have reproduced all he published resuls wih hese new implemenaions. As par of his process, he auhors have docked he implemenaions. The docking process was found o be challenging and ime consuming, bu ulimaely rewarding. The Beer Disribuion Game The Beer Disribuion Game ( Beer Game ) is a classic supply chain problem widely used in graduae business programs o each he conceps of supply chain managemen (Moseklide, Larsen, and Serman, 1991). I is well suied for his purpose since i is simple enough o be easily undersandable bu complex enough o be ineresing. In paricular, he Beer Game is noable for is abiliy o confound ypical human players (Serman, 1987, 1989). Many people who play he game find i difficul, if no impossible, o avoid he chaoic operaing regimes ha are he game s hallmark. The Beer Game consiss of a supply chain and five agens as shown in Figure 1. The cusomer places an order wih he reailer who fills he order if he reailer s invenory allows. The reailer file:///users/owen/xdocumens/beerdock/beerdock.hm Page 1

2 The Beer Dock :55: orders addiional iems from he wholesaler. There is a one-period delay in he order being received. The wholesaler fills he order if he wholesaler s invenory allows. There is a wo-period delay in iems being shipped and reaching heir desinaion. The wholesaler orders addiional iems from he disribuor and so on for he disribuor and facory. If he facory canno fill an order i places he order in producion. There is a hree-period producion delay. Iniially, he supply chain is in complee equilibrium in erms of demand, orders, supplies, and invenory. Figure 1: The Beer Disribuion Sysem In he Beer Game, each agen makes ordering decisions based only on locally available informaion. Orders are based on he following facors: Invenory Desired invenory level Iems in he pipeline (shipmens in-ransi from upsream agen and ousanding orders placed o agen upsream) Desired level of iems in he pipeline Curren demand Expeced demand Agens have he goal of achieving desired invenory and pipeline levels. Agens seek o close file:///users/owen/xdocumens/beerdock/beerdock.hm Page 2

3 The Beer Dock :55: he gap beween desired and acual levels in erms of heir sock and wha hey have in he pipeline. Sock adjusmens o invenory are deermined as: Sock Adjusmen o Invenory = a S (Desired Invenory - Invenory ), Sock adjusmens o he pipeline are deermined as: Sock Adjusmen o Pipeline = a SL (Desired Pipeline - Pipeline ). Ordering parameers a S and a SL represen he fracion of he gap o be closed beween desired and acual levels for each order. The ordering parameer b = (a SL / a S ) is he relaive weigh aached o pipeline versus sock discrepancies from desired levels. Agens adjus heir demand projecions each period. Agens weigh he curren demand and he expeced demand for his period o esimae he demand for he nex period: Expeced Demand +1 = q Demand + (1 - q) Expeced Demand where 0 <= q <= 1. Agens use an adapive behavioral decision rule o deermine orders: Indicaed Order = Expeced Demand + Sock Adjusmen Invenory + Sock Adjusmen Pipeline Tha is, Indicaed Order = ED + a S (Q - Invenory b * Iems in Pipeline ) where Q is a measure of desired invenory relaive o desired pipeline: Q = Desired Invenory + b * Desired Pipeline Line Finally since negaive orders are no allowed, he order he agen places o is upsream agen a is: Order = Max(0, Indicaed Order ) Arguably, his is a good, descripive, behavioral decision model for his conex (Serman 1987, Serman 1989). file:///users/owen/xdocumens/beerdock/beerdock.hm Page 3

4 The Beer Dock :55: The Beer Game sysem is compleely deerminisic. There are no random elemens in he model. If demand does no change, he sysem coninues forever in complee equilibrium: A every sage orders received equal orders sen For all sages he pipeline and invenory values are 12 The cusomer demand of four unis per period is always saisfied A ime five, a change occurs: Demand ramps up o 8 per period and says here A new ordering rule akes effec ha deermines how much agens a each sage of he supply chain will order, based on he locally available informaion The sysem is hen simulaed forward from his poin onward, ulimaely leading o deerminisic chaos in many cases. The Mahemaica Experimens To invesigae he Beer Game, an implemenaion was consruced using Wolfram Research s Mahemaicaâ. Mahemaica is an inegraed, highly programmable, echnical compuing environmen (Wolfram 2002). The goal was o duplicae he resuls of described in Mosekilde, Larsen, and Serman (MLS)(1991). MLS found ha varying he ordering parameers of he model resuls in various modes of long-run sysem behavior. They idenified behaviors as sable, chaoic, periodic (non-chaoic) and quasi-periodic (chaoic) over he range of parameer values. The MLS resuls were reproduced exacly for he case of a s = 0.3, b = 0.15, q = 0, Q = 15 as shown in Figures 2 and 3. Figure 3 is comparable o MLS Figure 5. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 4

5 The Beer Dock :55: Figure 2: The 60 Period Mahemaica Beer Game Invenory Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 Figure 3: The 60 Period Mahemaica Beer Game Order Rae Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 (Comparable o MLS Figure 5) Figures 4 and 5 plo he same case over 1,000 ime periods. Figure 5 is comparable o MLS Figure 6. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 5

6 The Beer Dock :55: Figure 5: The 1,000 Period Mahemaica Beer Game Invenory Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 Figure 6: The 1,000 Period Mahemaica Beer Game Order Rae Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 (Comparable o MLS Figure 6) file:///users/owen/xdocumens/beerdock/beerdock.hm Page 6

7 The Beer Dock :55: For he case of a s = 0.5, b = 0, q = 0.05, Q = 15, very similar bu no exac resuls were obained as shown in Figures 7 and 8. Figure 8 is comparable o MLS Figure 7. Figure 7: The 1,000 Period Mahemaica Beer Game Invenory Graph for a s = 0.5, b = 0, q = 0.05, Q = 15 Figure 8: The 1,000 Period Mahemaica Beer Game Order Rae Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 file:///users/owen/xdocumens/beerdock/beerdock.hm Page 7

8 The Beer Dock :55: (Comparable o MLS Figure 7) For he case of a s = 0.3, b = 0.65, q = 0.25, Q = 12, he same resuls were obained as shown in Figure 9. MLS classifies he resul as periodic, alhough echnically he series is non-repeaing and exhibis chaoic behavior. Figure 9 is comparable o MLS Figure 8. This is described by MLS as a periodic rajecory wih a ransien ha has a regular damped approach o a limi cycle. The Mahemaica analysis program was adjused o classify his series as quasi-periodic. Figure 9: The 1,000 Period Mahemaica Beer Game Invenory Graph for a s = 0.3, b = 0.65, q = 0.25, Q = 12 (Comparable o MLS Figure 8) For he case of a s = 0.425, b = 0.09, q = 0.25, Q = 17, somewha differen resuls were obained as shown in Figure 10. This Figure is comparable o MLS Figure 9. MLS classified his resuls as a period-4 limi cycle, bu he Mahemaica resuls are chaoic wih no clear limi cycles observed. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 8

9 The Beer Dock :55: Figure 10: The 1,000 Mahemaica Period Beer Game Invenory Graph for a s = 0.5, b = 0, q = 0.05, Q = 15 (Comparable o MLS Figure 9) The phase plo shown in Figure 11 is similar o ha obained by MLS bu does no exhibi heir period-4 limi cycle. Raher, Figure 11 exhibis a chaoic aracor. Figure 11: The Mahemaica Beer Game Phase Plo for a s = 0.425, b = 0.09, q = 0.25, Q = 17 Four behavior modes have been observed in he long-run ime series of he Mahemaica Beer file:///users/owen/xdocumens/beerdock/beerdock.hm Page 9

10 The Beer Dock :55: Game invenory levels: sable, chaoic, chaoic (quasi-periodic), and periodic. The chaoic modes are non-repeaing infinie sequences. In all cases observed, all ime series including all chaoic sequences were bounded. Transien behaviors leading up o hese long-run classificaions were mixed. Figure 12 illusraes he logic used for classifying he ime series. Figure 12: Mode Classificaion Scheme file:///users/owen/xdocumens/beerdock/beerdock.hm Page 10

11 The Beer Dock :55: A se of 100 x 100 simulaions were run over he parameer space (a s, b) ε [0, 1] x [0, 1] for q = 0.25, Q = 17. Each simulaion was run for 1,000 periods. The behavior mode for each facory invenory series was idenified based on he las 100 values in he series. The disribuion of modes shown in Figure compares favorably wih MLS Figure 10, allowing for he difference in resoluion beween he wo analyses (MLS conduced 200 x 200 simulaions for he same inerval). The Figure is read as follows: Ligh blue (medium gray) indicaes sable behavior Red (dark gray) indicaes chaoic behavior Dark blue (black) indicaes periodic (non-chaoic) behavior Yellow (ligh gray) indicaes quasi-periodic, chaoic behavior Figure 13: The Mahemaica Beer Game Behavior modes in he a S -b plane (Comparable o MLS Figure 10) The coss obained over he policy space are shown in Figure 14. Whie indicaes cos minimums. Ligher shades indicae cos near minimum. Darker shades indicae coss near maximum. As can be seen in he Figure, minimum cos policies are disribued hroughou he policy space, ofen residing in close proximiy o high-cos policies. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 11

12 The Beer Dock :55: Figure 14: The Mahemaica Beer Game Coss in he a S -b plane (Comparable o MLS Figure 12) Onward o RePas The original Beer Game was implemened as a sysems dynamics model. As previously discussed, he auhors have implemened he game using Mahemaica funcional programming. The Beer Game was hen implemened using he RePas agen-based modeling and simulaion (ABMS) oolki as shown in Figure 15. From he RePas web sie (Universiy of Chicago, 2002): The Universiy of Chicago's Social Science Research Compuing's RePas is a sofware framework for creaing agen based simulaions using he Java language (requires version Java 1.3 or greaer). I provides a library of classes for creaing, running, displaying and collecing daa from an agen based simulaion. In addiion, RePas can ake snapshos of running simulaions, and creae quickime movies of simulaions. RePas borrows much from he Swarm simulaion oolki and can properly be ermed "Swarm-like." In addiion, RePas includes such feaures as run-ime model manipulaion via gui widges firs found in he Ascape simulaion oolki. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 12

13 The Beer Dock :55: The resuling model run for a S = 0.3, b = 0.15, q = 0, Q = 15 is shown in Figures 16 and 17. Figure 15: The RePas Beer Game Implemenaion wih a s = 0.3, b = 0.15, q = 0, Q = 15 Figure 16: The 60 Period RePas Beer Game Invenory Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 file:///users/owen/xdocumens/beerdock/beerdock.hm Page 13

14 The Beer Dock :55: Figure 17: The 60 Period RePas Beer Game Order Rae Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 (Comparable o MLS Figure 5) Coninuing o Java Swarm Afer he RePas implemenaion, he Beer Game was implemened again using he Java Swarm ABMS oolki as shown in Figure 18. The resuling model run for a S = 0.3, b = 0.15, q = 0, Q = 15 is shown in Figures 19 and 20. The Java Swarm version of he Beer Game is couned as half an implemenaion since much of is code is shared wih he RePas implemenaion. According o he Swarm Developmen Group s web sie, Swarm is a sofware package for muli-agen simulaion of complex sysems (2002). file:///users/owen/xdocumens/beerdock/beerdock.hm Page 14

15 The Beer Dock :55: Figure 18: The Swarm Beer Game Implemenaion wih a s = 0.3, b = 0.15, q = 0, Q = 15 Figure 19: The 60 Period RePas Beer Game Invenory Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 file:///users/owen/xdocumens/beerdock/beerdock.hm Page 15

16 The Beer Dock :55: Figure 20: The 60 Period RePas Beer Game Order Rae Graph for a s = 0.3, b = 0.15, q = 0, Q = 15 (Comparable o MLS Figure 5) Docking According o Buron, docking is a compelling meaphor from space exploraion ha offers much promise o give simulaion modeling greaer validiy (1998). Laer Buron wroe he following abou docking any wo models (1999): Docking, or model alignmen, is an approach o validaion ha can give us greaer confidence in boh models. The ideas is o compare models in a basic way o see how hey are similar and differen and, more imporanly, o increase our confidence ha boh models can be used o say somehing abou he quesion under sudy. Docking is used o deermine wheher wo models can produce he same resuls, which in urn is he basis for criical experimens and for ess of wheher one model can subsume anoher (Axell, Axelrod, Epsein, and Cohen, 1996). Docking is hus a verificaion process ha seeks o find isomorphic relaions beween wo or more relaed models. Following his approach, he original sysems dynamics implemenaion, he Mahemaica implemenaion, he RePas implemenaion, and he Java Swarm implemenaions have where docked. In paricular, he key absracions of he models where compared and conrased. The original sysems dynamics version is couned as full implemenaion. Since his version relied on sysems dynamics i used direc mahemaical assignmen operaors o produce new values. This seems o have lead o unusual ordering rules ha depend only on he previous sae and ignore he curren knowledge. In paricular, using old pipeline and sock numbers o compue orders insead of he more reasonable approach of using he curren inpus o calculae he nex oupus is a key o maching he model oupu using oher echniques. The key absracions including he following: Scheduling is implemened using ime index variables. All of hese variables are updaed simulaneously, leading o he unusual ordering rules described above. Agens are represened as collecions of uncoupled variables. The Mahemaica version is a full implemenaion since i uses a funcional syle of programming ha is significanly differen from he original sysems dynamics approach. The key absracions including he following: Scheduling is implemened using simple ieraed funcion calls. Variables can be updaed sequenially wihin a ime sep. The original sysems dynamics model s unusual ordering rules mus be imposed by buffering values during a ime sep. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 16

17 The Beer Dock :55: Agens are represened as collecions of loosely coupled variables. Wriing he iniial Mahemaica model was quie easy. Docking he Mahemaica model o he original sysems dynamics by mapping corresponding absracions was exremely ime consuming since he only available informaion on he original model is given in he referenced papers (MLS 1991; Serman, 1987; Serman, 1989). Finding an exac mach beween he models required many repeaed runs o uncover and remove he suble variaions beween he wo models. The RePas version is a full implemenaion since i uses an objec-oriened syle of programming ha is significanly differen from he previous approaches. The key absracions including he following: Scheduling is implemened using a basic ABMS scheduling sysem. As wih Mahemaica, variables can be updaed sequenially wihin a ime sep. The original sysems dynamics model s unusual ordering rules mus be imposed by buffering values during a ime sep. Agens are represened as objecs. As wih wriing he Mahemaica model, creaing he iniial RePas model was easy. Docking he model was much more difficul. During he RePas docking a subsanial amoun of ime was spen maching he exac numeric oupus from he Mahemaica version. Afer a subsanial period of iniial work, an exac mach was found wih he firs weny or so Mahemaica model run seps. Laer ime seps seemed o slighly diverge from he Mahemaica values and he sysems dynamics chars. A firs his seemed o be he resul of numerical round off errors. However, increasing he number of Mahemaica model oupu digis from six o 17 revealed a sligh difference in he second ime sep. Afer a subsanial period of inspecion his difference was raced o he order of calculaion execuion. Correcing his order allowed an exac, full precision mach o be made and suggess an imporan observaion: Never blame numerical errors unless you can prove hey are he problem! As saed previously, he Java Swarm version of he Beer Game is couned as half an implemenaion since much of is objec-oriened Java code is shared wih he RePas implemenaion. Scheduling is implemened using a rich ABMS scheduling sysem. As wih Mahemaica and RePas, variables can be updaed sequenially wihin a ime sep. The original sysems dynamics model s unusual ordering rules mus be imposed by buffering values during a ime sep. Agens are represened as objecs. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 17

18 The Beer Dock :55: As wih wriing he Mahemaica model, creaing he iniial Java Swarm model was easy. The lesions learned from he RePas model work simplified he docking process enough o make i quie manageable. Conclusion The docking process was found o be challenging and ime consuming, bu ulimaely rewarding. All of he model implemenaions where able o produce he same resuls afer a subsanial amoun of effor was expended. The docking process was found o be paricularly valuable since i revealed hidden assumpions embedded in he underlying models. In paricular, i was found ha osensibly idenical models can have suble variaions ha are only revealed upon exremely close inspecion. References Axell, R., R. Axelrod, J. Epsein, and M. Cohen. (1996). Aligning Simulaion Models: A Case Sudy and Resuls, Compuaional and Mahemaical Organizaion Theory, 1(2): Richard Buron, (1998) Validaing and Docking: An Overview, Summary and Challenge, M. Prieula, K. Carley & L. Gasser, (ediors), Simulaing Socieies: Compuaional Models of Insiuions and Groups, AAAI/MIT Press, Cambridge, MA. Richard Buron, (Ocober 1999) The Challenge of Validaion and Docking, Proceedings of he 1999 Agen Simulaion Workshop, Universiy of Chicago, Chicago, IL. Moseklide, E., E. Larsen, and J. Serman. (1991). Coping wih Complexiy: Deerminisic Chaos in Human Decisionmaking Behavior, In Casi, J. L. and A. Karlqvis. (Eds.), Beyond Belief: Randomness, Predicion, and Explanaion in Science, CRC Press, Boson. Serman, J. D. (December 1987). Tesing Behavioral Simulaion Models by Direc Experimen, Managemen Science, 33(12): Serman, J. D. (March 1989). Modeling Managerial Behavior: Mispercepions of Feedback in a Dynamic Decision Making Experimen, Managemen Science, 35(3): Swarm Developmen Group (March 2002). Swarm Developmen Group Web Sie, Available as hp:// Universiy of Chicago (March 2002). RePas Web Sie, Available as hp://repas.sourceforge.ne /. file:///users/owen/xdocumens/beerdock/beerdock.hm Page 18

19 The Beer Dock :55: Wolfram Research (March 2002). Mahemaica Web Sie, Available as hp:// [1] Corresponding auhor address: Michael J. Norh, Argonne Naional Laboraory, 9700 S. Cass Avenue, Argonne, IL 60439; Telephone: (630) ; Facsimile: (630) file:///users/owen/xdocumens/beerdock/beerdock.hm Page 19