DID NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? SOME SENSITIVITY ANALYSIS AND OPTIMIZATION EXPERIMENTS USING SIMULATION BY VENSIM

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1 Elżbieta Kasperska Elwira Mateja-Losa Tomasz Bajon Rafał Marjasz Silesian University of Technology DID NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? SOME SENSITIVITY ANALYSIS AND OPTIMIZATION EXPERIMENTS USING SIMULATION BY VENSIM Introduction Analysis, modeling and of complex, non-linear, dynamic and multi-level systems have a long history, especially in area of famous System Dynamics method (see: [Coyl77, Coyl94, Coyl96, Coyl98, Coyl99, Forr61, Forr69, Forr71, Forr72, Forr75, Rado01, Ster00, Ster02]). The modern languages, such as Vensim [Vens09] can connect with optimization and in this way to estimate sensitive parameters in modeled objects and choose optimal decisions. The range of modeled objects is quite wide, from industrial models, to ecological or economics. The review of famous application of SD method Reader can find in monograph [Kasp09]. The application of SD method can answer such questions as: what if? (typical type prognostic), what structure conditions observed behavior of system over time? (type descriptive explanations study), how optimal in particulars conditions? (normative, optimization investigation). The latest are not so popular like, in our opinion, should be. The authors of this paper have some experience with so called embedding in optimization and vice-versa, working with family of models named DYN- BALANCE (see: [Kasp02, Kasp05, Kasp09, KaMa05, KaMa06, KaMS00,

2 98 Elżbieta Kasperska, Elwira Mateja-Losa, Tomasz Bajon, Rafał Marjasz KaMS01, KaMS03, KaMS06, KaSl05, KaSl05a, KaSl06]). In order to make popular possibilities of language Vensim in scope of sensitivity analysis and optimization, we decide to choose model of Waterloo Battle, created by prof. Coyle [Coyl96], with few authors modifications. The results are new and help to answer oretical question: wher Napoleon had to lose Waterloo Battle. Presentation of object of experiments and assumptions of The object of experiments is described in literature of fields (see: [Coyl96]). Now we summarized main assumptions of structure presented in Figure 1. Two enemies, coalition and Napoleon are on battlefield. The scenario is that both coalition and Napoleon have ces on frontier and ors in reserve, some distance away; both sides, if reserves are ordered into battle, re is a delay bee y can arrive and because of limitation in road network, re is a limit to rate at which reserves can be. Each side inflicts combat losses on or depending on rate at which each man fires, number of men in action and proportion of shot which hit a target. Napoleon wishes to achieve a speedy victory and ree commits his reserves as fast as possible; coalition commander hope to wait Allied ces to arrive from neighboring country and n to achieve an overwhelming victory. He ree commits reserves only when he needs to. His criterion of need is that he does not wish to be outnumbered, by more than 2 to 1 at point of combat.

3 DID D NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 99 Fig. 1. The structure of model Waterloo Battle in Vensim convention Source: Own idea.

4 100 Elżbieta Kasperska, Elwira Mateja-Losa, Tomasz Bajon, Rafał Marjasz Below we present list of variables and parameters and mamatical model of Waterloo Battle, interpreting structure from Figure 1. List of variables and parameters of this model: Coalition army reserves Coalition reserves engagement Coalition army marching toward battlefield Coalition army arrival rate Coalition army in battle Coalition army loss rate Coalition killed soldiers Coalition reserves trigger Napoleon killed soldiers Napoleon army loss rate Napoleon army in battle Napoleon army arrival rate Napoleon army marching towards battlefield Napoleon reserves engagement Napoleon army reserves Napoleon army transport time delay Napoleon transport capacity Coalition transport capacity Coalition army transport time delay Rate of fire per napoleon soldier Rate of fire per coalition soldier Napoleon soldier shot accuracy Coalition soldier shot accuracy Auxiliary number one 1 Auxiliary number two 2 STEP start of battle The mamatical model of Waterloo Battle: (1),, 1 0, 1 (3) 3 1, (4) (2)

5 DID NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? ,1 (5) (6) (7),0, 0 0, 0 (9), 0, 0 0, 0 (8) (10) (11) 3 2, (12) 2 1,1 (13) (14),, 0 0, 0 1, 2 0 0, 2 0 (15) (16) Now we present assumptions of basis in Table 1. Assumptions of Levels or parameters Initial value Dimension Car [Coalition Soldier] Camtb 0 [Coalition Soldier] Caib [Coalition Soldier] Cks 0 [Coalition Soldier] Nks 0 [Napoleon Soldier] Naib 9050 [Napoleon Soldier] Namtb 0 [Napoleon Soldier] Nar [Napoleon Soldier] Nssa [Coalition Soldier/Napoleon soldier shot] Cssa [Napoleon Soldier/Coalition soldier shot] Rofpns 150 [Napoleon soldier shot/hour/napoleon soldier] Rofpcs 150 [Coalition soldier shot/hour/coalition soldier] Cattd 6 [hour] Nattd 6 [hour] Ctc 4000 [Soldiers/hour] Ntc 2500 [Soldiers/hour] Table 1

6 102 Elżbieta Kasperska, Elwi ira Mate eja-losa, Tomasz Bajon, Rafał ł Marjasz The results of basis are presented in Figure 2 and Figure 3. We can observe that t at assumption model and at equations 1-16 Napo- leon could win batt tle, what is easily notice in values in Tab le 2, and Table 3. It will be interesting to suggest changes in mod del, specially stra ategy of coalition commander. If we change value 2 outnumbered coalition ces by Nap poleon ces, result of battle will be qui te opposite. We can observe this in Figure 4 and Figure 5. The authors have permed many experiments on model and noticed thatt model is very sensitive on values of parameters and initial values of levels. And bec ause of this we decide to execute sensitivity analysis and optimization on model on bases of possibilities of language Vensim [Vens09]. Fig. 2. The dynamics of variable: Coalition army in battl e (Coalition lose battle) Source: Own results.

7 DID D NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 103 End of Waterlooo Battle, values of variable Caib (observed) Table 2 Fig. 3. The dynamics of variable: Nap poleon army in battle (Napoleon win battle) Source: Own results.

8 104 Elżbieta Kasperska, Elwi ira Mate eja-losa, Tomasz Bajon, Rafał ł Marjasz End of Waterlooo Battle, values of variable Naib (observed) Table 3 Fig. 4. The dynamics of variable: Coalition army in battl e (Coalition win battle) Source: Own results.

9 DID D NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 105 End of Waterlooo Battle, values of variable Caib (observed) Table 4 Fig. 5. The dynamics of variable: Napoleon army in battl le (Napoleon lose battle) Source: Own results.

10 106 Elżbieta Kasperska, Elwi ira Mate eja-losa, Tomasz Bajon, Rafał ł Marjasz End of Waterlooo Battle, values of variable Naib (observed) Table 5 Some sensitivity analysiss and optimization experiments usi ng by Ven nsim The SD models containn usually many parameters. It is inte eresting to exam- ine effect on ir variation on output. We select some para ameters and assign max ximumm and minimumm values along with a rand dom distr ribution over which to vary m to see ir impact on model behavior. Vens sim has a meth hod of setting up such sensitivity. Monte Carlo mult ivariate sensitivity work ks by sampling a set of numbers from within boundedd domains. To per m one multivariate test, distribution each parameter specified is sampled, and resulting values used in a. When num mber of is set, example, at 200 0, this process will be repeated 2000 times. In order to do sensitivity you need to defi ine what kind of prob bability distribution valu ues each parameter will be drawn from. The simplest distribu- tion is Ran ndom Unim Distribution, in whic ch any number between minimum and maximum values is equally likely to occur. The Random Unim Distribution is suitable most sensitivity testing and is sele ected by default. Anor commonly-used distribution is Normal Distribution (or Bell Curve) in whic ch thatt values far from mean. Results of sensitivity testing can be dis-

11 DID D NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 107 played in different mats. Time graphs display behavior of a variable over a period of time e. The variables spread of values, at any period in time, are dis- played eir in terms of confidence bounds, or a separate values which combine to m individual traces. Now let present assumptions of sensitivity experiments: A) Firs st experiment will be rela ated to initial values of Coalition army reserves and Napoleon arm my reserves. The ey will be chosen (by Mont te Carlo method) from inter rvals: (25000, 35000) ICar and (30000, ) INar. The resu ults of s (2000 iterations) are presented in Figures 6, 7, 8. We can notice in m of confidence bou unds, how model is sensi- tive choice of initial value of variables: ICar, INar. Fig. 6. Confidence bounds variable Caib (at multi-variants ICar (25000, 35000), INar ( , 35000)) Fig. 7. Confidence bounds variable Naib (at multi-variants ICar (25000, 35000), INar ( , 35000))

12 108 Elżbieta Kasperska, Elwi ira Mate eja-losa, Tomasz Bajon, Rafał ł Marjasz Fig. 8. Confidencee bounds variable Crt (at multi-variants ICar (25000, 35000), INar ( , 35000)) B) Second expe eriment will be related to sensitivities of parameters Ctc and Ntc. They will be chos sen (by Monte Carlo method) from inter- vals s: ( 3000, 4000) and (200 00, 2500) ). The resultss of simu ulations are pre- sen nted in Figures 9, 10 and 11. C) Thi ird experiment will be related to sensitivities of Natt td and Cattd. They will be chosen (by Monte Carlo method) from inter- vals s: (4,6) Nat ttd,, (4,6) Cattd. The results of are pre- sen nted in Figures 12, 13 and 14. We can see in m of con nfidence bou unds, how model in sensitive choi ice of values pa- ram meters: Nattd, Cattd. Fig. 9. Confidence bounds variable Caib (at multi-variants Ctc (3000, 4000), Ntc (2000, 2500) ))

13 DID D NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 109 Fig. 10. Confidencee bounds variable Naib (at multi-variants Ctc (300 00, 400 0), Ntc (2000, 2500)) Fig. 11. Confidence bounds varia able Crt (at multi-variants Ctc (300 00, 400 0), Ntc (2000, 2500)) Fig. 12. Confidencee bounds variable Caib (at multi-variants Nattd (4, 6) ), and Cattd (4, 6))

14 110 Elżbieta Kasperska, Elwi ira Mate eja-losa, Tomasz Bajon, Rafał ł Marjasz D) Fourth experiment is most interesting because we decided in multi i- vari iants investigate results of choices: ICar (25000, 35000), INar (30000, 35000), Ctc (300 00, 4000) ), Ntc (2000, 2500), Nattd (5. 5,6) ), Cattd (5.5,6)) Fig. 13. Confidencee bounds variable Naib (at multi-variants Nattd (4, 6) ), Cattd ( 4, 6)) Fig. 14. Confidence bounds varia able Crt (at multi-variants ICar (25000, 35000), INar ( , 35000) ), Ctc (3000, 4000), Ntc (2000, 2500), Nattd (4,6) and Cattd (4,6)) In this case variations are all toger, and we can see, in Figu ures 15, 16 and 17 how model in sensitive such choi ices. This is entrance optimization experiments.

15 DID D NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 111 Fig. 15. Confidencee bounds variable Caib (at multi-variants ICar (25000, 35000), INar ( , 35000) ), Ctc (3000, 4000), Ntc (2000, 2500), Nattd (5.5,6) and Cattd (5.5,6)) Fig. 16. Confidencee bounds variable Naib (at multi-variants ICar (25000, 35000), INar ( , 35000) ), Ctc (3000, 4000), Ntc (2000, 2500), Nattd (5.5,6) and Cattd (5.5,6)) Fig. 17. Confidence bounds varia able Crt (at multi-variants ICar (25000, 35000), INar ( 30000, 35000) ), Ctc (3000, 4000), Ntc (2000, 2500), Nattd (5.5,6) and Cattd (5.5,6))

16 112 Elżbieta Kasperska, Elwira Mateja-Losa, Tomasz Bajon, Rafał Marjasz We can do optimization considering different point of view, example: Coalition commander wants to achieve maximum loses of Napoleon soldiers. Coalition commander wants to achieve minimum loses of own soldiers. Napoleon wish to maximize loses of Coalition soldiers. Napoleon wish to minimize own loses. Possibilities of language Vensim, allow to perm such experiments, using different so called payoff functions and choosing optimized parameters from given intervals. The intervals will be same like at sensitivity analysis, or sometimes modificated, consider real world circumstances. The results of experiments type optimization are presents in Table 6. On base of results of optimization, permed in Table 6, we can notice as follows: Napoleon reaches minimum of own killed soldiers as a result of specific partition of ces on frontier and in reserve at specific partition ces of coalition (see: experiment number two). On contrary maximum of coalition killed soldiers (what is advantageous from Napoleon point of view) is reached in experiment number 3. At assumption that only ces in reserve (both Napoleon and coalition) can be distributed from given intervals, we reached value of minimum Napoleon killed soldiers different than that in experiment 2, which is permed on position number 6. Similarly maximum of coalition killed soldiers is reach in experiment 7 (different than value in experiment 3). At assumption that it is possible to change transport capacity (both Napoleon s army and army of coalition), Napoleon can win in experiment 10 and 11, which is in that case quite obvious. At assumption that it is possible to change transport time delay (both Napoleon s army and army of coalition), Napoleon reaches minimum of own killed soldiers in experiment 14, and maximum of coalition killed soldiers of ces in reserves in experiment 15. In most complex version of experiments, we assume distribution of transport capacity and transport delays (both Napoleon s army and army of coalition). Napoleon reaches minimum of own killed soldiers in experiment 18, and maximum of coalition killed soldiers in experiment 19. The interesting experiments were permed, using ce ratio (recall value of criterion: 2 to 1 ratio of ces Napoleon to coalition at point of combat). The bases permed in chapter 2 of this paper, shown us that Napoleon can win battle when ce ratio has value 2, and lose battle at value 1,5 ( method we use n was trial and error ). Now, after optimization experiments it is reached that results are quite good, both in experiments with individual variable ce ratio as in with initial values of Napoleon and coalition reserves (see: experiments 21-28).

17 DID NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 113 The results of optimization on model Waterloo Battle Table 6

18 114 Elżbieta Kasperska, Elwira Mateja-Losa, Tomasz Bajon, Rafał Marjasz Table 6 cont.

19 Table 6 cont. DID NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 115

20 116 Elżbieta Kasperska, Elwira Mateja-Losa, Tomasz Bajon, Rafał Marjasz So in many circumstances Napoleon can win Waterloo Battle. But question is did that circumstances were possible and by what cost? The next question is that we used specific objective function, and by different criterion, example survivors on battlefield, results could be quite different. The dimension of paper doesn t allow to develop that aspect of research. Moreover it will be interesting to use more complex multivalued objective functions and functions with cost factors. Final conclusions The aim of this paper was presentation of some new results authors investigation in area of and optimization with use model of Waterloo Battle. We applied language Vensim which is very effective tool analysis, modelling and on models System Dynamics type. Many experiments we permed can answer oretical question: did Napoleon have to lose Waterloo Battle? Interesting visualization by so called confidence bounds help to estimate sensitivities of parameters, which are used in optimization experiments. It should be stress that possibilities of experimentation are practically unlimited. We can use many of objective functions, which cost elements and complex m (with weight factors and scaling). The dimension of that paper not allowed to develop many aspect of research. We hope that object of experiments help to cause concern with subject of sensitivity analysis and optimization, what is very interesting from methodical point of view. References [Coyl77] Coyle R.G.: Management System Dynamics. John Wiley & Sons, London [Coyl94] Coyle R.G.: Cosmic and Cosmos. User Manuals. The Cosmic Holding Co, London [Coyl96] Coyle R.G.: System Dynamics Modelling. A Practical Approach. Chapman & Hall, London [Coyl98] Coyle R.G.: The Practice System Dynamics, Milestones, Lessons and Ideas from 30 Years Experience. System Dynamics Review 1998, 14, pp [Coyl99] Coyle R.G.: Simulation by Repeated Optimization. J. Opt. R. S. 1999, 50, pp [Forr61] Forrester J.W.: Industrial Dynamics. MIT Press, Massachusetts [Forr69] Forrester J.W.: Urban Dynamics. MIT Press, Massachusetts 1969.

21 DID NAPOLEON HAVE TO LOSE THE WATERLOO BATTLE? 117 [Forr71] Forrester J.W.: World Dynamics. Wright-Allen Press, Massachusetts [Forr72] Forrester J.W.: Principles of Systems. Cambridge Press, Massachusetts [Forr75] Forrester J.W.: Collected Papers of Joy W. Forrester. Cambridge Press, Massachusetts [Gonc08] Goncalves G.: Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems. System Dynamics Review 2008, No 25. [Kasp02] Kasperska E.: Cybernetic Formulation of Some Functions of Mmanagement Types of Simulation and Optimization Approaches within System Dynamics Method. Proc. of 20th International Conference of System Dynamics Society. P.I. Davidsen, E. Mollona, V.G. Diker, R.S. Langer, J.I. Rowe (eds.), [Kasp05] Kasperska E: Some Remarks about Chosen Structural Aspect of System Dynamics Method. 6 éme Congrées Européen de Science des Systémes, AFSCET, Paris [Kasp09] Kasperska E.: Metodologia budowy i wykorzystania modeli ewolucyjnych w aspekcie uczenia się (w) organizacji społeczno-gospodarczej (in Polish). Wydawnictwo Politechniki Śląskiej, Gliwice [KaMa05] Kasperska E., Mateja-Losa E.: Simulation Embedded in Optimization A Key Effective Learning Process in (about) Complex, Dynamical Systems, ICCS LNCS 3516, Springer-Verlag, Heidelberg, Berlin [KaMa06] Kasperska E., Mateja-Losa E.: Extended Sensitivity Analysis of Parameters and Structure in System Dynamics Models Some Case Study. Proc. of 24th International Conference of System Dynamics Society, Nijmegen, The Nerlands. A. Grossler (ed.). New York [KaMS00] Kasperska E., Mateja-Losa E., Słota D.: Some Extension of System Dynamics Method Theoretical Aspects. Proc. of 16th IMACS World Congress. M. Deville, R. Owens (eds.). IMACS, [KaMS01] Kasperska E., Mateja-Losa E. Słota D.: Some Bynamics Dalance of Production via Optimization and Simulation within System Dynamics Method. Proc. of 19th International Conference of System Dynamics Society. J.H. Hines, V.G. Diker, R.S. Langer, J.I. Rowe (eds.), [KaMS03] Kasperska E., Mateja-Losa E. Słota D.: Optimal Dynamical Balance of Raw Materials Some Concept of Embedding Optimization in Simulation on System Dynamics Models and Vice Versa. Proc. of 20th International Conference of System Dynamics Society. P.I. Davidsen, E. Mollona, V.G. Diker, R.S. Langer, J.I. Rowe (eds.), [KaMS06] Kasperska E., Mateja-Losa E. Słota D.: Modelling and Simulation of Organizational Changes Using System Dynamics Method Some Case Study. Cybernetics and Systems Proc. of Eighteenth European Meeting on Cybernetics and Systems Research, Vienna, Austria, Vol. 2. R. Trappl (ed.). Austrian Society Cybernetics Studies, 2006.

22 118 Elżbieta Kasperska, Elwira Mateja-Losa, Tomasz Bajon, Rafał Marjasz [KaSl03] Kasperska E., Słota D.: Two Different Methods of Embedding Optimization in Simulation on Model Dynbalance (2-2). Proc. of 21st International Conference of System Dynamics Society. P.I. Davidsen, E. Mollona, V.G. Diker, R.S. Langer, J.I. Rowe (eds.). SDS, [KaSl05] Kasperska E., Słota D.: Modelling of Evolution in Structures, by Use of Hybrid Models on Base of System Dynamics. Proc. of 2005 Conference of System Dynamics and Management Science. Sustainable Development of Asia Pacific, Shanghai, China, November 4 to 6. Q. Wang (ed.). School of Economics and Management, Tongji University, Development Institute, Shanghai [KaSl05a] Kasperska E., Słota D.: Optimization Embedded in Simulation on Models Type System Dynamics Some Case Study, ICCS LNCS 3514, Springer-Verlag, Berlin-Heidelberg [KaSl06] Kasperska E., Słota D.: Parallel Dual Problem of Optimization Embedded in Some Model Type System Dynamics. Proc. of 24th International Conference of System Dynamics Society, Nijmegen, The Nerlands. A. Grossler et al. (eds.). The System Dynamics Society, New York [Rado01] Radosinski E.: Systemy inmatyczne w dynamicznej analizie decyzji. WN PWN, Warszawa [Ster00] Sterman J.D.: Business Dynamics System Thinking and Modeling a Complex World. McGraw-Hill, Boston [Ster02] Sterman J.D.: All Models Are Wrong: Reflections on Becoming a System Scientist. System Dynamics Review 2002, No 18. [Vens09] Vensim User s Guide Version 5. Ventana Simulation Environment, CZY NAPOLEON MUSIAŁ PRZEGRAĆ BITWĘ POD WATERLOO? ANALIZA WRAŻLIWOŚCI I OPTYMALIZACJA Z UŻYCIEM VENSIMA Streszczenie Problem analizy wrażliwości i optymalizacji nieliniowych, dynamicznych i wielopoziomowych systemów jest bardzo interesujący zarówno z punktu widzenia metodologii, jak i zastosowań praktycznych. Celem tego artykułu jest zaprezentowanie nowych wyników autorów w dziedzinie symulacji i optymalizacji z użyciem modelu typu SD (Dynamiki Systemowej). Modelowany system jest interesujący i prezentowany w prostszej wersji jako Bitwa pod Waterloo autorstwa profesora Coyla. Wykorzystywane przez autorów oprogramowanie Vensim umożliwia analizę wrażliwości (metodą Monte Carlo) i optymalizację z różnymi typami funkcji celu. Z kolei graficzna wizualizacja wyników zwana confidence bounds jest pomocna przy szacowaniu wrażliwych parametrów modelu i w ten sposób przy odpowiedzi na pytanie zawarte w tytule artykułu Czy Napoleon musiał przegrać bitwę pod Waterloo?.