Simulation with System Dynamics Models IE 680 Spring 2007

1. Simulation with System Dynamics ModelsIE 680 Spring 2007 Po-Ching C. DeLaurentis April 19, 2007

2. Outline • Systems Thinking • System Dynamics (SD) • Paradigm Comparison • Basics of System Dynamics • Quantification Challenges • Simulation with System Dynamics- An Example • Summary • System Dynamics Resources

3. Systems Thinking • Systems Thinking • Early 20th century physicists began to challenge Newtonian precepts; Werner Heisenberg, Norbert Weiner, Von Bertalanffy • “An approach for developing models to promote our understanding of events, patterns of behavior resulting in the events, and even more importantly, the underlying structure responsible for the patterns of behavior”* * http://www.systems-thinking.org

4. System Dynamics • System Dynamics • Introduced by Jay Forrester of MIT in 1958 • “The study of information-feedback characteristics of industry activity to show how organizational structure, amplification (in policies), and time delays (in decisions and actions) interact to influence the success of the enterprise” (Forrest 1958 & 1961)

5. High Abstraction Less Details Macro Level Strategic Level Aggregates, Global Casual Dependencies, Feedback Dynamics, … • System Dynamics • Levels (aggregates) • Stock-and-Flow Diagrams • Feedback loops • Agent Based • Active objects • Individual behavior rules • Direct or indirect interaction • Environment models • Discrete Event • Entities (passive objects) • Flowcharts and/or transport networks • Resources Middle Abstraction Medium Details Meso Level Tactical Level Low Abstraction More Details Micro Level Operational Level Mainly discrete Mainly continuous Individual objects, exact sizes, distances, velocities, timings, … Paradigm Comparison of System Dynamics, Discrete Event & Agent Based • Borshchev A , Filippov A. From System Dynamics and Discrete Event to Practical Agent Based Modeling: Reasons, Techniques, Tools. Proceedings of the 22nd International Conference, July 25-29, 2004, Oxford, England, UK.

6. Rate Stock A Stock B Decision Rules Basics of System Dynamics Stock-and-Flow Casual Loops

7. Basics of System Dynamics (cont’d) Brownies_in_Stomach(t) = Brownies_in_Stomach (t - dt) + (eating - digesting) * dt INIT Brownies_in_Stomach = 0 DOCUMENT: Initially Andy’s stomach is empty. UNITS: brownies eating = 1 DOCUMENT: Andy eats a brownie every hour. UNITS: brownies/hour digesting = 1/2 DOCUMENT: Andy digests 1 brownie every 2 hours. He therefore digests a half a brownie every hour. UNITS: brownies/hour

8. Quantification Challenges Basics of System Dynamics (cont’d) • Wide Range of System Dynamics Applications • Corporate planning and policy design • Economic behavior • Public management and policy • Biological and medical modeling • Energy and the environment • Supply chain management

9. Quantification Challenges • System dynamics is strategic in orientation and it is often seen to have ‘soft’ variables • Example (Coyle 2000): Consumer Satisfaction as an influence on New_Order_Inflow_Rate = Basic_Inflow * Satisfaction_Multiplier A variable that may range from 0 to an upper limit, and have a nonlinear relationship with Consumer Satisfaction

10. Quantification Challenges (Cont’d) • If it becomes New_Order_Inflow_Rate = Basic_Inflow * Satisfaction_Multiplier * Quality_Multiplier * Price_Multimplier * etc. The number of uncertainties becomes very large; Strong assumption that multipliers are multiplicative. EX: 0.5 * 0.5 * 0.5 = 0.125  only 25% of 0.5 0.510 = 0.000977  only 1/500 of 0.5

11. Quantification Challenges- Example

12. Simulation Example • Police & Driver– A System Dynamics Model for a Mixed Strategy Game (Kim & Kim, 1997) • System dynamics: dynamic fluctuations of a system • Game Theory • Players; preferences & strategies; payoff/utility functions • Players change decisions in response to other players’ actions • Finding equilibrium states in game situation • Dominant vs. mixed strategies

13. Police & Driver Game • “You are driving your car in a hurry…there are two states of the world: either the police are nearby or they are not. There are two actions to choose from: either to violate the speed limit or to abide by the law.” (Tsebelis 1989) q 1-q p 1-p Note: c1 > a1 b1 > d1 a2 > b2 d2 > c2

14. Police & Driver Game (Cont’d) • Mixed strategy equilibrium results: • p = prob. with which the driver chooses to speed • q = prob. with which the policeman decides to patrol • Solved for: p* = (d2-c2)/(a2-b2+d2-c2) q* = (b1-d1)/(b1-d1+c1-a1) • Observation • The probability of the driver’s law violation is not determined by the payoffs for the driver • Increase in penalty ( a1) • Argument • A contradiction to common sense: increase in penalty is conceived as one the most effective tools for policy implementation

15. Police & Driver Game with System Dynamics • Why does game theory produce theorems inconsistent with common sense? • Game theory applicability • Equilibrium applicability • How to model this game with SD? • Probability of players’ behaviors • Two independent players in the game • Population mixed-strategy game

16. System Dynamics Diagram of the Game parameters

17. Patrol to office Quitting Time Quit Patrol Difference eup eunp Policemen in Office Policemen in Patrolling prob p Go Patrol Office to Patrol dpcnp euno euv dpvnp eunv Patrolling Time eup dpvp dpcp ppvp ppvnp ppcnp ppcp Speed Down Time Violation to Conform Speed Down prob v Drivers in Violation Drivers in Conforming Difference euv eunv Speed Up Conform to Violation Speed Up Time System Dynamics Diagram of the Game (redrawn) parameters

18. Simulation Results • Oscillation Rather than Equilibrium p*=0.25 (2:prob p) q* = 0.5 (1:prob v) Tendency towards the equilibrium state But it takes a long time!!

19. Simulation Results (Cont’d) • The Effectiveness of Penalty Increase p*=0.25 (2:prob p) q* = 0.22 (1:prob v) Increase in Penalty Tendency towards the equilibrium state Prob. of violation reduced to < 0.25 for about 50 days

20. Simulation Results (Cont’d) • Considering Information Delay Between the Police & the Driver 10 days for Driver 5 days for Police Larger amplitude Not approaching steady state

21. Simulation Results (Cont’d) • The Effectiveness of Penalty Increase with Information Delay Increase in Penalty Prob. of violation is lightly reduced

22. Simulation Results (Cont’d) • Effectiveness of Automatic Penalty Management (without info delay) • Police change the amount of penalty in line with probability of changes in law violation Equilibrium reached after a short period of fluctuation Equations

23. Policy Implications • Simulation results suggest: • Temporarily reduce the violation tendency of drivers by changing the amount of penalty • Penalty management can decrease the amplitude of fluctuating behavior of drivers • For policy makers • Temporary reductions of speed limit violation will be a sufficient incentive for introduction of penalty increase • Penalty management can reduce the amplitude of fluctuation violations • Max. level of violation  probability of car accidents?

24. Insights • Information delay & policy interruptions exist in a dynamic system • Mixed-strategy equilibrium may be a poor guide • Equilibrium vs. Steady State • In the real world, players are usually myopic • System Dynamics: simulate evolutionary processes toward (non-)equilibrium states • Game Theory: a framework for modeling the world of competition and cooperation

25. Summary • System Dynamics • Stock-and-flow • Casual loops • Higher, strategic level modeling • Dynamic behaviors of a system • Transient processes • Challenges: quantification of influencing factors; underlying effects

26. System Dynamics Resources • System Dynamics Society • http://www.systemdynamics.org/ • Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World, McGraw-Hill/Irwin, 2000 • Software • VenSim– http://www.vensim.com/ • Stella– http://www.iseesystems.com/

27. Questions

28. Ref. Model Equations

29. System Dynamics Diagram of the Game (Cont’d) Parameters