Dynamics of Simple Structures: Positive Feedback Jayendran Venkateswaran IE604 Introduction p In a positive feedback process, a variable continually feeds back onto itself to reinforce its own growth or collapse n Example: Arms race, World population Threat perceived by + Nation B + + Weapons of Weapons of (+) Nation A (+) Nation B Net Birth Rate Population + + + Threat perceived by Nation A IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Characteristic Behavior p Consider an ordinary piece of paper n Say, thickness of paper is 0.1 mm n Fold it in half. Fold it again. How thick is it? n How thick will it be if we fold it 42 times? 100 times? p Exponential growth characterizes behavior of most positive feedback systems n Exponential collapse or accelerated decay is also possible IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Positive Feedback Loop p General structure Stock (S) Net Inflow Rate + + Fractional growth rate (g) p Example Principal n Interest rate 15% p.a. Net Interest n Income Compounded every year + + n For 15 years Interest IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Computation Steps p Initialize Time = 0, Stock0=initial_value 1. Calculate netrate from value of stock 2. Calculate increment = dt*netrate 3. Time = Time + dt 4. Add increment to stock 5. If Time <= EndTime, then Goto Step 1, else STOP. IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Example Computation p Time Level Rate Increment p 0 100 100x0.15=15 15x1=15 p 1 115 115x0.15=17.25 17.25 p 2 132.25 19.84 19.84 p 3 152.09 22.81 22.81 p … … … … p 15 813.75 122.06 180 900 800 160 700 140 600 120 Rate: Level: Principal 100 500 Net Interest Income 400 80 300 60 200 40 100 20 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Positive Feedback Loop p Exponential growth equation p Time constant (T) = 1/g n After passage of one time constant, stock value will be ____ times its initial value n Larger T (or smaller g) produces flatter growth curve. p Doubling time (Td) n Time interval required for an exponentially growing variable (stock) to double in value n Td=___*T IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Rate-Level graph p g can have any positive value … still will produce exponential growth 135 115 95 Rate 75 g 55 35 15 100 300 Level 500 700 900 IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Time horizon of exponential growth p The time horizon over which exponential growth occurs seem to alter the perception of growth even though underlying system remains the same. IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Time horizon of exponential growth (2) Source: Sterman, John D. Business Dynamics (Fig 8-5) IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran Super Exponential Growth p Many positive feedback systems are characterized by doubling times that decrease, rather than remain fixed, as value of system level increases n Such non-linear system has more pronounced growth than linear system Example: 135 World Population Nonlinear system 115 95 Linear system2100 Rate 75 g Super-Exponential growth 55 1600 35 1100 15 100 300 Level 500 700 900 600 Exponential growth 100 IEOR, IIT Bombay IE 604: System Dynamics0 Modelling1 2 & Analysis3 4 5 6 7 8 9Jayendran10 11 Venkateswaran12 13 14 15 Using Vensim® Rate Links Stock Auxiliary variables Outputs Download free software (student version) from www.vensim.com • Use Help Manuals to learn the software IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran.