Modeling Non-linear relations Jayendran Venkateswaran IEOR, IIT - PowerPoint PPT Presentation

Modeling Non-linear relations Jayendran Venkateswaran IEOR, IIT Bombay

Linear models • A process is linear if the process response is proportional to the input stimulus given – E.g. Saving’s account in bank; putting 10% more effort may get 10% more output, etc. – Linear systems are extensively studied since mathematical modeling of such systems is straightforward. – In fact, we use linear models even for real-life scenarios that are non-linear IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Non-Linear models • Nonlinear relationships are fundamental in dynamics of many systems – Price per unit reduces when we purchase in bulk – Company manufactures at desired rate, unless capacity is inadequate – Healthcare boost life expectancy… up to a point – Product sales must tend to zero as availability or quality falls to zero, no matter how cheap it is! – Parking charges at Malls – Fuel consumption vs. speed of vehicle – … IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Inventory Example • A small production unit had a fairly stable process, with 100 SKU per day of production. The finished goods are added to the end- inventory. The demand is satisfied from the end- inventory. The production continues independent of the actual demand. The initial inventory is 200 units. The average demand is 110 SKU per day. • Build a valid SD model of the above scenario IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Inventory Example (contd.) (1) Demand=110 Inventory Units: SKU/Day Sales rate Production rate (2) FINAL TIME = 100 The final time for the simulation. Demand Units: Day (3) INITIAL TIME = 0 The initial time for the simulation . Units: Day (4) Inventory= INTEG (Production rate-Sales rate, 200) Units: SKU (5) Production rate= 100 Do Units Check (Ctrl+U) Units: SKU/Day If you get error, fix it (6) Sales rate= Demand Units: SKU/Day (8) TIME STEP = 1 The time step for the simulation IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Inventory Example (contd.) • Although the model shows no error, it doesn’t seem to be realistic. The Factory Manager is unhappy that the stock of finished goods goes negative! How to fix this? • A simple way would be to use IF THEN ELSE or MIN function, where sales equals the demand only if we have enough inventory. • Do Units Check to see if there are errors. • Improve the model to ensure there are no unit errors. IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Table or Lookup Function • Table or Lookup function is a way to capture the nonlinear response function. • The nonlinear response function is specified using several pairs of points. • The simulation program then creates a curve through these points which is used to determine the necessary values to run the simulation. • Let’s see how to use a Table or Lookup function in Vensim with an example IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Rat Population Growth Model An experiment had been conducted on a population of rats. The rats were kept in controlled environment of area 11000 sq. ft. with sufficient food and water supplies. Soon, the population began to thrive. New rats were born. Old ones died after an average lifetime of 22 months. No migration or predation of population was allowed. The experiment found that population density affected infant mortality, which reduced the birth rate, while the death rate remained unaffected. Initial rat population was 10. Assume age doesn ’ t matter for reproduction and the male:female ratio is 1:1. Also, the normal rate fertility is 0.4 rats/ female/ month. Build & simulate a SFD of the above scenario Controlled experiment on population of Norway rats found that population density affected infant mortality, which reduced the birth rate (Based on Calhoun (1962)/ Goodman (1989) IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Rat Population Growth Model (2) Assumptions No migration or predation of population Controlled environment Ample & sufficient food supply Others for the simulation model Age doesn ’ t matter for reproduction Sex ratio à male:female::1:1 Infant survival multiplier varies with population density à Infant Survival multiplier Rat population density IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Rat Population Model Parameters (input, output) (0,1), (0.0025,1), Infant Survival multiplier (0.005,0.96), (0.0075,0.92), (0.01,0.82), (0.0125,0.7), (0.015,0.52), (0.0175,0.34), (0.02,0.2), (0.0225,0.14), Rat population density (0.025,0.1), (0.0275,0.1) We will model this in Vensim using Lookup IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Rat Population Model Download ratmod-class.mdl. Update equation as shown Rat Population Rat Birth Rate Rat Death Rate female rat Normal rat population fertility =0.5 sex ratio Average rat =22 months lifetime Infant Survival Rat Population Multiplier =11000 sq ft Area density d [Rat Pop] = Rat Birth Rate – Rat Death Rate Rat Death Rate = Rat Pop / Average lifetime Rat Birth Rate = Female rat pop * Normal Rat fertility * Infant Survival Multiplier Female pop = Rat pop * Sex Ratio Normal Rat fertility = 0.4 rats/ female/ month Initial Rat population = 10 rats TIMESTEP = 0.125; FINAL TIME = 50 Infant Survival Multiplier = f (density) = f (Rat pop/ Area) IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Use lookup function in Vensim 3 1 2 IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

In ‘As Graph’ dialog box, enter IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

In ‘As Graph’ dialog box, enter ß Add one more row with (0.0275,0.1) IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

Rat population model simulation Simulate rat pop model with initial population= 10 When does population reach stability? What is the stable population size? When is the ‘ inflection point ’ ? What if.. What behavior pattern do you observe? Initial population = 0 rats When does population reach Initial population = 150 rats stability? What is the stable population Initial population = 250 rats size? When is the ‘ inflection point ’ ? IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran

IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran