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

Modeling Non-linear relations

Jayendran Venkateswaran IEOR, IIT Bombay

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

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 Units: SKU/Day (2) FINAL TIME = 100 The final time for the simulation. 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 Units: SKU/Day (6) Sales rate= Demand Units: SKU/Day (8) TIME STEP = 1 The time step for the simulation

Inventory Production rate Sales rate Demand

Do Units Check (Ctrl+U) If you get error, fix it

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

• nly 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

Rat population density

Infant Survival multiplier

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

Rat Population Model Parameters

Rat population density

(input, output) (0,1), (0.0025,1), (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), (0.025,0.1), (0.0275,0.1) We will model this in Vensim using Lookup

Infant Survival multiplier

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

Rat Population Model

Download ratmod-class.mdl. Update equation as shown

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)

Rat Population Rat Birth Rate Rat Death Rate Rat Population density female rat population Infant Survival Multiplier Normal rat fertility Average rat lifetime Area sex ratio

=22 months =0.5 =11000 sq ft

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

Use lookup function in Vensim

1 2 3

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.. Initial population = 0 rats Initial population = 150 rats Initial population = 250 rats

What behavior pattern do you

• bserve?

When does population reach stability? What is the stable population size? When is the ‘inflection point’?

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