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Recitation

Lecture

Hands-on tutorial

Functions to stress the systemFunctions to stress the system

Three functions that can be used to “stress” the model of the system

STEP(,) PULSE(,) RANDOM UNIFORM(,,)

Which of these will produce Which of these will produce an input that looks like…an input that looks like…

Bringing in the Effects of Bringing in the Effects of Additional InformationAdditional Information

Jim Burns

REVIEW: Previously we have REVIEW: Previously we have used dimensionless ratios to used dimensionless ratios to do thisdo this

Natural gas model Gas usage / Gas usage normal

Rabbits model (Carrying capacity – rabbits) / carrying

capacity

x1:population

x2: gasreserves

remaining

r1: Birth rate r2: Death rate

r3: gasconsumption rate

p1: Birth rate percapita per year

p2: Death rate percapita per year

v2: Death rate due togas reserves multiplier

v1: Fraction ofreserves remaining

p4: Initial gasreserves

p3: Gas consumptionper capita per year

p5: Initial gasreserves

EquationsEquations

Fraction of Reserves Remaining = gas reserves remaining / Initial gas reserves

Gas Consumption Rate = Gas consumption per capita per year *

population *

Fraction of reserves remaining

Is this dimensionally consistent??

RabbitsNet Rabbit Birth rate

Effect of resourcesCarrying capacity

Normal Rabbit Growth Rate

B

R

Equations for Rabbits modelEquations for Rabbits model

Effect of Resources = (Carrying capacity - Rabbits)/Carrying capacity

Net Rabbit Growth rate = Normal Rabbit Growth Rate *

Rabbits *

Effect of resources

What if Dimensionless Ratios What if Dimensionless Ratios Don’t give us the effect we want?Don’t give us the effect we want?

Is there another way to pull in information?

Let’s look at the Forrester World Model

Population

birth rate normal

births crowdingmultiplier

births food multiplier

births deathsbirths material multiplier

births pollution multiplier

<Time>

death rate normal

deathscrowdingmultiplier deaths food multiplier

deaths material multiplier

deaths pollution multiplier

crowding

land areapopulation

density normalbirths

crowdingmult tab

deathscrowdingmult tab

food ratio

food coefficientfood coefficient 1

food crowding multiplier

food percapita normal

food per capita potential

food pollutionmultiplier

switch time 7<Time>

foodpollutionmult tab

<pollution ratio>

<capital ratio agriculture>

food per capitapotential tab

<foodcrowdingmult tab>

<Time>

capital agriculture fraction indicated

births foodmult tab

deaths foodmult tab

capital agriculturefraction indicated tab

<material standard ofliving>

deathsmaterialmult tab

birthspollutionmult tab

births materialmult tab

<materialstandard of

living>

deaths pollutionmult tab

<pollution ratio>

What we see here is the use of What we see here is the use of table functions—table functions—Dimensionless MultipliersDimensionless Multipliers

Births = Birth Rate Normal * Population *Births Material Multiplier *Births Pollution Multiplier *Births Food Multiplier *Births Crowding Multiplier

The last four multipliers are dimensionless table functions

Similarly for Death RateSimilarly for Death Rate

Deaths = Death Rate Normal * Population *Deaths Material Multiplier *

Deaths Pollution Multiplier *

Deaths Food Multiplier *

Deaths Crowding Multiplier

Here’s another view of the Here’s another view of the same basic ideasame basic idea

Quantifying the Un-quantifiedQuantifying the Un-quantified

How to IncludeHow to Include

Customer Satisfaction Market Attractiveness Quality of Life Consumer Confidence Faculty Morale Material Standard of Living IN YOUR MODEL

Often these are Un-quantifiedOften these are Un-quantified

Begin by defining what one unit of any of these would be

Consider Quality of Life In the Forrester World Model, one unit of Quality

of Life is the level of life quality enjoyed in the year 1970

Define this to be a Parameter called Quality of Life Normal

Quality of Life Normal = QLN = 1

What sort of things affect What sort of things affect Quality of Life on a global Quality of Life on a global scale?scale?

Pollution Material Standard of Living Food Population density

For each of these, construct a For each of these, construct a ratioratio

Pollution ratio = Pollution normal/Pollution Here pollution normal is the amount of

pollution experienced in the year 1970, in pollution units

MSL ratio = MSL/MSL normal Here, MSL normal is the amount of MSL

experienced in the year 1970, in MSL units

More ratiosMore ratios

Food ratio = Food/ Food normal Again, Food normal is the amount of food

available in the year 1970, in Food units Crowding ratio = Population density

normal/Population density again, Population density normal is the

population density in the year 1970, say

What about Units?What about Units?

For some of our soft variables the units are undefined Meaning no one has defined them

We have to define them For example, one unit of pollution could be defined

as “the average aggregate level of pollution experienced by a “typical” earthling in the year 1970”

One unit of Quality of Life could be “the average aggregate level of quality of life experienced by a ‘typical’ earthling in the year 1970.”

Under Normal Conditions,Under Normal Conditions,

What is true about all of these ratios?

What is the dimensionality of these ratios?

Under “Normal Conditions” the ratio assumes a value of “1”

The ratios are always dimensionless

We can now construct our We can now construct our Quality of Life FormulaQuality of Life Formula

Quality of Life = QLN * Pollution ratio * MSL ratio * Food ratio * Crowding ratio

Is this formula dimensionally consistent? Under normal conditions, Quality of Life = ?? If pollution gets higher than normal, what happens to

quality of life, assuming everything else remains the same?

If food is higher than normal, what happens to quality of life, assuming everything else is the same?

The VENSIM RepresentationThe VENSIM Representation

Quality of Life

Quality of Life Normal

Pollution Ratio

Food Ratio

MSL Ratio

Crowding ratio

What if we felt that Material What if we felt that Material Standard of Living affected Standard of Living affected birth and death rates?birth and death rates?

BR = BRN * POPULATION *MSL ratio MSL ratio = MSL / MSL Normal Does this change the dimensionality of the

BR formula? Under “normal” conditions what effect does

Material Standard of Living have on BR, birth rate?

Similarly for death rate

We could do something We could do something similar for foodsimilar for food

BR = BRN * POPULATION * MSL ratio * Food ratio

Suppose that we believe that Suppose that we believe that the effect of an increase in the effect of an increase in food is less than the ratio food is less than the ratio would suggestwould suggest

We can amplify or attenuate the effect of a rise above normal conditions with the use of TABLE FUNCTIONS

We call these multipliers They are also dimensionless

Quality of Life

Quality of Life Normal

Pollution Ratio

Food Ratio

MSL Ratio

Crowding ratio

Pollution Multiplier

Food Multiplier

MSL Multiplier

Crowding Multiplier

The new formula is:The new formula is:

Quality of Life = QLN * Pollution multiplier * MSL multiplier * Food multiplier * Crowding multiplier

It must be accompanied by the following equations Pollution multiplier = TABLE(pollution ratio) MSL multiplier = TABLE(MSL ratio) Food multiplier = TABLE(Food ratio) Crowding multiplier = TABLE(Crowding ratio)

In VENSIM these are writtenIn VENSIM these are written

Pollution multiplier = pollution multiplier tab(pollution ratio)

MSL multiplier = MSL multiplier tab(MSL ratio)

Food multiplier = food multiplier tab(Food ratio)

Crowding multiplier = crowding multiplier tab(Crowding ratio)

Here, the tables are defined as Here, the tables are defined as ….….

o pollution multiplier tabo MSL multiplier tabo food multiplier tabo crowding multiplier tab

o Are treated as ‘constants’ and defined using the AS GRAPH capability

Some Forrester ‘tricks’Some Forrester ‘tricks’

Almost all of Forrester’s tables contain the point (1,1) on them.

What does this mean? Under normal conditions, the ratio is 1 Under normal conditions, the impact of the multiplier is

‘nil’ That is to say, the multiplier neither enhances or

attenuates the rate it affects. So it has no effect.

Look at the Births (Birth Rate) Look at the Births (Birth Rate) EquationEquation

• Births = Population • * IF THEN ELSE ( Time• > switch time 1 ,• birth rate normal 1 ,• birth rate normal )• * births material multiplier• * births crowding multiplier• * births food multiplier• * births pollution multiplier

Look at the Births (Birth Rate) Look at the Births (Birth Rate) EquationEquation

Births = Population * birth rate normal * births material multiplier * births crowding multiplier * births food multiplier * births pollution multiplier

Population

birth rate normalbirth rate normal 1

population initial

births crowdingmultiplier

births food multiplier

births deathsbirths material multiplier

births pollution multiplier

switch time 1<Time>

death rate normaldeath rate normal 1

deathscrowdingmultiplier

deaths food multiplier

deaths material multiplier

deaths pollution multiplier

switch time 3

crowding

land area

populationdensity normal

Population & Food

birthscrowdingmult tab

deathscrowdingmult tab

food ratio

food coefficientfood coefficient 1

food crowding multiplier

food percapita normal

food per capita potential

food pollutionmultiplier

switch time 7

<Time>food

pollutionmult tab

<pollution ratio>

<capital ratio agriculture>

food per capitapotential tab

<foodcrowdingmult tab>

<Time>

capital agriculture fraction indicated

births foodmult tab

deaths foodmult tab

capital agriculturefraction indicated tab

<material standard ofliving>

deathsmaterialmult tab

birthspollutionmult tab

births materialmult tab

<materialstandard of

living>

deaths pollutionmult tab

<pollution ratio>

Let’s look at the World Model Let’s look at the World Model and try this!!and try this!!