TODAY: TODAY: Recitation Lecture Hands-on tutorial
TODAY:TODAY:
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!!