System Dynamics

7/14/2019 System Dynamics

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Andrea Rizzoli, IDSIA Modelling, Simulation, and Optimisation

System DynamicsAims, tools and concepts

(Reference: Sterman, Chapter 1)

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A brief philosophical motivation

Heraclitus said that you never bathe twice in the sameriver (panta rei)

If the world changes, we also have to change

Even when we do change, we often encounter anunexpected policy resistance

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A new way of thinking?

Systems thinking: the world is a complex entity

you can do just one thing

everything is connected

Thanks to Systems Dynamics we can improve ourunderstanding of complex systems

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Interdisciplinarity

Systems Dynamics theory is based on non-linear dynamicsand on control theory (mathematics, physics, engineering)

Applications require specialistic knowledge of engineering,economics, sociology, ecology and so on

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What we need

To learn and understand in and about dynamicsystems we need:

tools: to represent our mental models

formal models and simulations: to verify ourmodels

methods: to refine our scientific intuitions, andovercome our immune system

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Policy resistance

Often, willing to solve a problem, we make decisions thatworsen the state of things

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10

20

30

40

1966 1967 1968 1969 1970 1971

Crude

BirthRate

(Births/year/1000p

eople)

1 9 7 1 1 9 9 4

Birth rate in Romania

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What happened?

Illegal imports of contraceptives

Illegal abortions: three times the previous death rate

Very poor families could not afford children

Children ended up in orphanages

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Other examples

Limiting the choice of pharmaceuticals increases healthcosts (LA Times, 20/3/96)

Subsidies to farmers to limit exploitation of erosion-proneareas induces the opposite effect (U. Minn)

Low tar cigarettes are more dangerous

ABS may induce a more aggressive driving style

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More examples

More roads = more traffic

More household appliances reduce the free time

Computers increase the usage of paper

Civil engineering works to mitigate floods might worsenthe impact of floods in other locations

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and even more examples

The widespread usage of pesticides favoured thedevelopment of resistant pest species

Uncontrolled usage of antibiotics enabled tuberculosis tomutate and there are now antbiotic-resistant strains

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Andrea Rizzoli, DTI Modelling, Simulation, and Optimisation

Event oriented view of the world

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Situation

Goals

Problem Decision Results


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Goals ofOther

Agents Actions ofOthers

Side

EffectsGoals

Decisions

Environment

but also triggering side effects, delayed reactions, changes

in goals and interventions by others. These feedbacks maylead to unanticipated results and ineffective policies.

Our decisions alter our environment, leading to new decisions,

Goals

Decisions

Environment

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Feedback

The world is governed by feedbacks positive (reinforcing)

negative (correcting)

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Positive and negative feedback loops

Figure 1-5a

R ChickensEggs

+

+

A system!s feedback structure

generates its dynamics

Time

Eggs

Chickens

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Positive and negative feedback loops (cont.)

Figure 1-5b

BChickensRoad

Crossings

+

-

Structure:

Behavior:

Time

RoadCrossings

Chickens

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Dynamics of Multiple-Loop Systems

Figure 1-6 Dynamics arise from the interaction of multiple loops

BR ChickensEggs Road

Crossings

+

+

+

-

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Learning is a feedback process

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Real

World

Decisions InformationFeedback


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The role of information

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RealWorld

Strategy, Structure,Decision Rules

Mental Modelsof Real World

DecisionsInformationFeedback


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Figure 1-10 Mental model revealed by a diagram of a company!s supply chain

Current supply chain cycle time: 182 days

Goal: 50% reduction

Manufacturing

Lead Time

Order Fulfillment

Lead Time

Customer

Acceptance

Lead Time

75Days

85Days

22Days

182

Days

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Andrea Rizzoli, IDSIA Modelling, Simulation, and Optimisation21

Saul

Steinberg,1976


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RealWorld


Mental Modelsof Real World

Decisions

Information

Feedback

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Real World

Decisions

Implementation failure Game playing Inconsistency Performance is goal

Unknown structure Dynamic complexity Time delays Inability to conduct controlled experiments


Inability to infer dynamics from mental models

Mental Models

Misperceptions of feedback Unscientific reasoning Judgmental biases Defensive routines

Selective perception Missing feedback Delay Bias, distortion, error Ambiguity

Information Feedback

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Dynamic complexity

It does not depend on how many components there are,but on howthey interact

It is notcombinatorial complexity!

Long time delays create problems for learning

An evolving system is difficult to learn

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Systems are

dynamic, tightly interconnected, and ruled by feedbacks

nonlinear, affected by hysteresis (path-dependant) self-organising, adaptive

policy resistant, counterintuitive

thus, they are dynamically complex

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Limited information

All measures areobservations, they arefiltered

We see what we want /expectto see

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Before 1600 scurvy was the foremost reason for deathamong sailors (even more than shipwrecks, battles, firesaltogether!)

1601: Lancaster performs a controlled experiment ontwo ships belonging to the West Indies Company

Three tbsp of lemon per diem to a crew, nothing to theother ship

On arrival, on one ship 110 deaths on 278, on the othernone

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Fighting Scurvy


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1747: Dr Lind performs an experiment on some patientswith scurvy. Only patients receiving a lemon based potionrecover in few days.

1795: British navy starts delivering lemon to her sailors

1865: the Bristish mercantile navy imposes the use oflemon and scurvy is finally won

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Time spent in learning:264 years

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Ambiguity

We often receive ambiguous information since theeffects of our decisions are mixed with the effectsof external actions

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Limited rationality

Given that the world is too complex, we often makedecisions based on:

rules of thumb

rules of habit

extremely simplified models

We never take into account feedbacks

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In the beer distribution game users generate costlyoscillations, on average being 10 fold away from the

optimal solution

In simulated auctions, participants bid too high,creating bubbles, which eventually burst

In role plays for medical doctors more tests thannecessary are prescribed

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Problems in the cognitive maps

A cognitive map is a representation of our estimation ofcause-effect relations

In a study of Axelrod (1976) the author could not findfeedbacks in the cognitive maps of political leaders (alsoconfirmed by Hall (1976) and Drner (1996))

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Wrong inferences on the system dynamics

Even if the cognitive map is OK, the structure of thesystem can be too complex to be caught by our brain

Mental simulation quickly becomes impossible

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Partiality in our judgements

E K 7 4

How many cards shall I turn to prove that the rule if vowelthen even is true?

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Inability to interact

We need to be prepared to challenge our mental models Under scrutiny, we tend to assume a defensive behaviour

This is counterproductive in the long run

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Inability to implement

Even when we have a good policy, we can fail toimplement it

We need our decisions to be shared, so that they areaccepted and implemented

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The role of virtual worlds

Virtual worlds are formal models, simulations where thedecision makers can make experiments

They are labs for learning

impossible configurations and alternatives can be tested

Yet, the quality of simulations can be very high (or at leastwe can measure it)

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Figure 1-14 Idealized learning process

Virtual World

Known structure Variable level of complexity Controlled experiments


Simulation used to infer dynamics of mental models correctly

Real World Unknown structure Dynamic complexity Time delays Inability to conduct controlled experiments

Mental Models Mapping of feedback structure Disciplined application of scientific reasoning Discussability of group process, defensive behavior

Virtual World Perfect Implementation Consistent incentives Consistent application of decision rules Learning can be goal

Real WorldDecisions

Real World Selective perception Missing feedback Delay Bias, distortion, error Ambiguity

Information Feedback

Implementation failure Game playing Inconsistency Performance is goal

Complete, accurate, immediate feedback

Virtual World

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Modelling 101Problem definition, models: scope and applications(Reference: Sterman, Chapter 3)

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What is a system?

There are several different definitions, different level forformality

For a start: it is an object or a collection of objects whoseproperties we want to study (Ljung and Glad)

Examples: how long does it take for my bathtub to fill?How much energy do I need to heat my house?

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What is a model

Experimenting on the real-world can be expensive (orimpossible, or dangerous)

A model of a system is a tool we use to answerquestions about the system without having to do anexperiment (Ljung and Glad)

mental models

verbal models

physical models

mathematical models

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How to build a model

Physical modelling

from physical laws and experience (domain knowledge)

Identification

learn the model from data

map to known structure (parametric approach)

learn the structure (non-parametric approach)

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System

F=ma

u=RI

Physical

ModellingIdentification

Model

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Types of mathematical models

Deterministic - Stochastic

Dynamic - Static Continuous time - Discrete time

Lumped - Distributed

Change oriented - Discrete Event driven

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Modelling phases

1. Problem formulation

2. Design of model structure3. Implementation of a simulation

4. Testing and validation

5. Policy design and evaluation

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Problem formulation

What is the problem, and why it is a problem

What are the key variables of our problem? Are theyvariables or parameters?

Which is our temporal horizon

What is the past behaviour of our system. What could itbe the future one?

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Model structure design (i)

What hypotheses and assumptions are there?

The dynamic structure of the model must explainendogenously its behaviour

We draw some causal loop diagrams (using software tools

including data mining ones)

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Model structure design (ii)

Which tools can we use:

modular decomposition of the system

causal loop diagrams

stocks and flows

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Implementation

We translate the conceptual model in a simulation model:

specification of the structure and the decisional rules inthe control flow

we need to estimate the simulation model parametersand choose initial (boundary) conditions

we need to assess the models consistency with its aimand limitations

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Verification

Our tests must

compare the model and the observed (real world)behaviour

measure the robustness of the model under stress

measure its sensitivity to small changes in itsparameters and structure

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It is an iterative process

1. Problem Articulation(Boundary Selection)

3. Formulation4. Testing

5. PolicyFormulation& Evaluation

2. DynamicHypothesis

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An overview of themodelling process

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The importance of the model aim

The aim must be clear!

If we try to model too many things at once, we might loseourselves

A model as complex as the real world does not help

The model is the ockhams razor

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Reference modes

set of plots and graphs describing the behaviour of themodel over time

they define

the variables of interest

help setting the time horizon

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Variables and parameters

Everything changes, so everything is a variable

Yet, some variables change so slowly that they are seen asconstants: parameters

This is an artificial distinction, but often it is quite useful

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Time horizon

Very often we underestimate the length of the timehorizon

Often we think of an immediate relationship betweencause and effect

The choice of the temporal horizon affects our perceptionof the system

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Consumption


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0

6

12

1986 1988 1990 1992 1994 1996

MillionBarrels/Day

Alaska

Production,Lower 48 States

Imports

0

5

10

15

20

25

30

1986 1988 1990 1992 1994 1996

1990

$/bbl

US oil production, consumption, imports, and price over a 10-year time horizonSource: EIA (US Energy Information Agency) Annual Energy Review.

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0

6

12

1870 1890 1910 1930 1950 1970 1990

Millio

nBarrels/Day

Alaska

Production,Lower 48 States

Consumption

Imports

0

10

20

30

40

50

1870 1890 1910 1930 1950 1970 1990

1990

$/bbl

US oil production, consumption, imports, and price over a 130-year time horizonSource: Production & consumption: 1870-1949, Davidsen (1988); 1950-1966, EIA Annual Energy Review.1880-1968, Davidsen (1988); 1968-1996, EIA Annual Energy Review, Refiners Acquisition Cost.

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The fossil fuel era shown with a time horizon of 15 000 years


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The fossil fuel era shown with a time horizon of 15,000 years

Figure 3-5Source: Adapted from Hubbert (1962).

-10000 -5000 0 5000

FossilEner

gy

Production

Year

IronAge First Oil

ShockIndustrial

Revolution

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S i h bl


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Structuring the problem

We must formulate a theory explaining the behaviour ofthe system (the dynamic hypothesis)

We need to provide an endogenousexplanation

We need to map this explanation to the system structure

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M d l b d h


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Model boundary chart

extremely rare

extremely useful: it allows to instantaneously visualise keymodel assumptions

modellers fear these charts

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Endogenous Exogenous Excluded

GDP Population Inventories

Consumption Tech change International tradeInvestment Tax rates Environ. constraints

Savings Energy policies Nonenergy resourc.

Prices Interfuel subst.

Salaries

Inflation rate

Employment

Interest rate

Debt

Energy production

Energy demand

Energy imports

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S b d i i


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Subsystem decomposition

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Company Market

Mkt. Response to Quality

Mkt. Response to Suitability

Mkt. Response to Delivery Delay

Mkt. Response to Price

Orders

Payment

Delivery of Product

Price

Quality

Delivery Delay

Product Suitability

Sales Effort

M k t Competing


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Market

Market Share

Potential Market

Process

Improvement Quality Improvement

Process Improvement

Workforce Commitment

Management Support

ProductCosts

New Product Development

Breakthrough Products

Line Extensions

R&D Staff

Development Projects

NewProducts

CompetitorPrice

On TimeDelivery,Defects

Orders

ShipmentsPrice

The irm

Market Valueof the Firm

Yield,Cycle Time

Costs

Hiring/Firing

Investment

Utilization

ManagementAccounting

Cost of Goods Sold

Budget Variances

Unit Costs

FinancialStress

Operating Income

Labor Variances

Takeover Threat

Income Statement

Balance Sheet

Flow of Funds

Financial

AccountingR&D Spending

Forecasted Revenue

Desired Fraction to R&D

Share Price

Market Capitalization

Financial Markets

CompetingProducts

Competitor

Pricing

Manufacturing

Capital

Labor

Inventory

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C l l


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Causal loops

They describe the cause-effect relationships among systemvariables

They allow to represent the structure of feedback insystems

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St k d fl


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Stocks and flows

They focus on the structural characteristics of a system.They highlight exchanges: mass, energy, information,money, ..

Examples of stocks: inventories, populations, debt, ... wemake decisions on stocks

Stocks represent the stateof a system

Flows are the change rates of stocks

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Fr m m del t sim lati n


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From model to simulation

Formalising the knowledge sheds light on what we reallyunderstand

A computer does not work on hot air

Ambiguities must be removed

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Verification


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Verification

The simulated behaviour is compared to the observed one

Each variable must correspond to a real-world concept Each equation must be dimensionally correct

The sensitivity of the model is tested, both with respect tothe parameters values and to the structure of the model

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Verification


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Verification

Models must be verified under extreme conditions:

what happens to GNP if I set energy production to 0?

what happens to my sales if I increase the price by1000%. And what if I set orders to zero?

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Policy design


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Policy design

Only models that pass the verification stage can be usedto design a policy or a control law

The design of a policy might then require to change thestructure of the system, not only decisions

The model will also be used to test the robustnessof thepolicy itself

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Causal loopsA tool for systems thinking(Reference: Sterman, Chapter 5)

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Causal loop diagrams


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Causal loop diagrams

capture the hypotheses on the causes of the dynamicbehaviour of a system

elicit our mental models

communicate the feedback structure in our models

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Causal loop diagram notation


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BR

Fractional

Birth Rate

Average

Lifetime

Death Rate

-+ ++

-

Birth Rate Population

+

Key

Example

Birth Rate Population

+

Variable Variable

Causal Link

Link Polarity

Loop Identifier: Positive (Reinforcing) Loop

Loop Identifier: Negative (Balancing) Loop

or

or

R

B

+

-

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Symbol Interpreta t ion Mathematics Examples

X Y+

All else equal, if X increases(decreases), then Y increases

(decreases) above what it would havebeen.

In the case of accumulations, X adds

to Y.

!Y/!X > 0

In the case ofaccumulations,

Y = (X + ... )ds + Yt0

t0

t

Product

Quality

+Sales

Effort

+

Results

Births+Population

X Y-

All else equal, if X increases(decreases), then Y decreases

(increases) below what it would havebeen.

In the case of accumulations, X

subtracts from Y.

!Y/!X < 0

In the case ofaccumulations,

Y = (-X + ... )ds + Yt0t0

t

Product

Price

-Sales

Frustration

-

Results

Deaths

-

Population

Link polarity: definitions and examples

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Stocks


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Stocks

They are not represented by causal loops

There is no distinction between change rate andaccumulation

For instance, a CLD cannot tell us if a population isincreasing or decreasing (see the figure in the CLD

notation)

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CLD howto

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Caused /= Correlated


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Caused /= Correlated

Correlations do not represent the structure of the system

Correlations depend on the structure!

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Average

Temperature

Murder

RateIce Cream

Sales

+ +MurderRate

Ice CreamSales

+

CorrectIncorrect

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Label link polarities


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Label link polarities

R B

Customer

Loss Rate

Customer

Base

Sales from

Word of

Mouth

Customer

Loss Rate

Customer

Base

Sales from

Word of

Mouth

-

++

+

Incorrect

Correct

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How to establish the polarity of a loop?


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How to establish the polarity of a loop?

How to discover whether a loop is balancing orreinforcing?

Quick fix: count the number of negative links

Right way: follow the effects of changes in a loop

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Profits Number of

Competitors

Attractiveness

of Market

Price

Cumulative

Production

Price

Market

Share

Unit

Costs

Bank Cash

Reserves

Perceived

Solvency of

Bank

Net

Withdrawals

Cleanup

Effort

Environmental

Quality

Pressure to Clean

Up Environment

Identify and label the polarity of the links and loops in the examples shown.

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Calculating the open-loop gain of a loop


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Figure 5-6

x4

x3

x2

x1

x4

x3

x2

x1I x1

OBreak the loop

at any point

and trace the effect

of a change aroundthe loop.

Polarity = SGN(!x1O/!x1I)

!x1O/!x1

I= (!x1O/!x4)(!x4/!x3)(!x3/!x2)(!x2/!x1

I)

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Causal links must have unambiguous polarity

? (+ or -)

RevenuePrice

CorrectIncorrect

+

RevenuePrice

Sales+

-

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Labeling loops


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Labeling loops

Labels are an effective communication medium

They must bear a significance (simple numbers and digitswill not do)

They must be easy to remember!

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Name and number

TimeRemaining


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your loops toincrease diagram

clarity and providelabels.

Haste MakesWaste

R2

CornerCutting

B2

Burnout

R1

MidnightOil

B1

Error Rate

Time perTask

SchedulePressure

Overtime

Fatigue

Productivity

CompletionRate

Work

Remaining

-- -

-

+

-

+

-

+

+

-

+

Delay

Delay

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The role of delays


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The role of delays

Delays introduce inertia and oscillations

They are fundamental to understand short andlong term effects of our decisions

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Price Supply

+Delay

Gasoline

Price +


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Different time delaysin the response of

gasoline demand andexpenditures to price

Discretionary

Trips

Density of

Settlement Patterns,Development of NewMass Transit Routes

ExpectedShort-Term

Price

Efficiency ofCars on Road

Efficiencyof Cars on

Market

GasolineExpenditures

Demand forGasoline

-

+

+

-+

+-

+

+

-

+

+

+Car Pooling andUse of ExistingMass Transit

Gasoline Price

Gasoline Consumption

Expenditures on Gasoline

Time

Delay

Delay

Delay

Delay

Delay

Delay

ExpectedLong-Term

Price

Vehicle Milesper Year

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Variable names should be nouns or nounh


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phrases

Figure 5-12

Costs Price

++

CorrectIncorrect

Costs Rise Price Rises

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Variable names should have a clear sense of


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direction

Figure 5-13

MentalAttitude

+Feedback

from the

Boss

Praise fromthe Boss

Morale

+

CorrectIncorrect

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Choose variables whose normal sense of


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Figure 5-14

Choose variables whose normal sense ofdirection is positive.

Costs Profit

-

Costs Losses

+

Criticism Unhappiness

+

Criticism Happiness

-

CorrectIncorrect

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If your audience was confused by


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Figure 5-15 Make intermediate links explicit to clarify a causal relationship.

MarketShare

UnitCosts

-

Market

Share

Unit

Costs

Production

Volume -

++ Cumulative

ProductionExperience

If your audience was confused by

you might make the intermediate concepts explicit as follows:

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D i d

CorrectIncorrect


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Figure 5-16 Make the goals of negative loops explicit.

B

QualityImprovement

Programs

ProductQuality

-

+

B QualityShortfall

+

+

+-

DesiredProductQuality

ProductQuality

QualityImprovement

Programs

B

Cooling Rate +

CoffeeTemperature

B Temperature

Difference

+

-+

CoffeeTemperature

CoolingRate

RoomTemperature-

-

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Distinguish between actual and perceivedconditions.


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Figure 5-17

Bias inReporting

System

ManagementBias TowardHigh Quality

ManagementPerception of

Product Quality

ReportedProductQuality

B

QualityImprovement

Programs

QualityShortfall

DesiredProduct

Quality

ProductQuality

+

-

++

+

+

+

+

Delay

Delay

Delay

Delay

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Jackson Pollock, Number 1 (1948)

Too much information ...


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... is no information

our short term memory contains 7 +/- 2 elements

we are not Jackson Pollock

use small and handy diagrams

use a diagram for each fundamental loop

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The ant and thegrasshopper

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ek


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Business Dynamics

Figure 5-19Reference mode forthe ant strategy

Time (weeks of the semester)0 13

Assignment Rate Work Completion Rate

Tasksperwee

EnergyLevel(0-100%

)


Workweek

Energy Level

Grades

Workweek(hours/wee

k)

100

0

Grades(0-100)

100

0


Assignment Backlog

Task

s

Work Completion Rate

ek


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Business Dynamics

Figure 5-20Reference modefor the

grasshopperstrategy

.


Assignment Rate

Tasksperwee


Assignment Backlog

Tas

ks


Workweek

Energy Level

Grades

100

0

Workweek(hours/we

ek)

100

0

EnergyLevel(0-100%)

Grades(0-100)


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Business Dynamics

Figure 5-21 Basic control loopsfor the assignment backlog

AssignmentBacklog

AssignmentRate

WorkCompletion

Rate

Effort Devotedto Assignments

TimeRemaining

DueDate

Workweek

Productivity

B2

CornerCutting+

-

-+

+

+

+

-

-

-+

CalendarTime

WorkPressure

B1

MidnightOil

Assignment Work-+


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Business Dynamics

Figure 5-22 Theburnout loop

Assignment

Backlog

g

RateWork

Completion

Rate

Effort Devoted

to Assignments

Time

Remaining

Due

Date

Workweek

Productivity

B2

CornerCutting

R1

Burnout

+

-

-+

+

-+

+

+

-

-

Calendar

Time

Delay

Work

Pressure

Energy

Level

B1

MidnightOil

AssignmentRate

WorkC l i

-+


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Business Dynamics

Figure 5-23 The tootired to think loop

AssignmentBacklog

Rate CompletionRate


TimeRemaining

DueDate

Workweek

Productivity

Quality ofWork

Grades

B2

CornerCutting

R1

Burnout

R2

Too Tiredto Think

B3

QualityControl

-

+

-

-+

++

+

+

-+

+

+

-

-

CalendarTime

Delay

WorkPressure

EnergyLevel

B1

MidnightOil

AssignmentRate

Work-+


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Business Dynamics

Figure 5-24 My dog ate myhomeworkParkinson!s Law

AssignmentBacklog

Rate CompletionRate


TimeRemaining

DueDate

Workweek

Productivity

Quality ofWork

Grades

B2

CornerCutting

B1

MidnightOil

R1

Burnout

R2

Too Tiredto Think

B3

QualityControl

-

+

-

-+

++

+

+

-+

+

+

-

-

Requests forExtensions

+

+

B4

My Dog AteMy Homework

CalendarTime

Delay

WorkPressure

EnergyLevel

Work Pressure


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Business Dynamics

Figure 5-25 Making the goal of a loop explicit. Adding the desired GPA and itsdeterminants.

Satisfactionwith Grades

Energy Level

DesiredGPA

Pressure forAchievement

Effort Devoted

to Assignments

Quality ofWork

Grades

++

--

+

- +

+

+ Quality ControlGoal Erosion

B B

Traffic jams


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In Ticino:

129 cars every 100 families

22% of families has a public transport yearly ticket (44%in Switzerland)

34


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[0,-+P\CR + X]O $%%!N+)(70'(510,+ ](44 R'(,6 X B

35

The solution?


107/268


Build new roads!

Congestionand delays

Build new roads

This is an open loop!What really happens when we build new roads?

36

Road

ConstructionDelay


108/268


TravelTime

Highway

Capacity

TrafficVolume

+

+

-

Delay

37

R d +


109/268


TravelTime

Highway

Capacity

Traffic

Volume

RoadConstruction

DesiredTravel Time

Pressure to

ReduceCongestion

B1

CapacityExpansion

+

-+

+

+

-

Delay

38

Road

Construction

+

Delay


110/268


TravelTime

HighwayCapacity

Traffic

Volume

DesiredTravel Time

Pressure to

Reduce

Congestion

B1

Capacity

Expansion

Cars in

Region

Trips per

Day

Average

Trip Length

Attractiveness

of Driving

Populationand Economic

Activity ofRegion

Cars per

Person

Adequacy ofPublic Transit

Public

TransitRidership

Public

TransitFare

Extra Miles

Take theBus?

-+

+

+

-

+

++

-

+

+

++

-

-

+

+-

Delay

Delay

B2

B3

B4

DiscretionaryTrips

39

RoadConstruction

+

Delay


111/268


TravelTime

HighwayCapacity

TrafficVolume

DesiredTravel Time

Pressure toReduce

Congestion

B1

Capacity

Expansion

Cars inRegion

Trips perDay

AverageTrip Length

Attractivenessof Driving


Activity ofRegion

Cars perPerson

Adequacy of

Public Transit

PublicTransit

Ridership

PublicTransit

Fare

Move tothe Burbs

Extra Miles

Take theBus?

Open theHinterlands

-+

+

+

-

+

++

-

+

+

++

+

+

-

-

+

+- -

Delay

DelayDelay

R1

B3

B5

B4

Size of RegionWithin Desired

Travel Time

Discretionary

Trips

B2

40


112/268


TravelTime

Highway

Capacity

TrafficVolume

RoadConstruction

DesiredTravel Time

Pressure to

ReduceCongestion

B1

CapacityExpansion

Cars inRegion

Trips perDay

AverageTrip Length

Attractivenessof Driving


Activity ofRegion

Cars perPerson

Adequacy of

Public Transit

Public

TransitRidership Public

TransitRevenue

PublicTransit

Fare

PublicTransitCosts

PublicTransitDeficit

PublicTransit

NetworkMove to

the Burbs

Extra Miles

Take theBus?

Open theHinterlands

RouteExpansion

FareIncrease

CostCutting

Choke offRidership

+

-+

+

+

-

+

++

-

+

+

++

+

+

-

-+

-+

+

+

-+

+

+- -

+

Delay

DelayDelay

Delay

R1

B3

B5

B4

R3

R2

B7

B6

Size of RegionWithin DesiredTravel Time

DiscretionaryTrips

MT CapacityExpansion

B8

-

Delay

B2

41

Road

Construction

+

D l


113/268


TravelTime

Highway

Capacity

Traffic

Volume

Construction

Desired

Travel Time

Pressure to

Reduce

Congestion

B1

CapacityExpansion

Cars in

Region

Trips perDay

Average

Trip Length

Attractiveness

of Driving

Population

and EconomicActivity of

Region

Cars per

Person

Adequacy of

Public Transit

Public

Transit

Ridership Public

TransitRevenue

Public

Transit

Fare

Public

Transit

Costs

PublicTransit

Deficit

PublicTransit

NetworkMove tothe Burbs

Extra Miles

Take theBus?

Open theHinterlands

RouteExpansion

FareIncrease

CostCutting

Choke offRidership

Can't Get Thereon the Bus

-+

+

+

-

+

++

-

+

+

++

+

+

-

- + - +

+

+

-+

+

+- -

+

-

Delay

DelayDelay

Delay

B2

R1

B3

B5

B4

R3

R2

B7

R4

B6

Size of Region

Within Desired

Travel Time

DiscretionaryTrips

MT CapacityExpansion

B8

-

Delay

42


114/268


Stocks and FlowsEssentials of systems dynamics


1

Causal diagrams vs stocks and flows


115/268

Andrea Rizzoli, DTI Modelling, Simulation, and Optimisation2

CLD and dynamics


116/268


CLD are the right tool to represent processinterdependencies

They allow to capture mental models at the projectinception

They are good to communicate project outcomes

Yet, they do not capture the dynamics of the system

Why?

3

Stocks


117/268


CLD do not represent accumulation, Stocks do

Stocks characterise the state of the system (and we makedecisions based on the state)

E.G.: the inventory of a company, the # of employees, thebalance on your checking account

4

Flows


118/268


Flows change stocks

Inventory is changed by the shipping rate

# of employees is changed by hiring, layoffs andretirements

Very often there are problems is deciding what is a flowand what is a rate (what about inflation?)

5

Notation


119/268


Valve (regulates the flow)

Stock

Flow

Sink or source(external stocks)

6

Example


120/268


flow names

7

The hydraulic metaphor


121/268


From pics to math


122/268


Levels accumulate flows

9

[flow]=[stock/time]

d stock

dt = in(t) out(t)

stock(t) =

t0 [

in(s

) out

(s

)]ds

+stock

(t0)

Measurement units


123/268


Must be consistent:

[stock]

[flow]=[stock/time]

10

Stocks are for ...


124/268


characterising the state of the system

providing memory and inertia generating delays

decoupling flows rates and creating unbalances

11

Examples of stocks and flows


125/268


Domain Stocks Flows

Math, Physics, Engineering Integrals, state variables Derivatives, rates, flows

Chemistry Reactants, products of

reactions

Reaction rates

Manufacturing Buffers, inventories Throughput

Economics Levels Rates

Accounting Stocks, balance sheet items Cash flows, income items

Biology, physiology Compartments Diffusion rates, flows

Medicine, epidemiology Prevalence, reservoirs Incidence, infection

mordbidit and mortalit

12

The snapshot test


126/268


Have we selected the right stocks and flows?

Lets imagine we take a snapshot of our system in a snapshot we can count stocks

flows are invisible

13


127/268



128/268


Information as a stock


129/268


Feedback


130/268


dxdt

=!"x

17

Auxiliary variables


131/268


Stocks can be modified only by flows

Flows can be expressed as a function of auxiliary variables

Auxiliary variables are either function of stocks or ofother auxiliary variables

18


132/268


Pop(t) = INTEGRAL(Net birth

rate, Pop(t0))

Net birth rate = Pop *

f(Food/Pop)

Pop(t) = INTEGRAL Net birth

rate, Pop(t0))

Net birth rate = Pop *

Relative birth rate

Relative birth rate =

f(Food per capita)

Food per capita= Food /

Pop

19

Only flows change stocks


133/268



134/268



135/268

Andrea Rizzoli, IDSIA

Feedback dynamicsPart I - First order systems



136/268

Positive feedback andexponential growth

2


137/268


State of theSystem

Goal

Time

State of theSystem

Time

CorrectiveAction

BDiscrepancy

+

-

+

Goal(Desired

State of System)

State of theSystem

+

RNet

IncreaseRate

State of theSystem

+

+

3

Positive feedback


138/268


In a first order systemwe only have one stock (statevariable)

The stock accumulates net inflow

4

Analytical solution


139/268


S(t) = S(0)egt

dS

S

= gdt

dS

dt= gS

separate variables

integrate by parts

raise both sides

Z S(t)S(0)

dS

S=

Z t0

gdz

ln(S(t)

ln(S(0

)) =gt

5

Examples


140/268


.dS/dt = Net Inflow Rate = gS


141/268


Net

InflowRate

(u

nits/time)

State of the System (units)

1

g

0

0

UnstableEquilibrium

7


142/268


0

2

4

6

8

10

0 128 256 384 512 640 768 896 1024

Structure

NetInflow

(units/time)

State of System (units)

t = 1000

t = 900

t = 800

t = 700

0

128

256

384512

640

768

896

1024

0

1.28

2.56

3.845.12

6.4

7.68

8.96

10.24

0 200 400 600 800 1000

Behavior

StateoftheS

ystem

(Units)

NetInflow

(units/time)

State of the System(left scale)

Net Inflow(right scale)

Time

8

Grasping the power of exponential growth


143/268


A sheet of paper is approximately 0.1 mm thick

What is the thickness of the sheet after we have folded itonto itself for 40 times?

9

The 70s rule


144/268


As the stock increases, also the net inflow increases

Avalanche effect: the stock doubles in a fixedamount oftime

Doubling time: 2S(0) =S(0)egtd

td= ln(2)

gtd=

70

100g

10

2

Time Horizon = 0.1td

)

2

Time Horizon = 1td

)


145/268


0

1000

0 200 400 600 800 1000

Time Horizon = 10td

State

ofth

e

System

(units)

0

0 2 4 6 8 10

State

oftheSystem

(units)

0

0 20 40 60 80 100

StateoftheSy

stem

(units)

0

1 1030

0 2000 4000 6000 8000 10000

Time Horizon = 100td

Stateofth

e

System

(units)

11


146/268

Negative feedback andexponential decay

12

Negative feedback


147/268


Just one state variable....

Net outflow reduces the stock

Net outflow: -d*SS(t) = S(0)edt

13


148/268


Net Inflow Rate = - Net Outflow Rate = - dS


149/268


1

-d

State of the System (units)0

Stable

Equilibrium

NetIn

flowRate

(units/time)

15


150/268


-5

0

0 20 40 60 80 100

Structure

NetInflow

(units/time)

State of System (units)

t = 0

t = 10

t = 20

t = 3 0

t = 40

0

50

100

0

5

10

0 20 40 60 80 100

Behavior

NetInflow

(units/time)

State of the System(left scale)

Net Inflow(right scale)

Time

d=5% / time unit

S(0)=100 units

16

Explicit goal


151/268


[Discrepancy] = units [State] = units

[Inflow rate] = units/time

[Adj. time] = time

Dimensional analysis

17

I i l k h h i i i f h

Adjustment time


152/268


It is also known as the characteristic timeof the system

When the correction action is just a difference:

18

net inflow rate =

Characteristic time


153/268


It represents how quickly the system reacts to adiscrepancy

E.g.: desired inventory level: 100 SKU, current inventory 60SKU, discrepancy 40

Adjustment time 1 week. We will get 40 units per week

19

Net Inflow

SState of

the System

S*Desired State of

the System

dS/dt

General Structure


154/268


B

Rate

+

-

ATAdustment Time

-

Discrepancy(S* - S)

B

Net

ProductionRate

Inventory

+

+

DesiredInventory

-

Inventory

Shortfall

B

Net HiringRate

Labor

+

Desired

Labor Force

-

LaborShortfall

dS/dt = Net Inflow RatedS/dt= Discrepancy/ATdS/dt= (S* - S)/AT

Net Production Rate = Inventory Shortfall/AT = (Desired Inventory - Inventory)/AT

Net Hiring Rate = Labor Shortfall/AT = (Desired Labor - Labor)/AT

Examples

+

+

ATAdustment Time

-

AT

Adustment Time

-

20

Net Inflow Rate = - Net Outflow Rate = (S* - S)/AT


155/268


1

-1/AT

NetInflowRate

(units/time)

0

S*State of the System

(units)

Stable

Equilibrium

21


156/268


0

100

200

0 20 40 60 80 100

State

oftheS

ystem

(units)

22

Analytical solution


157/268


z= SS

dz

z=

dt

!

ln(z) + ln(z0) = t t0

!

z0

z= e

t

t0!

dSdt

=(S

S)!

S

S= (S

S(0))e tt0

S

(t

) =S

(S S

(0))e 0

23

Error is halved when:

Ha!f life

SS

5


158/268


Error is halved when:

.. but then:

solving it

we get

Its the Rule of 70! Half time occurs at 70% of the characteristic time

S S

SS(0)=0.5

0.5= e

tt0!

ln(2) = t t0

!

t=ln(2)!= 0.70!

24

..

Rate equation for first order linear negative loop system:

Net Inflow Rate = Net Outflow Rate = (S* S)/AT

Analytic Solution:


159/268


S(t) = S* (S* S(0)) * exp(t/AT)

State ofthe System

{InitialGap

{Fraction ofInitial GapRemaining

{Gap Remaining

DesiredState of

the System*=

State ofthe System

DesiredState of

the System

=

Time (multiples ofAT)

1 exp(1/AT)

exp(1/AT)

exp(3/AT)0

0.2

0.4

0.6

0.8

0 1 AT

Fractionof

InitialGapRemaining

1 - exp(3/AT)

exp(2/AT)

1 exp(2/AT)

2 AT 3 AT

1.0

25


160/268

Multi loop systems

26

System with positive AND negative feedback


161/268


Population is the integral of netinflow

Net inflow equals births less deaths

dP

dt= bPdP

PopolazioneTasso di nascita Tasso di mortalit

+ -++

Tasso dinascita relativo Tasso di

mortalit relativo

27

Mortalit ratePopulation

Fractional mortalityrateFractional birthrate

Birth rate

0Death

Rates

Net Birth Rate

Birth Rate1

b

1 b-dation

Structure (phase plot) Behavior (time domain)

b > d Exponential Growth


162/268


0

Population

Birth

and

D

0

Death Rate1

0Time0

Popul

0

Population

Birth

and

Death

Rat

es

0

Death Rate

Net Birth Rate

Birth Rate

0Time0

Population

0

Population

Birth

and

De

ath

Rates

0

Death Rate

Net Birth Rate

Birth Rate

0Time0

Popula

tion

b < d

b = d

Exponential Decay

Equilibrium

-d

28

Superposition property


163/268


It holds for linearsystems of every order and complexity

System behaviour is the sum (or superposition) of thebehaviour of each subsystem

29


164/268

Non linear systems: S-shaped growth

30

Limits to growth


165/268


Even the steepest growth are limited:

food availability for a population of bugs # of people infected by a virus

purchases of a new product

31

The models and its equations


166/268


dP

dt=b(

P

C)Pd(

P

C)P

PopolazioneTasso di nascita Tasso di mortalit

+ -++

Capacit portante

Popolazione /capacit portante

-

+

Tasso di

nascita relativo Mortalit relativa-

+

32

Mortalit ratePopulation

Birth rate

Fractional birth

rate

Fractional death

ratePopulation /

Carrying capacity

Carrying capacity

ates


167/268


0

FractionalBi

rthandDeath

Ra

(1/time)

Population/Carrying Capacity(dimensionless)

FractionalBirth Rate Fractional

Death Rate

FractionalNet Birth Rate

0 1

33

Loop dominance


168/268


Positive feedback dominates if the increment of a stock

with respect to itself is positive

Respectively, negative feedback dominates when:

!S

!S> 0

!S

!S< 0

34

Positive FeedbackDominant

Negative FeedbackDominant


169/268


0

BirthandD

eathRates

(individu

als/time)


Birth Rate

Death Rate

Net Birth Rate

0 StableEquilibriumUnstable

Equilibrium

(P/C)inf 1

35


170/268


0

1

2

0

Population

Net Birth Rate

0 Time

Population/Carryin

g

Capacity

Ne

tBirth

Rate

36

First order systems cannot oscillate


171/268


Look closely at the phase plot

Population grows up to a stable equilibrium

If population leaves this stable equilibrium, it is stillforced to return to it

We need two state variables (or a discrete differentialequation)

37


172/268

Modelling logisticgrowth

38

Logistic growth


173/268


Net growth rate becomesnegative for P> C

There are infinite curvesshowing an S-behaviour

Logistic growth wasdiscovered by Franois

Verhulst (1838)

0

BirthandDeathRa

tes

(individuals/time

)


Birth Rate

Death Rate

Net Birth Rate

0 StableEquilibriumUnstable

Equilibrium

Positive FeedbackDominant

(P/C)

inf 1

Negative FeedbackDominant

39

Mathematical model


174/268


Net growth rate is a function of population

40

Net growth rate = dPdt

=g(P, C)P =g

1 PC

P

Compliance to S-shaped growth


175/268


The relative growth raterequirements for S-shapedgrowth are

>0 for PC-70

-60

-50

-40

-30

-20

-10

0

10

20

0 0.5 1 1.5 2 2.5

P/C

Tasso

netto

dicrescita

41

dP

P =g

1

P

C

More features

The differential equation can be rewritten as:


176/268


q

Maximum growth rate? P=C/2

dPdt= g Pg P

2

C

42

Analytical solution


177/268


Z dP1

PC

P= Z

gdt

Z CdP

(CP)P=

Z1

P+

1

CP

dP=

Z gdt

ln(P) ln(CP) =gt+ c

ln(P) ln(CP) =gt+ ln(P(0)) ln[CP(0)]

PCP

= P(0)egtCP(0)

P(t) = C1+

CP(0) 1

egt

43

0

alNetGrowth

Rate

0 1

g*


178/268


Fractiona


0

NetGrowth

Rate


StableEquilibriumUnstable

Equilibrium

PositiveFeedbackDominant

(P/C)inf

= 0.5

NegativeFeedbackDominant

0 1

0.0

0.5

1.0

-4 -2 0 2 4

0

0.25

Population(Left Scale)

Net Growth Rate(Right Scale)

Time

Population/CarryingCapacity

(dimens

ionless)

NetBirthRate/CarryingCapacity

(1/time)

PC

= 11+exp[-g*(t -h)]

g*=1,h =0

44


179/268

Andrea Rizzoli, IDSIA

Feedback dynamicsPart II - Higher order systems: Growth and Collapse



180/268

Dynamics of epidemics

2

The SI model

Epidemics display an S-growth followed by a steep decline


181/268


Epidemics display an S-growth, followed by a steep decline

S: susceptibles I: infected

0

100

200

300

1/22 1/24 1/26 1/28 1/30 2/1 2/3

Patients

confined

to

bed

0

250

500

750

1000

0 5 10 15 20 25 30

Deaths

(people/week)

Weeks

Influenza in UK Plague in India (1905-6)

3

Model hypotheses


182/268


Births and deaths are unaccounted Migrations are unaccounted

Population is supposed to be homogeneous

Once sick, theres no healing Interactions in the community are not altered by the

epidemics

No quarantine

4

Model structure


183/268


SusceptiblePopulation S

InfectiousPopulation I

InfectionRate

+ +

Infectivity iContactRate c

++

B R

TotalPopulation N

-

Depletion Contagion

5

Model equations


184/268


dS

dt=(ciS)

I

N=IR

S+I= N

S: susceptible populationI: infected populationN: total

i: infectivity - prob. of falling ill after a contactc: contact rate [contacted persons / person * time unit]

Infection rate

6

Relationship to logistic growth


185/268


By substituting S = I - N:

that is the equation for logistic growth

while infected population grows logistically, infection rate(IR) grows and then collapses

7

dI

dt

= ci I1 I

N

Current

Infection Rate40

30

20


186/268


10

0Infectious Population I1,000

750

500

250

0Susceptible Population S

1,000

750

500

250

00 30 60 90 120

Time (Day)

Contact Rate cCurrent: 2

Infectivity i

Current: 0.5

Total Population N

Current: 10,000

8

The SIR model


187/268


Useful to model highly infective diseases

S: susceptibles I: infected

R: recovered

9

Model hypoteheses


188/268


Developed by Kermack e McKendrik (1927)

It relaxes the SI hypotheses that once ill no recovery ispossible

Some epidemics stop because the recovery rate is higherthan the infection rate

... or because the death rate is higher than the infectionrate ... (e.g. Ebola)

10

Model structure


189/268


SusceptiblePopulation S

InfectiousPopulation I

InfectionRate

+ +

Infectivity i

Contact Rate c

++

B R

RecoveredPopulation R

RecoveryRate

+

Average Duration

of Illness d

-

B

Total Population N

-

Depletion ContagionRecovery

Initial InfectiousPopulation

11

Model equations

dS= (ciS)

ISusceptible population


190/268


dt=(ciS)

N

RR=I

d

dI

dt = (ciS

)

I

N

I

d

dR

dt=

I

d

Susceptible population

Recovery rated: average illness duration

Infected population

Recovered population

S+I+R= N Total population

12

Model analysis


191/268


An epidemic is a disease that appears over a period in aproportion higher than expected

When can we observe an epidemic in our model?

It depends on the dominating loop In the positive loop the parameters are the contact rate

(c) and infectivity (i)

Also the illness duration has an impact: the longer, thehigher probability of an epidemic

13

The tipping point

2500

Rates


192/268


Which is the point beyond wehave an epidemic?

Which is the parametersetting that pushes oursystem away from stability(noepidemics) to instability(epidemic explosion)?

14

0

500

1000

1500

2000

0 4 8 12 16 20 24

Infection

and

Recovery

R

(people/day)

Days

InfectionRate

RecoveryRate

0

2500

5000

7500

10000

0 4 8 12 16 20 24

Suscep

tible

Population

(people)

Days

Susceptible

Infectious

Recovered

Computing the tipping point


193/268


Tipping occurs when the infection rate is greaterthan the recovery rate

IR > RR

ciS(1/N) > I/d

cid (S/N) > 1

contact number

reproduction rate

15

c < 2


194/268


0

2500

5000

7500

10000

0 10 20 30 40 50 60

Susceptible

Po

pulation

(people)

Days

c = 6

c = 3

c = 2.5c = 2

16

25


195/268


00 1

Contac

tNumber(cid)

(dim

ensionless)

Susceptible Fraction of Population (S/N)(dimensionless)

No Epidemic

(Stable; negative loops dominant)

Epidemic(Unstable; positive loop dominant)

cid(S/N) = 1

17


196/268

Innovation as infection

18

How do ideas spread?


197/268


Diffusion of new ideas and products often follow an S-shaped growth

We can imagine they spread like epidemics

But which is the structure of their dynamics? Whichfeedbacks are there?

Can we use System Dynamics to our advantage toimprove our marketing skills?

19

100US Cable Television Subscribers


198/268


0

2 5

5 0

7 5

100

1950 1960 1970 1980 1990 2000

% of Households with TVSubscribing to Cable

Cable Subscribers(Million Households)

20

Lets adapt the SI model


199/268


PotentialAdopters P

Adopters A

Adoption

Rate

+ +

Adoption

Fraction i

ContactRate c

+

+

B R

TotalPopulation N

-

MarketSaturation

Word ofMouth

21

The spread of VAX minicomputers

3000


200/268


0

1000

2000

3000

1981 1983 1984 1986 1988

Sales Rate

Units/Year

0

2000

4000

6000

8000

1981 1983 1984 1986 1988

Cumulative Sales(! Installed Base)

Units

22

Parameter estimation


201/268


C1989= 7600 Solution of the logistic growth model

from which:

but C=S+P, then C-P=S

we can estimate g using the least squares algorithm

P

CP=

P(0)egt

CP(0)

ln P

C

P=ln

P(0)

C

P(0

)+gt

lnP

S=ln

P(0)

S(0

)+gt Logit transformation

23

0 1

1

10

100

1000

s/PotentialAdopters

mensionless)

Data

Estimated Ratio of A/P(Adopters/Potential Adopters):

ln(A/P) = -5.45 + 1.52(t - 1981); R2= .99


202/268


0.001

0.01

0.1

1981 1983 1984 1986 1988

Adop

ters(di

Regression

0

2000

4000

6000

8000

1981 1983 1984 1986 1988

Cumulative Sales(!Installed Base)

Units

EstimatedInstalled Base

0

1000

2000

3000

1981 1983 1984 1986 1988

Sales Rate

Unit

s/Y

ear

Estimated

Sales Rate

24

A word of wisdom ...


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0

0.1

0.2

0.3

0 10 20 30 40 50 60

Frac

tionalGrowth

Rate(1/years)

Cable Subscribers(million households)

Data

Best Linear Fit

(g = 0.18 - 0.0024S; R2= .52)

0

25

50

75

100

1950 1960 1970 1980 1990 2000 2010

Estimated Cable

Subscribers

Cable Subscribers

Million

Hous

eholds

25

150


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0

50

100

150

1950 1960 1970 1980 1990 2000 2010 2020

Estimated CableSubscribers,

Gompertz Model

Cable Subscribers

Estimated CableSubscribers,

Logistic ModelMillionH

ouseholds

26

Bass diffusion model (1969)

In all growth models zero is an equilibrium. How is itpossible that a new product starts spreading?


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possible that a new product starts spreading?

Bass introduced the effect of advertisement (direct andindirect)

27

PotentialAdopters P

Adopters A

Adoption

Rate AR

AdoptionFraction i

ContactRate c

B R

TotalPopulation N

AdvertisingEffectiveness

a

MarketSaturation

Word ofMouth

Adoptionfrom

Advertising

Adoptionfrom Word of

Mouth

+

++

+

+

+

+-

+

B

MarketSaturation

Model equations


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Adoption rate:

AR = Ads + Word of Mouth

Advertisement = aP Word of mouth (infection) = ciP(A/N)

AR= aP + ciP(A/N)

if A=0 (no early adopters), the only input is advertisement

28

4000

6000

8000


Units

Logistic Model

Bass Model


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0

2000

1981 1983 1984 1986 1988

0

1000

2000

3000

1981 1983 1984 1986 1988

Sales Rate

Units/Year

Logistic Model

Bass Model

0

1000

2000

3000

1981 1983 1984 1986 1988

Sales Rate

Units/Year

Sales fromWord of Mouth

Sales from Advertising

29

If we use the Logistic growthmodel we need to initialisewith a value /= 0

2000

4000

6000

8000


Units

Logistic Model

Bass Model


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Logistic growthunderestimates sales at thebeginning (no ads!) andoverestimates the sales peak

Bass model has beensuccessfully applied to varioussectors

30

0

2000

1981 1983 1984 1986 1988

0

1000

2000

3000

1981 1983 1984 1986 1988

Sales Rate

Units/Year

Logistic Model

Bass Model

0

1000

2000

3000

1981 1983 1984 1986 1988

Sales Rate

Units/Yea

r

Sales fromWord of Mouth

Sales from Advertising

The Bass andlogistic diffusionmodels comparedto actual VAXsales


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Feedback DynamicsPart III - Higher order systems: delays and oscillations

(Reference: Sterman, Chapters 11 and 17)

1


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Modelling delaysFriends or foes?

2

Definition


211/268


A delayis a process the output of which is shifted in timewith respect to the input

3

DelayInput Output

-5.00

-3.75

-2.50

-1.25

0

1.25

2.50

3.75

5.00

time

Examples


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Materiale intransito

Lettere intransito

Tasso di afflusso Tasso di deflusso

Tasso di spedizione Tasso di consegna

General structure of a delay

A post office as a delay

4

Material intransit

Inflow rate Outflow rate

Letters intransit

Mailing rate Delivery rate

Structure of delays


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Delays model/approximate a dynamic process The outflow might depend on available resources Sometimes it can be assumed that the delay is

independentand therefore depends only on past inflows(pure delays)

Modelling a delay requires to determine:1. the average delay length

2. the distribution around the average?

5

Average delay length


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It is the average residence time: how long does it take foran element to go through the system

Examples:

First class postage (Posta A): ~ 1 day EMails: few seconds

Such data are obtained empirically

6

200

Distribution of the output


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0

50

100

150

0 1 2 3

%o

fUnitPulse/TimePeriod

Time (multiples of average delay time)

Inflow

B

C

D

Outflow A

This is actually computing the impulseresponse

7

Impulse response


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An impulseis analogous to mailing 1000 letters at once orto pouring a mass of material in a pulp processing system

and so on

Mathematically, in continuous time, is a Diracdelta

!(t,T) = limW0

!(t,T,W) =

0 per t T

1/Wpert T+W0 per t T+W

8

Impulse response

150

200

Period Inflow Outflow A


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In all cases the residence mean time is the same. Yet weobserve different behaviours

A. elements leave in the same order as they entered B, C, and D: various degrees of mixing

9

0

50

100

150

0 1 2 3

%o

fUnitPulse/Time

P


B

C

D

Service discipline


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It defines the way units are processed

FIFO LIFO Intermediate disciplines (even random)

10

Pipeline delay


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Also known as transport delay

Constant delay time = D

Materiale intransito

Tasso di afflusso Tasso di deflusso

Tempo mediodi ritardo

11

Material intransitInflow rate Outflow rate

Average delaylength

d(Material in transit)

dt = inflow(t) outflow(t)

outflow t = inflow t

D

First order delay

Mixing and changes in processing times of individualelements


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First order delays: perfect mixing

12

outflow = materia in transit

D

50

75

100Delay Inflow and Outflow

se/TimePeriod


221/268


0

25

50

0 1 2 3

%o

fUnitPuls


Inflow Outflow

0

25

50

75

100

0 1 2 3

Stock of Material in Transit

MaterialinTransit

(%o

fpulsequantity)

Documents

System Dynamics