System Dynamics 2

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System Dynamics 2

• CAP4800/5805
• Systems Simulation

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What we covered last time

• What is System Dynamics
• Causal Loop Diagram
• Augmenting Causal CLD
• Loop dominance
• Labeling link polarity
• Determining loop polarity
• Exogenous items and delays

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System Dynamics Modeling

• Identify a problem
• Develop a dynamic hypothesis explaining the cause
  of the problem
• Create a basic causal graph
• Augment the causal graph with more information
• Convert the augmented causal graph to a System
  Dynamics flow graph
• Translate a System Dynamics flow graph into
  DYNAMO programs or equations
• Simulate the DYNAMO programs or equations

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Casual-loops

• Provide insight into a system's structure
• Often difficult to infer the behavior of a system
  from its casual-loop representation
• Need to use computer simulation
• Simulation model flow diagrams, equations,
  simulation language
• DYNAMO (DYNAmic Models)
• Not a general-purpose language but special
  purpose language to aid in building computer
  models

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Flow Graph Symbols
Level
Rate
Flow arc
Auxiliary
Cause-and-effect arc
Source/Sink
Constant
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Level

• AKA stock, accumulation, or state variable
• A quantity that accumulates over time
• Change its value by accumulating or integrating
  rates
• Change continuously over time even when the rates
  are changing discontinuously

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Rate/Flow

• AKA flow, activity, movement
• Change the values of levels
• The value of a rate is
• Not dependent on previous values of that rate
• But dependent on the levels in a system along
  with exogenous influences

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Auxiliary

• Arise when the formulation of a levels influence
  on a rate involves one or more intermediate
  calculations
• Often useful in formulating complex rate
  equations
• Used for ease of communication and clarity
• Value changes immediately in response to changes
  in levels or exogenous influences

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Source and Sink

• Source represents systems of levels and rates
  outside the boundary of the model
• Sink is where flows terminate outside the system

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Example 1 (Population and birth)
Population
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Example 2 (Children and adults)
children
Adults
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DYNAMO

• Originally developed by Jack Pugh at MIT
• First system dynamics simulation language
• For a long time the language and the field were
  considered synonymous
• Provides an equation based development
  environment for system dynamics models
• DYNAMO today runs on PC compatibles under
  Dos/Windows.

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Time in DYNAMO

• LEVEL.K a level calculated at the present time
• LEVEL.J a level calculated one time interval
  earlier
• DT the length of the time interval between J and
  K

dt
dt
L future
J past
K present
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DYNAMO Program(Population and birth model)
Births
Population
Star statement

• Population Growth
• L POP.K POP.J DTBIRTH.JK
• N POP 10
• R BIRTH.KL (POP.K)(PAR)
• C PAR 0.1
• SPEC DT 1.0

Level statement
present
one time interval earlier
between J and K
Initial value statement
Rate statement
Constant statement
SPEC statement
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Integral/Differential Equations

• Diagran

R1
R2
L

• Integral Equation
• L(t) ? R1(s) R2(s) ds L(t0)

t
t0

• Differential Equation
• dL/dt Net Change in L R1(t) - R2(t)

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System Dynamics Algorithm

• Program Main
• We are given a concept graph with modes and arcs
• The arcs require sign (,-) labeling
• The nodes require labeling source, rate, level,
  constant, auxiliary
• For each level node (L) with an input rate node
  (R1) and and output rate node (R2) write
• dL/dt k1 R1 k2 R2 k1 and k2 are rate
  constants
• End for
• For all other nodes (N) write
• N(t) a linear function of all inset members of
  this node
• End for
• End Main

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From Causal Loop DiagramTo Simulation Models 1

• Flow Graph

• Causal Graph

R
L

• Equations
• dL/dt k1R(t)
• R(t) k2L(t)
• ? dL/dt k1k2L(t)

• Block Model

L
L
?
k1k2
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From Causal Loop DiagramTo Simulation Models 2

• Equations
• dL/dt R1 R2
• R2 k2L
• R1 k1
• ? dL/dt k1 - k2L

• Flow Graph

R1
R2
L

• Block Model

L1
L1
?
k2
k1
-
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From Causal Loop DiagramTo Simulation Models 3

• Equations
• dL1/dt R1 R2
• dL2/dt R2 R3
• R1 k1
• R2 K2 L1
• R3 K3 L2
• ? dL1/dt k1 k2L1
• ? dL2/dt k2L1 K3L2

• Flow Graph

R2
R3
R1
L2
L1

• Block Model

L1
L2
L2
L1
?
?
-
k2
k3
k1
-
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Building construction

• Problem statement
• Fixed area of available land for construction
• New buildings are constructed while old buildings
  are demolished
• Primary state variable will be the total number
  of buildings over time
• Causal Graph

-
-
-
-
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Simulation models

• Equations
• dBl/dt Cr Dr
• Cr f1(CF, Bl)
• Dr f2(AL,Bl)
• CF f3(FLO)
• FLO f4(LA,AA,Bl)

• Flow Graph

Construction (C)
Demolition (D)
Industrial Buildings (B)
Average lifetime for buildings (AL)
Construction fraction (CF)
Fraction of land occupied (FLO)
Land available for industrial buildings (LA)
Average area per building (AA)
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Next Class

• VenSim
• System Dynamics Simulation Tool
• http//www.vensim.com/

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References

• Simulation Model Design and Execution, Fishwick,
  Prentice-Hall, 1995 (Textbook)
• Introduction to Computer Simulation A system
  dynamics modeling approach, Nancy Roberts et al,
  Addison-wesley, 1983
• Business Dynamics Systems thinking and modeling
  for a complex world, John D. Sterman,
  McGraw-Hill,2000