VIII' Continuous Systems Simulation CSS

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VIII. Continuous Systems Simulation (CSS)

• M. Peter Jurkat
• CS452/Mgt532 Simulation for Managerial Decisions
• The Robert O. Anderson Schools of Management
• University of New Mexico

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How Various Models are Studied
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Approach

• Continuous models simulate situations which could
  be changing all the time usually described by
  differential equations (DEs) cannot calculate
  with DEs
• Approximate with Difference Equations (also
  DiffEs)
• Usually deterministic stochastic DE/DiffEs
  advanced topic
• Basic variable (time, distance, ) with constant
  step size and/or time/distance/ dependent inputs
  and/or parameters
• Often unknown and/or impractical analytic
  solutions - Solved by simulation
• Yields function which specifies state variables
  at all times/distances/ performance measure is
  a function of state variables
• Studied by DOE on parameter values

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CSS Examples

• Calculating space flight orbits trajectories due
  strength/direction of engine boosts and gravity
  of nearby heavenly bodies (sun, planets, moons,
  etc.)
• Calculating response of a controlled element due
  to control changes (e.g., machine tool path,
  flight simulators, refineries, atomic energy
  plants)
• Calculating best bid prices for financial
  options based on Black-Scholes payoff formulation
  (stochastic differential equations involving
  strike price, time, and price volatility/s.d.)

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CSS Example

• Fluid flow through complicated spaces
• Similar to flow though jet engines
• From http//www.uk.comsol/products/chem/applicatio
  ns.php

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CSS Example

• Finite Element Modeling to determine extend of
  trauma due to blows
• Mesh lines between nodes interpreted as springs
• Yields deformation vs. applied forces
• From Krabbel and Miller (?) DEVELOPMENT OF A
  FINITE ELEMENT MODEL OF THE HEAD USING THE
  VISIBLE HUMAN DATA, web site of Orthopedic
  Biomechanics Laboratory, Harvard Medical School,
  Boston

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Difference Equations

• Many models described by difference equations
• DEs are simulated by making the continuous
  underlying variable (time, distance) discrete by
  approximating continuous movement at set
  intervals, usually small
• Example Recall Example 2.4 (pp45-47) and
  Banks4Example2-4InvCosts.gps - unless each
  withdrawal and replenishment is to be tracked
  individually at its exact time, inventory systems
  can be well approximated by difference equations
  on a daily/monthly/yearly basis
• Balance equations at time period t is
• inventorylevelt inventorylevelt-1
  withdrawalst replenishmentst
• Need starting inventory level at time t 0, then
  get inventory level1, then inventory level2 ,

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Order of Difference Equations

• Level of largest delay from current time step in
  DE is order of system
• For instance yt yt-3 is a 3rd order system
  with no term has larger differences
• 1st order yt cyt-1
• 2nd order yt 2c(yt-1 yt-2)
• See DifferenceEquations.xls

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Population Dynamics

• Modern version is called the Leslie Model, after
  a quantitative biologist, and based on life
  tables developed in the 1700s by the Bernoullis,
  Euler, Halley, etc.
• Now formalized (e.g., Feller (1968) Intro to Prob
  Theory and its Apps)
• Basis of the entire life insurance industry
• Can also be applied to other populations, i.e.,
  any items that replicate themselves (e.g.,
  invested money)

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Simplest Model

• Let At number of adult breeding pairs during
  period t (often a generation)
• Let Yt number of young, non-breeding pairs
  during period t
• Then Yt1 f1Yt f2At
• At1 sYt
• where f1 and f2 are fertility rates and s is the
  survival rate
• All adults die out in each generation (at least
  are no longer involved in reproduction)
• Too simplistic fertility depends on age within
  reproductive periods need to introduce cohorts

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Multi-Cohort Leslie Model

• Assume k population cohorts subgroups
  distinguished by different reproduction and/or
  survival rates indexed by i 1, 2, , k
• Let Xi,t number of units is cohort k at time t
• Then X1,t1 f1X1,t f2X2,t fkXk.t
• X2,t1 s1X1.t
• Xk,t1 skXk,t
• where fi are fertility rates and si are survival
  rates
• Needs to be augmented by any in- and
  out-migration (e.g., inventory model)

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Examples of Population CohortsCull, Flavine, and
Robson (2005) Difference Equations From Rabbits
to Chaos, Springer, ISBN 9780387232348
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Assignment

• Use DifferenceEquations.xls to characterize both
  Leslie models and both the 1st and 2nd order
  equations for various values of their parameters.
  This can be a group effort.
• Try various values, both positive and negative,
  and show the plots of the solution that support
  your observations and conclusions. In
  particular, try to find values that make a
  qualitative difference.