Models, Gaming, and Simulation Session 4

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Models, Gaming, and Simulation - Session 4

• Aggregated Attrition Algorithms - Lanchester
  Models

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Topics

• Attrition in Aggregated Combat Models
• Lanchester Attrition Model
• Next Session Stochastic Lanchester model of
  attrition

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Attrition in Aggregated Combat Models

• Represents the results of engagements between
  military units (not individual weapons firing at
  individual targets). Example

Sensing
Command and Control
Movement
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Attrition in Aggregated Combat Models

• The following battlefield functions are sometimes
  combined and sometimes modeled by separate
  algorithms
• direct fire
• indirect fire
• air-to-ground fire
• ground-to-air fire
• air-to-air fire
• minefield attrition
• The following processes are directly or
  indirectly affected by the attrition process
• Opposing force strengths
• FEBA (forward edge of the battle area) movement
• Decision-making (including breakpoints)

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Attrition in Aggregated Combat Models

• Homogeneous Attrition Models
• DEFINITION a single scalar represents a units
  combat power
• Examples
• Force-ratio models
• Homogeneous Lanchester models
• Heterogeneous Attrition Models
• DEFINITION attrition is assessed by weapon type
  and target type
• Examples
• Heterogeneous Lanchester models
• ATCAL

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Lanchester Attrition Model

• CONCEPT describe the rate at which a force loses
  systems as a function of the size of the force
  and the size of the enemy force. This results in
  a system of differential equations in force sizes
  x and y.
• The solution to these equations as functions of
  x(t) and y(t) provide insights about battle
  outcome.
• History The British engineer F.W. Lanchester
  (1914) developed this theory based on World War I
  aircraft engagements to explain why concentration
  of forces was useful in modern warfare.
• The original formulation was in terms of two
  models, called the square law and the linear
  law. Many extensions and modifications have been
  proposed to add robustness and detail.
• This model underlies many low-resolution and
  medium-resolution combat models. Similar forms
  also apply to models of biological populations in
  ecology.

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Lanchester Attrition Model

• Original formulation gave two situations
• 1) "Ancient" or one-on-one warfare adding more
  troops to the battle was theorized not to affect
  the rate at which forces killed each other, since
  one soldier could supposedly only fight one other
  soldier. Mathematically,
• 2) "Modern" or many-on-many warfare, where
  several weapons can fire at a single target the
  rate at which troops of one side are killed is
  theorized to be proportional to the number of
  troops on the other side. Mathematically,

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Lanchester Attrition Model - Linear Law

• Integrating the equation that describes ancient
  warfare, where for x(t), we say that x(0) x0,
  we get the following state equation
• This is one form of Lanchester's "Linear Law",
  so-called because it is linear in both x(t) and
  y(t).
• Lanchester also theorized that "area fire" could
  be described by the equations
• The assumptions are that each side fires
  uniformly into an area occupied by the enemy.
  These equations also lead to the more common form
  of the Linear Law state equation

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Lanchester Attrition Model - Square Law

• Integrating the equations which describe modern
  warfare
• we get the following state equation, called
  Lanchester's "Square Law"
• These equations have also been postulated to
  describe "aimed fire".
• Observations
• measures battle intensity
• measures relative effectiveness

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Questions Addressed by Square and Linear Law
State Equations

• Who will win?
• What force ratio is required to gain victory?
• How many survivors will the winner have?
• (Assume for now that the loser will be
  annihilated, which is not usually a reasonable
  result.)
• How long will the battle last?
• How do force levels change over time?
• How do changes in parameters x0, y0, a, and b
  affect the outcome of battle?
• Is concentration of forces a good tactic?

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Lanchester Square Law - Force Levels Over Time

• After extensive derivation, the following
  expression for the X force level is derived as a
  function of time (the Y force level is
  equivalent)
• It can also be expressed using hyperbolic
  functions as

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Square Law - Force Levels Over Time

• Example

x(t) becomes zero at about t 14
hours. Surviving Y force is about y(14) 50.
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Square Law - Force Levels Over Time
How do kill rates affect outcome?

Now y(t) becomes zero at about t 24
hrs. Surviving X force is about x(24) 20.
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Square Law - Force Levels Over Time
Can Y overcome this disadvantage by adding forces?

Not by adding 30 (the initial size of X's whole
force).
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Square Law - Force Levels Over Time
What will it do to add a little more to Y?

This is enough to turn the tide decidedly in Y's
favor.
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Square Law - Who Wins a Fight-to-the-Finish?

• To determine who will win, each side must have
  victory conditions, i.e., we must have a "battle
  termination model". Assume both sides fight to
  annihilation.
• One of three outcomes at time tf, the end time of
  the battle
• X wins, i.e., x(tf) gt 0 and y(tf) 0
• Y wins, i.e., y(tf) gt 0 and x(tf) 0
• Draw, i.e., x(tf) 0 and y(tf) 0
• It can be shown that a Square-Law battle will be
  won by X if and only if

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Lanchester Square Law - Other Answers

• How many survivors are there when X wins a
  fight-to-the-finish?
• When X wins, how long does it take?

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Square Law - Breakpoint Battle Termination

• How long does
  it take
  if X wins?
• (Assume battle termination at x(t) xBP or
  y(t) yBP)
• In what case does X win?
  If and only if

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Linear Law - Some Answers

• What does it represent?
• Area, uniform fire, constant-area target, or
• Aimed fire, assuming target acquisition times
  are
• inversely proportional to number of targets,
• dominant part of attrition process
• Basic Equations
• State Equations
• X Force level as a function of time
• When does X win?
• How many X survivors when X wins?
• Duration of battle infinite

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Lanchester Extensions - Other Functional Forms

• Key F - depends on Firer size
• T - depends on Target size
• C - not dependent on firer,target
  (Constant)
• Form Name Comments
• FF Square Law aimed fire
• FTFT Linear Law area fire
• TT Logarithmic Law non-combat
  losses
• (FT)(FT) Morse-Kimball aimed w/ non-cbt
  losses
• FFT Mixed Combat ambush, prepared
  defense
• (F,FT,or T)(F,FT,or T)
  Helmbold
  Eqns diminishing returns,
  generalization of F, FT, and T
• 
  w 0 gtT
• 
  w 1/2 gt FT
• 
  w 1 gt F

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Heterogeneous, Discrete, and Stochastic
Lanchester Extensions

• Heterogeneous Models
• CONCEPT describe each type of system's strength
  as a function (usually sum of attritions) of all
  types of systems which kill it
• ASSUME additivity, i.e., no synergism can be
  relaxed with complex enhancements and
  proportionality, i.e., loss rate of Xi is
  proportional to number of Yj which engage it.
• No closed solutions, but can be solved
  numerically
• Discrete Models
• CONCEPT solve continuous equations with
  difference equations (Euler-Cauchy method)
• NOTE this solution method corresponds nicely
  with time-stepped combat models and so can be
  used with them. One version is called the
  Bonder-Farrell attrition algorithm, the algorithm
  used in Eagle. Many ad hoc attrition algorithms
  which build from first principles are in fact
  equivalent to the Euler-Cauchy method.
• Stochastic Models
• CONCEPT Model size of opposing forces as a
  continuous-parameter Markov chain, where each
  state is a pair of Blue strength/Red Strength,
  and attrition happens one casualty at a time
• It can be shown that for some important
  parameters, the Stochastic and Deterministic
  versions of Lanchester models are quite similar.