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Dynamical Systems MAT 5932

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Dynamical Systems MAT 5932

The Lanchester Equations of Warfare Explained

Larry L. Southard

Tuesday, November 18, 2014

- History of the Lanchester Equation Models
- Lanchester Attrition Model
- Deficiencies of the equations

- 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.
- Lanchester equations are taught and used at every major military college in the world.

- Both models work on the basis of attrition
- Homogeneous
- a single scalar represents a unit’s combat power
- Both sides are considered to have the same weapon effectiveness

- Heterogeneous
- attrition is assessed by weapon type and target type and other variability factors

- An “academic” model
- Useful for the review of ancient battles
- Not proper model for modern warfare

- 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

- More appropriate for “modern” battlefield.
- 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 measured in the heterogeneous model:
- Opposing force strengths
- FEBA (forward edge of the battle area) movement
- Decision-making (including breakpoints)

- Additional Areas of consideration to be applied:
- Training
- Morale
- Terrain (topographically quantifiable)
- Weapon Strength
- Armor capabilities

Attrition

Target Acquisition

Engagement Decision

Target Selection

Physical Attrition Process

Accuracy Assessment

Damage Perception by Firer

Damage Assessment

Sensing

Command and Control

Movement

- 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.

- This model underlies many low-resolution and medium-resolution combat models. Similar forms also apply to models of biological populations in ecology.

Mathematically it looks simple:

Integrating the equations which describe modern warfare

we get the following state equation, called Lanchester's "Square Law":

- measures battle intensity
- measures relative effectiveness

- Who will win?
- What force ratio is required to gain victory?
- How many survivors will the winner have?
- Basic assumption is that other side is annihilated (not usually true in real world battles)

- 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?

After extensive derivation, the following expression for the X force level is derived as a function of time (the Y force level is equivalent):

Example:

x(t) becomes zero at about t = 14 hours.

Surviving Y force is about y(14) = 50.

How do kill rates affect outcome?

Now y(t) becomes zero at about t =24 hrs.

Surviving X force is about x(24) = 20.

Can Y overcome this disadvantage by adding forces?

Not by adding 30 (the initial size of X's whole force).

What will it do to add a little more to Y?

This is enough to turn the tide decidedly in Y's favor.

- 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) > 0 and y(tf) = 0
- Y wins, i.e., y(tf) > 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:

How many survivors are there when X wins a fight-to-the-finish?

When X wins, how long does it take?

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: