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### Dynamical Systems MAT 5932

The Lanchester Equations of Warfare Explained

Larry L. Southard

Tuesday, November 18, 2014

Agenda

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

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

Two Types of Models

- 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

The Homogeneous Model

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

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

The Heterogeneous model

- 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 Heterogeneous model

- 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

Target Acquisition

Engagement Decision

Target Selection

Physical Attrition Process

Accuracy Assessment

Damage Perception by Firer

Damage Assessment

Decision Processing in Combat ModelingSensing

Command and Control

Movement

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.

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

The Lanchester Equation

Mathematically it looks simple:

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".

- measures battle intensity
- measures relative effectiveness

Questions Addressed by Square Law State Equation fire".

- 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?

Lanchester Square Law - Force Levels Over Time fire".

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

Square Law - Force Levels Over Time fire".

Example:

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

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

Square Law - Force Levels Over Time fire".

How do kill rates affect outcome?

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

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

Square Law - Force Levels Over Time fire".

Can Y overcome this disadvantage by adding forces?

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

Square Law - Force Levels Over Time fire".

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

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

Square Law - Who Wins a Fight-to-the-Finish? fire".

- 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:

Lanchester Square Law - Other Answers fire".

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

When X wins, how long does it take?

Square Law - Breakpoint Battle Termination fire".

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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