Dynamical Systems MAT 5932

The Lanchester Equations

of Warfare ExplainedLarry L. SouthardTuesday, April 11, 2023

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

Decision Processing in Combat Modeling

Movement

Command and Control

Sensing

Target Acquisition

Engagement Decision

Target Selection

Damage Perception by Firer

Physical Attrition Process

Accuracy Assessment

Damage Assessment

Attrition

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.

dx

dtf x y

dy

dtf x y 1 2, ,... , ,...

The Lanchester Equation

Mathematically it looks simple:

dx

dtay and

dy

dtbx

Lanchester Attrition Model - Square Law

Integrating the equations which describe modern warfare

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

b ax x y y( ) ( )0 0

2 2 2 2

dx

dtay and

dy

dtbx

These equations have also been postulated to describe "aimed fire".

ab measures battle

intensity

measures relative effectivenessa

b

Questions Addressed by Square Law State Equation

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

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

eyxeyxtx tab

b

atabb

a00002

1)(

Square Law - Force Levels Over Time

0

4

8

12

X ForceY Force

-10

0

10

20

30

40

50

60

Forc

e Lev

el

Time

X=30, Y=60, a=.04, b=.04 X=30, Y=60, a=.04, b=.04

-10

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time

Forc

e Lev

el

X Force

Y Force

Example:

x(t) becomes zero at about t = 14 hours.Surviving Y force is about y(14) = 50.

Square Law - Force Levels Over Time

05

10

15

20

25X Force

Y Force

-10

0

10

20

30

40

50

60

Force Levels

Time

Reduce a to .01, Increase b to .1

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

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

How do kill rates affect outcome?

Square Law - Force Levels Over Time

012

24

36

48

60X Force

Y Force

-10

0

10

20

30

40

50

60

70

80

90

Forc

e Lev

el

Time

Increase Y by 30

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

Can Y overcome this disadvantage by adding forces?

Square Law - Force Levels Over Time

0

19

38

57X Force

Y Force

-20

0

20

40

60

80

100

Forc

e Lev

el

Time

Increase Y by another 10 Increase Y by another 10

-20

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35 40 45 50 55

Time

Forc

e Lev

el

X Force

Y Force

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?

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: 0

0

xy

a

b

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?

fx xa

by

0

202

t xab

y

x

a

by

x

a

b

f

1

2

1

1

0

0

0

0

ln

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:

t y

abx

xifxy

a

b

ab

y y yb

ax

yb

ax

otherwiseBP

BP

BP BP

1

1

0 0

0

202

02

0 0

ln

ln

0

0

0

0

1

1

2

2

2

2

xy

a

b

y

y

x

x

BP

BP