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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
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